Boundary Value Problem for Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Impulses

Abdelkrim Salim,Mouffak Benchohra,Gaston N’Guérékata,Jamal Eddine Lazreg,Nasser-Eddine Tatar

doi:10.1155/2021/5592010

Abstract

This article deals with some existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer Fractional derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s and Schaefer’s fixed-point theorems.

Highlights

Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, finance, hydrology, electromagnetics, and viscoelasticity
A generalized Hilfer fractional derivative ρDαa+ of order α is defined by ðρDαa+
We are concerned with existence, uniqueness, and Ulam-Hyers-Rassias stability results for a class of boundary value problem for nonlinear implicit fractional differential equations with impulses and generalized Hilfer fractional derivative

Summary

Introduction

Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, finance, hydrology, electromagnetics, and viscoelasticity. There are numerous books and articles focused on linear and nonlinear initial and boundary value problems for fractional differential equations involving different kinds of fractional derivatives, see, for example, [1,2,3,4,5,6]. In [30], Harikrishnan et al investigated existence theory and different kinds of stability in the sense of Ulam, for the following initial value problem with nonlinear generalized Hilfer-type fractional differential equation and impulses:. Motivated by the works mentioned above, in this paper, we establish existence and uniqueness results to the boundary value problem with nonlinear implicit generalized Hilfer-type fractional differential equation and impulses: ρDαt+k,βu ðtÞ = f t, uðtÞ, ρDαt+k,βu ðtÞ , t ∈ Jk, k = 0, ⋯, m, ð2Þ ρ. We give examples to illustrate the applicability of our main results

Preliminaries

Existence of Solutions

Ulam-Hyers-Rassias Stability

Examples

Conclusion