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In this paper, we investigate some fixed point properties of nonexpansive mappings with respect to orbits in Banach spaces X equipped with abstract linear topologies $$\tau $$ for which the norm $$\Vert .\Vert _X$$ is sequentially lower semicontinuous relative to $$\tau $$ . The important situations where $$\tau $$ is the weak topology on X or $$\tau $$ is the topology of the convergence in measure on $$L^{1}(\mu )$$ will be considered. Next, we study the existence of common fixed points for actions of jointly continuous left amenable semitopological semigroups of nonexpansive and Kannan mappings with respect to orbits on compact convex subsets of 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 c...

In the present paper, we redefine the notion of $$T_f$$ -orbitally lower semi-continuous mapping, and obtain a hybrid coincidence point theorem satisfying new inequalities in the framework of $$\delta $$ -distance. We also obtain some sufficient conditions for the existence of common fixed points of a pair of asymptotically regular self-mappings on a complete metric space. Moreover, we show that the orbital continuity of a pair of non-commuting mappings of a complete metric space is equivalent to fixed point property. Several examples are given to illustrate the results. Our results extend several well-known fixed point theorems for both single valued and multi-valued mappings existing in the literature.

For a finite ground set X, this paper investigates properties of the set of orders with the fixed point property as a subset of the set ${\mathcal O} (X)$ of all orders on X, ordered by inclusion. In particular, it is shown that this set can have singleton components in the covering graph of ${\mathcal O} (X)$ , we identify longest possible chains of orders such that orders alternate between having and not having the fixed point property, and we give examples of nondismantlable orders with the fixed point property such that every upper cover in ${\mathcal O} (X)$ has the fixed point property, too.

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 (resp. typical $ k $-isomorphism) and the $ PN $-$ k $-continuity (resp. $ 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) $.

It is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

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

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.

In this short note we consider a sort of converse of the Banach fixed point theorem and prove that a metric space X is complete if and only if, for each closed subspace Y ⊆ X, any contraction f : Y → Y has a fixed point y ∈ Y.

In this paper, we present the fixed point property for nonexpansive type mappings in Banach spaces endowed with near-infinity concentrated norms. We also obtain a stability result. Finally, we present a nontrivial example to show the usefulness of these results.

The main purpose of this study is to characterize some matrix classes from classical sequence spaces into a newly introduced space and find the norm of some special matrix operators. Also, we give certain geometric properties of this space.

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.

In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.

In this paper we are dealing with the question on what amenability property should possess a semigroup to ensure it has the Schauder fixed point property. We provide a positive answer to this problem for the class $$\Sigma $$-ELA of all n-extremely left amenable semitopological semigroups. As an application, we derive partial answers (for $$\Sigma $$-ELA, locally finite groups, and semidirect product of $$\Sigma $$-ELA semigroups) to an open problem posted by Lau in 1976 (Lau, Fixed point theory and its applications, Academic Press, Orlando, 1976)

In this paper, we prove that every nonempty compact s-convex subset $$(0

Kovalevsky has developed an abstract cellular complex for representing digital images in the combinatorial grid. This paper initiates the notions of cellular homotopy and cellular path homotopy through the continuous connected preserving map and investigates some of their basic properties in the abstract cellular complex. Further, the concept of fundamental groups in the abstract cellular complex is introduced using the notion of cellular homotopy and some of their basic properties are investigated. Finally, the paper presents the concept of homology and fixed point property in the connected abstract cellular complex.

A Banach space X is said to have the fixed point property if for every closed convex bounded subset C ⊂ X and for every T : C → C nonexpansive, there is a fixed point. A Banach space is said to have the dual-fixed point property if its dual space X ⁎ verifies the fixed point property. It is a fact that ( C 0 ( K ) , ‖ ⋅ ‖ ∞ ) fails the dual-fixed point property for every locally compact Hausdorff set K. In this article we study the fulfilment of the dual-fixed point property under equivalent norms. We prove that C 0 ( K ) can be renormed to have the dual-fixed point property if and only if K is a countable set. In this case we prove that this renorming can be achieved as close to the standard norm on C 0 ( K ) as we like for the Banach-Mazur distance. We further obtain the corresponding weak⁎-fixed-point-property stability coefficient regarding the duality σ ( C 0 ( K ) ⁎ , C 0 ( K ) ) . It will be displayed that the value of this stability coefficient strongly depends on the distribution of the accumulation points of K. Finally, we prove that the dual-fixed point property is completely ‘‘unstable” for the Banach-Mazur distance in most of nonreflexive Banach spaces.

The present paper studies two topologies of the quotient spaces related to the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological plane. The topologies are exactly proved to be the cofinite particular point and the excluded two points topology. We also concern separation axioms and nowhere dense subset properties of the quotient spaces of the M-topological plane. Furthermore, we investigate both a particular point and an excluded point topological structure associated with the quotient spaces of the M-topological plane. Finally, we investigate the fixed point property of the compactification of the M-topological plane. Unlike the non-fixed point property of the Hausdorff compactification of the Euclidean topological plane, we prove that every continuous self-bijection of the compactification of the M-topological plane has a fixed point.

