Fixed Point Property Research Articles

Amenability properties and fixed point properties of affine representations of semigroups

Amenability properties and fixed point properties of affine representations of semigroups

General Higher-Order Lipschitz Mappings

In this paper, we introduce a new class of mappings and investigate their fixed point property. In the first direction, we prove a fixed point theorem for general higher-order contraction mappings in a given metric space and finally prove an approximate fixed point property for general higher-order nonexpansive mappings in a Banach space.

The almost fixed point property is not invariant under isometric renormings

In the present note we prove the non set-stability of the AFPP under isometric renormings in the setting of Banach spaces containing a complemented isomorphic copy of c_0 or ell _1

Isometric embedding of Banach spaces under optimal projection constants

Let X be a Banach space with separable dual. It is proved that for every ε∈(0,1), X embeds isometrically into a Banach space W with a shrinking basis (wn) which is (1+ε)-monotone. Moreover, if X has further an FDD (En) whose strong bimonotonicity projection constant is not larger than D, then (wn) has strong bimonotonicity projection constant not exceeding D(1+ε). Further, if (En) is C-unconditional then (wn) is C(1+ε)-unconditional. The proof uses renorming and skipped blocking decomposition techniques. As an application, we prove that every Banach space having a shrinking D-unconditional basis with D<6−1, has the weak fixed point property.

Strong and pure fixed point properties of mappings on normed spaces

In this paper, the concept of strong fixed point property of a mapping is defined. The relationship between an approximating fixed point sequence and compactness is studied. The notion of a center set for a mapping is introduced. Some necessary and sufficient conditions for existence of mappings admitting a center set are presented. The results proved herein extend, generalize and improve some well-known comparable results.

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 .

Unions of chainable continua with the fixed point property

We investigate the fixed point property for continua that are unions of chainable continua. We answer Question&nbsp;2 of our paper [Fund. Math. 231 (2013)] by showing the fixed point property is additive for chainable continua. We additionally show that (

The product property of the almost fixed point property for digital spaces

Consider two digital spaces $ (X_i, k_i), i \in \{1, 2\} $, (in the sense of Rosenfeld model ) satisfying the almost fixed point property( AFPP for brevity). Then, the problem of whether the AFPP for the digital spaces is, or is not necessarily invariant under Cartesian products plays an important role in digital topology, which remains open. Given a Cartesian product $ (X_1 \times X_2, k) $ with a certain $ k $-adjacency, after proving that the AFPP for digital spaces is not necessarily invariant under Cartesian products, the present paper proposes a certain condition of which the AFPP for digital spaces holds under Cartesian products. Indeed, we find that the product property of the AFPP is strongly related to both the sets $ X_i $ and the $ k_i $-adjacency, $ i \in \{1, 2\} $. Eventually, assume two $ k_i $-connected digital spaces $ (X_i, k_i), i \in \{1, 2\} $, and a digital product $ X_1 \times X_2 $ with a normal $ k $-adjacency such that $ N_k^\star(p, 1) = N_k(p, 1) $ for each point $ p \in X_1 \times X_2 $ (see Remark 4.2(1)). Then we obtain that each of $ (X_i, k_i), i \in \{1, 2\} $, has the AFPP if and only if $ (X_1 \times X_2, k) $ has the AFPP .

Digital products with $ PN_k $-adjacencies and the almost fixed point property in $ DTC_k^\blacktriangle $

