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.