IJOA VOL 01 ISSUE 02.Pdf

VOL 01 - ISSUE 02 2021 International Journal On Optimization and Applications

Vol 01 – Issue 02 2021

Editor in Chief Prof. Dr. Hanaa HACHIMI

ISSN : 2737 - 8314 FOREWORD

The International Journal on Optimization and Applications (IJOA) volume 01 - issue 02 is an open access, double blind peer- reviewed online journal aiming at publishing high-quality research in all areas of : Applied mathematics, Engineering science, Artificial intelligence, Numerical Methods, Embedded Systems, Electric, Electronic en-gineering, Telecommunication Engineering... the IJOA begins its publication from 2021. This journal is enriched by very important special manuscripts that deal with problems using the latest methods of optimization. It aims to develop new ideas and col-laborations, to be aware of the latest search trends in the optimi-zation techniques and their applications in the various fields.. Finally, I would like to thank all participants who have contributed to the achievement of this journal and in particular the authors who have greatly enriched it with their performing articles.

Prof. Dr. Hanaa HACHIMI Editor in chief Associate Professor in Applied Mathematics & Computer Science Systems Engineering Laboratory LGS Director, BOSS Team Sultan Moulay Slimane University TABLE OF CONTENTS

ARTICLE 1 -- EXISTENCE RESULT FOR A GENERAL NONLINEAR DEGENERATE ELLIPTIC PROBLEMS WITH MEASURE DATUM IN WEIGHTED SOBOLEV SPACES ...... 1 ARTICLE 2 - PERFORMANCE AND ANALYSES USING TWO ETL EXTRACTION SOFTWARE SOLUTIONS ...... 10 ARTICLE 3 – A NOVEL C/AG/SIO2 SONOGEL-CARBON ELECTRODE PREPARATION, CHARACTERIZATION, APPLICATION AND STATISTICAL VALIDATION AS AMPEROMETRIC SENSORS FOR CATHECOL DETERMINATION ...... 14 ARTICLE 4 – RENEWABLE ENERGY A NEW AVENUE FOR OPTIMIZING COSTS FOR COMPANIES ...... 20 ARTICLE 5 -- EXISTENCE OF SOLUTIONS FOR HYBRID EQUATION INVOLVING THE RIEMANN -LIOUVILLE DIFFERENTIAL OPERATORS ...... 22 ARTICLE 6 -- ON THE CAUCHY PROBLEM FOR THE FRACTIONAL DRIFT- DIFFUSION SYSTEM IN CRITICAL FOURIER-BESOV-MORREY SPACES ...... 28 ARTICLE 7 – EXISTANCE RESULTS FOR A NEUTRAL FUNCTIONNAL INTEGRODIFFERENTIAL INCLUSION WITH FINITE DELAY ...... 37 ARTICLE 8 – FUZZY FRACTIONAL DIFFERENTIAL WAVE EQUATION ...... 42 ARTICLE 9 – ULAM-HYERS-RASSIAS STABILITY FOR FUZZY FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS UNDER CAPUTO GH DIFFERENTIABILITY ...... 51 ARTICLE 10 – ECO-TAXATION AS AN INSTRUMENT TO FIGHT AGAINST CLIMATE CHANGE ...... 56

Existence Result for a General Nonlinear Degenerate Elliptic Problems with Measure Datum in Weighted Sobolev Spaces

Adil Abbassi Mohamed El Ouaarabi Laboratory LMACS, FST of Beni Mellal, Laboratory LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco. Sultan Moulay Slimane University, Morocco. [email protected] [email protected]

Chakir Allalou Laboratory LMACS, FST of Beni Mellal, Sultan Moulay Slimane University, Morocco. [email protected]

0 1 p0 1−p Abstract—In this paper, we study the Dirichlet problem asso- with f0 ∈ L (Ω) and for j = 1, ..., n, fj ∈ L (Ω, ω1 ). ciated to the degenerate nonlinear elliptic equations Furthermore, the functions A :Ω × Rn −→ R, B :Ω × R × ( n n Lu(x) = µ in Ω, R −→ R, g :Ω × R −→ R and H :Ω × R × R −→ R u(x) = 0 on ∂Ω, are Carateodory´ functions, who satisfying the assumptions of growth, ellipticity and monotonicity. where In the past decade, much attention has been devoted to h i Lu(x) = −div ω1(x)A(x, ∇u(x)) + ω2(x)B(x, u(x), ∇u(x)) nonlinear elliptic equations because of their wide application to physical models such as non-Newtonian fluids, boundary + ω (x)g(x, u(x)) + ω (x)H(x, u(x), ∇u(x)), 1 2 layer phenomena for viscous fluids, and chemical heteroge- is a second order degenerate elliptic operator, with A :Ω × nous model, we mention some works in this direction [1], [4], n n R −→ R, B :Ω × R × R −→ R, g :Ω × R −→ R and H : n [5], [7]. One of the motivations for studying (1) comes from Ω × R × R −→ R are Carateodory´ functions, who satisfies some conditions, and the right-hand side term µ belongs to L1(Ω) + applications to electrorheological fluids (see [19] for more n Q p0 1−p0 details) as an important class of non-Newtonian fluids. L (Ω, ω1 ), ω1 and ω2 are weight functions that will be k,p j=1 In general, the Sobolev spaces W (Ω) without weights defined in the preliminaries. occur as spaces of solutions for elliptic and parabolic par- Index Terms—Nonlinear degenerate elliptic equations, Dirich- tial difierential equations. For degenerate partial differential let problem, weighted Sobolev spaces, weak solution equations, where we have equations with various types of I.INTRODUCTION singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces [2], [4], [12], [13], [16]. Let Ω be a bounded open subset in n ( n ≥ 2), ∂Ω its R The type of a weight depends on the equation type. boundary and p > 1 and ω , ω are two weights functions 1 2 For ω ≡ ω ≡ 1 (the non weighted case) and A(x, ∇u) ≡ in Ω (ω and ω are measurable and strictly positive a.e. in 1 2 1 2 g ≡ 0, Equation of the from (1) have been widely studied in Ω). Let us consider the following nonlinear degenerate elliptic [10], where the authors obtain some existence results for the problem ( solutions (see also the references therein). Lu(x) = µ in Ω, (1) Boccardo et al. [6] considered the nonlinear boundary value u(x) = 0 on ∂Ω, problem where, L is a second order degenerate elliptic operator −div(a(x, u, ∇u)) + g(x, u, ∇u) = µ, h i 1 −1,p0 1 Lu(x) = −div ω1(x)A(x, ∇u(x)) + ω2(x)B(x, u(x), ∇u(x)) where µ ∈ L (Ω) + W (Ω) and g(x, u, ∇u) ∈ L (Ω). By combining the truncation technique with some delicate test + ω (x)g(x, u(x)) + ω (x)H(x, u(x), ∇u(x)), (2) 1 2 functions, the authors showed that the problem has a solution 1,p and n u ∈ W0 (Ω). Furthermore the degenerate case with difierent X conditions haven been studied by many authors (we refer to µ = f0 − Djfj, (3) j=1 [11], [22] for more details).

In [3], the authors proved the existence results, in the As a consequence, under conditions of Proposition 2.1, the p 1 setting of weighted Sobolev spaces, for quasilinear degener- convergence in L (Ω, ω) implies convergence in Lloc(Ω). ated elliptic problems associated with the following equation Moreover, every function in Lp(Ω, ω) has a distributional −div a(x, u, ∇u) + g(x, u, ∇u) = f − divF , where g derivatives. It thus makes sense to talk about distributional p satisfies the sign condition. derivatives of functions in L (Ω, ω). In [8] the author proved the existence of solutions for the A class of weights, which is particularly well understood, A problem (1) when ω1 ≡ ω2 and A(x, ∇u) ≡ g ≡ 0. When is the class of p-weight that was introduced by B. Mucken- H(x, u, ∇u) ≡ g ≡ 0 existence result for the Problem (1) houpt. have been shown in [9]. Definition 2.1: Let 1 ≤ p < ∞. A weight ω is said to be an Our objectif, in this paper, is to study equation (1) by adopt- Ap-weight, or ω belongs to the Muckenhoupt class, if there 1,p ing Sobolev spaces with weight W (Ω, ω1) (see Definition exists a positive constant C = C(p, ω) such that, for every 0 n 2.3). By apply the main theorem on monotone operators (see ball B ⊂ R p−1 Theorem 2.3), we show that the Problem (1) admits one and 1 Z 1 Z −1 1,p ω(x)dx (ω(x)) p−1 dx ≤ C if p > 1, only solution u ∈ W0 (Ω, ω1). |B| |B| The paper is organized as follows. In Section 2, we give B B some preliminaries and the definition of weighted Sobolev 1 Z 1 ω(x)dx ess sup ≤ C if p = 1, spaces and some technical lemmas needed in our peper. In |B| B x∈B ω(x) Section 3, we make precise all the assumptions on A, B, where |.| denotes the n-dimensional Lebesgue measure in n. g, H and we introduce the notion of weak solution for the R The infimum over all such constants C is called the A Problem (1). Our main result and his proof, the existence and p constant of ω. We denote by A , 1 ≤ p < ∞, the set of uniqueness of solution to Equation (1), are collected in Section p all A weights. 4. Section 5 is devoted to an example which illustrates our p 1 ≤ q ≤ p < ∞ A ⊂ A ⊂ A A main result. If , then 1 q p and the q constant of ω equals the Ap constant of ω (we refer to [15], II.PRELIMINARIES [16], [20] for more informations about Ap-weights). In this section, we present some definitions, and preliminar- Example 2.1: (Example of Ap-weights) ies facts which are used throughout this paper. (i) If ω is a weight and there exist two positive constants By a weight, we shall mean a locally integrable function C and D such that C ≤ ω(x) ≤ D for a.e. x ∈ Rn, n n ω on R such that ω(x) > 0 for a.e. x ∈ R . Every weight then ω ∈ Ap for 1 ≤ p < ∞. n η n ω gives rise to a measure on the measurable subsets on R (ii) Suppose that ω(x) = |x| , x ∈ R . Then ω ∈ Ap if through integration. This measure will also be denoted by ω. and only if −n < η < n(p − 1) for 1 ≤ p < ∞ Thus, (see Corollary 4.4, Chapter IX in [20]). Z (iii) Let Ω be an open subset of n. Then ω(x) = n R ω(E) = ω(x)dx for measurable subset E ⊂ . λϕ(x) 1,n R e ∈ A2, with ϕ ∈ W (Ω) and λ is sufficiently E small (see Corollary 2.18 in [15]). For 0 < p < ∞, we denote by Lp(Ω, ω) the space of Definition 2.2: A weight ω is said to be doubling, if there measurable functions f on Ω such that exists a positive constant C such that

1   p Z ω(2B) ≤ Cω(B), p ||f||Lp(Ω,ω) =  |f(x)| ω(x)dx < ∞, Z B = B(x, r) ⊂ n ω(B) = ω(x)dx E for every ball R , where B where ω is a weight, and Ω be open in Rn. and 2B denotes the ball with the same center as B which is It is a well-known fact that the space Lp(Ω, ω), endowed twice as large. The inﬁmum over all constants C is called the with this norm is a Banach space. We also have that the dual doubling constant of ω. 0 0 space of Lp(Ω, ω) is the space Lp (Ω, ω1−p ). It follows directly from the Ap condition and Holder¨ inequality We now determine conditions on the weight ω that guar- that an Ap-weight has the following strong doubling property. antee that functions in Lp(Ω, ω) are locally integrable on Ω. In particular, every Ap-weight is doubling (see Corollary 15.7 Proposition 2.1: [17], [18] Let 1 ≤ p < ∞. If the weight ω in [16]). is such that Proposition 2.2: [21] Let ω ∈ Ap with 1 ≤ p < ∞ and let n −1 1 E be a measurable subset of a ball B ⊂ R . Then ω p−1 ∈ L (Ω) if p > 1, loc |E|p ω(E) 1 ≤ C ess sup < +∞ if p = 1, |B| ω(B) x∈B ω(x) where C is the A constant of ω. for every ball B ⊂ Ω. Then, p Remark 2.1: If ω(E) = 0 then |E| = 0. The measure ω and p 1 L (Ω, ω) ⊂ Lloc(Ω). the Lebesgue measure |.| are mutually absolutely continuous,

that is they have the same zero sets ω(E) = 0 if and only (A1) For j = 1, ..., n, Bj, Aj, g and H are Carateodory´ if |E| = 0; so there is no need to specify the measure when functions. using the ubiquitous expression almost everywhere and almost (A2) There are positive functions ∞ every, both abbreviated a.e.. h1, h2, h3, h4, h5, h6 ∈ L (Ω) and 1,p The weighted Sobolev space W (Ω, ω) is defined as follows. p0 1 1 n K1,K4 ∈ L (Ω, ω1) with p + p0 = 1 and Definition 2.3: Let Ω ⊂ R be open, and let ω be an Ap- q0 1 1 weight, 1 ≤ p < ∞. We define the weighted Sobolev space K2,K3 ∈ L (Ω, ω2) with q + q0 = 1 such that : 1,p p W (Ω, ω) as the set of functions u ∈ L (Ω, ω) with weak p p p0 derivatives Dju ∈ L (Ω, ω), for j = 1, ..., n. The norm of u |A(x, ξ)| ≤ K1(x) + h1(x)|ξ| , 1,p q q in W (Ω, ω) is given by 0 0 |B(x, η, ξ)| ≤ K2(x) + h2(x)|η| q + h3(x)|ξ| q , Z n Z p X p p p 0 ||u||W 1,p(Ω,ω) = |u(x)| ω(x)dx+ |Dju(x)| ω(x)dx. |g(x, η)| ≤ K4(x) + h6(x)|η| p , Ω j=1 Ω and 1,p ∞ We also define W0 (Ω, ω) as the closure of C0 (Ω) in q q 1,p q0 q0 W (Ω, ω) with respect to the norm ||.||W 1,p(Ω,ω). Note that |H(x, η, ξ)| ≤ K3(x) + h4(x)|η| + h5(x)|ξ| . ∞ 1,p C0 (Ω) is dense in W0 (Ω, ω). (A3) There exists a constant α > 0 such that : Equipped by this norm, W 1,p(Ω, ω) and W 1,p(Ω, ω) are 0 0 0 0 separable and reflevixe Banach spaces see Proposition 2.1.2. hA(x, ξ) − A(x, ξ ), ξ − ξ i ≥ α|ξ − ξ |p, in [21] and see [18] for more informations about the spaces 0 0 0 1,p 1,p hB(x, η, ξ) − B(x, η , ξ ), ξ − ξ i ≥ 0, W (Ω, ω) . The dual of space W0 (Ω, ω) is the space 0 0 −1,p 1−p 0 0 W0 (Ω, ω ). g(x, η) − g(x, η ) η − η ≥ 0, Let us give the following theorems which are needed later. Theorem 2.1: [14] Let ω ∈ Ap, 1 ≤ p < ∞, and let Ω be a and n p bounded open set in R . If um −→ u in L (Ω, ω), then there 0 0 0 p H(x, η, ξ) − H(x, η , ξ ) η − η ≥ 0, exist a subsequence (umk ) and a function Φ ∈ L (Ω, ω) such that 0 whenever (η, ξ), (η0, ξ0) ∈ R × Rn with η 6= η and (i) u (x) −→ u(x), m −→ ∞, ω-a.e. on Ω. 0 mk k ξ 6= ξ where h., .i denotes here the usual inner (ii) |u (x)| ≤ Φ(x), ω-a.e. on Ω. mk Theorem 2.2: [11] (The weighted Sobolev inequality) Let product in Rn .

ω ∈ Ap, 1 ≤ p < ∞, and let Ω be a bounded open set in (A4) There are constants λ1, λ2, λ3, λ4 > 0 such that : n R . There exist constants CΩ and δ positive such that for all 1,p p n hA(x, ξ), ξi ≥ λ1|ξ| , u ∈ W0 (Ω, ω) and all θ satisfying 1 ≤ θ ≤ n−1 + δ, q q ||u||Lθp(Ω,ω) ≤ CΩ||∇u||Lp(Ω,ω), hB(x, η, ξ), ξi ≥ λ2|ξ| + λ3|η| , p where CΩ depends only on n, p, the Ap constant of ω and the g(x, η)η ≥ λ4|η| , diameter of Ω. and Theorem 2.3: [22] Let A : X −→ X∗ be a monotone, coercive and hemicontinuous operator on the real, separable, H(x, η, ξ)η ≥ 0. reflexive Banach space X. Then the following assertions hold: B. Notions of solutions ∗ (a) For each T ∈ X , the equation Au = T has a The definition of a weak solution for Problem (1) can be solution u ∈ X. stated as follows. (b) If the operator A is strictly monotone, then equation 1,p Definition 3.1: We say that an element u ∈ W0 (Ω, ω1) is Au = T has a unique solution u ∈ X. a weak solution of Problem (1) if : III.BASICASSUMPTIONSANDNOTIONOFSOLUTIONS Z Z hA(x, ∇u), ∇ϕiω dx + hB(x, u, ∇u), ∇ϕiω dx A. Basic assumptions 1 2 Ω Z Ω Z Let us now give the precise hypotheses on the Problem + g(x, u)ϕω1dx + H(x, u, ∇u)ϕω2dx (1), we assume that the following assumptions: Ω be a Ω Ω Z n Z bounded open subset of n( n ≥ 2), 1 < q < p < ∞, X R = f0ϕdx + fjDjϕdx, let ω1 and ω2 are two weights functions, and let Aj : Ω j=1 Ω n n Ω × R −→ R, Bj :Ω × R × R −→ R (j = 1,p for all ϕ ∈ W0 (Ω, ω1). 1, ..., n), with B(x, η, ξ) = B1(x, η, ξ), ..., Bn(x, η, ξ) and Remark 3.1: We seek to establish a relationship between A(x, ξ) = A1(x, ξ), ..., An(x, ξ) , g :Ω × R −→ R and ω1 and ω2, in order to ensure the existence and uniqueness H :Ω × × n −→ satisfying the following assumptions: of solution for our Problem (1). At first we notice if ω2 ∈ R R R ω1

0 0 s p −1,p 1−p L (Ω, ω1) where s = p−q , 1 < q < p < ∞ and ω1, ω2 ∈ Ap, We will show that T ∈ W0 (Ω, ω1 ) and B(u, .) is linear, 1,p then, by Holder¨ inequality we obtain for each u ∈ W0 (Ω, ω1).

q p (i) Using Holder¨ inequality and Theorem 2.2(with θ = 1), ||u||L (Ω,ω2) ≤ Cp,q||u||L (Ω,ω1), we obtain ω2 1/q where Cp,q = || || s . ω1 L (Ω,ω1) |T(ϕ)| IV. MAIN RESULT n Z X Z A. Result on the existence and uniqueness ≤ |f0| |ϕ| dx + |fj| |Djϕ| dx Ω j=1 Ω In this subsection we will state the existence and uniqueness  n  of solution to Problem (1) in Theorem 4.1. In the next X ≤ Cp,q||f0/ω2|| q0 + ||fj/ω1|| p0 ||ϕ|| 1,p .  L (Ω,ω2) L (Ω,ω1) W0 (Ω,ω1) subsections we will present the proof. j=1 Theorem 4.1: Let 1 < q < p < ∞ and assume that (A1) −

(A4) holds. If q0 0 0 According to f0/ω2 ∈ L (Ω, ω2) and f/ω1 ∈ q p 0 (i) f0/ω2 ∈ L (Ω, ω2) and fj/ω1 ∈ L (Ω, ω1)(j = p0 −1,p ∗ L (Ω, ω1), we deduce that T ∈ W0 (Ω, ω1 ). 1, ..., n).. 1,p ω2 s (ii) Let u ∈ W0 (Ω, ω1). We have (ii) ω1, ω2 ∈ Ap such that ∈ L (Ω, ω1), where s = ω1 p . p−q |B(u, ϕ)| ≤ |B (u, ϕ)| + |B (u, ϕ)| + |B (u, ϕ)| + |B (u, ϕ)|. u ∈ W 1,p(Ω, ω ) 1 2 3 4 Then, the Problem (1) has only one solution 0 1 . (4) B. Proof of Theorem 4.1 In (4), by (A2),Holder¨ inequality, Remark 3.1 and Theorem 2.2(with θ = 1), we have The basic idea of our proof is to reduce the Problem (1) to an operator equation Au = T and apply the Theorem 2.3. The Z proof of Theorem 4.1 will be divided into several steps. |B1(u, ϕ)| ≤ |A(x, ∇u)||∇ϕ|ω1dx Ω 1) Equivalent operator equation: In this subsection, we use Z p 0 the somme tools and the condition (A2) to prove an existence ≤ K1 + h1|∇u| p |∇ϕ|ω1dx Ω the operator A such that the Problem (1) is equivalent to the p p0 ≤ ||K || p0 + ||h || ∞ ||u|| 1,p ||ϕ|| 1,p , operator equation Au = T. We introduce the operators 1 L (Ω,ω1) 1 L (Ω) W (Ω,ω1) W0 (Ω,ω1) 0 1,p T : W0 (Ω, ω1) −→ R n Z X Z and ϕ −→ T(ϕ) = f0ϕdx + fjDjϕdx, Ω j=1 Ω Z |B2(u, ϕ)| ≤ |B(x, u, ∇u)||∇ϕ|ω2dx and Ω Z q q 0 0 1,p 1,p ≤ K2 + h2|u| q + h3|∇u| q |∇ϕ|ω2dx B : W0 (Ω, ω1) × W0 (Ω, ω1) −→ R Ω q q (u, ϕ) −→ B1(u, ϕ) + B2(u, ϕ) + B3(u, ϕ) + B4(u, ϕ), q0 q0 ≤ ||K2|| q0 Cp,q||∇ϕ|| p + ||h2|| ∞ Cp,q||u|| p L (Ω,ω2) L (Ω,ω1) L (Ω) L (Ω,ω1) q q where, B1, B2, B3 and B4 are deﬁned as follows q0 q0 p ∞ p Cp,q||∇ϕ||L (Ω,ω1) + ||h3||L (Ω)Cp,q||∇u||Lp(Ω,ω )Cp,q||∇ϕ||L (Ω,ω1) 1,p 1,p h 1 B1 : W0 (Ω, ω1) × W0 (Ω, ω1) −→ R q ≤ C ||K || q0 + C ||h || ∞ + ||h || ∞ Z p,q 2 L (Ω,ω2) p,q 2 L (Ω) 3 L (Ω) B1(u, ϕ) = hA(x, ∇u), ∇ϕiω1dx, q−1 i ||u|| 1,p ||ϕ|| 1,p . Ω W (Ω,ω1) W0 (Ω,ω1) 0 1,p 1,p B2 : W0 (Ω, ω1) × W0 (Ω, ω1) −→ R Z Analogously, we have B2(u, ϕ) = hB(x, u, ∇u), ∇ϕiω2dx, Ω Z : W 1,p(Ω, ω ) × W 1,p(Ω, ω ) −→ |B3(u, ϕ)| ≤ |g(x, u)||ϕ|ω1dx B3 0 1 0 1 R Ω Z p p0 B3(u, ϕ) = g(x, u)ϕω1dx. ≤ ||K || p0 + ||h || ∞ ||u|| 1,p ||ϕ|| 1,p , 4 L (Ω,ω1) 6 L (Ω) W (Ω,ω1) Ω W0 (Ω,ω1) 0 1,p 1,p B4 : W0 (Ω, ω1) × W0 (Ω, ω1) −→ R Z and B4(u, ϕ) = H(x, u, ∇u)ϕω2dx. Ω Z 1,p |B4(u, ϕ)| ≤ |H(x, u, ∇u)||ϕ|ω2dx Then u ∈ W0 (Ω, ω1) is a weak solution of Problem (1) if Ω h q and only if ≤ C ||K || q0 + C ||h || ∞ + ||h || ∞ p,q 3 L (Ω,ω2) p,q 4 L (Ω) 5 L (Ω) 1,p q−1 i ||u|| 1,p ||ϕ|| 1,p . B(u, ϕ) = T(ϕ), for all ϕ ∈ W0 (Ω, ω1). W (Ω,ω1) W0 (Ω,ω1) 0

1,p Hence, in (4) we obtain, for all u ∈ W0 (Ω, ω1), Hence, we obtain |B(u, ϕ)| hAu, ui p−1 ≥ min(λ1, λ4)kuk 1,p . h kuk 1,p W0 (Ω,ω1) ≤ kK k p0 + ||K || p0 + C kK k q0 W0 (Ω,ω1) 1 L (Ω,ω1) 4 L (Ω,ω1) p,q 3 L (Ω,ω2) p p0 Therefore, since p > 1, we have +kK k q0 + kh k ∞ + kh k ∞ kuk 2 L (Ω,ω2) 1 L (Ω) 6 L (Ω) 1,p W0 (Ω,ω1) q hAu, ui +Cp,q kh2kL∞(Ω) + kh3kL∞(Ω) + ||h4||L∞(Ω) + ||h5||L∞(Ω) −→ +∞ as kukW 1,p(Ω,ω ) −→ +∞, kuk 1,p 0 1 q−1 i W0 (Ω,ω1) kuk 1,p kϕk 1,p . W (Ω,ω ) W (Ω,ω1) 0 1 0 that is, A is coercive. Since B(u, .) is linear and continuous, for each u ∈ 3) Monotonicity of the operator A: The operator A is 1,p strictly monotone. In fact, for all u , u ∈ W 1,p(Ω, ω ) with W0 (Ω, ω1), there exists a linear and continuous 1 2 0 1 1,p operator denoted by A : W (Ω, ω1) −→ u1 6= u2, we have 0 0 0 W −1,p (Ω, ω1−p ) such that 0 1 hAu1 − Au2, u1 − u2i = B(u1, u1 − u2) − B(u2, u1 − u2) Z h u, ϕi = (u, ϕ), u, ϕ ∈ W 1,p(Ω, ω ), A B for all 0 1 = hA(x, ∇u1) − A(x, ∇u2), ∇(u1 − u2)iω1dx Ω where hf, xi denotes the value of the linear functional f Z at the point x . Moreover, we have + hB(x, u1, ∇u1) − B(x, u2, ∇u2), ∇(u1 − u2)iω2dx Ω Z kAuk∗ + g(x, u1) − g(x, u2) u1 − u2 ω1dx Ω ≤ kK k p0 + ||K || p0 + C kK k q0 1 L (Ω,ω1) 4 L (Ω,ω1) p,q 3 L (Ω,ω2) Z p p0 + H(x, u1, ∇u1) − H(x, u2, ∇u2) u1 − u2 ω2dx. +kK k q0 + kh k ∞ + kh k ∞ kuk 2 L (Ω,ω2) 1 L (Ω) 6 L (Ω) 1,p Ω W0 (Ω,ω1) q +Cp,q kh2kL∞(Ω) + kh3kL∞(Ω) + ||h4||L∞(Ω) + ||h5||L∞(Ω) Thanks to (A3), we obtain q−1 Z kuk 1,p , p W0 (Ω,ω1) hAu1 − Au2, u1 − u2i ≥ α|∇(u1 − u2)| ω1dx Ω p where ≥ αk∇(u1 − u2)k p , L (Ω,ω1) and by Theorem 2.2(with θ = 1), we conclude that kAuk∗ = sup |hAu, ϕi| = |B(u, ϕ)| : kϕkW 1,p(Ω,ω ) = 1 0 1 α hAu − Au , u − u i ≥ ku − u kp . 0 0 1 2 1 2 p 1 2 W 1,p(Ω,ω ) −1,p 1−p (CΩ + 1) 0 1 is the norm in W0 (Ω, ω1 ). Consequently, Problem (1) is equivalent to the operator equa- Therefore, the operator A is strictly monotone. tion 4) Continuity of the operator A: We need to show that 1,p the operator A is continuous. To this purpose let um −→ Au = T, u ∈ W0 (Ω, ω1). 1,p u in W0 (Ω, ω1) as m −→ ∞. Note that if um −→ u in 1,p p 2) Coercivity of the operator A: In this step, we prove W0 (Ω, ω1), then um −→ u in L (Ω, ω1) et ∇um −→ ∇u p n that the operator A is coercive. To this purpose let u ∈ in (L (Ω, ω1)) . Hence, thanks to Theorem 2.1, there exist 1,p p W0 (Ω, ω1), we have a subsequence (umk ), functions Φ1 ∈ L (Ω, ω1) and Φ2 ∈ p L (Ω, ω1) such that hAu, ui = B(u, u) Z Z umk (x) −→ u(x), ω1 − a.e. in Ω = hA(x, ∇u), ∇uiω1dx + hB(x, u, ∇u), ∇uiω2dx Ω Ω |u (x)| ≤ Φ (x), ω − a.e. Ω Z Z mk 1 1 in + g(x, u)uω1dx + H(x, u, ∇u)uω2dx. (5) Ω Ω ∇umk (x) −→ ∇u(x), ω1 − a.e. in Ω Moreover, from (A4) and Theorem 2.2(with θ = 1), we obtain |∇u (x)| ≤ Φ (x), ω − a.e. Ω. Z Z mk 2 1 in hAu, ui ≥ λ |∇u|pω dx + λ |∇u|qω dx 1 1 2 2 −1,p0 1−p0 ΩZ ZΩ We will show that Aum −→ Au in W0 (Ω, ω1 ). In q p + λ3 |u| ω2dx + λ4 |u| ω1dx order to prove this convergence we proceed in four steps. Ω Z Ω Z Step 1: p p ≥ min(λ1, λ4) |∇u| ω1dx + |u| ω1dx For j = 1, ..., n, we deﬁne the operator ΩZ ΩZ q q 1,p p0 + min(λ2, λ3) |∇u| ω2dx + |u| ω2dx Fj : W0 (Ω, ω1) −→ L (Ω, ω1) Ω Ω p q (Fju)(x) = Aj(x, ∇u(x)). = min(λ1, λ4)kuk 1,p + min(λ2, λ3)kuk 1,q W (Ω,ω1) W (Ω,ω2) p 0 0 ≥ min(λ1, λ4)kuk 1,p . We now show that the operator Fj is bounded and continuous. W0 (Ω,ω1)

1,p 1,p (i) Let u ∈ W0 (Ω, ω1). Using (A2) and Theorem 2.2(with (i) Let u ∈ W0 (Ω, ω1). Using (A2), Remark 3.1 and θ = 1), we obtain Theorem 2.2(with θ = 1), we obtain

