Chapter 39 - On some special classes of continuous maps

1. Let X be a Banach space. Then a self-mapping A of X is said to be nonexpansive provided that ‖AX − Ay‖≤‖X − y‖ holds for all x, y ∈ X. The class of nonexpansive mappings includes contraction mappings and is properly contained in the class of all continuous mappings. Keeping in view the fixed point theorems known for contraction mappings (e.g. Banach Contraction Principle) and also for continuous mappings (e.g. those of Brouwer, Schauderand Tychonoff), it seems desirable to obtain fixed point theorems for nonexpansive mappings defined on subsets with conditions weaker than compactness and convexity. Hypotheses of compactness was relaxed byBrowder [2] and Kirk [9] whereas Dotson [3] was able to relax both convexity and compactness by using the notion of so-called star-shaped subsets of a Banach space. On the other hand, Goebel and Zlotkiewicz [5] observed that the same result of Browder [2] canbe extended to mappings with nonexpansive iterates. In [6], Goebel-Kirk-Shimi obtainedfixed point theorems for a new class of mappings which is much wider than those of nonexpansive mappings, and mappings studied by Kannan [8]. More recently, Shimi [12] used the fixed point theorem of Goebel-Kirk-Shimi [6] to discuss results for approximating fixed points in Banach spaces.

Introduction. The characterization of the Lebesgue area as a functional 4) defined on a certain class of continuous mappings has attracted a great deal of attention. The conditions that are imposed upon a functional -I in this paper (see Section 3) are essentially those listed in L. Cesari [2; 7. 5]. Let Z be the class of all continuous mappings (T,A) from an admissible subset A of E2 into E, (see SectioIn 1), anld let L(T,A) be the Lebesgue area of (T; A). A subclass Z' of Z is considered which consists of all miappings (T, A) in Z admitting aii elementary shirinking approach (see Sectioni 4), and it is shown that Z' is the class of mappings on which each functional @ satisfying the conditions of Section 3 agrees with the Lebesgue area.

It is shown that there is no good answer to the question of the title, even if we restrict our attention to Set-based topological categories. Although very closely related, neither the natural notion of c-finality (designed in total analogy to c-initiality) nor the notion of c-quotient (modelled after the behaviour of topological quotient maps) provide universally satisfactory concepts. More dramatically, in the category Top with its natural Kuratowski closure operator k, the class of k-final maps cannot be described as the class of c-quotient maps for any closure operator c, and the class of k-quotients cannot be described as the class of c-final maps for any c. We also discuss the behaviour of c-final maps under crossing with an identity map, as in Whitehead's Theorem. In Top, this gives a new stability theorem for hereditary quotient maps.

Positive tree-like mapping classes

On limit stability of special classes of continuous maps

We consider families of dynamics that can be described in terms of Perron–Frobenius operators with exponential mixing properties. For piecewise C2 expanding interval maps we rigorously prove continuity properties of the drift J(λ) and of the diffusion coefficient D(λ) under parameter variation. Our main result is that D(λ) has a modulus of continuity of order , i.e. D(λ) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are quantified numerically for the latter class of maps by using exact series expansions for the transport coefficients that can be evaluated numerically. We numerically observe strong local variations of all continuity properties.

The purpose of this paper is to investigate some strong convergence as well as stability results of some iterative procedures for a special class of mappings. First, this class of mappings called weak Jungck -contractive mappings, which is a generalization of some known classes of Jungck-type contractive mappings, is introduced. Then, using an iterative procedure, we prove the existence of coincidence points for such mappings. Finally, we investigate the strong convergence of some iterative Jungck-type procedures and study stability and almost stability of these procedures. Our results improve and extend many known results in other spaces. MSC:47H06, 47H10, 54H25, 65D15.

Weak compactness and fixed point property for affine mappings

On isometrically universal spaces, mappings, and actions of groups

The continuous maps p : E → B for which p* : Top/B → Top/E reflects isomorphisms are shown to coincide with the universal quotient maps as characterized by Day and Kelly. Monadicity of p* turns out to be a local property. This is used to prove the main result of the paper, namely that p* is monadic for every locally sectionable map p : E → B. There are therefore important classes of maps p for which spaces over B are equivalently described as spaces over E which come equipped with a simple algebraic structure: local homeomorphisms, locally trivial quotient maps, surjective covering maps, etc. Finally, the monadic decomposition of p* is examined for arbitrary maps p.AMS Subject Classification18A3018C2018F1554C1055R10

In this article, we present the new class of neutrosophic continuous mappings called; -continuous map (WN -CM) strongly neutrosophic -continuous map (SN -CM), neutrosophic -weakly continuous map (N W-CM) and neutrosophic -weakly continuous map (N S-CM) includes an examination of the most notable traits and attributes associated with it. Moreover, we introduced the notions of -irresolute map (WN -CIM) strongly neutrosophic - irresolute map (SN -CIM), neutrosophic -weakly irresolute map (N W-CIM) and neutrosophic -weakly irresolute map (N S-CIM) in neutrosophic topological spaces (NTS) were investigated, coupled with a few characteristics that are connected to them. Finally, we studied the relationship among continuous and irresolute mappings for these kinds that have been presented

