This paper deals with a class of $p(x)$-Kirchhoff type problems involving the $p(x)$-Laplacian-like operators, arising from the capillarity phenomena, depending on two real parameters with Dirichlet boundary conditions. Using a topological degree for a class of demicontinuous operators of generalized $(S_{+})$ type and the theory of the variable exponent Sobolev spaces, we prove the existence of weak solutions of this problem.