0 Z 0 0 Z q p p0 q kGjuk q0 = |Bj(x, u, ∇u)| ω2dx kFjuk p0 = |Aj(x, ∇u)| ω1dx L (Ω,ω2) L (Ω,ω1) Ω Ω 0 0 Z q q q Z p p 0 0 0 ≤ K + h |u| q + h |∇u| q ω dx ≤ K1 + h1|∇u| p ω1dx 2 2 3 2 Ω Ω Z 0 0 0 Z 0 0 h q q q i p p p ≤ C K + h |u|q + h |∇u|q ω dx ≤ Cp K + h |∇u| ω1dx q 2 2 3 2 1 1 Ω Ω h 0 0 h p0 p0 p i q q q ≤ Cq kK2k 0 + kh2k ∞ kuk q ≤ Cp kK1k 0 + kh1k ∞ k∇uk p Lq (Ω,ω ) L (Ω) L (Ω,ω2) Lp (Ω,ω ) L (Ω) L (Ω,ω1) 2 1 0 i h p0 p0 p i q q +kh3k ∞ k∇uk q ≤ Cp kK1k p0 + kh1k ∞ kuk 1,p , L (Ω) L (Ω,ω2) L (Ω,ω1) L (Ω) W (Ω,ω1) 0 h q0 q0 q q ≤ Cq kK2k q0 + kh2k ∞ Cp,qkuk p L (Ω,ω2) L (Ω) L (Ω,ω1) where the constant Cp depends only on p. 0 i 1,p q q q +kh3k ∞ C k∇uk p (ii) Let um −→ u in W0 (Ω, ω1) as m −→ ∞. We need to L (Ω) p,q L (Ω,ω1) p0 h 0 0 show that Fjum −→ Fju in L (Ω, ω1). We will apply q q q ≤ Cq kK2k q0 + Cp,q kh2k ∞ L (Ω,ω2) L (Ω) the Lebesgue’s theorem and the convergence principle in q0 q i Banach spaces. +kh3kL∞(Ω) kuk 1,p , W0 (Ω,ω1) By (A2), we obtain where the constant Cq depends only on q. 0 1,p p (ii) Let um −→ u in W0 (Ω, ω1) as m −→ ∞. We will kF u − F uk 0 0 j mk j Lp (Ω,ω ) q Z 1 show that Gjum −→ Gju in L (Ω, ω2). p0 = |Fjumk (x) − Fju(x)| ω1dx According to (A2) and Remark 3.1, we obtain ZΩ p0 ≤ (|A (x, ∇u )| + |A (x, ∇u)|) ω dx 0 Z j mk j 1 q q0 Ω kGjumk − Gjuk q0 = |Gjumk (x) − Gju(x)| ω2dx Z L (Ω,ω2) p0 p0 Ω ≤ Cp |Aj(x, ∇umk )| + |Aj(x, ∇u)| ω1dx Z q0 Ω 0 0 ≤ |B (x, u , ∇u | + |B (x, u, ∇u)| ω dx Z p p p p j mk mk j 2 p0 p0 Ω ≤ Cp K1 + h1|∇umk | + K1 + h1|∇u| ω1dx Z 0 0 Ω Z q q 0 p0 p0 p ≤ Cq |Bj(x, umk , ∇umk )| + |Bj(x, u, ∇u)| ω2dx ≤ 2CpCp K1 + h1 Φ2 ω1dx Ω 0 Ω Z q q q 0 h p0 p0 p i 0 0 ≤ Cq K2 + h2|um | q + h3|∇um | q ω2dx ≤ 2CpCp kK1k p0 + kh1k ∞ kΦ2k p . k k L (Ω,ω1) L (Ω) L (Ω,ω1) Ω 0 Z q q q 0 0 Hence, thanks to (A1), we get, as k −→ ∞ + K2 + h2|u| q + h3|∇u| q ω2dx Ω 0 Z 0 0 0 Fjumk (x) = Aj(x, ∇umk (x)) −→ Aj(x, ∇u(x)) = Fju(x), q q q q q ≤ 2CqCq K2 + h2 Φ1 + h3 Φ2 ω2dx Ω for almost all x ∈ Ω. Therefore, by Lebesgue’s theorem, 0 h q0 q0 q ≤ 2CqCq kK2k q0 + kh2k ∞ kΦ1k q we obtain L (Ω,ω2) L (Ω) L (Ω,ω2) q0 q i +kh3k ∞ kΦ2k q kF u − F uk p0 −→ 0, L (Ω) L (Ω,ω2) j mk j L (Ω,ω1) 0 h q0 q q0 q ≤ 2CqCq kK2k q0 + Cp,qkh2k ∞ kΦ1k p that is, L (Ω,ω2) L (Ω) L (Ω,ω1) q q0 q i 0 +C kh3k ∞ kΦ2k p . p p,q L (Ω) L (Ω,ω1) Fjumk −→ Fju in L (Ω, ω1). Then, by (A1), we have, as k −→ ∞ Finally, in view to convergence principle in Banach

spaces, we have Gjumk (x) −→ Gju(x), a.e. x ∈ Ω.

p0 Therefore, in view to Lebesgue’s theorem, we have Fjum −→ Fju in L (Ω, ω1). (6)

kGjum − Gjuk q0 −→ 0, Step 2: k L (Ω,ω2) For j = 1, ..., n, we deﬁne the operator that is, 0 0 q 1,p q Gjumk −→ Gju in L (Ω, ω2). Gj : W0 (Ω, ω1) −→ L (Ω, ω2) (Gju)(x) = Bj(x, u(x), ∇u(x)). Hence, from the convergence principle in Banach spaces, we conclude that We also have that the operator Gj is continuous and bounded. q0 In fact, Gjum −→ Gju in L (Ω, ω2). (7)

1,p Step 3: (i) Let u ∈ W0 (Ω, ω1). Using (A2) and Remaek 3.1, we We deﬁne the operator obtain 1,p 0 p 0 Z 0 H : W0 (Ω, ω1) −→ L (Ω, ω1) ˜ q p kHuk q0 = |H(x, u(x), ∇u(x))| ω2dx (Hu)(x) = g(x, u(x)). L (Ω,ω2) Ω 0 Z q q q 0 0 In this step, we will show that the operator H is bounded and ≤ K3 + h4|u| q + h5|∇u| q ω2dx continuous. Ω Z h q0 q0 q q0 qi 1,p ≤ Cq K3 + h4 |u| + h5 |∇u| ω2dx (i) Let u ∈ W0 (Ω, ω1). Using (A2), we obtain Ω h 0 0 Z q q q 0 0 ≤ Cq kK3k q0 + kh4k ∞ kuk q p p L (Ω,ω2) L (Ω) L (Ω,ω2) kHuk p0 = |g(x, u)| ω1dx 0 L (Ω,ω1) q q i Ω 0 + kh5kL∞(Ω)k∇ukLq (Ω,ω ) Z p p 2 p0 h 0 0 ≤ K4 + h6|u| ω1dx q q q q ≤ Cq kK3k q0 + kh4k ∞ Cp,qkuk p Ω L (Ω,ω2) L (Ω) L (Ω,ω1) Z 0 p0 p0 p q q q i ≤ C K + h |u| ω dx + kh5k ∞ C k∇uk p p 4 6 1 L (Ω) p,q L (Ω,ω1) Ω h 0 0 h p0 p0 p i q q q ≤ Cq kK3k 0 + C kh4k ∞ ≤ Cp kK4k 0 + kh6k ∞ kuk p Lq (Ω,ω ) p,q L (Ω) Lp (Ω,ω ) L (Ω) L (Ω,ω1) 2 1 0 i h p0 p i q q +kh k ∞ kuk 1,p , ≤ C kK k p0 + kh k kuk , 5 L (Ω) p 3 L (Ω,ω1) 6 L∞(Ω) 1,p W0 (Ω,ω1) W0 (Ω,ω1)

where the constant Cp depends only on p. where the constant Cq depends only on q. 1,p 1,p (ii) Let um −→ u in W0 (Ω, ω1) as m −→ ∞. We need to (ii) Let um −→ u in W0 (Ω, ω1) as m −→ ∞. We need to p0 ˜ ˜ q0 show that Hum −→ Hu in L (Ω, ω1). show that Hum −→ Hu in L (Ω, ω2). By (A2) , we get According to (A2) and Remark 3.1, we have p0 0 Z 0 kHumk − Huk p0 q q L (Ω,ω1) ˜ ˜ ˜ ˜ kHumk − Huk q0 = |Humk (x) − Hu(x)| ω2dx Z 0 L (Ω,ω2) p Ω 0 = |Humk (x) − hu(x)| ω1dx Z q Ω ≤ |H(x, um , ∇um )| + |H(x, u, ∇u)| ω2dx Z 0 k k ≤ (|g(x, u )| + |g(x, u)|)p ω dx Ω mk 1 Z 0 0 Ω q q Z ≤ Cq |H(x, umk , ∇umk )| + |H(x, u, ∇u)| ω2dx p0 p0 Ω ≤ C |g(x, u )| + |g(x, u)| ω dx 0 p mk 1 Z q q q Ω q0 q0 ≤ Cq K3 + h4|um | + h5|∇um | ω2dx Z h p 0 p 0 i k k p0 p p0 p Ω ≤ Cp (K4 + h6|um | ) + (K4 + h6|u| ) ω1dx 0 k Z q q q Ω 0 0 Z + K3 + h4|u| q + h5|∇u| q ω2dx 0 p0 p0 p ≤ 2CpCp K4 + h6 Φ1 ω1dx Ω Z Ω 0 q0 q0 q q0 q 0 h p0 p0 p i ≤ 2CqCq K3 + h4 Φ1 + h5 Φ2 ω2dx ≤ 2CpC kK4k 0 + kh6k ∞ kΦ1k p , p Lp (Ω,ω ) L (Ω) L (Ω,ω1) Ω 1 0 h q0 q0 q ≤ 2CqCq kK3k q0 + kh4kL∞(Ω)kΦ1kLq (Ω,ω ) then, using condition (H1), we deduce, as k −→ ∞ L (Ω,ω2) 2 q0 q i + kh5k ∞ kΦ2k q L (Ω) L (Ω,ω2) Humk (x)) −→ Hu(x), a.e. x ∈ Ω. 0 h q0 q q0 q ≤ 2CqCq kK3k q0 + Cp,qkh4kL∞(Ω)kΦ1kLp(Ω,ω ) Therefore, by the Lebesgue’s theorem, we obtain L (Ω,ω2) 1 q q0 q i + C kh5k ∞ kΦ2k p . p,q L (Ω) L (Ω,ω1) kHu − Huk p0 −→ 0, mk L (Ω,ω1) that is, Hence, from (A1), we deduce, as k −→ ∞ p0 ˜ ˜ Humk −→ Hu in L (Ω, ω1). Humk (x) −→ Hu(x), a.e. x ∈ Ω. We conclude, from the convergence principle in Banach Therefore, by the the Lebesgue’s theorem, we obtain spaces, that

0 kHu˜ − Hu˜ k q0 −→ 0, p mk L (Ω,ω2) Hum −→ Hu in L (Ω, ω1). (8) Step 4: that is, We deﬁne the operator ˜ ˜ q0 Humk −→ Hu in L (Ω, ω2). ˜ 1,p q0 H : W0 (Ω, ω1) −→ L (Ω, ω2) (Hu˜ )(x) = H(x, u(x), ∇u(x)). Thanks to convergence principle in Banach spaces, we conclude that We now show that the operator H˜ is bounded and contin- q0 uous. Hu˜ m −→ Hu˜ in L (Ω, ω2). (9)

1,p 1,p Finally, let ϕ ∈ W0 (Ω, ω1) and using Holder¨ inequality, Hence, for all ϕ ∈ W0 (Ω, ω1), we have Theorem 2.2(with θ = 1) and Remark 3.1, we obtain |B(um, ϕ) − B(u, ϕ)| ≤ |B1(um, ϕ) − B1(u, ϕ)| + |B2(um, ϕ) − B2(u, ϕ)| |B1(um, ϕ) − B1(u, ϕ)| +|B3(um, ϕ) − B3(u, ϕ)| + |B4(um, ϕ) − B4(u, ϕ)| Z n = | hA(x, ∇um) − A(x, ∇u), ∇ϕiω1dx| h X Ω ≤ kFjum − Fjuk p0 + Cp,qkGjum − Gjuk q0 n L (Ω,ω1) L (Ω,ω2) X Z j=1 ≤ |Aj(x, ∇um) − Aj(x, ∇u)||Djϕ|ω1dx ˜ ˜ i Ω +kHum − Huk p0 + Cp,qkHum − Huk q0 kϕk 1,p . j=1 L (Ω,ω1) L (Ω,ω2) W0 (Ω,ω1) n X Z = |Fjum − Fju||Djϕ|ω1dx Then, we get j=1 Ω n kAum − Auk∗ X n ≤ kF u − F uk p0 kD ϕk p j m j L (Ω,ω1) j L (Ω,ω1) X ≤ kFjum − Fjuk p0 + Cp,qkGjum − Gjuk q0 j=1 L (Ω,ω1) L (Ω,ω2)   j=1 n ˜ ˜ X +kHum − Huk p0 + Cp,qkHum − Huk q0 . ≤ kFjum − Fjuk p0 kϕk 1,p , L (Ω,ω1) L (Ω,ω2)  L (Ω,ω1) W0 (Ω,ω1) j=1 Combining (6), (7), (8) and (9), we deduce that

kAum − Auk∗ −→ 0 as m −→ ∞,

|B2(um, ϕ) − B2(u, ϕ)| Z that is, A is continuous. = | hB(x, um, ∇um) − B(x, u, ∇u), ∇ϕiω2dx| Hence, the proof of the theorem 4.1 is completed. Ω n Z X V. EXAMPLE ≤ |Bj(x, um, ∇um) − Bj(x, u, ∇u)||Djϕ|ω2dx j=1 Ω 2 2 2 n Let Ω = {(x, y) ∈ R : x + y < 1}, and consider the X Z 2 2−1/2 = |Gjum − Gju||Djϕ|ω2dx weight functions ω1(x, y) = x + y and ω2(x, y) = −1/3 j=1 Ω x2 + y2 we have that ω , ω ∈ A , p = 4 and q = 3 ,   1 2 4 n 2 2 X and the functions Bj :Ω×R×R −→ R, Aj :Ω×R −→ R 0 q ≤  kGjum − Gjuk q  k∇ϕkL (Ω,ω2) 2 L (Ω,ω2) (j = 1, 2), g :Ω × R× −→ R and H :Ω × R × R −→ R j=1   deﬁned by n 3 X Aj((x, y), ξ) = h1(x, y)ξj , ≤ C kG u − G uk q0 k∇ϕk p p,q  j m j L (Ω,ω2) L (Ω,ω1) j=1 (x2+y2) where h1(x, y) = 2e ,  n  X ≤ Cp,q kGjum − Gjuk q0 kϕk 1,p , B ((x, y), η, ξ) = h (x, y)|ξ |ξ ,  L (Ω,ω2) W0 (Ω,ω1) j 3 j j j=1 2 2 where h3(x, y) = 2 + sin(x + y ),

3 g((x, y), η) = h6(x, y)|η| sgn(η), |B3(um, ϕ) − B3(u, ϕ)| Z 2 where h6(x, y) = 2 − sin (x + y), and ≤ |g(x, um) − g(x, u)||ϕ|ω1dx Ω 2 Z H((x, y), η, ξ) = h5(x, y)ξ sgn(η), = |Hum − Hu||ϕ|ω1dx Ω 2 where h5(x, y) = 2 − cos (xy). ≤ kHum − Huk p0 kϕkLp(Ω,ω ) L (Ω,ω1) 1 Let us consider the partial differential operator ≤ kHum − HukLp0 (Ω,ω )kϕkW 1,p(Ω,ω ), 1 0 1 h i Lu(x) = −div ω1(x)A(x, ∇u(x)) + ω2(x)B(x, u(x), ∇u(x)) and + ω1(x)g(x, u(x)) + ω2(x)H(x, u(x), ∇u(x)), (10) |B4(um, ϕ) − B4(u, ϕ)| Z Therefore, by Theorem 4.1, the problem ≤ |H(x, um, ∇um) − H(x, u, ∇u)||ϕ|ω2dx ZΩ ( cos(xy) ∂ sin(xy) ∂ sin(xy) Lu(x, y) = (x2+y2) − ∂x (x2+y2) − ∂y (x2+y2) in Ω, = |Hu˜ m − Hu˜ ||ϕ|ω2dx Ω u(x, y) = 0 on ∂Ω, ≤ kHu˜ − Hu˜ k q0 kϕk q m L (Ω,ω2) L (Ω,ω2) ˜ ˜ 1,4 ≤ Cp,qkHum − Huk q0 kϕk 1,p . admits one and only solution u ∈ W0 (Ω, ω1). L (Ω,ω2) W0 (Ω,ω1)

REFERENCES [1] A. Abbassi, E. Azroul, A. Barbara, Degenerate p(x)-elliptic equation with second membre in L1, Advances in Science, Technology and Engineering Systems Journal. Vol. 2, No. 5, 45-54, 2017. [2] A. Abbassi, C. Allalou and A. Kassidi. Existence of Entropy Solutions for Anisotropic Elliptic Nonlinear Problem in Weighted Sobolev Space. In : The International Congress of the Moroccan Society of Applied Mathematics. Springer, Cham, p. 102-122, 2019. [3] L. Aharouch , E.Azroul and A.Benkirane, Quasilinear degenerated equations with L1 datum and without coercivity in perturbation terms. Electronic Journal of Qualitative Theory of Differential Equations, vol no 19, p. 1-18, 2006. [4] Y. Akdim and Chakir Allalou. ”Existence and uniqueness of renormal- ized solution of nonlinear degenerated elliptic problems.” Analysis in Theory and Applications 30, 318-343, 2014. [5] L. Boccardo and T. Gallouett,¨ Strongly nonlinear elliptic equations having natural growth terms and L1, Nonlinear Analysis Theory methods and applications, 19(6), 573-579, 1992. [6] L. Boccardo, T. Gallouet,¨ and L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations,Journal Analyse Mathe- matique, vol. 73, pp. 203-223, 1997. [7] H. Brezis and W. Strauss, Semilinear second-order elliptic equations in L1, J. Math. Soc. Japan, 25(4), 565-590, 1973. [8] A.C. Cavalheiro. Existence of solutions for some degenerate quasilinear elliptic equations. Le Matematiche, vol. 63, no 2, p. 101-112, 2008. [9] A.C. Cavalheiro. Existence Results For A Class Of Nonlinear Degener- ate Elliptic Equations. Moroccan Journal of Pure and Applied Analysis, vol. 5, no 2, p. 164-178, 2019. [10] Chiarenza, Filippo. ”Regularity for solutions of quasilinear elliptic equations under minimal assumptions.” Potential Theory and Degenerate Partial Differential Operators. Springer, Dordrecht, 325-334, 1995. [11] Drbek, Pavel, Alois Kufner, and Francesco Nicolosi. Quasilinear elliptic equations with degenerations and singularities. Vol. 5. Walter de Gruyter, 2011. [12] Fabes, Eugene B., Carlos E. Kenig, and Raul P. Serapioni. ”The local regularity of solutions of degenerate elliptic equations.” Communications in Statistics-Theory and Methods 7.1, 77-116, 1982. [13] S. Fucik, O. John, and A. Kufner ”Function Spaces, Noordhoff In- ternational Publishing, Leyden. Academia, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1977.” [14] Garc´ıa-Cuerva, Jose,´ and J.L. Rubio De Francia. Weighted norm inequalities and related topics. Elsevier, North-Holland Mathematics Studies 116, Amsterdam, 1985. [15] J. Heinonen, T. Kilpelainen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monographs, Clarendon Press, Oxford,1993. [16] Kufner, Alois. Weighted sobolev spaces. Vol. 31. John Wiley and Sons Incorporated, 1985. [17] Kufner, Alois, and Bohum`ır Opic. ”How to deﬁne reasonably weighted Sobolev spaces.” Commentationes Mathematicae Universitatis Carolinae 25.3, 537-554, 1984. [18] Muckenhoupt, Benjamin. ”Weighted norm inequalities for the Hardy maximal function.” Transactions of the American Mathematical Society, 207-226, 1972. [19] M. Ruzicka,˙ Electrorheological Fluids: Modelling and Mathematical Theory, Springer, Berlin, 2000. [20] Turesson, Bengt O. Nonlinear potential theory and weighted Sobolev spaces. Vol. 1736. Springer Science and Business Media, 2000. [21] Xu, Xiangsheng. ”A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem.” Differential and Integral Equations 12.1, 83-100, 1999. [22] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol.II/B, Springer-Verlag, New York,1990.

Abdellah AMINE Rachid AIT DAOUD Belaid BOUIKHALENE

Sultan Moulay Slimane University, Sultan Moulay Slimane University, Sultan Moulay Slimane University, Beni Mellal, Morocco Beni Mellal, Morocco Beni Mellal, Morocco

[email protected] [email protected] [email protected]

Abstract—

In the prospect of doing a set of decision support onboards in II. RELATED WORK a public university, we will present a comparison of two ETL In the recent years, a number of different approaches have extraction based in a production databases of students' been suggested for the design, optimization, and automation information. For the deployment, we use Pentaho and Sql of ETL operations. In this section, we present a brief overview Server Tools and we demonstrate the application on the case of these several approaches [7]. Some of the leading data of Sultan Moulay Slimane University in Beni Mellal, integration vendors are IBM, Informatica, Oracle, Microsoft, Morocco Talend, Pentaho, Information Builders, etc. There are many available research papers that offer a Keywords—: Pentaho; Sql Server; Data Warehouse; comparative view of the leading ETL tools in the market, such Business Intelligence as [8-9]. They analyze in details the functionalities and features offered by these tools, and it can be deduced that all of them provide support for all the features that define data I. INTRODUCTION integration tools. Data warehouse (DWs) is delineated as "subject-oriented, Different variants of some approaches for integration of more integrated, timely-variant, and non-volatile collection of ETL tools with data warehouses have been proposes. Shaker data to support the management decision process" [1]. Data H. Ali ElSappagh tries to navigate through the efforts that warehouse emphasizes the collection of data from multiple have been made to use acronyms for ETL, DW, DM, OLAP, sources for useful analysis. Ion-line analytical processing.A data warehouse gives a set of At the center of DWs is the extraction-transformation-loading numerical values based on a set of input values in the form of (ETL) process. ETL is a process utilized to extract data from dimensions [10]. Li, Jain, overcame the limitations of the multiple sources, transform that data to the desired state traditional architecture of Extract, Transform, Load tools, and developed a three-layer architecture based on metadata. This through cleansing, and load it into a target database. The made the ETL process more flexible, versatile and efficient, deliverable is used to generate reports and for analysis. ETL and finally they designed and implemented a new ETL tool consumes up to 70% of all the resources [2-5]. for the drilling data warehouse [11]. A systematic review In the most professional field, the main approach before method was proposed to identify, extract, and analyze the selecting an ETL tool is to perform proofs of concept. main proposals for modeling the conceptual ETL process for However, it is almost impossible to perform proofs of concept data warehouse. The main proposals were identified and of all ETL tools available on the market. Then a pre-selection compared based on the characteristics, activities, and notation is made in the way that two ETL suites are kept for testing. of ETL processes, and the study was concluded by reflecting This pre-selection is generally based on criteria summarized on the studied approaches and providing an update skeleton as follows: the category of the tool, the cost, the type of ETL for future studies. project, and the proof of concepts. In this white paper, we will only look at the use of two ETL tools (Microsoft SQL Server Integration Services SSIS and Pentaho Kettle) [6] based on the generalized criteria for selecting the better tool.

  III. FEATURE COMPARISON BEWEEN PDI AND SSIS MOMS x  In this section, we are going to do a comparative study of  the features for the two extraction tools, especially the Pentaho Index   Data Integration and the Microsoft SQL Server Integration  Services POP   A. Access to data  We note for the triggering process by message, the Table 1. Access to data PDI tool is not suitable for this procedure, whereas for the features PDI SSIS trigger by type of polling the two tools are robust. Read the full table    Oracle is the only database that supports JMS natively in the form of Oracle Advanced Queueing. If the message receiver Complete view of   reading is not tookeen on thisJMS implementation, it is usually possible to find some sort of messaging bridge that will Calling stored   transform and forward messages from one JMS procedure implementation to another. Uploading clause   where/order by C. Data processing Query    Table 3. Data processing Query Builder   Features PDI SSIS  Transformation     functions of dates and Reading / writing all numbers simple and complex Statistical functions x  data types qualities

Read the full table Allows transcoding with x    a reference table  Heterogeneous joints  CSV   x    Supported modes of joint  Fixed / Limited   external    XML  Management of nested x    queries  Excel  Treatment options for a     programming language  Validity flat files  Added new   x  transformations and  business processes Validity of XML files   Mapping graphics     Drag and Drop   For the access to relational data, flat files and applications  of connectors, PDI and SSIS are good solutions for Graphical representation   thesefeatures.The two tools allow the analysis of data from of flow various sources to determine the transformations necessary to Viewing under x  perform aggregations, data deletions, automatic corrections of development data errors, etc.But for the validation of the flat files, the SSIS tool is more robust in comparaison to PDI. Impact analyses tools    B. Triggering pocess Debugging Tools    Table 2. Triggering process Generation of technical x  features PDI SSIS and functional CORBA x  documentation Viewing documentation x  through the web XML RPC  x  Management of For some   integration errors steps JMC x x

The two tools provide a mechanism of query directly Use of rights of a x x in SQL which allows to make all modes of joint and nested directory DBMS  queries.It ispossible with SQL Server to join data from an Security type active directory to data in a SQL Server and create a view of security which the joined data. For the treatment of the data, the two tools are not compatible for the transformations and calculations by contains the default, they are recommended for the manual transformations repository Security scenario   except for the generation of technical and functional creation documents. Security access to   metadata D. Advanced development and deployment/production start Safety manual task   launch Table 4.Advanced development and deployment/production Security Administration   start Console Features PDI SSIS Application   Programming We note that the PDI is not compatible for the Interface generation of specific log, the interfacage with Tools of Integration of   Supervision, the planning of integrated treatment and for the external functions security of the database management system that does not Crash recovery x x contain the repository. mechanism

Setting buffers /   OMPARATIVE TREATMENT TIMES indexes / caches IV. C Team Development   Management A. Test realization methodology Versioning x  Test n° Compilation x Yes for C# Descriptive treatments 1. Extracting data from an Excel file 2. Loading data into another Excelfile Type into Windows or Windows production Unix command 3. The input file contains 5 typed fields: command line  COD_IND [NUMBER] (Student Code) line  COD_NNE_IND [NUMBER] (National History x x visualization into ID of the student) production  DATE_NAI_IND [DATE] (Date of birth of the student) It was found that the two tools are not compatible for  LIB_NOM_PAT_IND [String] (Family the recovery mechanism on incident and for the history name of student) visualization into production,but generally they are used for the other properties of the advanced development and  LIB_PR_IND [String] (Student's first deployment of production setting. name)

B. Modeling in Pentaho Data Integration (PDI) [8]. [9]. E. Administration and security management

Table 5.Administration and security management Features PDI SSIS Administration Console  

Automated log   management Specific log generation x   Interfacing with x  monitoring tools  Integrated treatment x  planning tool Fig. 1: Extraction of 1000 rows with PDI

data flow and event driven architecture. It allows great flexibility to the developer to design the structure and flow the C. Modeling in SQL Server Integration Services ETL process.On the other side, PDI includes many more (SSIS)[10]. options to access outside data such as a Google Analytics and several options to access Web services. It can be used on either Windows or Linux operating systems. The choice between the SSIS ETL and PDI thus depends essentially on the typology of the project it leads.

REFERENCES

[1] Inmon W, Strauss D, Neushloss G. DW 2.0 the Architecture for the next generation of Data Warehousing. Morgan Kaufman. 2007. [2] Simitisis A, Vassiliadis P, Skiadopoulos S, Sellis T. Data Warehouse Refreshment. Data Warehouses and OLAP: Concepts, Architectures and Solutions. IRM Press. 2007: 111-134. [3] Kimball R, Caserta J. The Data Warehouse ETL Toolkit: Practical Techniques for Extracting, Cleaning, Conforming, and Delivering Data. Wiley Publishing, Inc. 2004. [4] Kabiri A, Wadjinny F, Chiadmi D. Towards a Framework for Fig. 2: Extraction of 1000 rows with SSIS Conceptual Modelling of ETL Processes. Proceedings of The first international conference on Innovative Computing Technology (INCT We performed the same work for 5000 and 10000 rows. 2011), Communications in Computer and Information Science. Heidelberg. 2011; 241: 146-160. Table 6.Processing time for both tools [5] Vassiliadis P, Simitsis A. Extraction-Transformation-loading, (ETL) Processes in Data Warehouse Environments. In Encyclopedia of Number of rows PDI SSIS Database Technologies and Applications, Idea Group. 2005. 1000 837 655 [6] Golfarelli M, Rizzi S. Data Warehouse Design, Modern Principles and 5000 1384 1014 Methodologies. McGraw Hill Companies, srl Publishing Group Italia. 10000 3009 1513 2009. [7] Jovanovic P, Theodorou V, Abelló A, Nakuçi E. Data generator for evaluating ETL process Quality (Science direct). In Press. 2016. [8] Thoo E, Friedman T, Beyer Mark A. Magic Quadrant for Data Integration Tools. Gartner RAS Core Research Note G. 2013. [9] Pall AS, Khaira JS. A comparative review of extraction, transformation and loading tools. Database Systems Journal BOARD. 2013; 4(2): 42-51. [10] Simitsis A, Vassiliadis P, Sellis T. State space optimization of ETL workflow. IEEE Transactions on Knowledge and Data Engineering. 2005; 17(10): 1404-1419. [11] Jian L, Bihua X. ETL tool research and implementation based on drilling data warehouse. Seventh International Conference on Fuzzy Systems and Knowledge Discovery. 2010; 6: 2567-2569.

Fig3: Comparison of the results obtained for the two tools

The performance of the treatment of time is an important criterion in the choice of an ETL, but from these results we cannot prejudge the actual performance in a production environment, since time of execution variesfollowing the typology of treatments. At the end of our comparative study, we can conclude that SSIS and PDI are two tools of ETL with their own specificities.These are real alternatives to the ETL owners as Informatica Power Centeror Oracle Warehouse Builder.These two tools offer all the features necessary for an ETL.

