Point Property For Nonexpansive Mappings Research Articles

The fixed point property and unbounded sets in spaces of negative curvature

Motivated by the well-known cases of the real Hilbert ball and complete ℝ-trees, both being particular cases of CAT(-1) spaces, we give an affirmative answer to the question of whether the geodesic boundedness property is a necessary and sufficient condition for a closed convex subset K of a complete CAT(κ) space, with κ < 0, to have the fixed point property for nonexpansive mappings.

Weak compactness is not equivalent to the fixed point property in c

We show that there exists a non-weakly compact, closed, bounded, convex subset W of the Banach space of convergent sequences (c,‖⋅‖∞), such that every nonexpansive mapping T:W⟶W has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett. This is also the first example of a non-weakly compact, closed, bounded, convex subset W of a Banach space X isomorphic to c0, for which W has the fixed point property for nonexpansive mappings. We also prove that the sets W may be perturbed to a large family of non-weakly compact, closed, bounded, convex subsets Wq of (c,‖⋅‖∞) with the fixed point property for nonexpansive mappings; and we discuss similarities and differences with work of Goebel and Kuczumow concerning analogous subsets of ℓ1.

New families of nonreflexive Banach spaces with the fixed point property

In this paper we define the concept of sequentially separating norm and we relate it to the fulfillment of the fixed point property for nonexpansive mappings. We develop a technique to construct families of nonreflexive Banach spaces with the fixed point property. Different examples will be exhibited. We also study the geometry of the Banach spaces that can be renormed with a sequentially separating norm.

Approximation of fixed points and variational solutions for pseudo-contractive mappings in banach spaces

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive mappings which is also a unique solution to variational inequality problem involving ϕ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, convex optimization problems, and split feasibility problems. Our result extends many recent important results.

An overview on the Prus-Szczepanik condition

Abstract In 2005 Prus and Sczcepanik introduced a large class of Banach spaces with the fixed point property for nonexpansive mappings. We say that this class satisfies the PSz condition. Checking that a given Banach space belongs to this class is not an easy task. Here we study the relationship between the PSz condition and other well-known geometrical properties of Banach spaces, and we give easier sufficient conditions for a Banach space to satisfy it.

Diversities, hyperconvexity and fixed points

Diversities have been recently introduced as a generalization of metrics for which a rich tight span theory could be stated. In this work we take up a number of questions about hyperconvexity, diversities and fixed points of nonexpansive mappings. Most of these questions are motivated by the study of the connection between a hyperconvex diversity and its induced metric space for which we provide some answers. Examples are given, for instance, showing that such a metric space need not be hyperconvex but still we prove, as our main result, that they enjoy the fixed point property for nonexpansive mappings provided the diversity is bounded and that this boundedness condition cannot be transferred from the diversity to the induced metric space.

THE FIXED POINT PROPERTY IN DIRECT SUMS AND MODULUS

AbstractWe show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.

The fixed point property for renormings of ℓ2

A famous open question in metric Fixed Point Theory is whether every Banach space which is isomorphic to the Hilbert space ℓ2 has the fixed point property for nonexpansive mappings. We give an overview about the state of the advances towards their solution.

Formula omitted] fixed point property and asymptotic centre for the Fourier–Stieltjes algebra of a locally compact group

In this paper we show that the Fourier–Stieltjes algebra B(G) of a non-compact locally compact group G cannot have the weak⁎ fixed point property for nonexpansive mappings. This answers two open problems posed at a conference in Marseille-Luminy in 1989. We also show that a locally compact group is compact exactly if the asymptotic centre of any non-empty weak⁎ closed bounded convex subset C in B(G) with respect to a decreasing net of bounded subsets is a non-empty norm compact subset. In particular, when G is compact, B(G) has the weak⁎ fixed point property for left reversible semigroups. This generalizes a classical result of T.C. Lim for the circle group. As a consequence of our main results we obtain that a number of properties, some of which were known to hold for compact groups, in fact characterize compact groups.

Remarks on the stability of the fixed point property for nonexpansive mappings

It is shown that in many instances the fixed point property for nonexpansive mappings actually implies the fixed point property for a strictly larger family of mappings. This paper is largely expository, but some of the observations are not readily available, and some appear here for the first time. Several related open questions in are discussed. The emphasis is on accessible problems, especially those that require little background. The problems themselves have been given little thought and may be trivial or difficult.

Distortion and stability of the fixed point property for non-expansive mappings

Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X. We prove that the subset of P(X) formed by the norms failing the fixed point property is dense in P(X) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of ℓ1 can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.

The super fixed point property for asymptotically nonexpansive mappings

We show that the super fixed point property for nonexpansive mappings and for asymptotically nonexpansive mappings in the intermediate sense are equivalent. As a consequence, we obtain fixed point theorems for asymptotically nonexpansive mappings in uniformly nonsquare and uniformly noncreasy Banach spaces. The results are generalized for commuting families of asymptotically nonexpansive mappings.

Renormings and the fixed point property in non-commutative [formula omitted]-spaces

Let M be a finite von Neumann algebra. It is known that L 1 ( M ) and every non-reflexive subspace of L 1 ( M ) fail to have the fixed point property for non-expansive mappings (FPP). We prove a new fixed point theorem for this class of mappings in non-commutative L 1 ( M ) Banach spaces which lets us obtain a sufficient condition such that a closed subspace of L 1 ( M ) can be renormed to satisfy the FPP. As a consequence, we deduce that the predual of every atomic finite von Neumann algebra can be renormed with the FPP.

On the fixed points of nonexpansive mappings in direct sums of Banach spaces

We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.

Failure of the FPP inside an asymptotically isometric-free copy of [formula omitted

We produce an equivalent renorming of c 0 which fails to contain an asymptotically isometric copy of c 0 , yet all of its infinite-dimensional subspaces fail to have the fixed point property for nonexpansive mappings on closed, bounded, convex subsets.

The Ptolemy and Zbăganu constants of normed spaces

In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖ x − y ‖ ‖ z − w ‖ ≤ ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ for any points w , x , y , z in H . It is known that for each normed space ( X , ‖ ⋅ ‖ ) , there exists a constant C such that for any w , x , y , z ∈ X , we have ‖ x − y ‖ ‖ z − w ‖ ≤ C ( ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ ) . The smallest such C is called the Ptolemy constant of X and is denoted by C P ( X ) . We study the relationships between this constant and the geometry of the space X , and hence with metric fixed point theory. In particular, we relate the Ptolemy constant C P to the Zbăganu constant C Z , and prove that if X is a Banach space with C Z ( X ) < ( 1 + 3 ) / 2 , then X has (uniform) normal structure and therefore the fixed point property for nonexpansive mappings. We derive general lower and upper bounds for both C P and C Z , and calculate the precise values of these two constants for several normed spaces. We also present a number of conjectures and open problems.

Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

Strong convergence theorems for a common zero of a countably infinite family of [formula omitted]-inverse strongly accretive mappings

Let E be a real reflexive Banach space which has a uniformly Gâteaux differentiable norm. Assume that every nonempty closed convex and bounded subset of E has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a common zero of a countably infinite family of α -inverse strongly accretive mappings are proved. Related results deal with strong convergence of theorems to a common fixed point of a countably infinite family of strictly pseudocontractive mappings.

Three-dimensional convexity and the fixed point property for nonexpansive mappings

We prove that the fixed point property holds in Banach spaces for which its unit sphere does not contain a triangle with sides of length larger than 4/3.

Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups

The main result of this paper is that a closed convex subset of a Banach space has the fixed point property for nonexpansive mappings if and only if it has the fixed point property for nonexpansive semigroups.