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This manuscript deals with a hyperbolic partial differential equation with fractional differential and fractional integral forcing functions. Semidiscretization method is used to establish a unique strong solution and also approximate solutions. Error estimates and continuous dependence of the strong solution on the given conditions have also been discussed. At the end, we illustrated the results with an example.

By utilizing two fractional integral identities and elementaryinequalities via geometric-geometric (GG for short) convex functions, we derive new type Hermite-Hadamard inequalities involving Hadamard fractional integrals. Some applications to special means of real numbers are given.

Aim of the present paper is to investigate a new integral inequality of Grüss type for a hypergeometric fractional integral. Two main results are proved, the first one deals with Grüss type inequality using the hypergeometric fractional integral. The second result states another inequality regarding two synchronous functions.

The purpose of the present article is to derive some subordination and superordination results for certain normalized analytic functions involving fractional integral operator. Moreover, this result is applied to find a relation between univalent solutions for fractional differential equation.

In this paper we consider the integral equation offractional order in sense of Riemann-Liouville operatorum(t) = a(t) Iα [b(t)u(t)]+f(t)with m ≥ 1, t ∈ [0, T], T < ∞ and 0< α

Lowndes has defined the new operators of fractional integration which generalize the operators due to Erdelyi and Kober. The purpose of this paper is to develop a theory of these fractional integral operators given by Lowndes for a certain class of functions.

In this paper, we study nonlocal boundary value problems of nonlinear Caputo fractional differential equations supplemented with Erdélyi-Kober type fractional integral boundary conditions. Existence results are obtained by applying the Mönch's fixed point theorem and the technique of measures of noncompactness. An example illustrating the main result is also constructed.

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. We have included some recent work on the topic dealing with sequential, Caputo-Fabrizio, Hadamard, Hilfer, Katugampola, hyprid, or nested fractional derivatives. We have also focused on Lyapunov-type inequalities for problems containing fractional derivatives with respect to a certain function, systems of fractional differential equations, and conformable derivatives. Furthermore, Lyapunov-type inequalities for fractional differential equations based on non Green function approach are discussed.

In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to Riemann–Liouville fractional integral boundary conditions. The uniqueness of solutions is established by Banach contraction principle, while the existence of solutions is derived by Leray–Schauder’s alternative. We also study the Hyers–Ulam stability of mentioned system. At the end, examples are also presented which illustrate our results.

A significantly large number of earlier works on the subject of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, et cetera). In the present paper, we study and develop the generalized fractional integral operators given by Saigo. First, we establish two Theorems that give the images of the product of H-function and a general class of polynomials inSaigo operators. On account of the general nature of the Saigo operators, H-function and a general class of polynomials a large number of new and known Images involving Riemann-Liouville and Erdélyi-Kober fractional integral operators and several special functions notably generalized Wright hypergeometric function, generalized Wright-Bessel function, the polylogarithm and Mittag-Leffler functions follow as special cases of our main findings.

The purpose of this paper is to establish some types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of implicit fractional-order differential equation.

In this paper, we derive several subordination results for a certain class of analytic functions defined by the generalized Al-Oboudi differential operator. Relevant connections of some of the results obtained with those in earlier works are also provided.

In this paper, we establish the existence and uniqueness of solution for a class of initial value problem for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle, Schauder' fixed point theorem and the nonlinear alternative of Leray-Schauder type. As applications, two examples are included to show the applicability of our results.

In this paper, we improve and further generalize some Ostrowski-Grüss type inequalities for the fractional integrals by using new Montogomery identities.

In this paper, we apply generalized operators of fractional integration involving Appell’s function F_3 (.) due to Marichev-Saigo-Maeda, to the Bessel function of first kind. The results are expressed in terms of generalized Wright function and hypergeometric functions _pF_q . Special cases involving this function are mentioned. Results given recently by Kilbas and Sebastian follow as special cases of the theorems establish here.

In this paper, we shall establish sufficient conditions for the existence of solutions for a first order boundary value problem for fractional differential equations.

Let Ω∈L∞(ℝn)×L2(Sn-1) be a homogeneous function of degree zero. In this article, we obtain some boundedness of the parameterized Littlewood-Paley operators with variable kernels on weighted Herz-Morrey spaces with variable exponent. As a supplement, the boundedness of fractional integral operators with variable kernel is also obtained on these spaces.