Hybrid and Component-wise Leggett-Williams type Fixed Point Theorems in Product Spaces with Applications

TL;DR

This paper develops new fixed point theorems in product spaces using Leggett-Williams and hybrid conditions, leading to multiple solutions for nonlinear boundary value problems.

Contribution

It introduces novel multiplicity fixed point theorems combining Leggett-Williams and Krasnosel'skii conditions in product spaces.

Findings

Existence of nine fixed points under Leggett-Williams conditions.

Identification of four coexistence fixed points with all components nontrivial.

Establishment of three fixed points using hybrid conditions.

Abstract

In this paper, we present new multiplicity fixed point theorems for operators acting on Cartesian products of two normed linear spaces. We show that Leggett-Williams type conditions in each component of the system guarantee the existence of nine distinct fixed points, of which four of them are coexistence fixed points, i.e., points with all components nontrivial. In addition, a hybrid approach combining Leggett-Williams conditions in one component with Krasnosel'skii compression-expansion conditions in the other allows us to obtain three fixed points. As an application, we establish the existence of multiple positive solutions for nonlinear systems of second-order equations with two-point boundary conditions.