&lt;abstract&gt;&lt;p&gt;Given two digital images $ (X_i, k_i), i \in \{1, 2\} $, first of all we establish a new $ PN_k $-adjacency relation in a digital product $ X_1 \times X_2 $ to obtain a relation set $ (X_1 \times X_2, PN_k) $, where the term $ ''$$ PN $" means $ ''$pseudo-normal". Indeed, a $ PN $-$ k $-adjacency is softer or broader than a normal $ k $-adjacency. Next, the present paper initially develops both $ PN $-$ k $-continuity and $ PN $-$ k $-isomorphism. Furthermore, it proves that these new concepts, the $ PN $-$ k $-continuity and $ PN $-$ k $-isomorphism, need not be equal to the typical $ k $-continuity and a $ k $-isomorphism, respectively. Precisely, we prove that none of the typical $ k $-continuity (&lt;italic&gt;resp.&lt;/italic&gt; typical $ k $-isomorphism) and the $ PN $-$ k $-continuity (&lt;italic&gt;resp.&lt;/italic&gt; $ PN $-$ k $-isomorphism) implies the other. Then we prove that for each $ i \in \{1, 2\} $, the typical projection map $ P_i: X_1 \times X_2 \to X_i $ preserves a $ PN_k $-adjacency relation in $ X_1 \times X_2 $ to the $ k_i $-adjacency relation in $ (X_i, k_i) $. In particular, using a $ PN $-$ k $-isomorphism, we can classify digital products with $ PN_k $-adjacencies. Furthermore, in the category of digital products with $ PN_k $-adjacencies and $ PN $-$ k $-continuous maps between two digital products with $ PN_k $-adjacencies, denoted by $ DTC_k^\blacktriangle $, we finally study the (almost) fixed point property of $ (X_1 \times X_2, PN_k) $.&lt;/p&gt;&lt;/abstract&gt;

On combining the zero bias transform and the empirical characteristic function to test normality

We propose a new powerful family of tests of univariate normality. These tests are based on an initial value problem in the space of characteristic functions originating from the fixed point property of the normal distribution in the zero bias transform. Limit distributions of the test statistics are provided under the null hypothesis, as well as under contiguous and fixed alternatives. Using the covariance structure of the limiting Gaussian process from the null distribution, we derive explicit formulas for the first four cumulants of the limiting random element and apply the results by fitting a distribution from the Pearson system. A comparative Monte Carlo power study shows that the new tests are serious competitors to the strongest well established tests.

Fixed point properties and reflexivity in variable Lebesgue spaces

In this paper the weak fixed point property (w-FPP) and the fixed point property (FPP) in Variable Lebesgue Spaces are studied. Given (Ω,Σ,μ) a σ-finite measure and p(⋅) a variable exponent function, the w-FPP is completely characterized for the variable Lebesgue space Lp(⋅)(Ω) in terms of the exponent function p(⋅) and the absence of an isometric copy of L1[0,1]. In particular, every reflexive Lp(⋅)(Ω) has the FPP and our results bring to light the existence of some nonreflexive variable Lebesgue spaces satisfying the w-FPP, in sharp contrast with the classic Lebesgue Lp-spaces. In connection with the FPP, we prove that Maurey's result for L1-spaces can be extended to the larger class of variable Lp(⋅)(Ω) spaces with order continuous norm, that is, every reflexive subspace of Lp(⋅)(Ω) has the FPP. Nevertheless, Maurey's converse does not longer hold in the variable setting, since some nonreflexive subspaces of Lp(⋅)(Ω) satisfying the FPP can be found. As a consequence, we discover that several nonreflexive Nakano sequence spaces ℓpn do have the FPP endowed with the Luxemburg norm. As far as the authors are concerned, this family of sequence spaces gives rise to the first known nonreflexive classic Banach spaces enjoying the FPP without requiring of any renorming procedure. The failure of asymptotically isometric copies of ℓ1 in Lp(⋅)(Ω) is also analyzed.

Some fixed point theorems for tree-like continua

We show that certain tree-like continua, that can be realized as inverse limits on [0,1] with set-valued bonding functions, must have the fixed point property. In this class of tree-like continua, one result involves inverse limits for which the bonding functions are interval-valued and their graphs are trees, and another result involves inverse limits whose partial graphs are dendrites.

Sectional category and the fixed point property

For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $\pi_{2,1}^X\colon F(X,2)\to X$, and the fixed point property (FPP) for self-maps on $X$. Explicitly, we demonstrate that a space $X$ has the FPP if and only if 2 is the minimal cardinality of open covers $\{U_i\}$ of $X$ such that each $U_i$ admits a continuous local section for $\pi_{2,1}^X$. This characterization connects a standard problem in fixed point theory to current research trends in topological robotics.