V. 5. CONCLUSION Both SSIS and PDI are robust solutions to perform ETL in a data warehouse. SSIS emphasizes configuration over coding; however, because of the limited amount of available transformation objects, coding will be required to process complex data. SSIS’s strength comes from its control flow, IJOA ©2021 13 A novel C/Ag/SiO2 Sonogel-Carbon electrode Preparation, characterization, application and statistical validation as amperometric sensors for cathecol determination

Redouan El Khamlichi Taoufiq El Harrouti Mohammed Lamarti Sefian

Electrochemistry & Interfacial Systems Laboratoire of Engineering Science Laboratoire des Sciences Appliquées et (ERESI) Team, Faculty of Sciences, National School of Applied Sciences Didactique (LaSAD), Ecole Normale Abdelmalek Essaâdi University, Ibn Tofail university Supérieure, Abdelmalek Essaâdi M’Hannech II, B.P. 2121, 93002, Kenitra, Morocco University, M’Hannech II, B.P. 2121, Tetouan, Morocco 93002, Tetouan, Morocco [email protected] [email protected] [email protected]

Sara Elliazidi Jalal Kassout Dounia Bouchta

Normal School of Applied Sciences, Department of Biology, Abdelmalek Electrochemistry & Interfacial Systems Abdelmalek Essaâdi University Essaâdi University Tetouan, Morocco. (ERESI) Team, Faculty of Sciences, Tetouan, Morocco Abdelmalek Essaâdi University, Tetouan, Morocco

[email protected] [email protected] [email protected]

R.Temsamani Khalid

Faculty of Science, University Abdelmalek Essaâdi Tetouan, Morocco

[email protected]

Abstract— The monitoring of the cathecol level is clinically material are the lowering of the potentials at which charge important. In this work a novel C/Ag/SiO2 Sonogel-Carbon transfer processes occur, the enhancement of the rele vant electrode was used for the sensitive voltametric determination current and the prevention of the passivation of the surface. of cathecolamine. A complete characterization of the electrodes The results of this paper have shown an analytical has been performed using scanning electron microscopy, performance and an efficient catalytic activity of the electrode Raman spectroscopy, cyclic voltammetry, and impedance for the electro-oxidation of cathecol. The advantage of spectroscopy. the novel electrode has shown an increase in the effective area of up to 70%, oxidation peaks and an excellent functional materials as an immobilization matrix for sensors electrocatalytic activity. The electrochemical response is due to high surface to volume ratio, the presence of reactive characteristics were investigated by cyclic and differential pulse groups on the surface, and fast electron transfer kinetics [1]. voltammetry, the limit of detection is estimated to be in the sub In recent years, nanostructured materials gained a very micromolar regime. important role in the development of amperometric sensors [2]. The high superficial area/volume ratio and the polyhedral statistical analysis of measurements performed in water shape induce a quite high number of defects to be present at samples has led to good apparent recovery. the electrode surface, imparting to the material high reactivity toward species in solution, suitable for the realization of Keywords— The statistical analysis, C/Ag/SiO2 Sonogel- electrocatalytic processes [3]. The result of our proposed Carbon electrode. Amperometric sensor. Cathecol. modified electrode as compared to other electrochemical methods reported in the literature [4,8] exhibit that our I. INTRODUCTION electrochemical sensor Seems to be very promising and they The application of sensors for clinical measurement are can be considered for quantification of cathecol. well recognised in the last ten years. In this work a C/Ag/SiO2 Sonogel-Carbon electrode is used for the sensitive pulse voltammetry determination of cathecol. The proper choice of the sensing material, in view of the specific application, is fundamental since it can impart to the device definite physicochemical properties and analytical peculiarities. The main advantages sought by adopting a specific electrode

Detection Modified materials limit Ref (µM)

0.02 [4] Q. Gou et al., N-doped carbon nanotubes 2016

Reduced graphene oxide 0.18 [5] H. Zhang et al., Glassy carbon electrode 2015 0.01 [6] H. Du et al., 2011 modified with graphene Scheme 1. Procedure for depositing silver nanoparticles on [7] H.Hammani et 0.05 Activated carbon al., 2019 silica spheres: KOH-activated graphene sheets 0.1 [8] L. Huang et al., 1. TEOS, 2. ethanol, 3. ammonia, 4. water, and 5. Ag NPs. 2016 carbon film 0.01 This work III. RESULTS AND DISCUSSION C/Ag/SiO2 Sonogel-Carbon A. Surface and Electrochemical characterization of modified electrodes II. EXPERIMENTAL Scanning microscopy (SEM) was used to explore the difference in structure between films of carbon paste alone, and those of A. Reagents and materials carbon paste in the presence of SiO2 (C / SiO2), and mixed Cathecol reagent grade ≥98% (HPLC) is purchased from compound of SiO2 and Ag (C / SiO2 / Ag). Sigma Aldrich (USA). KH2PO4, K2HPO4 for phosphate Analysis of the surface of the carbon-only paste electrode buffer and graphite powder (<20 microns) were purchased shows a granular structure of carbon. However, the from Fluka. Paraffin oil was purchased from a pharmacy. All incorporation of SiO2 into the carbon paste shows a more other chemicals were of reagent grade and used directly structured surface (cauliflower-shaped) with the appearance without further purification. Plastic capillary tubes, i.d. 2mm, of bright white particles. An even better organized were used as the bodies for the composite electrodes. morphology, when silver is added (C / Ag / SiO2 electrode), Solutions were prepared using deionized double-distilled is noted, corresponding to the C paste modified by silica-silver water with a measured resistance higher than 15 μS cm−1. nanoparticles. In this case, the film generated shows a better B. Instrumentation organization and an oriented structure as well as an increase in the specific surface (Figure 1). The presence of SiO2 and The cyclic voltammetry (CV), differential pulse Ag in the carbon paste therefore seems to have a favorable voltammetry (DPV) and electrochemical impedance (EI) were effect on the structure of the materials prepared. applied to study the behaviour of C/Ag/SiO2 Sonogel-Carbon electrode. They were all performed with a Voltalab®40, type PGZ301 from Radiometer (France). A conventional three- electrode cell (20 mL) was used at room temperature (25± 1°C), the counter electrode was a platinum wire and an SCE, 3M KCl electrode was used as the reference, the C/Ag/SiO2 electrodes were used as working electrode. The scanning electron microscope (SEM) image was obtained using a HITACHI X-650 SEM instrument. The statistical validation was carried out by the MATLAB statistical software. Figure 1. (A) SEM images obtained for C electrode. (B) SEM images obtained for C/SiO2 modified electrode. (C) C. Preparation of the C/Ag/SiO2 Sonogel-Carbon electrode SEM images obtained for C/Ag/SiO2 modified electrode To prepare the C/Ag/SiO2, the following procedure was used: 0.1g of silver nanoparticles silica Ag/SiO2 (Scheme 1) To characterize the interface features of the modified is dispersed in 0.5 M acetic acid solution, then 1g of Carbon electrode surface we have used the EIS method. The Figure 2 graphite powder was dispersed in the solution until obtaining a unique phase, and then the mixture was heated at 120°C to shows that the charge transfer resistance of C/Ag/Si O2 evaporate the acetic acid and water. In the next step, the electrode is much smaller than that of C/Si O2 electrode and carbon powder modified with Ag/SiO2, is dried and was the C electrode, suggesting that it is easier to transfer electrons mixed thoroughly in a mortar with 40% of paraffin oil until at C/Ag/SiO2, and this indicates that the incorporation of silver obtaining a homogenous paste. Thus, the plastic capillaries nanoparticles on silica spheres promote the electron transfer were filled, leaving a little extra mixture sticking out of the tube to facilitate the subsequent polishing. For establishing synergistically and accelerates the diffusion of ferricyanide electrical contact, a copper wire was inserted into the towards the modified electrode surface. capillary. Before usage the electrodes were polished with The active surface area of the modified electrode was 1/2 emery paper No1500, and were electrochemically cleaned by estimated according to the slope of the ip versus v plot, cyclic voltammetry until obtaining a stable cyclic based on the Randles–Sevcik equation [9,10]: voltammograms between -0,80 and 1,50 V in 0,005 mol.L-1 5 3/2 1/2 1/2 KCl. Lp = 2.69 × 10 n Aeff D C υ

Where Aeff is the effective surface area, n is the number of electrons transferred, D (= 7.6 × 10−6 cm2 s−1) is the diffusion coefficient of potassium ferricyanide (Gooding et al., 1998), and C are the concentration of potassium ferricyanide. The effective electrode area for C/Ag/SiO2 modified electrode is 2 2 C approximately 0.058 cm whereas 0.037cm for C/SiO2 and 0.032cm2 for C electrode.

Figure 3. (A) cyclic voltammetry of 10-5M of cathecol at C/Ag/SiO2 electrode and C/SiO2 electrode and Carbon electrode in PBS (0.05M), pH = 2, T = 25 ° C. (B) Cyclic voltammograms obtained at different scan rates from the C/Ag/SiO2 modified electrode in a PBS at pH 2 containing 2 μM of cathecol. Scan rates: 40, 60, 80,100, 120, 140,160 and -5 180 mV/s, (C) Nyquist plots of 10 M of cathecol at C/Ag/SiO2 electrode and C/SiO2 electrode and Carbon electrode in PBS (0.05M), pH = 2, T = 25 ° C.

Figure 2. Nyquist plots of EIS in 10-2 M potassium The Figure 3.A, shows the electrochemical behaviour of the ferricyanide prepared in 0.05 M KCl for C/Ag/SiO2 electrode cathecol at C/Ag/SiO2 electrode and C/SiO2 electrode and and C/SiO2 electrode and Carbon electrode, Amplitude: 5 Carbon electrode in PBS pH 2 using CV; First, the cathecol mV; frequency range: 100 kHz–10 mHz; potential: 0V. (pKa =9,5) presents an electroactive character that appears B. Electrochemical Behaviour of cathecol at the Modified with an oxidation peak in the studied potential ranges Electrode Use an unique style for units. (Tables.1), also we noticed the appearance of Epa peak corresponding to the oxidation of Ag incorporated in the paste of the modified electrode at 100 (mV)/ECS. The A relationship between the oxidation peak current (ipa) and the square root of the scan rate (ν1/2) (Figure 3.B) is linear with linear correlation coefficients R = 0,9974, indicates that the electrochemical process is controlled by diffusion. The (Figure 3.C), shows the Nyquist plots behaviour of the cathecol at C/Ag/SiO2 electrode and C/SiO2 electrode and Carbon electrode in PBS pH 2, the (Tables.2) presents the charge transfer resistance and the capacitance of the electrical layer at the electrode/solution interface, and the apparent rate of electron transfer at different modified electrodes.

Electrod Ipa Ipc Ipa/ Epa Epc ∆Ep

B e (µA/cm (µA/cm Ipc (mV)/EC (mV)/EC (mV)/EC 2) 2) S S S

Carbon 263.513 - 1.9 592 138 454 electrode 136.987 2

C/SiO2 254.479 -147..27 1.7 570 178 392 electrode 2 C/Ag/Si 373.19 - 1.8 506 200 306

O2 203.091 3 electrode

Tables.1 Electrochemical characterization of cathecol on three types of electrodes

Electrode Rtc Cdc Kapp 3.4 Determination of cathecol at C/Ag/SiO2 in Urine. (kohm .cm2) (µf/cm2) (cm /s) The objective of this study is the simultaneous detection of the cathecol in the presence of AA and AU in the Carbon electrode 17.04 10.45 1.56.10-5 urine. In this context, and in order to evaluate the -5 C/SiO2 electrode 7.239 19.69 3.68.10 applicability of the proposed method for the -4 C/Ag/SiO2 1.507 23.65 1.77.10 determination of the cathecol and AA and AU in urine, electrode the measurements were conducted in urine samples diluted 500% (with 0,05M PBS at pH 2) was then added to this mixture a deferred reports of AA, AU and cathecol Tables.2 the electrical parameters of the three types of (table 4). electrodes the AA and UA is the principal organic constituents of From the table (Tables.2) we notice that the charge transfer urine, the phenomenon of interference on the resistance (Rtc) decreases for the Ag / SiO2 carbon paste electrochemical response of cathecol in the presence of electrode, a remarkable increase in the capacitance of the the urinary AA and UA is one of the major problems that electric layer (Cdc) and an apparent speed increase of the hinders electrochemical detection of its substances in electron transfer These results show the efficiency of Ag / biological media, since the unmodified carbon electrode SiO2 carbon modified electrode. could not separate cathecol and AA and UA oxidation Given the results obtained in Figure 3 and Tables 1 and 2, peaks. The development of a simple and inexpensive the presence of SiO2 and Ag in the carbon paste therefore device for the simultaneous determination and separation seems to have a favorable effect on the structure of the of the electrochemical responses of these substances materials prepared. The modified electrode should promote remains the challenge of this work. the sensitivity and the selectivity of determination cathecol. The Tables.4 shows that the peak currents for cathecol As a result, C/Ag/SiO2 can accelerate the electron transfer increase linearly with increases their respective and decrease the overpotentials of cathecol oxidation at concentrations, without considerable effects on the other different levels of difusion modes, which is the key factor to peak currents of AA and UA while varying the adjust the problem of adsorptionat the electrod surface and concentration of cathecol from 10 to 100 µmol L−1. realize determination directe of cathecol. In addition, a various concentrations of AA from 20 to 3.3. Analytical Calibration Curves of determinations of 100 µmol L−1 in the presence of cathecol and UA exhibit cathecol. excellent responses to AA, AU, and cathecol without The DPV was used to obtain the calibration curve of cathecol any obvious intermolecular effects among them, the peak at the modified electrode in PBS pH2. the result in Figure 4 current of AA increased linearly with increased shows the linear relationship between the oxidation peak concentration. is also indicated that the peak current of current and cathecol concentrations. The peaks intensities are UA increased linearly with increases concentration of increased linearly in the range of 1–120 μM, the equation is UA, without considerable effects on the other peak Ipa (μA) = 0.3 C + 0.5 with a correlation coefficient of R² = currents while varying the concentration of UA from 30 0,9992 and the detection limit (S/N = 3) estimated to be 0.01 to 40 µmol L−1. μM in terms of signal to noise ratio of 3:1. These results confirm that the oxidation processes of cathecol, AA, UA at C/Ag/SiO2 electrode are independent from each other, this separation allows a simultaneous determination of AA, UA and cathecol in a mixture. The C/Ag/SiO2 possessed a higher active surface area and can separate cathecol and AA and UA oxidation peaks.

Figure 4. Calibration plots of cathecol (from 1 to 120 µmol L−1)

Table 3: Simultaneous determination of cathecol, AA and UA in mixtures synthesis samples (±SD; the standard deviation for n=3). Added (µmol/L) Found (µmol/L) Recovery (%) Sample CATHECOL AU AA CATHECOL AU AA CATHECOL AU AA 1 10 30 20 9.86±0.2 31±1.5 19.6±0.5 98.0% 103.3% 98.0% 2 30 40 40 28.4±1.3 41±2.7 38.3±1.5 94.7% 102.5% 95.8% 3 70 35 80 68.47±1.8 33.2±0.8 78.2±2.1 97.8% 100.6% 97.8% 4 100 40 100 98.5±1.7 40.3±1.3 98.7±1.2 98.5% 100.8% 98.7%

[1] L. Pigani, C. Rioli, D. López-Iglesias, C. Zanardi, B. The feasibility of the C/Ag/SiO2 sensor is demonstrated for Zanfrognini, L.M. Cubillana-Aguilera, J.M. analytical application, the recovery test was performed by the Palacios-Santander., 2020 Preparation and standard addition method (Table 3), with 4 different additions characterization of reusable Sonogel-Carbon of cathecol, AA and UA to the urine diluted samples, the electrodes containing carbon black: Application as obtained recoveries ranged from 94.7 to 98.5 for cathecol; amperometric sensors for determination of cathecol. 95.8 to 98,7 for AA and 100,6 to 103,3 for UA. This high Journal of Electroanalytical Chemistry. 877(2020) recovery and the perfect selectivity exerted by our C/Ag/SiO2 114653. http://dx.doi.org/10.1016/j.jelechem.2020.114653 electrochemical sensor looks very promising for the simultaneous detection of cathecol, AA and AU. So, an [2] Gooding, J.J., Praig, V.G., Hall, E.A.H., 1998. Platinum- effective sensor has been obtained for cathecol determination Catalyzed Enzyme Electrodes Immobilized on Gold in urine sample in this work. Using Self-Assembled Layers. Anal. Chem. 70, 2396– 2402. doi:10.1021/ac971035t C. Conclusions [3] Maringa, A., Mugadza, T., Antunes, E., Nyokong, T., 2013. The use of electrochemical techniques for the sensitive and Characterization and electrocatalytic behaviour of glassy selective determination of cathecol in urine by differential carbon electrode modified with pulse voltammetry using C/Ag/SiO2 modified electrode was shows that the C/Ag/SiO2 modified electrode present a perfect selectivity on the detection of the cathecol in presence of AA and UA in 0,05M PBS at pH 2, with a detection limit 0,01 µmol L-1 is obtained. This selectivity is maintained when the study is conducted in biological fluids such as in urine diluted 500% with 0,05M PBS at pH 2. Indeed, the results obtained were validated by the statistical validation methods and our electrochemical sensor looks very promising and they can be considered for early quantification of cathecol in clinical preparations.

nickel nanoparticles towards amitrole detection. J. Electroanal. Chem. 700, 86–92. doi:10.1016/j.jelechem.2013.04.022 [4] Q. Gou, M. Zhang, G. Zhou, L. Zhu, Y. Feng, H. Wang, B. Zhong, H. Hou, Highly sensitive simultaneous electrochemical detection of hydroquinone and cathecol with threedimensional n-doping carbon nanotube film electrode, J. Electroanal. Chem. 760 (2016) 15, https://doi.org/10.1016/j.jelechem.2015.11.034. [5] H. Zhang, X. Bo, L. Guo, Electrochemical preparation of porous graphene and its electrochemical application in the simultaneous determination of hydroquinone, cathecol, and resorcinol, Sensors Actuators B Chem. 220 (2015) 919, https://doi.org/10.1016/j.snb. 2015.06.035. [6] H. Du, J. Ye, J. Zhang, X. Huang, C. Yu, A voltammetric sensor based on graphenemodified electrode for simultaneous determination of cathecol and hydroquinone, J. Electroanal. Chem. 650 (2011) 209, https://doi.org/10.1016/j.jelechem.2010.10. 002. [7] H. Hammani, F. Laghrib, S. Lahrich, A. Farahi, M. Bakasse, A. Aboulkas, M.A. El Mhammedi, Effect of activated carbon in distinguishing the electrochemical activity of hydroquinone and cathecol at carbon paste electrode, Ionics 25 (2019) 2285, https://doi.org/10.1007/s11581- 018-2648-6 [8] L. Huang, Y. Cao, D. Diao, Electrochemical activation of graphene sheets embedded carbon films for high sensitivity simultaneous determination of hydroquinone, cathecol and resorcinol, Sensors Actuators B Chem. 305 (2020) 127495, https://doi.org/10. 1016/j.snb.2019.127495 [9] H.-Y. Tseng, V. Adamik, J. Parsons, S.-S. Lan, S. Malfesi, J. Lum, L. Shannon, B. Gray, Development of an electrochemical biosensor array for quantitative polymerase chain reaction utilizing three-metal printed circuit board technology, Sensors Actuators B Chem. 204 (2014) 459–466. doi:http://dx.doi.org/10.1016/j.snb.2014.07.123. [10] D. Salinas-Torres, F. Huerta, F. Montilla, E. Morallón, Study on electroactive and electrocatalytic surfaces of single walled carbon nanotube-modified electrodes, Electrochim. Acta. 56 (2011) 2464–2470. doi:http://dx.doi.org/10.1016/j.electacta.2010.1 1.02

Kamilia Filali Zoubida Benmamoun Hanaa Hachimi BOSS team BOSS team BOSS team Systems Engineering Laboratory GS laboratory Systems Engineering Laboratory Sultan Moulay Slimane University, Av ENSA University campus PO Box Sultan Moulay Slimane University, Av Med 242.kenitra14000 morocco Med 5, Po Box 591, Pc 23000, 5, Po Box 591, Pc 23000, [email protected] Beni Mellal, Morocco Beni Mellal, Morocco [email protected] [email protected] @univ-ibntofail.ac.ma

Abstract—IJOA Journal According to the economist, energy consumption increased Nowadays, Companies are carrying heavier and heavier loads than before, by +4.5 at the end of 2017, after + 1.9% at the end of 2016. considering the market requirement, competitors who have become more numerous in almost all sectors of activity, and above all more creative and According to the same newspaper, an increase of + 7.9% stronger on several fronts: marketing, marketing, productivity ... came from energy addressed to the national productive sector To win in the market, strong companies must have a strong management and + 3% concerning low voltage energy addressed mainly to system that helps them optimize their costs and differentiate themselves from households. In addition, the consumption of electricity went others to ensure a comfortable margin from 12,453 GWH to 37,446 GWH from 1998 to 2018. This article reviews the critical optimization problem that can make this This increase reflects the dynamism of our country both difference. This article will also present the different possible scenarios to economically and socially. And therefore; other solutions are optimize the significant costs of a company by proposing to opt for renewable needed to allow businesses to be more profitable. energies. A number of optimization tools will be discussed and analyzed in this article.

II. SOLAR ENERGY Keywords— Renewable energy, optimization, supply chain Our kingdom is one of the sunny countries most of the year, even in winter, something that cannot be found in the most I. INTRODUCTION developed countries in Europe. Nowadays, Companies are carrying heavier and heavier costs Moreover, the construction of several solar power plants in than before, there are some who spend more money on the various regions of Morocco provides for the realization of marketing to market their products and achieve their additional solar capacity for the years to come. objectives in Turnover, others prefer to invest in margin to have a competitive price compared to competitors ... Furthermore, solar energy has become the choice of a The methods to achieve the gain objective are known by Moroccan population that can be considered important, almost all companies, in terms of product marketing for but we don’t see companies that opt for this solution to reduce commercial companies, or to have a competitive cost price the electricity bill when their consumption far exceeds that of for production companies, or a cost of storage or low houses.[2] transportation for Logistics Company…[1] According to the economist: “The Noor Midelt I project, awarded to the EDF Renouvelables, Masdar and Green of on the other hand they all undergo very heavy loads which Africa consortium, should enable Ma¬roc to move from 3rd makes them lose all the gain which they had in their activities. to 2nd place in the world CSP market This plant will have an installed capacity of 800 MW, almost the equivalent of that of The purpose of this article is to propose solutions to optimize a conventional nuclear reactor (1,000 MW). part of the business expenses: energy expenses. It will have to feed 1.19 million inhabitants and produce a kwh at 0.68 DH. For the period 2019-2023, the equipment

plan provides for the realization of an additional solar 50% carbon dioxide (CO2) and some trace gases capacity of 2,015 MW (120 Noor PV Tafilalt in 2019, 200 (NH3, N2, H2S) MW Noor PV Atlas in 2020, 200 MW in Koudia Baida in 2023, 300 MW to be carried out in 2023 under the law 13- 2.2 advantages of recycling 09)”[3] this anaerobic digestion produces a double valuation of organic matter and energy, this is the specific interest in anaerobic digestion, compared to other sectors, III. BIOMASS decreases the amount of waste, On the other hand, all the companies that have organic waste also allows a reduction in greenhouse gas emissions by can use this waste to proscribe energy, citing all the food, replacing the use of fossil fuels or chemical fertilizers cardboard and paper companies, mass distribution like and can treat greasy or very wet organic waste. [4] hypermarkets and supermarkets 3.2 Agreed enterprise and biomass plant Biomass, this energy source not yet exploited in Morocco, and which can save the costs of the energy company by The idea is to question the possibility that a business exploiting its waste could benefit from energy from its own waste; have an agreement so that each company that gives its waste to a 2.1 definition of recycling biomass power plant benefits from energy according to the weight of the waste [5] Waste recycling is the direct reintroduction of a waste into the production cycle from which it comes, this means that the waste is transformed into a raw material which will be used REFERENCES to produce new consumer goods while avoiding to draw resources from the planet. [1] A.A. Rentizelas*, I.P. Tatsiopoulos, A. Tolis “ An optimization model for multi-biomass tri-generation energy supply,” These wastes can be used to produce energy, including [2] Peter McKendry” Energy production from biomass (part methanization, 1): overview of biomass. “ this technology based on degradation by microorganisms of [3] Weimin Wanga, Radu Zmeureanua,, HuguesRivard organic matter, under controlled conditions and in the “Applying multi-objective genetic algorithmsin green absence of oxygen, this degradation causes: building design optimization’  Digestate : a moist product, rich in partially [4] Zhi ZhouJuan Manuel De BedoutJohn Michael KernEmrah BiyikRamu Sharat Chandra “Energy stabilized organic matter optimization system”  Biogaz: gas mixture saturated with water at the [5] DAI LIN “THE DEVELOPMENT AND PROSPECTIVE OF outlet of the digester and composed of BIOENERGY TECHNOLOGY IN CHINA” approximately 50% to 70% methane (CH4), 20% to

Ahmed KAJOUNI Khalid HILAL Laboratory of Applied Mathematics & Scientific Calculus, Laboratory of Applied Mathematics & Scientific Calculus, Sultan Moulay Slimane University, BP 523, 23000, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco Beni Mellal, Morocco [email protected] [email protected]

Youssef ALLAOUI Laboratory of Applied Mathematics & Scientific Calculus, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco [email protected]

Abstract—In this paper, we study the existence of solutions for equations initiating the study of theory of such systems and the following fractional hybrid differential equations involving proved utilizing the theory of inequalities, its existence of Riemann-Liouville differential operators of order 1 < α ≤ 2. An extremal solutions and a comparaison results. existence theorem for fractional hybrid differential equations is proved under mixed Lipschitz and Caratheodory´ conditions and Zhao, Sun, Han and Li [11] have discussed the following using the Dhage point ﬁxe theorem. fractional hybrid differential equations involving Riemann- Index Terms—Fractional, Riemann, Hybrid Liouville differential operators  h x(t) i I.INTRODUCTION Dα = g(t, x(t)) a.e t ∈ J = [0,T ], R f(t, x(t)) (2) During the past decades, fractional differential equations x(0) = 0 , have attracted many authors [1], [4], [5], [7], [8], [9], [10], [11]. The differential equations involving fractional derivatives where f ∈ C1(J × R, R\{0}) and g ∈ Car(J × R, R) . in time, compared with those of integer order in time, are The authors of [11] established the existence theorem for more realistic to describe many phenomena in nature (for fractional hybrid differential equation and some fundamental instance, to describe the memory and hereditary properties of differential inequalities. They also established the existence of various materials and processes), the study of such equations extremal solutions. has become an object of extensive study during recent years. Hilal and Kajouni [5] studied boundary fractional hybrid The quadratic perturbations of nonlinear differential equa- differential equations involving Caputo differential operators tions have attracted much attention. We call such fractional of order 0 < α < 1 hybrid differential equations. There have been many works o  h x(t) i Dα = g(t, x(t)) a.e t ∈ J = [0,T ], n the theory of hybrid differential equations, and we refer the  C f(t, x(t)) readers to the articles [2], [3], [4], [5], [6], [7]. x(0) x(T ) (3) a + b = c, Dhage and Lakshmikantham [3] discussed the following  f(0, x(0)) f(T, x(T )) ﬁrst order hybrid differential equation where f ∈ C1(J × , \{0}) and g ∈ Car(J × , ) and a, b,  " # R R R R d x(t) c are real constants with a + b 6= 0. They proved the existence  = g(t, x(t)) a.e t ∈ J = [0, 1], dt f(t, x(t)) (1) result for boundary fractional hybrid differential equations  under mixed Lipschitz and Caratheodory´ conditions. Some x(t0) = x0, fundamental fractional differential inequalities are also estab- where f ∈ C1(J × R, R\{0}) and g ∈ Car(J × R, R). lished which are utilized to prove the existence of extremal (Car(J ×R, R) is called the Caratheodory´ class of functions). solutions. Necessary tools are considered and the comparaison They established the existence, uniqueness results and some principle is proved which will be useful for further study of fundamental differential inequalities for hybrid differential qualitative behavior of solutions.

In this paper we consider the fractional hybrid differential where n = [α] + 1, [α] denote the integer part of number α, equations with involving Riemann -Liouville differential op- Provided that the right side is pointwise defined on (0, ∞). erators of order 1 < α ≤ 2 From the definition of the Riemann-Liouville derivative, we  h x(t) i can obtain the following statement Dα = g(t, Bx(t)) a.e 0 ≤ t < 1, Lemma 3.1: [6] R f(t, Bx(t)) (4) Let α > 0 . If we assume x ∈ C(0, 1) ∩ L(0, 1), then the x(1) = x0(1) = 0, fractional differential equation where f ∈ C1(J × R, R\{0}), g ∈ Car(J × R, R). α RD0+ x(t) = 0 The term Bx(t) is given by: Bx(t) := R t K(t, s)x(s)ds where 0 α−1 α−2 α−n K ∈ C(D, R+), the set of all positive functions which are has x(t) = c1t +c2t +...+cnt , ci ∈ R, i = 1, ..., n, continuous on D := {(t, s) ∈ R2/0 ≤ s ≤ t ≤ T } and as unique solutions, where n is the smallest integer greater than or equal to α. Z t B∗ = sup K(t, s)ds < ∞ (5) Lemma 3.2: [6] t∈[0,1] 0 Assume x ∈ C(0, 1) ∩ L(0, 1) with a fractional derivative of α > 0 that belongs to C(0, 1) ∩ L(0, 1). Then Using the fixed point theorem, we give an existence theorem α α α−1 α−2 α−n of solutions for the boundary value problem of the above I0+ D0+ x(t) = x(t) + c1t + c2t + ... + cnt nonlinear fractional differential equation under both Lipschitz and Caratheodory´ conditions. We present two examples to for some ci ∈ R, i = 1, 2, ..., n where n is the smallest integer illustrate our results. greater than or equal to α. Lemma 3.3: II. MOTIVATION & METHODOLOGY Let h ∈ C[0, 1] et 1 < α ≤ 2. The unique solution of the problem A. Motivation ( Dα x(t) = h(t) a.e 0 ≤ t < 1 , III.PRELIMINARIES f(t,Bx(t)) (7) 0 In this section, we introduce notations, definitions, and x(1) = x (1) = 0, preliminaries facts which are used throughout this paper. is By C(J, ) we denote the Banach space of all continuous Z 1 R x(t) = f(t, Bx(t)) H(t, s)h(s)ds , (8) functions from J into R with the norm 0 kyk = sup{|y(t)|, t ∈ J} . where  (t−s)α−1−tα−1(1−s)α−1 s(1−t)tα−2(1−s)α−2  + , 0 ≤ s ≤ t ≤ 1 We denote by Car(J × R, R) the class of functions g : J ×  Γ(α) Γ(α−1) R −→ R such that H(t, s) = (i) the map t 7−→ g(t, x) is mesurable for each x ∈ and  −tα−1(1−s)α−1 s(1−t)tα−2(1−s)α−2 R  Γ(α) + Γ(α−1) , 0 ≤ t ≤ s ≤ 1 (ii) the map x 7−→ g(t, x) is is continuous for each∈ J. (9) The class Car(J × R, R) is called the Caratheodory´ class Preuve:: of functions on J × R which are Lebesgue integrable when Applying the Riemann-Liouville fractional integral of the bounded by a Lebesgue integrable function on J. order α for the equation (7), we obtain By L1(J, R) denote the space of Lebesgue integrable real- x(t) α α−1 α−2 valued functions on J endowed with the norm k . kL1 defined = I h(t) + c1t + c2t by f(t, Bx(t)) Z 1 for some c1, c2 ∈ R. k y kL1 = | y(s) | ds. 0 Consequently, the general solution of (7) is Z t Definition 3.1: [6] 1 α−1 α−1 α−2 x(t) = f(t, Bx(t)) (t−s) h(s)ds+c1t +c2t . The Riemann-Liouville fractional integral of the continuous Γ(α) 0 function h : (0, ∞) −→ R of order α > 0 is defined by (10) By x(1) = 0 then 1 Z t Iαh(t) = (t − s)α−1h(s)ds Z 1 1 α−1 Γ(α) 0 c1 + c2 = (1 − s) h(s)ds. Γ(α) 0 Provided that the right side is pointwise defined on (0, ∞) Definition 3.2: [6] From (10) we get The Riemann-Liouville fractional derivative of order α > 0 of x0(t)f(t, Bx(t)) − x(t)f (t, Bx(t)) t = the continuous function h : (0, ∞) −→ R is given by f 2(t, Bx(t)) n Z t Z t α 1 d n−α−1 1 α−2 α−2 α−3 0DRh(t) = n (t − s) h(s)ds, (6) (t−s) h(s)ds+(α−1)c1t +(α−2)c2t , Γ(n − α) dt 0 Γ(α − 1) 0

IJOA ©2021 23 International Journal on Optimization and Applications IJOA. Vol. 1, Issue No. 2, Year 2021, www.usms.ac.ma/ijoa Copyright ©2021 by International Journal on Optimization and Applications by x0(1) = 0 we have where q(t) = (1 − t)tα−2 and k(s) = s(1 − α−2 1 Z 1 s) . (α − 1)c + (α − 2)c = (1 − s)α−2h(s)ds. 1 2 Γ(α − 1) 0 IV. EXISTENCE RESULT Then  1 In this section, we prove the existence results for the hybrid c = 1 R ((1 − s)α−2 − (1 − s)α−1)h(s)ds  1 Γ(α−1) 0 differential equations with fractional order (4) on the closed   and bounded interval J = [0, 1] under mixed Lipschitz and 1 R 1 α−1 α−2 c2 = Γ(α−1) 0 ((1 − s) − (1 − s) )h(s)ds− Caratheodory´ conditions on the nonlinearities involved in it.  1  1 R (1 − s)α−1h(s)ds We defined the multiplication in X by (xy)(t) = x(t)y(t) for Γ(α) 0 x, y ∈ X. therefore Clearly X = C(J, R) is a Banach algebra with respect to 1 Z t above norm and multiplication in it. x(t) = f(t, Bx(t)) (t − s)α−1h(s)ds Γ(α) 0 Lemma 4.1: [2] Z 1 tα−1 Let S be a non-empty, closed convex and bounded subset of + ((1 − s)α−1 Γ(α − 1) the Banach algebra X and let A1 : X −→ X and A2 : X −→ 0 X be two operators such that tα−1 − (1 − s)α−2) − ((1 − s)α−1 Γ(α) (a) A1 is Lipschitzian with a Lipschitz constant L α−2 (b) B is completely continuous, t α−2 α−1 + ((1 − s) − (1 − s) ) h(s)ds (c) x = A1xA2y =⇒ x ∈ S for all y ∈ S , and Γ(α) (d) LM < 1 , where M = kA2(S)k = sup{kA2(x)k : x ∈ Z t 1 = f(t, Bx(t)) (t − s)α−1 S} Γ(α) 0 then the operator equation x = A1xA2y has a solution in S tα−1 + ((1 − s)α−1 − (1 − s)α−2) We make the following assumptions Γ(α − 1) x (H0) The function x 7−→ f(t,Bx) is increasing in R almost tα−1 every where for t ∈ J . − (1 − s)α−1 Γ(α) (H1) There exists a constant L > 0 such that tα−2 + ((1 − s)α−2 − (1 − s)α−1) h(s)ds | f(t, Bx) − f(t, By) |≤ LB∗|x − y| = L∗|x − y| , Γ(α − 1) Z 1 α−1 ∗ ∗ t α−1 α−2 for all t ∈ J and x, y ∈ R with L = LB . + ((1 − s) − (1 − s) ) (H ) There exists a function h ∈ L1(J, +) such that t Γ(α − 1) 2 R α−1 t α−1 − (1 − s) ∗ Γ(α) |g(t, Bx)| ≤ B h(t) a.e t ∈ J, tα−2 + ((1 − s)α−2 − (1 − s)α−1) h(s)ds for all x ∈ R . Γ(α − 1) For convenience we denote Z t (t − s)α−1 = f(t, Bx(t)) 1 Z 1 Γ(α) T = k(s)ds . (12) 0 Γ(α − 1) s(1 − t)tα−2(1 − s)α−2 0 + Γ(α − 1) Theorem 4.1: Assume that hypotheses (H1) and (H2) tα−1(1 − s)α−1 hold. Further, if − h(s)ds ∗ ∗ Γ(α) L B T khkL1 < 1, (13) Z 1 s(1 − t)tα−2(1 − s)α−2 + then the boundary value problem (4) has a solution define J. Γ(α − 1) t Preuve:: tα−1(1 − s)α−1 − h(s)ds We define a subset S of X by Γ(α) Z 1 S = {x ∈ X/kxk ≤ N} , = f(t, Bx(t)) H(t, s)h(s)ds , 0 where ∗ The proof is complete. B F T khk 1 N = 0 L , Lemma 3.4: ∗ ∗ 1 − B L T khkL1 The function H(t, s) defined by (9) satisfies the following conditions et F0 = sup|f(t, 0)| . Γ(α − 1)H(t, s) ≤ q(t)k(s) , (11) t∈J

It is clear that S satisﬁes hypothesis of lemma 4.1. for all t ∈ J. By application of Lemma 4.1, the equation (4) is equivalent Takin to sup from t, we obtain to the nonlinear hybrid integral equation ∗ Z 1 kA2xk ≤ TB khkL1 , x(t) = f(t, Bx(t)) H(t, s)g(s, Bx(s))ds , t ∈ J. 0 for all x ∈ S. (14) so A2 is uniformly bounded on S. Deﬁne two operators A1 : X −→ X and A2 : S −→ X by Next, we prove that A2(S) is an equi-continuous set on X. Given ε > 0 and let A1x(t) = f(t, Bx(t)), t ∈ J (15) n1 Γ(α + 1)εo δ < min , and 2 12khkL1 Z 1 A2x(t) = H(t, s)g(s, Bx(s))ds . (16) Let x ∈ S et t1, t2 ∈ [0, 1] with t1 < t2 , 0 < 0 t2 − t1 < δ . Then the hybrid integral equation (14) is transformed into the We have operator equation as Z 1

|A2x(t2) − A2x(t1)| = H(t2, s)g(s, Bx(s))ds x(t) = A1x(t)A2x(t) , t ∈ J. (17) 0 Z 1 We shall show that the operators A and A satisfy all the 1 2 − H(t1, s)g(s, Bx(s))ds conditions of Lemma 4.1. 0 Claim 1, Let x, y ∈ X then by hypothesis (H1), Z t2 α−1 α−1 α−1 ∗ (t2 − s) − t2 (1 − s) ≤ B khk 1 ds L Γ(α) |A1x(t) − A1y(t)| = |f(t, Bx(t)) − f(t, By(t))| 0 Z t2 s(1 − t )tα−2(1 − s)α−2 ≤ L∗|x(t) − y(t)| + 2 2 ds 0 Γ(α − 1) ∗ ≤ L kx − yk , Z 1 tα−1(1 − s)α−1 − 2 ds Γ(α) for all t ∈ J . t2 Taking supremum over t, we obtain t Z 1 s(1 − t )tα−2(1 − s)α−2 + 2 2 ds ∗ Γ(α − 1) kA1x − A1yk ≤ L kx − yk , t2 Z t1 (t − s)α−1 − tα−1(1 − s)α−1 for all x, y ∈ X. − 1 1 ds Claim 2, A2 is a continuous in S. 0 Γ(α) Let (x ) be a sequence in S converging to a point x ∈ S . t α−2 n Z 1 s(1 − t )t (1 − s)α−2 and Lebesgue dominated convergence theorem, we have − 1 1 ds 0 Γ(α − 1) Z 1 Z 1 α−1 α−1 lim A2xn(t) = lim H(t, s)g(s, Bxn(s))ds t1 (1 − s) n→∞ n→∞ + ds 0 Γ(α) Z 1 t1 = H(t, s) lim g(s, Bx (s))ds n Z 1 s(1 − t )tα−2(1 − s)α−2 0 n→∞ − 1 1 ds 1 Z t Γ(α − 1) = H(t, s)g(s, Bx(s))ds 1 0 then = A2x(t) , |A2x(t2) − A2x(t1)| ≤ for all t ∈ J. Z t2 α−1 This shows that A2 is a continuous operator on S. ∗ (t2 − s) B khkL1 ds Claim 3, A2 is compact operator on S . 0 Γ(α) First, we show that A2(S) is a uniformly bounded set in X. Z t1 (t − s)α−1 Z 1 (1 − s)α−1 Let x ∈ S be arbitrary. By Lemma 3.4, we have − 1 ds + (tα−1 − tα−1) ds Γ(α) 2 1 Γ(α) Z 1 0 0 |A2x(t)| = | H(t, s)g(s, Bx(s))ds| Z 1 α−2 α−2 α−2 (1 − s) 0 +(t2 − t1 ) ds 1 Z 1 0 Γ(α − 1) ≤ q(t) B∗ k(s)h(s)ds Z 1 α−2 Γ(α − 1) 0 α−1 α−1 (1 − s) ∗ +(t2 − t1 ) ds ≤ TB khkL1 , 0 Γ(α − 1) IJOA ©2021 25 International Journal on Optimization and Applications IJOA. Vol. 1, Issue No. 2, Year 2021, www.usms.ac.ma/ijoa Copyright ©2021 by International Journal on Optimization and Applications

α α α−1 α−1 ∗ t2 − t1 + t2 − t1 Taking supremum over t, ≤ B khkL1 Γ(α + 1) ∗ B F0T khkL1 kxk ≤ ∗ ∗ . tα−2 − tα−2 tα−1 − tα−1 1 − B L T khkL1 + 2 1 + 2 1 Γ(α) Γ(α) Then x ∈ S and the hypothesis (c) of Lemma 4.1 is satisfied. Finally, we have ∗ B khk 1 ∗ L α α α−1 M = kA2(S)k = sup{kA2xk : x ∈ S} ≤ B T khkL1 , ≤ t2 − t1 + (1 + α)(t2 Γ(α + 1) so, ∗ ∗ ∗ L M ≤ L B T khkL1 < 1 . −tα−1) + α(tα−2 − tα−2) 1 2 1 Thus, all the conditions of Lemma 4.1 are satisfied. ∗ Hence the operator equation A1xA2x = x has a solution in B khk 1 ≤ L tα − tα S. As a result, the boundary value problem (4) has a solution Γ(α + 1) 2 1 defined on J. This completes the proof.

α−1 α−1 α−2 α−2 +3(t2 − t1 ) + 2(t2 − t1 ) . V. EXEMPLES In this section, we will present two examples to illustrate α α α−1 α−1 In order to estimate t2 − t1 , t2 − t1 and the main results. α−2 α−2 t2 − t1 , we consider the following cases A. Exemple 1 Case 1: 0 ≤ t1 < δ , t2 < 2δ. we consider the fractional hybrid differential equation α α α α α t2 − t1 ≤ t2 < (2δ) ≤ 2 δ ≤ 4δ, ( 3 α−1 α−1 α−1 α−1 α−1 D 2 x(t) = sin x p.p. 0 ≤ t < 1 , t2 − t1 ≤ t2 < (2δ) ≤ 2 δ ≤ 2δ (18) α−2 α−2 α−2 α−2 α x(1) = x0(1) = 0 , t2 − t1 ≤ t2 < (2δ) ≤ 2 δ ≤ δ whetre f(t, x) = 1 , g(t, x) = sin x and h(t) = 1 . Case 2: 0 < t1 < t2 ≤ δ. Then hypothesis (H ) and (H ) hold. α α α α α−1 α−1 α−1 1 2 t2 − t1 ≤ t2 < δ ≤ αδ ≤ 4δ, t2 − t1 ≤ t2 < Since α−1 δ ≤ (α − 1)δ ≤ 2δ Z 1 α−2 α−2 α−2 α−2 1 t2 − t1 ≤ t2 < δ ≤ (α − 2)δ ≤ δ T = k(s)ds Γ(α − 1) 0 Z 1 Case 3: δ ≤ t < t ≤ 1. 1 1 1 2 2 α α α−1 α−1 = 1 s(1 − s) ds t2 − t1 ≤ αδ ≤ 4δ, t2 − t1 ≤ (α − 1)δ < 2δ Γ( ) 0 α−2 α−2 2 t2 − t1 ≤ (α − 2)δ < δ 4 = √ , 35 π we obtain choosing L = 1, then we have |A2x(t2) − A2x(t1)| < ε , LT khkL1 < 1 . for all t1, t2 ∈ J and all x ∈ X . This implies that A2(S) is an equi-continuous set in X. Therefore, the fractional hybrid differential equation (18) has Then by Arzela-Ascoli` theorem, A2 is a continuous and a solution. compact operator on S. B. Exemple 2 Claim 4, The hypothesis (c) of lemma 4.1 is satisﬁed. we consider the fractional hybrid differential equation Let x, y ∈ X such that x = A1xA2y. Then ( 3 h x(t) i 2 D sin x+2 = cos x p.p. 0 ≤ t < 1 |x(t)| = |A1x(t)||A2y(t)| (19) 0 = |f(t, Bx(t)) − f(t, 0) x(1) = x (1) = 0 Z 1 where f(t, x) = sin x + 2 , g(t, x) = cos x et + f(t, 0)|| H(t, s)g(s, Bx(s))ds| h(t) = 1. 0 Z 1 Then hypothesis (H1) and (H2) hold. ∗ ∗ 1 ≤ B [L |x(t)| + F0] q(t) k(s)h(s)ds Since Γ(α − 1) 0 4 ∗ ∗ T = √ . ≤ B [L |x(t)| + F0]T khkL1 . 35 π choosing L = 1, then

Thus, LT khkL1 < 1 . ∗ B F0T khkL1 Therefore, the fractional hybrid differential equation (19) has |x(t)| ≤ ∗ ∗ , 1 − B L T khkL1 a solution on [0, 1]. IJOA ©2021 26 International Journal on Optimization and Applications IJOA. Vol. 1, Issue No. 2, Year 2021, www.usms.ac.ma/ijoa Copyright ©2021 by International Journal on Optimization and Applications

REFERENCES [1] R.P. Agarwal. A propos d’une note de M. Pierre Humbert. C. R. Academie´ des Sciences 236, 2031-2032 (1953). [2] B.C. Dhage, On a ﬁxed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (2005) 273-280. [3] B.C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid 4 (2010) 414-424. [4] B.C. Dhage, V. Lakshmikantham, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Diff. Eq. et App. 2 (2010) 465-486 [5] K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order, Adv. Differ. Equ. (2015) 2015:183 [6] A. A. Kilbas, H. H. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam, 2006. [7] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008) 2677-2682. [8] V. Lakshmikantham, J.V. Devi, Theory of fractional differential equations in Banach Space, Eur. J. Pure Appl. Math. 1 (2008) 38-45. [9] K. B. Oldham, J. Spanier,The fractional calculus, Academic Press, New York, 1974. [10] I. Podlubny, Fractional Differential Equations, Mathematics in Sciences and Engineering,198, Academic Press, San Diego, 1999. [11] S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Electronic Journal of Differential Equations,vol. 36, pp. 1-12, 2006. [12] Y. Zhao, S. Suna, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Computers and Mathematics with Appl. 62 (2011) 1312-1324

Achraf Azanzal ∗, Adil Abbassi ∗ & Chakir Allalou ∗ ∗ Laboratory of Applied Mathematics and Scientiﬁc Computing, Sultan Moulay Slimane University, B.P523, Beni Mellal, Morocco, Email: [email protected], {abbassi91,chakir.allalou}@yahoo.fr

Abstract—In this paper, we study the Cauchy problem of the 2 < r ≤ ∞. Zhao, Liu, and Cui [21] established the existence fractional drift-diffusion system. By using the Fourier localization of global and local solution of the system (1) in critical Besov −2+ n argument and the Littlewood Paley theory, we get the local well- ˙ p n posedness for large initial data in critical Fourier-Besov-Morrey space Bp,r (R ) with 1 < p < 2n and 1 ≤ r ≤ ∞. n λ n λ 2−2α+ p0 + p 2−2α+ p0 + p We mention here that if w vanishes (w = 0) and for α = 2, space FNp,λ,q × FNp,λ,q , Moreover, if the initial data is sufficiently small then the solution is global. the system (1) becomes to the well-known Keller-Segel model Index Terms—Drift-diffusion, Local existence, Littlewood- of chemotaxis: Paley theory, Fourier-Besov-Morrey spaces.  n ∂tv = ∆v − ∇ · (v∇φ) in R × (0, ∞),  n I.INTRODUCTION ∆φ = v in R × (0, ∞), (3)  v(x, 0) = v (x), in n. In this paper, we consider the following Cauchy problem for 0 R n + the fractional drift-diffusion system in R ×R with fractional In the paper [4] the local well-posedness of the system (3) Laplacian has been proved in the three-dimensional case. Iwabuchi and  α n Nakamura [12], [13] get the global well-posednes of (3) for  ∂tv + (−∆) 2 v = −∇ · (v∇φ) in R × (0, ∞),  α n small initial data in the critical space  ∂tw + (−∆) 2 w = ∇ · (w∇φ) in R × (0, ∞), n (1) ∆φ = v − w in × (0, ∞), n  R −2+ p n  n B˙ ( )  v(x, 0) = v0(x), w(x, 0) = w0(x) in R , p,r R where the unknown functions v = v(x, t) and w = w(x, t) with 1 ≤ p < ∞ and 1 ≤ r ≤ ∞ Inspired by the work [21], denote densities of the electron and the hole in electrolytes, The purpose of this paper is to establish the existence of local respectively, φ = φ(x, t) denotes the electric potential, v0(x) solution to (1) for large initial data and global solution for and w0(x) are initial datum. Throughout this paper, we assume small initial data in the critical Fourrier-Besov-Morrey space that n ≥ 2 and 1 < α ≤ 2. 2−2α+ n + λ 2−2α+ n + λ Notice that the function φ is determined by the Poison equation p0 p p0 p FNp,λ,q × FNp,λ,q in the third equation of (1), and it’s given by: φ(x, t) = (−∆)−1(w − v)(x, t). Let us firstly recall the scaling property of the systems: if (v, w) solves (1) with initial data (v0, w0) (φ can be deter- So that the system (1) can be rewritten as the following system: mined by (v, w)), then (vγ , wγ ) with (vγ (x, t), wγ (x, t)) := α α α α  α −1 n +(γ v (γx, γ t) , γ w (γx, γ t)) is also a solution to (1) with ∂tv + (−∆) 2 v = −∇ · v∇(−∆) (w − v) in R × R  α −1 n +the initial data ∂tw + (−∆) 2 w = ∇ · w∇(−∆) (w − v) in R × R  n v(x, 0) = v0(x), w(x, 0) = w0(x) in R . (v (x), w (x)) := (γαv (γx), γαw (γx)) (4) (2) 0,γ 0,γ 0 0 Mathematical analysis of the Drift-diffusion system has (φγ can be determined by (vγ , wγ )). drawn much attention during the past three decades, we Definition 1.1: A critical space for initial data of the system refer the reader to see [1], [5] and the references therein (1) is any Banach space E ⊂ S0 ( n) whose norm is invariant for previous works on this system concerning existence of R under the scaling (4) for all γ > 0, i.e classical solutions and weak solutions. In the context of Besov spaces and for α = 2, Karch in [14] k(v0,γ (x), w0,γ (x))k ≈ k(v0(x), w0(x))kE . proved existence of global solution of the system (1) with E −2+ n p n small initial data in critical Besov space B˙p,∞ ( ) with Under these scalings, We can show that the space pair R n λ n λ n 2−2α+ p0 + p 2−2α+ p0 + p 2 ≤ p < n. After, Deng and Li [9] showed that the system FN × FN is critical for (1) see (Re- − 3 − 3 p,λ,q p,λ,q ˙ 2 2 ˙ 2 2 (1) is well-posed in B4,2 R , and ill-posed in B4,r R for mark 2.1 for details).

λ In order to solve the equation (1), we consider the following variation of the measure f on B (x0, r) and Mp as a subspace 0 p equivalent integral system of Radon measures. When λ = 0, we have Mp = L . The proofs of the results presented in this paper are based  α Z t α −t(−∆) 2 −(t−τ)(−∆) 2 on a dyadic partition of unity in the Fourier variables, the  v(t) =e v0 − e ∇ · (v∇φ(τ)) dτ  0 so-called, homogeneous Littlewood-Paley decomposition. We α Z t α recall briefly this construction below. For more detail, we refer  −t(−∆) 2 −(t−τ)(−∆) 2 w(t) =e w0 + e ∇ · (w∇φ(τ)) dτ. the reader to [2]. 0 (5) Let f ∈ S0 (Rn) . Define the Fourier transform as α α With F (−∆) 2 f (ξ) = |ξ| Ff(ξ). Z − n −ix·ξ Throughout this paper, we use FN s to denote the fˆ(ξ) = Ff(ξ) = (2π) 2 e f(x)dx p,λ,q n homogenous Fourier Besov-Morrey spaces, (v, w) ∈ X to R denote (v, w) ∈ X × X for a Banach space X the product and its inverse Fourier transform as Z −1 − n ix·ξ X ×X will be endowed with the usual norm k(v, w)kX×X := f˘(x) = F f(x) = (2π) 2 e f(ξ)dξ. n kvkX +kwkX , k(v, w)kX to denote k(v, w)kX×X , V . W R d means that there exists a constant C > 0 such that V ≤ CW, Let ϕ ∈ S R be such that 0 ≤ ϕ ≤ 1 and supp(ϕ) ⊂ 0 1 1 ξ ∈ d : 3 ≤ |ξ| ≤ 8 and and p is the conjugate of p satisfying p + p0 = 1 for R 4 3 1 ≤ p ≤ ∞. X ϕ 2−jξ = 1, for all ξ 6= 0. Now we present our main results as follows. j∈Z Theorem 1.1: Let n ≥ 2, 1 < α ≤ 2, ρ0 > α We denote α−1 , max{n − (n + 3 − 2α)p, 0} ≤ λ < n, 1 ≤ p < ∞, 2−2α+ n + λ p0 p 1 1 −j X q ∈ [1, ∞], (v0, w0) ∈ FN and + 0 = 1. ϕj(ξ) = ϕ 2 ξ , ψj(ξ) = ϕk(ξ) p,λ,q ρ0 ρ0 k≤j−1 Then there exists T ≥ 0 such that the system (1) has a and unique local solution h(x) = F −1ϕ(x), g(x) = F −1ψ(x). (v, w) ∈ XT , where We now present some frequency localization operators: n λ 2 2−2α+ n + λ + 2 2−2α+ + + 0 p0 p ρ0 ρ0 p0 p ρ0 ρ 0 XT = L 0,T ; FN ∩L 0 0,T ; FN , Z p,λ,q p,λ,q dj j ∆˙ jf = ϕj(D)f = 2 h 2 y f(x − y)dy d R and and 2−2α+ n + λ + 2 p0 p ρ0 Z (v, w) ∈ C 0,T ; FNp,λ,q . X dj j S˙jf = ∆˙ kf = ψj(D)f = 2 g 2 y f(x − y)dy. d k≤j−1 R Besides, there exixts K ≥ 0 such that if (v0, w0) satisfies: From the definition, one easily derives that k(v , w )k n λ ≤ K, then the above assertion holds 0 0 2−2α+ 0 + FN p p p,λ,q ∆˙ ∆˙ f = 0, |j − k| ≥ 2 for T = ∞; i.e, the solution (v, w) is global. j k if ∆˙ j S˙k−1f∆˙ kf = 0, if |j − k| ≥ 5. II.PRELIMINARIES In this section, we give some notations and recall basic The following Bony paraproduct decomposition will be properties about Fourier-Besov-Morrey spaces that will be applied throughout the paper. used throughout the paper. ˙ ˙ The Fourier-Besov-Morrey spaces were introduced in [10] are uv = Tuv + Tvu + R(u, v) P P constructed via a type of localization on Morrey spaces. where T˙uv = S˙j−1u∆˙ jv, R˙ (u, v) = ∆˙ ju∆˜ jv, λ j∈Z j∈Z We define the function spaces Mp . ˜ P ˙ ∆jv = |j0−j|≤1 ∆j0 v. Definition 2.1: [15] Let 1 ≤ p ≤ ∞ and 0 ≤ λ < n. The Lemma 2.1: [10] Let 1 ≤ p , p , p < ∞ and 0 ≤ λ 1 2 3 homogeneous Morrey space Mp is the set of all functions λ , λ , λ < n. p 1 2 3 f ∈ L (B (x0, r)) such that (i) (Holder’s¨ inequality) Let 1 = 1 + 1 and λ3 = λ1 + λ2 , p3 p1 p2 p3 p1 p2 − λ then we have λ p p kfkM = sup sup r kfkL (B(x0,r)) < ∞, (6) p n x0∈R r>0 kfgkMλ3 ≤ kfkMλ1 kgkMλ2 . (7) n p3 p1 p2 where B (x0, r) is the open ball in R centered at x0 and with (ii) (Young’s inequality) If ϕ ∈ L1 and g ∈ Mλ1 , then radius r > 0. p1 λ The space M endowed with the norm kfk λ is a Banach p M 1 p kϕ ∗ gkMλ1 ≤ kϕkL kgkMλ1 , (8) space. p1 p1 When p = 1, the L1 -norm in (6) is understood as the total where ∗ denotes the standard convolution operator.

Now, we recall the Bernstein type lemma in Fourier vari- which implies ables in Morrey spaces. n λ j(2−2α+ 0 − p ) k{2 p kf (ξ)k λ }k q Lemma 2.2: [10] Let 1 ≤ q ≤ p < ∞, 0 ≤ λ1, λ2 < j Mp l n−λ n−λ 1 2 j(2−2α+ n − λ ) log γ(2α−2− n + λ ) n, p ≤ q and let γ be a multi-index. If supp(fb) ⊂ p0 p 2 p0 p = k{2 2 kϕj−[log γ]u0(ξ)kM λ }klq {|ξ| ≤ A2j}, then there is a constant C > 0 independent of 2 c p ≈ ku0k 2−2α+ n − λ f and j such that p0 p FNp,λ,q

n−λ2 n−λ1 γ j|γ|+j( q − p ) k(iξ) fbk λ2 ≤ C2 kfbk λ1 . (9) and since Mq Mp X s n ϕ (ξ)u (ξ) = ϕ (ξ)f (ξ), Then, we define the function spaces FN p,λ,q(R ), see [10]. j d0,γ j k Definition 2.2: (Homogeneous Fourier-Besov-Morrey |k−j|≤2 spaces ) we can get Let 1 ≤ p, q ≤ ∞, 0 ≤ λ < n and s ∈ R. The Fourier- s Besov-Morrey space FNp,λ,q is defined as the set of all ku k n λ ≈ ku k n λ . 0 0,γ 2−2α+ 0 − 0 2−2α+ 0 + distributions f ∈ S \P, P is the set of all polynomials, such FN p p FN p p ˆ λ p,λ,q p,λ,q that ϕjf ∈ Mp , for all j ∈ Z, and Similary, we have  1 q q  P jsq ˆ  2 ϕ f for q < ∞ kw0,γ k 2−2α+ n + λ ≈ kw0k 2−2α+ n + λ . def  j∈ j 0 p 0 p Z Mλ FN p FN p kfkFN s = p p,λ,q p,λ,q p,λ,q js ˆ  sup 2 ϕjf for q = ∞.  j∈Z λ Mp Now, we give the definition of the mixed space-time spaces. (10) Definition 2.3: Let s ∈ R, 1 ≤ p < ∞, 1 ≤ q, ρ ≤ ∞, 0 ≤ s n Note that the space FN p,λ,q(R ) equipped with the λ < n, and I = [0,T ),T ∈ (0, ∞]. The space-time norm is 0 p norm (10) is a Banach space. Since Mp = L , we defined on u(t, x) by s s s s s have FNp,0,q = FBp,q , FN1,0,q = FB1,q = Bq and −1 1/q −1 s −1 n X jqs q o FN1,0,1 = χ where Bq is the Fourier-Herz space and χ s d˙ ku(t, x)kLρ(I,FN˙ ) = 2 k∆jukLρ(I,Mλ) , is the Lei-Lin space [18]. p,λ,q p j∈Z

2−2α+ n + λ ρ s Remark 2.1: The space pair FN p0 p × and denote by L (I, FN p,λ,q) the set of distributions in p,λ,q 0 n 2−2α+ n + λ S (R × R )/P with ﬁnite k.kLρ(I,FN s ) norm. FN p0 p p,λ,q p,λ,q is critical for (1). For this, According to Minkowski inequality, it is easy to verify that 2−2α set u0,γ (ξ) = γ u0(γξ), then its Fourier transform is u (ξ) = γ2−2α−nuˆ γ−1ξ . ρ s ρ s d0,γ 0 L I; FNp,λ,q ,→ L I, FNp,λ,q , if ρ ≤ q, Let ρ s ρ s L I, FNp,λ,q ,→ L I; FNp,λ,q , if ρ ≥ q, def f (ξ) = ϕ 2−j+[log2 γ]−log2 γ ξ u (ξ) s j d0,γ where ku(t, x)kLρ I; FN p,λ,q := 1/ρ −j+[log γ]−log γ 2−2α−n −1 R ρ = ϕ 2 2 2 ξ γ uˆ0 γ ξ ku(τ, ·)k s dτ . I FNp,λ,q At the end of this section we recall an existence and By change of variable, we get uniqueness result for an abstract operator equation in a Banach space, which will be used to prove Theorem 1.1 in the sequel. For the proof, we refer the reader to see [17] and [3]. kfjk λ Mp Lemma 2.3: Let X be a Banach space with norm k.kX and 2−2α−n −j+[log γ]−log γ −1 = γ ϕ 2 2 2 ξ uˆ0 γ ξ B : X × X 7−→ X be a bounded bilinear operator satisfying λ Mp 2−2α−n − λ = γ sup sup r p kB(u, v)kX ≤ ηkukX kvkX n x0∈R r>0 1 −j+[log γ]γ −1 −1 for all u, v ∈ X and a constant η > 0. Then, if 0 < ε < and ϕ 2 2 γ ξ u0 γ ξ 4η c p L (B(x0,r)) if y ∈ X such that kykX ≤ ε, the equation x := y + B(x, x) n −λ λ 2−2α−n −1 − p x X kxk ≤ 2ε = γ γ p γ p sup sup γ r has a solution in such that X . This solution n x0∈R r>0 is the only one in the ball B(0, 2ε). Moreover, the solution 0 0 −j+[log γ] depends continuously on y in the sense: if ky kX < ε, x = ϕ 2 2 η u0(η) c p −1 −1 0 0 0 0 L (B(γ x0,γ r)) y + B(x , x ), and kx kX ≤ 2ε, then 2−2α+ n − λ log γ p0 p 2 −j+[log2 γ] = 2 ϕ 2 η u0(η) , 0 1 0 c Mλ kx − x k ≤ ky − y k . p X 1 − 4εη X

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2−2α+ n + λ + 2 III.LINEAR ESTIMATES IN FOURIER-BESOV-MORREY ρ0 p0 p ρ0 for all f ∈ L I; FNp,λ,q ∩ SPACES 2−2α+ n + λ + 2 0 p0 p ρ0 ρ0 0 In this section, we will establish some crucial estimates in L I; FNp,λ,q n λ 2 the proof of Theorem 1.1. We now consider the following 0 2−α+ p0 + p + 0 ρ ρ0 t∆ and g ∈ L 0 I; FN ∩ linear estimates for the fractional heat semigroup e t≥0 . p,λ,q Lemma 3.1: Let I=(0, T), s ∈ R, p, q, ρ ∈ [1, ∞] and 0 2−α+ n + λ + 2 ρ0 p0 p ρ0 ≤ λ < n. There exists a constant C > 0 such that L I; FNp,λ,q

α Proof Applying Bony paraproduct decomposition and quasi- −t(−∆) 2 ke u k α ≤ Cku k s , (11) 0 s+ 0 FN p,λ,q orthogonality property for Littlewood-Paley decomposition, Lρ [0,T ),FN ρ p,λ,q for ﬁxed j, we obtain

s where u0 ∈ FNp,λ,q. n j−1 j+1 proof Since suppϕj ⊂ {ξ ∈ R : 2 ≤ |ξ| ≤ 2 }, we obtain

h α i α X X −t(−∆) 2 −t|ξ| ∆˙ (f∇g) = ∆˙ (S˙ f∆˙ ∇g) + ∆˙ (S˙ g∆˙ ∇f) F ∆je u0 = ϕje u0 j j k−1 k j k−1 k λ c λ Mp Mp |k−j|≤4 |k−j|≤4 −t2α(j−1) X ≤ e kϕju0k λ . ˙ ˙ ˙ b Mp + ∆j(∆kf∆e k∇g) k≥j−3 Then, by the Minkowski inequality, we have 1 2 3 = Ij + Ij + Ij α −t(−∆) 2 e u0 s+ α ρ ρ L I;FNp,λ,q  1  T ! ρ  α Z α(j−1)  j(s+ ρ ) −tρ2 ≤ 2 e dt kϕjub0kM λ  0 p  Then, by the triangle inequalities in M λ and in lq( ), we have `q p Z  1  −T ρ2(j−1)α ! ρ  j s+ α 1 − e  ≤ 2 ( ρ ) kϕ u k α(j−1) j b0 M λ  ρ2 p  `q

≤ C ku0k s . FNp,λ,q ! ! k∇. (f∇g) k 2−2α+ n + λ ≤ kf∇gk 3−2α+ n + λ 1 p0 p 1 p0 p L I;FNp,λ,q L I;FNp,λ,q Lemma 3.2: [8] Let I=(0, T), s ∈ R, p, q, ρ ∈ [1, ∞] 0 X n λ q 1 ≤ λ < n and 1 ≤ r ≤ ρ. j(3−2α+ p0 + p )q ˙ \ q ≤ { 2 k∆j(f∇g)kL1(I,M λ)} There exists a constant C > 0 such that p j∈Z Z t X j(3−2α+ n + λ )q q 1 (t−τ)∆ ≤ { 2 p0 p kIb1k } q e f(τ)dτ ! ≤ Ckfk 2 . (12) j L1(I,M λ) s+ 2 s−2+ p ρ r r 0 ρ L I;FN p,λ,q j∈ L I;FNp,λ,q Z n λ 1 X j(3−2α+ p0 + p )q 2 q q +{ 2 kIb k 1 λ } j L (I,Mp ) IV. BILINEAR ESTIMATES IN FOURIER-BESOV-MORREY j∈Z SPACES n λ 1 X j(3−2α+ p0 + p )q 3 q q +{ 2 kIb k 1 λ } j L (I,Mp ) Lemma 4.1: Let I = (0,T ), s ∈ , p, q ∈ [1, ∞], j∈ R α Z max{n − (n + 3 − 2α)p, 0} < λ < n, ρ > and := J + J + J 0 α − 1 1 2 3 1 1 + 0 = 1. There exists a constant C > 0 such that ρ0 ρ0

