On the Variational Method for the Existence of Solutions of Nonlinear Equations of Hammerstein Type

Abstract

Let Y be a real Banach space and X* its conjugate Banach space. Let A be an unbounded monotone linear mapping from A"to X* and Na potential mapping from X* to X. In this paper we establish the existence of a solution of the equation u + ANu = v for a given v in X* using variational method. Our method consists in using a splitting of A via an auxiliary Hubert space and solving an equivalent equation in this auxiliary Hubert space. In §2, we prove the same result in the case when Y is a Hubert space using the natural splitting of A in terms of its square root. We do this to compare and contrast the proofs in the two cases. Introduction. Let Y be a real Banach space and Y* denote its conjugate Banach space. Let A be an unbounded monotone linear mapping from .Yto X* and A/a nonlinear mapping from X* to Y satisfying no monotone hypothesis. In this paper we study the solvability of the equation