scholarly journals An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Nguyen Buong ◽  
Nguyen Thi Quynh Anh
Keyword(s):  
Variational Inequalities ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Iteration Method ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Finite Family ◽  
Implicit Iteration

scholarly journals Hybrid Mann Viscosity Implicit Iteration Methods for Triple Hierarchical Variational Inequalities, Systems of Variational Inequalities and Fixed Point Problems

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 142 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
General System ◽  
Inequality Constraint ◽  
Common Fixed Points ◽  
Common Solution ◽  
Implicit Iteration

In the present work, we introduce a hybrid Mann viscosity-like implicit iteration to find solutions of a monotone classical variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities and a problem of common fixed points of an asymptotically nonexpansive mapping and a countable of uniformly Lipschitzian pseudocontractive mappings in Hilbert spaces, which is called the triple hierarchical constrained variational inequality. Strong convergence of the proposed method to the unique solution of the problem is guaranteed under some suitable assumptions. As a sub-result, we provide an algorithm to solve problem of common fixed points of pseudocontractive, nonexpansive mappings, variational inequality problems and generalized mixed bifunction equilibrium problems in Hilbert spaces.

scholarly journals Approximation of common fixed points of a finite family of Ø - demicontractive mappings by an implicit iteration method.

2015 ◽  
Vol 12 (1) ◽  
pp. 31-38
Author(s):  
DI Igbokwe ◽  
UE Udofia
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Iteration Method ◽  
Iteration Process ◽  
Common Fixed Points ◽  
Finite Family ◽  
Implicit Iteration Process ◽  
Demicontractive Mappings ◽  
Implicit Iteration

We prove that the Implicit Iteration process of Xu and Ori (2001) converges strongly to the commonfixed pointsof a finite family of Ø - demicontractive mappings in real Hilbertand Banachspaces. Our results extend the results of Osilike (2004a) from strictl pseudocontractive maps to the much more general Ø - demicontractive maps;complement and generalize several others in the literature.KEY WORDS AND PHRASES: Ø - Demicontractive Maps, Implicit Iteration Process, Fixed Points,Strong Convergence.

scholarly journals Common Fixed Points Iteration Processes for a Finite Family of Asymptotically Nonexpansive Mappings

2004 ◽  
Vol 11 (1) ◽  
pp. 83-92
Author(s):  
Jui-Chi Huang
Keyword(s):  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive ◽  
Iteration Processes ◽  
Suitable Sequence

Abstract Let 𝐸 be a uniformly convex Banach space which satisfies Opial's condition or its dual 𝐸* has the Kadec–Klee property, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇𝑗 : 𝐶 → 𝐶 an asymptotically nonexpansive mapping for each 𝑗 = 1, 2, . . . , 𝑟. Suppose {𝑥𝑛} is generated iteratively by where 𝑈𝑛(0) = 𝐼, 𝐼 is the identity map and {α 𝑛(𝑗)} is a suitable sequence in [0, 1]. If the set of common fixed points of is nonempty, then weak convergence of {𝑥𝑛} to some is obtained.

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