On the Qualitative Behaviors of Volterra-Fredholm Integro Differential Equations With Multiple Time-Varying Delays

Abstract

This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.