Nonexpansive Semigroups Research Articles

Strong convergence theorems of strongly relatively nonexpansive semi-group in Banach spaces

In this paper, we prove some strong convergence approximation theorems of strongly relatively nonexpansive semi-group in the framework of Banach spaces. Using the concept of duality theorems, we obtain analogue results for strongly generalized nonexpansive semi-group and for semi-group of firmly generalized nonexpansive type. The results presented in this paper improve and extend some corresponding results announced by some authors. ∗

A New Iterative Method for Equilibrium Problems and Fixed Point Problems

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

Implicit Ishikawa Approximation Methods for Nonexpansive Semigroups in CAT(0) Spaces

This paper is devoted to the convergence of the implicit Ishikawa iteration processes for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain theΔ-convergence results of the implicit Ishikawa iteration sequences for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Ishikawa iteration sequences for nonexpansive semigroups in the CAT(0) spaces. The results presented in this paper extend and generalize some previous results.

Approximating a solution of an equilibrium problem by Viscosity iteration involving a nonexpansive semigroup

binayak12@yahoo.co.in, subhajit.math@gmail.comABSTRACTIn this paper we have defined a new iteration in order to solve an equilibrium problemin Hilbert spaces. The iteration we have introduced is a viscosity type iteration andinvolvesa semigroup of nonexpansiveoperators. We have established that depending onsome control conditions, our iteration stronglyconvergesto a solution of the equilibriumproblem.RESUMENEn este art´iculo hemos definido una iteraci´on nueva para resolver un problema de equi-librioen espaciosde Hilbert. Laiteraci´onqueintroducimoses detipo viscosoeinvolucraun semigrupo de operadores no expansivos. Hemos establecido que dependiendo de lascondiciones de control, nuestra iteraci´on converge fuertemente a una soluci´on de unproblema de equilibrio.Keywords and Phrases: Equilibrium problem, Nonexpansive semigroup, Viscosity iteration,Fixed point, Weak convergence, Hilbert space.2010 AMS Mathematics Subject Classification: 46C05, 47H10, 91B50.

Common solution of split feasibility, systems of equilibrium and fixed points problems in Hilbert spaces

The purpose of this paper is to introduce an iterative algorithm for finding finding a common element of the solution sets of split feasibility problems (SFP), systems of equilibrium problems (SEP) and fixed points (for nonexpansive semigroup) problems in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of that solution sets under mild conditions. Our main result improve and extend the recent ones announced by Ceng, Ansari and Yao [1] and many others.

Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup

Purpose: In this paper, we introduce and study an iterative method to approximate a common solution of a split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup in real Hilbert spaces. Methods: We prove a strong convergence theorem of the iterative algorithm in Hilbert spaces under certain mild conditions. Results: We obtain a strong convergence result for approximating a common solution of a split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup in real Hilbert spaces, which is a unique solution of a variational inequality problem. Further, we obtain some consequences of our main result. Conclusions: The results presented in this paper are the supplement, extension, and generalization of results in the study of Plubtieng and Punpaeng and that of Cianciaruso et al. The approach of the proof given in this paper is also different.

Approximation for the Hierarchical Constrained Variational Inequalities over the Fixed Points of Nonexpansive Semigroups

The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a pointx*such thatx*∈Ω,〈(A-γf)x*-(I-B)Sx*,x-x*〉≥0, ∀x∈Ω, whereΩis the set of the solutions of the following variational inequality:x*∈Ϝ,〈(A-S)x*,x-x*〉≥0, ∀x∈Ϝ, whereA,Bare two strongly positive bounded linear operators,fis aρ-contraction,Sis a nonexpansive mapping, andϜis the fixed points set of a nonexpansive semigroup{T(s)}s≥0. We present a double-net convergence hierarchical to some elements inϜwhich solves the above hierarchical constrained variational inequalities.

Viscosity iterative scheme for generalized mixed equilibrium problems and nonexpansive semigroups

In this paper, we propose a new general iterative scheme based on the viscosity approximation method for finding a common element of the set of solutions of the generalized mixed equilibrium problem and the set of all common fixed points of a finite family of nonexpansive semigroups. Then, we prove the strong convergence of the iterative scheme to find a unique solution of the variational inequality that is the optimality condition for the minimization problem. Our results extend and improve some recent results of Cianciaruso et al. (J. Optim. Theory Appl. 146:491–509, 2010), Kamraksa and Wangkeeree (J. Glob. Optim. 51:689–714, 2011), and many others.

Fixed point theorems for a semigroup of generalized asymptotically nonexpansive mappings in CAT(0) spaces

In this paper, we prove the existence of common fixed points for a generalized asymptotically nonexpansive semigroup { T s : s ∈ S } Open image in new window in CAT(0) spaces, when S is a left reversible semitopological semigroup. We also prove Δ- and strong convergence of such a semigroup when S is a right reversible semitopological semigroup. Our results improve and extend the corresponding results existing in the literature.

