Mappings between inverse limits of continua with multivalued bonding functions

We investigate the limit mappings between inverse limits of continua with upper semi-continuous bonding functions. Results are obtained when the coordinate mappings are surjective, one-to-one or homeomorphisms. We construct examples showing the hypothesis of the theorems are essential. Further, we construct an example showing that, unlike for the inverse limits with single valued maps, properties of being monotone, confluent or weakly confluent mappings between factor spaces are not preserved in the inverse limit map.     

Topological entropy of maps on regular curves

In [G.T. Seidler, The topological entropy of homeomorphisms on one-dimensional continua, Proc. Amer. Math. Soc. 108 (1990) 1025-1030], G.T. Seidler proved that the topological entropy of every homeomorphism on a regular curve is zero. Also, in [H. Kato, Topological entropy of monotone maps and confluent maps on regular curves, Topology Proc. 28 (2) (2004) 587-593] the topological entropy of confluent maps on regular curves was investigated. In particular, it was proved that the topological entropy of every monotone map on any regular curve is zero. In this paper, furthermore we investigate the topological entropy of more general maps on regular curves. We evaluate the topological entropy of maps f on regular curves X in terms of the growth of the number of components of f−n(y) (y∈X).     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

On dimension of inverse limits with upper semicontinuous set-valued bonding functions

We give results about the dimension of continua, obtained by combining inverse limits of inverse sequences of metric spaces and one-valued bonding maps with inverse limits of inverse sequences of metric spaces and upper semicontinuous set-valued bonding functions, by standard procedure introduced in [I. Bani?, Continua with kernels, Houston J. Math. (2006), in press].     

Multivalued generalized nonlinear contractive maps and fixed points

We introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps. Consequently, several known fixed point results are either improved or generalized including the corresponding recent fixed point results of Ciric [L.B. Ciric, Multivalued nonlinear contraction mappings, Nonlinear Anal. 71 (2009) 2716-2723], Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132-139], Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112] and Mizoguchi and Takahashi [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188].     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

Dimension zero at all scales

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.     

An extension of A. Ostrowski's Theorem on the round-off stability of iterations

Summary A. M. Ostrowski established the stability of the procedure of successive approximations for Banach contractive maps. In this paper we generalize the above result by using a more general contractive definition introduced by F. Browder. Further, we study the case of maps on metrically convex metric spaces and compact metric spaces, obtaining results relative to fixed point theorems of D. W. Boyd and J. S. W. Wong, and M. Edelstein. Finally, as a by-product of our basic lemma, we extend a recent result of T. Vidalis concerning the convergence of an iteration procedure involving an infinite sequence of maps.     

Banach operator pairs,common fixed-points,invariant approximations,and ∗ -nonexpansive multimaps

Common fixed-point results are proved for I$I$-nonexpansive maps, Ciric type maps, and ∗-nonexpansive multimaps. Invariant approximation results are obtained for these types of maps as applications. Our results extend or generalize several known results including the recent results of Chen and Li [J. Chen, Z. Li, Common fixed-points for Banach operators pairs in best approximation, J. Math. Anal. Appl. 336 (2007) 1466–1475]. In particular, we show that the results of Chen and Li on Banach operator pairs are particular cases of the results of Al-Thagafi and Shahzad [M.A. Al-Thagafi, N. Shahzad, Noncommuting selfmaps and invariant approximations, Nonlinear Anal. 64 (2006) 2778–2786].     

On freely decomposable maps

Freely decomposable and strongly freely decomposable maps were introduced by G.R. Gordh and C.B. Hughes as a generalization of monotone maps with the property that these maps preserve local connectedness in inverse limits. We study further these types of maps, generalize some of the results by Gordh and Hughes and present examples showing that no further generalization is possible.     

Some generalizations of Fedorchuk duality theorem—I

Generalizing duality theorem of V.V. Fedorchuk [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)], we prove Stone-type duality theorems for the following four categories: the objects of all of them are the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of compact Hausdorff spaces and open maps is obtained. Some equivalence theorems for these four categories are stated as well; two of them generalize the Fedorchuk equivalence theorem [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)].     

Fixed point results for set-valued contractions by altering distances in complete metric spaces

Nadler’s contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of set-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi–Takahashi, Feng–Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng–Liu and Klim–Wardowski theorems and the second result is different from the Reich and Mizoguchi–Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.     

Generalized uniform covering maps relative to subgroups

In “Rips complexes and covers in the uniform category” (Brodskiy et al., preprint [4]) the authors define, following James (1990) [5], covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper extends these results by investigating the existence of these covering maps relative to subgroups of the uniform fundamental group and the fundamental group of the base space.     

Normal covers of infinite products and normality of Σ-products

We give some characterizations for normal covers of infinite products of generalized metric spaces such as M-spaces, Σ-spaces and β-spaces. We prove them simultaneously in terms of β-spaces and perfect maps. Next, we give affirmative answers to two questions concerning the normality of Σ-products, which were raised by the author and Yamazaki, respectively. These results are stated in terms of Σ-products of β-spaces.     

Fixed point theorems for partially outward mappings

We generalize some classical theorems related to dimension. We extend Brouwer's fixed point theorem to a class of mappings whose images are not necessarily a subset of the domain. These results also generalize theorems of B.R. Halpern and G.M. Bergman. As applications, we prove some theorems for maps that pull absolute retracts outward into attached sphere collars. We note relationships to the relative Nielsen theory and show that certain of our applications can also be obtained using results of H. Schirmer.     

Maps and f-normal spaces

It has long been known that hyper-real maps preserve realcompactness. In this paper we show that hyper-real maps preserve nearly realcompactness as well. We will also introduce the concepts of ε-perfect maps and f-normal spaces and explore them in a way that mirrors Rayburn's 1978 study of δ-perfect maps and h-normal spaces.     

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.     

Convergence and existence results for best proximity points

We provide a positive answer to a question raised by Eldred and Veeramani [A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic φ$\phi$-contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic φ$\phi$-contraction maps are also obtained.     

Inverse limits of arcs and of simple closed curves with confluent bonding mappings

It is proved that Knaster's type continua and solenoids can be considered as inverse limits of arcs and of circles with confluent bonding mappings. Several other classes of bonding mappings, which are relative to confluent ones, also are discussed.