Existence and Ulam stability for implicit fractional q-difference equations

Mouffak Benchohra,Yong Zhou,Nadjet Laledj,Saïd Abbas

doi:10.1186/s13662-019-2411-y

Abstract

This paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

Highlights

1 Introduction Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics, and other applied sciences [33]
For some fundamental results in the theory of fractional calculus and fractional differential equations we refer the reader to the monographs [4,5,6, 23, 32, 39], the papers [24, 34, 36,37,38, 40] and the references therein
Considerable attention has been given to the existence of solutions of initial and boundary value problems for fractional differential equations and inclusions with Caputo fractional derivative; [5, 22]

Summary

Introduction

Fractional differential equations have recently been applied in various areas of engineering, mathematics, physics, and other applied sciences [33]. In this paper we discuss the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions for the following implicit fractional q-difference equation: cDαq u (t) = f t, u(t), cDαq u (t) , t ∈ I := [0, T], (1) Definition 2.4 ([8]) The Riemann–Liouville fractional q-integral of order α ∈ R+ := [0, ∞) of a function u : I → R is defined by (Iq0u)(t) = u(t), and

Results

Conclusion