Fixed point theorems for a sum of two mappings in locally convex spaces

A new class of almost asymptotically nonexpansive type mappings in Banach spaces is introduced, which includes a number of known classes of nonlinear Lipschitzian mappings and non-Lipschitzian mappings in Banach spaces as special cases; for example, the known classes of nonexpansive mappings, asymptotically nonexpansive mappings and asymptotically nonexpansive type mappings. The convergence problem of modified Ishikawa iterative sequences with errors for approximating fixed points of almost asymptotically nonexpansive type mappings is considered. Not only S. S. Chang's inequality but also H. K. Xu's one for the norms of Banach spaces are applied to make the error estimate between the exact fixed point and the approximate one. Moreover, Zhang Shi-sheng's method (Applied Mathematics and Mechanics (English Edition), 2001,22(1):25–34) for making the convergence analysis of modified Ishikawa iterative sequences with errors is extended to the case of almost asymptotically nonexpansive type mappings. The new convergence criteria of modified Ishikawa iterative sequences with errors for finding fixed points of almost asymptotically nonexpansive type mappings in uniformly convex Banach spaces are presented. Also, the new convergence criteria of modified Mann iterative sequences with errors for this class of mappings are immediately obtained from these criteria. The above results unify, improve and generalize Zhang Shi-sheng's ones on approximating fixed points of asymptotically nonexpansive type mappings by the modified Ishikawa and Mann iterative sequences with errors.

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings

Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces

Fixed point theorem for [formula omitted]-nonexpansive mappings in Banach spaces

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

In this paper, we introduce a new class of mappings called Bregman weak relatively nonexpansive mappings and propose new hybrid iterative algorithms for finding common fixed points of an infinite family of such mappings in Banach spaces. We prove strong convergence theorems for the sequences produced by the methods. Furthermore, we apply our method to prove strong convergence theorems of iterative algorithms for finding common fixed points of finitely many Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. These algorithms take into account possible computational errors. We also apply our main results to solve equilibrium problems in reflexive Banach spaces. Finally, we study hybrid iterative schemes for finding common solutions of an equilibrium problem, fixed points of an infinite family of Bregman weak relatively nonexpansive mappings and null spaces of a γ-inverse strongly monotone mapping in 2-uniformly convex Banach spaces. Some application of our results to the solution of equations of Hammerstein-type is presented. Our results improve and generalize many known results in the current literature. MSC:47H10, 37C25.

Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications

Strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in Banach spaces

A long time ago …

This is a survey chapter. We present a brief development of fixed point theory for nonexpansive type mappings in Banach spaces.

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces.

ABSTRACTWe introduce a new type of nonexpansive mappings and obtain a number of existence and convergence theorems. This new class of nonlinear mapping properly contains nonexpansive, Suzuki-type generalized nonexpansive mappings and partially extends firmly nonexpansive and α-nonexpansive mappings. Also, this class of mapping need not be continuous. Some useful examples are presented to illustrate facts. Some prominent iteration processes are also compared using numerical computations.

Strong convergence theorems by Halpern–Mann iterations for relatively nonexpansive mappings in Banach spaces

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications