Family Of Nonexpansive Self-mappings Research Articles

SEMI-ASYMPTOTIC NON-EXPANSIVE ACTIONS OF SEMI-TOPOLOGICAL SEMIGROUPS

Abstract. In this paper we extend Takahashi’s fixed point theorem ondiscrete semigroups to general semi-topological semigroups. Next we de-fine the semi-asymptotic non-expansive action of semi-topological semi-groups to give a partial affirmative answer to an open problem raised byA.T-M. Lau. 1. IntroductionA (not necessarily linear) self-mapping T : E → E on a Banach space Eis called non-expansive if kT(x) − T(y)k ≤ kx − yk for all x,y ∈ E. In 1963R. DeMarr proved the following common fixed point theorem for commutingfamilies of non-expansive self-mappings [6].Theorem 1.1. For any non-empty compact convex subset K of a Banach spaceE each commuting family of non-expansive self mappings on K has a commonfixed point in K.This generalizes the celebrated Markov-Kakutani fixed point theorem (forthe case of Banach spaces) on commuting families of continuous linear trans-formations on Hausdorff topological vector spaces leaving certain nonemptycompact convex subset invariant. DeMarr’s theorem has been generalized inseveral directions by Belluce and Kirk [2,3], Takahashi [16], Mitchell [14], Lauand Holmes [7,8].DeMarr theorem suggests that the action of certain commutative semigrouphas a fixed point. It is then natural to seek the same type offixed point propertyfor the actions of more general semigroups.Let S be a semi-topological semigroup, that is a semigroup with a Hausdorfftopology with separately continuous multiplication. We say that S is rightreversible if it has finite intersection property for closed left ideals. An actionof S on a topological space E is a mapping (s,x) → s(x) from S × E into

Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach spaceXwhich has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive inX. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rbe a finite family of nonexpansive self-mappings ofC. For a givenx1∈C, let{xn}and{xn(i)},i=1,2,…,r, be sequences definedxn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…, xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, whereani(j)∈[0,1]for allj∈{1,2,…,r},n∈ℕandi=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence{xn}to a common fixed point of a finite family of nonexpansive mappingsTi (i=1,2,…,r)are established under some certain control conditions.

Convergence of path and an iterative method for families of nonexpansive mappings

Let E be a real q-uniformly smooth Banach space with q ≥ 1 + d q . Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ ∈ (0, 1) define a family of nonexpansive maps by S i := (1 − δ)I + δT i where I is the identity map of K. Let Assume either at least one of the T i 's is demicompact or E admits weakly sequentially continuous duality map. It is proven that the fixed point sequence {z t n } converges strongly to a common fixed point of the family where and {t n } is a sequence in (0, 1), satisfying appropriate conditions. As an application, it is proven that the iterative sequence {x n } defined by: x 0 ∈ K, converges strongly to a common fixed point of the family where {α n } and {σ i,n } are sequences in (0, 1) satisfying appropriate conditions.