Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {T n (x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

Weak and strong convergence theorems and a nonlinear ergodic theorem for N-generalized hybrid mappings

In this paper, assuming an appropriate condition, we prove that [Formula: see text]-generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern–Ishikawa type and a nonlinear ergodic theorem for [Formula: see text]-generalized hybrid mappings have been proven in Hilbert spaces.

A UNIFORM QUANTITATIVE FORM OF SEQUENTIAL WEAK COMPACTNESS AND BAILLON'S NONLINEAR ERGODIC THEOREM

We apply proof-theoretic techniques of "proof mining" to obtain an effective uniform rate of metastability in the sense of Tao for Baillon's famous nonlinear ergodic theorem in Hilbert space. In fact, we analyze a proof due to Brézis and Browder of Baillon's theorem relative to the use of weak sequential compactness. Using previous results due to the author we show the existence of a bar recursive functional Ω* (using only lowest type bar recursion B0, 1) providing a uniform quantitative version of weak compactness. Primitive recursively in this functional (and hence in T0 + B0, 1) we then construct an explicit bound φ on for the metastable version of Baillon's theorem. From the type level of φ and another result of the author it follows that φ is primitive recursive in the extended sense of Gödel's T. In a subsequent paper also Ω* will be explicitly constructed leading to the refined complexity estimate φ ∈ T4.

Fixed Point Theorem and Nonlinear Ergodic Theorem for Nonexpansive Semigroups Without Convexity

We first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

A nonlinear ergodic theorem for asymptotically nonexpansive mappings

Let X be a real uniformly convex Banach space satisfying the Opial's condition, C a bounded closed convex subset of X, and T: C → C an asymptotically non-expansive mapping. Then we show that for each x in C, the sequence {Tnx} almost converges weakly to a fixed point y of T, that is,This implies that {Tnx} converges weakly to y if and only if T is weakly asymptotically regular at x, that is, weak- . We also present a weak convergence theorem for asymptotically nonexpansive semigroups.