Proceedings Book

PROCEEDINGS BOOK

OPERATORS IN GENERAL MORREY-TYPE SPACES AND APPLICATIONS OMTSA 2019

E d i t o r s VAGIF S. GULIYEV I.EKINCIOGLU

[email protected] http://omtsa.dpu.edu.tr ISBN NO: 978-605-69425-0-1 Operators in General Morrey-Type Spaces and Applications (OMTSA 2019), Kutahya Dumlupinar University, Kutahya, TURKEY, 16-20 July, 2019

Editors: Vagif S. GULIYEV and Ismail EKINCIOGLU

Associate Editor: Esra KAYA and Cansu KESKIN Copyright c OMTSA 2019 KUTAHYA D UMLUPINAR UNIVERSITY Published 17 December 2019 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

PREFACE

Operators in General Morrey-Type Spaces and Applications (OMTSA 2019) will be held on July 16-20, 2019 in Kutahya, Turkey. Firstly, we would like to thank all the invited speakers who have kindly accepted our invitation and have come to spend their precious time by sharing their ideas during the conference. The "Operators in General Morrey-Type Spaces and Applications" conference was held in 2011 and 2017 at Kirsehir Ahi Evran University. This year OMTSA 2019 has been held at Kutahya Dumlupinar University, Kutahya, Turkey on 16-20 July 2019 as an international conference. Kutahya has been home to many civilizations with its 7000 years of history and is an open air museum with its historical richness as the city of establishment and salvation, and the city of history and culture. Also, because of its location on the graben system of the Aegean Region and the cracks formed by this system Kutahya is one of the most important regions in terms of geothermal sources. Those sources have quite high thermal value and are important for health tourism. OMTSA 2019 International Conference on Operator Theory and its Applications and Harmonic Analysis, Partial Differential Equations and to explore interactions with Differential Geometry, Topology and Algebra aims to bring together leading academic scientists, researchers and research scholars to exchange and share their experiences and research results on all aspects of Operator Theory and Harmonic Analysis. It also provides a premier interdisciplinary platform for researchers, practitioners and educa- tors to present and discuss the most recent innovations, trends, and concerns as well as practical challenges encountered and solutions adopted in the above area related areas. The meeting will be devoted to various aspects of Theory Function Spaces, Opera- tor Theory of Function Spaces, Theory and applications of all function spaces related to Morrey spaces, Integral operators on Morrey type spaces, Regularity in Morrey type spaces of solutions to elliptic, parabolic and hypoelliptic equations, Approximation theory and Interpolation theory. Finally, we would like to convey our heartiest welcome to each of you who have come to attend this conference and we wish for an enjoyable high scientiﬁc level conference and hope to meet you again in the future. Best wishes and regards,

On behalf of the Organizing Committee Prof. Dr. Vagif S. GULIYEV Prof. Dr. Ismail EKINCIOGLU

iii OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

ORGANIZING COMMITTEE

Vagif S. GULIYEV (Kutahya Dumlupinar University, TURKEY) • Ismail EKINCIOGLU (Kutahya Dumlupinar University, TURKEY) • Victor BURENKOV (Nikolskii Institute of Mathematics, RUSSIA) • Maria Alessandra RAGUSA (Catania University & Accademia Gioenia Catania, ITALY) • Amiran GOGATISHVILI (Institute of Mathematics AS CR, CZECH REPUBLIC) • Salauddin UMARKHADZHIEV (Academy of Sciences of the Chechen Republic, RUSSIA) • Rovshan A. BANDALIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN) • Przemyslaw GORKA (Warsaw University of Technology, POLAND) • Ayhan SERBETCI (Cankiri Karatekin University, TURKEY) • Ali S. NAZLIPINAR (Kutahya Dumlupinar University, TURKEY) • Ali AKBULUT (Kirsehir Ahi Evran University, TURKEY) • Cansu KESKIN (Kutahya Dumlupinar University, TURKEY) • Mehriban N. OMAROVA (Baku State University, AZERBAIJAN) • Ahmet BOZ (Kutahya Dumlupinar University, TURKEY) • Ozgun GURMEN ALANSAL (Kutahya Dumlupinar University, TURKEY) • Ilkem TURHAN CETINKAYA (Kutahya Dumlupinar University, TURKEY) • Abdulhamit KUCUKARSLAN (Pamukkale University, TURKEY) • Canay AYKOL YUCE(Ankara University, TURKEY) • Fatih DERINGOZ (Kirsehir Ahi Evran University, TURKEY) • Esra KAYA (Kutahya Dumlupinar University, TURKEY) • Tugce Unver YILDIZ (Kırıkkale University, TURKEY) •

iv OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

SCIENTIFIC COMMITTEE Akbar B. ALIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN) • Alexey KARAPETYANTS (Southern Federal University, RUSSIA) • Alik M. NAJAFOV (Azerbaijan University of Architecture and Construction,AZERBAIJAN) • Amiran GOGATISHVILI (Institute of Mathematics AS CR, CZECH REPUBLIC) • Arash GHORBANALIZADEH (Institute for Advanced Studies in Basic Sciences, IRAN) • Ayhan SERBETCI (Cankiri Karatekin University, TURKEY) • Daniyal ISRAFILZADE (Balıkesir University, TURKEY) • Ekrem SAVAS (Usak University, TURKEY) • Elcin YUSUFOGLU (Usak University, TURKEY) • Fahreddin ABDULLAYEV (Mersin University, TURKEY) • Ismail EKINCIOGLU (Kutahya Dumlupinar University, TURKEY) • I. Naci CANGUL (Uludag University, TURKEY) • Kemal AYDIN (Selcuk University, TURKEY) • Lubomira SOFTOVA (Second University of Naples, ITALY) • Maria Alessandra RAGUSA (Catania University & Accademia Gioenia Catania, ITALY) • Michael RUZHANSKY (Imperial College, London, UK) • Mikhail GOLDMAN (Peoples’ Friendship University of Russia, RUSSIA) • Miloud ASSAL (Carthage University, TUNUSIA) • Misir C. MARDANOV (Institute of Mathematics and Mechanics, AZERBAIJAN) • Mubariz G. HACIBAYOV (National Academy, AZERBAIJAN) • Natasha SAMKO (Luleå University of Technology, SWEDEN ) • Pankaj JAIN (South Asian University, INDIA) • Przemyslaw GORKA (Warsaw University of Technology, POLAND) • Rabil AYAZOGLU (MASHIYEV) (Bayburt University, TURKEY) • Radouan DAHER (Faculty of Sciences Ain Chock University Hassan II, MOROCCO) • Rovshan A. BANDALIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN) • Salauddin UMARKHADZHIEV (Academy of Sciences of the Chechen Republic, RUSSIA) • Sergio POLIDORO (Univerita di Modena e Reggio Emilia, ITALY) • Stefan SAMKO (Universidade do Algarve, PORTUGAL) • Tahir S. HACIYEV (Institute of Mathematics and Mechanics, AZERBAIJAN) • Vagif S. GULIYEV (Kutahya Dumlupinar University, TURKEY) • Victor BURENKOV (Nikolskii Institute of Mathematics, RUSSIA) • Yagub Y. MAMMADOV (Nakhchivan Teacher-Training Institute, AZERBAIJAN) • Yoshihiro SAWANO (Tokyo Metropolitan University, JAPAN) •

v OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

CONTENTS

A contact problem in an inhomogeneous half-plane (Elcin YUSUFOGLU, llkem TURHAN CETINKAYA) ...... 1

Multivalent Harmonic Starlike Functions Deﬁned by Subordination (Sibel YALCIN TOKGOZ, Sahsene ALTINKAYA)...... 11

A New Local Smoothing Technique for Non-smooth Functions (Nurullah YILMAZ)...21

The (p, q)-Lucas Polynomial Coefﬁcient Estimates of a Bi-univalent Function Class with Respect to Symmetric Points (Sahsene ALTINKAYA, Sibel YALCIN TOKGOZ).... 30

Approximation of the Reachable Sets of an SEIR Control System and Comparison with Optimal Control Solution (A. Serdar NAZLIPINAR, Barbaros BASTURK)...... 40

On Totally Asymtotically Nonexpansive Mappings in Cat(0) Spaces (A. ABKAR, Mojtaba RASTGOO)...... 51

Boundary Value Problems for Convolution Differential Operator Equations on the Half Line (Hummet MUSAYEV)...... 57

Some Relations Between Partially q-Poly-Euler Polynomials of the Second Kind (Burak KURT)...... 61

Author Index...... 71

vi OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

A contact problem in an inhomogeneous half-plane Elcin YUSUFOGLU1, llkem TURHAN CETINKAYA2 1Usak University, TURKEY, 2Dumlupinar University, TURKEY

(Received: 18.09.2019, Accepted: 23.11.2019, Published Online: 17.12.2019)

Abstract. In this study, a frictionless contact problem for an elastic inhomogeneous half space is presented. The problem is reduced to a singular integral equation by using basic equations of the elasticity theory and Fourier transform technique. The singular integral equation is solved with the help of Chebyshev polynomials.

Keywords: Contact problems, Elasticity theory, Integral equations.

Introduction In elasticity theory, contact problems are very important subjects. A lot of studies in this ﬁeld have been done. Some of them are given below: Fabrikant and Sankar [1] studied a contact problem in an inhomogeneous half space whose elastic modulus is a power function of dept. The integral equation in polar coor- dinates is taken into consideration. An illustrative example is given. Kuo [2] consider two-dimensional contact problem and give the contact stress analysis of an elastic half plane. An axisymmetric contact problem for an elastic inhomogeneous half-space is presented for the case of presence of complete cohesion. The exact solution of the circular punch problem is obtained. Selvadurai and Lan [3] consider mixed boundary value problems result from axisymmetric contact, crack and inclusion problems in nonhomogeneous elastic medium. The problems are reduced to Fredhom integral equation and representative examples are given. Guler and Erdogan [4] consider a plane elasticity problem consisting of two mediums in which one of them is a homogeneous substrate, the other one is the functional graded material coating. The problem is reduced to system of Cauchy type singular integral equations and stress distributions are presented. Singh et.al. [5] examined the contact problem in which there are two nonhomogeneous medium. The problem is reformulated as Fredholm integral equations and stress analysis is done. Vasiliev et. al. [6] give an axisymmetric contact problem in which the punch’s base is ﬂat circular base. They obtained solutions in analytical form. The results are compared by the solutions obtained from other methods. In mechanics, the problems can be widely formulated as Cauchy type singular integral equations and their systems. One of the effective methods to approach the solutions is Chebyshev polynomials and quadrature technique.

1 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Some of the frictionless problems leads to first kind of singular integral equations are done Yan and Li [7] studied a double receding contact plane problem. By using the Fourier transform technique the problem is reduced to a system of singular integral equations of first kind and approached by Gauss-Chebyshev integration formulas. The results for the normalized contact half-widths with the other studies and effect of stiff- ness parameters are presented. Comez [8] obtain a Cauchy singular integral equation of first kind leading arising from moving contact problem. The collocation method is used to obtain the numerical results. El-Borgi et. al. [9] consider a receding contact plane problem which consist of an infinitely long functional graded layer and a homogeneous semi-infinite medium. By using Fourier transforms of the displacements, the problem is converted to singular integral equation of first kind. The singular integral equation is solved by Chebyshev polynomials and an iterative scheme. So, they could analyze the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact. The contact problems can be reformulated under the effect of the friction as the singular integral equations of second kind. Arslan and Dag [10] applied the singular integral equation technique and the finite elements methods to the sliding contact problem. The singular equation of the second is solved by Chebyshev polynomials. The numerical results for the flat punch and triangular punch are presented. Yilmaz et. al. [11] applied the Gauss-Jacobi integration formula to the system of singular integral equations of second kind resulted from the frictional receding contact problem. Also, by using the Finite element method, they showed that the obtained numerical and analytical results are in good agreement. Guler and Erdogan [12] considered a frictional contact problem in the case of rigid parabolic and cylindrical stamps. The detailed analysis is done on the stress intensity factors and stress distributions. In this study, the nonhomogeneous contact problem in [13] is considered. The problem is converted to a singular integral equation. Finally, the solution is approached by Chebyshev polynomials.

Formulation of the problem

In this section, the formulation of the contact problem on an elastic inhomogeneos half- space Ω is presented. As is seen from Figure 1, the punch length 2a is in contact with the elastic inhomogeneous coating and it is located symmetrically according to axis y . The punch is loaded to the coating length H with the force P and moment M. − The Lame parameters vary with depth as

§ Λ0(y), H y 0, Λ = − ≤ ≤ (1) Λ1 = Λ0( H), < y < H, − −∞ − § G0(y), H y 0, G = − ≤ ≤ G1 = G0( H), < y < H, − −∞ − 2 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Figure 1. Geometry of the contact problem for a half-space

Boundary conditions of the problem are of the form: § 0, x > a τ(1) 0; σ(1) ; ν(1) α xβ γ x f x , x a; x y y=0 = y y=0 = | | y=0 = ( + ( )) = ( ) | | q(x), x a, | − − − | | ≤ − | | ≤ (2)

(1) (2) (1) (2) u(1) u(2) (1) (2) τx y y= H = τx y y= H , σy y= H = σy y= H , = , ν = ν , (3) | − | − | − | − where σy and τx y are normal and shear stress components, u and ν are the displacement along the axis x and y, respectively and the function q(x) denotes the contact pressure distribution. Here and afterwards, the superscript denotes the layer. The relation between the contact pressure distrubiton q(x) and the force P and moment M is deﬁned as a a Z Z P = q(ξ)dξ, M = q(ξ)ξdξ. (4) a a − − Reduction of the problem to an integral equation by in the view of elasticity theory In this section, the problem is reduced to an integral equation in the view of elasticity theory. Equilibrium equations and the relation between the stresses and displacement are given as follows

∂ σ ∂ τx y ∂ τx y ∂ σy x 0, 0. (5) ∂ x + ∂ y = ∂ x + ∂ y =

∂ u ∂ ν ∂ u ∂ u ∂ ν ∂ ν σ Λ y 2G y , σ Λ y 2G y , (6) x = ( )[∂ x + ∂ y ] + ( )∂ x y = ( )[∂ x + ∂ y ] + ( )∂ x ∂ u ∂ ν τ G y , x y = ( )[∂ y + ∂ x ] respectively. The Lame parameters Λ, G and Young moduli E can be represented as

E Eν G(3Λ + 2G) Λ G = , Λ = , E = , ν = 2(1 + ν) 2(1 + ν)(1 2ν) Λ + G 2(Λ + G) − 3 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Let us express the displacements by Fourier transform as

1 Z∞ 1 Z∞ u x, y U α, y e iαx dα, ν x, y V α, y e iαx dα. (7) ( ) = 2π ( ) − ( ) = 2π ( ) −

−∞ −∞ By inserting the relation between the stresses and displacements into the equilibrium equations, using the Fourier transform (7) and applying inverse Fourier transform, the following system of ordinary differential equations is obtained:

00 2 G(y)U (α, y) + α(G(y) + Λ(y))Ve 0(α, y) α (2G(y) + Λ(y))U(α, y) (8) − G0 (y)U 0 (α, y) + αG0 (y)Ve(α, y) = 0,

00 0 2 (2G(y) + Λ(y))Ve (α, y) α(G(y) + Λ(y))U (α, y) α G(y)Ve(α, y) + (2G0(y) − − 0 + Λ0(y))Ve 0(α, y) αΛ (y)U(α, y) = 0, − where, V (α, y) = iVe(α, y). To solve the differential equation system above, the auxiliary functions is deﬁned as

Ui∗(α, y) = θ0 α Ui(α, y)/Q(α), (9) − | |

Vei∗(α, y) = θ0 α Vei∗(α, y)/Q(α), − | | θ0 = 2G(0)(Λ(0) + G(0))/(Λ(0) + 2G(0)), i = 1, 2, where, Q(α) is the inverse Fourier transform of the function q(x) : Z ∞ Q(α) = q(ξ)e(iαξ)dξ. (10)

−∞ By using the boundary conditions (2), (3), Eq. (7), auxiliary function (9) and Fourier transform technique, the solution of system of ordinary differential equations (8) are obtained as α y α y U2(α, y) = (c1 + c2 y)e| | + (c3 + c4 y)e−| | .

m1 α y m2 α y m3 α y m4 α y Ve2(α, y) = d1e | | + d2e | | + d3e | | + d4e | | . By using the boundedness condition while y , the solution can be rewritten as → −∞ α y U2(α, y) = (c1 + c2 y)e| | . The coefﬁcients will be determined from the conditions obtained from boundary conditions, Fourier transform of displacements and inverse Fourier transform technique as

G y U 0 V 0 y U y G y V 0 0( )( 1 + αe1 ) y=0 = 0, Λ0( )α 1 + (Λ0( ) + 2 0( ))e1 y=0 = α θ0, (11) | − | | | 4 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

U y U y V y V y 1(α, ) y= H = 2(α, ) y= H , e1(α, ) y= H = e2(α, ) y= H | − | − | − | − U 0 y U 0 y V 0 y V 0 y 1(α, ) y= H = 2(α, ) y= H , e1 (α, ) y= H = e2 (α, ) y= H | − | − | − | − The system of ordinary differential equations for the coating can be rewritten in matrix form as d w −→ = A w , H y 0, d y −→ − ≤ ≤ where ,   w1  0 1 0 0  w α2 2G Λ G G G Λ  2  ( + ) 0 0 + w = , A =  G G α G G  −→  w3  αΛ0 −G+Λ −2 G −2G0 +Λ0 w 2G Λ α 2G Λ α 2G Λ 2G Λ 4 + + + − + dU dV w U , w 1 , w V , w e1 , (12) 1 = 1 2 = d y 3 = e1 4 = d y The solution of the system of differential equations is obtained by [18] as w α, y d α a α, y e α y d α a α, y e α y . (13) −→k( ) = 1( )−→1k( ) | | + 2( )−→2k( ) | | From Cauchy problem, it is obtained that

d a −→ik = A a ik α a ik, H y 0, i = 1, 2, d y −→ − | |−→ − ≤ ≤ where the initial conditions obtained from Eqs. (11), (12) are

a α, y 1, α, 1, α , −→1k( ) y= H = ( ) | − Λ 3G Λ 3G a α, y αy, α α2 y, + αy, + α α2 y −→2k( ) y= H = ( + + + + ) y= H | − − Λ + G − Λ + G | − The function Ve1(α, 0) is obtained from Eqs. (11), (12),as

Ve1(α, 0) = d1(α)a13(α, 0) + d2(α)a23(α, 0). Now, let us consider the following integral equation obtained from Eq.(2) and Fourier transform tecnique Z 1 ∞ Ve1(α)Q(α) iαx e− dα = f (x), x a. (14) 2πθ0 α | | ≤ −∞ | | Inserting (10) into (14) and applying the transforms, it is obtained that Z Z 1 ∞ Ve1(α) ∞ iαξ iαx ( q(ξ)e dξ)e− dα = f (x), x a. 2πθ0 α | | ≤ −∞ | | −∞ 5 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

By using the transforms H αH u, λ = = a the following integral equation is obtained: Z Z 1 ∞ ∞ L(u) iu ξ x q(ξ)( e ( H− ))dξ = f (x), 2πθ0 u −∞ −∞ | | where, u L u V . ( ) = 1∗(H )

L(u) Considering the statement of the problem and the assumption that the function u is even, the integral equation can be rewritten as | |

a 1 Z ξ x q(ξ)k( − )dξ = f (x), 2πθ(0) a H − where, Z L(u) ∞ ξ x K(u) = , k(t) = 2K(u)cos(ut)du, t = − . u 0 H | | Now, applying the transform

ξ x H ξ0 , x 0 , λ , (15) = a = a = a the integral equation turns into the dimensionless form as

1 1 Z ξ x ϕ(ξ)k( − )dξ = f (x), (16) 2πθ0 1 λ − where, ξ x q ξ ϕ , f x fe . (17) ( ) = ( a ) ( ) = ( a ) Reduction of the integral equation to the singular integral equation In this section the problem will be converted to singular integral equation.

