Semilinear operator equations in real Hilbert spaces with Lipschitz nonlinearity

In this paper we establish an existence and uniqueness result for the semilinear equation Au + F(u) = f, making only the supposition that the nonlinearity F is Lipschitz operator. We use in this study a contractive method based on the Picard-Banach fixed point theorem (the method is frequently used in the study of the variational inequalities, see the proof of the Lions-Stampacchia theorem in [4], minimization methods for convex functionals in [3], a.s.o).