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On the Controllability of Non-Densely Defined Fractional Neutral Functional

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On the Controllability of Non-Densely Defined Fractional Neutral Functional International Journal of Pure and Applied Mathematics Volume 118 No. 11 2018, 257-276 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: 10.12732/ijpam.v118i11.32 Special Issue ijpam.eu On the controllability of non-densely defined fractional neutral functional differential equations in Banach spaces R. Subashini∗, S.Vimal Kumar†, S. Saranya‡ and C. Ravichandran§ Abstract In this paper, we investigates the controllability results for non-densely defined fractional neutral functional differential equations in Banach space by using fractional calculus theory and standard fixed point theorem. And also we showed the applications of abstract results. Keywords: Fractional differential equations; Neutral equations; Controllability; Fixed point theorem. 2010 Mathematics Subject Classification: 34A08 1 Introduction In the recent years, the controllability plays an important role in the fractional calculus the- ory and has been extensively studied, the authors [3, 5, 13, 15, 20, 21, 25]. This is due to the fact that fractional differential equations have various applications in engineering and scientific disciplines, for example, fluid dynamics, fractal theory, diffusion in porous media, fractional biological neurons, traffic flow, polymer rheology, neural network modeling, viscoelastic panel in super sonic gas flow, real system characterized by power laws, electrodynamics of complex medium, sandwich system identification, nonlinear oscillation of earthquake, models of pop- ulation growth, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, nuclear reactors and theory of population dynamics. The details on the theory and its applications may be found in [8, 10, 11, 12, 14, 18, 22, 23, 24, 27, 28, 30]. Non-densely defined functional differential systems in abstract space with control system studied many authors [4, 6, 7]. Neutral fractional differential equations arise in many areas of ∗Research and Development Center, Bharathiar University, Coimbatore - 641046, Tamilnadu, India and Department of Mathematics, GTN Arts College, Dindigul - 624004, Tamilnadu, India. E. Mail-: sub- [email protected] †Department of Mathematics, RVS Technical Campus, Coimbatore-641 402, India. E. Mail-: svimalku- [email protected] ‡Department of Mathematics, Vignan’s Foundation For Science, Technology and Research, Guntur-522213, Andra Pradesh, India. E. Mail: [email protected] §PG & Research Department of Mathematics, Kongunadu Arts & Science College, Coimbatore - 641 029, Tamil Nadu, India. Email: [email protected] 1 257 International Journal of Pure and Applied Mathematics Special Issue applied mathematics and for this reason these equations have received much attention in the last decades [1, 19, 26, 29]. But the literature related to neutral fractional differential equations with non-densely is very limited and we refer the reader to [9]. In this paper, we study the controllability of the integral solution for the fractional semi- linear functional differential equations of the form cDq[x(t) f(t, x )] = Ax(t) + Cu(t) + g(t, x ), t (0, b] (1.1) − t t ∈ x (s) + (h(x , x , ..., x ))(s) = φ(s), s [ r, 0]. 0 t1 t2 tn ∈ − where cDq is the Caputo fractional derivative of order 0 < q < 1, b > 0, r > 0. A : D(A) E ⊂ → E is a non-densely closed linear operator on E, E is a real Banach space with the norm . The |·| control function u( ) is given in L2([0, b],U), a Banach space of admissible control functions · with U as a Banach space. is a bounded linear operator from U to E. Let ([ r, 0],E) C C − denote the space of all continuous functions with the sup-norm φ c = maxs [ r,o] φ(s) , g, f : k k ∈ − | | [0, ) ([ r, 0],E) E and h :( ([ r, 0],E))n E are given functions are satisfying some ∞ × C − → C − → assumption, φ ([ r, 0],E) and define x ([ r, 0],E) by x (s) = x(t + s), s [ r, 0], for ∈ C − t ∈ C − t ∈ − t [0, b]. ∈ 2 Back ground results In this section, we introduce notations, definitions, and preliminary results which are used in the rest of the paper. We denoted by ([0, b],E) a Banach space of all continuous functions C from [0, b] into E with the norm x = sup y(t) : t [0, b] . B(E) denotes