Solvability and asymptotic stability of a nonlinear functional-integral equation

Abstract

The aim of this work is to study the nonlinear functional-integral equation

$$x\left(t\right)=f(t,x\left(t\right),{\int }_0^{t}u(t,s,x\left(a\left(s\right)\right),x\left(b\left(s\right)\right))ds)\text{,}\forall t\in {\mathbb{R}}_{+}\text{.}$$

Using the technique of the measure of noncompactness and the Darbo fixed-point theorem, we prove the existence and asymptotic stability of solutions for the equation. A nontrivial example is given to support our result.