System Of Equilibrium Problems Research Articles

Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces

Abstract In this paper, we introduce a new iterative scheme by a hybrid method and prove a strong convergence theorem of a common element in the set of fixed points of a finite family of closed quasi-Bregman strictly pseudocontractive mappings and common solutions to a system of equilibrium problems in reflexive Banach space. Our results extend important recent results announced by many authors. MSC:47H09, 47J25.

Solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings

In this paper, we introduce a hybrid viscosity approximation scheme for finding a common element of the solution set for a system of equilibrium problems, the solution set for a system of variational inequality problems and the common fixed point set for an infinite family of nonexpansive mappings in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme. Our results improve and extend the results announced by Ceng and Yao (2008) [4].

Iterative approximation of countable family of relatively nonexpansive mappings and system of equilibrium problems in Banach spaces

Our purpose in this paper is to construct a new iterative scheme and prove strong convergence theorem using the iterative scheme for approximation of a common fixed point of a countable family of relatively nonexpansive mappings, which is also a solution to a finite system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space. We apply our results to approximate fixed point of a nonexpansive mapping, which is also solution to a finite system of equilibrium problems in a real Hilbert space, and approximate a common solution to finite system of variational inequality problems and convex minimization problems which is also a common solution to countable family of relatively nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space. Our results extend many known recent results in the literature.

Strong convergence theorems of the Halpern-Mann’s mixed iteration for a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping in Banach spaces

In this paper, we introduce a class of totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping to modify the Halpern-Mann-type iteration algorithm for a totally quasi-ϕ-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. Our results are applied to study the approximation problem of solution to a system of equilibrium problems. Also, the results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012) and others.

Convergence analysis for system of equilibrium problems and left bregman strongly relatively nonexpansive mapping

In this paper, we prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

Uniformly closed replaced AKTT or ∗AKTT condition to get strong convergence theorems for a countable family of relatively quasi-nonexpansive mappings and systems of equilibrium problems

Abstract The purpose of this paper is to construct a new iterative scheme and to get a strong convergence theorem for a countable family of relatively quasi-nonexpansive mappings and a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. The notion of uniformly closed mappings is presented and an example will be given which is a countable family of uniformly closed relatively quasi-nonexpansive mappings but not a countable family of relatively nonexpansive mappings. Another example shall be given which is uniformly closed but does not satisfy condition AKTT and ∗AKTT. Our results can be applied to solve a convex minimization problem. In addition, this paper clarifies an ambiguity in a useful lemma. The results of this paper modify and improve many other important recent results. MSC:47H05, 47H09, 47H10.

Approximation of common solutions for system of equilibrium problems and fixed-point problems

In this paper, we complement the result of Zhu and Chang (J Ineq Appl 2013:146, 2013) by proving strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces using the Halpern–Mann’s iterations used in Zhu and Chang (J Ineq Appl 2013:146, 2013). We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

A modified Halpern-type iteration algorithm for quasi-ϕ -asymptotically nonexpansive mappings and applications

Abstract The purpose of this article is to modify the Halpern-type iteration algorithm for quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of Qin et al. [Convergence of a modified Halpern-type iterative algorithm for quasi-ϕ-nonexpansive mappings, Appl. Math. Lett. 22 (2009), 1051–1055], Wang et al. [A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011), 2364–2371], Su et al. [Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010), 3890–3906], Nartinez-Yanes et al}. [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), 2400–2411], and others.

A New Iterative Scheme for Common Solution of Equilibrium Problems, Variational Inequalities and Fixed Point of k-strictly Pseudo-contractive Mappings in Hilbert Spaces

In this paper, we present a new iterative scheme for finding a common point among the set of solution of equilibrium problems, the set of solution to a variational inequality problem and the fixed point set of k-strictly pseudo-contractive mappings in a real Hilbert space. We then prove that the proposed scheme converges strongly to a common element which is the solution of a variational inequality problem, system of equilibrium problems, and a fixed point of k-strictly pseudo-contractive mappings. These results improve and generalize recent works in this direction.