In 2011 A.I. EL Maghrabi and A.M. Mubaraki introduced and studied the notions of Y – open and Y – closed sets in general topology as well as presented some characterizations of these notions. We introduce and investigate several properties and characterizations of a new class of maps between topological spaces called Y – open maps, Y – closed maps, Y – continuous maps and Y – irresolute maps. We also introduce slightly Y – continuous, totally Y – continuous and almost Y – continuous maps between topological spaces and establish several characterizations of these new forms of maps. Furthermore, we introduce and study the notions of Y – separated sets and Y – connectedness in topological spaces.

In this paper, we introduce a new class of sets called supra generalized locally closed sets and new class of maps called supra generalized locally continuous functions. Furthermore, we obtain some of their properties. KEYWORDS S-GLC sets, S-GLC* sets, S-GLC** sets, S-GL- continuous, S-GL*- continuous, S-GL**- continuous, S-GL- Let A be a subset (X, µ). irresolute, S-GL*- irresolute and S-GL**- irresolute 1. INTRODUCTION In 1921, Kuratowski and Sierpinski [8] considered the difference of two closed subsets of an n-dimensional Euclidean space. Implicit in their work is the notion of a locally closed subset of a topological space. Bourbaki [2] defined a subset of space (X, τ) is called locally closed, if it is the intersection of an open set and a closed set. Stone [11] has used the term FG for a locally closed subset. Ganster and Reilly [4] and [5], Balachandran et al. [1] and J. H. Park et al. [6] introduced the concept of LC-continuous functions, GLC continuous functions and SGLC*-continuous functions respectively. Mashhour et al. [9] introduced the supra topological spaces and studied S-continuous functions and S*-continuous functions. In 2008, Devi et al. [3] introduced and studied a class of sets and maps between topological spaces called supra –open sets and supra –continuous maps, respectively. In 2010, O.R. Sayed et al.[12], introduced and studied a class of sets and a class of maps between topological spaces called supra b–open sets and supra b–continuous maps, respectively. Supra g-closed sets, supra g-continous function and supra g-closed maps are introduced and investigated by Ravi et al. [10]. In this paper we introduce the concept of supra generalized locally closed sets and study its basic properties. Also we introduce the concepts of supra generalized locally continuous functions and investigate some of the basic properties for this class of functions. implies s

Cain and Nashed generalized to locally convex spaces a well known fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The class of asymptotically nonexpansive mappings includes properly the class of nonexpansive mappings as well as the class of contraction mappings. In this paper, we prove by using the same method some results concerning the existence of fixed points for a sum of nonexpansive and continuous mappings and also a sum of asymptotically nonexpansive and continuous mappings in locally convex spaces. These results extend a result of Cain and Nashed.

The main focus of this study is to introduce a new category of generalized closed sets, referred to as \(\mathcal{I}_\mathcal{p}\)- closed sets, within the framework of ideal topological spaces. By using a few instances, we demonstrate \(\mathcal{I}_\mathcal{p}\)- closed sets and establish some fundamental properties of \(\mathcal{I}_\mathcal{p}\)-closed sets. We also investigate the relationship between \(\mathcal{I}_\mathcal{p}\)-closed sets and other classes of generalized closed sets in ideal topological spaces, such as (\mathcal{I}_\mathcal{g}\)- closed sets, \(\alpha\)\(\mathcal{I}_\mathcal{g}\)-closed sets, and (\mathcal{I}_\mathcal{r}\)\mathcal{g}\)-closed sets. Then, we focus on the topological implications of \(\mathcal{I}_\mathcal{p}\)-closed sets and investigate how they relate to the concepts of \(\mathcal{I}_\mathcal{p}\)-continuous map, (\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. First and foremost, we define the \(\mathcal{I}_\mathcal{p}\)-continuous map, investigate the behavior of \(\mathcal{I}_\mathcal{p}\)- continuous map with respect to \(\mathcal{I}_\mathcal{p}\)-closed sets, and derive several important properties of \(\mathcal{I}_\mathcal{p}\)-continuous map. Further, we studied their relationships with other classes of continuous maps in ideal topological spaces. Nevertheless, we defined the definitions of \(\mathcal{I}_\mathcal{p}\)-irresolute maps and strongly \(\mathcal{I}_\mathcal{p}\)-continuous maps in ideal topological spaces. We explored the connections with the notions of \(\mathcal{I}_\mathcal{p}\)-continuous map, (\mathcal{I}_\mathcal{p}\)-irresolute map, and a strongly \(\mathcal{I}_\mathcal{p}\)-continuous map. Our results provide new insights into the study of ideal topological spaces.