Nonexpansive Semigroups In Hilbert Spaces Research Articles

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces

In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.

General Iterative Methods for Semigroups of Nonexpansive Mappings Related to Optimization Problems

In this article, we introduce two general iterative methods for a certain optimization problem of which the constrained set is the set of the solution set of the variational inequality problem for the fixed point set of nonexpansive semigroups in Hilbert spaces. Under some control conditions, we establish the strong convergence of the proposed methods to the fixed point set, which is the unique solution of a certain optimization problem. Applications to solutions of equilibrium problems are also presented.

Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces

Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

Regularization Methods for Nonexpansive Semigroups in Hilbert Spaces

The purpose of this paper is to present a regularization method for finding a common fixed point of a nonexpansive semigroup in a real Hilbert space. A combination of the considered regularization scheme with the proximal point algorithm and another one of the regularization method with an iterative scheme is studied in this research. We also discuss an application of the proposed methods for an initial value problem in Hilbert spaces.

On convergence of iteration processes for nonexpansive semigroups in uniformly convex and uniformly smooth Banach spaces

Let C be a closed, bounded and convex subset of a uniformly convex and uniformly smooth Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equationsxk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions, the sequence {xk} converges weakly to a common fixed point of the semigroup {Tt}t≥0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property. However, many important spaces like Lp for 1≤p≠2 do not possess the Opial property. In this paper, we do not assume the Opial property. We do assume instead that X is uniformly convex and uniformly smooth. Lp for p>1 are prime examples of such spaces.

Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces

In this paper, we introduce a general iterative method for a split variational inclusion and nonexpansive semigroups in Hilbert spaces. We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Moreover, a numerical example is given, to illustrate our methods and results, which may be viewed as a refinement and improvement of the previously known results announced by many other authors.

Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces

Let \(C\) be a convex compact subset of a uniformly convex Banach space. Let \(\{T_t\}_{t \geq0}\) be a strongly-continuous nonexpansive semigroup on \(C\). Consider the iterative process defined by the sequence of equations $$x_{k+1} =c_k T_{t_{k+1}}(x_{k+1})+(1-c_k)x_k.$$ We prove that, under certain conditions on \(\{c_k\}\) and \(\{t_k\}\), the sequence \(\{x_k\}_{n=1}^\infty\) converges strongly to a common fixed point of the semigroup \(\{T_t\}_{t \geq0}\). There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak convergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.

An Iterative Method for Equilibrium, Variational Inequality, and Fixed Point Problems for a Nonexpansive Semigroup in Hilbert Spaces

The purpose of this paper is to present a new iteration method based on the hybrid method in mathematical programming, extragradient method, and Mann’s method for finding a common element of the solution set of equilibrium problems, the solution set of variational inequality problems for a monotone, Lipschitz continuous mapping and the set of fixed points for a nonexpansive semigroup in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by this process. The results in this paper generalize and extend some well-known strong convergence theorems in the literature.

Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup

In this paper, we introduce and study an implicit iterative method to approximate a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup in real Hilbert spaces. Further, we prove that the nets generated by the implicit iterative method converge strongly to the common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. Furthermore, we justify our main result through a numerical example. The results presented in this paper extend and generalize the corresponding results given by Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48 (2008) 279–286] and Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space, J. Optim. Theory Appl. 146 (2010) 491–509].

Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

Abstract In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others. MSC:47H05, 47H10, 47H17.

Algorithms for solving generalized equilibrium problems and fixed points of nonexpansive semigroups in Hilbert spaces

In this article, strong convergence theorems by the modified viscosity approximation method associated with Meir–Keeler contractions are proved for solving fixed point problems of a nonexpansive semigroup and generalized equilibrium problems in a Hilbert space.

Approximation of Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

LetHbe a real Hilbert space. Consider onHa nonexpansive semigroupS={T(s):0≤s<∞}with a common fixed point, a contractionfwith the coefficient0<α<1, and a strongly positive linear bounded self-adjoint operatorAwith the coefficientγ¯> 0. Let0<γ<γ¯/α. It is proved that the sequence{xn}generated by the iterative methodx0∈H, xn+1=αnγf(xn)+βnxn+((1-βn)I-αnA)(1/sn)∫0snT(s)xnds, n≥0converges strongly to a common fixed pointx*∈F(S), whereF(S)denotes the common fixed point of the nonexpansive semigroup. The pointx*solves the variational inequality〈(γf-A)x*,x-x*〉≤0for allx∈F(S).

Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065---3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491---509, 2010) and many others.

STRONG CONVERGENCE OF HYBRID METHOD FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS AND SEMIGROUPS

In this paper, some strong convergence theorems are obtained for hybrid method for modifled Ishikawa iteration process of asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups in Hilbert spaces. The results presented in this article generalize and improve results of Tae-Hwa Kim and Hong-Kun Xu and others. The convergence rate of the iteration process presented in this article is faster than hybrid method of Tae-Hwa Kim and Hong-Kun Xu and others.

General Viscosity Approximation Methods for Common Fixed Points of Nonexpansive Semigroups in Hilbert Spaces

This paper is devoted to the strong convergence of two kinds of general viscosity iteration processes for approximating common fixed points of a nonexpansive semigroup in Hilbert spaces. The results presented in this paper improve and generalize some corresponding results in (X. Li et al., 2009, S. Li et al., 2009, and Marino and Xu, 2006).

A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups

In this paper, we introduce a composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. We prove strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions. Our result extends and improves those announced by Li et al [General iterative methods for a one-parameter nonexpansive semigroup in Hilbert spaces, Nonlinear Anal. 70 (2009) 3065–3071], Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Modelling 48 (2008) 279–286], Plubtieng and Wangkeeree [S. Plubtieng, R. Wangkeeree, A general viscosity approximation method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Bull. Korean Math. Soc. 45 (4) (2008) 717–728] and many others.

Strong convergence theorems for a semigroup of nonexpansive mappings in Banach spaces

Let E be a real reflexive Banach space, which admits a weakly sequentially continuous duality mapping of E into E*, and C be a nonempty closed convex subset of E. Let {T(t):t≥0} be a semigroup of nonexpansive self-mappings on C such that F:=∩ t≥0Fix(T(t))≠∅, where Fix(T(t))={x∈C: x=T(t)x}, and let f: C→C be a fixed contractive mapping. If {α n }, {β n }, {t n } satisfy some appropriate conditions, then a iterative process {x n } in C, defined by converges strongly to q∈F, and q is the unique solution in F to the following variational inequality: ⟨ (I−f)q, j(q−u)⟩≤0 for all u∈F. Our results extend and improve corresponding ones of Suzuki [T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces, Proc. Amer. Math. Soc. 131 (2002), pp. 2133–2136.], Xu [H.K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (2005), pp. 371–379.] and Chen and He [R.D. Chen and H. He, Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space, Appl. Math. Lett. 20 (2007), pp. 751–757.].