Theorem 1 The function K(t) deﬁned by Z ∞ L(u) K(t) = cos(ut)du. 0 u

6 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University can be rewritten for tε( , ) as ∀ −∞ ∞ K(t) = ln t F(t), − | | − where, Z u ∞ (1 L(u))cos(ut) e− F(t) = − − du 0 u [17]. By using the theorem above in Eq. (16) and deriving with respect to x,

Z 1 Z 1 ϕ(ξ) 1 dξ ϕ(ξ)M(ξ, x)dξ = f 0 (x)πG0(0) (18) 1 ξ x − λ 1 − − − is obtained, where, Z ∞ ξ x M(ξ, x) = (1 L(u))sin( − u)du 0 − λ The additional condition can be obtained from Eq. (4) by using the transform (15) and the notation (17), 1 Z P ϕ(ξ)dξ = (19) 1 a − Numerical Approximation to the singular integral equation In this section, the singular integral equation will be approached by Chebyshev poyno- mials. The technique leads to the following solution in the view of index theory: [14- 16] 1 ϕ(t) = w(t)φ(t), w(t) = . (20) p1 t2 Substituting Eq. (20) into Eq. (18), −

Z 1 Z 1 1 φ(ξ) 1 φ(ξ) dξ M(ξ, x)dξ = f 0 (x)G0(0). π p 2 πλ p 2 1 (ξ x) 1 ξ − 1 1 ξ − − − − − is obtained. Let us assume that

p X φ(ξ) = Bm Tm(ξ), m=0 where Tm(ξ) are the ﬁrst kind Chebyshev polynomials of order m. The ﬁrst kind and second kind Chebyshev polynomials has the following relation [16]:

7 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Z 1 1 Tm(ξ) dξ = Um 1(xi). π p 2 1 (ξ x) 1 ξ − − − − n § X Tj(ξk) 0, j = 0, = n ξ x Uj 1 xr , 0 < j < n. k 1 ( k r ) ( ) = − − By using these relations and

π 2k 1 T ξ 0, ξ cos ( ) , k 1, ..., n, n( k) = k = ( 2n− ) =

πr Un 1(xr ) = 0, xr = cos( ), r = 1, ..., n 1, − n − the singular integral in Eq. (18) is obtained as

Z 1 n 1 φ(ξ) X φ(ξk) p dξ = . π ξ x 1 ξ2 n ξk x r 1 ( ) k=1 ( ) − − − − So, under the consideration of the case of ﬂat base punches, singular integral equation (16) turns into n X φ ξ 1 1 ( k) M ξ , x 0 (21) n (ξ x λ ( k r )) = k 1 k r − = − By applying the same procedure to Eq. (19),

n X π P φ ξ (22) n ( k) = a k=1 is obtained. By solving linear equation system (21), (22), the function φ(ξk) is obtained. Using the interpolation polynomial obtained by φ(ξk) in Eq. (20), the function ϕ(ξ) is obtained. So q(ξ) is founded. In the following studies, the numerical results can be obtained and analyzed in detailed in the view of the mechanics.

8 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

References

[1] Fabrikant V.I.,Sankar T.S., On contact problems in an inhomogeneous half-space, In- ternational Journal of Solids and Structures, 20(2), 1984, 159-166. [2] Kuo C.H., Contact stress analysis of an elastic half-plane containing multiple inclu- sions, International Journal of Solids and Structures, 45, 2008, 4562-4573. [3] Selvadurai A.P.S.,Lan Q., Axisymmetric mixed boundary value problems for an elastic halfspace with a periodic nonhomogeneity, Int. J. Solids Siruciures, 35(15), 1998, 1813- 1826. [4] Guler M.A., Erdogan F., Contact mechanics of two deformable elastic solids with graded coatings, Mechanics of Materials, 38, 2006, 633-647. [5] Singh B.M., Rokne J., Dhaliwal R.S., Vrbik J., Contact problem for bonded nonhomogeneous materials under shear loading, IJMMS, 29, 2003, 1821-1832. [6] Vasiliev A., Volkov S., Aizikovich S., Jeng Y.R., Axisymmetric contact problems of the theory of elasticity for inhomogeneous layers, Journal of Applied Mathematics and Me- chanics, 94(9), 2014, 705-712. [7] Yan J., Li X., Double receding contact plane problem between a functionally graded layer and an elastic layer, European Journal of Mechanics A/Solids, 53, 2015, 143-150. [8] Comez I., Contact problem for a functionally graded layer indented by a moving punch, International Journal of Mechanical Sciences, 100, 2015, 339-344. [9] El-Borgi S., Abdelmoula R., Keer L., A receding contact plane problem between a functionally graded layer and a homogeneous substrate, International Journal of Solids and Structures, 43, 2006, 658-674. [10] Arslan O., Dag S., Contact mechanics problem between an orthotropic graded coating and a rigid punch of an arbitrary proﬁle, International Journal of Mechanical Sciences, 135, 2018, 541-554. [11] Yilmaz K.B., Comez I., Yildirim B., Güler M.A., El-Borgi S., Frictional receding contact problem for a graded bilayer system indented by a rigid punch, International Journal of Mechanical Sciences, 141, 2018, 127-142. [12] Guler M.A., Erdogan F., The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings, International Journal of Mechanical Sciences, 49, 2007, 161-182 [13] Aizikovich S.M, Aleksandrov, V.M.,Belokon A.V.,Krenev, L.I, Trubchik I.S., Contact problems of the theory of elasticity for non-homogeneous medium, Fizmatli, 2006. (In Russian). [14] Muskheleshvili N.I., Singular Integral Equations, Edited by J.R.M. Rodok, Noord- hoff International publishing Leyden, 1997. [15] Erdogan F., Gupta G.D., Cook T.S., Numerical solution of singular integral equations, In: Sih GC, editor. Method of analysis and solution of crack problems, Leyden: Noordhoff International Publishing, 1973. [16] Erdogan F., Gupta G.D., On the numerical solution of singular integral equations,

9 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Quaterly of Applied Mathematics, 30, 1972, 525-534. [17] Aleksandrov V.M., Smetanin B.I., Sobol B.V., Thin Stress Concentrators in Elastic Solids, Nauka, Moskow, 1993. [18] Babeshko V.A., Glushkov E.V., Glushkova N.V., Methods for constructing the Green function of a stratiﬁed elastic half-space, Zh. Vychisl. Mat. Mat. Fiz., 27(1), 1987, 93-101.

10 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Multivalent Harmonic Starlike Functions Deﬁned by Subordination Sibel YALCIN TOKGOZ1, Sahsene ALTINKAYA1 1Bursa Uludag University, TURKEY

(Received: 31.07.2019, Accepted: 07.10.2019, Published Online: 17.12.2019)

Abstract. We have introduced a generalized class of complex-valued multivalent harmonic starlike functions deﬁned by subordination. We study some properties of our class. The results obtained here include a number of known and new results as their special cases.

Keywords: Harmonic functions, Multivalent functions, Starlike functions, Subordina- tion.

Introduction

A continuous complex-valued function f = u+iv deﬁned in a simply connected complex domain D C is said to be harmonic in D if both u and v are real harmonic in D. Consider the⊂ functions U and V analytic in D so that u = ReU and v = ImV . Then the harmonic function f can be expressed by

f (z) = h(z) + g(z), z D, ∈ where h = (U + V )/2 and g = (U V )/2. We call h the analytic part and g co-analytic part of f . If g is identically zero then− f reduces to the analytic case. For a fixed positive integer p 1, let H(p) denote the class of all multivalent harmonic functions f = h + g which≥ are sense-preserving in the open unit disk E = z : z C and z < 1 and are of the form { ∈ | | } p X∞ n p X∞ f z z a z + 1 b zn+p 1, b < 1. (23) ( ) = + n+p 1 − + n+p 1 − p − − n=2 n=1 | | Recent interest in the study of multivalent harmonic function prompted the publication of several articles such as [1, 2, 6, 7, 8, 9]. It is known, (see Clunie and Sheil-Small [3]), that harmonic function is sense- preserving in E is that g0(z) < h0(z) , z E. Note that the class H(p) for p = 1 was defined and studied| by Clunie| | and| Sheil-Small∈ [3]. We say that f H(p) is a multivalent harmonic starlike of order α, 0 α < 1 if f satisfies the condition∈ ≤ § ª z fz(z) z fz¯(z) − pα, ℜ f (z) ≥ 11 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications for each z E. Denote this class of multivalent harmonic starlike functions of order α ∈ by SH∗ (p, α). The classes SH∗ (p, α) and SH∗ (1, α) were studied in [1] and [5]. We say that f H(p) is subordinate to a function F H(p), and write f (z) F(z) if there exists a complex-valued∈ function w which maps∈E into oneself with w(0≺) = 0, such that f (z) = F(w(z)) (z E). ∈ Denote by SH∗ (p, α, A, B) the subclass of H(p) consisting of functions f of the form (73) that satisfy the condition

zh z zg z p pB p α A B z 0( ) 0( ) + [ + ( )( )] , (24) − 1 −Bz − h(z) + g(z) ≺ + where 1 B B < A 1 and 0 α < p . − ≤ ≤ − ≤ ≤ By suitably specializing the parameters, the classes SH∗ (p, α, A, B) reduces to the various subclasses of harmonic univalent functions. Such as,

(i) SH∗ (1, 0, A, B) = SH∗ (A, B) ([4]]), (ii) SH∗ (p, α, 1, 1) = SH∗ (p, α),([1]), − (iii) SH∗ (1, α, 1, 1) = SH∗ (α),([5]), − (iv) SH∗ (1, 0, 1, 1) = SH∗ ([5]). − Finally, we deﬁne the family TSH∗ (p, α, A, B) SH∗ (p, α, A, B) TH(p), where TH(p), p 1 denote the class of functions f = h + g in H(p≡) so that h and∩g are of the form ≥

X X h z zp ∞ a zn+p 1 g z ∞ b zn+p 1 b ( ) = n+p 1 − and ( ) = n+p 1 − , p < 1. (25) − − − n=2 | | n=1 | | | |

Making use of the techniques and methodology used by Dziok (see [4]), in this paper, we find necessary and sufficient conditions, distortion bounds, compactness and extreme points for the above defined class TSH∗ (p, α, A, B). Main Results

For functions f1 and f2 H(p) of the form ∈

p X∞ n p X∞ f z z a z + 1 b zn+p 1, k 1, 2 , k( ) = + k,n+p 1 − + k,n+p 1 − ( = ) − − n=2 n=1 we deﬁne the Hadamard product of f1 and f2 by

p X∞ n p X∞ f f z z a a z + 1 b b zn+p 1. ( 1 2)( ) = + 1,n+p 1 2,n+p 1 − + 1,n+p 1 2,n+p 1 − − − − − ∗ n=2 n=1 First we state and prove the necessary and sufﬁcient conditions for harmonic functions in SH∗ (p, α, A, B).

12 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Theorem 2 Let f H(p). Then f SH∗ (p, α, A, B) if and only if ∈ ∈ f (z) φ(z; ζ) = 0, (ζ C, ζ = 1, z E 0 ), ∗ 6 ∈ | | ∈ \{ } where

φ z; ζ zp (p α)(A B)ζ z [B+(p α)(A B)]ζz ( ) = − − −(1− z)2 − − −

zp 2p(1+Bζ)+(p α)(A B)ζ+[(1 2p)(1+Bζ)+(p α)(A B)ζ]z . + − − (1 −z)2 − − −

Proof. Let f H(p) be of the form (73). Then f SH∗ (p, α, A, B) if and only if it satisﬁes (74) or equivalently∈ ∈

zh z zg z p pB p α A B ζ 0( ) 0( ) + [ + ( )( )] , (26) − = 1 −B − h(z) + g(z) 6 + ζ where ζ C, ζ = 1 and z E 0 . Since ∈ | | ∈ \{ } zp zp h(z) = h(z) , g(z) = g(z) , ∗ 1 z ∗ 1 z − − and zp p 1 p z zp p 1 p z zh z h z [ + ( ) ], zg z g z [ + ( ) ] , 0( ) = ( ) −2 0( ) = ( ) −2 ∗ (1 z) ∗ (1 z) the inequality (77) yields − − p pB p α A B ζ h z g z 1 Bζ zh z zg z ( + [ + ( )( )] )( ( ) + ( )) ( + ) 0( ) 0( ) − − − − § zp zp p 1 p z ª h z p pB p α A B ζ 1 Bζ [ + ( ) ] = ( ) ( + [ + ( )( )] ) ( + ) −2 ∗ − − 1 z − (1 z) − − zp zp p 1 p z g z p pB p α A B ζ 1 Bζ [ + ( ) ] + ( ) ( + [ + ( )( )] ) + ( + ) −2 ∗ − − 1 z¯ (1 z) − − p α A B ζ z B p α A B ζz h z zp ( )( ) [ + ( )( )] = ( ) − − − − 2 − − ∗ (1 z) − g z zp 2p(1+Bζ)+(p α)(A B)ζ+[(1 2p)(1+Bζ)+(p α)(A B)ζ]z + ( ) − − (1 −z)2 − − ∗ − = f (z) φ(z; ζ) = 0. ∗ 6 Next we give the sufﬁcient coefﬁcient bound for functions in SH∗ (p, α, A, B). Theorem 3 Let f be of the form (73). If 1 B B < A 1, 0 α < p, p 1 and − ≤ ≤ − ≤ ≤ ≥ X∞ p A B a p A B b p A B Φn( , α, , ) n+p 1 + Ψn( , α, , ) n+p 1 2( α)( ) (27) − − n=1 ≤ − −

13 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications where

Φn(p, α, A, B) = (1 B)(n 1) + (p α)(A B) (28) − − − −

and

Ψn(p, α, A, B) = (1 B)(n + 2p 1) (p α)(A B) (29) − − − − −

then f SH∗ (p, α, A, B). ∈

Proof. We need to show that if (74) holds then f SH∗ (p, α, A, B). By deﬁnition of ∈ subordination, f SH∗ (p, α, A, B) if and only if there exists a complex valued function w; w(0) = 0, w(z∈) < 1 (z E) such that | | ∈

zh z zg z p pB p α A B w z 0( ) 0( ) + [ + ( )( )] ( ) − = 1 −Bw z − h(z) + g(z) + ( )

or equivalently

zh z zg z ph z pg z 0( ) 0( ) ( ) ( ) − − − < 1. (30) B zh z zg z pB p α A B h z g z ( 0( ) 0( )) [ + ( )( )]( ( ) + ( )) − − − − 14 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Substituting for zh0(z), zg0(z), h(z) and g(z) in (81), we obtain

B zh z zg z pB p α A B h z g z ( 0( ) 0( )) [ + ( )( )]( ( ) + ( )) − − − − zh z zg z ph z pg z 0( ) 0( ) ( ) ( ) − − − −

p = (p α)(A B) z | − − X∞ B n p 1 pB p α A B a zn+p 1 + [ ( + ) + + ( )( )] n+p 1 − − n=2 − − − −

X∞ B n p 1 pB p α A B b zn+p 1 [ ( + ) ( )( )] n+p 1 − − − n=1 − − − − − −

X∞ n p X∞ n 1 a z + 1 n 2p 1 b zn+p 1 ( ) n+p 1 − ( + ) n+p 1 − − − − n=2 − − n=1 −

p (p α)(A B) z ≥ − − | | X ∞ B n p p A B a z n+p 1 [(1 )( + 1) + ( α)( )] n+p 1 − − − n=2 − − − − | || | X ∞ B n p p A B b z n+p 1 [(1 )( + 2 1) ( α)( )] n+p 1 − − − n=1 − − − − − | || | p > z (p α)(A B) | | { − − « X ∞ B n p p A B a [(1 )( + 1) + ( α)( )] n+p 1 − − n=2 − − − − | | « X ∞ B n p p A B b [(1 )( + 2 1) ( α)( )] n+p 1 − − n=1 − − − − − | | 0, ≥ by (78). The harmonic functions

X p α A B X ∞ ( )( ) n+p 1 ∞ (p α)(A B) n p 1 f z z xn p z − − yn p z + , (31) ( ) = + − − + 1 − + Ψn(p,α,A,B) + 1 − Φn p, α, A, B − − n=2 ( ) n=1

P P where ∞ x ∞ y 1, show that the coefﬁcient bound given by in n=2 n+p 1 + n=1 n+p 1 = Theorem 2.2| is sharp.− | | − |

15 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Since

X∞ X∞ p A B a p A B b Φn( , α, , ) n+p 1 + Ψn( , α, , ) n+p 1 − − n=2 n=1

X∞ X∞ = (p α)(A B) xn+p 1 + (p α)(A B) yn+p 1 − − − − n=2 | | − − n=1 | | = (p α)(A B), − − the functions of the form (82) are in SH∗ (p, α, A, B). Next we show that the bound (78) is also necessary for TSH∗ (p, α, A, B).

