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.

Strong convergence of a composite Halpern-type iteration for a family of nonexpansive mappings in CAT(0) spaces

In this paper, we prove the strong convergence of the composite Halpern-type iteration for a family of nonexpansive mappings in CAT(0) spaces and compare our results with several recent results in this subject. Also, the inexact version of the Halpern iteration is studied in CAT(0) spaces.

Strong convergence of Halpern iteration for products of finitely many resolvents of maximal monotone operators in Banach spaces

In this paper, using Bregman functions, we introduce a new Halpern-type iterative algorithm for finding common zeros of finitely many maximal monotone operators and obtain a strongly convergent iterative sequence to the common zeros of these operators in a reflexive Banach space. Furthermore, we study Halpern-type iterative schemes for finding common solutions of a finite system of equilibrium problems and null spaces of a -inverse strongly monotone mapping in a 2-uniformly convex Banach space. Some applications of our results to the solution of equations of Hammerstein-type are presented. Our scheme has an advantage that we do not use any projection of a point on the intersection of closed and convex sets which creates some difficulties in a practical calculation of the iterative sequence. So the simple construction of Halpern iteration provides more flexibility in defining the algorithm parameters which is important from the numerical implementation perspective. Presented results improve and generalize many known results in the current literature.

Effective results on nonlinear ergodic averages in CAT spaces

In this paper we apply proof mining techniques to compute, in the setting of CAT$(\unicode[STIX]{x1D705})$ spaces (with $\unicode[STIX]{x1D705}>0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.