ON PERTURBED FRACTIONAL DIFFERENTIAL EQUATIONS AND INCLUSIONS WITH GENERALIZED RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL BOUNDARY CONDITIONS

B Ahmad,S.K Ntouyas

doi:10.12732/ijpam.v109i1.4

Abstract

In this paper, we present the sufficient criteria for the existence of solutions for perturbed fractional differential equations and inclusions with generalized Riemann-Liouville fractional integral boundary conditions. We make use of a nonlinear alternative, which deals with the sum of completely continuous and contractive single-valued or multi-valued operators, to obtain the desired results.

Highlights

The subject of fractional differential equations has emerged as an interesting and popular field of research in view of its extensive applications in applied and technical sciences
The significance of fractional derivatives owes to the fact that they serve as an excellent tool for the description of memory and hereditary properties of various materials and processes
One can notice that fractional derivatives are defined via fractional integrals

Summary

Introduction

The subject of fractional differential equations has emerged as an interesting and popular field of research in view of its extensive applications in applied and technical sciences. The Riemann-Liouville fractional integral of order q > 0 of a continuous function f : (0, ∞) → R is defined by. The Riemann-Liouville fractional derivative of order q > 0 of a continuous function f : (0, ∞) → R is defined by. [20] The generalized Riemann-Liouville fractional integral of order q > 0 and ρ > 0, of a function f (t), for all 0 < t < ∞, is defined as ρIqf (t). Substituting the values of c0 and c1 in (8), we obtain (6) It can be shown by direct computation that x given by the integral equation (6) satisfies the problem (5).

Consequently we have x

Thus x