Theorem 4 Let f = h+ g with h and g of the form (74). Then f TSH∗ (p, α, A, B) if and only if the condition (78) holds. ∈

Proof. In view of Theorem 2.2, we only need to show that f / TSH∗ (p, α, A, B) if condition (78) does not hold. We note that a necessary and sufficient∈ condition for f = h + g given by (74) to be in TSH∗ (p, α, A, B) is that the coefficient condition (78) to be satisfied. Equivalently, we must have

H(z) < 1. G(z) where

X∞ n p X∞ H z n 1 a z + 1 n 2p 1 b zn+p 1 ( ) = ( ) n+p 1 − + ( + ) n+p 1 − − − n=2 − n=1 − and

p G(z) = (p α)(A B) z − − X ∞ n+p 1 [ B(n 1) + (p α)(A B)] an+p 1 z − − − n=2 − − − − X∞ B n 2p 1 p α A B b zn+p 1. [ ( + ) ( )( )] n+p 1 − − − n=1 − − − − − For z = r < 1 we obtain

P∞ n 1 P∞ n 1 (n 1) an+p 1 r − + (n+2p 1) bn+p 1 r − n=2 − | − | n=1 − | − | < 1. (32) p α A B P∞ B n 1 p α A B a rn 1 P∞ B n 2p 1 p α A B b rn 1 ( )( )+ [ ( ) ( )( )] n+p 1 − + [ ( + )+( )( )] n+p 1 − − − n=2 − − − − | − | n=1 − − − | − | If condition (78) does not hold then condition (84) does not hold for r sufﬁciently close to 1. Thus there exists z0 = r0 in (0, 1) for which the quotient (84) is greater than 1. This contradicts the required condition for f TSH∗ (p, α, A, B) and so the proof is complete. ∈

16 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Theorem 5 Let f TSH∗ (p, α, A, B). Then for z = r < 1, we have ∈ | | f z 1 b r p (p α)(A B) [2p(1 B) (p α)(A B)] b r p+1, ( ) + p + 1 B+(−p α)(−A B) 1 −B+(−p α−)(A B−) p | | ≤ | | − − − − − − − | | and f z 1 b r p (p α)(A B) [2p(1 B) (p α)(A B)] b r p+1. ( ) p 1 B+(−p α)(−A B) 1 −B+(−p α−)(A B−) p | | ≥ − | | − − − − − − − − | | Proof. We only prove the left hand inequality. The proof for the right hand inequality is similar and will be omitted. Let f TSH∗ (p, α, A, B). Taking the absolute value of f we have ∈

X∞ f z b r p a b r n+p 1 ( ) 1 p n+p 1 + n+p 1 − − − | | ≥ − | | − n=2

Φ p, α, A, B a Ψ p, α, A, B b p X∞ n( ) n+p 1 + n( ) n+p 1 p 1 1 b r − − r + p 1 B p A B ≥ − | | − n 2 + ( α)( ) = − − − p α A B 2p 1 B p α A B bp p ( )( ) [ ( ) ( )( )] p+1 1 bp r − − − − − − − | | r . ≥ − | | − 1 B + (p α)(A B) − − −

The following covering result follows from the left hand inequality in Theorem 2.4.

Corollary 1 Let f = h + g with h and g of the form (74). If f TSH∗ (p, α, A, B) then ∈ 1 B + [2p(1 B) (p α)(A B)] bp w : w < − − − − − | | f (E). | | 1 B + (p α)(A B) ⊂ − − − Theorem 6 Set p α A B h z zp, h z zp ( )( )zn+p 1, n 2, 3, ... , p( ) = n+p 1( ) = − − − ( = ) − − Φn(p, α, A, B) and p (p α)(A B) g z z zn+p 1, n 1, 2, ... . n+p 1( ) = + − − − ( = ) − Ψn(p, α, A, B)

Then f TSH∗ (p, α, A, B) if and only if it can be expressed as ∈ X f z ∞ x h z y g z ( ) = n+p 1 n+p 1( ) + n+p 1 n+p 1( ) − − − − n=1 where x y and P∞ x y In particular, the extreme n+p 1 0, n+p 1 0 ( n+p 1 + n+p 1) = 1. − ≥ − ≥ n=1 − − points of TS p A B are h and g H∗ ( , α, , ) n+p 1 n+p 1 . { − } { − } 17 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Proof. Suppose

X f z ∞ x h z y g z ( ) = n+p 1 n+p 1( ) + n+p 1 n+p 1( ) − − − − n=1 X X p α A B ∞ x y zp ∞ ( )( ) x zn+p 1 = ( n+p 1 + n+p 1) − − n+p 1 − − − Φn p, α, A, B − n=1 − n=2 ( ) X∞ (p α)(A B) y zn+p 1. + − − n+p 1 − Ψn p, α, A, B − n=1 ( ) Then

X∞ X∞ p A B a p A B b Φn( , α, , ) n+p 1 + Ψn( , α, , ) n+p 1 − − n=2 n=1 X X p A B ∞ x p A B ∞ y = ( α)( ) n+p 1 + ( α)( ) n+p 1 − − − − n=2 − − n=1 = (p α)(A B)(1 xp) (p α)(A B) − − − ≤ − − and so f TSH∗ (p, α, A, B). Conversely, if f TSH∗ (p, α, A, B), then ∈ ∈ p α A B p α A B a ( )( ) and b ( )( ). n+p 1 − − n+p 1 − − | − | ≤ Φn(p, α, A, B) − ≤ Ψn(p, α, A, B) Set Φ p, α, A, B x n( ) a n n+p 1 = n+p 1 ( = 2, 3, ...) − (p α)(A B)| − | and − − Ψ p, α, A, B y n( ) b n n+p 1 = n+p 1 ( = 1, 2, ...). − (p α)(A B)| − | Then note by Theorem 2.3, − −

x n y n 0 n+p 1 1( = 2, 3, ...) and 0 n+p 1 1 ( = 1, 2, ...). ≤ − ≤ ≤ − ≤ We deﬁne X X x ∞ x ∞ y p = 1 n+p 1 n+p 1 − − − n=2 − n=1 and note that by Theorem 2.3, xp 0. Consequently, we obtain ≥ X f z ∞ x h z y g z ( ) = n+p 1 n+p 1( ) + n+p 1 n+p 1( ) − − − − n=1 as required.

Now we show that TSH∗ (p, α, A, B) is closed under convex combinations of its mem- bers.

18 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Theorem 7 The class TSH∗ (p, α, A, B) is closed under convex combination.

Proof. For k = 1, 2, 3, ... let fk TSH∗ (p, α, A, B), where fk is given by ∈

p X∞ n p X∞ f z z a z + 1 b zn+p 1. k( ) = k,n+p 1 − + k,n+p 1 − − − − n=2 | | n=1 | |

Then by (79),

X ∞ p A B a p A B b p A B Φn( , α, , ) k,n+p 1 + Ψn( , α, , ) k,n+p 1 2( α)( ) . (33) − − n=1 | | | | ≤ − − P For ∞k 1 tk = 1, 0 tk 1, the convex combination of fk may be written as = ≤ ≤ X X X ∞ t f z zp ∞ ∞ t a zn+p 1 k k( ) = k k,n+p 1 − − k=1 − n=2 k=1 | | X∞ X∞ t b zn+p 1. + k k,n+p 1 − − n=1 k=1 | |

Then by (85),

X X X ∞ p A B ∞ t a p A B ∞ t b Φn( , α, , ) k k,n+p 1 + Ψn( , α, , ) k k,n+p 1 − − n=1 k=1 | | k=1 | | X X ∞ t ∞ p A B a p A B b = k Φn( , α, , ) k,n+p 1 + Ψn( , α, , ) k,n+p 1 − − k=1 n=1 | | | |

X∞ 2(p α)(A B) tk = 2(p α)(A B). ≤ − − k=1 − − P This is the condition required by (79) and so ∞k 1 tk fk(z) TSH∗ (p, α, A, B). = ∈

19 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

References

[1] Ahuja O.P.,Jahangiri J.M., Multivalent harmonic starlike functions, Ann. Univ. Mariae Cruie Sklod. Sec. A., 55(1), 2001, 1-13. [2] Ahuja O.P., Jahangiri J.M., On a linear combination of classes of multivalently harmonic functions, Kyungpook Math. J., 42(1), 2002, 61-70. [3] Clunie J., Sheil-Small T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 9, 1984, 3-25. [4] Dziok J., Jahangiri J.M., Silverman H., Harmonic functions with varying coefﬁcients, Journal of Inequalities and Applications, 139, 2016, 1-12. [5] Jahangiri J.M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235, 1999, 470-477. [6] Jahangiri J.M., Seker B., Sumer Eker S., Salagean-type harmonic multivalent functions, Acta Univ. Apulensis, 18, 2016, 233-244. [7] Yasar E., Yalcin S., On a new subclass of Ruscheweyh-type harmonic multivalent functions, Journal of Inequalities and Applications, 271, 2013, 1-15. [8] Yasar E., Yalcin S., Partial sums of starlike harmonic multivalent functions, Afrika Mathematica, 26(1-2), 2015, 53-63. [9] Yasar E., Yalcin S., Neighborhoods of a new class of harmonic multivalent functions, Comput. Math. Appl., 62(1), 2011, 462-473.

20 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

A New Local Smoothing Technique for Non-smooth Functions Nurullah YILMAZ Suleyman Demirel University, TURKEY

(Received: 12.09.2019, Accepted: 08.11.2019, Published Online: 17.12.2019)

Abstract. In this study, we propose a new local smoothing technique for some subclasses of the non-smooth functions. We introduce useful properties of this new smoothing technique. Finally, the application of the technique is illustrated numerically on non-smooth test problems for global optimization.

Keywords: Non-smooth analysis, Optimization, Smoothing technique. Introduction The non-smooth problems have been arisen in many practical problems of engineering, finance, medical and other sciences. They have been transformed into well-known optimization problems such as regularization, eigenvalue, min-max, min-sum-min problems [1-6]. The methods those are used in solving such optimization problems are generally based on gradient and smoothness of the problem. Since the non-smooth functions are not completely smooth on the domain, most of the efficient gradient- based approaches become unusable in their optimization process. To overcome these deficiencies, smoothing techniques have been proposed for some subclasses of non- smooth functions. The smoothing is employed by modifying the objective function or approximating the objective function by smooth functions. The smoothing studies start with the Bert- sekas’s paper [7] and it rises its importance by proposing different types of efficient smoothing techniques [8-11]. In recent years, it protects its popularity for solving the non-smooth optimization problems [12,13]. In fact, the smoothing approach is generally used in min-max, l1 minimization problems and exact penalty methods for constrained optimization [14-16− ]. In this study, we propose a new smoothing approach for non-smooth functions. Moreover, we apply this new smoothing technique on some numerical examples to illustrate the efficiency of it. Basic Definitions and Preliminaries The continuously differentiability of the functions provide an important advantageous in determining of the conditions for optimality. The differentiability of the given objective function is an important foundation for optimization process. The smoothing techniques make the function differentiable then, the rich theory for smooth problems is available for the non-smooth problems. The smoothing function is defined as follows:

21 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Let f n be a continuous function. The function f˜ n Deﬁnition 1 (4) : R R : R R+ R ˜ n is called a smoothing function→ of f (x), if f ( , β) is continuously differentiable× in R→for n any ﬁxed β, and for any x R , · ∈ lim f˜ z, β f x . z x,β 0 ( ) = ( ) → → The smoothing studies starts with the smoothing of the following “max” function:

q(t) = max t, 0 { } for t R. The first smoothing function (defined by Bertsekas [7]) is the following: ∈ 2  t (1 y) , (1 y) t,  −2c −c − 1 2 y ≤ (1 y) q˜(t, y, c) = y t + 2 ct , −c t −c ,  y2 ≤ y ≤ −2c , t −c , ≤ where y and c parameters with 0 < y < 1 and c > 0. One of the useful smoothing approaches was proposed by Zang [8] and sometimes it is called local smoothing. This smoothing approach is based on the smoothing of the objective function in a neighborhood of the kink points. The smoothing function for the function q is the following: § q(t), t β, q˜(β, t) = b(β, t), |t| ≥ β, | | ≤ 1 2 1 1 where β > 0 and b(β, t) = 4β t + 2 t + 4 β. The max function and its smoothing functions proposed by Bertsekas [7] and Zang [8] are illustrated in Fig. 2 (a) and (b). These smoothing approaches are used for different types of problems such as min- max-min problems, exact penalty methods, regularization problems and etc [4,17]. Moreover making some modification on the function q˜ some important smoothing approach is proposed for non-Lipschitz problems by Chen [18]. In the first smoothing approach, the smoothing function take the same value at the kink point. This is the advantageous side of this smoothing approach but in the remain part the smoothing function move away from the original function. In the second smoothing approach, the original function is changed by smooth function only in a neighborhood of kink point and this is the advantageous of this smoothing approach but it does not take the same value at the kink point. In fact, both approaches have advantageous and disadvantageous. Moreover, these smoothing techniques do not use for non-Lipschitz functions without modifications. A New Smoothing Technique In this section, we present a new smoothing approach for non-smooth problems in which their non-smoothness depend on presence of the "max” or " ” operators. The | · | 22 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

3 max function max function Zang smoothing Bertsekas smoothing 2 2.5

2 1.5

1.5 1

y−axis y−axis 1 0.5 0.5

0 0

−0.5 −0.5 −3 −2 −1 0 1 2 −3 −2 −1 0 1 2 3 x−axis x−axis (a) (b)

Figure 2. (a) The green and solid one is the graph q(t) = max t, 0 , the blue and dashed one is the graph of Bertsekas type smoothing (b) the green{ and} solid one is the graph q(t), the blue and dashed one is the graph of Zang type smoothing approach. smoothing method is constructed on smoothing out the kink points in the β neighborhood. The smoothing is realized by substituting "max” operator for ν(β, t) in that neighborhood. Namely, for a function q(t) = max t, 0 , the smoothing function is deﬁned by { } § q(t), t β, q˜(t, β) = (34) ν(t, β), |t|< ≥ β. | | where ν(t, β) continuously differentiable with respect to t and it is expressed by the following formula: 2 2βq (t) ν(t, β) = , β 2 + q2(t) where β > 0 real number. This smoothing approach have the following properties.

Theorem 8 Let β > 0, then we have

i. The function q˜(t, β) continuously differentiable,

ii. t R, q(t) q˜(t, β), ∀ ∈ ≥ iii. q˜(t, β) approaches q(t), when β 0. → Proof.

i. Since, q(t) is differentiable outside of the β neighborhood of the kink point and ν(t, β) is differentiable inside of the β neighborhood of the kink point, it is only

23 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

need to check the differentiability of q˜(t, β) at the end points of the β, for β > 0. For t = β − lim q˜0 t, β lim q0 t 0, t β ( ) = t β ( ) = →− − →− − and

lim q˜0 (t, β) = lim ν0 (t) t β+ t β → →− − 3 4β q(t)q0 (t) = lim t β β 2 q2 t 2 →− − ( + ( )) = 0

and for t = β lim q˜0 t, β lim q0 t 1, t β ( ) = t β ( ) = →− − →− − and

lim q˜0 (t, β) = lim ν0 (t) t β+ t β → →− − 3 4β q(t)q0 (t) = lim t β β 2 q2 t 2 →− − ( + ( )) = 1.

ii. Since outside of the β neighborhood of the kink point of the functions q˜(x, β) and q(t) are equal, we only check the for only the case β t β − ≤ ≤ q t q t β 2 q t q˜ t, β ( )( ( ) ) 0. ( ) ( ) = 2 2− − β + q (t) ≥

iii. By considering the above inequality, we have

q(t) q˜(t, β) < β, − when β 0, the result is obtained. →

Corollary 2 A point xk∗ is an optimal solution of q if and only if it is an optimal solution of q.˜ Moreover,

q(x ∗) = q˜(x ∗, β), for all β > 0.

Proof. The proof is straightforward. Our smoothing approach is controlled by the parameter β and it has a same value with the original function at the optimal point (see Fig. 3 (a) and (b)).

