Existence of solutions to nonlocal weighted Kirchhoff-type problems via topological degree theory

This paper investigates a weighted Kirchhoff-type equation driven by the fractional $p(x)-$Laplacian operator. The proposed model captures both nonlocal interactions and variable exponent effects, which naturally arise in several applied contexts. Under suitable structural assumptions, we analyze the associated nonlocal boundary value problem with Dirichlet conditions. By employing topologicaldegree methods, we prove the existence of weak solutions, thereby extending and enriching existing results on Kirchhoff-type problems involving fractional and variable exponent operators.