Teck-Cheong Lim

A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space

where H(A, B) denotes the Hausdorff distance between A and B. A point x e C is called a fixed point of T if x e Tx. Fixed point theorems for such mappings T have been established by Mar kin [11] for Hubert spaces, by Browder [2] for spaces having weakly continuous duality mapping, and by Lami Dozo [7] for spaces satisfying OpiaPs condition. Lami Dozos result is also generalized by Assad and Kirk [1]. By making use of Edelsteins asymptotic center [4], [5], we are able to prove Theorem 1. Let C be a closed convex subset of a uniformly convex Banach space and let {wj be a bounded sequence in C. The asymptotic center x of {wj in (or with respect to) C is the unique point in C such that

On fixed point stability for set-valued contractive mappings with applications to generalized differential equations

where d(x, C) = inf,..,. d(x, c). A set-valued mapping T: X-+ C(X) is a cI < 1 and for x, y E X. If X is complete, then every set-valued contraction has a fixed point, i.e., a point x with x E TX. The set of fixed points of T will be denoted by F(T). Stability of fixed points of set-valued contractions was investigated in [12] and [lo]. In [lo], a stability theorem was proved under some rather restrictive conditions, including (i) the domain of the maps being a closed convex-bounded subset of a Hilbert space, (ii) the image of each point under each map being a closed convex subset. Theorem 1 below removes these conditions.

On characterizations of Meir-Keeler contractive maps

Let (X, d) be a metric space. A map T : X → X is called Meir-Keeler contractive if ∀ > 0 ∃δ > 0 such that ≤ d(x, y) < + δ ⇒ d(Tx, Ty) < We introduce ”L-functions” and characterize Meir-Keeler contractive maps as maps that satisfy d(Tx, Ty) < φ(d(x, y)) for some L-function φ. This characterization makes it easy to compare such maps with those satisfying the Boyd-Wong’s condition.

Characterizations of normal structure

The notions of asymptotic center for a decreasing net of sets and asymptotic normal structure are defined and several characterizations of normal structure are proved. Among these, the problem of whether complete normal structure is equivalent to normal structure is answered in the affirmative.

On the normal structure coefficient and the bounded sequence coefficient

The two notions of normal structure coefficient and bounded sequence coefficient introduced by Bynum are shown to be the same. A lower bound for the normal structure coefficient in LP, p > 2, is also given. Let X be a Banach space and C a closed convex bounded subset of X. For each x in C, let R(x, C) =sup{ I x y I: y in C) and let R(C) denote the Chebyshev radius of C [2, p. 178]: R(C) = inf{R(x, C): x in C). Let D(C) denote the diameter of C, D(C) = sup{ 1 x -yH 1: x, y C). For a bounded sequence {xj} in X, the asymptotic diameter A({x,}) of {xj} is defined to be limn supt H Xk Xm 1: m > n, k > n}. In [1], Bynum introduced the following two coefficients of X, called the normal structure coefficient and the bounded sequence coefficient respectively: N(X) = inf{D(C)/R(C): C closed convex bounded nonempty subsets of Xwith I Cl> 1), BS(X) = sup M: for every bounded sequence {xnj in X, there existsy in Co (xn) such that Mlim supIIxn-YII A({xx})J. n Another coefficient relating to the asymptotic radius of a sequence (see e.g. [3]) can be defined as follows: Let {xnj be a bounded sequence in X. For each x in X, define r(x, {xn}) = limsup H1Xn X1. n The number r({xn}) = inf{f(x, {xn}): x ECo(xn)} will be called the asymptotic radius of {xnj, or more precisely, the asymptotic radius of {xnj w.r.t. Co(xn). We shall denote the coefficient inf{A({xn})/r({xn}): {Xn} bounded nonconvergent sequences in X} by A(X). Received by the editors April 13, 1982 and, in revised form, July 23, 1982. 1980 Mathematics Subject Classification. Primary 46B20, 47H09, 47H10; Secondary 52A05.

