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New algebras of functions on topological groups arising from G-spaces

E. Glasner, M. Megrelishvili (2008)

Fundamenta Mathematicae

For a topological group G we introduce the algebra SUC(G) of strongly uniformly continuous functions. We show that SUC(G) contains the algebra WAP(G) of weakly almost periodic functions as well as the algebras LE(G) and Asp(G) of locally equicontinuous and Asplund functions respectively. For the Polish groups of order preserving homeomorphisms of the unit interval and of isometries of the Urysohn space of diameter 1, we show that SUC(G) is trivial. We introduce the notion of fixed point on a class...

New fixed point theorems for mappings satisfying a generalized weakly contractive condition with weaker control functions

Hemant Kumar Nashine (2012)

Annales Polonici Mathematici

The purpose of this paper is to derive new common fixed point theorems for a pair of mappings satisfying a more general weakly contractive condition with weaker control functions in a complete metric space. Applications to new fixed point results with conditions of integral type are also given. We furnish an example to demonstrate that these results improve the previously existing ones.

New versions on Nikaidô's coincidence theorem

Liang-Ju Chu, Ching-Yan Lin (2002)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In 1959, Nikaidô established a remarkable coincidence theorem in a compact Hausdorff topological space, to generalize and to give a unified treatment to the results of Gale regarding the existence of economic equilibrium and the theorems in game problems. The main purpose of the present paper is to deduce several generalized key results based on this very powerful result, together with some KKM property. Indeed, we shall simplify and reformulate a few coincidence theorems on acyclic multifunctions,...

Non-additivity of the fixed point property for tree-like continua

C. L. Hagopian, M. M. Marsh (2015)

Fundamenta Mathematicae

We investigate the fixed point property for tree-like continua that are unions of tree-like continua. We obtain a positive result if finitely many tree-like continua with the fixed point property have dendrites for pairwise intersections. Using Bellamy's seminal example, we define (i) a countable wedge X̂ of tree-like continua, each having the fpp, and X̂ admitting a fixed-point-free homeomorphism, and (ii) two tree-like continua H and K such that H, K, and H∩ K have the fixed point property, but...

Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities

Chris Good, Robin Knight, Brian Raines (2006)

Fundamenta Mathematicae

We examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional nonhyperbolic sets (inverse limits of finite graphs), particularly when those inhomogeneities form a countable set. Using tools from descriptive set theory we prove a surprising restriction on the topological structure of these invariant sets if the map satisfies a weak repelling or...

Noninvertible minimal maps

Sergiĭ Kolyada, L'ubomír Snoha, Sergeĭ Trofimchuk (2001)

Fundamenta Mathematicae

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 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 with A those topological properties which describe how large a set is. Using these results...

Nonlinear contractive conditions: A comparison and related problems

Jacek Jachymski, Izabela Jóźwik (2007)

Banach Center Publications

We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations,...

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