24 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

2 2.5

1.5 2

1 1.5 y−axis y−axis

0.5 1

0 0.5

−0.5 0 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 x−axis x−axis (a) (b)

Figure 3. (a) The green and solid one is the graph q(x) = max x, 0 ,the blue and dashed one is the graph of q˜(x, 1). (b) The green and solid one is the{ graph} h(x) = x , ˜ the blue and dashed one is the graph of h(x, 1). | |

n Remark 1 Suppose that g is a smooth function defined on R to R. If t replaced by g(x) in the function q(t) defined as above, the smoothing function of max g(x), 0 is obtained. Moreover, it is sufficient to consider a smoothing function just for max{ g(x)}, 0 { } since max g1(x), g2(x) = g1(x) + max g2(x) g1(x), 0 for smooth functions g1 and g2. { } { − }

Remark 2 Since min g(x), 0 = max g(x), 0 and f (x) = max g(x), 0 +max g(x), 0 , it is not needed to present{ the} smoothing− {− results for} the operators| | min{and .} {− } | · |

We can conclude that, the above smoothing process is applicable for l1 minimization and some of exact penalty approach. Thus, it is possible to consider the above smoothing approach as another smoothing method for non-smooth, Lipschitzian problems. Numerical Examples In this section, we present numerical results on some test problems. We apply our smoothing technique to smooth out these non-smooth problems and we apply the Es- thetic Delving Algorithm (EDA) proposed in [19] and we compare our results with the results obtained from the Global Descent Algorithm (GDA) proposed in [20]. Numeri- cal results is obtained by using the program Matlab R2015A on PC with conﬁguration of Intel Core i3, 8GB RAM. The detailed results are presented in the Table 1 for all problems. For the table we use some symbols in order to abbreviate the expressions. The 0 meaning of these symbols are as k for the problem number x for the starting point x ∗ is the global minimizer point of the corresponding problem and fg∗ for the value of global

25 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications minimum point of the corresponding problem. We consider 3 different non-smooth test problems which are described as follows:

Problem 1 (20)

x 1 x 1 min f (x) = + sin π 1 + + 1 −4 −4 s.t. 10 x 10. − ≤ ≤

The global minimizer of the function is x ∗ = 1 and the global minimum value is f (x ∗) = 1. The graph of the function f is presented in Fig 4 (a) and the graph of the smoothing function of f is presented in 4 (b).

5 5

4.5 4.5

4 4

3.5 3.5

3 3 y−axis y−axis

2.5 2.5

2 2

1.5 1.5

1 1 −10 −5 0 5 10 −10 −5 0 5 10 x−axis x−axis (a) (b)

Figure 4. (a) The graph of the function f (b) The graph of the smoothed version of the function f .

Problem 2 (20)

min f (x) = x 1 (1 + 10 sin(x + 1) ) + 1 s.t. | 10− | x 10.| | − ≤ ≤

The global minimizer of the function is x ∗ = 1 and the global minimum value is f (x ∗) = 1.

Problem 3 (20)

min f (x) = max fi(x) + max f j i=1,2,3 j=4,5,6 s.t. 10 x1, x2 10. − ≤ ≤ 26 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University where

4 2 2 2 x1+x2 f1(x) = x1 + x2, f2(x) = (2 x1) + (2 x2) , f3(x) = 2e− 2 2 − − 2 2 f4(x) = x1 2x1 + x2 4x2 + 4, f5(x) = 2x1 5x1 + x2 2x2 + 4, − − 2 2 − − f6(x) = x1 + 2x2 4x2 + 1. −

The global minimizer of the function is x ∗ = (1, 1) and the global minimum value is f (x ∗) = 2.

Table 1. The numerical results.

EDA GDA

k x0 x ∗g fg∗ Time(sec) xk∗ fk∗ Time(sec) 1 10 1.0000 1.0000 0.1250 1.0000 1.0000 0.1554 2 10 1.0000 1.0000 0.1094 1.0000 1.0000 0.3779 3 (10, 10)(1.0000, 1.0000) 2.0000 1.0469 (10, 10) 2.0000 2.4738

Conclusion We have introduced a new smoothing technique for non-smooth functions. The presented smoothing process can be useful for the problem which contain any of “max, min and operators. Our smoothing technique can be easily controlled by only one parameter.| · | This parameter give a possibility to obtain sensitive approximation to the original non-smooth function. It can be observed from the numerical results that the smoothing is useful in solving global optimization problems of non-smooth functions. By the help of the this new smoothing approach, the global optimization methods ﬁnds same local minimizer and global minimum value as the original objective function. For future work, we plan to generalize the above smoothing technique to solve the non-Lipschitz type problems.

27 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

References

[1] Bagirov A., Hyperbolic smoothing functions for nonsmooth minimization, Optimiza- tion, 62(6), 2013, 759-782. [2] Lewis A.S., Overton M.L., Eigenvalue optimization, Acta Numerica, 5, 1996, 149- 190. [3] Alfonso M.V.,Bioucas-Dias J. M., Figueiredo M.A.T., An augmented Langrangian approach to the constarained optimization formulation of imaging inverse problems, IEEE Trans on Image Process, 20(3), 2011, 681-695. [4] Chen X., Smoothing methods for nonsmooth, nonconvex minimization, Math. Pro- gram., Ser. (B), 134, 2012, 71-99. [5] Taylan P.,Weber G-W., Yerlikaya-Ozkurt F., A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization, TOP,18, 2010, 377-395. [6] Chen X., Zhou W., Convergence of reweighted l1 minimization algorithm for l2 lp minimization, Comput. Optim. Appl., 59, 2014, 47-61. − [7] Bertsekas D., Nondifferentiable optimization via approximation, Mathematical Pro- gramming Study, 3, 1975, 1-25. [8] Zang I., A smooting out technique for min-max optimization, Math. Program, 19, 1980, 61-77. [9] Ben-Tal A., Teboule M., Smoothing technique for nondifferentiable optimization problems, lecture notes in mathematics, 1405, Springer-Verlag, Heidelberg, 1989, 1-11. [10] Chen C., Mangasarian O.L., A Class of Smoothing Functions for Nonlinear and Mixed Complementarity Problem, Comput. Optim. Appl., 5, 1996, 97-138. [11] Xavier A.E., The hyperbolic smoothing clustering method, Patt. Recog., 43, 2010, 731-737. [12] Sahiner A., Yilmaz N., Ibrahem S.A., Smoothing approximations to non-smooth functions, Journal of Multidisciplinary Modeling and Optimization, 1(2), 2018, 69-74. [13] Yilmaz N., Sahiner A., New smoothing approximations to piecewise smooth functions and applications, Numer. Func. Anal. Opt., 40(5), 2019, 523-534. [14] Bagirov A.M., Sultanova N., Nuamiat A. Al and Taheri S., Solving minimax problems: local smoothing versus global smoothing, Numerical Analysis and Optimization, Springer Proceedings in Mathematics an Statistic, 235, 2018, 23-43. [15] Lian S.J., Smoothing approximation to l1 exact penalty for inequality constrained optimization, Appl. Math. Comput., 219, 2012, 3113-3121. [16] Grossmann C., Smoothing techniques for exact penalty function methods, Contem- porary Mathematics, In book: Panorama of Mathematics: Pure and Applied, 658, 2016, 249-265. [17] Chen X., Ng M.K., Zhang C., Non-lipschitz lp-regularization and box constrained model for image processing, IEEE Trans. on Image Process., 21(12), 2012, 4709-4721. [18] Chen X., Niu L., Yuan Y., Optimality conditions and smoothing trust region method

28 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University for nonlipschitz optimization, SIAM J. Optim., 23 (3), 2013, 1528-1552. [19] Sahiner A., Yilmaz N., Kapusuz G., A descent global optimization method based on smoothing techniques via Bezier curve, Carpathian J. Math., 33(3), 2017, 373-380. [20] Ketﬁ-Cherif A, Ziadi A., Global descent method for constrained continuous global optimization, Appl Math Comput, 2014, 244:209-221.

29 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

The (p, q)-Lucas Polynomial Coefﬁcient Estimates of a Bi-univalent Function Class with Respect to Symmetric Points Sahsene ALTINKAYA1, Sibel YALCIN TOKGOZ1 1Bursa Uludag University, TURKEY

(Received: 31.07.2019, Accepted: 28.09.2019, Published Online: 17.12.2019)

Abstract. The main idea of this current paper stems from the work of Lee and Asci [8]. By using the (p, q)-Lucas polynomials, we aim to build a bridge between the theory of geometric functions and special functions, which are usually considered as different ﬁelds.

Keywords: Bi-univalent functions, Coefﬁcient bounds, (p, q)-Lucas polynomials.

Introduction, preliminaries and known results The polynomials deﬁned recursively over the integers, such as the Dickson polynomials, Chebychev polynomials, Fibonacci polynomials and Lucas polynomials, have been extensively studied. Most of these polynomials share numerous interesting properties. They have been also found to be topics of interest in many different areas of pure and applied science; see, for example, [9, 10, 13] and [18]. The classical Lucas polynomials Ln(x) studied by M. Bicknell in 1970 are introduced as follows

Ln(x) = x Ln 1(x) + Ln 2(x)(n 2), − − ≥ with the initial condition

L0(x) = 2,

L1(x) = x. Since the above classical Lucas polynomials appeared, some authors have explored their different extensions. For example, the (p, q)-Lucas polynomials with some properties introduced by Lee and A¸sçı [8] as follows.

Definition 2 (see [8]) Let p(x) and q(x) be polynomials with real coefficients. The (p, q)- Lucas polynomials Lp,q,n(x) are defined by the recurrence relation

Lp,q,n(x) = p(x)Lp,q,n 1(x) + q(x)Lp,q,n 2(x)(n 2), − − ≥ 30 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University from which the ﬁrst few Lucas polynomials can be found as

Lp,q,0(x) = 2,

Lp,q,1(x) = p(x),

2 (35) Lp,q,2(x) = p (x) + 2q(x),

3 Lp,q,3(x) = p (x) + 3p(x)q(x), . .

For the special cases of p(x) and q(x), we can get the polynomials given in Table 1.

Table 1: Special cases of the Lp,q,n(x) with given initial conditions are given. p(x) q(x) Lp,q,n(x) x 1 Lucas polynomials Ln(x) 2x 1 Pell-Lucas polynomials Dn(x) 1 2x Jacobsthal-Lucas polynomials jn(x) 3x -2 Fermat-Lucas polynomials fn(x) 2x -1 Chebyshev polynomials ﬁrst kind Tn(x)

Theorem 9 (see 8 ) Let G z be the generating function of the p, q -Lucas poly- [ ] Lp,q,n(x) ( ) ( ) nomial sequence Lp,q,n(x). Then{ }

X∞ n 2 p(x)z G L x (z) = Lp,q,n(x)z = . p,q,n( ) p x− z q x z2 n 0 1 ( ) ( ) { } = − − Let A be the class of functions f of the form

2 3 f (z) = z + a2z + a3z + , (36) ··· which are analytic in the open unit disk ∆ = z C : z < 1 and normalized under the conditions { ∈ | | } f (0) = 0,

f 0(0) = 1. Further, by S we shall denote the class of all functions in A which are univalent in ∆. Here, we recall some deﬁnitions and concepts of classes of analytic functions. By

S∗ (φ) and C (φ) we indicate the following classes of functions § ª z f 0 (z) S∗ (φ) = f : f A and φ (z) , z ∆ ∈ f (z) ≺ ∈ 31 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications and § z f z ª C φ f : f A and 1 00 ( ) φ z , z ∆ . ( ) = + f z ( ) ∈ 0 ( ) ≺ ∈

The classes S∗ (φ) and C (φ) are the extensions of a classical sets of a starlike and convex functions and in a such form were deﬁned and studied by Ma and Minda [10, 11]. In [15], Sakaguchi introduced the class SS∗ of starlike functions with respect to sym- z f z metric points in ∆, consisting of functions f A that satisfy the condition 0 ( ) > ∈ ℜ f (z) f ( z) 0, z ∆. Similarly, in [19], Wang et al. introduced the class CS of convex functions− − with respect∈ to symmetric points in ∆, consisting of functions f A that satisfy the z f z 0 ∈ condition ( 0( )) > 0, z ∆. In the style of Ma and Minda, Ravichandran (see f (z)+f ( z) ℜ 0 0 − ∈ [14]) deﬁned the class SS∗ (φ) and CS (φ) , as follows: A function f A is in the class SS∗ (φ) if ∈ 2z f 0 (z) φ (z)(z ∆) f (z) f ( z) ≺ ∈ − − and in the class CS (φ) if 2 z f z ( 0 ( ))0 φ z z ∆ . f z f z ( )( ) 0 ( ) + 0 ( ) ≺ ∈ − With a view to recalling the principle of subordination between analytic functions, let the functions f and g be analytic in ∆. Given functions f , g A, f is subordinate to g if there exists a Schwarz function w Λ, where ∈ ∈ Λ = w : w (0) = 0, w (z) < 1, z ∆ , { | | ∈ } such that f (z) = g (w (z)) (z ∆) . ∈ We denote this subordination by

f g or f (z) g (z)(z ∆) . ≺ ≺ ∈ In particular, if the function g is univalent in ∆, the above subordination is equivalent to f (0) = g(0), f (∆) g(∆). ⊂ According to the Koebe-One Quarter Theorem [5], it ensures that the image of ∆ under every univalent function f A contains a disc of radius 1/4. Thus every univalent ∈ 1 1 1 function f A has an inverse f − satisfying f − (f (z)) = z and f f − (w) = w ∈ 1 w < r0 (f ) , r0 (f ) 4 , where | | ≥ 1 2 2 3 3 4 f − (w) = w a2w + 2a2 a3 w 5a2 5a2a3 + a4 w + . (37) − − − − ··· 32 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

1 A function f A is said to be bi-univalent in ∆ if both f and f − are univalent in ∆. Let σ denote the∈ class of bi-univalent functions in ∆ given by (36). For a brief history and interesting examples in the class σ, see [16] (see also[1-4, 7, 12, 17, 20]). We want to remark explicitly that, in our article, by using the Lp,q,n(x), our methodology builds a bridge, to our knowledge not previously well known, between the theory of geometric functions and that of special functions, which are usually considered as very different ﬁelds.

Deﬁnition 3 A function f Σ is said to be in the class ∈ £S,Σ (α; x)(0 α 1, z, w ∆) ≤ ≤ ∈ if the following subordinations are satisﬁed: α 1 α 2z f 0 (z) 2 (z f 0 (z))0 − G L x (z) 1 f z f z f z f z p,q,n( ) ( ) ( ) 0 ( ) + 0 ( ) ≺ − − − − { } and α 1 α 2wg0 (w) 2 (wg0 (w))0 − G L x (w) 1 g w g w g w g w p,q,n( ) ( ) ( ) 0 ( ) + 0 ( ) ≺ − − − − { } where the function g is given by (37).

Example 1 For α = 1, a function f Σ is said to be in the class ∈ SΣ∗ (x)(z, w ∆) ∈ if the following conditions are satisﬁed: 2z f z 0 ( ) G z 1 Lp,q,n(x) ( ) f (z) f ( z) ≺ − − − { } 2wg0 (w) G L x (w) 1 g w g w p,q,n( ) ( ) ( ) ≺ { } − where the function g is given by (37).− −

Example 2 For α = 0, a function f Σ is said to be in the class ∈ CΣ (x)(z, w ∆) ∈ if the following conditions are satisﬁed:

2 (z f 0 (z))0 G L x (z) 1 f z f z p,q,n( ) 0 ( ) + 0 ( ) ≺ − − { } 2 (wg0 (w))0 G L x (w) 1 g w g w p,q,n( ) 0 ( ) + 0 ( ) ≺ { } − where the function g is given by (37). −

33 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Coefﬁcient Estimates

We begin this section by finding the estimates on the coefficients a2 and a3 for func- | | | | tions in the class £S,Σ (α; x) proposed by Definition 1.2.

Theorem 10 Let f given by (36) be in the class £S,Σ (α; x) . Then p p(x) p(x) a2 q | | | | 2 | | ≤ 2 (5α α2 5) p2(x) 4 (2 α) q(x) − − − − and p2 x p x a ( ) ( ) . 3 2 + | | | | ≤ 4 (2 α) 2 (3 2α) − − Proof. Let f £S,Σ (α; x) . From Deﬁnition 1.2, for some analytic functions Φ, Ψ such that Φ(0) = Ψ(0)∈ = 0 and Φ(z) < 1, Ψ(w) < 1 for all z, w ∆, we can write | | | | ∈ α 1 α 2z f 0 (z) 2 (z f 0 (z))0 − = G L x (Φ(z)) 1 f z f z f z f z p,q,n( ) ( ) ( ) 0 ( ) + 0 ( ) − − − − { } and α 1 α 2wg0 (w) 2 (wg0 (w))0 − = G L x (Ψ(w)) 1, g w g w g w g w p,q,n( ) ( ) ( ) 0 ( ) + 0 ( ) { } − or equivalently − − −

α 1 α 2z f z 2 z f z 0 0( ) ( 0( )) − f (z) f ( z) f (z)+f ( z) − − 0 0 −

2 = 1 + Lp,q,0(x) + Lp,q,1(x)Φ(z) + Lp,q,2(x)Φ (z) + , − ··· (38) α 1 α 2wg w 2 wg w 0 0( ) ( 0( )) − g(w) g( w) g (w)+g ( w) − − 0 0 −

2 = 1 + Lp,q,0(x) + Lp,q,1(x)Ψ(w) + Lp,q,2(x)Ψ (w) + . − ··· From the equalities in (38), we obtain that

α 1 α 2z f z 2 z f z 0 0( ) ( 0( )) − f (z) f ( z) f (z)+f ( z) − − 0 0 − (39)

2 2 = 1 + Lp,q,1(x)t1z + Lp,q,1(x)t2 + Lp,q,2(x)t1 z + ··· and α 1 α 2wg w 2 wg w 0 0( ) ( 0( )) − g(w) g( w) g (w)+g ( w) − − 0 0 − (40)

2 2 = 1 + Lp,q,1(x)s1w + Lp,q,1(x)s2 + Lp,q,2(x)s1 w + . ··· 34 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Additionally, it is fairly well-known that if 2 3 Φ(z) = t1z + t2z + t3z + < 1 (z ∆) | | ··· ∈ and 2 3 Ψ(w) = s1w + s2w + s3w + < 1 (w ∆), | | ··· ∈ then tk 1 and sk 1 (k N). | | ≤ | | ≤ ∈ Thus, upon comparing the corresponding coefﬁcients in (39) and (40), we have

2 (2 α) a2 = Lp,q,1(x)t1, (41) −

2 2 2 (3 2α) a3 2α (1 α) a2 = Lp,q,1(x)t2 + Lp,q,2(x)t1, (42) − − −

2 (2 α) a2 = Lp,q,1(x)s1 (43) − − and 2 2 2 2 (3 2α) 2a2 a3 2α (1 α) a2 = Lp,q,1(x)s2 + Lp,q,2(x)s1. (44) − − − − From the equations (41) and (43), we can easily see that

t1 = s1, (45) −

2 2 2 2 2 8 (2 α) a2 = Lp,q,1(x) t1 + s1 . (46) − If we add (42) to (44), we get 2 2 2 2 4 3 3α + α a2 = Lp,q,1(x)(t2 + s2) + Lp,q,2(x) t1 + s1 . (47) − By using (46) in the equality (47), we have

2 2 2 2 3 4 3 3α + α Lp,q,1(x) 2 (2 α) Lp,q,2(x) a2 = Lp,q,1(x)(t2 + s2) . (48) − − − which gives p p(x) p(x) a2 . q | | | | 2 | | ≤ 2 (5α α2 5) p2(x) 4 (2 α) q(x) − − − − Moreover, if we subtract (44) from (42), we obtain 2 2 2 4 (3 2α)(a3 a2) = Lp,q,1(x)(t2 s2) + Lp,q,2(x) t1 s1 . (49) − − − − Then, in view of (45) and (46), also (49) L2 x L x p,q,1( ) 2 2 p,q,1( ) a3 = 2 t1 + s1 + (t2 s2) . 8 (2 α) 4 (3 2α) − − − Then, with the help of (35), we ﬁnally deduce p2 x p x a ( ) ( ) . 3 2 + | | | | ≤ 4 (2 α) 2 (3 2α) − − 35 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Corollary 4 Let f given by (36) be in the class CΣ (x) . Then p p(x) p(x) a2 p | | | | | | ≤ 2 5p2(x) + 16q(x) | | and 2 p (x) p(x) a3 + | |. | | ≤ 16 6 Fekete-Szego¨ Problem The classical Fekete-Szegö inequality, presented by means of Loewner’s method, for the coefﬁcients of f S is ∈ 2 a3 ϑa2 1 + 2 exp( 2ϑ/(1 ϑ)) for ϑ [0, 1) . − ≤ − − ∈ 2 As ϑ 1−, we have the elementary inequality a3 a2 1. Moreover, the coefﬁcient functional→ − ≤ 2 bϑ(f ) = a3 ϑa2 − on the normalized analytic functions f in the unit disk ∆ plays an important role in function theory. The problem of maximizing the absolute value of the functional bϑ(f ) is called the Fekete-Szegö problem, see [6]. In this section, we aim to provide Fekete-Szegö inequalities for functions in the class £S,Σ (α; x). These inequalities are given in the following theorem.

