Point Property For Nonexpansive Mappings Research Articles

Fixed points of $G$-monotone mappings in metric and modular spaces

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

“Lion–man” game and the fixed point property in Hilbert spaces

The main goal of the paper is to show that in the setting of a Hilbert space a closed and convex K has the fixed point property for nonexpansive mappings if and only if the lion always wins the Lion–Man game played in K.

A fixed point theorem in strictly convex $ b $-fuzzy metric spaces

<abstract><p>The main motivation for this paper is to investigate the fixed point property for non-expansive mappings defined on $ b $-fuzzy metric spaces. First, following the idea of S. Ješić's result from 2009, we introduce convex, strictly convex and normal structures for sets in $ b $-fuzzy metric spaces. By using topological methods and these notions, we prove the existence of fixed points for self-mappings defined on $ b $-fuzzy metric spaces satisfying a nonlinear type condition. This result generalizes and improves many previously known results, such as W. Takahashi's result on metric spaces from 1970. A representative example illustrating the main result is provided.</p></abstract>

A large class in Köthe-Toeplitz duals of generalized Cesàro difference sequence spaces with fixed point property for nonexpansive mappings

In 1970, Ces?ro sequence spaces was introduced by Shiue. In 1981, K?zmaz defined difference sequence spaces for ??, c0 and c. Then, in 1983, Orhan introduced Ces?ro difference sequence spaces. Both works used difference operator and investigated K?the-Toeplitz duals for the new Banach spaces they introduced. Later, various authors generalized these new spaces, especially the one introduced by Orhan. In this study, first we discuss the fixed point property for these spaces. Then, we recall that Goebel and Kuczumow showed that there exists a very large class of closed, bounded, convex subsets in Banach space of absolutely summable scalar sequences, ?1 with fixed point property for nonexpansive mappings. So we consider a Goebel and Kuczumow analogue result for a K?the-Toeplitz dual of a generalized Ces?ro difference sequence space. We show that there exists a large class of closed, bounded and convex subsets of these spaces with fixed point property for nonexpansive mappings.

Stability of the fixed point property for Hilbert spaces

We prove that if X is a Banach space whose Banach-Mazur distance to a Hilbert space is less than 2412+72+6(19+122), then X has the fixed point property for nonexpansive mappings. Also, a renorming of ℓ2 is constructed in order to show the limitations of the usual techniques.

On Certain Reversed Mean Lipschitzian Mappings in Banach Spaces

We introduce reversed mean Lipschitzian mappings and find the bound for reversed mean Lipschitz constants of its iterates under certain conditions. Moreover, we prove that if a bounded closed and convex subset C of Banach space has fixed point property for nonexpansive mappings, then C also has fixed point property for certain reversed -Lipschitzian mappings such that

Weak* fixed point property and the space of affine functions

First we prove that if a separable Banach space X X contains an isometric copy of an infinite-dimensional space A ( S ) A(S) of affine continuous functions on a Choquet simplex S S , then its dual X ∗ X^* lacks the weak ∗ ^* fixed point property for nonexpansive mappings. Then, we show that the dual of a separable L 1 L_1 -predual X X fails the weak ∗ ^* fixed point property for nonexpansive mappings if and only if X X has a quotient isometric to some infinite-dimensional space A ( S ) A(S) . Moreover, we provide an example showing that “quotient” cannot be replaced by “subspace”. Finally, it is worth mentioning that in our characterization the space A ( S ) A(S) cannot be substituted by any space C ( K ) \mathcal {C}(K) of continuous functions on a compact Hausdorff K K .

The nonexpansive and mean nonexpansive fixed point properties are equivalent for affine mappings

Let C be a convex subset of a Banach space X and let T be a mapping from C into C. Fix $$\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)$$ a multi-index in $${\mathbb {R}}^n$$ such that $$\alpha _i\ge 0$$ ( $$1\le i\le n$$ ), $$\sum _{i=1}^n\alpha _i=1$$ , $$\alpha _1,\alpha _n>0$$ , and consider the mapping $$T_\alpha :C\rightarrow C$$ given by $$T_\alpha =\sum _{i=1}^n \alpha _i T^i$$ . Every fixed point of T is a fixed point for $$T_\alpha $$ but the converse does not hold in general. In this paper we study necessary and sufficient conditions to assure the existence of fixed points for T in terms of the existence of fixed points of $$T_\alpha $$ and the behaviour of the T-orbits of the points in the domain of T. As a consequence, we prove that the fixed point property for nonexpansive mappings and the fixed point property for mean nonexpansive mappings are equivalent conditions when the involved mappings are affine. Some extensions for more general classes of mappings are also achieved.

Asymptotically isometric copies of $\ell^{1\boxplus 0}$

Using James' Distortion Theorems, researchers have inquired relations between spaces containing nice copies of $c_0$ or $\ell^1$ and the failure of the fixed point property for nonexpansive mappings especially after the fact that every classical nonreflexive Banach space contains an isometric copy of either $\ell^1$ or $c_0$. For instance, finding asymptotically isometric (ai) copies of $\ell^1$ or $c_0$ inside a Banach space reveals the space's failure of the fixed point property for nonexpansive mappings. There has been many researches done using these tools developed by James and followed by Dowling, Lennard, and Turett mainly to see if a Banach space can be renormed to have the fixed point property for nonexpansive mappings when there is failure.In this paper, we introduce the concept of Banach spaces containing ai copies of $\ell^{1\boxplus 0}$ and give alternative methods of detecting them. We show the relationsbetween spaces containing these copies and the failure of the fixed point property for nonexpansive mappings. Finally, we give some remarks and examples pointing our vital result: if a Banach space contains an ai copy of $\ell^{1\boxplus 0}$, then it contains an ai copy of $\ell^1$ but the converse does not hold.

