Finite Family Of Nonexpansive Mappings Research Articles

Viscosity iteration method for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of solutions of generalized equilibrium problem and the set of common fixed point of family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of viscosity iterative algorithm to common element of the set of generalized equilibrium problem and the set of common fixed point of family of nonexpansive mappings. The result improve the recent ones of Ceng and Yao (2008) [L.C. Ceng, J.C. Yao, Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Appl. Math. Comput. 198 (2008) 729-741].

Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems

Abstract Recently, Colao et al. (J Math Anal Appl 344:340-352, 2008) introduced a hybrid viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings in a real Hilbert space. In this paper, by combining Colao, Marino and Xu's hybrid viscosity approximation method and Yamada's hybrid steepest-descent method, we propose a hybrid iterative method for finding a common element of the set GMEP of solutions of a generalized mixed equilibrium problem and the set ⋂ i = 1 N Fix ( S i ) of fixed points of a finite family of nonexpansive mappings { S i } i = 1 N in a real Hilbert space. We prove the strong convergence of the proposed iterative algorithm to an element of ⋂ i = 1 N Fix ( S i ) ∩ G M E P , which is the unique solution of a variational inequality. AMS subject classifications: 49J40; 47J20; 47H09.

Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E. Let be a sequence of nonexpansive mappings from K to itself with F :={x ∈ K :T i x = x, ∀i ≥ 1}≠ ∅. For an arbitrary initial point x1 ∈ K, the modified hybrid iteration scheme {x n } is defined as follows: , where A: K → K is an L-Lipschitzian mapping, with i satisfying: n = [(k-i+1)(i+k)/2]+[1+(i-1)(i+2)/2],k ≥ i-1(i = 1,2,...),{λ n } ⊂ [0,1), and {α n } is a sequence in [a, 1 - a] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of {x n } to a common fixed point of the nonexpansive mappings are obtained. The results in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings. Mathematics Subject Classifications 2000: 47H09; 47J25.

Implicit Iterative Method for Hierarchical Variational Inequalities

We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.

Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces

We study the strong convergence of a regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. 2010 Mathematics Subject Classification: 47H09; 47J25; 47J30.

Relaxed hybrid steepest-descent methods with variable parameters for triple-hierarchical variational inequalities

In this article, we consider a monotone variational inequality with variational inequality constraint over the set of fixed points of a nonexpansive mapping, which is called a triple-hierarchical variational inequality (THVI). A relaxed hybrid steepest-descent method with variable parameters is introduced for solving the THVI. Strong convergence of the method to a unique solution of the THVI is studied under certain assumptions. We also investigate another monotone variational inequality with the variational inequality constraint over the set of common fixed points of a finite family of nonexpansive mappings, and present an iterative algorithm for solving such kind of problems. It is proven that under mild conditions the sequence generated by the proposed algorithm converges strongly to a unique solution of later problem.

Regularization Proximal Point Algorithm for Common Fixed Points of Nonexpansive Mappings in Banach Spaces

We study the strong convergence of a regularization proximal point algorithm for the problem of finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space E.

Convergence of hybrid steepest descent method for variational inequalities in Banach spaces

Let E be a q-uniformly smooth real Banach space with constant d q , q > 1. Let T i : E → E, i = 1, 2, … , r be a finite family of nonexpansive mappings with K ≔ ∩ i = 1 r Fix ( T i ) ≠ ∅ and K = Fix( T r T r−1 … T 1) = Fix( T 1 T r … T 2) = ⋯ = Fix( T r−1 T r−2 … T r ). Let G : E → E be an η-strongly accretive map which is also κ-Lipschitzian. A hybrid steepest descent method introduced by Yamada [25] and studied by various authors is proved to converge strongly to the unique solution of the variational inequality problem VI( G, K) in q-uniformly smooth real Banach space, in particular, in L p spaces 1 < p < ∞.

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

We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.

Strong Convergence of an Iterative Method for Generalized Mixed Equilibrium Problems and Fixed Point Problems

In this paper, we introduce a hybrid iterative method for finding a common element of the set of common solutions of generalized mixed equilibrium problems and the set of common fixed points of an finite family of nonexpansive mappings. Furthermore, we show a strong convergence theorem under some mild conditions.

Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces

We introduce an iterative algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality. As an application, we use the result of this paper to solve a multiobjective optimization problem. Our result extends and improves the ones of Colao et al. (2008) and some others.

A GENERAL ITERATIVE ALGORITHM FOR EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS IN A HILBERT SPACE

In this paper, we introduce a general iterative algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a Hilbert space. Then, we prove that the iterative sequence converges strongly to a common element of the three sets. The results obtained in this paper extend and improve the several recent results in this area.

Strong convergence theorems for a general system of variational inequality problems, mixed equilibrium problems and fixed points problems with applications

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Using the demi-closedness principle for nonexpansive mapping, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions. Our main result extends a recent result of Ceng, Wang and Yao [L.C. Ceng, C.Y. Wang and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008) 375–390].

Convergence of an implicit algorithm for two families of nonexpansive mappings

In this paper, we study an implicit iterative algorithm for two nonexpansive mappings and two finite families of nonexpansive mappings in Banach spaces. We prove some weak and strong convergence theorems for these iterative algorithms. Our results extend some existing results.

The common minimal-norm fixed point of a finite family of nonexpansive mappings

Abstract Consider a finite family of nonexpansive mappings which are defined on a closed convex subset of a Hilbert space H . Suppose the set of common fixed points of this family is nonempty. We then address the problem of finding the minimum-norm element of this common fixed point set. We introduce both cyclic and parallel iteration methods to find this minimal-norm element.

A New General Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces

We introduce a new general iterative method by using the -mapping for finding a common fixed point of a finite family of nonexpansive mappings in the framework of Hilbert spaces. A strong convergence theorem of the purposed iterative method is established under some certain control conditions. Our results improve and extend the results announced by many others.

Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings

We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by using -mapping generated by a finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence theorem of the proposed iteration under some control condition. By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings.

A composite iterative algorithm for common fixed points for a finite family of nonexpansive mappings and applications

This paper is concerned with a new composite iteration approximating to common fixed points for a finite family of nonexpansive mappings in Banach spaces which have a uniformly Gâteaux differentiable norm. Utilizing the iterative algorithm, we obtain the strong convergence theorems for a finite family of nonexpansive mappings. Furthermore, the problem of image recovery is considered in the above result. Our results extend and improve the corresponding results.

A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings

The purpose of this paper is to consider a new hybrid relaxed extragradient method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of variational inequalities for an inverse-strongly monotone mapping in Hilbert spaces. Then, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions. Our results extend and improve the recent results of Cai and Hu [G. Cai, C.S. Hu, A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping, Nonlinear Anal. Hybrid Syst., 3(2009) 395–407], Kangtunyakarn and Suantai [A. Kangtunyakarn, S. Suantai, A new mapping for finding common solution of equilibrium problems and fixed point problems of finite family of nonexpansive mappings, Nonlinear Anal., 71(2009) 4448–4460] and Thianwan [S. Thianwan, Strong convergence theorems by hybrid methods for a finite family of nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. Hybrid Syst., 3(2009) 605–614] and many others.

Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method

Strong convergence theorems are obtained for a finite family of nonexpansive mappings and semigroups by the hybrid method.