Goebel and Kirk fixed point theorem for multivalued asymptotically nonexpansive mappings

2017 ◽  
Vol 33 (3) ◽  
pp. 335-342
Author(s):  
M. A. KHAMSI ◽  
◽  
A. R. KHAN ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Nonexpansive Mapping ◽  
Point Theorem ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Mann Iteration ◽  
Asymptotically Nonexpansive ◽  
Mann Iteration Process

We introduce the concept of a multivalued asymptotically nonexpansive mapping and establish Goebel and Kirk fixed point theorem for these mappings in uniformly hyperbolic metric spaces. We also define a modified Mann iteration process for this class of mappings and obtain an extension of some well-known results for singlevalued mappings defined on linear as well as nonlinear domains.

scholarly journals Mann iteration process for monotone nonexpansive mappings with a graph

2019 ◽  
Vol 26 (4) ◽  
pp. 629-636
Author(s):  
Monther Rashed Alfuraidan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Mann Iteration ◽  
Iteration Sequence ◽  
Mann Iteration Process ◽  
Monotone Nonexpansive Mapping

Abstract Let {(X,\lVert\,\cdot\,\rVert)} be a Banach space. Let C be a nonempty, bounded, closed and convex subset of X and let {T:C\rightarrow C} be a G-monotone nonexpansive mapping. In this work, it is shown that the Mann iteration sequence defined by x_{n+1}=t_{n}T(x_{n})+(1-t_{n})x_{n},\quad n=1,2,\dots, proves the existence of a fixed point of G-monotone nonexpansive mappings.

scholarly journals Existence and approximation of fixed points in convex metric spaces

2014 ◽  
Vol 30 (2) ◽  
pp. 175-185
Author(s):  
HAFIZ FUKHAR-UD-DIN ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Uniformly Convex Banach Spaces ◽  
Convex Metric Spaces ◽  
Asymptotically Nonexpansive ◽  
Generalized Nonexpansive Mapping ◽  
The Fixed Point Theorem

A fixed point theorem for a generalized nonexpansive mapping is established in a convex metric space introduced by Takahashi [A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142–149]. Our theorem generalizes simultaneously the fixed point theorem of Bose and Laskar [Fixed point theorems for certain class of mappings, Jour. Math. Phy. Sci., 19 (1985), 503–509] and the well-known fixed point theorem of Goebel and Kirk [A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174] on a nonlinear domain. The fixed point obtained is approximated by averaging Krasnosel’skii iterations of the mapping. Our results substantially improve and extend several known results in uniformly convex Banach spaces and CAT(0) spaces.

scholarly journals Fixed points of Osilike-Berinde-G-nonexpansive mappings in metric spaces endowed with graphs

2021 ◽  
Vol 37 (2) ◽  
pp. 311-323
Author(s):  
A. KAEWKHAO ◽  
C. KLANGPRAPHAN ◽  
B. PANYANAK
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Nonexpansive Mapping ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Fixed Point Set ◽  
Quasinonexpansive Mapping ◽  
Ishikawa Iteration Process ◽  
Point Set

"In this paper, we introduce the notion of Osilike-Berinde-G-nonexpansive mappings in metric spaces and show that every Osilike-Berinde-G-nonexpansive mapping with nonempty fixed point set is a G-quasinonexpansive mapping. We also prove the demiclosed principle and apply it to obtain a fixed point theorem for Osilike-Berinde-G-nonexpansive mappings. Strong and \Delta-convergence theorems of the Ishikawa iteration process for G-quasinonexpansive mappings are also discussed."

scholarly journals Fibonacci–Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 481 ◽  
Author(s):  
Buthinah Dehaish ◽  
Mohamed Khamsi
Keyword(s):  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Modular Function ◽  
Function Spaces ◽  
Mann Iteration ◽  
Modular Spaces ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Modular Function Spaces ◽  
Mann Iteration Process

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by

scholarly journals Fibonacci-Mann Iteration for Monotone Asymptotically Nonexpansive Mappings in Modular Spaces

2018 ◽  
Author(s):  
Buthinah A. Bin Dehaish ◽  
Mohamed A Khamsi
Keyword(s):  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Modular Function ◽  
Function Spaces ◽  
Mann Iteration ◽  
Modular Spaces ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Modular Function Spaces ◽  
Mann Iteration Process

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci-Mann iteration process defined by $$x_{n+1} = t_n T^{\phi(n)}(x_n) + (1-t_n)x_n,$$ for $n \in \mathbb{N}$, when $T$ is a monotone asymptotically nonexpansive self-mapping.

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