Some new equivalents of the Brouwer fixed point theorem

PARK, Sehie

doi:10.31197/atnaa.1086232

Cited by 13 publications

(30 citation statements)

References 27 publications

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“…Proof Apply Theorem 4.4 with p = 1, this completes the proof. Theorem 4.4 improves or unifies corresponding results given by Askoura and Godet-Thobie [6], Cauty [21], Cauty [22], Chen [29], Isac [53], Li [70], Nhu [77], Okon [79], Park [92], Reich [100], Smart [115], Yuan [133], Theorem 3.3 of Ennassik and Taoudi [33], Theorem 3.14 of Gholizadeh et al [41], Xiao and Lu [123], Xiao and Zhu [124,125] under the framework of topological vector spaces.…”

Section: Theorem 43supporting

confidence: 77%

“…Here we first give some notion and a brief introduction to the abstract convex spaces, which play an important role in the development of the KKM principle and related applications. Once again, for the corresponding comprehensive discussion on the KKM theory and its various applications to nonlinear analysis and related topics, we refer to Agarwal et al [1], Alghamdi et al [5], Balaj [8], Mauldin [75], Granas and Dugundji [48], Park [91] and [92], Yuan [133], and related comprehensive references therein.…”

Section: The Kkm Principle In Abstract Convex Spacesmentioning

confidence: 99%

“…Secondly, the the single-valued version of Theorem 4.3 which was first given by Ennassik and Taoudi (Theorem 3.3 of [34]) for 0 < p ≤ 1 indeed provides an answer to Schauder's conjecture under the TVS. Here we also mention a number of related works and discussion by authors in this drection, see Mauldin [75], Granas and Dugundji [48], Park [91,92] and the references therein.…”

Section: Theorem 43mentioning

confidence: 99%

“…A generalization of Schauder's theorem from a normed space to general topological vector spaces is an old conjecture in fixed point theory, which is explained by Problem 54 of the book "The Scottish Book" by Mauldin [75] and stated as Schauder's conjecture: "Every nonempty compact convex set in a topological vector space has the fixed point property, or in its analytic statement, does a continuous function defined on a compact convex subset of a topological vector space to itself have a fixed point?" Recently, this question has been answered by the work of Ennassik and Taoudi [34] by using the p-seminorm methods under locally p-convex spaces; see also related contribution given by Cauty [22], plus the works by Askoura and Godet-Thobie [6], Cauty [21], Chang [23], Chang et al [24], Chen [29], Dobrowolski [32], Gholizadeh et al [41], Isac [53], Li [70], Li et al [69], Liu [72], Nhu [77], Okon [79], Park [90][91][92], Reich [100], Smart [115], Weber [121,122], Xiao and Lu [123], Xiao and Zhu [124,125], Xu [129], Xu et al [130], Yuan [132,133] in both TVS, LCS and related references therein under the general framework of p-vector spaces for nonself set-valued or single-valued mappings (0 < p ≤ 1).…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear analysis by applying best approximation method in p-vector spaces

Yuan

2022

Fixed Point Theory Algorithms Sci Eng

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It is known that the class of p-vector spaces $(0 < p \leq 1)$ ( 0 < p ≤ 1 ) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but 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 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p-convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for $0 < p \leq 1$ 0 < p ≤ 1 . Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 . In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p-convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p-convex spaces for $0 < p \leq 1$ 0 < p ≤ 1 seems even more important, and can be done by the approach established in this paper.

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Section: Theorem 43supporting

confidence: 77%

Section: The Kkm Principle In Abstract Convex Spacesmentioning

confidence: 99%

Section: Theorem 43mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear analysis by applying best approximation method in p-vector spaces

Yuan

2022

Fixed Point Theory Algorithms Sci Eng

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“…provides an answer to Schauder's conjecture under the TVS. Here we also mention a number of related works and discussion by authors in this drection, see Mauldin [74], Granas and Dugundji [46], Park [90,91] and the references therein.…”

Section: Fixed Point Theorems For Condensing Mappings In P-vector Spacesmentioning

confidence: 99%

Nonlinear Analysis in p-Vector Spaces for Singe-Valued 1-Set Contractive Mappings

Yuan

2022

Preprint

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The goal of this paper is to develop some fundamental and important nonlinear analysis for single-valued mappings under the framework of p-vector spaces, in particular, for locally p-convex spaces for 0 < p ≤ 1. More precisely, based on the fixed point theorem of single-valued continuous condensing mapping in locally p-convex spaces as the starting point, we first establish best approximation results for (single-valued) continuous condensing mappings which are then used to develop new results for three classes of nonlinear mappings consisting of 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive mappings in locally p-convex spaces. Next they are used to establish general principle for nonlinear alternative, Leray - Schauder alternative, fixed points for non-self mappings with different boundary conditions for nonlinear mappings from locally p-convex spaces, to nonexpansive mappings in uniformly convex Banach spaces, or locally convex spaces with Opial condition. The results given by this paper not only include the corresponding ones in the existing literature as special cases, but also expected to be useful tools for the development of new theory in nonlinear functional analysis and applications to the study of related nonlinear problems arising from practice under the general framework of p-vector spaces for 0 < p ≤ 1. In addition, the work presented by this paper focuses on the development of nonlinear analysis for single-valued (instead of set-valued) mappings for locally p-convex spaces, essentially, is indeed the continuation of the associated work given recently by Yuan [134] therein, the attention is given to the study of nonlinear analysis for set-valued mappings in locally p-convex spaces for 0 < p ≤ 1.

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