scholarly journals Existence of fixed points of weak enriched nonexpansive mappings in Banach spaces

2021 ◽  
Vol 37 (2) ◽  
pp. 287-294
Author(s):  
SUTHEP SUANTAI ◽  
DAWAN CHUMPUNGAM ◽  
PANITARN SARNMETA
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
New Class ◽  
Nonlinear Mappings

In this work, we introduce and study a new class of weak enriched nonexpasive mappings which is a generalization of enriched nonexpansive mappings provided by Berinde [Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces], Carpathian J. Math., 35 (2019), No. 3, 293–304]. This class of mappings generalizes several important classes of nonlinear mappings. We prove some fixed point theorems regarding this kind of mappings which extend some important results in [Berinde, V., Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304]. Moreover, some examples, to ensure the existence of these mappings and support our main theorems, are also given.

scholarly journals Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition

2020 ◽  
Vol 36 (1) ◽  
pp. 27-34 ◽  
Author(s):  
VASILE BERINDE
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Displacement Condition ◽  
Nonlinear Mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

scholarly journals Fixed Point Problems for Nonexpansive Mappings in Bounded Sets of Banach Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Xianbing Wu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Process ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Fixed Point Problems ◽  
Bounded Sets

It is well known that nonexpansive mappings do not always have fixed points for bounded sets in Banach space. The purpose of this paper is to establish fixed point theorems of nonexpansive mappings for bounded sets in Banach spaces. We study the existence of fixed points for nonexpansive mappings in bounded sets, and we present the iterative process to approximate fixed points. Some examples are given to support our results.

scholarly journals Fixed points and their approximation in Banach spaces for certain commuting mappings

1982 ◽  
Vol 23 (1) ◽  
pp. 1-6
Author(s):  
M. S. Khan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Contraction Mappings ◽  
Continuous Mappings ◽  
Banach Contraction ◽  
The Fixed Point Theorem

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.

scholarly journals Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces

2019 ◽  
Vol 35 (3) ◽  
pp. 293-304
Author(s):  
VASILE BERINDE ◽  
◽  
Keyword(s):  
Fixed Points ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Strong And Weak Convergence ◽  
New Class ◽  
Contractive Mappings ◽  
Contractive Type ◽  
First Time ◽  
Nonlinear Mappings

Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, presented here for the first time, we introduce and study the class of enriched nonexpansive mappings in Hilbert spaces. In order to approximate the fixed points of enriched nonexpansive mappings we use the Krasnoselskij iteration for which we prove strong and weak convergence theorems. Examples to illustrate the richness of the new class of contractive mappings are also given. Our results in this paper extend some classical convergence theorems established by Browder and Petryshyn in [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228] from the case of nonexpansive mappings to that of enriched nonexpansive mappings,thus including many other important related results from literature as particular cases.

scholarly journals Approximating Fixed Points of Bregman Generalized α-Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 709 ◽  
Author(s):  
Kanikar Muangchoo ◽  
Poom Kumam ◽  
Yeol Je Cho ◽  
Sompomg Dhompongsa ◽  
Sakulbuth Ekvittayaniphon
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Weak And Strong Convergence ◽  
Convergence Theorems ◽  
Iterative Schemes ◽  
Numerical Example ◽  
New Class

In this paper, we introduce a new class of Bregman generalized α -nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized α -nonexpansive mappings in Banach spaces. A numerical example is given to illustrate the main results of fixed point approximation using Halpern’s algorithm.

scholarly journals Fixed point theorems of discontinuous increasing operators and applications to nonlinear integro-differential equations

2000 ◽  
Vol 158 ◽  
pp. 73-86
Author(s):  
Jinqing Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theorems ◽  
Existence Theorems ◽  
Maximal And Minimal Solutions

AbstractIn this paper, we obtain some new existence theorems of the maximal and minimal fixed points for discontinuous increasing operators in C[I,E], where E is a Banach space. As applications, we consider the maximal and minimal solutions of nonlinear integro-differential equations with discontinuous terms in Banach spaces.

scholarly journals Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

scholarly journals Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Eskandar Naraghirad ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Fixed Point Theorems ◽  
Weak And Strong Convergence ◽  
Bregman Distances ◽  
Opial Property ◽  
Nonspreading Mappings

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.

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