" ! ! k∇. (f∇g) k 2−2α+ n + λ ≤ C kfk 2−2α+ n + λ + 2 p0 p p0 p ρ 1 ρ0 0 L I;FNp,λ,q L I;FNp,λ,q By using the Young’s inequality in Morrey spaces and Bernstein-type inequality with |γ| = 0, we have ×kgk  2−α+ n + λ + 2  0 p0 p ρ0 ρ0 0 L I;FNp,λ,q  #

+kgk 2−α+ n + λ + 2 ! × kfk  2−2α+ n + λ + 2  p0 p ρ 0 p0 p 0 Lρ0 I;FN 0 ρ ρ0 n λ p,λ,q L 0 I;FN ˆ j( p0 + p ) ˆ  p,λ,q  ϕjf ≤ C2 ϕjf 1 λ L Mp

Then inequality, to get

1 X 3 X ˙ \˙ kIb k 1 λ ≤ k(S˙ \f∆˙ ∇g)k 1 λ kIb k 1 λ ≤ k(∆kf∆e k∇g)k 1 λ j L (I,Mp ) k−1 k L (I,Mp ) j L (I,Mp ) L (I,Mp ) |k−j|≤4 k≥j−3 X X ≤ kϕj∇gk ρ0 kϕlfk ρ0 1 X \ c L 0 (I,M λ) b L (I,L ) = k([∆˙ f ∗ ∆e˙ ∇g)k 1 λ p k k L (I,Mp ) |k−j|≤4 l≤k−2 k≥j−3 X k ≤ C 2 kϕ gk 0 X X k ρ0 λ ≤ kϕ fk 0 kϕ ∇gk ρ 1 b L (I,Mp ) k b ρ0 λ l c L 0 (I,L ) L (I,Mp ) |k−j|≤4 k≥j−3 |l−k|≤1 n λ X ( 0 + p )l n λ p ρ λ X X l l( + ) 2 kϕlfbkL 0 (I,M ) ≤ C kϕ fk 0 2 2 p0 p kϕ gk ρ λ p k b ρ0 λ l L 0 (I,M ) L (I,Mp ) b p l≤k−2 k≥j−3 |l−k|≤1 X k 1 ≤ C 2 kϕ gk 0 α 0 0 j b Lρ0 (I,M λ) X X l(α−1− )q q p ≤ C kϕ fk 0 2 ρ0 k b ρ0 λ |k−j|≤4 L (I,Mp ) k≥j−3 |l−k|≤1 X (2−2α+ n + λ + α )l 2 p0 p ρ0 kgk 2−α+ n + λ + α p0 p ρ l≤k−2 ρ0 0 L (I,FN p,λ,q ) α (2α−2− ρ )l 0 ρ λ 2 kϕlfbkL 0 (I,M ) ≤ Ckgk 2−α+ n + λ + 2 p p0 p ρ ρ0 0 L (I,FN p,λ,q ) ≤ Ckfk 2−2α+ n + λ + α α p0 p ρ ρ0 0 X k(α−1− ρ ) L (I,FN p,λ,q ) 2 0 kϕ fk 0 k b Lρ0 (I,M λ) 1 p α 0 0 k≥j−3 X k X l(2α−2− )q q 2 2 ρ0 kϕ gk 0 k ρ0 λ b L (I,Mp ) |k−j|≤4 l≤k−2 Then, applying the Holder¨ inequality for series, and noticing n λ that λ > n − (n + 3 − 2α)p implies that 3 − 2α + p0 + p > 0 ≤ Ckfk 2−2α+ n + λ + α p0 p ρ ρ0 0 , we obtain L (I,FN p,λ,q ) X k(2α−1− α ) ρ0 0 n λ α 2 kϕkgk ρ λ , J3 ≤ Ckgk 2−α+ + + b L 0 (I,M ) p0 p ρ p Lρ0 (I,FN 0 ) |k−j|≤4 p,λ,q X j(3−2α+ n + λ )q X 2 p0 p α where we have used the fact that ρ0 > in the last j∈Z k≥j−3 α − 1 1 inequality. k(α−1− α ) q q 2 ρ0 kϕ fk 0 k b Lρ0 (I,M λ) Thus, by using the Young inequality, we have p

≤ Ckgk 2−α+ n + λ + α p0 p ρ ρ0 0 L (I,FN p,λ,q ) J1 ≤ Ckfk 2−2α+ n + λ + α p0 p ρ Lρ0 (I,FN 0 ) X X (j−k)(3−2α+ n + λ ) p,λ,q 2 p0 p X j(3−2α+ n + λ )q X 2 p0 p j∈Z k≥j−3 1 j∈ |k−j|≤4 n λ α q Z k(2−α+ 0 + + ) q 2 p p ρ0 kϕ fk 0 1 k b Lρ0 (I,M λ) k(2α−1− α ) q q p 2 ρ0 kϕ gk 0 k ρ0 λ b L (I,Mp ) ≤ Ckgk 2−α+ n + λ + α p0 p ρ ρ0 0 L (I,FN p,λ,q ) ≤ Ckfk 2−2α+ n + λ + α p0 p ρ n λ ρ0 0 X i(3−2α+ + ) L (I,FN p,λ,q ) p0 p kfk −2+ n + λ + 2 2 0 p0 p ρ0 X X (j−k)(−1+ n + λ )q Lρ0 (I,FN 0 ) 2 p0 p p,λ,q i≤3

j∈ |k−j|≤4 ≤ Ckgk 2−α+ n + λ + α kfk 2−2α+ n + λ + α . Z 0 p ρ 0 p 0 ρ p 0 ρ0 p ρ 1 L 0 (I,FN ) 0 0 p,λ,q L (I,FN p,λ,q ) k(2+ n + λ − α ) q q 2 p0 p ρ0 kϕ gk 0 k ρ0 λ b L (I,Mp ) Thus, we ﬁnished the proof of Lemma 4.1.

≤ Ckfk n λ α kgk n λ α , 2−2α+ 0 + + 2−α+ 0 + + 0 ρ p p ρ0 0 p p ρ L 0 (I,FN ) ρ0 0 V. PROOFOF THEOREM 1.1 p,λ,q L (I,FN p,λ,q ) To ensure the existence of the global and local solution of 1 1 where we have used + 0 = 1. the system (1), we will use Lemma 2.3 with the linear and ρ0 ρ0 Similary, we get bilinear estimate that we have established in section 3 and 4. α 1 1 Let ρ0 > be any given real number and ρ + ρ0 = 1. α − 1 0 0 J ≤ Ckgk n λ α kfk n λ α . 2 2−α+ 0 + + 2−2α+ 0 + + 0 Note that the space XT deﬁned in Theorem 1.1 is a Banach ρ p p ρ0 0 p p ρ L 0 (I,FN ) Lρ0 (I,FN 0 ) p,λ,q p,λ,q space equipped with the norm

For J3, ﬁrst we use the Young inequality in Morrey spaces, kuk = kuk n λ α +kuk n λ α . XT 2−2α+ 0 + + 2−2α+ 0 + + 0 ρ p p ρ0 0 p p ρ the Bernstein inequality (|γ| = 0) together with the Holder¨ L 0 (I,FN ) ρ0 0 p,λ,q L (I,FN p,λ,q )

n λ We ﬁrst prove global existence for small initial data. For Applying Lemma 3.2 with s = 2 − 2α + p0 + p and ρ1 = 1, this purpose we choose T = ∞. and Lemma 4.1, we obtain Set kB1(v, w)k 2−2α+ n + λ + α ! p0 p ρ ρ0 0 L 0,∞;FNp,λ,q

Z t α Z t α −(t−τ)(−∆) 2 −(t−τ)(−∆) 2 −1 = k e ∇ B1(v, w) := − e ∇· v∇(−∆) (w − v) (τ)dτ, 0 0 −1 · v∇(−∆) (w − v) (τ)dτk 2−2α+ n + λ + α ! p0 p ρ ρ0 0 L 0,∞;FNp,λ,q

−1 ! Z t α . k∇ · v∇(−∆) (w − v) k 2−2α+ n + λ −(t−τ)(−∆) 2 −1 L1 0,∞;FN p0 p B2(v, w) := e ∇· w∇(−∆) (w − v) (τ)dτ, p,λ,q 0 kvk 2−2α+ n + λ + α ! . p0 p ρ ρ0 0 L 0,∞;FNp,λ,q Then the equivalent integral system (1.2) can be rewritten as −1 ×k(−∆) (w − v)k  2−α+ n + λ + α  0 p0 p ρ0 ρ0 0 L 0,∞;FNp,λ,q 

α α −t(−∆) 2 −t(−∆) 2 (v(t), w(t)) = (e v0, e w0)+(B1(v, w),B2(v, w)) . −1 (13) +k(−∆) (w − v)k 2−α+ n + λ + α ! p0 p ρ ρ0 0 L 0,∞;FNp,λ,q n λ According to Lemma 3.1 with s = 2−2α+ 0 + ,I = [0, ∞) p p   0 ×kvk 2−2α+ n + λ + α and ρ = ρ (or ρ ), we obtain 0 p0 p ρ0 0 0 ρ0 0 L 0,∞;FNp,λ,q 

α 2 kvk 2−2α+ n + λ + α ! −t(−∆) . p0 p ρ ke v0k 2−2α+ n + λ + α ! ≤ ρ0 0 p0 p ρ L 0,∞;FNp,λ,q ρ0 0 L 0,∞;FNp,λ,q

×kw − vk  2−2α+ n + λ + α  p0 p 0 C kv k n λ ρ0 ρ 0 0 2−2α+ 0 + 0 0 p p L 0,∞;FNp,λ,q  FNp,λ,q

+kw − vk 2−2α+ n + λ + α ! p0 p ρ Lρ0 0,∞;FN 0 and p,λ,q

×kvk  2−2α+ n + λ + α  α 0 p0 p ρ0 −t(−∆) 2 ρ0 0 L 0,∞;FNp,λ,q  ke v0k  2−2α+ n + λ + α  0 p0 p ρ0 Lρ0 0,∞;FN 0  p,λ,q  ≤ C3 k(v, w)k 2−2α+ n + λ + α ! p0 p ρ Lρ0 0,∞;FN 0 ≤ C0 kv0k 2−2α+ n + λ , p,λ,q FN p0 p p,λ,q ×k(v, w)k  2−2α+ n + λ + α  0 p0 p ρ0 Lρ0 0,∞;FN 0 which implies  p,λ,q  ≤ C k(v, w)k2 3 X∞ α −t(−∆) 2 ke v0kX ≤ 2C0 kv0k 2−2α+ n + λ ∞ p0 p Analogously, we get FNp,λ,q 2 kB (v, w)k  n λ α  ≤ C k(v, w)k 1 2−2α+ + + 3 X∞ 0 p0 p ρ0 Lρ0 0,∞;FN 0 Similary,  p,λ,q 

α Thus, we obtain −t(−∆) 2 ke w0kX∞ ≤ 2C1 kw0k 2−2α+ n + λ FN p0 p 2 p,λ,q kB (v, w)k ≤ 2C k(v, w)k 1 X∞ 3 X∞ Similary, Thus kB (v, w)k ≤ 2C k(v, w)k2 2 X∞ 4 X∞ α α −t(−∆) 2 −t(−∆) 2 k(e v0, e w0)kX∞ ≤ Finally,

C2 k(v0, w0)k 2−2α+ n + λ (14) 2 FN p0 p k(B (v, w),B (v, w))k ≤ Ck(v, w)k p,λ,q 1 2 X∞ X∞

By Lemma 2.3, we know that if Using the fact that |ξ| ≈ 2j for all j ∈ , we have α α Z −t(−∆) 2 −t(−∆) 2 1 k(e v0, e w0)kX∞ ≤ ε with ε = 4C , α α then the system (1) has a unique global solution in −t(−∆) 2 −t(−∆) 2 e v0,1, e w0,1 2−2α+ n + λ + α ¯ p0 p ρ B(0, 2ε) = {x ∈ X∞ kxkX∞ ≤ 2ε} . To prove ρ0 0 α α L (I,FN p,λ,q ) −t(−∆) 2 −t(−∆) 2 k(e v , e w )k ≤ ε, according to (14) we 1/q 0 0 X∞ n X j(2−2α+ n + λ + α )q α q o p0 p ρ0 −t(\−∆) 2 = 2 kϕje v0,1k ρ0 λ have L (I,Mp ) j∈Z α α −t(−∆) 2 −t(−∆) 2 n n λ α α o1/q X j(2−2α+ 0 + p + ρ )q −t(−\∆) 2 q k(e v0, e w0)kX∞ ≤ p 0 + 2 kϕje w0,1k ρ0 λ L (I,Mp ) C2 k(v0, w0)k 2−2α+ n + λ j∈Z p0 p FNp,λ,q 1/q n X j(2−2α+ n + λ )q j( α )q α q o p0 p ρ0 = 2 2 kϕj|ξ| χB(0,δ)vˆ0k ρ0 λ L (I,Mp ) 1 j∈Z So, if k(v0, w0)k 2−2α+ n + λ ≤ K with K = , then p0 p 4CC2 FN n λ α 1/q p,λ,q n X j(2−2α+ + )q j( )q α q o p0 p ρ0 + 2 2 kϕj|ξ| χB(0,δ)wˆ0k ρ0 λ (1) has a unique global solution (v, w) ∈ X∞ satisfying L (I,Mp ) j∈Z 1/q α+ α X j(2−2α+ n + λ )q q 1 ρ0 p0 p k(v, w)k ≤ . δ ( 2 kϕjvˆ0kLρ0 (I,Mλ) X∞ 2C p j∈Z 1/q n j(2−2α+ n + λ )q o X p0 p q For the local existence, we shall decompose the initial data v0 + 2 kϕjwˆ0k ρ0 λ L (I,Mp ) into two terms j∈Z α 1 α+ ρ ρ ≤ C δ 0 T 0 k(v , w )k n λ 5 0 0 2−2α+ 0 + −1 −1 FN p p v0 = F χB(0,δ)vˆ0 + F χBC (0,δ)vˆ0 := v0,1 + v0,2, p,λ,q where δ = δ (v0) > 0 is a real number. Similary, we Thus decompose w0 α α −t(−∆) 2 −t(−∆) 2 e v0,1, e w0,1 2−2α+ n + λ + α ≤ p0 p ρ −1 −1 Lρ0 (I,FN 0 ) w0 = F χB(0,δ)wˆ0 + F χBC (0,δ)wˆ0 := w0,1 + w0,2. p,λ,q α 1 α+ ρ ρ C δ 0 T 0 k(v , w )k n λ 5 0 0 2−2α+ 0 + FN p p Since p,λ,q

 n λ 2−2α+ p0 + p Similary,  v0,2 −→ 0 in FNp,λ,q when δ → +∞, 2−2α+ n + λ p0 p α α  w0,2 −→ 0 in FNp,λ,q when δ → +∞, −t(−∆) 2 −t(−∆) 2 e v0,1, e w0,1 2−2α+ n + λ + α ≤ 0 p0 p ρ0 ρ0 0 L (I,FN p,λ,q ) then, there exists δ large enough such that α+ α 1 ρ0 ρ0 C δ 0 T 0 k(v , w )k n λ . 5 0 0 2−2α+ 0 + FN p p ε p,λ,q C k(v , w )k n λ ≤ . 2 0,2 0,2 2−2α+ 0 + FN p p 2 p,λ,q Hence,

We get α α −t(−∆) 2 −t(−∆) 2 e v0,1, e w0,1 ≤ α α XT −t(−∆) 2 −t(−∆) 2 ε e v0, e w0 ≤ α+ α 1 ρ0 ρ0 XT 2 C5δ T k(v0, w0)k 2−2α+ n + λ 0 p α α FN p −t(−∆) 2 −t(−∆) 2 p,λ,q + e v0,1, e w0,1 α 1 X α+ 0 0 T ρ0 ρ0 +C5δ T k(v0, w0)k 2−2α+ n + λ p0 p FNp,λ,q We have

α α Then, if we choose T small enough such that −t(−∆) 2 −t(−∆) 2 e v0,1, e w0,1 = XT α α  α+ α 1 −t(−∆) 2 −t(−∆) 2 ρ0 ρ0 ε n λ α  C5δ T k(v0, w0)k 2−2α+ n + λ ≤ e v0,1, e w0,1 2−2α+ + +  p0 p 4 p0 p ρ0 Lρ0 (I,FN )  FNp,λ,q p,λ,q  α α and −t(−∆) 2 −t(−∆) 2 + e v , e w n λ α α+ α 1 0,1 0,1 2−2α+ 0 + + 0  0 0 ε 0 p p ρ  ρ0 ρ0 Lρ0 (I,FN 0 )  C5δ T k(v0, w0)k 2−2α+ n + λ ≤ . p,λ,q  p0 p 4 FNp,λ,q

IJOA ©2021 34 International Journal on Optimization and Applications IJOA. Vol. 1, Issue No. 2, Year 2021, www.usms.ac.ma/ijoa Copyright ©2021 by International Journal on Optimization and Applications i.e, According to Taylor’s formula and using the same arguments as [ [21], Proposition 2.3], we get   ρ0 q  X j(2−2α+ n + λ )  p0 p   ε  2 kvˆj (t1) − vˆj (t2)kMλ  T ≤ p   α+ α  j≤N  ρ0  4C5δ k(v0, w0)k 2−2α+ n + λ   p0 p X n λ q  FNp,λ,q q j(2−2α+ p0 + p )q ˆ  . |t1 − t2| 2 (∂tu)j L1(I,Mλ) and j≤N p ρ0    0 q q  |t1 − t2| × k∂tuk !  . 2−2α+ n + λ  ε L1 0,T ;FN p0 p    p,λ,q  T ≤  α+ α   ρ0  4C δ 0 k(v , w )k n λ   5 0 0 2−2α+ 0 + q q  p p |t − t | × k∆vk ! FNp,λ,q . 1 2 2−2α+ n + λ 1 p0 p L 0,T ;FNp,λ,q So, if we choose q + k∇ · (v∇φ)k ! 2−2α+ n + λ ε ρ0 1 p0 p L 0,T ;FNp,λ,q T ≤ min α+ α , ρ0 4C5δ k(v0, w0)k 2−2α+ n + λ p0 p q q FNp,λ,q |t − t | × kvk ! . 1 2 2−α+ n + λ 0 L1 0,T ;FN p0 p ε ρ0 p,λ,q α+ α ρ0 q 4C δ 0 k(v , w )k n λ 5 0 0 2−2α+ 0 + + k∇ · (v∇φ)k ! p p 2−2α+ n + λ FNp,λ,q 1 p0 p L 0,T ;FNp,λ,q then q |t − t |q × kv k . 1 2 0 2−2α+ n + λ α α FN p0 p −t(−∆) 2 −t(−∆) 2 ε p,λ,q e v0,1, e w0,1 ≤ . X q T 2 +2 k∇ · (v∇φ)k ! . 2−2α+ n + λ L1 0,T ;FN p0 p This result with (5.2) yields that p,λ,q Thus, we obtain the continuity of v in time t. α α −t(−∆) 2 −t(−∆) 2 Similary, we use the same discusion to get the continuity of e v0, e w0 ≤ ε. XT w in time t. n λ 2−2α+ p0 + p n λ 2−2α+ p0 + p Hence (v, w) ∈ C 0,T ; FNp,λ,q , and we are done. Thus for any arbitrary (v0, w0) ∈ FNp,λ,q , (1) has a ¯ unique local solution in B(0, 2ε) = {x ∈ XT : kxkXT ≤ REFERENCES 2ε} . [1] A., Naoufel Ben, F. Mehats,´ and N. Vauchelet, A note on the long Regularity: time behavior for the drift-diffusion-Poisson system, Comptes Rendus Mathematique, Vol.339, pp.683-688, 2004. We know if (v, w) ∈ XT × XT is a solution of (1), then [2] B., Hajer, The Littlewood-Paley theory: a common thread of many words we can show that in nonlinear analysis, 2019. [3] H. Bahouri, J. Chemin, and R. Danchin. Fourier analysis and nonlinear n λ 1 2−2α+ p0 + p partial differential equations, Springer Science and Business Media, ∇ · (v∇φ) , ∇ · (w∇φ) ∈ L 0,T ; FNp,λ,q . Vol.343, 2011. [4] Biler, Piotr, Existence and asymptotics of solutions for a parabolic- elliptic system with nonlinear no-ﬂux boundary conditions, Nonlinear By using the deﬁnition of the Fourier-Besov-Morrey spaces, Analysis: Theory, Methods and Applications Vol.19, pp.1121-1136, we have 1992. [5] B., Piotr, and J. Dolbeault, Long time behavior of solutions to Nernst- Planck and Debye-Huckel¨ drift-diffusion systems, Annales Henri, Vol.1, No.3, 2000. q [6] B., Piotr, D. Hilhorst, and T. Nadzieja, Existence and nonexistence kv (t1) − v (t2)k 2−2α+ n + λ p0 p FNp,λ,q of solutions for a model of gravitational interaction of particles, II., q Colloquium Mathematicum, Vol.67, Instytut Matematyczny Polskiej X j(2−2α+ n + λ ) p0 p Akademii Nauk, 1994. ≤ 2 kvˆj (t1) − vˆj (t2)kMλ p [7] Biler, Piotr, Waldemar Hebisch, and Tadeusz Nadzieja, The Debye sys- j≤N tem: existence and large time behavior of solutions, Nonlinear Analysis: n λ q X j(2−2α+ p0 + p ) Theory, Methods and Applications, Vol.23, pp.1189-1209, 1994. +2 2 kvˆj(t)k ∞ λ , L (I,Mp ) [8] C., Xiaoli, Well-Posedness of the Keller–Segel System in Fourier– j>N Besov–Morrey Spaces, Zeitschrift fur¨ Analysis und ihre Anwendungen, Vol.37, pp.417-434, 2018. where vˆj = ϕjv.ˆ For any small constant ε > 0, let N be large [9] D., Chao, and C. Li, Endpoint bilinear estimates and applications to the enough such that two-dimensional Poisson–Nernst–Planck system, Nonlinearity, Vol.26, 2013. [10] F., Lucas CF, and L. SM Lima, Self-similar solutions for active scalar X j(2−2α+ n + λ )q q ε p0 p equations in Fourier–Besov–Morrey spaces, Monatshefte fur¨ Mathe- 2 kvˆj(t)k ∞ λ ≤ . L (I,Mp ) 4 j>N matik, Vol.175, pp.491-509, 2014.

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Khalid HILAL ∗, Ahmed KAJOUNI ∗ & Hamid LMOU ∗ ∗ Laboratory of Applied Mathematics and Scientiﬁc Computing, Sultan Moulay Slimane University, B.P523, Beni Mellal, Morocco, Email: [email protected], {kajjouni, lmou.hamid}@yahoo.fr

Abstract— This paper is mainly concerned with existence of prove the existence of mild solution for our probleme (1), mild solution for a neutral functionnal integrodiferential inclusion relying on a fixed point theorem for condensing maps due to with finit delay. The results are bobtained by using a fixed point Martelli [4]. theoreme for condensing multivalued maps. II.PRELIMINARIES: Index Terms—integrodifferentional inclusion, Selection, Fixed point theory, neutral functional differential and integrodifferen- In this section, we introduce some basic definitions, tial inclusion, convex multivalued map. notations, and lemmas that are used throughout this paper. C(J, X) is the Banach space of continuous functions from I.INTRODUCTION: J into X with the norm : In this paper we prove the existence of mild solution, for a neutral functional integrodifferentional inclusion with kuk∞ := sup{|u(t)|; t ∈ J} finite delay. In section 2 we will recall briefly some basic A measurable function u : J −→ X is Bochner integrable definitions and preliminary facts which will be used in the if and only if |u| is Lebesgue integrable (For properties of the following section. Section 3 deals with the existence of mild Bochner integral see Yosida [5]). solution for a neutral functional integrodifferentional inclusion L1(J, X) denotes the Banach space of continuous functions with finite delay of the forme : u : J −→ X which are Bochner integrable normed by : Z T  Z t 1  d kukL1 := |u(t)|dt for all u ∈ L (J, X)  F(t, ut) ∈ AF(t, ut) + B(t − s)F(s, us)ds 0 dt 0 (1) Lemma 2.1: : +G(t, ut) for t ∈ [0, b]  Let (X, k.k) be a Banach space. A multivalued map G : u = ϕ(θ) for θ ∈ J = [−r, 0], 0 0 X −→ 2X is convex closed, if G(x) is convex closed, for where A, D(A) is the infinitesimal generator of a all x ∈ X; and G is bounded on bounded sets, if G(B) = compact resolvant operator R(t), t ≥ 0, in Banach space ∪ G(x) is bounded in X, for any bounded set B of X. x∈B X, for t ≥ 0 B(t) is a closed linear operator with domain Theorem 2.1: : X D(B), such that D(A) ⊂ D(B). G : J × C(J0,X) −→ 2 G is said to be completely continuous if G(B) is rela- (J0 = [−r, 0]), is a bounded, closed, convex, multivalued map tively compact, for every bounded subset B ⊂ X. and X a real Banach space. Theorem 2.2: : For any continus function u defined on J1 = [−r, b], and G is called upper semi-continuous (u.s.c) on X, if for any t ∈ J, we denot by ut the element of C(J0,X) defined each x ∈ X, the set G(x) is a nonempty, closed subset of X, by: and if for each open set B of X containing G(x), there exists an open neighborhood V of x such that G(V ) ∈ B. ut(θ) = u(t + θ) = ϕ(θ), θ ∈ J0 = [−r, 0], Lemma 2.2: : Here ut(.) represents the history of the state from time t−r, If the multivalued map G is completely continuous with up to the present time t, and F : J ×C(J0,X) −→ X defined nonempty compact values, then G is u.s.c. if and only if G has by : a closed graph i.e xn −→ x, yn −→ y; yn ∈ G(xn) imply F(t, ϕ) = ϕ(0) − F (t, ϕ) = u(t) − F (t, ut), ∀(t, ϕ) ∈ y ∈ G(x) . J × C(J ,X), 0 Definition 2.1: : Where F : J × C(J0,X) −→ X. an upper semi-continuous multivalued map G : X −→ Wen B = 0 we refer to the paper of K.HILAL and K.EZZINBI X is said to be condensing if for any subset B ⊂ X with [1] and the paper of K.EZZINBI and X.FU [2]. α(B) 6= 0, we have αG(B) < αB , where α denotes the This paper is motivated by the recents results of [1] and Kuratowski measure of noncompactness [6]. BENCHOHRA [3]. Here we compose the above results and Lemma 2.3: :

1 A completely continuous multivalued map is a condens- Where g ∈ SG,u = {g ∈ L (J, X): g(t) ∈ G(t, ut); t ∈ ing map J} Theorem 2.3: :(Arzela–Ascoli’s theorem) Deﬁnition 3.1: : Let K be a compact space and (E, d) a metric space. A function u ∈ C([−r, b],X) is called a mild solution of A ⊂ C(K,E) is relatively compact (i.e. included in a compact) (1) if : if and only if, for any x of K: 1- u(0) = ϕ(θ); θ ∈ [−r, 0].

• A is equicontinuous in x, i.e. for everything ε > 0, there 2- There exist a function g ∈ SG,u such that : exist a neighborhood V of x such that : ∀f ∈ A, ∀y ∈ u(t) = V d(f(x), f(y)) < ε Z t R(t)F(0, ϕ)+F (t, ut)+ R(t−s)g(s)ds t ∈ [0, b] • The set A(x) = {f(x); f ∈ A} is relatively compact. 0 In the following BCC(X) denotes the set of all nonempty Where, F(0, ϕ) = ϕ(θ) − F (t, ϕ) bounded, closed and convex subsets of X Assume that : Theorem 2.4: :(Leray-schauder’s fixed point) (H1)- A is the infinitesimal generator of a compact resolvent Let X be a Banach space and N : X −→ BCC(X) an operator R(t) in X such that : u.s.c condensing map. If the set : kR(t)k ≤ M1 for some M1 ≥ 1 ; t ∈ J Ω := {u ∈ X : λu ∈ Nu for λ > 1} (H2)- There exists constants 0 ≤ c1 < 1 and c2 ≥ 0 such that : is bounded, then N has a fixed point. |F (t, u)| ≤ c kuk + c ; t ∈ J u ∈ C(J ,X) Definition 2.2: :(Finite delay differential equation) 1 2 0 (H3)- ϕ ∈ C([−r, 0],X) is completely continuous and there Let r > 0; and C = C[−r, 0], n, the Banach space r R exists a constant M such that: of continuous functions, 2 n kϕk ≤ M2 ϕ :[−r, 0] −→ R with kϕk∞ = sup kϕ(θ)k. We denot θ∈[−r,0] (H4)- G : J × C(J0,X) −→ BCC(X) ; (t, u) −→ G(t, u) by ut element of Cr defined by : is measurable with respect to t for each u ∈ C(J0,X), ut(θ) = u(t + θ) = ϕ(θ), θ ∈ J0 = [−r, 0], u.s.c with respect to u for each t ∈ J; and for each fixed Let f : + × C −→ , a general form of the finit-delay R r R u ∈ C(J0,X) the set : differential equation is : S = {g ∈ L1(J, X): g(t) ∈ G(t, u ); t ∈ J} d G,u t u(t) = f(t, u ) dt t is nonempty. Definition 2.3: :(Resolvent operator [2] ) (H5)- kG(t, u)k := sup{|g| : g ∈ G(t, u)} ≤ p(t)Ψ(kuk) for 1 + A family of bounded linear operators R(t) ∈ B(X), all t ∈ J and all u ∈ C(J0,X), where p ∈ L (J, R ) + B(X) is the Banach space of all linear bounded operator and Ψ: R −→ [0, +∞) is continuous and increasing from X into X, for t ∈ J is called a resolvent operator for : with : du Z t Z b Z ∞ dτ = Au(t) + f(t − s)u(t)ds ω(s)ds < dt 0 0 c τ + Ψ(τ) If: Where 1 1- R(0) = I, the identity operator on X, and kR(t)k 6 M c = {M1 M2(1 + c1) + c2 + c2} and ω(s) = 1 − c1 with M > 1. 1 2- For all u ∈ X; R(t)u is continuous for t ∈ J M1p(t) 1 − c1 3- R(t) ∈ B(Y ); t ∈ J; where Y is the Banach space (H6)- The function F is completly continuous and for any formed from D(A), for y ∈ Y,R(.)y ∈ C1(J, X) ∩ bounded set B ⊆ C(J1,X) the set {t −→ F (t, ut): C(J, Y ) and : u ∈ B} is equicontinuous in C. t 0 Z R (t)y = AR(t)y + f(t − s)R(s)yds = The following lemma is crucial in the proof of our 0 existence results. Z t R(t)Ay + R(t − s)f(s)yds. Lemma 3.1: : 0 Let I be a compact real interval and X be a Banach III. EXISTANCE RESULTS : space. Let G be a multivalued map satisfying (H4). And let Γ be a linear continuous mapping from L1(I,X) to C(J, X). In order to define the concept of mild solution for (1), Then the operator : by comparaison with the evolution problem Γ ◦ SG : C(I,X) −→ BCC C(I,X) ; u −→ Z t dv Γ ◦ SG (u) = Γ SG = Av(t) + f(t − s)v(t)ds + h(t); v(0) = a Is closed graph operator in C(I,X) × C(I,X) dt 0 Our main result may be presented as the following We asssociate (1) to the integral equation : theorem. Z t Theorem 3.1: : u(t) = R(t)F(0, ϕ) + F (t, ut) + R(t − s)g(s)ds t ∈ Assume that hypotheses (H1) − (H6) hold, then the 0 [0, b] problem (1) has at least one mild solution on J1.

Proof 3.1: : By (H1) − (H3) , and (H5) we have for each t ∈ J : Let C := C(J1,X) be the Banach space of continuous Z t function from J1 into X endowed with the sup-norm : |h1(t)| ≤ kR(t)F(0, ϕ)k + kR(t − s)g(s)kds 0 kuk∞ := sup{|u| : t ∈ [−r, b]}; for u ∈ C ≤ M [c M + c ] Transform the problem into a ﬁxed point problem. Con- 1 1 2 2 t sider the multivalued map, N : C −→ 2C deﬁned by : Z ( + M1 sup Ψ(u) p(s)ds u∈[0,q] 0 N u := h ∈ C : Then for each h ∈ N (B ):  1 q ϕ(t); t ∈ J0 )  kh1(t)k∞ ≤ M1[c1M2 + c2] h(t) = R(t)F(0, ϕ) + F (t, u ) t Z b  R t + 0 R(t − s)g(s)ds; t ∈ J + M1 sup Ψ(u) p(s)ds 1 u∈[0,q] 0 Where : g ∈ SG,u = {g ∈ L (J, X): g(t) ∈ G(t, ut); t ∈ J} Then N1 is bounded.

We have that the fixed points of N are mild solutions to (1). ii- N1 maps bounded set into equicontinuous sets of C: Now we shall prove that N is a completely continuous multivalued map, u.s.c, with convex closed values. The proof Let τ1, τ2 ∈ J; τ1 < τ2, and Bq be bounded set of C; for will be given in several steps. each u ∈ Bq and h1 ∈ N1u; there exist g ∈ SG,u such that : Step 1 : N u is convex for each u ∈ C. Z t Indeed, if h1, h2 belong to N u, then there exist g1, g2 ∈ SG,u such that for each t ∈ J we have: h1(t) = R(t)F(0, ϕ) + R(t − s)g(s)ds; t ∈ J 0 Z t Thus, h1(t) = R(t)F(0, ϕ) + F (t, ut) + R(t − s)g1(s)ds 0 and h1(τ2) − h1(τ1) = R(τ2)F(0, ϕ) Z τ2 Z t + R(τ2 − s)g(s)ds − R(τ1)F(0, ϕ) h2(t) = R(t)F(0, ϕ) + F (t, ut) + R(t − s)g2(s)ds 0 0 Z τ1 − R(τ1 − s)g(s)ds Let 0 ≤ k ≤ 1. Then for each t ∈ J we have : 0 = R(τ2) − R(τ1) F(0, ϕ) kh1 + (1 − k)h2 (t) = R(t)F(0, ϕ) + F (t, ut) + Z τ1 Z t + R(τ2 − s) − R(τ1 − s) g(s)ds R(t − s) kg1(s) + (1 − k)g2(s) ds 0 0 Z τ2 + R(τ − s)g(s)ds Thus kh1 + (1 − k)h2 ∈ N u because S is convex , 2 G,u τ then N u is convex for each u ∈ C 1 Step 2 : We will prove that N is a completely continuous Then operator. Using (H6) it suffices to show that the operator kh (τ ) − h (τ )k ≤ kR(τ ) − R(τ )k ( 1 2 1 1 2 1 τ1 C Z N1 : C −→ 2 defined by : N1u := h1 ∈ C : + kR(τ2 − s)  0 ϕ(t); t ∈ J0 ) − R(τ − s)kkg(s)kds  1 h (t) = R(t)F(0, ϕ) Z τ2 1 + kR(τ − s)kkg(s)kds  R t 2 + 0 R(t − s)g(s)ds; t ∈ J τ1 is completly continuous . As τ2 −→ τ1 the right-hand side of the above inequality tends to zero, implies that N1u is equicontinuous on J1 i- N1 map bounded set into bounded set in C : iii- V (t) = {h1(t); h1 ∈ N1(Bq)} is relatively compact Indeed, it is enough to show that there exists a positive constant on X: l such that for each h1 ∈ N1u; u ∈ Bq = {u ∈ C : kuk∞ ≤ By (H4) V (t) is relatively compact for t = 0; let 0 ≤ t ≤ b q} we have kh1k∞ ≤ l . be fixed and let ε be a real number satisfying 0 ≤ ε < t for If h1 ∈ N1u then there exist g ∈ S , such that for every G,u u ∈ Bq and g ∈ S such that : t ∈ J we have : G,u Z t Z t h1(t) = R(t)F(0, ϕ) + R(t − s)g(s)ds h1(t) = R(t)F(0, ϕ) + R(t − s)g(s)ds; t ∈ J 0 0

and, It follows that g ∈ SG,u such that Z t−ε h(t) = R(t)F(0, ϕ) + F (t, ut) h1,ε(t) = R(t)F(0, ϕ) + R(t − s)g(s)ds; t ∈ J t 0 Z + R(t − s)g(s)ds; t ∈ J The set Vε(t) = {h1,ε(t); h1,ε ∈ N1(Bq)} is relatively 0 compact beceuse R(t) is compact then; From the Step 1, step 2 and step 3 we deduce that N is u.s.c, completely continuous then by lemma (2.3), N is a Z t condensing, bounded, closed and convex operator. In order to kh1(t) − h1,ε(t)k = R(t − s)g(s)k prove that N has a ﬁxed point, we need one more step. t−ε Step 4 : The set Ω := {u ∈ X : λu ∈ N u for λ > 1} Z t ≤ M1 sup Ψ(u) p(s)ds.ε is bounded. Let u ∈ Ω. Then λu ∈ N u, thus there exists u∈[0,q] 0 g ∈ SG,u such that : ≤ rε −1 −1 u(t) = λ R(t)F(0, ϕ) + λ F (t, ut) Z b Z t −1 With; r = M1 sup Ψ(u) p(s)ds, this implies that + λ R(t − s)g(s)ds u∈[0,q] 0 0 V (t) is relativement compact thus by (i), (ii), (iii) and by By (H1) − (H3)and (H5) we have : Arzela-Ascoli theorem, we can deduce that N1 is completely continuous then N : C −→ 2C is completely continuous. |u(t)| ≤ M (1 + c )M + c + c ku k + c Step 3 : N has a closed graph. 1 1 2 2 1 t 2 Let un −→ u, hn ∈ N un and hn −→ h, we shall prove that Z t h ∈ N u. + M1 p(s)Ψ(kusk)ds 0 hn ∈ N un then there exists gn ∈ SG,u such that n Consider the function deﬁned by : hn(t) = R(t)F(0, ϕ) + F (t, unt) Z t µ(t) = sup{|u(s)| : −r ≤ s ≤ t}; 0 ≤ t ≤ b + R(t − s)gn(s)ds; t ∈ J ∗ ∗ 0 Let t ∈ [−r, t] be such that µ(t) = |u(t )|. ∗ ? If t ∈ J0 = [−r, 0] then : We should prove that g ∈ SG,u such that for each t ∈ J : µ(t) ≤ kφk ≤ M2

∗ h(t) = R(t)F(0, ϕ) + F (t, ut) ?? If t ∈ J = [0, b] then : Z t + R(t − s)g(s)ds; t ∈ J µ(t) ≤ M1 (1 + c1)M2 + c2 + c1µ(t) + c2 0 Z t Since F is continuous, we have that: + M1 p(s)Ψ µ(s) ds 0 k hn(t) − R(t)F(0, ϕ) − F (t, unt) then;

1 − h(t) − R(t)F(0, ϕ) − F (t, ut) k∞ −→ 0 µ(t) ≤ M1 (1 + c1)M2 + c2 + c2 1 − c1 As n −→ ∞. Z t ! Consider the linear operator : + M1 p(s)Ψ µ(s) ds 0 Γ: L1(J, X) −→ C(J, X) since M1 ≥ 1 Let us take the right-hand side of the above Z t inequality as ν(t) . Then we have. g −→ Γ(g)(t) = R(t − s)g(s)ds ! 0 1 c = ν(0) = M1 M2(1 + c1) + c2 + c2 and 1 − c1 From Lemma 3.1; Γ ◦ SG is a closed graph operator then we have that : µ(t) ≤ ν(t); ∀t ∈ J then, 1 h (t) − R(t)F(0, ϕ) − F (t, u ) ∈ Γ S 0 n nt G,un ν (t) = M1p(t)Ψ µ(t) 1 − c1 Since un −→ u, and by the lemma 3.1 : By using H(5) we get :

0 h(t) − R(t)F(0, ϕ) − F (t, ut) ∈ Γ SG,u ν < ω(t)Ψ ν(t)

This implies that Z ν(t) dτ Z b Z ∞ dτ < ω(s)ds < ν(0)=c Ψ(τ) 0 c τ + Ψ(τ) This implies that there exists a constant K such that ν ≤ K, t ∈ J and µ ≤ K, t ∈ J. Since for every t ∈ J we have kutk ≤ µ(t) then

kuk∞ := sup{|u(t)|; −r ≤ t ≤ b} ≤ K Where K depends only on b and on the functions p and Ψ. This shows that Ω is bounded. As a consequence of theorem 2.4 (Leray-schauders’s ﬁxed point) we deduce that N has a ﬁxed point which is a solution of (1).

REFERENCES [1] K. HILLAL and K. EZZINBI, Nonlocal differential inclusion with nondense domain and multivalued perturbations in Banach spaces. [2] K. EZZINBI and X. FU, Existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions. [3] M. BENCHOHRA and S. K., NTOUYAS Nonlocal Cauchy Problems for Neutral Functional Differential and Integrodifferential Inclusions in Banach Spaces. [4] M. Martelli, A Rothe’s type theorem for non-compact acyclic-valued map, Boll. Un. Mat. Ital. [5] K. Yosida, Functional Analysis. 6th ed. Springer, Berlin, 1980. [6] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Dekker, New York, 1980.

Said Melliani ∗, M’hamed Elomari ∗, Khalid Hilal ∗ & Manal Menchih ∗ ∗ Laboratory of Applied Mathematics and Scientiﬁc Computing, Sultan Moulay Slimane University, Morocco Email: [email protected], [email protected], {hilalkhalid2005, manalmenchih97}@yahoo.fr

Abstract—This work is related to investigate integral Riemann-Liouville differentiability notion as the base solution of wave equation with fuzzy initial data under to deﬁne the concept of fuzzy fractional DEs. After generalized fuzzy Caputo derivative. For the concerned that, they proved the existence of solutions of fuzzy investigation, we use the Fourier transform. The exact fractional integral equations (IEs) under compactness solution is given in the case of γ = 2. Some examples are presented to illustrate the results. type conditions using the Hausdorff measure of non- Index Terms—Generalized fuzzy derivative, Caputo compactness in the paper [2]. Allahviranloo et all in fractional derivative, Hukuhara difference, fuzzy fourier [3] presented two new results on the existence of two transform. kinds of gH−weak solutions of these problems and indicated the boundedness and continuous depen- I. Introduction dence of solutions on the initial data of the problems. The present paper investigate the analytic solution In [5] the authors prove the existence and uniqueness of the following problem theorems for non-linear fuzzy fractional Fredholm  2  Dγu(t, x) − c2 ∂ u(t, x) = 0, integro-differential equations under fractional gen- gH t g ∂x2  eralized Hukuhara derivatives in the Caputo sense. −∞ < x < ∞, t ≥ 0, 1 < γ < 2 From the idea of [5] we will try to prove the existence u(0, x) = a(x)  and uniqueness of fuzzy fractional wave equation.   ∂ u( x) = b(x) ∂t 0, This paper is organized as follows. In section 2 where a and b are two absolutely valued-functions we recall some concepts concerning the fuzzy metric in E1. −g is the generalized Hukuhara difference. space. the generalized derivative take place in the gH D is the generalized fuzzy fractional caputo’s section 3. In section 4 we give the concept of fuzzy derivative. Fourier transform and we presented some properties. We presented the solution of the fuzzy wave equation In 1965 L.Zadeh [13] introduced the basic ideas of in section 5. Finally in section 6 two examples are the fuzzy set theory, as an extension of the classical given to illustrate the usefulness of our main results. notion of set. The authors in [6] give a generaliza- tion of the Hukuhara difference which guaranteed II. preliminaries the existence of this is for two segments in R. As consequance in the same work Bede and Stefanini In this section, we present some deﬁnitions and presented the generalized derivative of a set valued- introduce the necessary notation, which will be used functions. Agarwal et al. [1] are the pioneers work- throughout the paper. ing in fuzzy fractional (DEs). They formulated the

We denote E1 the class of function defined as 2) zα, α ∈ (0, 1] is open, bounded, connected and follows: simply connected; 3) z1 is non-empty, compact, arcwise connected and n o simply connected. E1 = u : R → [0, 1], u satisfies (1 − 4) below We denote the set of all fuzzy complex number by C1. 1) u is normal, i.e. there is a x0 ∈ R such that u(x0) = 1; 2) u is a fuzzy convex set; Definition II.3 [6] The generalized Hukuhara difference 1 3) u is upper semi-continuous; of two fuzzy numbers u, v ∈ E is defined as follows n  4) u closure of {x ∈ R , u(x) > 0} is compact u = v + w 1 u −g v = w ⇔ For all α ∈ (0, 1] the α-cut of an element of E is or v = u + (−1)w defined by In terms of α-levels we have

α n o u = x ∈ R, u(x) ≥ α α h u −g v = min {u(α) − v(α), u(α) − v(α)} , By the previous properties we can write i max {u(α) − v(α), u(α) − v(α)} α u = [u(α), u(α)] and the conditions for the existence of w = u −g v ∈ E1 are By the extension principal of Zadeh we have  w(α) = u(α) − v(α) and w(α) = u(α) − v(α)  α case (i) with w(α) increasing, (u + v) = uα + vα;   α w(α) decreasing, w(α) ≤ w(α) (λu) = λuα  w(α) = u(α) − v(α) and w(α) = u(α) − v(α)  For all u, v ∈ E1 and λ ∈ R  case (ii) with w(α) increasing, 1 The distance between two element of E is given  w(α) decreasing, w(α) ≤ w(α) by (see [4]) for all α ∈ [0, 1]. n o d(u, v) = sup max |u(α) − v(α)|, |u(α) − v(α)| α∈(0,1] Throughout the rest of this paper, we assume that 1 u −g v ∈ E The metric space (E1, d) is complete, separable and locally compact and the following properties for Proposition II.4 [11] metric d are valid: ku −g vk = d(u, v) 1) d(u + v, u + w) = d(u, v); Since k.k is a norm on En and by the proposition 2) d(λu, λv) = |λ|d(u, v); (II.4) we have 3) d(u + v, w + z) ≤ d(u, w) + d(v, z); Proposition II.5 Remark II.1 The space (E1, d) is a linear normed space with kuk = d(u, 0). kλu −g µu, 0k = |λ − µ|kuk R 1 Deﬁnition II.2 [10] A complex fuzzy number is a Let f : [a, b] ⊂ → E a fuzzy-valued function. mapping z : C → [0, 1] with the following properties: The α-level of f is given by h i 1) z is continuous; f (x, α) = f (x, α), f (x, α) , ∀x ∈ [a, b], ∀α ∈ [0, 1].

Definition II.6 [6] Let x0 ∈ (a, b) and h be such that neighborhood V of x0 there exist points x1 < x0 < x2 x0 + h ∈ (a, b), then the generalized Hukuhara derivative such that ( ) → 1 of a fuzzy value function f : a, b E at x0 is defined 1) type (1). at x1 (II.2) holds while (II.3) does not hold as and at x2 (II.3) holds and (II.2) does not hold, or f (x + h) − f (x ) 0 g 0 0 2) type (2). at x1 (II.3) holds while (II.2) does not hold lim −g fgH(x0) = 0 (II.1) h→0 h and at x2 (II.2) holds and (II.3) does not hold. If f (x ) ∈ E1 satisfying (II.1) exists, we say that f is gH 0 Definition II.11 [3] Let f : (a, b) → E1. We say that generalized Hukuhara differentiable (gH-differentiable for f (x) is gH-differentiable of the 2nd-order at x0 whenever short) at x . 0 the function f (x) is gH-differentiable of the order i, i = 1 (i) 1 Definition II.7 [6] Let f : [a, b] → E and x0 ∈ (a, b), 0, 1, at x0, (( f (x0))gH ∈ E ), moreover there isn’t any 00 with f (x, α) and f (x, α) both differentiable at x0. switching point on (a, b). Then there exists ( f )gH(x0) ∈ We say that E1 such that f 0(x + h) − f 0(x ) 1) f is [(i) − gH]-differentiable at x0 if 0 g 0 00 lim , fgH(x0) = 0 h→0 h 0 h 0 0 i fi,gH(x0) = f (x, α), f (x, α) (II.2) 1 0 Definition II.12 [3] Let f : [a, b] → E and fg H(x) be 2) f is [(ii) − gH]-differentiable at x0 if gH-differentiable at x0 ∈ (a, b), moreover there isn’t any 0 h 0 0 i switching point on (a, b) and f (x, α) and f (x, α) both fii,gH(x0) = f (x, α), f (x, α) (II.3) differentiable at x0. We say that ⊂ R → 1 → R 0 Theorem II.8 Let f : J E and g : J and • f is [(i) − gH]-differentiable at x0 if x ∈ J. Suppose that g(x) is differentiable function at x h 00 i f 00 (x ) = f 00(x, α), f (x, α) and the fuzzy-valued function f (x) is gH-differentiable i,gH 0 0 at x. So • f is [(ii) − gH]-differentiable at x0 if 0 0 0 ( f g)gH = ( f g)gH + ( f g )gH 00 h 00 00 i fii,gH(x0) = f (x, α), f (x, α) Proof Using (II.5), for h enough small we get Definition II.13 [8] Let f : [a, b] → E1. We say that f (x + h)g(x + h) − f (x)g(x) g f (x) is fuzzy Riemann integrable to I ∈ E1 if for any

h > > 0 0 e 0, there exists δ 0 such that for any division −g ( f (x)g(x))gH + ( f (x)g (x))gH P = {[u, v]; ξ} with the norms ∆(P) < δ, we have = f (x+h)g(x+h)−g f (x)g(x+h)+ f (x)g(x+h)−g f (x)g(x) h ∗ !

0 0 d (v − u) f (ξ),I < e −g ( f (x)g(x))gH + ( f (x)g (x))gH ∑ p = ( f (x+h)−g f (x))g(x+h)+ f (x)(g(x+h)−g g(x)) h ∗ where ∑p denotes the fuzzy summation. We choose to 0 0 b −g ( f (x)g(x))gH + ( f (x)g (x))gH R write I = a f (x)dx. ( f (x+h)− f (x))g(x+h) ≤ g − ( f 0(x)g(x)) h g gH Theorem II.14 [6] If f is gH-differentiable with no

+ ( f (x)(g(x+h)−g g(x)) − ( ( ) 0( )) h g f x g x gH switching point in the interval [a, b] then we have ( ( + )− ( )) ≤ f x h g f x ( + ) − ( 0( ) ( )) Z b h g x h g f x g x gH f (t)dt = f (b) −g f (a) + ( ) ((g(x+h)−g g(x)) − ( ( ) 0( )) a f x h g f x g x gH Theorem II.15 [12] Let f (x) be a fuzzy-valued func- which complet the proof by passing to limit. tion on (−∞, ∞) and it is represented by f (x, α) = h Deﬁnition II.10 [6] We say that a point x0 ∈ (a, b), is f (x, α), f (x, α)]for any ﬁxed α ∈ [0, 1]. Assume a switching point for the differentiability of f , if in any that | f (x, α)| and | f (x, α)| are Riemann integrable on

(−∞, ∞) for all α ∈ [0, 1]. Then f (x) is improper fuzzy Theorem II.22 Let f : [a, b] → E1 and g : [a, b] → R Riemann-integrable on (−∞, ∞) and the improper fuzzy are two differentiable functions ( f is gH-differentiable), Riemann integral is a fuzzy number. Furthermore, we then Z b have 0 fgH(x)g(x)dx = f (b)g(b) −g f (a)g(a) Z ∞ Z ∞ Z ∞ a h i Z b f (x)dx = f (x, α)dx, f (x, α)dx 0 −∞ −∞ −∞ −g f (x)g (x)dx a

1 From this theorem we can discuss the Fuzzy Rie- Remark II.23 If f , g ∈ AE with lim f (x) = 0, |x|→∞ mann’s improper integral lim|x|→∞ g(x) = 0 then + 1 Z ∞ Z ∞ Lemma II.16 Let f : R × R → E , given by 0 0 + fgH(x)g(x)dx = f (x)g (x)dx f (x, t; α) = [ f (x, t; α), f (x, t; α)], and let a ∈ R −∞ −∞ R ∞ R ∞ If a f (x, t; α)dt and a f (x, t; α)dt are converges then III. Fuzzy generalized Hukuhara partial differentiation Z ∞ 1 f (x, t; α)dt ∈ E + 1 a In this section f : D ⊂ R × R → E is called the two variable fuzzy-valued function. The parametric Proof Just use the conditions (II.1). representation of the fuzzy-valued function fis ex- h i Theorem II.18 Let f : R × R+ → E1 be fuzzy-valued pressed by f (x, t, α) = f (x, t, α), f (x, t, α) function such that f (x t ) = [ f (x t ) f (x t )]. , ; α , ; α , , ; α Definition III.1 [3] Let f : D ⊂ R × R+ → E1 Suppose that for each x ∈ [a, ∞), the fuzzy integral and (x0, t0) ∈ D. Then first generalized Hukuhara R ∞ f (x t)dt is convergent and moreover R ∞ f (x t)dx as c , a , partial derivative ([gH − p]-derivative for short) of f with a function of t is convergent on [c, ∞). Then respect to variables x, t are the functions ∂xgH f (x0, t0) and

Z ∞ Z ∞ Z ∞ Z ∞ ∂tgH f (x0, t0) given by f (x, t)dxdt = f (x, t)dtdx c a a a f (x + h, t ) − f (x , t ) 0 0 g 0 0 lim −g ∂xgH f (x0, t0) = 0 h→0 h Proof Applying the theorem of Fubini-Tonelli [7] to these two functions f (x, t; α) and f (x, t; α), and use the condi- and f (x , t + h) − f (x , t ) tions (II.1) 0 0 g 0 0 lim , ∂xgH f (x0, t0) = 0 h→0 h ( ) ( ) Theorem II.20 Suppose both, f x, t and ∂xgH f x, t , 1 provided that ∂x f (x0, t0), ∂t f (x0, t0) ∈ E . are fuzzy continuous in [a, b] × [c, ∞). Suppose also that gH gH 1 the integral converges for x ∈ R, and the integral Deﬁnition III.2 [3] Let f (x, t) : D → E , (x0, t0) ∈ R ∞ D and f (x, t; α) and f (x, t; α) both partial differentiable c f (x, t)dt converges uniformly on [a, b]. Then F is gH- differentiable on [a, b] and w.r.t. t at (x0, t0). We say that • ( ) [( ) − ] ( ) Z ∞ f x, t is i p -differentiable w.r.t. t at x0, t0 if 0 FgH(x) = ∂xgH f (x, t)dt c h i ∂ti,gH f (x0, t0) = ∂t f (x0, t0; α), ∂t f (x0, t0; α) Proof The continuity of ∂x f (x, t) on [a, b] by the con- gH (III.1) vergence domainee theorem of to f (x, t; α) and f (x, t; α) and use the condition (II.1). h i ∂ f (x , t ) = ∂ f (x , t ; α), ∂ f (x , t ; α) According to the theorem (II.8) we get tii,gH 0 0 t 0 0 t 0 0 (III.2)

We inspired of the definition (II.11) we presented IV. Generalized fuzzy fractional derivative the following definition We present generalized fuzzy fractional derivative Definition III.3 f : R × R+R → E1. We say that and their properties. 1 the function t = h(x), is switching boundary for the Definition IV.1 [5] Let f ∈ AE ([a, b]). The fuzzy differentiability of f (x, t) with respect to t, if for all x Riemann-Liouville integral of fuzzy-valued function f is + belongs to domain of h(x) and for all t ∈ R , there exist defined as following: points t0 < t1 < t2 such that 1 Z t Iq f (t) = (t − s)q−1 f (s)ds, 1) at (x, t1) (III.1) holds while (III.2) does not hold and Γ(1 − q) a at (x, t2) (III.2) holds and (III.1) does not hold, or a < s < t, 0 < q < 1. 2) at (x, t ) (III.2) holds while (III.1) does not hold and 1 Definition IV.2 [5] Let f (x, t; α) = at (x, t ) (III.1) holds and (III.2) does not hold. 2 [ f (x, t; α), f (x, t; α)] be a valued-fuzzy function. Theorem III.4 Consider f : R × R+ → E1 and u : The fuzzy Riemann-Liouville integral of f is defined as R → E1 are fuzzy-valued functions such that u(x; α) = following: Z t [u(x; α), u(x; α)]. Suppose that h : R → Rand p : R × q 1 q 0 gH Dt f (t, x; α) = (t − s) fgH(s)ds, R+ → R+ is a differentiable function w.r.t. t and Γ(1 − q) a  a < s < t, 0 < q < 1 ∂t p(x, t) ≥ 0, h1(t) < x < h2(t); ∂ p(x, t) = t Also we say that f is [(i) − gH]-differentiable at t0 if ∂t p(x, t) < 0, h2(t) < x < h3(t) q q gH Dt f (x, t; α) = [D f (x, t; α), f (x, t; α)] and f (x, t) = p(x, t)u(x). Then ∂tgH f (x, t) exists and and f is [(ii) − gH]-differentiable at t if  0 ∂ti gH p(x, t) ≥ 0, h1(t) < x < h2(t); q q , gH D f (x, t; α) = [D f (x, t; α), f (x, t; α)] ∂tgH p(x, t) = t ∂t p(x, t) < 0, h2(t) < x < h3(t) 1 ii,gH Lemma IV.3 Let f ∈ AE and r ∈ (0, 1),then

In fact, the function h2(t) is switching boundary type 1 ( ) r for differentiability of f x, t with respect to t. 1) If f is [(i) − gH]-differentiable at t0 then D f is [( ) − ] Proof Since p is valued in R+ then we can set i gH -differentiable at t0. [( ) − ] r f (x, t; α) = p(x, t)[u(x; α), u(x; α)], which implies that 2) If f is ii gH -differentiable at t0 then D f is [(ii) − gH]-differentiable at t0 Proof Note that ∂tgH = ∂t p(x, t)[u(x; α), u(x; α)] Z t q 1 −q 0 gH D f (t) = (t − s) fgH(s)ds . Γ(1 − q) 0 If h1(t) < x < h2(t) then 1 ( − )−q Since Γ(1−q) t s is a nonegative quantity whenever 0 < t < s. ∂ = [∂ p(x, t)u(x; α), ∂ p(x, t)u(x; α)] 1 tgH t t Theorem IV.5 Let f ∈ AE and q ∈ (1, 2),then then f (x t) is [(i)-differentiable] by report at t. In the q q−1 0 , gH D f (t) = gH D fgH(t) same if h2(t) < x < h3(t) we get Proof We set f (t) = [ f (t; α), f (t; α)]and use lemma (IV.3) = [ p(x t)u(x ) p(x t)u(x )] ∂tgH ∂t , ; α , ∂t , ; α If f is [(i)-differentiable] then 0 thus f (x, t) is [(ii)-differentiable] by report at t f (t)0 = [ f 0(t; α), f (t; α)]

and is well deﬁned 0 Dq−1 f (t)0 = [Dq−1 f 0(t; α), Dq−1 f (t; α)] Proof We have

−iωw If f is [(i)-differentiable] then f (x)e = f (x)

0 0 0 1 1 f (t) = [ f (t; α), f (t; α)] Since f ∈ AE then f (x)e−iωw ∈ AC , which complet and the proof.

