More on minimal invariant sets for nonexpansive mappings

Fixed Point Theory and Applications, 2010

We prove that the set of common fixed points of a given countable family of relatively nonexpansive mappings is identical to the fixed-point set of a single strongly relatively nonexpansive mapping. This answers Kohsaka and Takahashi's question in positive. We also introduce the concept of strongly generalized nonexpansive mappings and prove the analogue version of the result above for Ibaraki-Takahashi's generalized nonexpansive mappings. The duality theorem for two classes of strongly relatively nonexpansive mappings and of strongly generalized nonexpansive mappings is proved.

Fundamenta Mathematicae, 2001

For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is even open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f (A) and f -1 (A) share those topological properties with A which describe how large a set is. Using these results it is proved that any minimal map in a compact metric space is almost one-to-one and, moreover, when restricted to a suitable invariant residual set it becomes a minimal homeomorphism. Finally, two kinds of examples of noninvertible minimal maps on the torus are given -these are obtained either as a factor or as an extension of an appropriate minimal homeomorphism of the torus.

Bulletin of the Australian Mathematical Society, 1999

Journal of Applied Mathematics, 2013

A common fixed point theorem for a pair of maps satisfying condition (C) is proved under certain conditions. We extend the well-knownDeMarr's fixed point theorem to the case of noncommuting family of maps satisfying condition (C). As for an application, an invariant approximation theorem is also derived.

Journal of Mathematical Analysis and Applications, 2006

Minimal maps in compact metric spaces are known to be almost one-to-one. Thus, the set of points with more than one preimage is of first category. In the present paper we study the measure of this set with respect to the invariant measures of the considered minimal map. Among others, we give an example of a minimal self-mapping of a continuum such that the set of points with more than one preimage has positive measure for every invariant measure.

Abstract and Applied Analysis, 2011

Abstract and Applied Analysis, 1998

LetXbe a Banach space andτa topology onX. We say thatXhas theτ-fixed point property (τ-FPP) if every nonexpansive mappingTdefined from a bounded convexτ-sequentially compact subsetCofXintoChas a fixed point. Whenτsatisfies certain regularity conditions, we show that normal structure assures theτ-FPP and Goebel-Karlovitz's Lemma still holds. We use this results to study two geometrical properties which imply theτ-FPP: theτ-GGLD andM(τ)properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of theτ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach spaceXsuch that theτ-FPP is shared by any isomorphic Banach spaceYsatisfying that the Banach-Mazur distance betweenXandYis less than some of these constants.

Mathematics, 2020

In the paper, we show that some results related to Reich and Chatterjea type nonexpansive mappings are still valid if we relax or remove some hypotheses.

Journal of Fixed Point Theory and Applications, 2019

We introduce two types of mappings, namely Reich type nonexpansive and Chatterjea type nonexpansive mappings, and derive some sufficient conditions under which these two types of mappings possess an approximate fixed point sequence (AFPS). We obtain the desired AFPS using the well-known Schäef er iteration method. Along with these, we check some special properties of the fixed point sets of these mappings, such as closedness, convexity, remotality, unique remotality, etc. We also derive a nice interrelation between AFPS and maximizing sequence for both types of mappings. Finally, we will get some sufficient conditions under which the class of Reich type nonexpansive mappings reduces to that of nonexpansive maps.

Bulletin of the Australian Mathematical Society, 2015

We define a class of nonlinear mappings which is properly larger than the class of nonexpansive mappings. We also give a fixed point theorem for this new class of mappings.

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2007

The homotopic invariance of fixed points of set-valued contractions and nonexpansive mappings is studied. As application, nonlinear alternative principles are given. A LeraySchauder alternative and an antipodal theorem for set-valued nonexpansive mappings are also included.

