scholarly journals Strong Convergence of an Inertial Iterative Algorithm for Generalized Mixed Variational-like Inequality Problem and Bregman Relatively Nonexpansive Mapping in Reflexive Banach Space

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Saud Fahad Aldosary ◽  
Watcharaporn Cholamjiak ◽  
Rehan Ali ◽  
Mohammad Farid
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Reflexive Banach Space ◽  
Existence Of A Solution ◽  
Relatively Nonexpansive Mapping ◽  
Inequality Problem ◽  
Relatively Nonexpansive ◽  
Hybrid Iterative Method

In this paper, we consider a generalized mixed variational-like inequality problem and prove a Minty-type lemma for its related auxiliary problems in a real Banach space. We prove the existence of a solution of these auxiliary problems and also prove some properties for the solution set of generalized mixed variational-like inequality problem. Furthermore, we introduce and study an inertial hybrid iterative method for solving the generalized mixed variational-like inequality problem involving Bregman relatively nonexpansive mapping in Banach space. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational examples to emphasize the efficiency and relevancy of the main result.

The Strong Convergence of a New Iterative Algorithm for Asymptotically Nonexpansive Mappings

2013 ◽  
Vol 756-759 ◽  
pp. 3628-3633
Author(s):  
Yuan Heng Wang ◽  
Wei Wei Sun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In a real Banach space E with a uniformly differentiable norm, we prove that a new iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. The results in this paper improve and extend some recent results of other authors.

scholarly journals Approximation of Fixed Points of Weak Bregman Relatively Nonexpansive Mappings in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Jiawei Chen ◽  
Zhongping Wan ◽  
Liuyang Yuan ◽  
Yue Zheng
Keyword(s):  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Relatively Nonexpansive Mapping ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Projection Techniques

We introduce a concept of weak Bregman relatively nonexpansive mapping which is distinct from Bregman relatively nonexpansive mapping. By using projection techniques, we construct several modification of Mann type iterative algorithms with errors and Halpern-type iterative algorithms with errors to find fixed points of weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings in Banach spaces. The strong convergence theorems for weak Bregman relatively nonexpansive mappings and Bregman relatively nonexpansive mappings are derived under some suitable assumptions. The main results in this paper develop, extend, and improve the corresponding results of Matsushita and Takahashi (2005) and Qin and Su (2007).

scholarly journals Strong Convergence of a Modified Ishikawa Iterative Sequence for Asymptotically Quasi-Pseudo-Contractive-Type Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liuhong Li ◽  
Yuanheng Wang
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Iterative Sequence ◽  
Convergence Problem ◽  
Type Mapping ◽  
Ishikawa Iterative Sequence ◽  
Asymptotically Nonexpansive ◽  
Contractive Type

The purpose of this paper is to investigate the strong convergence problem of a modified mixed Ishikawa iterative sequence with errors for approximating the fixed points of an asymptotically nonexpansive mapping in the intermediate sense and an asymptotically quasi-pseudo-contractive-type mapping in an arbitrary real Banach space. The results here improve and extend the corresponding results reported by some other authors recently.

scholarly journals A Halpern-Mann Type Iteration for Fixed Point Problems of a Relatively Nonexpansive Mapping and a System of Equilibrium Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-22
Author(s):  
Utith Inprasit ◽  
Weerayuth Nilsrakoo
Keyword(s):  
Nonexpansive Mapping ◽  
Real Banach Space ◽  
Equilibrium Problems ◽  
Common Element ◽  
Uniformly Convex ◽  
Relatively Nonexpansive Mapping ◽  
Uniformly Smooth ◽  
Relatively Nonexpansive ◽  
System Of Equilibrium Problems ◽  
Convergence Of The Scheme

A new modified Halpern-Mann type iterative method is constructed. Strong convergence of the scheme to a common element of the set of fixed points of a relatively nonexpansive mapping and the set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. The results presented in this work improve on the corresponding ones announced by many others.

scholarly journals Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space

2020 ◽  
Vol 21 (2) ◽  
pp. 397-412 ◽  
Author(s):  
H.A. Abass ◽  
◽  
C. Izuchukwu ◽  
O.T. Mewomo ◽  
Q.L. Dong ◽  
...  
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Real Banach Space ◽  
Splitting Method ◽  
Accretive Operators

scholarly journals Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Moosa Gabeleh ◽  
Naseer Shahzad
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Sufficient Conditions ◽  
Weakly Compact ◽  
Relatively Nonexpansive Mapping ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Contractive Type

LetAandBbe two nonempty subsets of a Banach spaceX. A mappingT:A∪B→A∪Bis said to be cyclic relatively nonexpansive ifT(A)⊆BandT(B)⊆AandTx-Ty≤x-yfor all (x,y)∈A×B. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach spaceX. It is shown that if (A,B) is a nonempty, weakly compact, and convex pair and (A,B) has seminormal structure, then a cyclic relatively nonexpansive mappingT:A∪B→A∪Bhas a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.Erratum to “Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings”

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