Left Amenable Semigroup Research Articles

On the Schauder fixed point property

In this paper we are dealing with the question on what amenability property should possess a semigroup to ensure it has the Schauder fixed point property. We provide a positive answer to this problem for the class \(\Sigma \)-ELA of all n-extremely left amenable semitopological semigroups. As an application, we derive partial answers (for \(\Sigma \)-ELA, locally finite groups, and semidirect product of \(\Sigma \)-ELA semigroups) to an open problem posted by Lau in 1976 (Lau, Fixed point theory and its applications, Academic Press, Orlando, 1976)

AMENABLE SEMIGROUPS OF NONLINEAR OPERATORS IN UNIFORMLY CONVEX BANACH SPACES

In 1965, Browder proved the existence of a common fixed point for commuting families of nonexpansive mappings acting on nonempty bounded closed convex subsets of uniformly convex Banach spaces. The purpose of this paper is to extend this result to left amenable semigroups of nonexpansive mappings.

CQ method for approximating fixed points of nonexpansive semigroups and strictly pseudo-contractive mappings

We use the CQ method for approximating a common fixed point of a left amenable semigroup of nonexpansive mappings, an infinite family of strictly pseudo-contraction mappings and the set of solutions of variational inequalities for monotone, Lipschitz-continuous mappings in a real Hilbert space. Our results are a generalization of a result announced by Nadezhkina and Takahashi [N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230-1241] and some other recent results.

Set amenability for semigroups

In this paper a concept of amenability for an arbitrary subset A of discrete semigroup S called A-amenable is introduced and studied. This concept is characterized by several equivalent statements which are analogues of properties characterizing left amenable semigroups. We also obtain the relationship between this version of amenability and Folner’s condition.

A Fixed Point Theorem for Left Amenable Semi-Topological Semigroups

In this note, we extend and improve the corresponding result of Takahashi7. Fixed point theorem for amenable semi group of non-expansive mappings.

A Fixed Point Theorem for Left Amenable Semi-Topological Semi Groups

In this note, we extend and improve the corresponding result of Takahashi. Explanation of DeMarr’s theorem is further generalized for some semi groups of non-expansive self- maps on K by the following considerations which are explained in the paper. The application of Zorn’s lemma and its application are explained. An application of Zorn’s lemma shows that there exists a minimal non-empty compact convex and S-invariant subset.

An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups

We prove an analogue of Fekete’s lemma for subadditive right-subinvariant functions defined on the finite subsets of a cancellative left-amenable semigroup. This extends results previously obtained in the case of amenable groups by E. Lindenstrauss and B. Weiss and by M. Gromov.

The Myhill property for cellular automata on amenable semigroups

Let $S$ be a cancellative left-amenable semigroup and let $A$ be a finite set. We prove that every pre-injective cellular automaton $\tau \colon A^S \to A^S$ is surjective.

Necessary and sufficient conditions for a subsemigroup of a cancellative left amenable semigroup to be left amenable

Let T be a subsemigroup of a cancellative, left amenable semigroup S. We give conditions which are both necessary and sufficient for the subsemigroup T to be left amenable.

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.

Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces

Abstract In this paper, we provide the existence and convergence theorems of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in general Banach spaces, which extend and improve many recent results in this area. MSC:47H09, 47H10, 47H20.

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.

CONVERGENCE THEOREMS FOR VARIATIONAL INEQUALITIES EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS BY HYBRID METHOD

In this paper, we introduce iterative schemes for finding a common element of the set of common fixed points for a left amenable semigroup of nonexpansive mappings, the set of solutions of the variational inequalities for a family of $\alpha$-inverse-strongly monotone mappings and the set of solutions of a system of equilibrium problems in a Hilbert space. We establish weak and strong convergence theorems for the sequences generated by our proposed schemes. Moreover, we present various applications to the additive semigroup of nonnegative real numbers and families of strictly pseudocontractive mappings.