Journal of Fourier Analysis and Applications

A sufficient condition for a set Ω ⊂ L 1 ([0, 1] m) to be invariant Kminimal with respect to the couple L 1 ([0, 1] m) , L ∞ ([0, 1] m) is established. Through this condition, different examples of invariant K-minimal sets are constructed. In particular, it is shown that the L 1-closure of the image of the L ∞-ball of smooth vector fields with support in (0, 1) m under the divergence operator is an invariant K-minimal set. The constructed examples have finite-dimensional analogues in terms of invariant K-minimal sets with respect to the couple ℓ 1 , ℓ ∞ on R n. These finite-dimensional analogues are interesting in themselves and connected to applications where the element with minimal K-functional is important. We provide a convergent algorithm for computing the element with minimal Kfunctional in these and other finite-dimensional invariant K-minimal sets. 2010 Mathematics Subject Classification. Primary 46E30 Secondary 46N10. Key words and phrases. Invariant K-minimal sets, taut strings, real interpolation. NATAN KRUGLYAK AND ERIC SETTERQVIST for all g ∈ Ω and every t > 0. Recall that for a general Banach couple (X 0 , X 1), the K-functional is given by K (t, x; X 0 , X 1) := inf x=x0+x1 x 0 X0 + t x 1 X1 for x ∈ X 0 + X 1 and t > 0. We refer to [2] or [4] for an introduction to the theory of real interpolation. To put our results in a general framework, we introduce the notion of invariant K-minimal sets: Definition 1.1. Given a Banach couple (X 0 , X 1), a set Ω ⊂ X 0 + X 1 is called invariant K-minimal with respect to (X 0 , X 1) if for every a ∈ X 0 + X 1 there exists an element x * ,a ∈ Ω such that K(t, x * ,a − a; X 0 , X 1) ≤ K(t, x − a; X 0 , X 1) holds for all x ∈ Ω and every t > 0. From Definition 1.1 it follows that x * ,a is the nearest element of a in Ω with respect to the norms of all exact interpolation spaces of (X 0 , X 1) generated by the K-method of the theory of real interpolation. In particular, x * ,a is the nearest element of a in Ω with respect to the norms of all interpolation spaces (X 0 , X 1) θ,q , 0 < θ < 1, 1 ≤ q ≤ ∞. As an example, consider the specific couple (X 0 , X 1) = L 1 , L ∞. Then the element x * ,a is the nearest element of a in Ω with respect to the norms of all L p-spaces, 1 < p < ∞, i.e. element of best approximation of a in Ω exists and is invariant with respect to all L p-norms, 1 < p < ∞. Note further from Definition 1.1 that if the set Ω ⊂ X 0 + X 1 is invariant Kminimal, then the set Ω + a, for any a ∈ X 0 + X 1 , is also invariant K-minimal. This is the reason for calling the set invariant K-minimal. Our motivation for introducing invariant K-minimal sets was the taut string problem considered by Dantzig, see [7], in connection with problems in optimal control. Taut string problems have since then appeared in a broad range of applications including statistics, see [1] and [15], image processing, see [19], stochastic processes, see [13], and communication theory, see [18] and [20]. A brief presentation of the taut string problem is given in Section 2.2. Now, let us recall some notion and results from [12].

Discrete Mathematics, 2006

We describe a canonical form for continuous functions : [N] ∞ → [N] ∞ that commute with the shift map X → X\{min X}. Then we investigate in which cases such a function satisfies that for every A ∈ [N] ∞ , there is X ∈ [N] ∞ such that [X] ∞ ⊆ [A] ∞ . This will lead us to solution of The family [N] ∞ of infinite sets of non-negative integers is a prototype of a Ramsey space described long ago in papers of Galvin-Prikry [5], Silver [14] and Ellentuck [2]. It is perhaps less known that Nash-Williams [8] proved the first infinite-dimensional version of Ramsey Theorem in order to handle the shift graph on [N] ∞ (or more precisely on [N] <N ). Recall that the shift map S : [N] ∞ → [N] ∞ is defined by S(A) = A\{min A}. It is therefore quite natural to investigate how much of the infinite-dimensional Ramsey Theorem is captured by the chromatic properties of the shift graph ([N] ∞ , S). Another motivation for the present note is the study initiated in [4] of the chromatic number theory for Borel colorings of Borel graphs. Note that the Galvin-Prikry Theorem shows that the Borel chromatic number of the shift graph ([N] ∞ , S) is infinite. A problem from [7] asks for a characterization of those Borel subsets of [N] ∞ on which the shift graph has infinite Borel chromatic number. We shall address this question here by showing that not all infinitely Borel chromatic subgraphs of ([N] ∞ , S) contain subgraphs of the form [X] ∞ for X ∈ [N] ∞ . We do this by first describing a canonical form of continuous maps : [N] ∞ → [N] ∞ that commute with the shift map S, and then showing that there are maps : [N] ∞ → [N] ∞ that commute with S whose ranges do not contain any set of the form [X] ∞ , X ∈ [N] ∞ . It turns out that the canonical form for shift-invariant continuous maps : [N] ∞ → [N] ∞ is in complexity somewhere between the canonical forms of arbitrary continuous maps of the form : [N] ∞ → N and : [N] ∞ → [N] ∞ described by Pudlak-Rödl [12] and Promel-Voigt [11], respectively. Some notation. We denote by [N] ∞ the set of all infinite subsets of N, the set of natural numbers. [N] ∞ can be seen as a subspace of the space 2 N equipped with the product topology. Given an infinite A ⊆ N, [A] ∞ denotes the collection of infinite subsets of A. The map S : [N] ∞ → [N] ∞ defined by S(A) = A\{min A} is the shift map on [N] ∞

Journal of Geometric Mechanics, 2012

Dedicated to Tudor Ratiu on the occasion of his sixtieth birthday.