Qualitative analysis of nonlinear implicit neutral differential equation of fractional order

H. H. G. Hashem; Hessah O. Alrashidi; H. H. G. Hashem; Hessah O. Alrashidi

doi:10.3934/math.2021220

AIMS Mathematics

2021, Volume 6, Issue 4: 3703-3719. doi: 10.3934/math.2021220

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Research article

Qualitative analysis of nonlinear implicit neutral differential equation of fractional order

H. H. G. Hashem ^1,2,
Hessah O. Alrashidi ^{2
,
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1.
Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt
2.
Department of mathematics, College of Science, Qassim University, P. O. Box 6644 Buraidah 51452, Saudi Arabia

Received: 03 December 2020 Accepted: 14 January 2021 Published: 25 January 2021
MSC : 26A33, 34K37, 35B35

In this paper, we discuss sufficient conditions for the existence of solutions for a class of Initial value problem for an neutral differential equation involving Caputo fractional derivatives. Also, we discuss some types of Ulam stability for this class of implicit fractional-order differential equation. Some applications and particular cases are presented. Finally, the existence of at least one mild solution for this class of implicit fractional-order differential equation on an infinite interval by applying Schauder fixed point theorem and the local attractivity of solutions are proved.
- neutral differential equation,
- Caputo fractional derivative,
- Ulam stability,
- Ulam-Hyers stability,
- mild solution,
- attractivity of solutions
Citation: H. H. G. Hashem, Hessah O. Alrashidi. Qualitative analysis of nonlinear implicit neutral differential equation of fractional order[J]. AIMS Mathematics, 2021, 6(4): 3703-3719. doi: 10.3934/math.2021220

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Abstract

In this paper, we discuss sufficient conditions for the existence of solutions for a class of Initial value problem for an neutral differential equation involving Caputo fractional derivatives. Also, we discuss some types of Ulam stability for this class of implicit fractional-order differential equation. Some applications and particular cases are presented. Finally, the existence of at least one mild solution for this class of implicit fractional-order differential equation on an infinite interval by applying Schauder fixed point theorem and the local attractivity of solutions are proved.

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