Nonexpansive Semigroups Research Articles

The Hybrid Projection Methods for Pseudocontractive, Nonexpansive Semigroup, and Monotone Mapping

We modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

An iterative algorithm for finding the solution of a general equilibrium problem system

In this paper, we introduce a new iterative method for finding a common element of the set of solution of a general equilibrium problem system (GEPS) and the set of fixed points of a nonexpansive semigroup. Furthermore, we present some numerical examples (by using MATLsoftware) to guarantee the main result of this paper.

Convergence Theorem of Hybrid Iterative Algorithm for Equilibrium Problems and Fixed Point Problems of Finite Families of Uniformly Asymptotically Nonexpansive Semigroups

Throughout this paper, we introduce a new hybrid iterative algorithm for finding a common element of the set of common fixed points of a finite family of uniformly asymptotically nonexpansive semigroups and the set of solutions of an equilibrium problem in the framework of Hilbert spaces. We then prove the strong convergence theorem with respect to the proposed iterative algorithm. Our results in this paper extend and improve some recent known results.

Common Fixed Point Theorems for Totally Quasi-G-Asymptotically Nonexpansive Semigroups with the Generalized f-Projection

In this paper, we introduce some new classes of the totally quasi-G-asymptotically nonexpansive mappings and the totally quasi-G-asymptotically nonexpansive semigroups. Then, with the generalized f-projection operator, we prove some strong convergence theorems of a new modified Halpern type hybrid iterative algorithm for the totally quasi-G-asymptotically nonexpansive semigroups in Banach space. The results presented in this paper extend and improve some corresponding ones by many others.

Convergence theorems of modified Mann iterations

In this paper, we introduce the modified iterations of Mann’s type for nonexpansive mapping and asymptotically nonexpansive mapping to have the strong and weak convergence in a uniformly convex Banach space. We also proved strong convergence theorems of our modified Mann’s iteration processes for nonexpansive semigroups and asymptotically nonexpansive semigroups. The results presented in the paper give a partially affirmative answer to the open question raised by Kim and Xu (Nonlinear Anal. 64:1140-1152, 2006). Applications to the accretive operators are also included. MSC:47H09, 47H10, 47J25.

Strong convergence theorems for fixed point problems of a nonexpansive semigroup in a Banach space

Abstract In this paper, we study the implicit and explicit viscosity iteration schemes for a nonexpansive semigroup in a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition. Our results improve and generalize the corresponding results given by Yao et al. (Fixed Point Theory Appl. 2013, doi:10.1186/1687-1812-2013-31) and many others. MSC:47H05, 47H10, 47H17.

A Solution of a General Equilibrium Problem

Under the framework of a real Hilbert space, we introduce a new iterative method for finding a common element of the set of solution of a general equilibrium problem and the set of fixed points of a nonexpansive semigroup. Moreover, a numerical example is presented. This example grantee the main result of the paper.

Strong convergence theorems of total quasi-ϕ-asymptotically nonexpansive semi-groups in Banach spaces

The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive semigroups which contains many kinds of semi-groups as its special cases, and then to modify the Halpern–Mann-type iteration algorithm for this kind of semi-groups 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.

Fixed point of asymptotic pointwise nonexpansive semigroups in metric spaces

Let C be a bounded, closed, convex subset of a uniformly convex metric space ( M , d ) Open image in new window. In this paper, we introduce the concept of asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t : C → C Open image in new window, i.e., a family such that T 0 ( x ) = x Open image in new window, T s + t = T s ( T t ( x ) ) Open image in new window, and d ( T t ( x ) , T t ( y ) ) ≤ α t ( x ) d ( x , y ) Open image in new window, where lim sup t → ∞ α t ( x ) ≤ 1 Open image in new window for every x ∈ C Open image in new window. Then we investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups. The proof is based on the concept of types extended to one parameter family of points.

Viscosity Approximation Method for Nonexpansive Semigroups in Banach Spaces

Let C be a nonempty closed convex subset of a reflexive Banach space E which admits a weakly sequentially continuous duality mapping from E to E∗ and {T(t):t>0} be a nonexpansive semigroup on C such that F=⋂t>0F(T(t))≠∅,f:C→C, be a fixed contractive mapping. With some appropriate conditions on {αn} and {tn}, two strongly convergent theorem for the following implicit and explicit viscosity iterative schemes {xn} are proved: xn=αnf(xn)+(1−αn)T(tn)xn, for \(n\in \mathbb{N} \), xn+1=αnf(xn)+(1−αn)T(tn)xn, for \(n\in \mathbb{N} \), and the cluster point of {xn} is the unique solution in F to the following variational inequality: 〈(I−f)p,j(p−x)〉≤0, ∀x∈F. The idea and method presented in this paper are especially based on the ones of Chen and He (Appl. Math. Lett. 20:751–757, 2007).

A general iterative algorithm for semigroups of nonexpansive mappings with generalized contractive mapping

The purpose of this paper is to study the strong convergence of the implicit and composite viscosity iteration schemes to a unique common fixed point of nonexpansive semigroups T, which is a solution of some variational inequality under certain conditions. As a application, we apply the proposed iterative algorithm to the minimization problem of finding a optimality condition. The results of this paper extend some recent results.

Common fixed points for pointwise Lipschitzian semigroups in modular function spaces

Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space L ρ . We investigate the existence of common fixed points for asymptotic pointwise nonexpansive semigroups of nonlinear mappings T t :C→C, i.e. a family such that T 0 (f)=f, T s + t (f)= T s ∘ T t (f) and ρ ( T ( f ) − T ( g ) ) ≤ α t (f)ρ(f−g), where lim sup t → ∞ α t (f)≤1 for every f∈C. In particular, we prove that if L ρ is uniformly convex, then the common fixed point is nonempty ρ-closed and convex.MSC:47H09, 46B20, 47H10, 47E10.

Viscosity approximation methods for nonexpansive semigroups in CAT ( 0 ) spaces

In this paper, we study the strong convergence of Moudafi’s viscosity approximation methods for approximating a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings in spaces. We prove that the proposed iterative scheme converges strongly to a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings which is also a unique solution of the variational inequality. The results presented in this paper extend and enrich the existing literature.

The modified general iterative methods for asymptotically nonexpansive semigroups in Banach spaces

In this paper, we introduce the modified general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping and uniform normal structure. The main result extends various results existing in the current 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].

Multi-valued totally quasi-ϕ-asymptotically nonexpansive semi-groups and strong convergence theorems in banach spaces

The purpose of this article is first to introduce the concept of multi-valued totally Quasiϕ-asymptotically nonexpansive semi-groups, which contains many kinds of semi-groups as its special cases, and then to modify the Halpern-Mann-type iteration algorithm for multi-valued totally Quasiϕ-asymptotically nonexpansive semi-groups to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in this article improve and extend the corresponding results announced by many authors recently.

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.

An Iterative Method for Approximating the Common Solutions of Nonexpansive Semigroups, a System of Variational Inequalities, Variational Inequalities and Equilibrium Problems

In this work, we consider an general iterative method for finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup problem, a system of variational inequality problems, a variational inequality problem and the set of solutions of a suitable equilibrium problem in a real Hilbert space. Further we establish a strong convergence theorem based on this method. Our results improve and extend corresponding ones announced by many others.

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.

A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.