Stability of isometries in $p$-Banach spaces

Tabor, Jacek; Żołdak, Marek

doi:10.7169/facm/1229624655

Cited by 8 publications

(5 citation statements)

References 8 publications

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“…Jarchow [54], Kalton [55,56], Kalton et al [57], Machrafi and Oubbi [73], Park [90], Qiu and Rolewicz [99], Rolewicz [103], Silva et al [112], Simons [113], Tabor et al [116], Tan [117], Wang [120], Xiao and Lu [123], Xiao and Zhu [124,125], Yuan [133], and many others). However, to the best of our knowledge, the corresponding basic tools and associated results in the category of nonlinear functional analysis have not been well developed.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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-normed spaces (0 < p ≤ 1) is an important generalization of usual normed spaces, and it has a rich topological and geometrical structure, and related study has received a lot of attention (e.g., see Balachandran [5], Bayoumi [6], Bayoumi et al [7], Bernuées and Pena [9], Ding [23], Gal and Goldstein [32], Gholizadeh et al [33], Jarchow [37], Kalton [39]- [40], Kalton et al [41], Part [61], Qiu and Rolewicz [66], Rolewicz [69], Simons [75], Tabor et al [79], Tan [80], Wang [82], Xiao and Lu [85], Xiao and Zhu [86]- [87], Yuan [91], and many others).…”

Section: Introductionmentioning

confidence: 99%

The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces

Yuan¹

2022

Preprint

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It is known that the class of p-normed spaces (0 < p ≤ 1) is an important generalization of usual normed spaces with rich topological and geometrical structure, but the most of tools and general principles in nature with nonlinearity have not been developed yet, thus the main goal of this paper is to develop tools for nonlinear analysis under the framework of p-vector spaces. In particular, we first develop the general fixed point theorems which provide solutions to answer Schauder conjecture since 1930’s in the affirmative for p-vector spaces when p = 1 (which is just general topological vector spaces); then the one best approximation result for upper semi-continuous mappings is given, which is used as a powerful tool to establish fixed points for non-self set-valued mappings with either inward or outward set conditions; and finally we establish comprehensive existence results of solutions for Birkhoff-Kellogg Problems, and the general principle of nonlinear alternative by including Leray-Schauder alternative and related results. The results given in this paper 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.

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“…Ennassik and Taoudi [31], Ennassik et al [32], Gal and Goldstein [38], Gholizadeh et al [39], Jarchow [51], Kalton [53]- [54], Kalton et al [55], Machrafi and Oubbi [72], Park [89], Qiu and Rolewicz [98], Rolewicz [102], Silva et al. [113], Simons [110], Tabor et al [115], Tan [116], Wang [119], Xiao and Lu [122], Xiao and Zhu [124]- [123], Yuan [134], and many others. However, to the best of our knowledge, the corresponding basic tools and associated results in the category of nonlinear functional analysis for p-vector spaces have not been well developed, in particular for the three classes of (single-valued) continuous nonlinear mappings which are: 1) condensing; 2) 1-set contractive; and 3) semiclosed 1-set contractive operators under locally p-convex spaces.…”

Section: Introductionmentioning

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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Stability of isometries in $p$-Banach spaces

Cited by 8 publications

References 8 publications

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

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

The Nonlinear Analysis Related to Schauder Fixed Points, Best Approximation, Birkhoff-Kellogg Problems and Principle of Leray-Schauder Alternatives in p-Vector Spaces

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

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