0 0 Remark V.3 In the same the map and under same as- Dq−1 f (t)0 = [Dq−1 f (t; α), Dq−1 f (t; α)] symption 1 Proposition IV.7 Let f : LE . 1 γ−1 0 F : R 7−→ C If D f (t) = g(t), then f (t) = f (0) + t fgH(0) + ∞ γ−1 R iωw I g(t) ω → −∞ f (x)e dx

Proof We set f (t) = [ f (t; α), f (t; α)] and is well defined g(t) = [g(t ) g(t )]. ; α , ; α By the previous lemma and remark we can give a 1) If f is [(i)-differentiable] by theorem (IV.5) definition of the fuzzy Fourier transform γ−1 γ−1 γ−1 D f (t) = [D f (t; α), D f (t; α)] Definition V.4 Let f : R → E1 a fuzzy-valued function. = [g(t; α), g(t; α)] The fuzzy Fourier transform of f , denote F( f ) : R → C1, is given by Z ∞ 1 −iωw Which implies that F ( f (x)) = √ f (x)e dx = F(ω) 2π −∞  Dγ−1 f (t; α) = g(t; α) Also the fuzzy inverse Fourier transform of F(ω) is given Dγ−1 f (t; α) = g(t; α) by 1 Z ∞ By [9] we get F −1 (F(ω)) = √ f (x)eiωwdx = f (x)  2π −∞ 0 γ−1  f (t; α) = f (0; α) + t f (0; α) + I g(t; α) By the conditions (II.1) we have 0 γ−1  f (t; α) = f (0; α) + t f (0; α) + I g(t; α) 1 Remark V.5 Let f ∈ AC . in the same if f is [(ii)-differentiable] then If f (x, t; α) = [ f (x, t; α), f (x, t; α)], then we can denote  0 γ−1 h i  f (t; α) = f (0; α) + t f (0; α) + I g(t; α) F ( f (x, t; α)) = F f (x, t; α) , F f (x, t; α)  f (t; α) = f (0; α) + t f 0(0; α) + Iγ−1g(t; α) with Thus [z1, z2] = [Re(z1), Re(z2)] × [Im(z1), Im(z2)] 0 γ−1 f (t) = f (0) + t fgH(0) + I g(t) and V. Fuzzy Fourier transform h i F −1 ( f (x, t; α)) = F −1 f (x, t; α) , F −1 f (x, t; α) In this section we discuss the Fourier transform in the fuzzy case Using the conditions (II.1) and the linearity of

1 Fourier transform on a "crisp" function we get for Lemma V.1 If f ∈ AE then the map all a, b > 0 F : R 7−→ C1 aF ( f (x, t; α)) + bF (g(x, t; α)) = F (a f (x, t; α) + bg(x, t; α)) R ∞ −iωw ω → −∞ f (x)e dx

1 Theorem V.6 Let f ∈ AE such that lim f (x) = 0. It follows from the corollary (V.8) that |x|→∞ 1 2 0 E 2 ∂ 2 2 suppose that fgH ∈ A . Then F c u(t, x) = −c ω U(ω, t) ∂x2 F f 0 (x) = iωF ( f (x)) γ γ gH F gH Dt u(t, x) = Dt U(ω, t) Proof Using theorem (II.22) we get We get γ 2 0 gH Dt U(ω, t) = −c U(ω, t) F fgH(x) = 1 h iωx ∞ R ∞ iωx i √ [ f (x)e ] −g (−iω) f (x)e dx It follows that 2π −∞ −∞ γ−1 0 2 Using the limite lim f (x) = 0 we get the result. gH Dt UgH(ω, t) = −c U(ω, t) |x|→∞ Thus we have the following problem (k) ∈ E1 (k)(x) = Corollary V.8 If fgH A and lim f 0 − |x|→∞ γ 1 0 2 gH D UgH(ω, t) = −c U(ω, t) (VI.2) for k = 0, 1, 2, then t ( ) = F ( ( )) U ω, 0 a x (VI.3) F f 00 (x) = −ω2F ( f (x)) ∂ gH U(ω, 0) = F (b(x)) (VI.4) ∂t By the theorems (II.20) and (IV.5) we have by lemma 3.2 [5] this problem has a unique solution given Theorem V.9 by

γ γ ∂ F D f (x, t) = D F ( f (x, t)) U(ω, t) =U(ω, 0) + t U(ω, 0)− gH t gH t ∂t g c2 Z t Z s VI. The solution of the fuzzy fractional wave (s − τ)γ−2U(ω, τ)dτds ( − ) equation Γ γ 1 0 0 if u0 is [(i)-differentiable], and In this section consider the following problem ∂  2 U(ω, t) =U(ω, 0) + t U(ω, 0)+  Dγu(t, x) − c2 ∂ u(t, x) = 0 ∂t  gH t g ∂x2  c2 Z t Z s 0 < x, t < 1, 0 < γ < 1 (s − τ)γ−2U(ω, τ)dτds (VI.1) Γ(γ − 1) 0 0 u(0, x) = a(x), 0  if u is [(ii)-differentiable].  ∂  ∂t u(0, x) = b(x) Which implies the existence and uniqueness of the solution 1 of the problem (VI.2) and by the inverse of Fourier where a and b are belongs to AE , transform we get the existence and uniqueness of the Proposition VI.1 the problem (VI.1) has a unique solu- solution of (VI.1). tion. VII. Case γ = 2 Proof Let u(x, t) is fuzzy absolutely integrable, we deﬁne In this section we set the fuzzy Fourier transform of u(x, t) and its inverse by u(x, t; α) = [u(x, t; α), u(x, t; α)] 1 Z ∞ F (u(x, t)) = √ u(x, t)e−iωtdx = U(ω, t) 2π −∞ a(x; α) = [a(x; α), u(x; α)] 1 Z ∞ F −1 (U(ω, t)) = √ U(ω, t)eiωtdω = u(x, t) b(x; α) = [b(x; α), b(x; α)] 2π −∞ 0 γ If u is [(i)-differentiable] then If Dt u(x, t), ∂xgH u(x, t) and ∂xxgH u(x, t) are fuzzy gH ∂2 ∂2 absolutely integrable in (−∞, ∞) by using u(x, t; α) = c2 u(x, t; α) ∂t2 ∂x2 ∂2 2 2 γ 2 ∂ 2 ∂ F gH Dt u(t, x) −g F c u(t, x) = 0 u(x, t; α) = c u(x, t; α) ∂x2 ∂t2 ∂x2

IJOA ©2021 48 International Journal on Optimization and Applications IJOA. Vol. 1, Issue No. 2, Year 2021, www.usms.ac.ma/ijoa Copyright ©2021 by International Journal on Optimization and Applications which implies

u(x, t; α) = F(x − ct; α) + G(x + ct; α) u(x, t; α) = F(x − ct; α) + G(x + ct; α) where

a(x; α) = F(x − ct; α) + G(x + ct; α) (VII.1)

a(x, t; α) = F(x − ct; α) + G(x + ct; α) (VII.2) and

b(x; α) = F0(x − ct; α) + G0(x + ct; α) (VII.3)

b(x, t; α) = F0(x − ct; α) + G0(x + ct; α) (VII.4)

By the conditions (II.1) the solution is given by

u(x, t) = F(x − ct) + G(x + ct) where F and G are given by the above formula (7.1) − (7.4). Fig. 1. Lower and upper branch of u(x, t)with α = 1

VIII. Examples the solution is given by In this section we will give some examples to α u(x, t) = [α, 1 − ]e−x cosh(ct) illustrate the previous results. 2 Example VIII.1

 3 2  D 2 u(t, x) − c2 ∂ u(t, x) = 0  gH t g ∂x2  0 < x, t < 1, 0 < γ < 1 IX. Conclusions (VIII.1) −x2 −x2 u(0, x; α) = [(1 + α)e , (3 − α)e ], This study makes it possible to explain the wave   ∂ phenomena with uncertainty in experimental data.  ∂t u(0, x) = 0 the solution is given by u(x, t) = F −1 (U(ω, t)) with References

h 2 2 i U(ω, t) = α√+1 e−ω , −√α+3 e−ω + [1] R.P. Agarwal, S. Arshad, D. O’Regan, V. Lupulescu, Fuzzy 2 2 fractional integral equations under compactness type condi- Z t Z s c2 − 1 (s − τ) 4 U(ω, τ)dτds tion, Fract. Calc. Appl. Anal. 15 (2012) 572-590. ( 1 ) Γ 2 0 0 [2] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept Example VIII.2 of solution for fractional differential equations with uncertainty, Nonlinear Anal. 72 (2010) 2859-2862.  2  2 ( ) − 2 ∂ ( ) = [3] T. Allahviranloo, Z. Gouyandeh, A. Armand, A. Hasanoglu,  gH Dt u t, x g c x2 u t, x 0  ∂ On fuzzy solutions for heat equation based on generalized  0 < x, t < 1, 0 < γ < 1 hukuhara differentiability, Fuzzy Sets Syst. 265 (2015) 1-23. 2 2 (VIII.2) u(x, 0; α) = [αe−x , (2 − α)e−x ], [4] G.A. Anastassiou, Fuzzy Mathematics: Approximation The-   ory. Studies in Fuzziness and Soft Computing, Springer,  ∂ Berlin Heidelberg, 2010.  ∂t u(x, 0) = 0

Fig. 2. Lower and upper branch of u(x, t)with α = 0.5

[5] A. Armand, Z. Gouyandeh, Fuzzy fractional integrodifferential equations under generalized Caputo differentiability. Annals of Fuzzy Mathematics and Informatics Volume 10, No. 5, (November 2015), pp. 789798. [6] B. Bede, L. Stefanini, Generalized differentiability of fuzzy- valued functions, Fuzzy Sets Syst. 230(0) (2013) 119-141. [7] Brezis.H. (2010) Functional Analysis, Sobolev Spaces and Partial Differential Equations,Springer. [8] W. Congxin, M. Ming, Embedding problem of fuzzy number space: Part III, Fuzzy Sets Syst. 46(2) (1992) 281-286. [9] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, New York, 2006 [10] J. Buckley, Fuzzy complex numbers, Fuzzy Sets Syst. 33(3) (1989) 333-345. [11] L. Stefanini, B. Bede, Generalized hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71(34) (2009) 1311-1328. [12] H.C. Wu, The improper fuzzy Riemann integral and its numerical integration, Inf. Sci. 111(14) (1998) 109-137. [13] Zadeh L. A., Fuzzy sets, Information and Control, 8, (1965), 338-353

Said Melliani ∗, Elhoussain Arhrrabi ∗, M’hamed Elomari ∗ & Lalla Saadia Chadli ∗ ∗ Laboratory of Applied Mathematics and Scientiﬁc Computing, Sultan Moulay Slimane University, Beni Mellal, Morocco Email: {said.melliani, arhrrabi.elhoussain, mhamedmaster}@gmail.com , [email protected]

Abstract—In this paper, we establish the Ulam-Hyers stability II.PRELIMINARIES and Ulam-Hyers-Rassias stability for fuzzy integrodifferential equations under Caputo gH-differentiability by using the fixed In this section, we introduce some definitions, theorems point method. and lemmas which are used in this paper. For more details, Index Terms—Fuzzy Ulam-Hyers-Rassias stability, Caputo we can see papers [3] [9] [12]. fractional derivatives, fuzzy fractional integrodifferential equations, fixed point theory. Definition 2.1: A function d : X × X −→ [0, +∞) is called a generalized metric on X if and only if d satisfies: I.INTRODUCTION (1) d(x, y) = 0 if and only if x = y, In this paper, we will propose fuzzy Ulam-Hyers-Rassias stability for the two kinds of fuzzy fractional integrodifferen- (2) d(x, y) = d(y, x) for all x, y ∈ X, tial equations of order α ∈ (0, 1) with generalized Hukuhara derivative under form (3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. C α R t Theorem 2.2: (Banach) Let d : X × X −→ [0, +∞) gH Da+ u(t) = f(t, u(t)) + a g(t, s, u(s))ds, t ∈ [0, a], d be a generalized metric on X and (X, d) is a generalized u(0) = u0 ∈ E . (1) complete metric space. Assume that T : X −→ X is a C α strictly contractive operator with the Lipschitz constant Where gH Da+ is the Caputo’s generalized Hukuhara deriva- f : [0, a] × Ed −→ Ed [0, a] L < 1. If there exists a nonnegative integer n such that tive, , is continuous on and n+1 n g : [0, a] × [0, a] × Ed −→ E is continuous on [0, a] × [0, a]. d(T x, T x) < ∞ for some x ∈ X, then the following are We wish to mention that the theory of fuzzy fractional integral true: and differential equations have recently been the subject of n ∗ important studies (see e.g [1]–[11] ). In [12], Shen et al (i) the sequence T x converges to a fixed point x of T , studied the Ulam stability problems of the first order linear ∗ fuzzy differential equations under some suitable conditions, (ii) x is the unique fixed point of T in ∗ n and in [13], Diaz et al has introduced a fixed point theorem of X = {y ∈ X | d(T x, y) < ∞}, the alternative for contractions on a generalized metric space, ∗ ∗ 1 with which Shen et al in [14] proved the Ulam stability of (iii) if y ∈ X , then we have d(y, x ) ≤ 1−L d(T y, y). fuzzy differential equations. Since the number of documents Lemma 2.3: Let ϕ : J −→ [0, +∞) be a continuous dealing with the stability of Ulam for fuzzy fractional inte- function. We define the set grodifferential equations (FFIEs) is rather limited compared := {x : J −→ | x is continuous function on J}, to the number of publications concerning FFIEs, we decide X RF to study by using the fixed point technique, the Ulam-Hyers- where RF is the space of fuzzy sets, equipped with the metric Rassias stability for FFIEs. d(x, y) = inf{η ∈ [0, +∞) ∪ {+∞} | D(x(t), y(t)) ≤ Our results are inspired by the one in [15] where the fuzzy ηϕ(t), ∀t ∈ J}. Ulam-Hyers-Rassias stability of FFIEs is studied. The rest of Then, (X, d) is a complete generalized metric space. this paper is organized as follows: In section 2, we recall some d Let Kc(R ) denote the family of all nonempty, compact and notations of the fuzzy number space, the fixed point theorem d convex subsets of R . The addition and scalar multilplication and the basic notations of the Riemann-Liouville fractional d d in Kc(R ) are defined as usual i.e, for A, B ∈ Kc(R ) and integral and Caputo fractional derivative for fuzzy functions. λR, The Ulam-Hyers-Rassias stability for fuzzy fractional integrodifferential equations are discussed in Sections 3. A + B = {a + b | a ∈ A, b ∈ B}, λA = {λa | a ∈ A}

Let Ed denote the set of fuzzy subsets of the real axis, if Let t ∈ (a, b) and h such that t + h ∈ (a, b), then the ω : Rd −→ [0, 1], satisfying the following properties: generalized Hukuhara derivative of fuzzy-valued function x : d d (i) ω is normal, that is, there exists z0 ∈ R such that (a, b) −→ E at t is defined as ω(z0) = 1, x(t + h) gH x(t) DgH x(t) = lim . (ii) ω is fuzzy convex, that is, for 0 ≤ λ ≤ 1 h−→0 h d d If DgH x(t) ∈ E satisfying last inequality, we say that x ω(λz1+(1−λ)z2) ≥ min{ω(z1), ω(z2)}, for any z1, z2 ∈ R , is generalized Hukuhara differentiable (gH-differentiable for (iii) ω is upper semicontinous on Rd, short) at t. Definition 2.6: Let x :[a, b] −→ Ed, the fuzzy Rieman- (iv)[ω]0 = cl{z ∈ Rd : ω(z) > 0} is compact, where cl Liouville integral of fuzzy-valued function x is defined as denotes the closure in (Rd, | . |). follows: d Then E is called the space of fuzzy number. For Z t r d α 1 α−1 r ∈ (0, 1], we denote [ω] = {z ∈ R | ω(z) ≥ r} (J + x)(t) = (t − s) x(s)ds. a Γ(α) and [ω]0 = {z ∈ Rd | ω(z) > 0}. From the conditions a r (i) to (iv), it follows that the r − level set of ω, [ω] , is 1 For a ≤ t, and 0 < α ≤ 1. For =α = 1, we set Ja = I, the a nonempty compact interval, for all r ∈ [0, 1] and any ω ∈ E. identity operator. Definition 2.7: Let D ∈ C([a, b],Ed) ∩ L([a, b],Ed). r gH The notation [ω] = [ω(r), ω(r)], denotes explicitly the The fuzzy gH-fractional Caputo diffentiability of fuzzy-valued r − level set of ω, for r ∈ [0, 1]. We refer to ω and ω C function x ([gH]a − differentiable for short) is defined as the lower and upper branches of ω, respectively. For as following: d ω ∈ E , we define the lengh of the r − level set of ω as C α 1−α 1 R t D + x(t) = J + (DgH x)(t) = (t − len([ω]r) = ω(r)−ω(r) gH a a Γ(1−α) a . For addition and scalar multiplication s)−α(D x)(s)ds, in fuzzy set space Ed, we have [ω + ω ]r = [ω ]r + [ω ]r, gH 1 2 1 2 where 0 < α ≤ 1, t > a. [λω]r = λ[ω]r. Lemma 2.8: Suppose that x :[a, b] −→ Ed be a fuzzy function and D x(t) ∈ C([a, b],Ed) ∩ L([a, b],Ed). Then The Hausdorff distance between fuzzy numbers is given by gH α C α Ja+ (gH Da+ x)(t) = x(t) gH x(a). D0[ω1, ω2] = sup {| ω1(r) − ω2(r) |, | ω1(r) − ω2(r) |}. 0≤r≤1 Lemma 2.9: Let u : [0, a] −→ Ed be a continuous function d The metric space (E ,D0) is complet metric space and the on [0, a] and let α ∈ (0, 1), then the FFIE (1) is equivalent to following properties of the metric D0 are valid. the following integral equation:

D0[ω1 + ω3, ω2 + ω3] = D0[ω1, ω2], (1) If u is ω-increasing on [0, a], then

D0[λω1, λω2] =| λ | D0[ω1, ω2], 1 Z t D0[ω1, ω2] ≤ D0[ω1, ω3] + D0[ω3, ω2], u(t) = ϕ(0) + (t − s)α−1(f(s, u(s)) Γ(α) a d d d for all ω1, ω2, ω3 ∈ E and λ ∈ R . Let ω1, ω2 ∈ E , if there Z s d exists ω3 ∈ E such that ω1 = ω2 + ω3 then ω3 is called the + g(s, r, u(r))dr)ds, (2) a H-difference of ω1, ω2. We denote the ω3 by ω1 ω2. Let us remark that ω1 ω2 6= ω1 + (−1)ω2. (2) If u is ω-decreasing on [0, a], then Definition 2.4: The generalized Hukuhara difference of two fuzzy numbers ω , ω ∈ Ed (gH-difference for short) is 1 2 (−1) Z t defined as follows: u(t) = ϕ(0) (t − s)α−1(f(s, u(s)) Γ(α) a (i) ω1 = ω2 + ω3, Z s ω1 gH ω2 = ω3 ⇔ or (ii) ω2 = ω1 + (−1)ω3. + g(s, r, u(r))dr)ds. (3) a Let [0, a] be a compact interval in +. Denote by diam[u(t)]r R III.MAIN RESULTS the diameter of fuzzy set u, for t ∈ [0, a]. A function u : [0, a] −→ Ed is called ω-increasing (ω-decreasing) on In the sequel, our aim score is to present the results for the [0, a] if for every r ∈ [0, 1] the function t 7−→ diam[u(t)]r is existence and the stability of the problem (1). The methods nondecreasing (nonincreasing) on [0, a]. If u is ω-increasing to solve these problems are quite similar. However, since the or ω-decreasing on [0, a], then we say that u is ω-monotone conditions for the existence of solutions of fuzzy fractional on [0, a]. integrodifferential equations (2) and (3) are dissimilar, we shall Definition 2.5: present the two kinds (2) and (3) in two separate subsections.

A. Fuzzy Ulam-Hyers-Rassias stability for FFIEs (2) for any t ∈ [0, a], then there exists a unique u˜ : [0, a] −→ Ed Firstly, we present the definitions of fuzzy Ulam-Hyers of (2.2) such that stability and fuzzy Ulam-Hyers-Rassias stability. Definition 3.1: We say that the problem (2) is fuzzy Ulam- Z t Z s 1 α−1 Hyers stable, if there exists a constant Kf > 0 such that for u˜(t) = u0+ (t−s) (f(s, u˜(s))+ g(s, r, u˜(r))dr)ds, each ε > 0 and for each solution v ∈ C1([0, a],Ed) of the Γ(α) a a (8) following inequality and hC α R t i D gH Da+ v(t), f(t, v(t)) + a g(t, s, v(s))ds ≤ ε, ∀t ∈ [0, a], 1 then, there exists a solution u ∈ C1([0, a],Ed) of problem (2) d[˜u(t), u(t)] ≤ , ∀t ∈ [0, a]. (9) with 1 − LfgK(1 + C) D [v(t), u(t)] ≤ Kf ε, for all t ∈ [0, a]. We call Kf a Ulam-Hyers stability constant Proof: of (2). Definition 3.2: We say that the problem (2) is fuzzy Ulam- Let us consider the space of all continuous fuzzy function Hyers-Rassias stable, if there exists a constant C > 0 such f u : [0, a] −→ Ed by that for each ε > 0 and for each solution v ∈ C1([0, a],Ed) of the following inequality h i d C α R t X = {u : [0, a] −→ E | u is continuous on [0, a]}, D gH Da+ v(t), f(t, v(t)) + a g(t, s, v(s))ds ≤ ϕ(t), ∀t ∈ [0, a], then, there exists a solution u ∈ C1([0, a],Ed) of problem (2) equipped by the metric with D [v(t), u(t)] ≤ Cf ϕ(t), d(u, v) = inf{C ∈ [0, +∞) ∪ {+∞} | D[u(t), v(t)] ≤ Cϕ(t)}, ∀t ∈ [0, a]. for all t ∈ [0, a] and for some nonnegative function ϕ defined on [0, a]. Remark 3.3: We observe that definition 3.2 ⇒ definition By lemma 2.3, we observe that (X, d) is also a complete 3.1. generalized metric space. We define an operator Q : X −→ X In the following, we shall prove that the FFIEs (2) is fuzzy by Ulam-Hyers-Rassias stable on bounded interval by the fixed point theorem. Theorem 3.4: Assume that f : [0, a] × Ed −→ Ed and Z t g : [0, a] × [0, a] × Ed −→ Ed are continuous functions 1 α−1 (Qu)(t) = u0 + (t − s) (f(s, u(s)) satisfying the following conditions: Γ(α) a Z s + g(s, r, u(r))dr)ds, ∀t ∈ [0, a]. (10) (i) There exists a constant Lfg > 0 such that: a max {D[f(t, u), f(t, v)]; D[g(t, s, u), g(t, s, v)]} ≤ LfgD[u, v], (4) Because f and g are a continuous fuzzy functions, the right d for all each (t, s, u), (t, s, v) ∈ [0, a] × [0, a] × E . hand side of (10) is also continuous on [0, a]. This yields that Qu is continuous on [0, a]. So, the operator Q is well-defined. (ii) There exists a constant K, C > 0 such that 0 < To apply theorem 2.2 in the proof of this theorem, we need LfgK(1+C) < 1 and let ϕ : [0, a] −→ [0, ∞) be a continuous the operator Q to be strict contractive on X. For any u, v ∈ X function and increasing on [0, a] with: and let Cuv ∈ [0, +∞) ∪ {+∞} such that Z t ϕ(s)ds ≤ C.ϕ(t), ∀t ∈ [0, a], (5) a d(u, v) ≤ Cuv, ∀t ∈ [0, a]. and Z t 1 α−1 (t − s) ϕ(s)ds ≤ Kϕ(t), ∀t ∈ [0, a], (6) Then, by the definition of d, we have Γ(α) a If a continuously ω-increasing function u : [0, a] −→ Ed satisfies the following inequality D[u(t), v(t)] ≤ Cuvϕ(t), ∀t ∈ [0, a]. (11) Z t C α D gH Da+ u(t), f(t, u(t)) + g(t, s, u(s))ds ≤ ϕ(t), a (7) From the definition of the operator Q and assumption (4)-

(6), we have the following estimation Finally, by theorem 2.2 − (iii) and from the estimation (13), it implies that 1 Z t D[(Qu)(t), (Qv)(t)] = D[u + (t − s)α−1(f(s, u(s)) 0 Γ(α) d(u, Qu) 1 a d(˜u(t), u(t)) ≤ ≤ , Z s Z t 1 α−1 1 − LfgK(1 + C) 1 − LfgK(1 + C) + g(s, r, u(r))dr)ds, u0 + (t − s) (f(s, v(s)) a Γ(α) a Z s which means the estimation (9) holds true for any t ∈ [0, a]. + g(s, r, v(r))dr)ds], This completes the proof. a 1 Z t ≤ (t − s)α−1D[f(s, u(s)), f(s, v(s))]ds B. Fuzzy Ulam-Hyers-Rassias stability for FFIEs (3) Γ(α) a 1 Z t Z s Theorem 3.5: Suppose that the functions f, g and ϕ satisfy + (t − s)α−1( D[g(s, r, u(r)), g(s, r, v(r))]dr)ds, all conditions as in theorem 3.4. Assume that for each t ∈ Γ(α) a a [0, a] and for each continuous fuzzy function z : [0, a] −→ Ed, L Z t ≤ fg (t − s)α−1D[u(s), v(s)]ds if the Hukuhara difference (−1) t α−1 s Γ(α) a z(0) R (t − s) f(s, z(s)) + R g(s, r, z(r))dr ds, Z t Z s Γ(α) a a Lfg α−1 exists and a continuously ω-nonincreasing function v : + (t − s) ( D[u(r), v(r)]dr)ds, d Γ(α) a a [0, a] −→ E satisﬁes L C Z t ≤ fg uv (t − s)α−1ϕ(s)ds (−1) Z t Γ(α) D[v(t), v (t − s)α−1(f(s, v(s)) a 0 Γ(α) L C Z t Z s a + fg uv (t − s)α−1( ϕ(r)dr)ds, Z s Γ(α) a a + g(s, r, v(r))dr)ds] ≤ ϕ(t), (14) a ≤ LfgCuvKϕ(t) + LfgCuvCKϕ(t)

= LfgK(1 + C)Cuvϕ(t). for any t ∈ [0, a], where v0 = u0, then there exists a unique solution uˆ : [0, a] −→ Ed of the problem (3) which satisfies Hence Z t (−1) α−1 D[(Qu)(t), (Qv)(t)] ≤ LfgK(1 + C)Cuvϕ(t). (12) uˆ(t) = u0 (t − s) (f(s, uˆ(s)) Γ(α) a Z s So, by the definition of metric d, we get + g(s, r, uˆ(r))dr)ds, (15) d a d(Qu, Qv) ≤ LfgK(1 + C)d(u, v), for all u, v ∈ E . and Where 0 < LfgK(1 + C) < 1, hence the operator Q is 1 d[û(t), v(t)] ≤ , (16) strictly contractive mapping on X. 1 − LfgK(1 + C) For an arbitrary ω ∈ X and from the definition of X and Q, it follows that there exists a constant 0 < Cω < ∞ such that: for any t ∈ [0, a]. Proof: 1 R t α−1 D[(Qω)(t), ω(t)] = D[u0 + Γ(α) a (t − s) (f(s, ω(s)) We consider the complete generalized space (X, d) defined as s in the proof of theorem 2. Define the operator P : X −→ X + R g(s, r, ω(r))dr)ds, ω(t)] ≤ C ϕ(t), a ω as follows: for any t ∈ [0, a], since f, g and ω are bounded on [0, a], and Z t (−1) α−1 the minimum of ϕ(t) > 0 on t ∈ [0, a]. Then, we infer that (P u)(t) = u0 (t − s) (f(s, u(s)) d(Qω, ω) ≤ C < ∞. Therefore, according to (i) and (ii) of Γ(α) a ω Z s theorem 2.2, there exists a continuously function u˜ : [0, a] −→ + g(s, r, u(r))dr)ds, t ∈ [0, a]. (17) d n E such that Q ω −→ u˜ in the space (X, d) as n −→ ∞ and a Qu˜ =u ˜, that u˜ satisfies the problem (8) for any t ∈ [0, a]. ∗ Since the function f and g is continuous on [0, a] Now, we shall confirm that {u ∈ X | d(ω, u) < ∞} = X . For (−1) t and the Hukuhara difference u R (t − an arbitrary u ∈ Ed, since u and ω are bounded on [0, a] and 0 Γ(α) a s)α−1 f(s, u(s)) + R s g(s, r, u(r))dr ds exists, similary to mint∈[0,a] ϕ(t) > 0, there exists a constant 0 < Cu < ∞ such a 1 P u [0, a] P u that D[ω(t), u(t)] ≤ Cuϕ(t) for any t ∈ [0, a]. Therefore, we theorem , it follows that is well-defined on or have d(ω, u) < ∞ for any u ∈ Ed, that is {u ∈ X | d(ω, u) < is continuous on [0, a]. Now, we observe that the operator ∞} = X∗. By theorem 2.2-(ii), we conclude that u˜ is the P is strictly contractive on X. Indeed, for any u, v ∈ X and unique fixed point of Q on X. let Cuv ∈ [0, +∞) ∪ {+∞} be an arbitrary constant with On the other hand, from the inequality (7) it follows that d(u, v) ≤ Cuv for t ∈ [0, a], that is, let us assume that

d(u, Qu) ≤ 1. (13) D[u(t), v(t)] ≤ Cuvϕ(t), (18)

IJOA ©2021 54 International Journal on Optimization and Applications IJOA. Vol. 1, Issue No. 2, Year 2021, www.usms.ac.ma/ijoa Copyright ©2021 by International Journal on Optimization and Applications for t ∈ [0, a]. Furthermore, from (17), (18) and by the Lips- [2] T. V. An, H. Vu, N. V. Hoa, Applications of contractive-like mapping chitz condition of f and g, we have the following estimation: principles to interval-valued fractional integro-differential equations, Journal of Fixed Point Theory and Applications, 19(4) (2017), 2577- Z t 2599. (−1) α−1 D[(P u)(t), (P v)(t)] = D[u0 (t − s) (f(s, u(s)) [3] T. V. An, H. Vu, N. V. Hoa, Hadamard-type fractional calculus for fuzzy Γ(α) a functions and existence theory for fuzzy fractional functional integro- Z s Z t differential equations, Journal of Intelligent and Fuzzy Systems, 36(4) (−1) α−1 + g(s, r, u(r))dr)ds, u0 (t − s) (f(s, v(s)) (2019), 3591-3605. a Γ(α) a [4] N. V. Hoa, Fuzzy fractional functional differential equations under Z s Caputo gH-differentiability, Communications in Nonlinear Science and + g(s, r, v(r))dr)ds], Numerical Simulation, 22(1) (2015), 1134-1157. a [5] N. V. Hoa, Fuzzy fractional functional integral and differential equations, 1 Z t Fuzzy Sets and Systems, 280 (2015), 58-90. ≤ (t − s)α−1D[f(s, u(s)), f(s, v(s))]ds [6] N. V. Hoa, V. Lupulescu, D. Oregan, Solving interval-valued frac- Γ(α) a tional initial value problems under Caputo gH-fractional differentiability, 1 Z t Z s Fuzzy Sets and Systems, 309 (2017), 1-34. + (t − s)α−1( D[g(s, r, u(r)), g(s, r, v(r))]dr)ds, [7] N. V. Hoa, H. Vu, T. Minh Duc, Fuzzy fractional differential equations Γ(α) a a under Caputo-Katugampola fractional derivative approach, Fuzzy Sets L Z t and Systems, 375(15) (2019), 70-99. ≤ fg (t − s)α−1D[u(s), v(s)]ds [8] H. V. Long, N. T. K. Son, N. V. Hoa, Fuzzy fractional partial differential Γ(α) a equationd in partially ordered metric spaces, Iranian Journal of Fuzzy Systems, 14(2) (2017), 107-126. L Z t Z s + fg (t − s)α−1( D[u(r), v(r)]dr)ds, [9] M. T. Malinowski, Random fuzzy fractional integral equations- theoretical foundations, Fuzzy Sets and Systems, 265 (2015), 39-62. Γ(α) a a [10] N. Ngoc Phung, B. Quoc Ta, H. Vu, Hyers-Ulam stability and Hyers- ≤ LfgCuvKϕ(t) + LfgCuvCKϕ(t) Ulam-Rassias stability for fuzzy integrodifferential equation, Complex- = L K(1 + C)C ϕ(t). ity, 2019 (2019), fg uv [11] H. Vu, V. Lupulescu, N. Van Hoa, Existence of extremal solutions to interval-valued delay fractional differential equations via monotone Hence iterative technique, Journal of Intelligent and Fuzzy Systems, 34(4) (2018), 2177-2195. D[(P u)(t), (P v)(t)] ≤ LfgK(1 + C)Cuvϕ(t). (19) [12] Y. Shen, On the Ulam stability of rst order linear fuzzy differential equations under generalized differen- This means that d(P u, P v) ≤ LfgK(1 + C)d(u, v). Hence, tiability, Fuzzy Sets and Systems, 280(C) (2015), 27-57. the operator P is a strictly contractive mapping on X by the [13] J. B. Diaz, B. Margolis, A assumption 0 < L K(1 + C) < 1. Simalar to the theorem fixed point theorem of the alternative, for contractions on a generalized fg complete metric space, Bulletin of the American Mathematical Society, 3.4, we can show that for each ω ∈ X satisfies d(P ω, ω) < ∞. 74(2) (1968), 305-309. Hence, by theorem 1, it implies that there exists a continuously [14] Y. Shen, On the Ulam stability of first order linear fuzzy differential function uˆ : [0, a] −→ Ed such that P nω −→ uˆ in ( , d) as equations under generalized differentiability, Fuzzy Sets and Systems, X 280(C) (2015), 27-57. n −→ ∞, and such that P uˆ =u ˆ, that is uˆ satisfies (4.15) for [15] H. Vu, J. M. Rassias and N. Van Hoa, Ulam-Hyers-Rassias stability for t ∈ [0, a]. Similar to the proof of theorem 3.4, we observe that fuzzy fractional integral equations, Iranian Journal of Fuzzy Systems Volume 17, Number 2, (2020), pp. 17-27. there exists a constant Cω > 0 such that D[ω(t), u(t)] ≤ Cω, for any t ∈ [0, a]. This means that d(ω, u) < ∞ for each u ∈ Ed, or equivalently, {u ∈ X | d(ω, u) < ∞} = X∗. Furthermore, by theorem 2.2, we imply that uˆ is a unique continuous function which satisfies (15). Moreover, by theorem 2.2, we also obtain d(u, P u) 1 d(û(t), u(t)) ≤ ≤ , 1 − LfgK(1 + C) 1 − LfgK(1 + C) which means the estimation (16) holds true for any t ∈ [0, a]. This completes the proof.

IV. CONCLUSION In this study, we are studied the Ulam-Hyers-Rassias stability for fuzzy intergodifferential equation via the ﬁxed point technique. This result can be used to study fractional fuzzy differential equations with other types of derivative concepts in fuzzy setting, for example, Riemann-Liouville and Hadamard generalized Hukuhara differentiability.

