A Unified Topological Analysis of Variable Growth Kirchhoff-Type Equations

TL;DR

This paper develops a unified topological framework for analyzing Kirchhoff-type equations with variable growth, establishing the existence of positive solutions across different growth regimes using fixed point theory.

Contribution

It introduces a novel approach that handles variable exponent growth in nonlocal Kirchhoff equations, unifying convex, concave, and mixed growth cases.

Findings

Existence of at least one positive solution for variable growth Kirchhoff equations.

Unified treatment of convex, concave, and mixed growth regimes.

Application of topological fixed point theory to nonlocal problems.

Abstract

We consider a nonlocal differential equation of Kirchhoff type with a convolution coefficient involving variable growth. The novelty of our work lies in allowing a variable exponent in the nonlocal term. By relating the variable growth problem to a corresponding constant growth problem, we establish the existence of at least one positive solution subject to boundary conditions. Our approach relies on topological fixed point theory. The results treat convex, concave, and mixed growth regimes, providing a unified framework for one-dimensional Kirchhoff-type problems.