Applications of some basic theorems in the KKM theory

Park, Sehie

doi:10.1186/1687-1812-2011-98

Cited by 5 publications

(8 citation statements)

References 41 publications

(47 reference statements)

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“…In our KKM theory on abstract convex spaces given in [40], [49], there exist some basic theorems from which we can deduce several equivalent formulations and useful applications. In this section, we introduce some of such basic theorems in [40].…”

Section: Basic Theorems In the Kkm Theorymentioning

confidence: 99%

“…In our previous work [43], we reviewed applications of our fixed point theorems for the multimap class of compact compositions of acyclic maps and, in [48], we collected most of fixed point theorems related to the KKM theory due to the author. Moreover, applications of our versions of the Fan-Browder fixed point theorem were introduced in [49]. Furthermore, in a later work [53], we reviewed applications of our fixed point theorems and our multimap classes, appeared mainly in other authors' works.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, in a later work [53], we reviewed applications of our fixed point theorems and our multimap classes, appeared mainly in other authors' works. Most of them are not treated in [43], [48], [49].…”

Section: Introductionmentioning

confidence: 99%

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On various multimap classes in the KKM theory and their applications

Park¹

2018

Advances in the Theory of Nonlinear Analysis and Its Application

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Fixed point theory of convex-valued multimaps are closely related to the KKM theory from the beginning. In the last twenty-five years, we introduced the acyclic multimap class, the admissible multimap class A κ c , the better admissible class B, and the KKM admissible classes KC, KO in the frame of the KKM theory. Our aim in this review is to collect the basic properties of our multimap classes and some mutual relations among them in general topological spaces or our abstract convex spaces. We add some new remarks and further comments to improve many of those results, and introduce some recent applications of our multimap classes.

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Section: Basic Theorems In the Kkm Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On various multimap classes in the KKM theory and their applications

Park¹

2018

Advances in the Theory of Nonlinear Analysis and Its Application

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“…Assume that, Many authors applied Proposition 16 or particular forms of Proposition 16 to study various nonlinear problems in topological vector spaces, for example, generalized Minty vector variational inequality problems, generalized variational inequality problems, generalized vector equilibrium problems, and the constrained or the competitive Nash type equilibrium problems. For more details, see Park [29] and the references therein.…”

Section: Particular Fixed Point Theoremsmentioning

confidence: 99%

“…As it is well known, -convex spaces are typical example of abstract convex spaces. The following extension of the Fan-Browder fixed point theorem to -convex spaces is a particular form of Theorem 3.3 in Park [29], and it includes the fixed point theorems mentioned previously as special cases. (c) there exists a nonempty compact subset of such that, for each ∈ ⟨ ⟩, there exists a compactconvex subset of ( ; Γ) containing such that…”

Section: By Using Proposition 21 Cubiotti and Nordomentioning

confidence: 99%

A Collectively Fixed Point Theorem in Abstract Convex Spaces and Its Applications

2013

Journal of Function Spaces and Applications

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The Nonlinear Analysis in p-Vector Spaces by Applying Best Approximation Method

YUAN

2022

Preprint

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It is known that the class of p-vector spaces (0 < p ≤ 1) is an important generalization of usual norm spaces with rich topological and geometrical structure, but the most tools and general principles with nature in nonlinearity have not been developed yet, the goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of semiclosed 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop the general fixed point theorems of condensing mappings which provide answer to Schauder conjecture in 1930’s in the affirmative way under the setting of p-vector spaces by taking p = 1 for a p-vector space being a topological vector space. Then one best approximation result for upper semi-continuous and 1-set contractive set-valued is established, which is used as a useful tool to establish fixed points of non-self set-valued mappings with either inward or outward set conditions. Finally, we develop fixed points and related principle of nonlinear alterative for the classes of semiclosed set-valued mappings including nonexpansive set-valued mappings as special cases under uniformly convex Banach spaces, or locally convex topological spaces with Opial condition. The results in this paper under the category of nonlinear analysis not only include the corresponding results in the existing literature as special cases, but also expected to be useful tools for the study of nonlinear problems arising from theory to practice under the general framework of p-vector spaces for 0 < p ≤ 1.

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Applications of some basic theorems in the KKM theory

Cited by 5 publications

References 41 publications

On various multimap classes in the KKM theory and their applications

On various multimap classes in the KKM theory and their applications

A Collectively Fixed Point Theorem in Abstract Convex Spaces and Its Applications

The Nonlinear Analysis in p-Vector Spaces by Applying Best Approximation Method

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