Fixed point theorems for mappings of nonexpansive type

Constructive fixed point theorems for single-valued and compact-valued nonexpansive mappings, which map a closed convex subset C of a Banach space X into X and send the boundary of C relative to X into C, are given. Mappings for which the method of asymptotic center applies are also considered. In this note we consider fixed point theorems for mappings f: C -X of the following types: (1) f is single-valued, nonexpansive and f (aC) c C. (2)f is multivalued, nonexpansive andf(x) c C for x E aC. (3) f is single-valued and satisfies the condition lim sup lim sup lfm (x) _ffl(y)l 0. In the sequel, unless otherwise stated, X denotes a uniformly convex Banach space, C a closed convex nonempty bounded subset of X and aC the boundary of C relative to X. If f: C -* X is a mapping such that f(3C) c C, we define F: C -C by putting F(x) = f (x) if f (x) E C, = the point where the line segment [ x, f (x) ] and aC intersect if f(x) E C. Iff is a contraction, it is known that f has a fixed point (Assad and Kirk [1]). 1ff is nonexpansive, by considering the contractions fA (x) =Xx0 + (I1 X)f(x), 0 < X < 1, x0 EC, we get for each X, a fixed point xA of fx, and it follows that II x f(xx) II 0 as X -0. The asymptotic center w.r.t. C of a bounded sequence tx,j in X is the unique point in C at which the function r(x) = lim sup llx-xnll, xEC, n attains its minimum (Edelstein [2]). r(x) is clearly a convex function. We begin with the following Received by the editors November 5, 1976. AMS (MOS) subject classifications (1970). Primary 47H 10, 46B99. ( Aniericani Maithematical Society 1977

On weak compactness and countable weak compactness in fixed point theory

We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some fixed point theorems proved recently are consequences of a classical theorem of Kirk. Let (X, d) be a metric space. Let F be the family of subsets of X consisting of X and sets which are complements of closed balls of X . The weak topology (also called ball topology) on X is the topology whose open sets are generated by F , i.e. F forms a subbase for the open sets of X . If X is a subset of a Banach space, the weak topology defined in this sense is generally weaker than the usual weak topology on X . For details on this topology in a Banach space setting, see [1] and [2]. We say X is weakly compact if it is compact in the weak topology. A subset of X is called a ball-intersection if it is an intersection of closed balls. X is said to have normal structure if every ball-intersection D of X containing more than one point contains a non-diametral point, i.e. a point r ∈ D such that sup{d(r, x) : x ∈ D} < diam(D). It is clear that X is weakly compact if and only if every nonempty family of ballintersections with the finite intersection property has a nonempty intersection. And by the Alexander subbase theorem, one can replace ‘ball-intersections’ with ‘closed balls’. X is countably weakly compact if every decreasing sequence of nonempty ball-intersections in X has a nonempty intersection. Obviously every weakly compact metric space is countably weakly compact. In this paper we show that the converse is false. However, if the metric space has normal structure, the converse is true. This shows that the assumption of countable weak compactness in some theorems in the literature is not really weaker than that of weak compactness. It also shows that every complete metric space with uniform normal structure is weakly compact, thereby a fixed point theorem of Khamsi [3] actually follows from that of Kirk [4]. Throughout this paper, Br(p) will denote the closed ball centered at p with radius r. Received by the editors January 3, 1995 and, in revised form, March 27, 1995. 1991 Mathematics Subject Classification. Primary 47H10, 47H09; Secondary 54E50, 54D30.

On Some LP Inequalities in Best Approximation Theory

h(x) = ,JxP ’ - p - (Ax - p)” ~ ’ is strictly increasing in the interval p/J O; thus h(x)=0 has a unique solution x(p). (Note that x( l/2) = 1.) x(p) is strictly increasing in p and x(0 + ) is the unique solution of the equation (p-2)xP-‘+(p- l).F2= 1 in the interval 0 6 x < 1. (1.1) In [ 11, we proved the inequality

Fixed points of isometries on weakly compact convex sets

In this paper, we prove that every isometry from a nonempty weakly compact convex set K into itself fixes a point in the Chebyshev center of K, provided K satisfies the hereditary fixed point property for isometries. In particular, all isometries from a nonempty bounded closed convex subset of a uniformly convex Banach space into itself have the Chebyshev center as a common fixed point.

Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations

We characterize (i) matrices which are nonexpansive with respect to some matrix norms, and (ii) matrices whose average iterates approach zero or are bounded. Then we apply these results to iterative solutions of a system of linear equations.