Theorem 11 Let f given by (36) be in the class £S,Σ (α; x) and ϑ R. Then ∈ 2 a3 ϑa2 −  p(x) 1 2 2 q(x)  , ϑ 1 5α α 5 4 2 α 2  2(|3 2α| ) 3 2α ( ) p (x)  − | − | ≤ − − − − − . 3 ≤  1 ϑ p (x) 1 2 2 q(x)  | − | 2 , ϑ 1 5α α 5 4 2 α 2  2 (5α α2 5)p2(|x) 4(|2 α) q(x) 3 2α ( ) p (x) | − | ≥ − − − − − | − − − − | 36 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Proof. From (48) and (49)

3 L x 1 ϑ t2 s2 2 p,q,1( )( )( + ) a3 ϑa2 = − 2 2 2 − 4 (3 3α + α ) Lp,q,1(x) 2 (2 α) Lp,q,2(x) − − −

Lp,q,1(x)(t2 s2) + − 4 (3 2α) − Lp,q,1(x) 1 1 = h (ϑ) + t2 + h (ϑ) s2 4 3 2α − 3 2α − − where 2 Lp,q,1(x)(1 ϑ) h ϑ − . ( ) = 2 2 2 (3 3α + α ) Lp,q,1(x) 2 (2 α) Lp,q,2(x) − − − Then, in view of (35), we conclude that  p(x) 1  , 0 h ϑ  2 |3 2|α ( ) 3 2α 2  ( ) ≤ | | ≤ a3 ϑa2 − − . − ≤  p(x) h (ϑ) 1  | | | |, h (ϑ) 2 | | ≥ 3 2α − Corollary 5 Let f given by (36) be in the class SΣ∗ (x) and ϑ R. Then ∈ 2 p(x) a3 ϑa2 | |. − ≤ 2 Corollary 6 Let f given by (36) be in the class CΣ (x) and ϑ R. Then ∈ 2 p(x) a3 ϑa2 . − ≤ 6 If we choose ϑ = 1, we get the next corollaries.

Corollary 7 If f £S,Σ (α; x) , then ∈ 2 p(x) a3 a2 | | . − ≤ 2 (3 2α) − Corollary 8 If f SΣ∗ (x) , then ∈ 2 p(x) a3 a2 | |. − ≤ 2 Corollary 9 If f CΣ (x) , then ∈ 2 p(x) a3 a2 | |. − ≤ 6

37 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

References

[1] Altinkaya S., Yalcin S., Faber polynomial coefficient bounds for f subclass of bi-univalent functions, C. R. Acad. Sci. Paris, Ser. I, 353(12), 2015, 1075-1080. [2] Altinkaya S., Yalcin S., Faber polynomial coefficient estimates for bi-univalent functions of complex order based on subordinate conditions involving of the Jackson (p, q)- derivative, Miskolc Mathematical Notes, 17(2), 2017, 1075-1080. [3] Brannan D.A., Clunie J.G., Aspects of contemporary Complex Analysis, Proceedings of the NATO Advanced Study Institute Held at University of Durham, 1979. [4] Brannan D.A., Taha T.S., On some classes of bi-univalent functions, Studia Universi- tatis Babes-Bolyai Mathematica, 31(2), 1986, 70-77. [5] Duren P.L., Univalent functions, Grundlehren der Mathematischen Wissenschaften, Springer, 1983. [6] Fekete M., Szego G., Eine Bemerkung Uber ungerade Schlichte funktionen, J. London Math. Soc., 1-8(2), 1983, 85-89. [7] Lewin M., On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18(1), 1967, 63-68. [8] Lee G.Y., Asci M., Some properties of the (p, q)-Fibonacci and (p, q)-Lucas polynomials, Journal of Applied Mathematics, 2012, 1-18. [9] Lupas A., A guide Of Fibonacciand Lucas polynomials, Octagon Mathematics Maga- zine, 7, 1999, 2-12. [10] Ma R., Zhang W., Several identities involving the Fibonacci numbers and Lucas numbers, Fibonacci Q, 45, 2007, 164-170. [11] Ma W.C.,MindaD., A unified treatment of some special classes of univalent functions, Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge, MA, 1994. [12] Netahyahu E., The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in z < 1, Archive for Rational Mechanics and Analysis, 32, 1969, 100-112. | | [13] Filipponi P.,Horadam A.F., Derivative sequences of Fibonacci and Lucas polynomials, Applications of Fibonacci Numbers, 4, 1991, 99-108. [14] Ravichandran V., Starlike and convex functions with respect to conjugatepoints, Acta Math. Acad. Paedagog. Nyireg., 20, 2004, 31-37. [15] Sakaguchi K., On a certain univalent mapping, J. Math. Soc. Japan, 11, 1959, 72-75. [16] Srivastava H.M., Mishra A.K., Gochhayat P., Certain subclasses of analytic and bi- univalent functions, Applied Mathematics Letters, 23, 2010, 1188-1192. [17] Srivastava H.M., Murugusundaramoorthy G., Magesh N., Certain subclasses of bi- univalent functions associated with the Hohlov Operator, Applied Mathematics Letters, 1, 2013, 67-73. [18] Wang T., Zhang W., Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roum., 55, 2012, 95-103.

38 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

[19] Wang Z.G. Gao C.Y., Yuan S.M., On certain subclasses of close-to-convex and quasi- convex functions with respect to k-symmetric points, J. Math. Anal. Appl., 322, 2006, 97-106. [20] Zireh A., Adegani E.A., Bulut S., Faber polynomial coefﬁcient estimatesfor a compre- hensive subclass of analytic bi-univalent functions deﬁned by subordination, Bull. Belg. Math. Soc. Simon Stevin, 23, 2016, 487-504.

39 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Approximation of the Reachable Sets of an SEIR Control System and Comparison with Optimal Control Solution A. Serdar NAZLIPINAR1, Barbaros BASTURK1 1Dumlupinar University, TURKEY

(Received: 28.08.2019, Accepted: 19.11.2019, Published Online: 17.12.2019)

Abstract. In this paper the controllable spread of Susceptible-Exposed-Infectives- Recovered (SEIR) epidemiological model which is described by the 4 dimensional nonlinear ordinary differential equations system, is considered. Vaccination− and treatment are accepted as control parameters of the system. It is assumed that vaccination and treatment stocks are limited. Reachable sets of the system are approximately calculated with some initial conditions and for different control stocks. Graphical simulations are presented and results are compared with the solutions of an apropriate optimal control problem.

Keywords: Attainable set, Control system, Integral constraint, SEIR model.

Introduction There are several mathematical models describing the spread of infectious diseases and these models have been applied for studying of many diseases [2, 3, 4, 5, 6]. One of primitive models of epidemiology is the model Kermack-McKendrick published in 1927 [1]. Since then, models have emerged for several epidemic disease. One of these models is the SEIR models. The SEIR model is used in epidemiology to com- pute the amount of susceptible(S), exposed to disease but not yet infectious(E), infectious(I),recovered(R) people in a population. SEIR models might represent several infectious diseases, for example chicken pox, ﬂu, measles etc. There are also some examples on the use of SEIR model for controlling of infectious diseases. These studies are generally related with optimal control theory and their aims are to ﬁnd optimal control strategies to effect the spread of diseases [7, 8, 9]. Consider the SEIR system

 I t S t · ( ) ( )  S(t) = µN(t) λ N t µ∗S(t),  ( )  I t S t − −  · ( ) ( ) E(t) = λ N t (" + µ∗)E(t), ( ) − (50) I· t E t I t ,  ( ) = " ( ) (γ + µ∗) ( )  −  · R(t) = γI(t) µ∗R(t) , − 40 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

where S(t) : the number of susceptible individuals E(t) : the number of individuals exposed to the disease but not yet infected I(t) : the number of infected individuals R(t) : the number of recovered individuals who are recovered from disease with permanent immunity to reinfection. N(t) : the total population i.e., N(t) = S(t) + E(t) + I(t) + R(t). Moreover, in the model λ : the number of contacts each individual has made with other individuals in the population per unit of time, " :the fraction of citizens who have natural immunity against to infection, γ: the recover rate of infected person per unit of time,

µ∗ : natural death rate, µ : birth rate are all positive constants. S E I We can use fractions instead of numbers by using new varibles s = N , e = N , i = N , R r = N then the population is now normalised i.e s + e + i + r = 1. Also, it notices that the compartment r = r(t) does not appear in the ﬁrst three equations of (50).It can be determined r from r = 1 s e i or from the last equation of the system (50). Therefore, we can consider the− sub-system− − given by fractions,

  s· (t) = µ λi(t)s(t) µs(t),  − − e· (t) = λi(t)s(t) (" + µ)e(t), (51)  −  i· (t) = "e(t) (γ + µ)i(t). − The feasible region of (51) is 3 Ω = x = (s, e, i) R 0 s + e + i 1 , (52) ∈ +| ≤ ≤ which can be verified positively invariant(i.e. for given initial point x 3 , the trajec- R+ tory lies in Ω). Hence, the system is both mathematically and epidemiologially∈ well- posed. Thus, we can restrict our attention to the region Ω. If the system (51) is solved numerically with high contact rate and low recover rate then the evaluation of the system is shown in the figure below: It is noticed in the figure that the fraction of the infected individuals are getting increased. In this circumstances, spreading of the disease must be controlled for the sake of society. Let us introduce two available exterior effort to control the spread of disease (u1(t) and u2(t)) to the system (51)

  s· (t) = µ λi(t)s(t) µs(t) u1(t) s(t),  − − − | | e· (t) = λi(t)s(t) (" + µ)e(t), (53) −  ·  i(t) = "e(t) (γ + µ)i(t) u2(t) i(t), − − | | 41 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

No control effect with initial values S(0)=0.7, E(0)=0.2, I(0)=0.1 1 S(t) E(t) 0.9 I(t) 0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10

Figure 5. (s0, e0, i0) = (0.7, 0.2, 0.1) µ = 0.2; γ = 0.03; " = 0.53; λ = 0.9

Here, u1(t) is symbolized the vaccination of the susceptible individuals and u2(t) is the treatment of the infected ones at the instant of time t. 2 We assume that the control function u( ) = (u1( ), u2( )) : [0, 1] R satisﬁes the inequality: · · · →

1 1 Z Z 1 2 2 2 2 2 u(t) d t = u1 (t) + u2 (t) d t µ0. (54) 0 k k 0 ≤

This inequality means that the whole stock of vaccines and medicines which can be consumed to effect the spread of disease, is µ0. We want to calculate the all possible points that the system can be reached at any given terminal time under the effect of this controls. The Algorithm For the Approximate Calculation of Reach- able Set In this section, ﬁrstly reachable sets notion is given for a general control system with integral constraints on control function. Then, we mention the approximate calculation method for the reachable sets. Detailed information and algorithm about this method is given in [10, 11, 12, 13]. The System and Assumptions

x· (t) = f (t, x(t), u(t)), x(t0) X0 (55) ∈ 42 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

n m where x R is the phase state vector of the system, u R is the control vector, ∈ n ∈ t [t0, θ] is the time and X0 R is a compact set. ∈ ⊂ Let p > 1, µ0 > 0. We deﬁne the set,

1  θ  p Z m p Up = u( ) Lp [t0, θ], R : u( ) p =  u(t) d t µ0 (56) · ∈ k · k k k ≤ t0

where denotes the Euclidean norm. It is assumedk·k that the right hand side of the system (55) satisﬁes the following conditions:

n m n 1.A. The function f ( ) : [t0, θ] R R R is continuous; · × × → n 1.B. For any bounded set D [t0, θ] R there exist constants L1 = L1(D) > 0, ⊂ × L2 = L2(D) > 0 and L3 = L3(D) > 0 such that

f (t, x1, u1) f (t, x2, u2) [L1 + L2 ( u1 + u2 )] x1 x2 k − k ≤ k k k k k − k +L3 u1 u2 k − k m m for any (t, x1) D, (t, x2) D, u1 R and u2 R ; ∈ ∈ ∈ ∈ 1.C. There exists a constant c > 0 such that

f (t, x, u) c(1 + x )(1 + u ) k k ≤ k k k k n m for every (t, x, u) [t0, θ] R R . ∈ × × By the symbol x( ; t0, x0, u( )) we denote the solution of the system (55) with initial · · condition x(t0) = x0, which is generated by the admissible control function u( ). It is obvious that the conditions 1.A, 1.B and 1.C are guaranteed the existence, uniqueness· and the extendability of the solutions over the whole interval [t0, θ]. We deﬁne the sets, X p(t0, X0) = x ( ; t0, x0, u( )) : x0 X0, u( ) Up , · · ∈ · ∈

n X p(t; t0, X0) = x(t) R : x( ) X p(t0, X0) ∈ · ∈ where t [t0, θ]. ∈ The set X p(t; t0, X0) is called the reachable set of the system (55) at the instant of time t.

43 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Figure 6. Reachable set of the system at the instant of times t and θ.

n n The Hausdorff distance between the sets A R and E R is denoted by h(A, E) and is deﬁned as ⊂ ⊂ h (A, E) = max sup d(x, E), sup d(y, A) , x A y E ∈ ∈ where d(x, E) = inf x y : y E . For given σ > 0,{k let − k ∈ }

Sσ = s0, s1, s2,..., sK { } m be a ﬁnite σ-net of unit sphere S = u R : u = 1 . { ∈ k k } Let Γ = t0 < t1 < ... < tN = θ be a uniform partition of the interval [t0, θ], ∆ = t t i{ N } i+1 i ( = 0, 1, . . . 1). − − Γ ∗ = 0 = y0 < y1 < ... < ya = H be a uniform partition of the segment [0, H] and y { y j a } ∆ = j+1 j ( = 0, 1, . . . 1). ∗ − − ¦ Setting, U H u L 0, 1 ; m : u t y s , t t , t , y Γ , s p,∆,∆ ,σ = ( ) p ([ ] R ) ( ) = ji li [ i i+1) ji ∗ li PN 1∗ p p ·©∈ ∈ ∈ ∈ Sσ and ∆ i −0 yj µ0 · = i ≤ H we deﬁne a new control functions set and it is clear that Up,∆,∆ ,σ Up. ∗ ⊂ Since Γ ∗ = 0 = y0 < y1 < ... < ya = H is the uniform partition of the segment 0, H and the diameter{ of is , then y } can be represented as [ ] Γ ∗ ∆ ji Γ ∗ ∗ ∈ y j , (57) ji = i∆ ∗ where 0 j a is an integer. Since the the numbers y , i 0, 1, . . . , N 1, satisfy i ji Γ = ≤ ≤ N 1 ∈ ∗ − X− the inequality ∆ y p µp, then the integers 0 j a, i 0, 1, . . . , N 1, have to ji 0 i = · i=0 ≤ ≤ ≤ −

44 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University satisfy the inequality

N 1 p X µ − j p 0 . (58) ( i) ∆ ∆ p i=0 ≤ ( ) ∗ H Taking into consideration (57) and (58), we can redeﬁne the set Up,∆,∆ ,σ as ∗

¦ U H u L 0, 1 ; m : u t ∆ j s , t t , t , p,∆,∆ ,σ = ( ) p ([ ] R ) ( ) = i li [ i i+1) ∗ ∗ · ∈ N 1 ∈ p X− µ © 0 j a, s S , jp 0 . i li σ i ∆ ∆ p ≤ ≤ ∈ i=0 ≤ ( ) ∗ where i = 0, 1, . . . , N 1 H − By Zp,∆,∆ ,σ (θ; t0, x0), we denote the set of all points z(θ) = z(tN ) calculated by the recurrent formula∗

z t z t t t f t , z t , ∆ j s , z t x (59) ( i+1) = ( i) + ( i+1 i) i ( i) i li ( 0) = 0 − ∗ where s S and the integers 0 j a, i 0, 1, . . . , N 1, satisfy the inequality li σ i = (58). ∈ ≤ ≤ − H H Actually, Zp,∆,∆ ,σ (1; 0, x0) is the euler approximation of the attainable set X p,∆,∆ ,σ (1; 0, x0) ∗ H ∗ whose control functions are chosen from the set Up,∆,∆ ,σ ∗ H The relation beetween the attainable set X p(θ, t0, x0) and the set Zp,∆,∆ ,σ (θ; t0, x0) is studied in the papers [10, 11, 12, 13]. ∗ Following theorem characterizes the Hausdorff distance between the reachable set H of the system (55) with constraint (56) and the set Zp,∆,∆ ,σ (θ; t0, x0) which consists of ﬁnite number of points. ∗

Theorem 12 ( [14],[15]) For each given " > 0 there exists H (") (0, ), ∆∗(") > 0, ∆ (") > 0 and σ(") > 0 such that the inequality ∈ ∞ ∗ h X θ; t , x , Z H(") θ; t , x < " (60) p ( 0 0) p,∆,∆ ("),σ(") ( 0 0) ∗ holds for every ∆ ∆∗("). ≤

For given " > 0, the determination of the numbers H (") (0, ), ∆∗(") > 0, ∆ (") > 0 and σ(") > 0 in theorem can be found in [14]. ∈ ∞ ∗ Approximate Calculation of the Reachable Sets of SEIR System In the previous section, approximate calculation method for a general control system is given. Now we adapt the method for the approximate calculation of reachable set of the SEIR system given by (51) by choosing t0 = 0, θ = 1, p = 2.

