
Popa, Valeriu; Patriciu, AlinaMihaela
In this paper, two general fixed point theorem for a sequence of mappings satisfying implicit relations in Gp  complete metric spaces are proved.

Protasov, Igor V.; Slobodianiuk, Sergii
We consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$thin, sparse and scattered subsets.For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$thin if $g_0A\cap\dots\cap g_nA$ is finite for all distinct $g_0,\dots,g_n\in G$.Each $n$thin subset of a group of cardinality $\aleph_0$ can be partitioned into $n$ $1$thin subsets but there is a $2$thin subset in some Abelian group of cardinality $\aleph_2$ which cannot be partitioned into two $1$thin subsets. We eliminate the gap between $\aleph_0$ and $\aleph_2$...

Comfort, Wistar; Gould, Franklin R.
This paper derives from and extends selected portions of theDoctoral Dissertation [19],written at Wesleyan University (Middletown, Connecticut,USA) by the secondlisted coauthor under the guidance of the firstlisted coauthor.

Elghaoui, Mohamed; Ayadi, Adlene
In this paper, we give an explicit criterion to decide thedensity of finitely generated additive subgroups of R^n and C^n.

Haouati, Afef; Lazaar, Sami
In [3], the author has introduced the notion of primal spaces.The present paper is devoted to shedding some light on relations between quasihomeomorphisms and primal spaces.Given a quasihomeomorphism q from X to Y , where X and Y are principal spaces, we are concerned specically with a main problem: what additional conditions have to be imposed on q in order to render X (resp.Y ) primal when Y (resp.X) is primal.

Karamzadeh, O. A. S.; Namdari, M.; Soltanpour, S.
Let $C_c(X)=\{f\in C(X) : f(X)\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): f(X)<\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $f(U)\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered.We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that...

Gabeleh, Moosa
In this article, we prove a fixed point theorem for cyclic relatively nonexpansive mappings in the setting of generalized semimetric spaces by using a geometric notion of seminormal structure and then we conclude some results in uniformly convex Banach spaces. We also discuss on the stability of seminormal structure in generalized semimetric spaces.

Ozcag, Selma
This is a continuation of the work where the notions of Lebesgue uniformity and Lebesgue quasi uniformity in a texture space were introduced. It is well known that the quasi uniform space with a compact topology has the Lebesgue property. This result is extended to direlational quasi uniformities and dual dicovering quasi uniformities. Additionally we discuss the completeness of lebesgue diuniformities and dual dicovering lebesgue diuniformities.

Elfard, Ali Sayed
Let FP(X) be the free paratopological group on a topological space X in the sense of Markov. In this paper, we study the group FP(X) on a $P_\alpha$space $X$ where $\alpha$ is an infinite cardinal and then we prove that the group FP(X) is an Alexandroff space if X is an Alexandroff space. Moreover, we introduce a~neighborhood base at the identity of the group FP(X) when the space X is Alexandroff and then we give some properties of this neighborhood base. As applications of these, we prove that the group FP(X) is T_0 if X is T_0, we characterize the...

Ivansic, Ivan; Rubin, Leonard R.
Let Z, H be spaces. In previous work, we introduced the direct (inclusion) system induced by the set of maps between the spaces Z and H. Its direct limit is a subset of Z × H, but its topology is different from the relative topology. We found that many of the spaces constructed from this method are pseudocompact and Tychonoff. We are going to show herein that these spaces are typically not sequentially compact and we will explore conditions under which a finite product of them will be pseudocompact.

Felicit, J. Maria; Eldred, A. Anthony
We consider pcyclic mappings and prove an analogous result to Edelstien contractive theorem for best proximity points. Also we give similar results satisfying BoydWong and Geraghty contractive conditions.

Nazam, Muhammad; Arshad, Muhammad; Abbas, Mujahid
In this paper, we introduce a new contraction called dualistic contraction of rational type and obtain some fixed point results. These results generalize various comparable results appeared in the literature. We provide an example to show the superiority of our results over corresponding fixed point results proved in metric spaces.

Almontashery, Khulod; Kalantan, Lutfi
In this paper, we present some new results about the Alexandroff Duplicate Space. We prove that if a space X has the property P, then its Alexandroff Duplicate space A(X) may not have P, where P is one of the following properties: extremally disconnected, weakly extremally disconnected, quasinormal, pseudo compact. We prove that if X is $\alpha$normal, epinormal, or has property $\omega D$, then so is A(X). We prove almost normality is preserved by A(X) under special conditions.

Buzyakova, Raushan
We show that for any continuous monotonic bijection $f$ on a $\sigma$compact subgroup $G\subset \mathbb R$ there exists a binary operation $+_f$ such that $\langle G, +_f\rangle$ is a topological group topologically isomorphic to $\langle G, +\rangle$ and $f$ is a shift with respect to $+_f$. We then show that monotonicity cannot be replaced by a periodicpoint free continuous bijections. We explore a few routes leading to generalizations and counterexamples

Sharma, Puneet
In this paper, we study the dynamics induced by finite commutative relation on the hyperspaces. We prove that the dynamics induced on the hyperspace by a nontrivial commutative family of continuous self maps cannot be transitive and hence cannot exhibit higher degrees of mixing. We also prove that the dynamics induced on the hyperspace by such a collection cannot have dense set of periodic points. We also give example to show that the induced dynamics in this case may or may not be sensitive.

Kannan, V; Gopal, Sharan
The sets of periodic points of self homeomorphisms on an ordinal of finite derived length are characterised, thus characterising the same for homeomorphisms on compact metric spaces with finite derived length. A partition of ordinal is introduced to study this problem which is also used to solve two more problems: one about an equivalence relation and the other about a group action, both on an ordinal of finite derived length.

Komal, Somayya; Kumam, Poom
The purpose of this article is to establish the global optimization with partial orders for the pair of nonself mappings, by introducing new type of contractions like $\alpha$ordered contractions and $\alpha$ordered proximal contraction in the frame work of complete metric spaces. Also calculates some fixed point theorems with the help of these generalized contractions. In addition, established an example to show the validity of our main result. These results extended and unify many existing results in the literature.

Boxer, Laurence; Staecker, P. Christopher
The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but dierent notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.E....

Boxer, Laurence; Ege, Ozgur; Karaca, Ismet; Lopez, Jonathan; Louwsma, Joel
A. Rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).

Komal, Somayya; Kumam, Poom; Gopal, Dhananjay
In this article, we introduced the best proximity point theorems for $\mathcal{Z}$contraction and Suzuki type $\mathcal{Z}$contraction in the setting of complete metric spaces. Also by the help of weak $P$property and $P$property, we proved existence and uniqueness of best proximity point. There is a simple example to show the validity of our results. Our results extended and unify many existing results in the literature. Moreover, an application to fractional order functional differential equation is discussed.