In this article, we introduce the concepts of multivalued (DL)-type and multivalued $$\alpha $$-nonexpansive mappings in the Banach spaces. We show that these two classes of mappings properly contain some important classes of nonlinear mappings. Moreover, we compare the relationship between such classes of mappings and obtain some fixed point results. In addition, we give partial answer to the open question posed by Reich in 1983, about the relationship between fixed point property of multivalued and single-valued nonexpansive mappings. This contribution generalizes and improves some recent results in this context.

We classify elements of a cluster modular group into three types. We characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble. The characterization gives an analogue of the Nielsen-Thurston classification theory on the mapping class group of a surface.

[EN] 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 {Tnx} 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., The authors would like to thank the anonymous referee for the comments and suggestions. The first author acknowledges the University Grants Commission, New Delhi, for providing financial support in the form of project fellow through Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai.

The present paper introduces and estimates the complexity of the fixed point property of a digital image (X, k) for any k − D C -self-map f of (X, k), where a k − D C -self-map f of (X, k) means a digitally k-continuous self-map of (X, k) with a digital version of the Banach contraction principle. To do this work, we need to study some properties of iterations of a k − D C -self-map f of (X, k) and to establish the notion of complexity of (X, k) denote by C♯(X, k) (see Definition 7 in the present paper). According to C♯(X, k), we can estimate complexity of the fixed point property of (X, k) for any k − D C -self-map f of (X, k). Based on this approach, the present paper investigates some relationships between the k-adjacency of (X, k) and C♯(X, k). Furthermore, we prove that C♯(X, k) is not a digital topological invariant. Besides, we develop the notions of uniform k-connectedness and strict k-connectivity to calculate C♯(X, k) for some digital images (X, k). In the paper each (X, k) is assumed to be a k-connected and non-empty set and 2 ≤ | X |≨∞, where | X | means the cardinal number of the given set X.

We propose two families of tests for the classical goodness-of-fit problem to univariate normality. The new procedures are based on $L^2$-distances of the empirical zero-bias transformation to the normal distribution or the empirical distribution of the data, respectively. Weak convergence results are derived under the null hypothesis, under fixed alternatives as well as under contiguous alternatives. Empirical critical values are provided and a comparative finite-sample power study shows the competitiveness to classical procedures.

In this paper, we show that orbital continuity of a pair of non-commuting mappings of a complete metric space is equivalent to fixed point property under the Proinov type condition. Furthermore, we establish a situation in which orbital continuity turns out to be a necessary and sufficient condition for the existence of a common fixed point of a pair of mappings yet the mappings are not necessarily continuous at the common fixed point.

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 $$c_0$$ c 0 or $$\ell _1$$ ℓ 1

[EN] This article introduces free group representations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer., The first author has been supported by the Natural Sciences & Engineering Research Council of Canada (NSERC) discovery grant 185986 and Instituto Nazionale di Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036 and Scientific and Technological Research Council of Turkey (TUBITAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211301223.

In this paper, we discuss the fixed point property for an infinite family of order-preserving mappings which satisfy the Lipschitz condition on comparable pairs. The underlying framework of our main results is a metric space of any global upper curvature bound \(\kappa \in \mathbb {R}\), i.e., a \(\mathrm CAT(\kappa )\) space. In particular, we prove the existence of a fixed point for a nonexpansive semigroup on comparable pairs. Then, we propose and analyze two algorithms to approximate such a fixed point.

The present paper, as a survey paper, studies the fixed point property (FPP, for brevity) and the almost fixed point property (AFPP, for short) for digital spaces whose structures are induced by a digital graph in terms of the Rosenfeld model (or digital metric space), the Khalimsky (K-, for brevity), or the (extended) Marcus-Wyse (M-, for short) topology. Furthermore, we also investigate various properties of digital isomorphic (or homeomorphic), digital homotopic, retract, and product properties of the FPP and the AFPP of them. This approach can be used in applied sciences such as some areas of pure and applied topologies, applied analysis, and computer science such as computer graphics, image processing, pattern recognition, mathematical morphology, artificial intelligence, and so forth. All digital spaces are assumed to be connected (or k-connected) unless stated otherwise.

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.

We study a fixed point property for linear actions of discrete groups on weakly complete convex proper cones in locally convex topological vector spaces. We search to understand the class of discrete groups which enjoys this property and we try to generalize it for Hausdorff topological groups.

We prove that if X is a Banach space whose Banach-Mazur distance to a Hilbert space is less than 2 4 12 + 7 2 + 6 ( 19 + 12 2 ) , then X has the fixed point property for nonexpansive mappings. Also, a renorming of l 2 is constructed in order to show the limitations of the usual techniques.

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.

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 L p ( ⋅ ) ( Ω ) in terms of the exponent function p ( ⋅ ) and the absence of an isometric copy of L 1 [ 0 , 1 ] . In particular, every reflexive L p ( ⋅ ) ( 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 L p -spaces. In connection with the FPP, we prove that Maurey's result for L 1 -spaces can be extended to the larger class of variable L p ( ⋅ ) ( Ω ) spaces with order continuous norm, that is, every reflexive subspace of L p ( ⋅ ) ( Ω ) has the FPP. Nevertheless, Maurey's converse does not longer hold in the variable setting, since some nonreflexive subspaces of L p ( ⋅ ) ( Ω ) satisfying the FPP can be found. As a consequence, we discover that several nonreflexive Nakano sequence spaces l p n 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 l 1 in L p ( ⋅ ) ( Ω ) is also analyzed.

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.