Convergence of a hybrid iterative scheme for fixed points of nonexpansive maps, solutions of equilibrium, and variational inequalities problems
type: Item
instance of: scholarly article
title: Convergence of a hybrid iterative scheme for fixed points of nonexpansive maps, solutions of equilibrium, and variational inequalities problems
zbMATH Open document ID: 1268.47079
DOI: 10.1155/2013/370143
author: Bashir Ali
published in: Journal of Mathematics
publication date: 2013-07-25T00:00:00Z
review text: Summary: Let \(K\) be a closed, convex, and nonempty subset of a real \(q\)-uniformly smooth Banach space \(E\) which is also uniformly convex. For some \(\kappa > 0\), let \(T_i : K \to E\), \(i \in \mathbb N\), be a family of nonexpansive maps and \(A : K \to E\) be a \(\kappa\)-inverse strongly accretive map. Let \(G : K \times K \to \mathbb R\) be a bifunction satisfying some conditions. Let \(P_K\) be a nonexpansive projection of \(E\) onto \(K\). For some fixed real numbers \(\delta \in (0, 1)\), \(\lambda \in (0, (q\kappa/d_q)^{1/(q-1)})\), and arbitrary but fixed vectors \(x_1, u \in E\), let \(\{x_n\}\) and \(\{y_n\}\) be sequences generated by \(G(y_n, \eta) + (1/r)\langle \eta - y_n, j_q(y_n - x_n)\rangle \geq 0\) for all \(\eta \in K\), \(x_{n+1} = \alpha_nu + (1 - \delta)(1 - \alpha_n)x_n + \delta \sum_{i \geq 1} \sigma_{in}T_iP_K(y_n - \lambda Ay_n)\), \(n \geq 1\), where \(r \in (0, 1)\) is fixed, and \(\{\alpha_n\}, \{\sigma_{i, n}\} \subset (0, 1)\) are sequences satisfying appropriate conditions. If \(F := [\cap^\infty_{i=1}F(T_i)] \cap \text{VI}(K, A) \cap \text{EP}(G) \neq \emptyset\), under some mild conditions, we prove that the sequences \(\{x_n\}\) and \(\{y_n\}\) converge strongly to some element in \(F\).
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 6191145
zbMATH Keywords: real \(q\)-uniformly smooth Banach space
zbMATH Keywords: family of nonexpansive mappings
zbMATH Keywords: \(\kappa\)-inverse strongly accretive map
zbMATH Keywords: fixed point iteration
zbMATH Keywords: strong convergence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/370143
OpenAlex ID: W2044322179
Wikidata QID: Q59015368
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Nonexpansive projections on subsets of Banach spaces
cites work: Q3515662
cites work: Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces
cites work: A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems
cites work: Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems
cites work: Strong convergence theorems for variational inequalities and fixed point problems of asymptotically strict pseudocontractive mappings in the intermediate sense
cites work: Mann type hybrid extragradient method for variational inequalities, variational inclusions and fixed point problems
cites work: Q4681445
cites work: An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings
cites work: A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings
cites work: Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization
cites work: Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems
cites work: Q3097058
cites work: Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points
cites work: A Unified Hybrid Iterative Method for Solving Variational Inequalities Involving Generalized Pseudocontractive Mappings
cites work: A new hybrid general iterative algorithm for common solutions of generalized mixed equilibrium problems and variational inclusions
cites work: General iterative methods for generalized equilibrium problems and fixed point problems of \(k\)-strict pseudo-contractions
cites work: Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings
cites work: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings
cites work: Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space
cites work: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces
cites work: Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings
cites work: A characterization of strict convexity of Banach spaces and other uses of duality mappings
cites work: Q3998720
cites work: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals
cites work: Iterative Algorithms for Nonlinear Operators
cites work: Inequalities in Banach spaces with applications
cites work: Strong Convergence of a Proximal-Type Algorithm in a Banach Space
cites work: Q4366516
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description: scientific article; zbMATH DE number 6191145