The solution by iteration of nonlinear equations in Hilbert spaces

Abstract

The weak and strong convergence of the iterates generated by x k + 1 = ( 1 − t k ) x k + t k T x k ( t k ∈ R ) {x_{k + 1}} = (1 - {t_k}){x_k} + {t_k}T{x_k}({t_k} \in R) to a fixed point of the mapping T : C → C T:C \to C are investigated, where C is a closed convex subset of a real Hilbert space. The basic assumptions are that T has at least one fixed point in C, and that I − T I - T is demiclosed at 0 and satisfies a certain condition of monotony. Some applications are given.