• Honam Mathematical Journal
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• v.35 no.4
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• pp.757-764
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• 2013

In this paper, we give sufficient conditions for a map with nonconvex values to have a continuous selection and the selection extension property in LC-metric spaces under the one-point extension property. And we apply it to weakly lower semicontinuous maps and generalize previous results. We also get a continuous selection theorem for almost lower semicontinuous maps with closed sub-admissible values in $\mathbb{R}$-trees.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1363-1371
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• 2016

This paper is concerned with best proximity points for multimaps in normed spaces and in hyperconvex metric spaces. Using the generalized KKM theorem, we deduce new best proximity pair theorems for a family of multimaps with unionly open fibers in normed spaces. And we prove a new best proximity point theorem for quasi-lower semicontinuous multimaps in hyperconvex metric spaces.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.681-691
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• 2002

We study categorical properties of the category SWQOS (S$\^$U/WQOS, S$\^$L/WQOS, resp) of (upper, lower, resp.) semi-continuous quasi-ordered spaces and the subcategory SWPOS (S$\^$U/WPOS, S$\^$L/WPOS, resp.) of (upper, lower, resp.) semicontinuous partially ordered spaces. We show that the categories S$\^$U/WPOS, S$\^$L/WPOS and SWPOS are closed under the formation of initial mono-sources in the category TQOS of topological quasi-ordered spaces, and they we mono-topological, complete and cocomplete epireflective subcategories of the category TQOS. We obtain their MacNeille and universal initial completions, as well as those of subcatego.ies S$\^$U/WQOS(S$\^$L/WQOS) and SWQOS, in which the topologies are T$\_$0/ and T$_1$, respectively.

• Nonlinear Functional Analysis and Applications
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• v.26 no.4
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• pp.749-780
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• 2021

In this paper, we give the notion of M(., .)-𝜂-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. As an application, we introduce and investigate a new system of variational-like inclusions in Banach spaces. By means of M(., .)-𝜂-proximal mapping method, we give the existence of solution for the system of variational inclusions. Further, propose an iterative algorithm for finding the approximate solution of this class of variational inclusions. Furthermore, we discuss the convergence and stability analysis of the iterative algorithm. The results presented in this paper may be further expolited to solve some more important classes of problems in this direction.