New Method for Computing zeros of monotone maps in Lebesgue spaces with applications to integral equations, fixed points, optimization, and variational inequality problems

TL;DR

This paper introduces a new algorithm for finding zeros of monotone maps in Lebesgue spaces and demonstrates its applications to integral equations, fixed points, optimization, and variational inequalities, supported by numerical examples.

Contribution

The paper presents a novel algorithm for approximating zeros of monotone maps in Lebesgue spaces and applies it to various mathematical problems with numerical validation.

Findings

Effective approximation of zeros in Lebesgue spaces.

Successful application to integral equations and optimization problems.

Numerical examples demonstrating the algorithm's performance.

Abstract

Let $E = L_p, \; 1<p\leq 2,$ and $A : E \to E^*$ be a bounded monotone map such that $0 \in R(A)$. In this paper, we introduce and study an algorithm for approximating zeros of $A$. Furthermore, we study the application of this algorithm to the approximation of Hammerstein integral equations, fixed points, convex optimization, and variational inequality problems. Finally, we present numerical and illustrative examples of our results and their applications.