VISCOSITY APPROXIMATION METHODS FOR NONEXPANSIVE SEMINGROUPS AND MONOTONE MAPPPINGS

Abstract. Let C be a nonempty closed convex subset of real Hilbertspace Hand F = fS(t) : t0ga nonexpansive self-mapping semigroupof C, and f: C!Cis a xed contractive mapping. Consider the processfx n g:(x n+1 = n x n + (1 n )z n z n = n f(x n ) + (1 n )S(t n )P C (x n r n Ax n ):It is shown that fx n gconverges strongly to a common element of the setof xed points of nonexpansive semigroups and the set of solutions of thevariational inequality for an inverse strongly-monotone mapping whichsolves some variational inequality. 1. IntroductionLet H be a real Hilbert space with inner product h;iand induced normkk, Cbe a nonempty closed convex subset of H. S: C!Cis nonexpansiveif kS(x) S(y)kkx yk;for all x;y2C. The set of xed points of SisF(S) = fx2C: x= Sxg. We assume that F(S) 6= ˚, it is well known thatF(S) is closed convex.A nonexpansive semigroup is a family F = fS(t) : t0gof self-mapping ofCif the following conditions are satis ed:(a) S(0)x= xfor all x2C;(b) S(s+ t) = S(s)S(t) for all s;t0;(c) For each t>0, kS(t)x S(t)ykkx yk;x;y2C:(d) For each x2C, the mapping S()xis continuous.In this paper, we use Fto denote the set of common xed points of F; that is,F= fx2C: S(t)x= x;t0g=

The general iterative scheme for semigroups of nonexpansive mappings and variational inequalities with applications

In this paper, we introduce the implicit and explicit viscosity iteration schemes for nonexpansive semigroups {T(t):t≥0}. Additionally, it proves that the proposed iterative schemes converge strongly to a unique common fixed point of {T(t):t≥0} in the framework of reflexive and strictly convex Banach space, which solves some variational inequality. The main results of this paper improve and generalize recent known results in the current literature.

A viscosity scheme for mixed equilibrium problems, variational inequality problems and fixed point problems

In this paper, we study a new general iterative scheme for finding a common element of the set of solutions of finite general mixed equilibrium problems, the set of solutions of finite variational inequalities for cocoercive mappings, the set of solutions of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of fixed points of a nonexpansive semigroup in Hilbert space. Strong convergence theorems are obtained under appropriate conditions.

Total quasi-ϕ-asymptotically nonexpansive semigroups and strong convergence theorems in Banach spaces

Abstract The purpose of this article is to modify the Halpern-Mann-type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups 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 recent results announced by many authors. MSC:47J05, 47H09, 49J25.

Implicit Mann approximation with perturbations for nonexpansive semigroups in CAT(0) spaces

In this paper, we study the convergence of implicit Mann iteration processes with bounded perturbations for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the △-convergence results of implicit Mann iteration schemes with bounded perturbations for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Mann iteration schemes with bounded perturbations for nonexpansive semigroups in CAT(0) spaces. The results presented in this paper extend and enrich the existing literature.MSC:47H05, 47H10, 47J25.

Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces

The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.MSC:47H05, 47H09, 49M05.

Convergence results for the zero-finding problem and fixed points of nonexpansive semigroups and strict pseudocontractions

In this work, we establish strong convergence theorems for solving the fixed point problem of nonexpansive semigroups and strict pseudocontractions, and the zero-finding problem of maximal monotone operators in a Hilbert space. We further apply our result to the convex minimization problem and commutative semigroups. MSC: 47H09; 47H10

Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings

Abstract In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors. MSC:47H05, 47H10, 47J25.

Strong convergence theorem for nonexpansive semigroups and systems of equilibrium problems

Our purpose in this paper is to prove strong convergence theorem for finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup and the set of solutions to a system of equilibrium problems in a real Hilbert space using a new iterative method. Finally, we give an application of our result in Hilbert spaces.

An iterative algorithm for common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings with optimization problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a system of mixed equilibrium problems, the set of solutions of a variational inclusion problems for inverse strongly monotone mappings, the set of common fixed points for nonexpansive semigroups and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in Hilbert spaces. Furthermore, we prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm under some suitable conditions which solves some optimization problems. Our results extend and improve the recent results of Chang et al. (Appl Math Comput 216:51---60, 2010), Hao (Appl Math Comput 217(7):3000---3010, 2010), Jaiboon and Kumam (Nonlinear Anal 73:1180---1202, 2010) and many others.

Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups

In this paper, we introduce new iterative algorithms for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we prove the strong convergence theorems of the sequence generated by these iterative algorithms under some suitable conditions. The results obtained in this paper extend the recent ones announced by many others.Mathematics Subject Classification (2010): 47H09, 47J05, 47J25, 49J40, 65J15