On the asymptotics of (c)-mapping iterations

In this paper, we give an investigation of the asymptotics for the iterations associated with (c)-mappings acting on unbounded closed convex subsets of Banach spaces. In particular, the almost fixed point property (in short; AFPP) and conditions ensuring an ergodic type theorem for this class of mappings are established.

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.

Weak compactness and fixed point property for affine bi-Lipschitz maps

Let X be a Banach space and let C be a closed convex bounded subset of X. It is proved that C is weakly compact if, and only if, C has the generic fixed point property (G-FPP) for the class of L-bi-Lipschitz affine mappings for every L>1. It is also proved that if X has Pełczyński's property (u), then either C is weakly compact, contains an ℓ1-sequence or a c0-summing basic sequence. In this case, weak compactness of C is equivalent to the G-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We also introduce a generalized form of property (u), called property(su), and use it to prove that if X has property (su) then either C is weakly compact or contains a wide-(s) sequence which is uniformly shift equivalent. In this case, weak compactness in such spaces can also be characterized in terms of the G-FPP for affine uniformly bi-Lipschitz mappings. It is also proved that every Banach space with a spreading basis has property (su), thus property (su) is stronger than property (u). These results yield a significant strengthening of an important theorem of Benavides, Japón-Pineda and Prus published in 2004.

Groups of piecewise linear homeomorphisms of flows

To every dynamical system$(X,\varphi )$over a totally disconnected compact space, we associate a left-orderable group$T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of$(X,\varphi )$which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system$(X,\varphi )$, the group$T(\varphi )$does not have infinite subgroups with Kazhdan's property$(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if$(X,\varphi )$has a dense orbit, then the isomorphism type of the group$T(\varphi )$is a complete invariant of flow equivalence of the pair$\{\varphi , \varphi ^{-1}\}$.

On isomorphic embeddings of c into L1-preduals and some applications

First we study the distortion ‖T‖‖T−1‖ of an isomorphic embedding T of the space c of convergent sequences into an infinite-dimensional L1-predual X such that (extBX⁎)′⊂rBX⁎ for some r∈(0,1), that is, the set of all weak⁎ cluster points of the set of all extreme points of the closed unit ball of the dual space X⁎ is contained in the closed ball of radius r∈(0,1). We prove that ‖T‖‖T−1‖≥(3−r)/(1+r). Moreover, we give some examples showing that for every r∈(0,1) our bound is optimal. Then, we apply our theorem to provide stability results for polyhedrality and extendability of compact operators in the setting of L1-preduals. To achieve the latter goal, we additionally give a new characterization of infinite-dimensional Banach spaces having the compact norm-preserving extension property for compact operators. Furthermore, in the framework of ℓ1-preduals such that the standard basis in ℓ1 is weak⁎ convergent, we provide precise values of stability constants for the weak⁎ fixed point property.

Invariant means on weakly almost periodic functions and generalized fixed point properties

In this paper, we prove common fixed point theorems for Generalized Suzuki Contractions (abbreviated as GSC) involving two semi-topological semigroups of self-mappings $$S_{1}$$ and $$S_{2}$$ , besides establishing the existence of a left invariant mean (abbreviated as LIM) on the space of all weakly almost periodic functions on $$S_{1}\cap S_{2}$$ (abbreviated as $$WAP(S_{1}\cap S_{2})$$ ).

On Local Generalized Ulam–Hyers Stability for Nonlinear Fractional Functional Differential Equation

We discuss the existence of positive solution for a class of nonlinear fractional differential equations with delay involving Caputo derivative. Well-known Leray–Schauder theorem, Arzela–Ascoli theorem, and Banach contraction principle are used for the fixed point property and existence of a solution. We establish local generalized Ulam–Hyers stability and local generalized Ulam–Hyers–Rassias stability for the same class of nonlinear fractional neutral differential equations. The simulation of an example is also given to show the applicability of our results.