Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem

Mohammad Esmael Samei,Jehad Alzabut,Ahmad Ahmadi,A George Maria Selvam,Shahram Rezapour

doi:10.1186/s13662-021-03631-2

Abstract

In this paper, we propose the conditions on which a class of boundary value problems, presented by fractional q-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution by means of the Schauder fixed point theorem. Then, the stability of solution will be discussed under the perturbations of boundary condition, a function existing in the problem, and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Highlights

In many applications fractional differential equations present more accurate models of phenomena than the ordinary differential equations
There have appeared many papers dealing with the existence of solutions for different types of fractional boundary value problems; see, for example, [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]
We investigate the conditions on which the fractional q-differential equation

Summary

Introduction

In many applications fractional differential equations present more accurate models of phenomena than the ordinary differential equations. There are several papers dealing with the existence and uniqueness of solutions to initial and boundary value problem of fractional order in Caputo or. In [56], authors studied the existence and uniqueness of solution for the fractional differential equation Dσ [y](t) = w(t, y(t), Dς [y](t)), where 2 < σ < 3, ς ∈ J0, via sum boundary conditions y(0) = 0, m–2. We recall that a problem is said to be well-posed if it has a uniqueness solution and this solution depends on a parameter in a continuous way This parameter, in the classical order differential equations, is dependent on the initial conditions and the function exists in the problem; whereas in the FDEs this dependency and the stability solution with respect to the perturbation of fractional order derivative should be taken into the account too [58].

Preliminaries and lemmas

Conclusion