A Novel Fixed Point Framework for Positive Solutions to Coupled Undamped Sturm-Liouville Systems

Abstract

This study addresses the existence of at least one positive and nondecreasing solution to a system of secondorder undamped Sturm-Liouville Boundary Value Problems (BVPs). By carefully analyzing the properties of the associated Green functions and applying a fixed-point theorem in a Banach space, we establish new existence results under broadly nonlinear and coupled boundary settings. To the best of our knowledge, this is the first work to derive such findings for coupled undamped Sturm-Liouville systems using this innovative analytical strategy. The proposed approach extends classical methods to effectively accommodate nonlinear interdependencies and atypical boundary conditions, thereby providing a novel perspective for the theory of BVPs. Unlike previous studies that mainly address damped systems or single-equation models, this work fills an important gap by considering the more challenging undamped coupled case. The findings extend classical methods and contribute to a deeper understanding of nonlinear BVPs under nonstandard and interdependent boundary structures.