  In this paper, we prove the existence of a weak solution for the Dirichlet
boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla
u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using
the degree theory after turning the problem into a Hammerstein equation. The
right hand side $g$ is a possibly discontinuous function in the second variable
satisfying some non-standard growth conditions..
