The aim of this paper is to study the existence of solutions of multi- valued nonlinear elliptic problems involving the natural growth term, measure data and the general p(.)-Leray-Lions type operator. Using a decomposition of Radon diffuse measure due to Nyanquini et al.(see [26]) and approximation method, we construct an approximate problem and solve it by the technic of maximal monotone operators in Banach spaces. We prove that the approximate solutions converge to the solution of the studied problem