REFERENCES [1] T. V. An, H. Vu, N. V. Hoa, A new technique to solve the initial value problems for fractional fuzzy delay differential equations, Advances in Difference Equations, 181 (2017),

Mohamed Amine ERROCHDI Mohammed EL HADDAD PhD student at the Laboratory of Studies Head of the Management Sciences Laboratory, and Research in Management Sciences, Faculty of Legal, Economic and Social Faculty of Legal, Economic and Social Sciences- Agdal Sciences- Agdal Mohammed V University – Rabat, Morocco Mohammed V University – Rabat, Morocco [email protected]

Abstract— Taxation is one of the instruments for changing financing of measures to prevent, reduce and fight against the behavior of agents, either by encouraging or discouraging pollution, via their tax contribution. certain behavior’s considered good or bad, for instance certain investments with negative or positive impacts that arouse In this contribution, we will analyze all the levies aimed government’s concerns. at combating existing climate change in Morocco and in certain European countries, our problematic will be as Eco-taxation seeks to change the behavior of agents follows: to what extent does eco-taxation effectively by using the instrument of taxation to discourage, especially, contribute to change the behavior of economic players? negligent behavior leading to more pollution and climate change. Eco-taxation also aims to implement the polluter-pays We will try to answer this problem through three axes: principle, that is, to make the polluter bear the cost of his Axis I: Theoretical overview on Eco-taxation; environmental damage, as well as allowing the State to Axis II: Eco-Taxation in Europe; generate tax revenues that can be directed towards financing Axis III: Eco-Taxation in Morocco. appropriate policies for a transition towards a more environmentally friendly economy. Eco-taxation can thus be Axis I: Theoretical overview on Eco-taxation. used to limit the production and consumption of goods and services that are harmful to the environment. Eco-taxation aims to solve the following problems:  Fight against global warming; This research will compare all the tax levies existing in Europe and in Africa (the case of Morocco), as these  Reduction of pollution; instruments are designed to meet the challenges of climate  Rational use of resources; change, and to show how coherent eco-taxation contribute  Preservation of natural environments of effectively to changing the behavior of all economic actors. biodiversity.

Keywords— Eco-taxation, environmental taxation, ecological Indeed, Eco-taxation, is a mode of production, a product, or a taxation, climate change. service damaging the environment (“evils”), makes it I. INTRODUCTION possible to limit the attacks only in the interest of allowing the public authorities to finance the damages of the public Earth's climate is regulated by the ability of the expenses. atmosphere to partially retain the energy reflected from the The main difference between taxation and Eco- earth. This physical phenomenon is called the greenhouse taxation stems is its objective, where taxation is defined as effect, because it is similar to that encountered in a glass greenhouse. It is a natural phenomenon essential to the the means for the State to collect the resources which will development of life on earth; in its absence, there would be enable it to finance these expenses, or more generally public no liquid water, because the average temperature on earth goods (education , defense, health, etc.) or to ensure a certain would reach degrees much lower than the current redistribution of income; while Eco-taxation aims to modify temperature. the behavior of economic agents in order to prevent them from making decisions that they might regret in the future, or These problems of economic changes, posed by the thus to limit consumption to protection of the Environment . environment were partly behind the creation of certain taxes Several terms refer to the concept of Eco-taxation, and levies, because companies when they buy, sell or fix the such as: Green Taxation, Environmental Taxation, price of products, do not directly integrate the cost of the Ecological Taxation, Energy Taxation, and Eco Tax... damages that 'they cause the environment, and the future scarcity of energies and raw materials. The notion of Eco-taxation is a challenge, which has given rise to numerous debates on the most relevant Eco-taxation aims to integrate the environmental cost it perimeter to be retained, which bring us to the definition of causes into the cost price; it is thus a means of changing the the OECD [1], “all taxes and fees whose base is constituted behavior of economic agents in a way that is favorable to the by a pollutant or, more generally, by a product or a service environment. which deteriorates the environment or which results in a levy The use of Eco-taxation is also justified by the “polluter on natural resources ”. pays” principle, that is to say, the polluter participates in the

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According to this definition, Eco-taxation goes This increase in prices encourages consumers and buyers to beyond the sole taxation explicitly designed to fight against change their decisions vis-à-vis this type of polluting goods pollution, but incorporates another main purpose is the and service, by choosing other non-polluting ones, this financing of public services; also it has the effect of helping decision change is desired precisely because it comes from to limit pollution. correct the behavior of agents against the environment The tax council in France in 2005[2]; announces resulting from climate change. that “The situation can be considered as paradoxical: the As a conclusion, environmental taxation aims to integrate most important environmental effects are due to taxes and additional costs in the form of environmental taxes (called fees for services rendered, created long before the emergence “externalities”) into the cost borne by each of the economic of public policies in favor of the environment; fiscal parties. in addition to the regulatory approach, and stimulates measures directly inspired by environmental concerns have innovation in the medium term respects the polluter pays only a limited effect, whether it concerns the various principle, defined by the Organization for Economic components of the tax on polluting activities (TGAP [3]) or Cooperation and Development (OECD) in 1972, which derogatory tax measures ” assumes that the costs resulting from pollution prevention, This announcement is considered as a broad reduction and control measures must be borne by the approach, eco-taxation is an important instrument to face the polluter. challenges of climate change, by using only energy taxation, Axis II: Eco-Taxation in Europe_ case of France_ the objective is to change the behavior of all agents for a Eco-taxation is the set of taxes, fees and charges that are transition to an economy low in carbon and more imposed on polluting taxpayers, more generally, through a environmentally friendly. product or service that damages the environment. It was This approach was confirmed by Thierry Wahl, introduced to limit the effect on the climate of the Inspector General of Finance and responsible for a report on consumption of polluting goods and services, and thus fight the subject, "the most successful expression of the polluter against global warming by promoting energy savings and pays principle". It is intended to encourage virtuous behavior less polluting energies. The “polluter pays” principle remains in environmental matters and to deter bad behavior. It can the basis of Eco-taxation. This principle consists of take several forms: tax, royalties, tax credit, exemption or pollutants contributing to limiting pollution and damage to even direct aid. [4] the environment. Environmental taxation also allows the Environmental taxation occupies an important place in the internalization of external costs, that is to say to pass on in tax policies of member countries of the European Union. the price of goods and services, certain environmental costs This part provides an overview of environmental taxation in which are currently unduly supported by future generations, Europe in general, based on numerous taxes that are part of the objective is to push the agents to make financially and the list of taxes included in ecological taxation. Recourse to environmentally sound decisions. European experiences, in particular the case of France, The externality or external effects appearing when constructs an evaluation of the efficiency and consistency of the decisions of an economic agent affect the well-being of ecological taxation. other agents, involuntarily, despite the absence of any market In recent years, environmental taxation has evolved to transaction between them. support the ecological transition. The government in France The external effects can be positive (beneficial has put in place an arsenal of regulations with stakeholders to influence) or negative (deterioration of the situation). support these developments. We will present the main taxes Pollution constitutes a negative externality therefore, to get made up of Eco taxation in France, and an analysis of the polluters to take into account the external cost of their performance of the French experience in ecological taxation. activity, the regulatory world proposed by Arthur C. Pigou II-1 The main taxes constituted Eco-taxation in France: [5] consists of the implementation of a tax called " Environmental taxation in France can be divided into four Pigovienne Tax ”, whose unit tax must be equal to the categories depending on the function of the tax [6]. marginal damage caused by polluting emissions at their 1- The taxes themselves, which are compulsory levies optimal level (that is to say the level which maximizes social without compensation and the basis of which is a polluting welfare, the marginal damage suffered by the victims is product. This is the case, for example, with the tax on equal to marginal cost of cleaning up the polluting sector) it polluting activities (TGAP), based on emissions to the air or provides the price signal which ensures the internalization of pesticides, and the TIPP. externalities. 2- Charges that cover costs for environmental services, At this level, all polluters want to minimize their mainly in the areas of water and waste. costs either by motivating innovation to seek less polluting 3- So-called positive measures such as tax credits which solutions to reduce their production costs or by offering less seek in particular to orient investment choices in a more polluting products by taking advantage of the opportunities favorable direction for the environment. provided by environmental regulations. 4-Tax incentives (exemptions, deductions, rate cuts) which In general, the increase in the price of the polluting good or also seek to orient behavior in favor of the environment. service due to the environmental tax results in an increase in Eco-taxation in France can be divided into four categories its price compared to other goods and services. depending on the type of tax; 4 following categories:

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• Energy taxes; Tax on registration Tax power of the vehicle • Taxes on transport; certificates (gray cards) • Pollution taxes; proportional contribution to Additional tax on motor • Resource taxes. insurance premiums on motor vehicle insurance the In this part we will present a list of all the environmental vehicles taxes in force in France, based on the Eurostat nomenclature. Number of vehicles detained For each tax, its base is specified in 2016. individuals or leased by, or held by employees of the Tax on company cars TABLE 1. The Main Environmental Taxes in France. company and the miles driven which are subject to Tax Base reimbursement fees Energy Tax due by motorway Number of kilometers Internal tax on the Petroleum products used as concessionaires traveled by users consumption of energy fuels or fuels Number of passengers and products - TICPE Civil aviation tax mass of freight and mail Carbon component Fossil energies whose loaded in France (integrated into ICT rates) combustion emits CO2 Tax on the purchase of the Payable on the most polluting Contribution to the public Prorate of the quantity of most polluting new private passenger cars, when they are electricity service (CSPE) electricity consumed vehicles (penalty of first registered in France Local taxes on electricity purchase) (Internal tax on final Quantity of electricity The number of passengers on Solidarity contribution on electricity consumption subscribed (TICFE if greater board, excluding passengers plane tickets TICFE + Tax on final than or equal to 250 kilo, in transit electricity consumption otherwise TCFE) Truck with an authorized TCFE)volt-amperes weight equal to or greater 9 components: Wind turbines than 12 tones, registered in and tidal turbines, nuclear or France or in a third country Axle tax thermal installation, (outside the EU) that has not photovoltaic or hydroelectric concluded a reciprocal installation, electrical Flat-rate tax on network exemption agreement with transformers, radio stations, companies (IFER) France gas installations, SNCF Hydraulic works and Hydraulic Tax railway equipment, RATP hydroelectric works railway equipment, and Turnover relating to certain telephone switching operations carried out for equipment Territorial solidarity passenger rail transport Internal tax on the contribution services and commercial consumption of natural gas Natural gas used as fuel services directly related to –maintenance. TICGN them Fuel tax in the overseas Petroleum products used as Gross revenue from the sale departments fuel Tax on ski lifts of transport tickets ski lifts in Tax for the professional Costs of constitution and mountain areas committee of strategic conservation during an Andes Tax on pleasure boats Ownership of a displaced petroleum stocks strategic stocks (annual francization and vessel Contribution of low- navigation rights) voltage electrical energy The number of passengers distributors low-voltage Receipts from boarding in the Corsica and Tax due by public air and electrical energy Guadeloupe regions , sea transport companies distributors Guyana, Martinique and Pylons supporting power lines Annual flat-rate tax on Reunion whose voltage is at least equal pylons Part of the infrastructure use to 200 kilovolts Fee owed by the railway charge paid to SNCF signal General tax on polluting Release for consumption of undertakings for the within the limit of 5 activities ( TGAP) fuels fuels regulatory authority for thousandths + € 0.10 / km TIC on coal, lignite and Quantity of energy delivered railway activities traveled on other lines of the coke expressed in kWh rail network Transport Tax on maritime transport Number of passengers on

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Copyright © 2021 by International Journal on Optimization and Applications at tination of protected board destined for protected TGAP waste: installations for natural areas natural areas (list fixed by the elimination of household decree) and similar waste Annual tax on the Vehicles emitting more than (incineration or storage) and possession of polluting 245gCO2 / km (240gCO2 / special industrial waste: private vehicles (annual km for vehicles registered variation in rates depending penalty) since 2012) on the environmental Vehicles Used vehicles performance of the facilities Purchase tax on the most emitting more than 200gCO2 (between € 4 / ton and € 150 / polluting second-hand / km or having a fiscal power ton) TGAP emission pollutants in the atmosphere greater than 10 horsepower General tax on polluting Additional tax on the issuance of certain substances: Tax intended to finance the activities (TGAP) of vehicle registration between € 5 / t (benzene) and development of vocational (excluding fuel TGAP) documents for goods € 1,000 / t (mercury) TGAP training actions in road transport and public installations classified for transport passenger transport environmental protection: Pollution / resources between € 501.61 and € charges The domestic 2,525.35 per year and per pollution charge is based on TGAP installation lubricants the quantity of drinking water oils and lubricating consumed: € 0.3 / m3 preparations: 44.02 € / t TGAP detergents: between The fee for the modernization 39.51 and 283.65 € / t TGAP of wastewater collection extraction materials: 0.20 € / t networks is based on the fees Annual volume of water quantities of water used and Water withdrawal sent to the collection withdrawal , expressed in m2 Quantities of products networks: 0.5 € / m3 Municipal and extracted from mines, mines The diffuse pollution charge departmental mining concerns phytosanitary or quarries products The fee for the protection of (phytopharmaceuticals) and aquatic environments is based takes into account the on fishing cards: € 10 per toxicity, the dangerousness adult who engages in fishing, for the environment of the for one year Water pollution substances they contain: The fee for water storage between 2 and 5 , 1 € / kg during low flow: the base is The charge for water the volume of water stored during low flow; the ceiling pollution of non-domestic Fees on other water uses rate of 0.01 € / m3 stored. The origin is based on the annual fee for obstacles on pollution discharged into the watercourses is due on natural environment and relates to 10 constituent structures constituting an elements of pollution obstacle on watercourses, blocking sediment transit and The charge for water fish migration; the rate is set pollution by livestock is by the water agency for a based on the Large Cattle maximum of 150 € / m of Unit (UGB) and takes into height difference. account the stocking rate Value of the production of (number of animals per Royalty due by the liquid or gaseous hectare), by promoting operators of liquid hydrocarbons at the start of extensive breeding: 3 € / hydrocarbon mines the field (does not apply to UGB deposits at sea) products Fishery products Tax on sea landed in France

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Is environmental taxation a tool for protecting the contributions by country. Relative to GDP, in 2015 the environment? Gilles Rotillon In Different perspectives on the revenue from environmental taxes in the European Union economy 2007/1 (n ° 1), page 109. reached the highest value in Croatia (4.1%), followed by Source : CGDD, d’après les annexes au PLF 2017, Tome 1 Denmark with a ratio of 4.0%, Slovenia (3 , 9%) and Greece de l’Évaluation des voies et moyens et rapport Agences de (3.7%). The lowest ratios between the product of l’eau, et d’après le fichier Évaluation de recettes de la environmental taxes and GDP (below 2%) were recorded in DGDDI. six Member States (Slovakia, Lithuania, Luxembourg, Spain, II-2 the performance of the French experience in Ireland and Germany) ecological taxation: At the end of 2016 the highest tax revenue / GDP ratio in We can analyze the performance of Eco taxation in France Europe was that of France, Belgium and Denmark The tax by several forms and methods, in our work we choose to revenue / GDP ratio in France (48.4%), Belgium (47.3%) analyze it through their contribution to revenue budget, we and Denmark (46.5%).[7] At the level of the OECD, the will take as a reference the year 2016 data available for ranking of France is slightly better (among the 20th out of 31 analysis. member countries for which data is available). In France the internal tax on the consumption of energy products TABLE 2. Total from environmental taxes, by tax category. constitutes the fifth tax revenue, behind the value added tax (VAT), income tax, corporation tax and property tax. as a% of in% of As a% of the total Total TABLE 3 : The main environmental taxes in 2016 In millions Gross taxes and environ euros Domestic social Revenue 2016 mental Eurostat Product contributi Name of the tax (in millions of taxes classification ons euros) Total des Taxes Internal tax on the 359 294 100,00 2,4 6,3 environnementales consumption of energy 28 456 Energy Taxes sur l'énergie 275 392 76,6 1,9 4,8 products (TICPE) Contribution to public Taxes sur les 8264 71 269 19,8 0,5 1,3 service electricity (CSPE) transports Local taxes on electricity 1588 Taxes sur la Flat-rate tax on network 1592 pollution / les 12 633 3,5 0,1 0,2 companies (IFER) Domestic consumption tax ressources 1104 EU-28, in 2015 on natural gas (TICGN) Source: Eurostat (envactax) Ecotaxes Other energy taxes 1 310 Tax on registration 2 187 Transport The environmental taxes in the European Union In 2015, the certificates (gray cards) total public revenues arising from environmental taxes in the Additional tax on 996 EU- 28 was 359.3 billion euros, or 2.4% of gross domestic automobile insurance product (GDP) and 6.3% of total government revenue from Tax due by motorway 599 compulsory levies. concessionaires Tax on company vehicles 542 Fig. 1. Total product of environmental taxes, 2015 (in%) Civil aviation 410 Other taxes on transport 1167 charges Water pollution 1960 Pollution General tax on polluting activities (TGAP) (waste, 654 atmospheric pollution, etc.) excluding fuel TGAP Water levy fees 385 Resource Other resource taxes 22 Total (Eurostat field) 51 235 . Source: Volume I of ways and means of the 2018 finance bill, data Source: Eurostat (envactax) from the General Directorate of Customs and indirect duties. Source: Eurostat (envactax), press release in November 2018 The product of environmental taxes for 2015 compared to GDP and to the total product of all taxes and social

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According to these data, we see that environmental taxes (in TABLE 4 : The damage caused by the cost of environmental the sense of Eurostat) represent 51 billion euros in 2016, we degradation in Morocco. see that ecological taxation constitutes a transfer tax, where the revenue collected under an environmental tax will be Average Upper allocated to the financing of an environmental public policy. value % We take, for example, the case of fees collected by water Lower Terminal Average Gross agencies, which are allocated to policies for managing water bound Billions value resources and improving their ecological and sanitary Domestic of dh condition. So we will conclude that several environmental Product taxes have proven their effectiveness through their contribution to the financing of several public policies and to Water 11.10 12.20 11.70 1.26% return a capital which makes it possible to revive activity and Air 6.30 13.10 9.70 1.05% to make the economic system more efficient overall. Soils 4.60 5.30 5.00 0.54% Axis III: Eco-Taxation in Morocco Waste (including 3.70 3.70 3.70 0.40% Morocco has drawn up a negative balance sheet in terms of hazardous waste) the environment, which calls for the economic priorities of Littoral 2.50 2.50 2.50 0.27% the Moroccan State, to face this problem through economic Drills 0.00 0.00 0.00 0.00% and political instruments, and have taken several forms, costs for Moroccan notably legal, financial or fiscal, and depending on the nature 28.30 36.80 32.50 3.52% of the environmental dimension (water, air, soil, waste, etc.). society Among these main forms that promote environmental Carbon emissions 4.60 25.40 15.00 1.62% protection, environmental taxation. This part will be an essay Cost for the global of an analysis of environmental taxation in the Moroccan tax 4.60 25.40 15.00 1.62% system. It wonders about the state of play of environmental environment taxation in Morocco? And to know is the Moroccan Source: World Bank World Bank Report, The Cost of Environmental Degradation in environmental taxation a viable solution to fight against Morocco Lelia Croitoru and Maria Sarraf (editors), January 2017. climate change? The concept of environmental taxation: Among the national costs, water pollution (1.26% of GDP) is the The concept of environmental taxation was first mentioned main vector of water degradation, followed by air pollution (1.05% in Morocco in 2014 in a publication in the official bulletin N of GDP. Land degradation also entails considerable costs (0.54% ° 6240 of 20-06-2014 of du Dahir n ° 1-14-09 of 4 joumada l of GDP), in particular due to erosion of cultivated land, land in 1435 (March 6, 2014) promulgating the framework law n clearing and the desertification of rangelands. Waste represents a ° 99-12 on the national charter of the environment and relatively high cost (0.4% of GDP).This critical situation shown by sustainable development, that the concept of environmental these studies to assess the cost of environmental degradation, in particular the studies based on carried out by the World Bank in taxation, and this aforementioned by article 30 "Is instituted 2006 and 2017, urges the Moroccan State to establish an effective an environmental taxation system composed of ecological environmental policy. taxes and fees imposed on activities characterized by a high level of pollution and consumption of natural resources. Moroccan regulations on green taxation: These taxes and fees can be applied to any characterized The Moroccan tax system is made up of an arsenal of taxes behavior, individual or collective, harming the environment and duties having an impact on the environment. n the effects and infringing the principles and rules of sustainable of the environment and exemptions and tax incentives development”. encouraging the protection of the environment, we will The state of play of environmental taxation in Morocco: include the main laws and codes: The industrial evolution of Morocco generated by the • The General Tax Code (CGI) provides for measures massive exploitation of the means of production having to be exempt from the Annual Motor Vehicle Tax (TSAVA); negative side effects, and contributes to the acceleration of vehicles intended for the public transport of person’s whose the degradation of the environment. Statistics show this total laden weight or the maximum total towed laden weight negative balance of the cost of environmental degradation in is less than or equal to 3,000 kilograms and electric motor Morocco, in particular in 2000, the World Bank conducted a vehicles and hybrid motor vehicles (electric and thermal); the study entitled “Assessment of the cost of environmental subjection of sales of solar water heaters to value added tax degradation in Morocco” (CDE). This study evaluated, for (VAT) at the reduced rate of 10%. the first time, the cost of environmental degradation which • The Code of Customs and Indirect Taxes (CDII) was estimated, for the year 2000, at 3.7% of GDP.[8]in 2014 requires the royalty on the exploitation of phosphates which The cost of environmental degradation for Moroccan society was abolished from January 1, 2008 (article 7 of the finance was estimated at nearly 32.5 billion dirhams, or 3.52% of law for the year 2008); internal taxes on energy products; the GDP. ecological tax on plastics; the special cement tax; the Special Tax on Concrete Iron; the Special Tax on Sand.

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• Law 47-06 relating to the taxation of local environmental degradation and promoting sustainable authorities constituted by the tax on the extraction of quarry development. products; the parking fee; driver's license tax; tax on taxi and coach licenses; the tax for checking vehicles over 5 years General conclusion: old; the tax on motorcycles with a cylinder capacity equal to To conclude this work, we can notice that or greater than 125 cm3. Established the rental value as the environmental taxation is a tool and instrument to deal with basis for calculating the business tax is capped at 50 million climate change, but remains a difficult procedure to Dhs (this ceiling was 100 million Dhs from July 1, 1998 to implement like all tax regulations, green taxation was 2001). introduced in the payroll of the European Union since the These measures are intended to limit large polluting 1970s, this aspect has been developed either at the level of industrial companies. To confirm the objective of national the differentiation of taxes and levies, and to broaden the tax strategies against pollution and the effects of climate change. revenue, or at the level of the expenditure target by the “Green, national and territorial taxation to help make financing of actions environmental protection. Morocco a regional green factory ”Morocco has adopted an Morocco is following a journey to implement environmental ambitious policy of environmental protection and clean taxation and then generalize it in the Moroccan tax system. energy production. This policy is certainly a source of We note that Morocco has taken a big step forward in opportunities, both for developing the country's establishing procedures and laws for the establishment of attractiveness and for generating new activities. Anticipation taxes related to environmental protection, such as framework of this problem when planning activity zones, by providing law n ° 99-12 on the National Charter Environment and them with the most advanced environmental characteristics, Sustainable Development encourages, but the real debate would make it possible to benefit from a rationalization of which aims to analyze the revenues of environmental costs through the scale effect and at the same time from taxation, either at the level of fundraising, or at the level of reduce investment procedures and improve Morocco's targeted expenditure. attractiveness for the launch of new economic projects, as Moroccan companies accept the introduction of Eco taxes part of an integrated vision of 'Morocco Regional Green but they ask that the funds collected should be paid into Factory', Taxation, with its two dimensions, national and actions for the protection of the environment and not to be local , should play a direct role, in support of this ambition, paid into the state budget. At this level, we really have to which is likely to create activities with high added value and question the efficiency of the management of these funds. quality jobs ” [9],. The Main Environmental Taxes in Morocco: REFERENCES The Moroccan tax system made up of a number of taxes that [1] The Organization for Economic Cooperation and are favorable to the environment, we will mainly present: Development. (L’Organisation de Coopération et de • Taxes and taxes in favor of the environment Développement Economiques). • The tax for checking vehicles over 5 years old; [2] Taxation And Environment Twenty-Third Report • The tax on motorcycles with certain engine To The President Of The Republic, 2005, Page 13. capacity; [3] The General Tax On Polluting Activities . (la taxe • Internal taxes on energy products; générale sur les activités polluantes.) • Taxes on the extraction of quarry products; [4] • The wastewater treatment charge; https://www.boursorama.com/patrimoine/actualites/ • The tipping fee; l-ecofiscalite-qu-est-ce-que-c-est • The special cement tax; c46ffc1e77e863445a8658a10c9ac0 Consulté le 23-07-2020 • The ecological tax on plastics; à12h. • The Special Tax on Concrete Iron; [5] arthur cecil pigou,m.a, the macmillan and co., • The tax on the deterioration of pavements; limited st. martin’s streety, london,1920. • The Special Tax on Sand, ... [6] Is environmental taxation a tool for protecting the Main tax exemptions and reductions favorable to the environment? Gilles Rotillon In Perspectives on the environment: economy 2007/1 (n ° 1), page 109. • Reduction of VAT on the rental of water and [7] Eurostat (Envactax), Press Release In November electricity meters; 2018 • The reduction of VAT on the economy car; and [8] The World Bank's Report, The Cost Of solar water heaters Environmental Degradation In Morocco Lelia Croitoru And • The suspension of import VAT on butane gas; Maria Sarraf (Editors), January 2017. Etc. [9] Report of the Economic, Social and Environmental It has been observed that the measures of environmental Council, “A Fiscal System, pillar for the New Development taxation either in the institutional or legal plan, are new Model” prepared by the Permanent Commission in charge of adopted or require an effort to further explain these Economic Affairs and Strategic Projects Page 103 measures, but it should be noted that Morocco has taken a big step to establish taxation with the objective of combating

Abdellah AMINE Khalid TEMSAMANI Achraf AZANZAL Adil Lalla Saadia CHADLI ABBASSI M’hamed ELOMARI Ahmed KAJOUNI Manal MENCHIH Belaid Mhamed ELOMARI BOUIKHALENE Mohamed Amine ERROCHDI Chakir ALLALOU Mohamed El OUAARABI Dounia BOUCHTA Mohammed EL HADDAD Elhoussain ARHRRABI Mohammed LAMARTI Hamid LMOU SEFIAN Rachid AIT DAOUD Hanaa HACHIMI Redouan EL KHAMLICHI Said Jalal KASSOUT MELLIANI Kamilia FILALI Khalid Sara ELLIAZIDI HILAL Taoufiq El HARROUTI Youssef ALLAOUI Zoubida BENMAMOUN ABOUT THE EDITOR IN CHIEF

Prof. Dr. Hanaa Hachimi, Ph.D in Applied Mathematics & Computer Science and a Ph.D in Mechanics & Systems Reliability, Secretary General of Sultan Moulay Slimane University in Beni Mellal. President of the Moroccan Society of Engineering Sciences and Technology (MSEST). I am Associate Professor at the Sultan Moulay Slimane University (USMS) of Beni Mellal, Morocco. I am Director of the Systems Engineering Laboratory (LGS) and IEEE Senior Member, precisely I am affiliated at the Big Data, Optimization, Service and Security (BOSS) team at USMS. I am Lecture and Keynote Speaker of the courses: Optimization & Operational Research, Graph Theory, Statistics, Probability, Reliability and Scientific Computing. I am Member of the Moroccan Society of Applied Mathematics (SM2A).

Previously Associate Professor at Ibn Tofail University, National School of Applied Sciences of Kenitra, Morocco.

for more information, visit our website : http://www.hanaahachimi.com/

http://www.usms.ac.ma/ijoa EDITORIAL BOARD

Editorial bord: Prof. Dr. Abou El Majd Badr (FS, Rabat, Morocco) Prof. Dr. Addaim Adnane (EMI, Morocco) Prof. Dr. Amine Abdellah (USMS, Beni Mellal, Morocco) Prof. Dr. Amlan Chakrabarti (Director A.K. Choudhury School Of I.T., University Of Calcutta, India) Prof. Dr. Assif Safaa (ENSA, El Jadida, Morocco) Prof. Dr. Bakhadach Idris ( USMS, Beni Mellal, Morocco ) Prof. Dr. Belhouideg Soufiane ( FP, Beni Mellal, Morocco ) Prof. Dr. Ben Maissa Yann (INPT, Rabat, Morocco (Ieee Senior Member)) Prof. Dr. Benterki Djamel (Setif University, Algeria) Prof. Dr. Bouloiz Hafida (ENSA, Agadir, Morocco) Prof. Dr. Boutalline Mohammed (USMS, Beni Mellal, Morocco) Prof. Dr. Chadli Lalla Saadia ( FST, Beni Mellal, Morocco ) Prof. Dr. Chaoui Habiba (ENSA, Kenitra, Morocco)Prof. Dr. Saidi Rajaa (Insea, Rabat, Morocco) Prof. Dr. Darouichi Aziz (FST, Marrakech, Morocco) Prof. Dr. Driss Mentagui (Faculte Des Sciences Universite Ibn Tofail, Kenitra) Prof. Dr. El Abbadi Laila (ENSA, Kenitra, Morocco) Prof. Dr. El Hami Abdelkhalek (INSA, Rouen, France) Prof. Dr. El Hissi Youmna (USMS, Beni Mellal, Morocco) Prof. Dr. El Mokhi Chakib (UIT, Kenitra, Morocco) Prof. Dr. Ellaia Rachid (EMI, Rabat, Morocco) Prof. Dr. Farouk Yalaoui (Directeur Du Laboratoire D'optimisation Des Systemes De Production (Losi) Prof. Dr. G. Suseendran (Vistas, India) Prof. Dr. Hanaa Hachimi (USMS, Morocco (Ieee Senior Member (Ieee Senior Member)) Prof. Dr. Hammadi Nait Charif (Universite De Bournemouth, Royaume-Uni) Prof. Dr. Hmina Nabil (USMS, Morocco) Prof. Dr. Ibrahim Rosziati (Uthm University, Malaysia) Prof. Dr. Ing. Andrei Victor Sandu (Gheorghe Asachi Technical University Of Iasi, Romania) Prof. Dr. Jensen Nils (Ostfalia, Wolfenbüttel, Germany) Prof. Dr. Jihane Farahat (President Of Egyptian Center Of Innovation And Invention) Prof. Dr. Jraifi Abdelilah (ENSA, Safi, Morocco) Prof. Dr. Kaicer Mohammed (FS, Kenitra, Morocco) Prof. Dr. Lakhouit Abderrahim (Usherbrooke, Canada) Prof. Dr. Madini-Zouine Zhour (ENSA, Kenitra, Morocco) Prof. Dr. Maslouhi Mustapha (ENSA, Kenitra, Morocco) Prof. Dr. Masulli Francesco ( University Of Genova, Italy ) Prof. Dr. Mehar Chand (Guru Kashi University bathinda, India) Prof. Dr. Mejri Mohammed (Ulaval, Quebec, Canada) Prof. Dr. Melliani Said (FST, Usms)

http://www.usms.ac.ma/ijoa Prof. Dr. Mraoua Mohammed (HEC, Montreal, Canada) Prof. Dr. Nayyar Anand (Duy Tan University, Vietnam) Prof. Dr. Obaid Abdelatif (UQAM, Canada) Prof. Dr. Oscar Castillo (Tijuana Institute Technology, Mexico) Prof. Dr. Petrica Vizureanu Gheorghe Asachi Technical University Of Iasi Romania Prof. Dr. Ribaudo Marina (University of Genoa, Italy) Prof. Dr. Rokia Missaoui (Universite Du Quebec En Outaouais, Canada) Prof. Dr. Rovetta Stefano (Unig, Genova, Italy) Prof. Dr. El Ghazali Talbi (University Of Lille, France) Prof. Dr. Rovetta Stefano (Unige University, Genova, Italy) Prof. Dr. Rui Lopes (Electrical Engineering Department Fct Nova And Uninova - Cts) Prof. Dr. Semlali Naoual (EMI, Rabat, Morocco) Prof. Dr. Soulaymani Abdelmajid (FSK, Kenitra, Morocco) Prof. Dr. Xin She Yang (National Physical Laboratory, Universite D’oxford Royaume- Uni) Prof. Dr. Zhoure Madini (Ibn Tofail University, Morocco) Prof. Dr. Zouine Youness (ENSA, Kenitra, Morocco)

http://www.usms.ac.ma/ijoa