Asymptotic stability for mixed fractional delay differential equations

Abdelouaheb Ardjouni̇,Hamid Boulares,Ahmed Hallaci̇

doi:10.31197/atnaa.607459

Abstract

This paper is concerned with the stability analysis of nonlinear mixed fractional delay differential equations using Krasnoselskii's fixed point theorem in a weighted Banach space.

Highlights

Fractional differential equations have become an important field of applied mathematics and modeling of many physical phenomena associated to very rapid and very short changes, for more details we refer to the books ([7, 11, 15, 16, 20, 21]) and the references therein
Initial value problems and boundary value problems related to the qualitative theory of the existence, uniqueness and stability of solutions for fractional differential equations have been mainly discussed by a lot of authors especially in last three decades or so
Motivated by the works mentioned above and the papers [1, 2, 3, 8, 10, 12, 13, 14, 17, 18, 19, 23, 24] and the references therein, We aim to enrich the field of differential equations by talking about the analysis of qualitative theory of the subjects of the stability and asymptotic stability of the zero solution to the following initial value problem of mixed Riemann-Liouville-Caputo fractional differential equations with delay on unbounded interval

Summary

Introduction

Fractional differential equations have become an important field of applied mathematics and modeling of many physical phenomena associated to very rapid and very short changes, for more details we refer to the books ([7, 11, 15, 16, 20, 21]) and the references therein. Agarwal et al [4], investigated an existence result for the following Caputo fractional order functional differential equations with delay using the Krasnoselskii’s fixed point theorem. Ge and Kou [9], by utilizing the Krasnoselskii’s fixed point theorem, discussed the stability and asymptotic stability of the zero solution to the following Caputo type fractional differential equation. Motivated by the works mentioned above and the papers [1, 2, 3, 8, 10, 12, 13, 14, 17, 18, 19, 23, 24] and the references therein, We aim to enrich the field of differential equations by talking about the analysis of qualitative theory of the subjects of the stability and asymptotic stability of the zero solution to the following initial value problem of mixed Riemann-Liouville-Caputo fractional differential equations with delay on unbounded interval. M is relatively compact in Cλ if the following conditions are satisfied i) e−λtu(t) : u ∈ M is uniformly bounded, ii) e−λtu(t) : u ∈ M is equicontinuous on any compact interval of R, iii) e−λtu(t) : u ∈ M is equiconvergent at infinity, i.e. for any given > 0, there exists a T0 > 0 such that for all u ∈ M and t1, t2 > T0, it holds e−λt u(t2) − e−λt u(t1)

Main results

Conclusion