Geodesic rays, the “Lion-Man” game, and the fixed point property

This paper focuses on the relation among the existence of different types of curves (such as directional ones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings, and a discrete lion and man game. Our main result holds in the setting of CAT(0) spaces that are additionally Gromov hyperbolic.

Fixed point properties for Lorentz sequence spaces $$\ell _{\rho ,\infty }^0$$ and $$\ell _{\rho ,1}$$

In this paper, we work on Lorentz sequence spaces and explore the fixed point property for $$\ell _{\rho ,\infty }^0$$ and $$\ell _{\rho ,1}$$ space. We prove that both spaces enjoy the weak fixed point property for nonexpansive mappings. To prove the weak fixed point property for $$\ell _{\rho ,1}$$, we provide alternative methods such that one gives exact value of its Riesz angle. We show that both spaces fail the fixed point property for nonexpansive mappings since $$\ell _{\rho ,1}$$ contains an asymptotically isometric copy of $$\ell ^1$$ and $$\ell _{\rho ,\infty }^0$$ contains an asymptotically isometric copy of $$c_0$$. We also show that there exists a non-empty, closed, bounded and convex subset C of $$\ell _{\rho ,1}$$ and a fixed point-free affine, nonexpansive mapping $$T:C\longrightarrow C$$ and so $$\ell _{\rho ,1}$$ fails to have the fixed point property for affine nonexpansive mappings. Contrary to this, in the final section, we get a Goebel and Kuczumow analogy for $$\ell _{\rho ,1}$$ by proving that there exists a large class of non-weak* compact, closed, bounded and convex subsets of $$\ell _{\rho ,1}$$ with the fixed point property for affine nonexpansive mappings.

Some renormings of Banach spaces with the weak fixed point property for nonexpansive mappings

Some renormings of Banach spaces with the weak fixed point property for nonexpansive mappings

Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings

<p>Kirk introduced the notion of pointwise eventually asymptotically non-expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically non expansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings when ever X has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space X has the fixed point property for pointwise eventually asymptotically nonexpansive maps if X has uniform normal structure or X is uniformly convex in every direction with the Maluta constant D(X) < 1. Also, we study the asymptotic behavior of the sequence {T<sup>n</sup>x} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map.</p>

Generalized trend constants of Lipschitz mappings

In 2015, Goebel and Bolibok defined the initial trend coefficient of a mapping and the class of initially nonexpansive mappings. They proved that the fixed point property for nonexpansive mappings implies the fixed point property for initially nonexpansive mappings. We generalize the above concepts and prove an analogous fixed point theorem. We also study the initial trend coefficient more deeply.

A norm inequality in James’ space and stability of the fixed point property

In this paper we prove a norm inequality in James’ space J, and use it to show that the fixed point property for nonexpansive mappings is passed on from J to those Banach spaces X whose Banach–Mazur distance to J satisfies $$d(X,J)<\sqrt{\frac{17+\sqrt{97}}{12}}$$ .

The generalized Day norm. Part II. Applications

In this paper we prove that for each \(1&amp;lt; p, \tilde{p} &amp;lt; \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.

Stability constants of the weak⁎ fixed point property for the space ℓ1

The main aim of the paper is to study some quantitative aspects of the stability of the weak⁎ fixed point property for nonexpansive mappings in ℓ1 (shortly, w⁎-fpp). We focus on two complementary approaches to this topic. First, given a predual X of ℓ1 such that the σ(ℓ1,X)-fpp holds, we precisely establish how far, with respect to the Banach–Mazur distance, we can move from X without losing the w⁎-fpp. The interesting point to note here is that our estimate depends only on the smallest radius of the ball in ℓ1 containing all σ(ℓ1,X)-cluster points of the extreme points of the unit ball. Second, we pass to consider the stability of the w⁎-fpp in the restricted framework of preduals of ℓ1. Namely, we show that every predual X of ℓ1 with a distance from c0 strictly less than 3, induces a weak⁎ topology on ℓ1 such that the σ(ℓ1,X)-fpp holds.

On existence of fixed points for pointwise eventually nonexpansive mappings

Kirk and Xu introduced the concept of pointwise eventually nonexpansive maps and raised the question: “Does a Banach space X have the fixed point property for pointwise eventually nonexpansive mappings whenever X has the fixed point property for nonexpansive mappings?” Butsan et al. proved that nearly uniform convex Banach spaces have fixed point property for pointwise eventually nonexpansive mappings. In this paper, we prove that X has the fixed point property for pointwise eventually nonexpansive mappings if X has uniform normal structure or X is uniformly convex in every direction.

Separable Lindenstrauss spaces whose duals lack the weak$^*$ fixed point property for nonexpansive mappings

In this paper we study the $w^*$-fixed point property for nonexpansive mappings. First we show that the dual space $X^*$ lacks the $w^*$-fixed point property whenever $X$ contains an isometric copy of the space $c$. Then, the main result of our paper provides several characterizations of weak-star topologies that fail the fixed point property for nonexpansive mappings in $\ell_1$ space. This result allows us to obtain a characterization of all separable Lindenstrauss spaces $X$ inducing the failure of $w^*$-fixed point property in $X^*$.

Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces

In this paper, we first show that a Banach space X has weak normal structure if and only if X has the weak fixed point property for nonexpansive mappings with respect to (wrt) orbits. Then, we give a counterexample to show that the Goebel–Karlovitz lemma does not hold for minimal invariant sets of nonexpansive mappings wrt orbits, and we present a modified version of the Goebel–Karlovitz lemma.