Nonexpansive Mappings In Hilbert Spaces Research Articles

General alternative regularization methods for nonexpansive mappings in Hilbert spaces

Let C be a nonempty closed convex subset of a real Hilbert space H with the inner product and the norm . Let be a nonexpansive mapping with a nonempty set of fixed points and let be a Lipschitzian strong pseudo-contraction. We first point out that the sequence generated by the usual viscosity approximation method may not converge to a fixed point of T, even not bounded. Secondly, we prove that if the sequence satisfies the conditions: (i) , (ii) and (iii) or , then the sequence generated by a general alternative regularization method: converges strongly to a fixed point of T, which also solves the variational inequality problem: finding an element such that for all . Furthermore, we prove that if T is replaced with the sequence of average mappings ( ) such that , where and are two positive constants, then the same convergence result holds provided conditions (i) and (ii) are satisfied. Finally, an algorithm for finding a common fixed point of a family of finite nonexpansive mappings is also proposed and its strong convergence is proved. Our results in this paper extend and improve the alternative regularization methods proposed by HK Xu. MSC:47H09, 47H10, 65K10.

An Iterative Method For Semigroups Of Nonexpansive Mappings

We introduce an iterative method for finding a common fixed point of a semigroup of infinite family of nonexpansive mappings in Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real valued functions of the semigroup. we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

Parallel algorithms for variational inclusions and fixed points with applications

In this paper, we introduce two parallel algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive mapping in Hilbert spaces and prove some strong convergence theorems of the proposed algorithms. As special cases, we can approach the minimum-norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive mapping without using the metric projection. Further, we give some applications of our main results. MSC:49J40, 47J20, 47H09, 65J15.

A STRONGLY CONVERGENT SHRINKING DESCENT-LIKE HALPERN’S METHOD FOR MONOTONE VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming, the descent-like method and the Halpern’s iteration method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a countably infinite family of nonexpansive mappings in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature for a more general problem.

A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings

In this paper, we introduce a new explicit iterative algorithm for finding a solution for a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the variational inequality. Our result improves and extends the corresponding results announced by many others. At the end of the paper, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings.

Iterative algorithms based on hybrid method and Cesàro mean of asymptotically nonexpansive mappings for equilibrium problems

Using Cesaro means of a mapping, we modify the progress of Mann’s iteration in hybrid method for asymptotically nonexpansive mappings in Hilbert spaces. Under suitable conditions, we prove that the iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. We also introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of asymptotically nonexpansive mappings and the set of solutions of an equilibrium problem in Hilbert spaces. MSC:47H10, 47J25, 90C33.

Weak convergence theorem by a new extragradient method for fixed point problems and variational inequality problems

We introduce a new extragradient iterative process, motivated and inspired by [S. H. Khan, A Picard-Mann Hybrid Iterative Process, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2013-69], for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a Hilbert space. Using this process, we prove a weak convergence theorem for the class of nonexpansive mappings in Hilbert spaces. Finally, as an application, we give some theorems by using resolvent operator and strictly pseudocontractive mapping.

Convergence Theorem for Equilibrium and Variational Inequality Problems and a Family of Infinitely Nonexpansive Mappings in Hilbert Space

We introduce a hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the varitional inequality problem and the equilibrium problem in Hilbert space. Under suitable conditions, some strong convergence theorems are obtained. Our results improve and extend the corresponding results in (Chang et al. (2009), Min and Chang (2012), Plubtieng and Punpaeng (2007), S. Takahashi and W. Takahashi (2007), Tada and Takahashi (2007), Gang and Changsong (2009), Ying (2013), Y. Yao and J. C. Yao (2007), and Yong-Cho and Kang (2012)).

Hybrid iterative algorithms for nonexpansive and nonspreading mappings in Hilbert spaces

Recently, Iemoto and Takahashi considered a weak convergence iterative scheme for a nonspreading mapping and a nonexpansive mapping in Hilbert spaces. In this paper, we suggest two hybrid iterative algorithms by modifying Iemoto and Takahashi’s iterative scheme for a countable family of nonspreading mappings and a nonexpansive mapping in Hilbert spaces.MSC:47H05, 47H09.

