This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a ν-th (0 < ν ≤ 1) order fractional difference equation is defined. A nonlinear problem wi...

In this paper, we focus on the existence of Hilfer fractional stochastic differential systems via almost sectorial operators. The main results are obtained by using concepts and ideas from calculus, multivalued maps, semigroup theory, operators, fixed-point technique. We start confirming mild solution Dhage’s theorem. Finally, an example is provided to demonstrate considered Hilferr theory.

In this paper‎, ‎a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly‎. ‎First‎, ‎the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs)‎. ‎Then‎, ‎the unknown functions are approximated by the hybrid functions‎, ‎including Bernoulli polynomials and Block-pulse functions based o...

Abstract Fractional calculus operators play a very important role in generalizing concepts of used diverse fields science. In this paper, we use Riemann–Liouville fractional integrals to establish generalized identities, which are further applied obtain midpoint and trapezoidal inequalities for convex function with respect strictly monotone function. These reproduce convex, harmonic p -convex, ...

The aim of this paper is to establish the existence of solutions of boundary value problems of nonlinear fractional integro-differential equations involving Caputo fractional derivative by using the techniques such as fractional calculus, H"{o}lder inequality, Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type. Examples are exhibited to illustrate the main resu...

Given a weighted ℓ 2 space with weights associated an entire function, we consider pairs of shift operators, whose commutators are diagonal when considered as operators over general Fock space. We establish calculus for the algebra these and apply it to case Gelfond–Leontiev derivatives. This class includes many known examples, such classic fractional derivatives Dunkl operators. allows us fram...