CONVERGENCE THEOREMS FOR VARIATIONAL INEQUALITIES EQUILIBRIUM PROBLEMS AND NONEXPANSIVE MAPPINGS BY HYBRID METHOD

In this paper, we introduce iterative schemes for finding a common element of the set of common fixed points for a left amenable semigroup of nonexpansive mappings, the set of solutions of the variational inequalities for a family of $\alpha$-inverse-strongly monotone mappings and the set of solutions of a system of equilibrium problems in a Hilbert space. We establish weak and strong convergence theorems for the sequences generated by our proposed schemes. Moreover, we present various applications to the additive semigroup of nonnegative real numbers and families of strictly pseudocontractive mappings.

Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings

In this paper, we introduce hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family and left amenable semigroup of non-expansive mappings in the frame work of Hilbert spaces. Our goal is to prove a result of strong convergence for hybrid pseudo-viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non-expansive mappings. The results presented in this paper can be treated as an extension and improvement of the corresponding results announced by Ceng et al. [L.C. Ceng, Q.H. Ansari, and J.C. Yao, Hybrid pseudo-viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non-expansive mappings, Nonlinear Analysis 4 (2010) 743–754] and many others.

Strong convergence for a minimization problem on solutions of systems of equilibrium problems and common fixed points of an infinite family and semigroup of nonexpansive mappings

In this paper, we introduce a suitable Mann type algorithm for finding a common element of the set of solutions of systems of equilibrium problems and the set of common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme to the unique solution of the minimization problem on the solution of systems of equilibrium problems and the common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings. Our results extend and improve the recent result of Colao and Marino [V. Colao and G. Marino, Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings, Nonlinear Anal. 73 (2010) 3513–3524] and many others.

Density and invariant means in left amenable semigroups

A left cancellative and left amenable semigroup S satisfies the Strong Følner Condition. That is, given any finite subset H of S and any ϵ > 0 , there is a finite nonempty subset F of S such that for each s ∈ H , | s F △ F | < ϵ ⋅ | F | . This condition is useful in defining a very well behaved notion of density, which we call Følner density, via the notion of a left Følner net, that is a net 〈 F α 〉 α ∈ D of finite nonempty subsets of S such that for each s ∈ S , ( | s F α △ F α | ) / | F α | converges to 0. Motivated by a desire to show that this density behaves as it should on cartesian products, we were led to consider the set LIM 0 ( S ) which is the set of left invariant means which are weak ∗ limits in l ∞ ( S ) ∗ of left Følner nets. We show that the set of all left invariant means is the weak ∗ closure of the convex hull of LIM 0 ( S ) . (If S is a left amenable group, this is a relatively old result of C. Chou.) We obtain our desired density result as a corollary. We also show that the set of left invariant means on ( N , + ) is actually equal to LIM 0 ( N ) . We also derive some properties of the extreme points of the set of left invariant means on S, regarded as measures on βS, and investigate the algebraic implications of the assumption that there is a left invariant mean on S which is non-zero on some singleton subset of βS.

Iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings

We introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a Hilbert space. 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. Our results extend, for example, the recent result of [V. Colao, G. Marino, H.K. Xu, An Iterative Method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl. 344 (2008) 340–352] to systems of equilibrium problems.

Approximating Common Fixed Points of Lipschitzian Semigroup in Smooth Banach Spaces

Let be a left amenable semigroup, let be a representation of as Lipschitzian mappings from a nonempty compact convex subset of a smooth Banach space E into C with a uniform Lipschitzian condition, let be a strongly left regular sequence of means defined on an -stable subspace of , let be a contraction on , and let be sequences in (0, 1) such that , for all n. Let , for all . Then, under suitable hypotheses on the constants, we show that converges strongly to some in , the set of common fixed points of , which is the unique solution of the variational inequality , for all .

Fixed points of non-expansive mappings associated with invariant means in a Banach space

In this paper, we study the fixed point set of the non-expansive mapping T μ for a Banach space with uniformly Gâteaux differentiable norm when μ is a multiplicative left invariant mean on l ∞ ( S ) . As an application, we establish nonlinear ergodic properties for an extremely amenable semigroup of non-expansive mappings in a Banach space with uniformly Gâteaux differentiable norm. Furthermore, we improve a recent result of Atsushiba and Takahashi [S. Atsushiba, W. Takahashi, Weak and strong convergence theorems for non-expansive semigroups in a Banach spaces satisfying Opial’s condition, Sci. Math. Jpn. (in press)] on the fixed point set of non-expansive mappings associated with a left invariant mean on a left amenable semigroup.