Nonlinear Implicit Generalized Hilfer-Type Fractional Differential Equations with Non-Instantaneous Impulses in Banach Spaces

Abdelkrim Sali̇m,Jamal Eddine Lazreg,Mouffak Benchohra,Johnny Henderson

doi:10.31197/atnaa.825294

Abstract

In the present article, we prove some results concerning the existence of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivative in Banach spaces. The results are based on fixed point theorems of Darbo and Monch associated with the technique of measure of noncompactness. An example is included to show the applicability of our results. .

Highlights

Fractional derivatives and fractional integrals generalize to noninteger order the derivative and the integral of a function
We prove some results concerning the existence of solutions for a class of initial value problem for nonlinear implicit fractional dierential equations with non-instantaneous impulses and generalized Hilfer fractional derivative in Banach spaces
The results are based on xed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness

Summary

Introduction

Fractional derivatives and fractional integrals generalize to noninteger order the derivative and the integral of a function. Focused on linear and nonlinear problems for fractional dierential equations involving dierent kinds of fractional derivatives, see, for example, [3, 7, 8, 10, 11, 13]. The class of problems for fractional dierential equations with abrupt and instantaneous impulses is vastly studied, and dierent topics on the existence and qualitative properties of solutions are considered, [15, 18, 28]. Motivated by the works mentioned above, in this paper, we establish existence results for the initial value problem of a nonlinear implicit generalized Hilfer-type fractional dierential equation with non-instantaneous impulses, ρDsα+k,βu (t) = f t, u(t), ρDsα+k,βu (t) ; t ∈ Ik, k = 0, . In the last section, we give an example to illustrate the applicability of our results

Preliminaries

Existence of Solutions

Example