We investigate a class of variable growth nonlocal differential equations of Kirchhoff-type having the general form \(\begin{aligned} -A\!\left( \int _0^1 b(1-s)\big (u(s)\big )^{p(s)}\,ds\right) u''(t) = \lambda f\big (t,u(t)\big ), \quad t\in (0,1), \end{aligned}\) 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 sharper localisation of the solution to the nonlocal problem. Moreover, conditions imposed on both \(\lambda \) and f are appreciably weakened by analysing the problem within the Luxemburg norm framework. An example explicitly demonstrates both the qualitative and the quantitative advantages over earlier techniques.
