An Algorithm for Solutions of Hammerstein Integral Equations with Monotone Operators

Abstract

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X ∗ . Let F : X → X ∗ and K : X ∗ → X be bounded monotone mappings such that the Hammerstein equation u + KF u = 0 has a solution in X. An explicit iteration sequence is constructed and proved to converge strongly to a solution of the equation. This is achieved by combining geometric properties of uniformly convex and uniformly smooth real Banach spaces recently introduced by Alber with our method of proof which is also of independent interest.