Integrable solutions of a generalized mixed-type functional integral equation

TL;DR

This paper establishes the existence of integrable solutions for a generalized nonlinear mixed-type functional integral equation on an unbounded interval using a fixed point theorem, with an illustrative example provided.

Contribution

It introduces new existence results for solutions of a complex class of functional integral equations using Krasnosel'skii's fixed point theorem.

Findings

Proves existence of solutions under certain conditions.

Extends previous results to a more general equation.

Provides an example illustrating the theoretical results.

Abstract

In this work, we prove the existence of integrable solutions for the following generalized mixed-type nonlinear functional integral equation $$x(t)=g\left(t,(Tx)(t)\right)+f\left(t,\int_0^t k(t,s)u(t,s,(Qx)(s))\;ds\right),\;t\in[0,\infty).$$ Our result is established by means of a Krasnosel'skii type fixed point theorem proved in [M.A. Taoudi: \textit{Integrable solutions of a nonlinear functional integral equation on an unbounded interval}, Nonlinear Anal. 71 (2009) 4131-4136]. In the last section we give an example to illustrate our result.