Some problems in metric fixed point theory

Goebel, Kazimierz; Kirk, W. A.

doi:10.1007/s11784-008-0076-3

Cited by 16 publications

(15 citation statements)

References 35 publications

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“…The notion of the so-called Opial condition was first given by Opial [80], and it says that a Banach space X is said to satisfy Opial's condition if lim inf n→∞ w nw < lim inf n→∞ w np whenever (w n ) is a sequence in X weakly convergent to w and p = w. We know that Opial's condition plays an important role in the fixed point theory, e.g., see Lami Dozo [66], Goebel and Kirk [44], Xu [127], and the references therein. The following result shows that nonexpansive set-valued mappings in Banach spaces with Opial's condition (see Lami Dozo [66]) satisfy condition (H1).…”

Section: Theorem 71 Let U Be a Bounded Open P-convex Subset Of A P-(s...mentioning

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: Theorem 71 Let U Be a Bounded Open P-convex Subset Of A P-(s...mentioning

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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“…We hope however that this survey article highlights the development of the fixed point theory for 1-set contractions and also reports in a satisfactory way some recent advances in this theory. Note finally that some open problems, in connection with metric fixed point theory can be found in [52] while a recent survey is given in [6,Chapter 1].…”

Section: Resultsmentioning

confidence: 99%

Fixed Point Theory for 1-Set Contractions: a Survey

Djebali¹

2015

Applied Mathematics in Tunisia

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In this report, we first review some classical results concerning the fixed point theory for an important class of mappings for which the Banach contraction principle fails, namely nonexpansive mappings. Both metric and topological fixed point theory will be surveyed. We will also discuss some known results regarding the extension to nonlinear contractions and to˛-contractive mappings with respect to some measure of noncompactness˛. The second part of this survey paper will be devoted to some recent progress and development of the fixed point theory of 1-set contractions that have been achieved during the last couple of years. The theory for different boundary conditions and when the corresponding space is endowed with the weak topology are also discussed. Finally, some applications to equations of Krasnosels'kȋi type and to the solvability of nonlinear integral equations of Volterra type are presented.

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“…We first note that the "(H1) Condition" above is actually the "Condition (C)" used by Theorem 1 of Petryshyn 1450 [93]. Indeed, by following Goebel and Kirk [42] (see also Xu [127] and reference therein), Browder [15] (see also [16], p. 103) proved that if K is a closed and convex subset of a uniformly convex Banach space X, and if T : K → X is nonexpansive, then the mapping f := I − T is demiclosed on X. This result, known as Browder's demiclosedness principle (Browder's proof, which was inspired by the technique of Göhde in [43]), is one of the fundamental results in the theory of nonexpansive mappings, which satisfies the "(H1) condition".…”

Section: Nonlinear Alternatives Principle For the Class Of Semiclosed...mentioning

confidence: 94%

“…Now let K(X) denote the family of all non-empty compact convex subsets of topological vector space X. The notion of the so-called " Opial's condition" first given by Opial [79], which says that a Banach space X is said to satisfy Opial's condition if lim inf n→∞ w n − w < lim inf n→∞ w n − p whenever (w n ) is a sequence in X weakly convergent to w and p = w, we know that Opial's condition plays an important role in the fixed point theory, e.g., see Lami Dozo [64], Goebel and Kirk [42], Xu [127] and references where. The following result shows that there nonexpansive set-valued mappings in Banach spaces with Opial's condition (see Lami Dozo [64] satisfying the condition (H1).…”

Section: Fixed Points For the Class Of 1-set Contractive Mappingsmentioning

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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Some problems in metric fixed point theory

Cited by 16 publications

References 35 publications

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

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

Fixed Point Theory for 1-Set Contractions: a Survey

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

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