On the variational method for the existence of solutions of nonlinear equations of Hammerstein type
Djairo Guedes De Figueiredo, Chaitan P. Gupta · Proceedings of the American Mathematical Society · 1973
Let X X be a real Banach space and X ∗ {X^ \ast } its conjugate Banach space. Let A A be an unbounded monotone linear mapping from X X to X ∗ {X^ \ast } and N N a potential mapping from X ∗ {X^ \ast } to X X . In this paper we establish the existence of a solution of the equation u + A N u = v u + ANu = v for a given v v in X ∗ {X^ \ast } using variational method. Our method consists in using a splitting of A A via an auxiliary Hilbert space and solving an equivalent equation in this auxiliary Hilbert space. In §2, we prove the same result in the case when X X is a Hilbert space using the natural splitting of A A in terms of its square root. We do this to compare and contrast the proofs in the two cases.