
Nazam, Muhammad; Arshad, Muhammad; Abbas, Mujahid
[EN] 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
[EN] 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
[EN] 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
[EN] 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
[EN] 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
[EN] 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
[EN] 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...

Boxer, Laurence; Ege, Ozgur; Karaca, Ismet; Lopez, Jonathan; Louwsma, Joel
[EN] A. Rosenfeld 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
[EN] 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.

Shah, Sejal; Das, Ruchi; Das, Tarun
[EN] We prove that if a uniformly continuous selfmap $f$ of a uniform space has topological specification property then the map $f$ has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having specification property. An example is also provided to justify that the converse is not true.

Grecova, Svetlana; Sostak, Alexander; Uljane, Ingrida
[EN] After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzytype topologies induced by fuzzy metrics. In particular, in the paper (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems, 6938 DOI...