Convergence theorems for a finite family of relatively quasi-nonexpansive mappings and system of equilibrium problems

Convergence theorems for a finite family of relatively quasi-nonexpansive mappings and system of equilibrium problems

Approximating Common Fixed Points of Bregman Weakly Relatively Nonexpansive Mappings in Banach Spaces

Using Bregman functions, we introduce a new hybrid iterative scheme for finding common fixed points of an infinite family of Bregman weakly relatively nonexpansive mappings in Banach spaces. We prove a strong convergence theorem for the sequence produced by the method. No closedness assumption is imposed on a mappingT:C→C, whereCis a closed and convex subset of a reflexive Banach spaceE. Furthermore, we apply our method to solve a system of equilibrium problems in reflexive Banach spaces. Some application of our results to the problem of finding a minimizer of a continuously Fréchet differentiable and convex function in a Banach space is presented. Our results improve and generalize many known results in the current literature.

Strong convergence theorems for Bregman totally quasi-asymptotically nonexpansive mappings in reflexive Banach spaces

The purpose of this article is by using the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for Bregman total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive Banach spaces. As applications, we apply our results to a system of equilibrium problems and zero point problem of maximal monotone mappings in reflexive Banach spaces. The results presented in the paper improve and extend the corresponding results of Reich and Sabach (2010) [12], Suantai et al. (2012) [13], Nilsrakoo and Saejung (2011) [11], Qin et al. (2009) [5], Wang et al. (2011) [6], Su et al. (2010) [7], Martinez-Yanes and Xu (2006) [3] and others.

Algorithms for treating equilibrium and fixed point problems

Abstract In this paper, a common solution problem is investigated based on a projection algorithm. Strong convergence theorems for common solutions of a system of equilibrium problems and a family of asymptotically quasi-ϕ-nonexpansive mappings are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

Strong convergence theorems for fixed point problems of infinite family of asymptotically quasi-ϕ-nonexpansive mappings and a system of equilibrium problems

Abstract In this paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of infinite family of asymptotically quasi-ϕ-nonexpansive mappings and of the set of solutions for finite equilibrium problems in a real Banach space. Our results are the generalization of the results (Shehu in Comput. Math. Appl. 63:1089-1103, 2012; Kim in Fixed Point Theory Appl., 2011, doi:10.1186/1687-1812-2011-10) and (Kim and Buong in Fixed Point Theory Appl., 2011, doi:10.1155/2011/780764), and improvement of the result (Yang et al. in Appl. Math. Comput. 218:6072-6082, 2012). MSC:47H09, 47H10, 47H17.

Convergence theorems for a system of equilibrium problems and fixed point problems of a strongly nonexpansive sequence

The purpose of this paper is to prove a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence. Moreover, in the last section, by using our main result, we obtain a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of a finite family of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence in a Hilbert space, and we also give some examples to support our main result.

WEAK AND STRONG CONVERGENCE THEOREMS FOR A SYSTEM OF MIXED EQUILIBRIUM PROBLEMS AND A NONEXPANSIVE MAPPING IN HILBERT SPACES

In this paper, we introduce an iterative sequence for finding solution of a system of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Then, the weak and strong convergence theorems are proved under some parameters control- ling conditions. Moreover, we apply our result to fixed point problems, system of equilibrium problems, general system of variational inequal- ities, mixed equilibrium problem, equilibrium problem and variational inequality.

Strong convergence theorems for modifying Halpern iterations for a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping in reflexive Banach spaces

In this paper, we discuss an iterative sequence for a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping for modifying Halpern’s iterations and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:6489-6497, 2012).MSC:47J05, 47H09, 49J25.

Weak convergence theorems for common solutions of a system of equilibrium problems and operator equations involving nonexpansive mappings

In this paper, a monotone variational inequality, a system of equilibrium problems, and nonexpansive mappings are investigated based on an iterative algorithm. Weak convergence theorems of common solutions are established in Hilbert spaces.

Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups

In this work, strong convergence theorems by the viscosity approximation method associated with Meir---Keeler contractions are established for solving fixed point problems of a nonexpansive semigroup, a system of equilibrium problems and variational inequality problems in a real Hilbert space. Further, applications related to commutative semigroup are obtained.

Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012) and many others.