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 $-A\text{\hspace{0.17em}⁣}({\int }_0^{1}b(1-s)(u(s){)}^{p(s)}ds)u^{\mathrm{\prime }\mathrm{\prime }}(t)=\lambda f(t,u(t))$ 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.