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 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 Approximating Fixed Points of Nonself Contractive Type Mappings in Banach Spaces Endowed with a Graph

2016 ◽  
Vol 24 (2) ◽  
pp. 27-43 ◽  
Author(s):  
Laszlo Balog ◽  
Vasile Berinde ◽  
Mădălina Păcurar
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Closed Subset ◽  
Functional Equations ◽  
Fixed Point Theorems ◽  
Contraction Principle ◽  
Nonlinear Functional ◽  
Contractive Type

Abstract Let K be a non-empty closed subset of a Banach space X endowed with a graph G. We obtain fixed point theorems for nonself G-contractions of Chatterjea type. Our new results complement and extend recent related results [Berinde, V., Păcurar, M., The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936; Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press)] and thus provide more general and flexible tools for studying nonlinear functional equations.

scholarly journals The concept of cone b-Banach space and fixed point theorems

2021 ◽  
Vol 19 (1) ◽  
pp. 1187-1196
Author(s):  
Chen Yang ◽  
Xiaolin Zhu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Fixed Point Theorems

Abstract In this article, the concepts of cone b-norm and cone b-Banach space are given. Some new fixed point theorems in cone b-Banach spaces are established. The new results improve some fixed point theorems in cone Banach spaces. Furthermore, we also investigate the uniqueness of fixed points.

scholarly journals Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets

2003 ◽  
Vol 2003 (2) ◽  
pp. 83-91 ◽  
Author(s):  
Wieslawa Kaczor
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Pairwise Disjoint ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Nonconvex Sets ◽  
Opial’S Property ◽  
Asymptotically Regular

It is shown that ifXis a Banach space andCis a union of finitely many nonempty, pairwise disjoint, closed, and connected subsets{Ci:1≤i≤n }ofX, and eachCihas the fixed-point property (FPP) for asymptotically regular nonexpansive mappings, then any asymptotically regular nonexpansive self-mapping ofChas a fixed point. We also generalize the Goebel-Schöneberg theorem to some Banach spaces with Opial's property.

scholarly journals Common fixed points of two nonexpansive mappings in Banach spaces

2004 ◽  
Vol 69 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Tomonari Suzuki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

In this paper, we discuss a necessary and sufficient condition for common fixed points of two nonexpansive mappings. We then prove a convergence theorem to a common fixed point. Finally, we discuss the existence of a nonexpansive retraction onto the set of common fixed points of nonexpansive mappings. In these theorems, we do not assume the strict (uniform) convexity of the norm of the Banach space.

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.

Some convergence results of M iterative process in Banach spaces

2019 ◽  
pp. 2150017 ◽  
Author(s):  
Kifayat Ullah ◽  
Faiza Ayaz ◽  
Junaid Ahmad
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Process ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Weak And Strong Convergence ◽  
Convergence Results

In this paper, we prove some weak and strong convergence results for generalized [Formula: see text]-nonexpansive mappings using [Formula: see text] iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196].

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