Christopher S. Goodrich

arXiv2026

A Unified Topological Analysis of Variable Growth Kirchhoff-Type Equations

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.

math.AP↗

arXiv2025

Luxemburg Norm Localisation for Nonlocal Differential Equations in Variable Exponent Lebesgue Spaces

We investigate a class of variable growth nonlocal differential equations of Kirchhoff-type having the general form \(-A\!\left(\int_0^1 b(1-s)\,\big(u(s)\big)^{p(s)}\,ds\right)\,u''(t) = \lambda\,f(t,u(t))\) for \(t\in(0,1)\), where \(A\) is a possibly sign-changing function. Our analysis is carried out in the variable-exponent Lebesgue space \(L^{p(\cdot)}([0,1])\) under the standing hypothesis \(p(t)>1\). We demonstrate that using the Luxemburg norm allows for a much sharper localisation of the solution to the nonlocal problem. Moreover, the conditions imposed on both \(\lambda\) and \(f\) are appreciably weakened when the problem is analysed within the Luxemburg norm framework. An example explicitly demonstrates both the qualitative and quantitative advantages over earlier techniques.

math.GM↗