We approximate a locally unique solution of a generalized equations in a Banach space setting using a new midpoint methods (see (1.2) and (3.10)). An existence-convergence theorem and a radius of convergence are given under Lipschitz and center-Lipschitz conditions on the first order Frechet derivative and Lipschitz-like continuity property of set-valued mappings. We show that our method (1.2) is locally quadratically convergent using a fixed points theorem [10]. Motivated by optimization considerations [3], [4] related to the resolution on nonlinear equations, a smaller ratio and a larger radius of convergence are also provided. Our methods extend the midpoint method related to the resolution of nonlinear equations [7].