Algorithms for Hammerstein inclusions in certain Banach spaces

Abstract

Let E be a reflexive smooth and strictly convex real Banach space. Let F : E → 2E and K : E∗ → E be bounded maximal monotone mappings such that D(F) = E and R(F) = D(K) = E∗. Suppose that the Hammerstein inclusion 0 ∈ u+ KFu has a solution in E. We present in this paper a new algorithm for approximating solutions of the inclusion 0 ∈ u+ KFu. Then we prove strong convergence theorems. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings. Furthermore, our technique of proof is of independent interest.