Existence theorems for generalized Hammerstein equations

Abstract

In this paper we obtain existence theorems for generalized Hammerstein-type equations K ( u ) N u + u = 0 K(u)Nu + u = 0 , where for each u in the dual X ∗ {X^\ast } of a real reflexive Banach space X , K ( u ) : X → X ∗ X,K(u):X \to {X^\ast } is a bounded linear map and N : X ∗ → X N:{X^\ast } \to X is any map (possibly nonlinear). The method we adopt is totally different from the methods adopted so far in solving these equations. Our results in the reflexive space generalize corresponding results of Petry and Schillings.