Ulam\u2013Hyers stability of impulsive integrodifferential equations with Riemann\u2013Liouville boundary conditions

Akbar Zada,Jehad Alzabut,Hira Waheed,Ioan-Lucian Popa

doi:10.1186/s13662-020-2534-1

Abstract

This paper is concerned with a class of impulsive implicit fractional integrodifferential equations having the boundary value problem with mixed Riemann–Liouville fractional integral boundary conditions. We establish some existence and uniqueness results for the given problem by applying the tools of fixed point theory. Furthermore, we investigate different kinds of stability such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we give two examples to demonstrate the validity of main results.

Highlights

1 Introduction During the last few decades, boundary value problems of fractional differential equations have been utilized in different problems of applied nature; for example, we can find it in analytical formulations of systems and processes
Due to a more accurate behavior of fractional differential equations, it got the interest of research community in various applied fields of sciences such as chemistry, engineering, mechanics, physics, and so on
Integral boundary conditions are very important in the solutions of many practical systems [1, 51]

Summary

Introduction

During the last few decades, boundary value problems of fractional differential equations have been utilized in different problems of applied nature; for example, we can find it in analytical formulations of systems and processes. 2.2 Existence and uniqueness solution for system (1.4) we consider the coupled system of nonlinear implicit fractional differential equation with impulsive conditions from (1.4). Definition 3.16 Problem (1.4) is said to be generalized Hyers–Ulam–Rassias stable with respect to ψr,p = (ψr, ψp) ∈ C1(J , R) if there exists a constant Cψr,ψp = max(Cψr , Cψp ) > 0 such that, for each solution (ω, y) ∈ X × Y of (3.9), there exists a solution (ω∗, y∗) ∈ X × Y of (1.4) with (ω, y)(τ ) – ω∗, y∗ (τ ) X ×Y ≤ Cψr,ψp ψr,p for all τ ∈ J. For all ω, ω ∈ M, y, y ∈ R, and τ ∈ [0, 1], we have

Υ1 ω 3

Υj y 4