45 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Let, x0 = (s(0), e(0), i(0)) = (S0, E0, I0) 1. At ﬁrst, it can be easily seen that the right hand side of the system (51),

  f1(s, e, i) f (s, e, i) =  f2(s, e, i)  f3(s, e, i)   µ λi(t)s(t) µs(t) u1(t) s(t) =  λi−(t)s(t) ("−+ µ)e(t−) | |  − "e(t) (γ + µ)i(t) u2(t) i(t) − − | | satisties the conditions 1.A, 1.B and 1.C

Denote ˜ 2 U2 = u( ) L2 [0, 1]; R : u( ) 2 µ0 . · ∈ k · k ≤ ˜ So, the set of control functions U2 consists of Lebesgue measurable functions u( ) : 2 [0, 1] R such that the inequality (54) is satisfied. · → ˜ The set of trajectories of the system (51) generated by the control function u ( ) U2 ∗ · ∈ and satisfying the initial condition (S(0), E(0), I(0)) = (S0, E0, I0) is denoted by symbol (S ( ; 0, S0, u ( )) , E ( ; 0, E0, u ( )) , I ( ; 0, I0, u ( ))). Let · ∗ · · ∗ · · ∗ · ˜ X p (t; 0, (S0, E0, I0)) = ˜ = (S (t; 0, S0, u( )) , E (t; 0, E0, u( )) , I (t; 0, I0, u( ))) : u( ) Up · · · · ∈ ˜ Thus, the set X p (t; 0, (S0, E0, I0)) is reachable set of the system (51) at instant of time t from initial position (0, (S0, E0, I0)) , where the control functions satisfy the inequality (54). 2. Since the control functions of our SEIR control system are u( ) = (u1( ), u2( )) : 2 [0, 1] R , for given σ > 0 we construct the set · · · → Sσ = s0, s1, s2,..., sK { } 2 which is finite σ-net of the unit sphere S = u = (u1, u2) R : u = 1 ∈ k k 2 In R the σ-net Sσ can be defined as

Sσ = (sin iθ, cos iθ) : i = 0, 1, . . . , r , (61) { } where

2 σ 2π θ , r = , (62) ≤ 2 θ

46 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

[ ] means the integer part. | · |(detailed algorithm for determining a σ-net on unit sphere S can be found in [15]). ˜ H By Zp,∆,∆ ,σ (1; 0, (s0, e0, i0)), we denote the set of all points (s(1), e(1), i(1)) = (s(tN ), e(tN ), i(tN )) calculated by∗ the recurrent formula

s· t s t i t s t s t j l s t ( i+1) = ( i) + ∆ [(µ λ ( i) ( i) µ ( i) ∆ i sin iθ ( i)] − − − ∗ | | e· t e t i t s t e t ( i+1) = ( i) + ∆ [λ ( i) ( i) (" + µ) ( i)] − i· t i t e t i t j l i t ( i+1) = ( i) + ∆[" ( i) (γ + µ) ( i) ∆ i cos iθ ( i)] − − ∗ | | where 0 li r, 0 ji a for every i = 0, 1, . . . , N 1, the integers ji, i = 0, 1, . . . , N 1, satisfy≤ the≤ inequality≤ ≤ (58). − − H 3. In calculation of the set Zp,∆,∆ ,σ (1; 0, x0), the integers j0, j1,..., jN 1 satisfying ∗ − the inequality (58) are chosen, where 0 ji a, i = 0, 1, . . . , N 1. ≤ ≤ − After choosing the integers j , j ,..., j and elements s , s ,..., s from S , 0 1 N 1 l0 l1 lN σ H − the points of the set Zp,∆,∆ ,σ (1; 0, x0) are calculated by formula (59) ∗

Graphical Representations and Conclusions In this section ﬁrstly we consider an optimal control problem for the system (51) and the reachable set of the system which can be calculated by using the algorithm which is mentioned in previous section. Moreover, numerical solution of optimal control problem and approximately calculated reachable set of the system (51) will be graphically represented. Let us consider the optimal control problem, we intend to ﬁnd control funcions u1(t), u2(t) for t [0, 1], to minimize ∈ 1 Z 2 2 J[u1, u2] = (w1u1 + w2u2 ) d t (63)

0 subject to the SEIR system given by (51), where u1(t) and u2(t) are measurable functions which belong to the set

U = (u1(t), u2(t)) 0 u1(t) 1, 0 u2(t) 1, t [0, 1] , { | ≤ ≤ ≤ ≤ ∈ } w1 and w2 are weight constants. Also we calculate the set Z˜ H(") 1; 0, S , E , I which is an approximation p,∆,∆ ("),σ(") ( ( 0 0 0)) ˜ ∗ of the reachable set X p (1; 0, (S0, E0, I0)) of the system (51) at instant of time t = 1. In the calculation processes it is accepted that p = 2, i.e. the admissible control functions 2 are chosen from the space L2 [0, 1] ; R . Numerical solutions of the optimal control problem (51)-(63) and approximate calculations of Reachable Sets are executed using MATLAB with the following realistic hypothetical parameter values and initial conditions:

47 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Parameters µ λ " γ S0 E0 I0 w1 w2 µ0 Values 0.2 0.9 0.53 0.03 0.7 0.2 0.1 0.5 0.5 1

Note that it is represented in the figure 5, if u1(t) = 0, u2(t) = 0 (no control) for initial state (s0, e0, i0) = (0.7, 0.2, 0.1) the fraction of infected individuals is reached 40-percent in unit time. By using vaccination and treatment controls, spreading of the disease might be controlled in significant levels depending on control stock. The solution of optimal control problem (51)-(63) and the cross sections of approx- ˜ imately calculated set X2 (1; 0, (s0, e0, i0)) with respect to s(t) is displayed in following figures.

Optimal Control Solution w =1 w =1 1 2 0.8 S(t) 0.7 E(t) I(t)

0.6

0.5

0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10

Figure 7

By looking optimal control solution(ﬁgure 7), it is understood that the spreading of the disease is able to controlled and the fraction of infected individuals can be held around 15-percent. However, it might be obtained better results by changing weight parameters of cost functional. At the other hand, we can see the extremal capacities of the system by calculating reachable sets if the costs of control functions are not important. It comes out in ﬁgure 8, it is possible to get better results in disease control. The fraction of infected individuals in population can be steered to 0.05 percent in unit time by using whole control stock.

48 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Figure 8

References

[1] Kermack W.O., Mckendric A.G., Contributions to the mathematical theory of epi- demics, part i, Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 115 (772), 1927, 700-721. [2] Hethcote H.W., The mathematics of infectious diseases, SIAM Review, 42(4), 2000, 599–653. [3] Hoppensteadt F.C., Mathematical methods in population biology, Cambridge Univer- sity Press, Cambridge, 1982. [4]Anderson R.M., Population dynamics of infectious diseases: Theory and applications, Chapman and Hall, London, 1982. [5] Grassly N.C., Fraser C., Mathematical models of infectious disease transmission, Na- ture Reviews Microbiology 6, 2008, 477-487. doi:10.1038/nrmicro1845. [6] Keeling M.J., Danon L., Mathematical modelling of infectious diseases, Br Med Bull, 92(1), 2009, 33-42. doi: 10.1093/bmb/ldp038. [7] Biswas M.H.A., Paiva L.T., MdR de Pinho. A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11(4), 2014, 761- 784. doi:10.3934/mbe.2014.11.761. [8] Neilan R.M., Lenhart S., An Introduction to Optimal Control with an Application in Disease Modeling, Modeling Paradigms and Analysis of Disease Trasmission Models, 2010, 67-82.

49 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

[9] Gaff H., Schaffer E., Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Bio. Sci. Eng. (MBE), 6, 2009, 469-492. [10]Guseinov Kh.G., Nazlipinar A.S., On the continuity property of Lp balls and an application, J.Math. Anal. Appl., 335, 2007, 1347-1359. [11] Guseinov Kh.G., Nazlipinar A.S., On the continuity properties of attainable sets of nonlinear control systems with integral constraint on controls, Abstr. Appl. Anal., 2008, p.14. [12] Guseinov KH.G., Approximation of the attainable sets of the nonlinear control systems with integral constraints on control, Nonlinear Analysis, TMA, 71, 2009, 622-645. [13] Guseinov Kh.G., Nazlipinar A.S., An algorithm for approximate calculation of the attainable sets of the nonlinear control systems with integral constraint on controls, Comp. Math. Appl., 62(4), 2011, 1887-1895. [14] Krasovskii N.N., Subbotin A.I., Game-theoretical control problems, Springer, New York, 1988.

50 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

On Totally Asymtotically Nonexpansive Mappings in Cat(0) Spaces A. ABKAR1, Mojtaba RASTGOO1 1Imam Khomeini International University, IRAN

(Received: 02.08.2019, Accepted: 19.10.2019, Published Online: 17.12.2019)

Abstract. In this article, we prove the existence of ﬁxed points for totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature.

Keywords: Iterative algorithm, Totally asymptotically quasi-nonexpansive mapping, ∆-convergence, CAT(0) space.

Introduction Fixed point theory in Cartan-Alexandrov-Toponogov spaces, or briefly in CAT(0) spaces, was first studied by W. A. Kirk (see [1]). Let (X , d) be a metric space and x, y be two fixed elements in X such that d(x, y) = l. A geodesic path from x to y is an isometry c : [0, l] c([0, 1]) X such that c(0) = x, c(l) = y. The image of a geodesic path between two→ points is called⊂ a geodesic segment. A metric space (X , d) is called a geodesic space if every two points of X are joined by a geodesic segment. A geodesic triangle represented by (x, y, z) in a geodesic space consists of three points x, y, z and the three segments4 joining each pair of the points. A comparison triangle of a geodesic triangle (x, y, z), denoted by 42 (x, y, z) or (x, y, z), is a triangle in the Euclidean space R such that d(x, y) = 4d 2 x, y , d x4, z d 2 x, z , and d y, z d 2 y, z . This is obtainable by using the R ( ) ( ) = R ( ) ( ) = R ( ) 2 triangle inequality, and it is unique up to isometry on R . Bridson and Haefliger [2] have shown that such a triangle always exists. A geodesic segment joining two points x, y in a geodesic space X is represented by [x, y]. Every point z in the segment is represented by αx (1 α)y, where α [0, 1], that is, [x, y] := αx (1 α)y : α [0, 1] .A subset⊕ of− a metric space X∈is called convex if for all x{, y ⊕ ,−[x, y] .∈ A geodesic} space isC called a CAT(0) space if for every geodesic triangle∈ C and its⊂ comparison C , the following inequality is satisfied: d x, y d 2 x, y for all4 x, y and x, y 4. ( ) R ( ) Examples of CAT(0) spaces include the R-tree,≤ Hadamard manifold,∈ 4 and Hilbert∈ ball4 equipped with hyperbolic metric.

Deﬁnition 4 Let xn be a bounded sequence in a CAT(0) space (X , d). { } 51 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

1. The asymptotic radius r( xn ) of xn is given by { } { } r xn : inf r x, xn , ( ) = x X ( ) { } ∈ { { } } where r(x, xn ) := lim supn d(xn, x). { } →∞ 2. The asymptotic center A( xn ) of xn is the set { } { } A( xn ) := x X : r(x, xn ) = r( xn ) . { } { ∈ { } { } }

In 2006, Dhompongsa et al [5] observed that for a bounded sequence xn in a { } CAT(0) space, A( xn ) is a singleton. { } Deﬁnition 5 Let be a nonempty, closed convex subset of a CAT(0) space (X , d).A mapping T : C is said to be uniformly L-Lipschitzian if there exists a constant L 0 such that C → C ≥

n n d(T x, T y) L d(x, y), x, y , and n N. ≤ ∀ ∈ C ∈ It is now time to recall the concept of -convergence in CAT(0) spaces. 4 Deﬁnition 6 Let (X , d) be a CAT(0) space. A sequence xn in X is said to -converge to x X if and only if x is the unique asymptotic center of{ all} subsequences of4 xn . In this ∈ { } case, we write limn xn = x and x is called the -limit of xn . 4 − →∞ 4 { } In the following, we recall some basics for nonlinear mappings on CAT(0) spaces. Let be a nonempty subset of a CAT(0) spaces (X , d). A self-mapping T : is calledC nonexpansive if d(T x, T y) d(x, y) for all x, y and is calledC quasi- → C nonexpansive if F ix(T) = x : T≤ x = x = and d(T x, p∈) C d(x, p) for all x and p F ix(T). The class of{ quasi-nonexpansive∈ C } 6 ; mappings properly≤ contains the class∈ C of nonexpansive∈ mappings with ﬁxed points; see, for example, [6]. A mapping T is called asymptotically nonexpansive [7] if there exists a sequence kn [1, ) such { } ⊂ ∞ that kn 1 as n and, for every n N, → → ∞ ∈ n n d(T x, T y) knd(x, y), x, y . ≤ ∀ ∈ C If F ix(T) = and there exists a sequence kn [1, ) such that kn 1 as n and, for every6 ; n N : { } ⊂ ∞ → → ∞ ∈ n d(T x, p) knd(x, p), x , and p F ix(T), ≤ ∀ ∈ C ∈ then T is called an asymptotically quasi-nonexpansive mapping. A mapping T is called totally asymptotically nonexpansive if there exist null real sequences u and v n ∞n=1 n ∞n=1 of nonnegative numbers (i.e., un, vn 0 as n ) and a strictly increasing{ } function{ } ψ : [0, ) [0, ) with ψ(0) = 0→ such that:→ ∞ ∞ → ∞ n n d(T x, T y) d(x, y) + unψ d(x, y) + vn, x, y . ≤ ∀ ∈ C 52 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

A mapping T is called totally asymptotically quasi-nonexpansive if F ix(T) = 6 ; and there exist null real sequences un ∞n 1 and vn ∞n 1 of nonnegative numbers (i.e., { } = { } = un, vn 0 as n ) and a strictly increasing function ψ : [0, ) [0, ) with ψ(0) =→0 such that→ ∞ ∞ → ∞

n d(T x, p) d(x, p) + unψ d(x, p) + vn, x , and p F ix(T). ≤ ∀ ∈ C ∈ We recall that the concept of asymptotically nonexpansive mappings was first introduced by Goebel and Kirk [7]. Then Alber et al. [8] introduced the class of totally asymptotically nonexpansive mappings that generalizes several classes of maps that are extensions of asymptotically nonexpansive mappings. We remark that according to the Example 1 of [10], the class of totally asymptotically nonexpansive mappings properly contains the class of asymptotically nonexpansive mappings. We now turn to recall some well-known iteration processes. The Mann iteration process is defined by the sequence xn , { } ¨ x1 , x ∈ C x T x n n+1 = (1 αn) n + αn ( n), 1, − ≥ where α is a sequence in 0, 1 . n ∞n=1 ( ) Further,{ } the Ishikawa iteration process is defined as the sequence xn , { } x ,  1 x ∈ C x T y n+1 = (1 αn) n + αn ( n),  − yn = (1 βn)xn + βn T(xn), n 1, − ≥ where α and β are some sequences in 0, 1 . n ∞n=1 n ∞n=1 ( ) In 2016,{ } Huang{ in [}11], introduced the following algorithm for a family of nonexpansive mappings in a CAT(0) space: ¨ x , 1 (64) x ∈ C f x L T x n n+1 = αn ( n) (1 αn) n( n), 1, − ≥ where α is a sequence in 0, 1 and f is a φ-weak contraction on . n ∞n=1 ( ) Further,{ } in 2016, Balwant Singh Thakur, Dipti Thakur and Mihai PostolacheC in [12], introduced the following algorithm for nonexpansive mappings in uniformly convex Banach spaces:  x1  z ∈ C1 β x β T x n = ( n) n + n ( n) (65) y T−1 α x α z  n = ( n) n + n n x T y− n+1 = ( n),

53 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications where α and β are real sequences in 0, 1 . n ∞n=1 n ∞n=1 ( ) In this{ paper,} inspired{ } by the algorithms (64) and (65), we introduce a new iterative algorithm for approximating ﬁxed points of totally asymptotically nonexpansive mappings on CAT(0) spaces. The results we obtain in this paper improve and extend several results stated by many others; they also complement many known recent results in the literature. Throughout this paper, we denote by N the set of positive integers and by R the set of real numbers. We write x * x to indicate that the sequence x converges n n ∞n=1 weakly to x, and x x to indicate that the sequence x converges{ } strongly to x. n n ∞n=1 We begin by recalling→ some known facts on the space CAT(0).{ }

Lemma 1 ([13], Lemma 2) Let an , $n and ξn be sequences of nonnegative real P P numbers such that a 1 $ {a } ξ{ , for} all n{ 1.} If ∞ $ < and ∞ ξ < n+1 ( + n) n + n n=1 n n=1 n , then lim a exists.≤ Moreover, if there exists≥ a subsequence a ∞of a such that n n nj n ∞a 0 as j →∞ , then a 0 as n . { } { } nj n → → ∞ → → ∞ Lemma 2 ([14], Lemma 4.5) Let x be a given point in a CAT(0) space (X , d) and tn be 1 { } a sequence in a closed interval [a, b] with 0 < a b < 1 and 0 < a(1 b) . Suppose ≤ − ≤ 2 that xn and yn are two sequences in X such that { } { } 1. lim supn d(xn, x) r, →∞ ≤ 2. lim supn d(yn, x) r, →∞ ≤ 3. lim supn d (1 tn)xn tn yn, x = r →∞ − ⊕ for some r 0. Then limn d(xn, yn) = 0. ≥ →∞ Lemma 3 The following assertions in a CAT(0) space hold:

( 1) Every bounded sequence in a complete CAT(0) space has a -convergent subse- A quence [3]. 4

( 2) If xn is a bounded sequence in a closed convex subset of a complete CAT(0) A { } C space (X , d), then the asymptotic center of xn is in [15]. { } C ( 3) If xn is a bounded sequence in a complete CAT(0) space (X , d) with A( xn ) = p , A { } { } { } νn is a subsequence of xn with A( νn ) = ν , and the sequence d(xn, ν) {converges,} then p = ν [4].{ } { } { } { } Lemma 4 ([9], Theorem 2.8) Let be a nonempty bounded closed convex subset of complete CAT(0) space (X , d) and TC : be a totally asymptotically nonexpansive and uniformly L-Lipschitzian mapping.C → If Cxn is a bounded sequence in such that { } C limn d(xn, T(xn)) = 0 and limn xn = p, then T(p) = p. →∞ 4 − →∞ Theorem 13 ([16], Corollary 3.2) Let be a nonempty bounded closed convex subset of a complete CAT(0) space (X , d) and TC: be a continuous totally asymptotically nonexpansive mapping. Then T has a ﬁxedC point. → C

54 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Main result In this section we shall only state the main result of our paper without providing the details of the proof.