Hybrid methods for a mixed equilibrium problem and fixed points of a countable family of multivalued nonexpansive mappings

In this paper, we prove a strong convergence theorem for a new hybrid method, using shrinking projection method introduced by Takahashi and a fixed point method for finding a common element of the set of solutions of mixed equilibrium problem and the set of common fixed points of a countable family of multivalued nonexpansive mappings in Hilbert spaces. We also apply our main result to the convex minimization problem and the fixed point problem of a countable family of multivalued nonexpansive mappings.

A New Hybrid Method for Variational Inequality and Fixed Point Problems

In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming, the descent-like method and the Halpern iteration method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a countably infinite family of nonexpansive mappings in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature.

A viscosity projection method for class T mappings

Abstract In this paper, we firstly introduce a viscosity projection method for the class T mappings xn+1=αnPH(xn, Snxn) f(xn) + (1-αn)Snxn, where Sn = (1 - w)I + wTn, w ∈ (0; 1); Tn ∈ T and prove strong convergence theorems of the proposed method. It is verified that the viscosity projection method converges locally faster than the viscosity method. Furthermore, we present a viscosity projection method for a quasi-nonexpansive and nonexpansive mappings in Hilbert spaces. A numerical test provided in the paper shows that the viscosity projection method converges faster than the viscosity method.

The general iterative methods for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces

In this paper, the researcher introduces the general iterative scheme for finding a common element of the set of equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others. MSC: 47H10; 47H09

Convergence in Norm of Projection Regularized Krasnoselski-Mann Iterations for Fixed Points of Cutters

In this article, we introduce a projection regularized Krasnoselski-Mann iteration for cutters. The proposed algorithm ensures the strong convergence of the generated sequence toward the least norm element of the set of fixed points of the cutter. It is verified that the projection regularized Krasnoselski-Mann iteration converges locally faster than the regularized Krasnoselski-Mann iteration introduced by Maingé and Maruster [11]. Furthermore, we present projection regularized Krasnoselski-Mann iterations for quasi-nonexpansive and nonexpansive mappings in Hilbert spaces.

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.

A Resolvent Algorithm for System of General Mixed Variational Inequalities

In this paper, we suggest and analyze a new resolvent algorithm for finding the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point of a nonexpansive mapping in Hilbert spaces. We also prove the convergence analysis of the proposed algorithm under some suitable mild conditions. In this respect, our results present a refinement and improvement of the previously known results.

Iterative method for a finite family of nonexpansive mappings in Hilbert spaces with applications

In this paper, we introduce a new iterative method 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. Furthermore, we prove that the studied iterative method converges strongly to a common element of these three sets. Consequently, we apply our main result to the problem of approximating a zero of a finite family of maximal monotone mappings in Hilbert spaces. The theorems presented in this paper, improve and extend the corresponding results of Takahashi and Toyoda [18] and many others.

Convergence to common solutions of various problems for nonexpansive mappings in Hilbert spaces

In this paper, motivated and inspired by Ceng and Yao (J. Comput. Appl. Math. 214(1):186-201, 2008), Iiduka and Takahashi (Nonlinear Anal. 61(3):341-350, 2005), Jaiboon and Kumam (Nonlinear Anal. 73(5):1180-1202, 2010), Kim (Nonlinear Anal. 73:3413-3419, 2010), Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006) and Saeidi (Nonlinear Anal. 70:4195-4208, 2009), we introduce a new iterative scheme for finding a common element of the set of solutions of a mixed equilibrium problem for an equilibrium bifunction, the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of some variational inequality problem, and the set of fixed points of a left amenable semigroup of nonexpansive mappings with respect to W-mappings and a left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroup S. Furthermore, we prove that the iterative scheme converges strongly to a common element of the above four sets. Our results extend and improve the corresponding results of many others. MSC:43A65, 47H05, 47H09, 47H10, 47J20, 47J25, 74G40.

General iterative methods for equilibrium and constrained convex minimization problem

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. Based on Marino and Xu's method [G. Marino and H.-K. Xu, A general method for nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 318 (2006), pp. 43–52], we combine GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained convex minimization problem for the first time in this article. Under suitable conditions, strong convergence theorems are obtained.

Approximating fixed points of amenable semigroup and infinite family of nonexpansive mappings and solving systems of variational inequalities and systems of equilibrium problems

We introduce an iterative scheme for finding a common element of the set of solutions for systems of equilibrium problems and systems of variational inequalities and the set of common fixed points for an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. The results presented in this paper mainly extend and improved some well-known results in the literature. Mathematics Subject Classification (2000): 47H09; 47H10; 47H20; 43A07; 47J25.