Existence and uniqueness for delay fractional differential equations with mixed fractional derivatives

Ahmed Hallaci,Abdelouaheb Ardjouni,Hamid Boulares

doi:10.30538/psrp-oma2020.0059

Ahmed Hallaci, Abdelouaheb Ardjouni + Show 1 more

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https://doi.org/10.30538/psrp-oma2020.0059

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Abstract

Existence and uniqueness for delay fractional differential equations with mixed fractional derivatives

Highlights

The fractional differential equations is a hot topic of research due to its various applications in many scientific disciplines such as physics, chemistry, biology, engineering, viscoelasticity, signal processing, electrotechnical, electrochemistry and controllability, see [1–6] and the references therein
To the best of our knowledge, the use of mixed fractional derivative in neutral fractional differential equations which is an important type of fractional differential equations is still not sufficiently generalized
Motivated and inspired by above mentioned works, in this paper we investigate the existence and uniqueness of solutions for the following initial value problem of the mixed Riemann-Liouville and Caputo fractional functional differential equation with delay

Summary

Introduction

The fractional differential equations is a hot topic of research due to its various applications in many scientific disciplines such as physics, chemistry, biology, engineering, viscoelasticity, signal processing, electrotechnical, electrochemistry and controllability, see [1–6] and the references therein. Benchohra et al [7], investigated the existence of solutions for the following Riemann-Liouville fractional order functional differential equation with infinite delay. C Dα[u(t) − g(t, ut)] = f (t, ut), t ∈ (t0, ∞) , t0 ≥ 0, 0 < α < 1, ut0 = φ, and established the existence results of solutions of this problem by using Krasnoselskii’s fixed point theorem. In [9], Ahmad et al studied the existence and uniqueness of solutions to the following boundary value problem. Motivated and inspired by above mentioned works, in this paper we investigate the existence and uniqueness of solutions for the following initial value problem of the mixed Riemann-Liouville and Caputo fractional functional differential equation with delay. To show the existence of solutions, we transform (1) into an integral equation and use Krasnoselskii’s fixed point theorem.

Preliminaries

Main results