Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

Abstract

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

“…This together with (24) gives that lim k→∞ T i x k − x k = 0 for all i = 1, • • • , m. Therefore, we also get (10). Further, from (21) with p replaced by p * , it follows (17). Thus, by Lemma 2.2, we reach that lim k→∞ x k − p * = 0.…”

“…where {t k } satisfies the condition (t), formulated bellow, with an additional assumption ∞ k=0 |t k − t k+m | < +∞ and either T = T m T m−1 ...T 1 or T = i∈L ω i T i with ω i ∈ (0, 1) and i∈L ω i = 1. Several results, related with method (2) in the case that C is not only the intersection of fixed point sets of nonexpansive mappings but also the set of common zeros of a finite family of nonlinear monotone-type mappings, are presented in [7]- [17], [45], [50], [51] and references therein.…”