Fixed point theorems for -nonexpansive mappings

Fixed Point Theory and Applications, 2012

We introduce a condition on mappings, namely condition (K). In a uniformly convex Banach space, the condition is weaker than quasi-nonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition (K) and a multivalued mapping satisfying conditions (E) and (C λ ) for some λ ∈ (0, 1).

Fixed Point Theory and Applications, 2011

Bruck [Pac. J. Math. 53, 59-71 1974 Theorem 1] proved that for a nonempty closed convex subset E of a Banach space X, if E is weakly compact or bounded and separable and suppose that E has both (FPP) and (CFPP), then for any commuting family S of nonexpansive self-mappings of E, the set F(S) of common fixed points of S is a nonempty nonexpansive retract of E. In this paper, we extend the above result when one of its elements in S is multivalued. The result extends previously known results (on common fixed points of a pair of single valued and multivalued commuting mappings) to infinite number of mappings and to a wider class of spaces.

arXiv (Cornell University), 2016

Let C be a nonempty, bounded, closed, and convex subset of a Banach space X and T : C → C be a monotone asymptotic nonexpansive mapping. In this paper, we investigate the existence of fixed points of T. In particular, we establish an analogue to the original Goebel and Kirk's fixed point theorem for asymptotic nonexpansive mappings.

Glasgow Mathematical Journal, 1982

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 obtainedfix...

Abstract and Applied Analysis, 1998

LetXbe a Banach space andτa topology onX. We say thatXhas theτ-fixed point property (τ-FPP) if every nonexpansive mappingTdefined from a bounded convexτ-sequentially compact subsetCofXintoChas a fixed point. Whenτsatisfies certain regularity conditions, we show that normal structure assures theτ-FPP and Goebel-Karlovitz's Lemma still holds. We use this results to study two geometrical properties which imply theτ-FPP: theτ-GGLD andM(τ)properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of theτ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach spaceXsuch that theτ-FPP is shared by any isomorphic Banach spaceYsatisfying that the Banach-Mazur distance betweenXandYis less than some of these constants.

2012

In this thesis we study the existence of a retraction of a closed subset of a Banach space. Then we introduce and study a three-step iterative process with viscosity to approximate common fixed points for asymptotically quasi-nonexpansive nonself mappings in Banach spaces. Criteria for strong convergence of such iteration is given. We also introduce and study a multi-step iterative schemes with viscosity to approximate of common fixed points of finite family for asymptotically quasi-nonexpansive nonself mappings in Banach spaces. Finally, weak and strong convergence theorems for such iteration in uniformly convex Banach spaces are established under some sufficient conditions.

Abstract and Applied …, 2003

Let X be a Banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict Opial condition. Let C be a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C, and T a nonexpansive mapping from C into KC(C). We prove that T has a fixed point. The nonstrict Opial condition can be removed if, in addition, T is a 1-χcontractive mapping.

Journal of Applied Mathematics, 2012

Proceedings of the American Mathematical Society, 2018

Let C be a nonempty, bounded, closed, and convex subset of a Banach space X and T : C → C be a monotone asymptotically nonexpansive mapping. In this paper, we investigate the existence of fixed points of T. In particular, we establish an analogue to the original Goebel and Kirk's fixed point theorem for asymptotically nonexpansive mappings.

Arabian Journal of Mathematics, 2012

Recall that a Banach space X has the weak fixed point property if for any nonempty weakly compact subset C of X and any nonexpansive mapping T : C→C, T has at least one fixed point. In this article, we present three recent results using the ultraproduct technique. We also provide some open problems in this area.

Nonlinear Analysis: Theory, Methods & Applications, 2007

In the present paper some common fixed point theorems for a sequence and a pair of nonself-mappings in complete metrically convex metric spaces are proved which generalize such results due to Khan et al. Some fixed point theorems in metrically convex spaces, Georgian Math. J. 7 (3) (2000) 523-530], Assad [N.A. Assad, On a fixed point theorem of Kannan in Banach spaces, Tamkang J. Math. 7 (1976) 91-94], Chatterjea [S.K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci. 25 (1972) 727-730] and several others. Some related results are also discussed.

Fixed Point Theory and Algorithms for Sciences and Engineering, 2024

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces.

Fixed Point Theory and Applications, 2010

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

2015

In this paper, we established some weak and strong convergence theorems for common fixed points of three nonself asymptotically Banach spaces. Our results extended and improve the result announed by Wang[6] [Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323(2006)550-557.] and WeiQiDeng, Lin Wang and Yi-Juan Chen[13] [Strong and Weak Convergence Theorems for common fixed points of two asymptotically nonexpansive mappings in Banach spaces, International Mathematical Forum, Vol. 7, 2012, no. 9, 407 – 417.] For a smooth banach space E, let us assume that K is a nonempty closed convex subset of with P as a sunny nonexpansive retraction. Let,T1, T2, T3 : K → E be three weakly inward nonself asymptotically nonexpansive mappings with respect to P with three sequences kn i ∁ [1,∞) satisfying (kn i ∞ n=1 −1) < ∞ ,(i=1,2,3) and F(T1)∩ F(T2)∩ F T3 = xεk, T1x = T2x = T3x = x respectively . For any given x1 ∈...

Journal of Functional Analysis, 2006

It is shown that if the modulus X of nearly uniform smoothness of a reflexive Banach space satisfies X (0) < 1, then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory.