Hybrid approximation of solutions of integral equations of the hammerstein type

Abstract

Let X be a uniformly convex and uniformly smooth real Banach space with dual X∗. Let F : X → X∗ and K : X∗ → X be continuous monotone operators. Suppose that the Hammerstein equation u + KFu = 0 has a solution in X. It is proved that a hybrid-type approximation sequence converges strongly to u∗, where u∗ is a solution of the equation u + KFu = 0. In our theorems, the operator K or F need not be defined on a compact subset of X and no invertibility assumption is imposed on K.