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This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

For the Riesz potential of variable order over bounded domains in Euclidean space, we prove the boundedness result from variable exponent Morrey spaces to variable exponent Campanato spaces. A special attention is paid to weaken assumptions on variability of the Riesz potential. Russian Foundation for Basic ResearchRussian Foundation for Basic Research (RFBR) [18-01-00094-a, 19-01-00223] United Arab Emirates University [G00002994]

WOS: 000434090900002 Let L = -Delta + V be a Schrodinger operator on R-n, where n >= 3 and the nonnegative potential V belongs to the reverse Holder class RHq1 for some q(1) > n/2. Let b belong to a new Campanato space Lambda(theta)(nu) (rho) and I-beta(L) be the fractional integral operator associated with L. In this paper, we study the boundedness of the commutators [b, I-beta(L)] with b is an element of Lambda(theta)(nu) (rho) on local generalized Morrey spaces LM rho,phi alpha,V,{x0} generalized Morrey spaces M-rho,phi(alpha,V) and vanishing generalized Morrey spaces VM rho,phi alpha,V associated with Schrodinger operator, respectively. When b belongs to Lambda(theta)(nu) (rho) with theta > 0, 0 < nu < 1 and (phi(1),phi(2)) satisfies some conditions, we show that the commutator operator [b,I-beta(L)] are bounded from LM rho,phi 1 alpha,V,{x0} to LM rho,phi 2 alpha,V,{x0} from LMq,phi 1 alpha,V to VMq,phi 2 alpha,V and from VM rho,phi 1 alpha,V to VM rho,phi 2 alpha,V, 1/p - 1/q = ( beta +nu)/n. Ahi Evran University Scientific Research ProjectAhi Evran University [FEF.A4.17.020]; 1st Azerbaijan-Russia Joint Grant Competition [18-51-06005]; Ministry of Education and Science of the Russian FederationMinistry of Education and Science, Russian Federation [02.a03.21.0008] The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.17.020). The research of V.S. Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A4.17.008), by the grant of 1st Azerbaijan-Russia Joint Grant Competition (the Agreement number No. 18-51-06005) and by the Ministry of Education and Science of the Russian Federation (Agreement number: 02.a03.21.0008).

In this paper, we investigate four different types of Ulam stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of nonlinear implicit fractional differential equations with non-instantaneous integral impulses and nonlinear integral boundary condition. We also establish certain conditions for the existence and uniqueness of solutions for such a class of fractional differential equations using Caputo fractional derivative. The arguments are based on generalized Diaz-Margolis’s fixed point theorem. We provide two examples, which shows the validity of our main results.

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

In this paper some new inequalities of Simpson-type are established for the classes of functions whose derivatives of absolute values are convex functions via Riemann-Liouville integrals. Also, by special selections of n;we give some reduced results. © 2017, University of Nis. All rights reserved., Firat University Scientific Research Projects Management Unit British Association for Psychopharmacology, 2010 Mathematics Subject Classification. Primary 26D15 Keywords. convex functions, Simpson-Type Inequalities,Riemann-Liouville fractional integrals Received: 10 March 2016; Accepted: 29 March 2016 Communicated by Dragan S. Djordjevi? This research is supported by Ordu University Scientific Research Projects Coordinations Unit (BAP). Project Number: YKD-280 Email addresses: erhanset@yahoo.com (Erhan Set), ahmetakdemir@agri.edu.tr (Ahmet Ocak Akdemir), mozdemirr@yahoo.com (M. Emin Özdemir)

In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann&ndash, Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions. Our results rely on modern tools of functional analysis. We also demonstrate the application of the obtained results with the aid of examples.

In this paper we prove several fractional quantum integral inequalities for the new q-shifting operator ${_{a}}\Phi_{q}(m) = qm + (1-q)a$ introduced in Tariboon et al. (Adv. Differ. Equ. 2015:18, 2015), such as: the q-Holder inequality, the q-Hermite-Hadamard inequality, the q-Cauchy-Bunyakovsky-Schawrz integral inequality, the q-Gruss integral inequality, the q-Gruss-Cebysev integral inequality, and the q-Polya-Szego integral inequality.

In the paper we study fractional systems with generalized Riemann-Liouville derivatives. A theorem on the existence and uniqueness of a solution to a fractional nonlinear ordinary Cauchy problem is proved. Next a formula for the solution to a linear problem of such a type is presented.

This paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

In this paper, we study the existence of solutions for a new nonlocal boundary value problem of integro-differential equations involving mixed left and right Caputo and Riemann&ndash, Liouville fractional derivatives and Riemann&ndash, Liouville fractional integrals of different orders. Our results rely on the standard tools of functional analysis. Examples are constructed to demonstrate the application of the derived results.

We develop the existence criteria for solutions of Liouville&ndash, Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are also presented. It is imperative to mention that our results correspond to the ones for a symmetric second-order nonlocal multipoint integral boundary value problem under suitable conditions (see the last section).

The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer fractional derivative and fractional integral operator with generalized Mittag-Leffler function in its kernel. We also discuss one dimensional cases of our related results. As a special case of our general results we obtain the results of Iqbal et al. (2017). Moreover, the refinement of Hardy-type inequalities for Hilfer fractional derivative is also included. Keywords: Convex function, Kernel, Hilfer fractional derivatives, Fractional integral

In this paper a certain class of convex functions in an Opial-type integral inequality is considered. Cauchy type mean value theorems are proved and used in studying Stolarsky type means defined by the observed integral inequality. Also, a method of producing n- exponentially convex and exponentially convex functions is applied. Some new Opial-type inequalities are given for different types of fractional integrals and fractional derivatives as applications.

In this paper we give generalization of Opial-type inequalities by using generalized fractional integral operator involving generalized Mittag–Leffler function. We deduce some results which already have been proved. Also we consider n-exponential convexity of some non-negative differences of inequalities involving Mittag-Leffler function and deduce their exponential convexity and log-convexity.

In this article we present applications of Hardy- type and refined Hardy-type inequalities for a generalized fractional integral operator involving the Mittag-Leffler function in its kernel and for the Hilfer fractional derivative using convex and monotone convex functions.

In the present paper, the notion of MTm-preinvex function is introduced and some new integral inequalities involving MTm-preinvex functions along with beta function are given. Moreover, some generalizations of Hermite-Hadamard and Ostrowski type inequalities for MTm-preinvexfunctions via classical integrals and Riemann-Liouville fractional integrals are established. These results not only extends the results appeared in the literature (see [10], [11], [12]), but also provide new estimates on these types.

We investigate a new kind of nonlocal boundary value problems of nonlinear Caputo fractional differential equations supplemented with integral boundary conditions involving Erdelyi-Kober and generalized Riemann-Liouville fractional integrals. Existence and uniqueness results for the given problem are obtained by means of standard fixed point theorems. Examples illustrating the main results are also discussed.

The aim of this research paper is to establish the Hardy-type inequalities for Hilfer fractional derivative and generalized fractional integral involving Mittag-Leffler function in its kernel using convex and increasing functions.

We study the existence of solution to a periodic boundary value problem for nonlinear impulsive fractional differential equations by using Schaeffer’s fixed point theorem. The research of J. J. Nieto and R. Rodríguez-López is partially supported by Ministerio de Economía y Competitividad, project MTM2010-15314, and co-financed by the European Community fund FEDER SI

Fractional delay differential equations (FDDEs) are widely used in ecology, physiology, physical Sciences and many other areas of applied science. Fractional Delay differential equations usually do not have analytic solutions and can only be solved by some numeric methods. In this paper, a new method, which is generalized from finite difference method, has been provided to solve the delay differential equations. It has been used for numerical solution of such models and used for solving a number of problems such as the problem of the impact of food on the population changes of an area [1], the problem of the fluctuations of the population of adults in an area per time [2], the problem of the number of blood cells in humans [3] and the problem of the effect of noise on light which is reflected from laser to mirror [4]. The proposed method, besides being simple, is so exact and sensible in the solved problems.

Bu çalışma, 21-23 Nisan 2016 tarihlerinde Antalya[Türkiye]’düzenlenen International Conference on Advances in Natural and Applied Sciences (ICANAS) Kongresi‘nde bildiri olarak sunulmuştur. In this note, some new Simpson-type inequalities for functions whose derivatives of absolute value are quasi-convex functions with the help of Riemann-Liouville fractional integrals are obtained. Also by special selections of n and alpha we give some reduced results. Ordu Üniversitesi British Association for Psychopharmacology

The electrostatics properties of composite materials with fractal geometry are studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integro-differentiation. The fractional Maxwell equations are obtained, and methods of fractional calculus are employed to obtain analytical expressions of the electric field inside the fractal composite structures.