• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.669-677
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• 2013

In this paper we introduce the notion of NI near-rings similar to the notion introduced in rings. We give topological properties of collection of strongly prime ideals in NI near-rings. We have shown that if N is a NI and weakly pm near-ring, then $Max(N)$ is a compact Hausdorff space. We have also shown that if N is a NI near-ring, then for every $a{\in}N$, $cl(D(a))=V(N^*(N)_a)=Supp(a)=SSpec(N){\setminus}int\;V(a)$.

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.803-829
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• 1998

We give general fixed point theorems for compact multimaps in the "better" admissible class $B^{K}$ defined on admissible convex subsets (in the sense of Klee) of a topological vector space not necessarily locally convex. Those theorems are used to obtain results for $\Phi$-condensing maps. Our new theorems subsume more than seventy known or possible particular forms, and generalize them in terms of the involving spaces and the multimaps as well. Further topics closely related to our new theorems are discussed and some related problems are given in the last section.n.

• Korean Journal of Mathematics
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• v.29 no.1
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• pp.25-40
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• 2021

In this article we introduce tribonacci sequence spaces c0(T) and c(T) derived by the domain of a newly defined regular tribonacci matrix T. We give some topological properties, inclusion relations, obtain the Schauder basis and determine ��-, ��- and ��- duals of the spaces c0(T) and c(T). We characterize certain matrix classes (c0(T), Y) and (c(T), Y), where Y is any of the spaces c0, c or ℓ∞. Finally, using Hausdorff measure of non-compactness we characterize certain class of compact operators on the space c0(T).

• Journal of the Korean Mathematical Society
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• v.59 no.1
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• pp.105-127
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• 2022

It is known that the maximal invariant set of a continuous iterative dynamical system in a compact Hausdorff space is equal to the intersection of its forward image sets, which we will call the first minimal image set. In this article, we investigate the corresponding relation for a class of discontinuous self maps that are on the verge of continuity, or topologically almost continuous endomorphisms. We prove that the iterative dynamics of a topologically almost continuous endomorphisms yields a chain of minimal image sets that attains a unique transfinite length, which we call the maximal invariance order, as it stabilizes itself at the maximal invariant set. We prove the converse, too. Given ordinal number ξ, there exists a topologically almost continuous endomorphism f on a compact Hausdorff space X with the maximal invariance order ξ. We also discuss some further results regarding the maximal invariance order as more layers of topological restrictions are added.

• The Mathematical Education
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• v.16 no.2
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• pp.22-26
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• 1978

It is shown that a map having an extension to an open map between the Alex-androff base compactifications of its domain and range has a unique such extension. J.S. Wasileski has introduced the Alexandroff base compactifications of Hausdorff spaces endowed with Alexandroff bases. We introduce a definition of morphism between such spaces to obtain a category which we denote by ABC. We prove that the Alexandroff base compactification on objects can be extended to a functor on ABC and that the compact objects give an epireflective subcategory of ABC. For each topological space X there exists a completely regular space $\alpha$X and a surjective continuous function $\alpha$$_{x}$ : Xlongrightarrow$\alpha$X such that for each completely regular space Z and g$\in$C (X, Z) there exists a unique g$\in$C($\alpha$X, 2) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\alpha$$_{x}$, $\alpha$X) is called a completely regularization of X. Let TOP be the category of topological spaces and continuous functions and let CREG be the category of completely regular spaces and continuous functions. The functor $\alpha$ : TOPlongrightarrowCREG is a completely regular reflection functor. For each topological space X there exists a compact Hausdorff space $\beta$X and a dense continuous function $\beta$x : Xlongrightarrow$\beta$X such that for each compact Hausdorff space K and g$\in$C (X, K) there exists a uniqueg$\in$C($\beta$X, K) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\beta$$_{x}$, $\beta$X) is called a Stone-Cech compactification of X. Let COMPT$_2$ be the category of compact Hausdorff spaces and continuous functions. The functor $\beta$ : TOPlongrightarrowCOMPT$_2$ is a compact reflection functor. For each topological space X there exists a realcompact space (equation omitted) and a dense continuous function (equation omitted) such that for each realcompact space Z and g$\in$C(X, 2) there exists a unique g$\in$C (equation omitted) with g=g$^{\circ}$(equation omitted). Such a pair (equation omitted) is called a Hewitt's realcompactification of X. Let RCOM be the category of realcompact spaces and continuous functions. The functor (equation omitted) : TOPlongrightarrowRCOM is a realcompact refection functor. In [2], D. Harris established the existence of a category of spaces and maps on which the Wallman compactification is an epirefiective functor. H. L. Bentley and S. A. Naimpally [1] generalized the result of Harris concerning the functorial properties of the Wallman compactification of a T$_1$-space. J. S. Wasileski [5] constructed a new compactification called Alexandroff base compactification. In order to fix our notations and for the sake of convenience. we begin with recalling reflection and Alexandroff base compactification.

• The Pure and Applied Mathematics
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• v.18 no.3
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• pp.261-268
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• 2011

For a compact Hausdorff space X with its p-fold covering map ${\sigma}$, we construct its corresponding topological groupoid ${\Gamma}$, and show that there is a strong relation between the dynamical structures of (X, ${\sigma}$) and the groupoid structures of ${\Gamma}$.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.91-103
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• 2007

In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder ${\varphi}$ is defined by $f(x{\ast}y)+f(x{\ast}y^{-1})-2g(x)-2g(y)={\varphi}(x,y)$, $f(x{\ast}y)+g(x{\ast}y^{-1})-2h(x)-2k(y)={\varphi}(x,y)$, where (G, *) is a group, X is a real or complex Hausdorff topological vector space and f, g, h, k are functions from G into X.

• The Pure and Applied Mathematics
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• v.2 no.2
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• pp.111-114
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• 1995

A C$\^$$\infty$/ manifold is a pair (M, C) where a) M is a Hausdorff topological space such that every point $\chi$$\in$M has a neighborhood homeomorphic to an open subset of R$^n$. b) C is a collection of these homeomorphisms whose domains cover M. If ø, $\psi$ $\in$ C then ø o $\psi$$\^$-1/ is C$\^$$\infty$/. c) C is maximal with respect to (b).(omitted)

• The Pure and Applied Mathematics
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• v.13 no.3 s.33
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• pp.227-236
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• 2006

The purpose of this paper is to establish a random fixed point theorem for nonconvex valued random multivalued operators, which generalize known results in the literature. We also derive a random coincidence fixed point theorem in the noncompart setting.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.4
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• pp.279-288
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• 2023

We obtain a Chandrabhan type fixed point theorem for a multimap having a non-compact domain and a weakly closed graph, and taking convex values only on a quasi-convex subset of Hausdorff locally convex topological vector space. We introduce the definition of Chandrabhan-set and find a sufficient condition for every countably condensing multimap to have a relatively compact Chandrabhan-set. Finally, we establish a new version of Sadovskii fixed point theorem for multimaps.