Theorem 14 Let (X , d) be a complete CAT(0) space, be a nonempty, closed convex subset of (X , d) and T : be a uniformly L-LipschitzianC and totally asymptotically C → C P quasi-nonexpansive mapping with sequences un ∞n 1, vn ∞n 1 satisfying ∞n 1 un < P { } = { } = = ∞ and ∞ v < , and strictly increasing mapping ψ : 0, 0, with ψ 0 0. n=1 n [ ) [ ) ( ) = Let α , β∞ and γ be sequences in 0, 1 and∞ suppose→ ∞ that the following n ∞n=1 n ∞n=1 n ∞n=1 ( ) conditions{ } are{ satisﬁed:} { }

(C1) there exist constants a, b such that 0 < a αn b < 1 for all n N, ≤ ≤ ∈ (C2) there exists a constant M such that ψ(r) M r for all r 0. ≤ ≥

Then xn ∞n 1 which is deﬁned by { } =  x1  z ∈ CT n 1 α x α T n x n = ( n) n n ( n) (66) y T n 1− β z ⊕ β T n z  n = ( n) n n ( n) x T n − T⊕n y T n z n+1 = (1 γn) ( n) γn ( n) , − ⊕ is -convergent to some p F ix(T). 4 ∈ Remark 3 We note that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence kn := 1 for all n N and each asymptotically nonexpansive { } ∈ mapping is a ( un , vn , ψ) totally asymptotically nonexpansive mapping with two se- { } { } − quences vn := kn 1 and un =: 0 for all n N and ψ being the identity mapping. Also, we{ see that each− asymptotically} { nonexpansive} ∈ mapping is a uniformly L Lipschitzian mapping with L : sup k . − = n N n ∈ { }

55 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

References

[1] Kirk W.A., Seminar of Mathematical Analysis: Geodesic Geometry and Fixed Point Theory, Seville, Spain. University of Malaga and Seville, Spain, September 2002-February, 2003, 195–225. [2] Bridson M., Haefliger A., Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin, Heidelberg, 1999. [3] Kirk W.A., Panyanak B., A concept of convergence in geodesic spaces, Nonlinear Anal., TMA, 68, 2008, 3689-3696. [4] Dhompongsa S., Panyanak B., On ∆ convergence theorems in CAT(0) spaces, Com- put. Math. Appl., 56, 2008, 2572-2579.− [5] Dhompongsa S., Kirk W.A., Sims B., Fixed points of uniformly lipschitzian mappings, Nonlinear Anal.,TMA, 65, 2006, 762-772. [6] Dotson W. D. Fixed points of quasi-nonexpansive mappings, J. Aust. Math. Soc., 13, 1972, 167-170. [7] Goebel K.; Kirk W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc., 35, 1972, 171-174. [8] Alber Y. I., Chidume C. E., Zegeye H., Approximating fixed points of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications, 2006, 2006, 10673. [9] Chang S.S., Wang, L., Lee H.W.J., Chan C.k., Yang L., Demiclosed principle and - convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces4, Appl. Math. Comput., 219, 2012, 2611-2617. [10] Ofoedu E.U., Nnubia A.C., Approximation of minimum-norm fixed point of total asymptotically nonexpansive mappings, Afr. Math., 26, 2015, 699-715. [11] Huang, S. Viscosity approximations with weak contractions in geodesic metric spaces of nonpositive curvature, J. Nonlinear Convex Anal., 17, 2016, 77-91. [12] Thakur, B. S.; Thakur, D.; Postolache, M. A new iterative scheme for numerical reck- oning fixed points of Suzuki’s generalized nonexpansive mappings, Applied Mathematics and Computation, 275, 2016, 147-155. [13] Liu, Q. Iterative sequences for asymptotically quasi-nonexpansive mappings with er- ror member, J. Math. Anal, 259, 2001, 18-24. [14] Nanjaras, B.; Panyanak, B. Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory and Applications, 2010, 268780. [15] Dhompongsa, S.; Kirk, W. A.; Panyanak, B. Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear and Convex Anal., 8, 2007, 35-45. [16] Panyanak, B. On total asymptotically nonexpansive mappings in CAT(κ) spaces, J. Inequal. Appl., 2014, 2014, 336.

56 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Boundary Value Problems for Convolution Differential Operator Equations on the Half Line Hummet K. MUSAEV Baku State University, AZERBAIJAN

(Received: 10.08.2019, Accepted: 14.11.2019, Published Online: 17.12.2019)

Abstract. This paper focuses uniform separability properties of parameter dependent convolution-differential equations. The equations and boundary conditions contain certain small and spectral parameters. Here the explicit formula for the solution is given and behavior of solution is derived when the small parameter approaches zero. It used to obtain singular perturbation result for the convolution parabolic equation.

Convolution-differential operator equations (CDOEs) with small parameters have sig- niﬁcant applications to the developed theory to problems in mathematical physics. The main aim of this paper is to show the uniform separability properties of boundary value problems (BVPs) for the following CDOE with parameters.

1/2 "u00(t) + Aλu(t) + " (aA1 u0)(t) + (A0 u)(t) = f (t), (67) − ∗ ∗ where Aλ = A + λI, A, A1 = A1(t), A0 = A0(t) are linear operators in a Banach space E, a = a(t) is a scalar valued function on (0; ), " is a small and λ is a complex parameter, u(t) = u(", t). ∞ We derive the representation of solution involving semigroup of operator A which allows to obtain the maximal regularity properties of DOEs and the sharp coercive Lp estimates of solution uniformly with respect to small and spectral parameter. It can be used for singular perturbation problem for the integro-differential parabolic equation

§ A1(t)v0(t) + Aλ v(t) + (A0 v)(t) = f (t), ∗ v(0) = v0. The treatment of the singular perturbation problem for the abstract integro-differential equations studied e.g in [4-8] (see also the references therein). In contrast to these, here uniform separability properties of the problem (67) is derived in Lp(0, ; E). ∞ Let E0 and E be two Banach spaces and E0 is continuously and densely embedded m into E. Let m is a positive integer. Wp (0, ; E0, E) denotes the collection of E-valued ∞ (m) functions u Lp (0, ; E0) that have the generalized derivatives u Lp (0, ; E) with the norm∈ ∞ ∈ ∞

57 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

(m) u W m u W m 0, ;E ,E u L 0, ;E u < . p = p ( 0 ) = p( 0) + Lp(0, ;E) k k k k ∞ k k ∞ ∞ ∞ m For E0 = E it denotes by Wp (Ω; E). Let " is a positive parameter. We deﬁne in the m space Wp (0, ; E0, E) the following parametrized norm ∞ (m) u W m 0, ;E ,E u L 0, ;E "u . p,"( 0 ) = p( 0) + Lp(0, ;E) k k ∞ k k ∞ ∞ Consider the following BVP for convolution-differential elliptic equation with small parameters

1/2 Lu = "u00(t) + Aλu(t) + " (aA1 u0)(t)+, (A0 u)(t) = f (t), t (0; ) , p+1 1 − 2p 2p ∗ ∗ ∈ ∞ L1u = " αu0 (0) + " βu(0) = f0. (68) where u (t) = u (", t) is a solution of (68), A, A1 = A1(t), A0 = A0(t) are linear operators in a Banach space E, Aλ = A + λI, a = a(t) is a scalar valued function on (0; ), ∞ f0 Ep = (E(A), E)θ,p, here (E(A), E)θ,p denotes the real interpolation space between ∈ 1+p E(A) and E, p (1, ), θ = 2p , α, β are complex numbers, " is a small positive, and λ is a complex∈ parameters,∞ i.e. " (0, 1). Let us ﬁrstly consider the corresponding∈ homogeneus problem § "u00(t) + Aλu(t) = 0, − (69) L1u = f0. Theorem 15 Assume the following conditions are satisﬁed:

1. E is a Banach space satisfying the uniformly multiplier condition for p (1, ); ∈ ∞ 2. A is a R positive operator in E for 0 and 1 S , 0 . ϕ < π βα− ϕ1 ϕ1 + ϕ < π − ≤ − ∈ ≤2 Then the problem (69) for f0 Ep has a unique solution u(t) Wp (R+; E(A), E) and the coercive estimate ∀ ∈ ∈

2 X 1 i i i 1 θ λ 2 " 2 u( ) t Au t C λ f f (70) − ( ) + ( ) Lp ;E − 0 E + 0 Ep Lp(R+;E) (R+ ) i=0 | | k k ≤ | | k k k k

holds uniformly with respect to " and λ Sϕ with sufficiently large λ . ∈ | | For investigation of main problem first all of, consider the leading part of (68), i.e. § "u00(t) + Aλu(t) = f (t), − (71) L1u = f0. f t L E Assume that the all conditions of Theorem 15 are satisfied. Then for ( ) p (R+; ), f0 Ep and λ, with sufficiently large λ , the problem (71) has a unique∈ solution ∀u W∈ 2 E A E | | p (R+; ( ), ) and the following coercive uniformly estimate holds ∈ 58 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

2 X 1 i i i 1 θ λ 2 " 2 u( ) t Au t C f λ f f . − ( ) + ( ) Lp ;E Lp ;E + − 0 E + 0 Ep Lp(R+;E) (R+ ) (R+ ) i=0 | | k k ≤ k k | | k k k k

Now, we consider the problem (68).

Let L" and L0" are operators in Lp (R+; E) generated by (68) and (71) for λ = 0, respectively i.e.

1 2 L"u = "u00 + Au + " aA1 u0 + (A0 u) , − ∗ ∗

D L W 2 E A E L L u u Au ( 0") = p (R+; ( ), , 1) , 0" = " 00 + . − Under the conditions Theorem 15 we prove that the problem (71) has a unique u W 2 E A E f L E solution p (R+; ( ), ) for all p (R+; ) and the following uniform coercive estimate holds∈ ∈

2 X 1 i i i λ 2 " 2 u( ) Au C f . (72) − + Lp ;E Lp ;E Lp(R+;E) (R+ ) (R+ ) i=0 | | k k ≤ k k

The estimate (72) implies that the operator L0" generated by (71) is bounded from L E W 2 E A E p (R+; ) into p (R+; ( ), ) and

2 X i i 1 2 2 1 λ − " (L0" + λ)− B L ;E C. ( p(R+ )) i=0 | | ≤

Theorem 16 Assume that the all conditions of the Theorem 15 are satisﬁed, and a(t) 1 µ 1 µ 1∈ L ,A t A ( 2 1) L B E ,A t A ( 2) L B E for 1(R+) 1( ) − − (R+; ( )) 0( ) − − (R+; ( )) 0 < µ1 < , ∈ ∞ ∈ ∞ 2 . Then for all f L E and sufﬁciently large there exists 0 < µ2 < 1 p (R+; ) λ > λ0 > 0 a unique solution u W 2 ∈ E A E of the problem (68) and| | the following coercive p (R+; ( ), ) uniform estimate holds∈

2 X i 1 i i " 2 λ 2 u( ) Au c f . − L ;E + Lp( ;E) p(R+ ) R+ Lp(R+;E) i=0 | | k k ≤ k k

59 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

References

[1] Agarwal R., O’ Regan D., Shakhmurov V.B., Separable anisotropic differential operators in weighted abstract spaces and applications, J. Math. Anal. Appl., 338, 2008, 970-983. [2] Ashyralyev A., Akturk S., Positivity of a one-dimensional difference operator in the half-line and its applications, Appl. and Comput. Math., 14(2), 2015, 204-220. [3] Denk R., Hieber M., Prüss J., R-boundedness Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166(788), 2003. [4] Guliev V.S., To the theory of multipliers of Fourier integrals for functions with values in Banach spaces, Trudy Math. Inst., Steklov, 214(17), 1996, 164-181. [5] Favini A., Shakhmurov V.,Yakubov Y., Regular boundary value problems for complete second order elliptic differential operator equations in UMD Banach spaces, Semigroup Forum, 79, 2009, 22-54. [6] Shakhmurov V.B., Ekincioglu I., Linear and nonlinear convolution-elliptic equations, Boundary Value Problems, 2013(211), 2013. [7] Shakhmurov V.B., Musaev H.K., Separability properties of convolution-differential operator equations in weighted Lp spaces, Appl. and Comput. Math., 14(2), 2015, pp. 221-233. [8] Shakhmurov V.B., Shahmurov R., Sectorial operators with convolution term, Math. Inequal. Appl., 13(2), 2010, 387-404. [9] Weis L., Operator-valued Fourier multiplier theorems and maximal Lp regularity, Math. Ann., 319, 2001, 735-758.

60 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Some Relations Between Partially q-Poly-Euler Polynomials of the Second Kind Burak KURT Akdeniz University, TURKEY

(Received: 18.08.2019, Accepted: 22.11.2019, Published Online: 17.12.2019)

Abstract. In recent years, many researchers studied on the Euler, poly-Euler and q-poly- Euler polynomials and numbers. They introduced and investigated some properties of these polynomials including several identities for them. In this paper, we deﬁne the partially q-poly-Euler polynomials of the second kind. We also prove some relations between the partially q-poly-Euler polynomials of the second kind and the q-Bernoulli polynomials Bn,q(x, y). By using these polynomials and numbers. In addition, we obtain many identities relations including the Roger-Szégo ö polynomials, the partially q-poly-Euler polynomials of the second kind En,q (x, y) and q-Hermite polynomials Hn,q(x)

Keywords: Euler polynomials and numbers, q-Euler polynomials of second kind, q- poly Bernoulli polynomials, Poly-logarithm function, q-Stirling numbers of the second kind. Introduction and Notation The classical Bernoulli polynomials, the classical Euler polynomials and the classical Genocchi polynomials are deﬁned by the following generating functions, respectively;

n X∞ t t B x ex t , t < 2π, (73) n( )n! = et 1 n 0 | | = − n X∞ t 2 E x ex t , t < π (74) n( )n! = et 1 n=0 + | | and n X∞ t 2t G x ex t , t < π. (75) n( )n! = et 1 n=0 + | | Also, let Bn = Bn(0), En = En(0) and Gn = Gn(0) where Bn, En and Gn are respectively, the Bernoulli numbers, the Euler numbers and the Genocchi numbers.

