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

TL;DR

This paper introduces a Luxemburg norm-based approach to localize solutions of nonlocal Kirchhoff-type differential equations with variable exponents, resulting in sharper analysis and weaker conditions compared to previous methods.

Contribution

It develops a novel Luxemburg norm framework for variable exponent spaces, improving solution localization and relaxing conditions on parameters and functions.

Findings

Sharper solution localization achieved

Weaker conditions on nd f for existence results

Explicit example demonstrating advantages

Abstract

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.