A hybrid Krasnosel'ski\u{i}-Schauder fixed point theorem for systems

TL;DR

This paper introduces a new fixed point theorem combining Krasnosel'ski
21 and Schauder methods to localize solutions of nonlinear operator systems, proving multiple solutions and applying results to integral equations and boundary value problems.

Contribution

It develops a hybrid fixed point theorem that improves solution localization for nonlinear systems, extending existing methods with new theoretical and numerical insights.

Findings

Established a localization of solutions within a conical shell and convex set.

Proved the existence of multiple solutions for the system.

Provided numerical solutions consistent with theoretical results.

Abstract

We provide new results regarding the localization of the solutions of nonlinear operator systems. We make use of a combination of Krasnosel'ski\u{\i} cone compression-expansion type methodologies and Schauder-type ones. In particular we establish a localization of the solution of the system within the product of a conical shell and of a closed convex set. By iterating this procedure we prove the existence of multiple solutions. We illustrate our theoretical results by applying them to the solvability of systems of Hammerstein integral equations. In the case of two specific boundary value problems and with given nonlinearities, we are also able to obtain a numerical solution, consistent with our theoretical results.