61 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

k Z, k > 1, then k-th polylogarithm is deﬁned by ([?], [?], [?]) as ∈ X zn Li z ∞ . (76) k( ) = nk n=1

This function is convergent for z < 1, when k = 1 | | Li1(z) = log(1 z). (77) − − The q-numbers and q-factorial are deﬁned by

1 qn [n]q = − , q = 1, [n]q! = [n]q [n 1]q [1]q , 1 q 6 − ··· − n N, q C, respectively where [0]q! = 1. ∈The q∈-polynomials coefﬁcients are deﬁned by

n (q : q)n k = q (q : q)k (q : q)n k − 2 n where (q : q)n = (1 q) 1 q (1 q ). The q-binomial formula− − is known··· as−

n 1 n n Y− j X n k k k a : q 1 a 1 q a q(2) 1 a . ( )n = ( )q = = k ( ) q − j=0 − k=0 −

n The q-analogue of the function (x + y)q is deﬁned by

n n X n k n k k x y : q(2)x y , n . (78) ( + )q = k − N q k=0 ∈ The q-exponential functions are given by

n X∞ z Y∞ 1 1 e z , 0 < q < 1, z < (79) q( ) = n ! = 1 1 q qkz 1 q n 0 [ ]q k 0 ( ( ) ) | | | | = = − − | − | and n X∞ n z Y∞ 2 k Eq(z) = q( ) = 1 + (1 q) q z , 0 < q < 1, z C. (80) n q! n=0 [ ] k=0 − | | ∈

From here, we easily see that eq(z)Eq( z) = 1. Moreover − Dqeq(z) = eq(z) and Dq Eq(z) = Eq(qz)

62 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

where Dq is deﬁned by

f (qz) f (z) Dq f (z) = − , 0 < q < 1, 0 = z C. qz z | | 6 ∈ − The above q-notation can be found in [?]. (α) Mahmudov ([?], [?]) deﬁned the q-Bernoulli polynomials n,q (x, y) of order α, the (α) B (α) q-Euler polynomials n,q (x, y) of order α and the q-Genocchi polynomials n,q (x, y) of order α respectively,E the following generating functions G

n (α) X∞ t t (α) x, y e t x E t y , t < 2π, (81) n,q ( ) n! = e t 1 q ( ) q ( ) n 0 B q ( ) | | = − α X t n 2 ( ) ∞ (α) n,q (x, y) = eq (t x) Eq (t y) , t < π (82) n! eq t 1 n=0 E ( ) + | | and α X t n 2t ( ) ∞ (α) n,q (x, y) = eq (t x) Eq (t y) , t < π (83) n! eq t 1 n=0 G ( ) + | | where q C, α N and 0 < q < 1. It is obvious∈ ∈ that | |

(α) : (α) 0, 0 , lim (α) x, y B(α) x y , lim (α) B(α) n,q = n,q ( ) q 1 n,q ( ) = n ( + ) q 1 n,q = n B B −→ −B −→ −B (α) : (α) 0, 0 , lim (α) x, y E(α) x y , lim (α) E(α) n,q = n,q ( ) q 1 n,q ( ) = n ( + ) q 1 n,q = n E E −→ −E −→ −E and (α) : (α) 0, 0 , lim (α) x, y G(α) x y , lim (α) G(α). n,q = n,q ( ) q 1 n,q ( ) = n ( + ) q 1 n,q = n G G −→ −G −→ −G (α) (α) (α) (α) Dq,x n,q (x, y) = [n]q n 1,q(x, y), Dq,y Bn,q (x, y) = [n]q n 1,q(x, q y), B B − B − (α) (α) (α) (α) Dq,x n,q (x, y) = [n]q n 1,q(x, y), Dq,y n,q (x, y) = [n]q n 1,q(x, q y), E E − E E − (α) (α) (α) (α) Dq,x n,q (x, y) = [n]q n 1,q(x, y), Dq,y n,q (x, y) = [n]q n 1,q(x, q y) G G − G G − and Dqeq(x t) = xeq(x t), Dq Eq(y t) = y Eq(q y t). Hamahata et al. [?] deﬁned poly-Euler polynomials by

X t n 2Li 1 e t ∞ E(k) x k ( − )ex t . n ( ) n! = t et −1 n=0 ( + )

(1) For k = 1, we get En (x) = En (x).

63 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

The q-analogue of the Stirling numbers of the second kind S2,q (n, k) is deﬁned as k n X∞ t eq (t) 1 S2,q (n, k) = − . (84) n q! k q! n=0 [ ] [ ]

The q-Hermite polynomials Hn,q(x) is defined by Mahmudov in [?] as 2 n t X∞ t eq (t x) Eq2 = Hn,q(x) . (85) 2 q n q! −[ ] n=0 [ ] It is clear that t2 lim Hn,q(x) = exp t x . q 1 2 −→ − − (a) The Roger-Szégo polynomials Hn(x : q) and the Al-Salam Carlitz polynomials Un (x : q) in [?] are defined by the generating functions n X∞ t eq(t)eq(x t) = Hn(x : q) (86) n q! n=0 [ ] and n eq x t X t ( ) ∞ (a) = Un (x : q) . (87) eq t eq at n q! ( ) ( ) n=0 [ ] ö The classical Euler numbers of the second kind En and the classical Euler polyno- ö mials of the second kind En (x) are defined in [?] by means of the following generating functions, respectively

n n X ö t 2 X ö t 2 ∞ E and ∞ E x ex t . n n! = et e t n ( ) n! = et e t n=0 + − n=0 + − Agarwal et al. in [?] deﬁned the q-Euler polynomials of second kind in two parameters as: n X ö t 2 ∞ E x, y e x t E t y (88) n,q ( ) n = e t e t q( ) q( ) n 0 [ ]q! q( ) + q( ) = − where x, y C. ∈ ö [k,α] Kurt is deﬁned the generalized partially q-poly-Euler numbers En,q of the second ö [k,α] kind of order α and the generalized partially q-poly-Euler polynomials En,q (x, y) of the second kind of order α as follows in [?], respectively α n t [k,α] X∞ ö t 2Lik (1 e− ) En,q = − (89) n q! t e t e t n=0 [ ] q( ) + q( ) − and α n t [k,α] X∞ ö t 2Lik (1 e− ) En,q (x, y) = − eq(x t)Eq(y t). (90) n q! t e t e t n=0 [ ] q( ) + q( ) − 64 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Main Theorems In this section, we give explicit relations for these polynomials. Furthermore, we give some relations between the partially q-poly-Euler polynomials of the second kind, the q-Stirling numbers of the second kind, the two variable Bernoulli polynomials and so. Further, we prove the relation between these polynomials and q-Hermite polynomials Hn,q(x). Theorem 17 The following relation holds true:

¨ n n ö ö m X X n j 1 n E[k,α] x, y E[k,α] 0, y G mx, 0 [ ]q n 1,q ( ) = j r j n j,q ( ) j r,q ( ) 2 q q m − − r=0 r=j − n « X n 1 ö E[k,α] 0, y G mx, 0 (91) + p mp n p,q ( ) p,q ( ) p=0 q − Proof. By (90), we write as α X ö t n 2Li 1 e t ∞ E[k,α] x, y k ( − ) E y t n,q ( ) = − q ( ) n q! t e t e t n=0 [ ] q ( ) + q ( ) t −2t eq m + 1 t m e mx 2t t q m × m eq m + 1 ¨ m X ö t n X 1 t n ∞ E[k,α] 0, y ∞ 1 = n,q ( ) n + 2t n q! m n q! n=0 [ ] n=0 [ ] n « X∞ t Gp,q (mx, 0) . n q! × n=0 [ ] Making mathematical operations and use the Cauchy product and comparing the coef- ﬁcients, we have (91).

Theorem 18 There is the following relation for the partially q-poly-Euler polynomials of the second kind as: n X 1 m m 1 ! X n ∞ ( ) ( + ) x y l S n l, m 1 − k ( + ) 2 ( + ) m 1 l m=0 ( + ) l=0 − n 1 X− n 1 ö l = n − En 1,l (x, y) 1 + ( 1) . (92) l − l=0 − Proof. By (78) and (90), for α = k = 1, we write

n X∞ ö t t En,q (x, y) t eq(t) + eq( t) = 2eq(x t)Eq(y t)Lik 1 e− n q! n=0 [ ] − −

65 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

X t l X 1 m+1 m 1 ! e t 1 m+1 2 ∞ x y l ∞ ( ) ( + ) ( − ) = ( + )q − k − l q! m 1 m 1 ! l=0 [ ] m=0 ( + ) ( + ) X t l X 1 m+1 m 1 ! X t p 2 ∞ x y l ∞ ( ) ( + ) ∞ S p, m 1 . = ( + )q − k 2 ( + ) l q! m 1 p! l=0 [ ] m=0 ( + ) p=0 On the other hand, the left hand of this equality

m l X∞ ö t X∞ l t t Em,q (x, y) 1 + ( 1) m q! l q! m=0 [ ] l=0 − [ ] by using Cauchy product and get to q 1−, we have (92). By the polylogarithm deﬁnition, we−→ write the equation (90), for α = 1

k,1 n X∞ ö [ ] t En,q (x, y) n q! n=0 [ ] Z t Z y Z y 2eq(t x)Eq(t y) 1 1 1 y d y d y. (93) = e y 1 et 1 et 1 e y 1 t eq(t) + eq( t) 0 0 ··· 0 ··· − | − − {z − − } (k 2) times − We take k = 2 in (93)

n m Z t X ö [2,1] t X ö t y ∞ E x, y ∞ E x, y d y n,q ( ) = m,q ( ) y n q! m q! e 1 n=0 [ ] m=0 [ ] 0 m − l X∞ ö t X∞ l!Bl t = t Em,q (x, y) . (94) m q! l 1 l! m=0 [ ] l=0 +

ö [2,1] ö Since lim En,q x, y En x y . q 1 ( ) = ( + ) − We−→ write (94) as

2,1 n m l X∞ ö [ ] t X∞ ö t X∞ l!Bl t lim En,q (x, y) = t (x + y) Em . q 1 n q! m! l 1 l! −→ − n=0 [ ] m=0 l=0 +

tn By using Cauchy product and comparing the coefﬁcients of the n! , we have the following theorem.

Theorem 19 For n 0, x, y C, we have ≥ ∈ n 1 [2,1] ö X− n 1 ö l!Bl (x + y) En = n − (x + y) En 1 l . l − − l 1 l=0 ( + )

66 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

Theorem 20 The following relations holds true ¨ n r [k,α] [k,α] ö m X n ö X r r En 1,q (x, y) = En r,q (0, y) m− − n q r − s [ ] r=0 q s=0 q n « [k,α] X n ö s En s,q (0, y) m− Bs,q(mx, 0). (95) s − − s=0 q Proof. By (81) and (90) α n t t t X ö [k,α] t 2Li 1 e eq 1 t ∞ E x, y k ( − ) E y t m m e mx n,q ( ) = − q( ) t − t q n q! t e t e t e 1 m n=0 [ ] q ( ) + q ( ) m q m − − ¨ α t m 2Li 1 e t t t k ( − ) E y t e m e mx = t − q( ) q m t q m t eq (t) + eq ( t) eq m 1 − α t − « 2Li 1 e t t k ( − ) E y t m e mx . − q( ) t q m − t eq (t) + eq ( t) eq m 1 − − k,α n+1 X∞ ö [ ] t En,q (x, y) = m (I1 I2) . (96) n q! n=0 [ ] −

p l s X ö [k,α] t X t X t I ∞ E 0, y ∞ ∞ B mx, 0 1 = p,q ( ) l s,q ( ) s p q! m l q! m s q! p=0 [ ] l=0 [ ] s=0 [ ] n r n [k,α] X∞ X n ö X r r t = En r,q (0, y) m− Bs,q(mx, 0) . (97) r − s n q! n=0 r=0 q s=0 q [ ]

p s X ö [k,α] t X t I ∞ E 0, y ∞ B mx, 0 2 = p,q ( ) s,q ( ) s p q! m s q! p=0 [ ] s=0 [ ] n n [k,α] X∞ X n ö s t = En s,q (0, y) Bs,q(mx, 0)m− . (98) s − n q! n=0 s=0 q [ ] From (96), (97) and (98), we obtain (95). Theorem 21 There is the following relation between the partially q-poly-Euler polynomi- ö [k,α] als of the second kind En,q (x, y) and q-Euler polynomials En,q (x, y): ¨ n r [k,α] [k,α] ö 1 X n ö X r r En,q (x, y) = En r,q (0, y) m− 2 r − s r=0 q s=0 q n « [k,α] X n ö s En s,q (0, y) m− Es,q (mx, 0) . (99) s − − s=0 q

67 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications

Proof. By (82) and (90),

α t k,α n t X∞ ö [ ] t 2Li 1 e eq m + 1 2 t E x, y k ( − ) E y t e mx n,q ( ) = − q( ) t q n q! t e t e t 2 e 1 m n=0 [ ] q ( ) + q ( ) q m + −

¨ α 1 2Li 1 e t t 2 t k ( − ) E y t e e mx = 2 − q( ) q m t q m t eq (t) + eq ( t) eq m + 1 − α « 2Li 1 e t 2 t k ( − ) E y t e mx + − q( ) t q m t eq (t) + eq ( t) eq m + 1 − k,α n X∞ ö [ ] t 1 En,q (x, y) = (I1 + I2) . (100) n q! 2 n=0 [ ]

p l s X ö [k,α] t X t X t I ∞ E 0, y ∞ ∞ E mx, 0 1 = p,q ( ) l s,q ( ) s p q! m l q! m s q! p=0 [ ] l=0 [ ] s=0 [ ] ¨ n r « n [k,α] X∞ X n ö X r r t = En r,q (0, y) m− Es,q(mx, 0) . (101) r − s n q! n=0 r=0 q s=0 q [ ]

¨ n « n [k,α] X∞ X n ö s t I2 = En s,q (0, y) Es,q(mx, 0)m− . (102) s − n q! n=0 s=0 q [ ] From (100), (101) and (102), we obtain (99).

Theorem 22 There is an explicit relation between q-Hermite polynomials Hn,q(x) and the partially q-poly-Euler polynomials of the second kind:

  n   2 n 2p ö [k,α] X X− n 2p ö [k,α] [n]q! E x, y E 0, y H x . n,q ( ) = −j j,q ( ) n 2p j,q( ) p q − − 2 p q2 ! n 2p q! p=0 j=0 [ ]q [ ] [ ] − (103)

Proof. By (85) and (90),

α n t [k,α] X∞ ö t 2Lik (1 e− ) En,q (x, y) = − eq (x t) Eq(y t) n q! t e t e t n=0 [ ] q ( ) + q ( ) − α t 2 2 2Lik (1 e− ) t t E y t e x t E 2 e 2 = − q( ) q ( ) q 2 q 2 t eq (t) + eq ( t) −[ ]q [ ]q − 68 OMTSA 2019 Operators in General Morrey-Type Spaces and Applications Dumlupınar University

k,α n n 2n X∞ ö [ ] t X∞ t X∞ t = En,q (0, y) Hn,q(x) n n q! n q! 2 n q2 ! n=0 [ ] n=0 [ ] n=0 [ ]q [ ]

n n 2n X X n ö [k,α] t X t ∞ E 0, y H x ∞ = j j,q ( ) n j,q( ) n q − n q! 2 n q2 ! n=0 j=0 [ ] n=0 [ ]q [ ]

m m 2p X X m ö [k,α] t X t 1 ∞ E 0, y H x ∞ = j j,q ( ) m j,q( ) p q − m q! 2 p q2 ! m=0 j=0 [ ] p=0 [ ]q [ ]   n   2 n 2p k,α n X∞ X X− n 2j ö [ ] [n]q! t = − Ej,q (0, y) Hn 2p j,q(x) p . j 2 p 2 ! n 2p ! n ! n 0 p 0 j 0 q − − [ ]q [ ]q [ ]q [ ]q = = = − From here, we have (103). Conclusions In this work, we have ﬁrstly the partially q–poly-Euler polynomials of the second kind, the q-Stirling numbers of the second kind and the q-Hermite polynomials Hn,q(x) . Uti- lizing these generalizations, we have constructed and proved some recurrence relations and identities. By using some calculations, we have derived several new and interesting identities for these polynomials.

References

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69 OMTSA 2019 Dumlupınar University Operators in General Morrey-Type Spaces and Applications and q-Genocchi polynomials, Abstract and Appl. Analy., vol. 2013, Article ID:293532. [10] D. Kim and T. Kim, A note on poly-Bernoulli and higher order poly-Bernoulli polynomials, Russian J. of Math. Physics, 22(1), (2015), 26-33. [11] T. Kim, V.S. Jang and J. J. Seo, A note on poly-Genocchi numbers and polynomials, Appl. Math. Sci., 8(96), (2014), 4775-4781. [12] V. Kurt, Some identities and recurrence relations for the q-Euler polynomials, Hacettepe J. of Math. and Statistics, 44(6), (2015), 1397-1404. [13] V.Kurt, New identities and relations derived from the generalized Bernoulli polynomials, Euler and Genocchi polynomials, Advanced in Diff. Equa., 2014, 2014.5. [14] V.Kurt, On the generalized q-Poly-Euler polynomials of the second kind, Filomat, accepted, (2019). [15] G. Liu, Generating functions and generalized Euler numbers, Proc. Japon. Acad. 84 SerA, (2008), 29-34. [15] Q.-M. Luo, Some recursion formulae and relations for Bernoulli numbers and Eu- ler numbers of higher order, Adv. Stud. in Contem. Math., 10(1), (2005), 63-70. [16] N. I. Mahmudov, q-Analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pinter addition theorem, Discrete Dyn. in Nature and Society, 2012, Article ID: 169348. [18] N. I. Mahmudov, On a class of q-Bernoulli and q-Euler polynomials, Advances in Diff. Equ., 2013, 2013.108. [18] N. I. Mahmudov and M. E. Keleshteri, q-Extensions for the Apostol type polynomials, J. of Applied Math., 2014, Article ID:868167. [19] N. I. Mahmudov, Difference equations of q-Appell polynomials, Applied Math. and Comp., 245 (2014), 530-545. [20] C. S. Ryoo and R. P. Agarwal, Some identities involving q-poly-tangent numbers and polynomials and distribution of their zeros, Adv. in Diff., (2017), 2017.213. [21] P.N. Sadjang, q-addition theorems for the q-Appell polynomials and the associated classes of q-polynomials expansion, J. Korean Math. Soc., 85(5), (2018), 1179-1192. [22] H. M. Srivastava, Some generalization and basic (or q-) extensition of the Bernoulli, Euler and Genocchi polynomials, App. Math. ˙Inform. Sci., (5), (2011), 390-444. [23] H. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluver Academic Pub., Dordrect, Boston and London, (2001). [24] C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, Journal of Inequalities and Appl., (2015), 2015.219.

70 Author Index

A. llkem ABKAR, 51 TURHAN CETINKAYA, 1 A. Serdar NAZLIPINAR, 40 Mojtaba Barbaros RASTGOO, 51 BASTURK, 40 Burak Nurullah KURT, 61 YILMAZ, 21 Elçin Sahsene YUSUFOGLU, 1 ALTINKAYA, 11, 30 Hummet K. Sibel MUSAEV,57 YALCIN TOKGOZ, 11, 30