scholarly journals On convergence theorems for single-valued and multi-valued mappings in p-uniformly convex metric spaces

2021 ◽  
Vol 37 (3) ◽  
pp. 513-527
Author(s):  
JENJIRA PUIWONG ◽  
◽  
SATIT SAEJUNG ◽  
◽  
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Fixed Point Set ◽  
Convergence Theorems ◽  
Convex Metric Spaces ◽  
Strict Fixed Point ◽  
Point Set

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

scholarly journals Fixed point iterations for Prešić-Kannan nonexpansive mappings in product convex metric spaces

2018 ◽  
Vol 10 (1) ◽  
pp. 56-69
Author(s):  
Hafiz Fukhar-ud-din ◽  
Vasile Berinde
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Metric Spaces ◽  
Unique Fixed Point ◽  
Nonexpansive Mappings ◽  
Uniformly Convex ◽  
Product Spaces ◽  
Convex Metric Spaces ◽  
Fixed Point Iterations

Abstract We introduce Prešić-Kannan nonexpansive mappings on the product spaces and show that they have a unique fixed point in uniformly convex metric spaces. Moreover, we approximate this fixed point by Mann iterations. Our results are new in the literature and are valid in Hilbert spaces, CAT(0) spaces and Banach spaces simultaneously.

scholarly journals Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Aunyarat Bunyawat ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Method ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convex Banach Space ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Uniformly Convex Banach Spaces

We introduce an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping{Ti}in a uniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed point sequence of eachTi. Some strong convergence theorems of the proposed method are also obtained for the following cases: allTiare continuous and one ofTiis hemicompact, and the domainKis compact.

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."

Convergence theorems for generalized hemicontractive mapping in p-uniformly convex metric space

2020 ◽  
Vol 26 (2) ◽  
pp. 221-229
Author(s):  
Godwin C. Ugwunnadi ◽  
Chinedu Izuchukwu ◽  
Oluwatosin T. Mewomo
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Metric Spaces ◽  
Iteration Process ◽  
Uniformly Convex ◽  
Convex Metric Space ◽  
Convergence Theorems ◽  
Uniformly Convex Metric Space ◽  
Convex Metric Spaces ◽  
Hemicontractive Mappings

AbstractIn this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in 𝑝-uniformly convex metric spaces, and prove both Δ-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete 𝑝-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.

scholarly journals A Mean Ergodic Theorem for Affine Nonexpansive Mappings in Nonpositive Curvature Metric Spaces

2021 ◽  
Vol 29 (1) ◽  
pp. 111-125
Author(s):  
Hadi Khatibzadeh ◽  
Hadi Pouladi
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Ergodic Theorem ◽  
Tauberian Theorem ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Nonpositive Curvature ◽  
Fixed Point Set ◽  
Von Neumann ◽  
Point Set

Abstract In this paper, we consider the orbits of an affine nonexpansive mapping in Hadamard (nonpositive curvature metric) spaces and prove an ergodic theorem for the inductive mean, which extends the von Neumann linear ergodic theorem. The main result shows that the sequence given by the inductive means of iterations of an affine nonexpansive mapping with a nonempty fixed point set converges strongly to a fixed point of the mapping. A Tauberian theorem is also proved in order to ensure convergence of the iterations.

scholarly journals Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Withun Phuengrattana ◽  
Suthep Suantai
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convex Metric Space ◽  
Convergence Theorems ◽  
Convex Metric Spaces

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings{Tn}in convex metric spaces. We prove that the sequence{xn}generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when{Tn}satisfies the AKTT-condition, and strong convergence theorems of the proposed iteration to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0) spaces are established under AKTT-condition and the SZ-condition. We also generalize the concept ofW-mapping for a countable infinite family of nonexpansive mappings from a Banach space setting to a convex metric space and give some properties concerning the common fixed point set of this family in convex metric spaces. Moreover, by using the concept ofW-mappings, we give an example of a sequence of nonexpansive mappings defined on a convex metric space which satisfies the AKTT-condition. Our results generalize and refine many known results in the current literature.

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Preeyalak Chuadchawna ◽  
Ali Farajzadeh ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Hyperbolic Space ◽  
Hyperbolic Spaces ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive ◽  
Uniformly Convex Hyperbolic Space

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

Sign in / Sign up

Export Citation Format

Share Document