Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions

Bashir Ahmad,Ahmed Alsaedi,Sotiris K. Ntouyas

doi:10.3390/fractalfract3020021

Abstract

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–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.

Highlights

Fractional differential equations have gained much importance due to their widespread applications in various disciplines of social and natural sciences, and engineering
There has been a remarkable development in fractional calculus and fractional differential equations; for instance, see the monographs by Kilbas et al [1], Lakshmikantham et al [2], Miller and Ross [3], Podlubny [4], Samko et al [5], Diethelm [6], Ahmad et al [7] and the papers [8,9,10,11,12,13,14,15,16]
We prove the existence of solutions for the boundary value problem (6) with a non-convex-valued right-hand side by applying a fixed-point theorem for multi-valued map due to Covitz and Nadler [27]

Summary

Introduction

Fractional differential equations have gained much importance due to their widespread applications in various disciplines of social and natural sciences, and engineering. One can find many works on boundary value problems containing mixed fractional derivatives of different types. In [17] the authors studied a new class of nonlinear differential equations with Caputo-type fractional derivatives of different orders, and Caputo-type integro-differential boundary conditions: Dα[Dβx(t) − g(t, x(t))] = f (t, x(t)), t∈ J := [0, T], (1). In [18] the authors considered two Caputo–Hadamard type fractional derivatives in a neutral-type differential equation supplemented with Dirichlet boundary conditions: Dω[Dκy(t) − h(t, y(t))] = f (t, y(t)), t ∈ J := [1, T], T > 1,

Results

Discussion

Conclusion