Simeon Reich

Weak convergence theorems for nonexpansive mappings in Banach spaces

Let C be a closed convex subset of a Banach space E, and let T: C C be nonexpansive (that is, j TX Ty 1 < 1 N y 1 for all x and y in C). J.-B. Baillon [l] has recently shown that if E = D, 1 < p < co, and T has a fixed point, then for each x in C the Cesaro means of the iterates [T”s} convegre weakly to a fixed point of T. The purpose of this note is to point out that his ideas also lead to the following results. Recall that a sequence {x,} C E is weakly almost convergent (cf. [9]) toy E E if (~~~~ xick)/n y uniformly in k, and that an operator A C E x E is said to be m-accretive if R(.Z + -4) = E and /.x-x~] <~~~-~~+v(y,-y~)~ for ally,EAZx,, i=l,2, andr>O.

Asymptotic behavior of contractions in Banach spaces

In this note we study more general iteration processes in certain Banach spaces with the purpose of extending this theorem. It turns out that sometimes our aim can indeed be (partially) achieved (see, for example, Theorem 2.10). Several related results, as well as some open problems, are also included. In Section 1 we relate the boundedness of the sequence of iterates with the existence of a fixed point. Section 2 is devoted to a discussion of the crucial “minimum property” of cl(R(A)) ( see the definition below). Section 3 contains a result concerning the convergence of a certain sequence of iterates towards a fixed point of T. We shall consider here only real normed linear spaces. This restriction does not cause any loss of generality.

Iterations of paracontractions and firmaly nonexpansive operators with applications to feasibility and optimization

A generalized “measure of distance” defined by . is generated from any member f of the class of Bregman functions. Although it is not. technically speaking. a distance function. it has been used in the past to define and study projection operators. In this paper we give new definitions of paracont ractions. convex combinations. and firmly nonexpansivc operators. based on Df (x,y), and study sequential and simultaneous iterative algorithms employing them for the solution of the problem of finding a common asymptotic fixed point of a family of operators. Applications to the convex feasibility problem. to optimization and to monotone operator theory are also included.

Nonexpansive iterations in hyperbolic spaces

ONE OF THE most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings. Most of the results, however, have been obtained in normed linear spaces. It is natural, therefore, to try to develop a theory of nonexpansive iterations in more general infinite-dimensional manifolds. This is the purpose of the present paper. More specifically, we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general, and of iterative processes for nonexpansive mappings in particular. This class of metric spaces, which is defined in Section 2, includes all normed linear spaces and Hadamard manifolds, as well as the Hilbert ball and the Cartesian product of Hilbert balls. In Section 3 we introduce co-accretive operators and their resolvents, and present some of their properties. In the fourth section we discuss the concept of uniform convexity for hyperbolic spaces. Section 5 is devoted to two new geometric properties of (infinite-dimensional) Banach spaces. Theorem 5.6 provides a characterization of Banach spaces having these properties in terms of nonlinear accretive operators. In Sections 6, 7 and 8 we study explicit, implict and continuous iterations, repectively, using the same approach in all three sections. We illustrate this common approach with the following special case. Let C be a closed convex subset of a hyperbolic space (X, p), let T: C --f C be a nonexpansive mapping, and let x be a point in C. In order to study the iteration (T”x: n = 0, 1,2, . . .), we set z,, = (1 (l/n))x 0 (l/n)T”x, K = clco(zj;j I l), and d = inf(p(y, Ty): y E C). The first step is to show that p(x, K) = lim p(x, T”x)/n = d. This leads to the convergence “+m of lz,) when X is uniformly convex and to the weak convergence of (z,,] when X is a Banach space which is reflexive and strictly convex. When T is an averaged mapping we are also able to establish the following triple equality. For all k 2 1,

Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces

Abstract Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function ƒ on X. We prove (under certain assumptions on ƒ) that the iterates of a generic mapping in these spaces converge strongly to a retraction onto F.

Approximate selections, best approximations, fixed points, and invariant sets

In the first section of this note we prove an approximate selection theorem for upper semicontinuous mappings defined on paracompact subsets of a Hausdorff topological vector space. In Section 2 we study continuity properties of setvalued metric projections. In the third section we combine the results obtained in the previous sections and establish some new fixed point theorems. In [36] we presented a set-valued version of Fan’s fixed point theorem for inward singlevalued mappings [ 171. We assumed there that the set-valued mapping in question was continuous. Now we are able to show that the same result is true for upper semicontinuous mappings. We also improve recent theorems due to Fitzpatrick and Petryshyn [19] and extend a theorem of Lim’s [27]. In the last section we consider invariance criteria that are similar to the inwardness conditions used in Section 3. In particular, we relate Nagumo’s subtangency condition [31] with Browder’s local support cones [6].

Algorithms for the Split Variational Inequality Problem

We propose a prototypical Split Inverse Problem (SIP) and a new variational problem, called the Split Variational Inequality Problem (SVIP), which is a SIP. It entails finding a solution of one inverse problem (e.g., a Variational Inequality Problem (VIP)), the image of which under a given bounded linear transformation is a solution of another inverse problem such as a VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert space and then discuss special cases, some of which are new even in Euclidean space.

Some remarks concerning contraction mappings

The following result is proved in [1, p. 6]. Theorem 1. Let X be a complete metric space, and let T and T n (n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k n respectively. Suppose that lim n → ∞ Tn(x) = T(x) for every x ∊ X. Then lim n → ∞ un = u .

Proximinal Retracts and Best Proximity Pair Theorems

Abstract This note is concerned with proximinality and best proximity pair theorems in hyperconvex metric spaces and in Hilbert spaces. Given two subsets A and B of a metric space and a mapping best proximity pair theorems provide sufficient conditions that ensure the existence of an such that Thus such theorems provide optimal approximate solutions in the case that the mapping T does not have fixed points.

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.