Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings

2011 ◽  
pp. 345-390 ◽  
Author(s):  
Isao Yamada ◽  
Masahiro Yukawa ◽  
Masao Yamagishi
Keyword(s):  
Fixed Point ◽  
Convex Functions ◽  
Nonexpansive Mappings ◽  
Moreau Envelope ◽  
Fixed Point Set ◽  
Point Set

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 Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Zhanfei Zuo
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Fixed Point Set ◽  
Common Fixed Point Problem ◽  
The Family ◽  
Point Set ◽  
The Common

It is our purpose in this paper to prove two convergents of viscosity approximation scheme to a common fixed point of a family of multivalued nonexpansive mappings in Banach spaces. Moreover, it is the unique solution in to a certain variational inequality, where stands for the common fixed-point set of the family of multivalued nonexpansive mapping .

Asymptotically invariant net and fixed point set for semigroup of nonexpansive mappings

1997 ◽  
Vol 29 (5) ◽  
pp. 539-550 ◽  
Author(s):  
Osamu Kada ◽  
Anthony T.M. Lau ◽  
Wataru Takahashi
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Set ◽  
Point Set ◽  
Semigroup Of Nonexpansive Mappings

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 OF ITERATIVE SCHEMES FOR NONEXPANSIVE MAPPINGS

2011 ◽  
Vol 04 (04) ◽  
pp. 671-682 ◽  
Author(s):  
Mengistu Goa Sangago
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Initial Point ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Fixed Point Set ◽  
Iteration Formula ◽  
Point Set ◽  
Implicit Iteration

Let K be a nonempty closed convex subset of a real uniformly convex Banach space X and suppose T : K → K is a nonexpansive mapping with the nonempty fixed point set Fix(T). Let [Formula: see text], [Formula: see text] and [Formula: see text] be sequences in [0, 1] such that [Formula: see text][Formula: see text][Formula: see text] for some constants a, b, α, β, and γ. Let x0 ∈ K be any initial point. Then it is proved that the implicit iteration [Formula: see text] defined by [Formula: see text] converges weakly to a fixed point of T. Furthermore, it is generalized that if [Formula: see text] is a finite family of nonexpansive self-mappings of K with the nonempty common fixed points set [Formula: see text] and if the parameters [Formula: see text], [Formula: see text], and [Formula: see text] satisfy the conditions (0.1), (0.2), (0.3), and [Formula: see text] for some constant c , then the modified implicit iteration [Formula: see text] defined by [Formula: see text] where Tn = Tn(modN), converges weakly to a common fixed point of the family [Formula: see text]. The results presented in this paper improve and extend the corresponding results of Z. Opial [11], S. Reich [13], H.-K. Xu and R. G. Ori [20], and Zhao et al. [21].

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