the Banach || || {| | ∈ } space of bounded linear operators from E into E, with the norm N = sup N(y) : y = 1 . || || {| | | | } Assume that J R, and 1 p . For measurable functions m : J R, define the norm ⊂ ≤ ≤ ∞ → 1 p p J m(t) dt , 1 p < m p = | | ≤ ∞ || ||L J infR p(J)=0 supt J J¯ m(t) , p = , ∈ − | | ∞ where µ(J¯) is the Lebesgue measure on J¯. Let LP (J, R) be the Banach space of all Lebesgue measurable functions m : J R with p < . → || · ||L J ∞ 1 1 r Lemma 2.1. (Holder’s inequality). Assume that r, p 1, and r + p = 1. If l L (J, R), then 1 ≥ ∈ for 1 p , lm L (J, R) and lm 1 l r m p . ≤ ≤ ∞ ∈ k kL J ≤ k kL J k kL J Lemma 2.2. (Bochner Theorem). A measurable function H : [0, b] E is Bochner’s integrable → if H is Lebesgue integrable. | | Lemma 2.3. (Schauder’s Fixed Point Theorem). If is a closed bounded and convex subset B of a Banach space E and F : is completely continuous, then F has a fixed point in . B → B B Lemma 2.1 - 2.3 are classical, which can be found in many books. 2 258 International Journal of Pure and Applied Mathematics Special Issue Definition 2.1. [2] Let E be a Banach space. An integrated semigroup is a family of operators S(t) t 0 of bounded linear operators S(t) on E with the following properties: { } ≥ (i) S(0) = 0; (ii) t S(t) is strongly continuous; → (iii) S(s)S(t) = s(S(t + r) S(r))dr for all t, s 0. 0 − ≥ Definition 2.2. R[10] An operator A is called a generator of an integrated semigroup, if there exists ω R such that (ω, + ) ρ(A) and there exists a strongly continuous exponentially ∈ ∞ ⊂ 1 bounded family (S(t))t 0 of linear bounded operator such that S(0) = 0 and (λI A)− = ≥ λt − λ 0∞ e− S(t)dt for all λ > ω. DefinitionR 2.3. [2] Let A be the generator of an integrated semigroup (S(t))t 0. Then for all ≥ x E and t 0, t S(s)xds D(A) and S(t)x = A t S(s)xds + tx. ∈ ≥ 0 ∈ 0 Definition 2.4. [10]R We say that linear operator A satisfiesR the Hille-Yosida condition if there exist M 0 and ω R such that (ω, + ) ρ(A) and sup (λ ω)n R(λ, A)n , n N, λ > ω ≥ ∈ ∞ ⊂ { − k k ∈ } M. Here and hereafter, we assume that A satisfies the Hille-Yosida condition. Let us intro- ≤ duce the part A0 of A in D(A): A0 = A on D(A0) = x D(A); Ax D(A) . Let (S(t))t 0 ∈ ∈ ≥ be the integrated semigroup generated by A. We note thatn (S0(t))t 0 is a C0-semigroupo on D(A) ≥ generated by A and S (t) Meωt, t 0, where M and ω are the constants considered in the 0 k 0 k ≤ ≥ Hille-Yosida condition. Semiflows generated by Lipschitz perturbations of non-densely defined operators. Let B = λR(λ, A) := λ(λI A) 1, Then for all x D(A), B x x as λ . Also from λ − − ∈ λ → → ∞ the Hille-Yosida condition it is easy to see that limλ Bλx M x . →∞ | | ≤ | | Definition 2.5. [18] The fractional integral of order α R+ with the lower limit zero for a ∈ function f : R+ E is defined as → 1 t f(s) Iαf(t) = ds, t > 0, 0 < α < 1, Γ(α) (t s)1 α Z0 − − provided the right side is point-wise defined on R+, where Γ( ) is the Gamma function. · Definition 2.6. [18] Riemann-Liouville derivative of order α R+ with the lower limit zero ∈ for a function f : R+ E is defined by → 1 d t f(s) LDαf(t) = ds, t > 0, 0 < α < 1, Γ(n α) dt (t s)α − Z0 − provided the right side is point-wise defined on R+, where Γ( ) is the Gamma function. · Definition 2.7. [18] The Caputo derivative of order 0 < α < 1 of a continuous function f : R+ E is defined by → 1 t f 0 (s) cDαf(t) = ds, t > 0. Γ(n α) (t s)α − Z0 − 3 259 International Journal of Pure and Applied Mathematics Special Issue 3 Main results Definition 3.1. Let φ ([ r, 0],E), a function x :[ r, b] E be an integral solution of ∈ C − − → problem (1.1) if (i) x is continuous on the interval [0,b]; (ii) 1 t(t s)q 1x(s)ds D(A) for t [0, b]; Γ(q) 0 − − ∈ ∈ (iii) R φ(0) (h(x , x , ..., x ))(0) f(0, x ) + f(s, x ) − t1 t2 tn − 0 s x(t) = + 1 A t(t s)q 1x(s)ds + 1 t(t s)q 1[Cu(s) + g(s, x )]ds, t [0, b], Γ(q) 0 − Γ(q) 0 − s − − ∈ φ(t), t R[ r, 0], R ∈ − Lemma 3.1. If x is an integral solution of (1.1), then for all t [0, b], x(t) D(A). In ∈ ∈ particular, x(0) = φ(0) D(A). ∈ Proof. The proof is similar to the proof of the Lemma 2.7 in [17]. 1 n 1 qn 1 Γ(nq+1) + Lemma 3.2. [16] Let ψ (θ) = ∞ ( 1) θ sin(nπq), θ R , then ψ (θ) is q π n=1 − − − − n! ∈ q a one-sided stable probability densityP function and its Laplace transform is given by ∞ pθ pq e− ψ (θ)dθ = e− , q (0, 1), p > 0.
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