Existence and stability analysis of solution for fractional delay differential equations

TL;DR

This paper investigates the existence, uniqueness, and stability of solutions for fractional delay differential equations, employing Bielecki norm and Burton's method, and notably omits typical contraction conditions.

Contribution

It provides new proofs for solution existence, uniqueness, and Hyers-Ulam stability of fractional delay differential equations without requiring contraction constant conditions.

Findings

Proved existence and uniqueness of solutions using Bielecki norm.

Established solution uniqueness for constant delay form via Burton's method.

Demonstrated Hyers-Ulam stability for the considered equations.

Abstract

In this article, we give some results for fractional-order delay differential equations. In the first result, we prove the existence and uniqueness of solution by using Bielecki norm effectively. In the second result, we consider a constant delay form of this problem. Then we apply Burton's method to this special form to prove that there is only one solution. Finally, we prove a result regarding the Hyers-Ulam stability of this problem. Moreover, in these results, we omit the conditions for contraction constants seen in many papers.