Existence, uniqueness, and approximation for solutions of a functional-integral equation in $L^p$ spaces

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to a general functional-integral equation in $L^p$ spaces, and demonstrates approximation methods with numerical examples.

Contribution

It provides new existence and uniqueness criteria for solutions in $L^p$ spaces and applies fixed point and approximation methods with numerical validation.

Findings

Existence and uniqueness conditions in $L^p$ spaces.

Application of Banach Fixed Point Theorem and successive approximation.

Numerical examples illustrating the theoretical results.

Abstract

In this work we consider the general functional-integral equation: \begin{equation*} y(t) = f\left(t, \int_{a}^{b} k(t,s)g(s,y(s))ds\right), \qquad t\in [a,b], \end{equation*} and give conditions that guarantee existence and uniqueness of solution in $L^p([a,b])$, with $1<p<\infty$. We use Banach Fixed Point Theorem} and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.