Collectively fixed point theorems in noncompact abstract convex spaces with applications

Haishu Lu; Kai Zhang; Rong Li; Haishu Lu; Kai Zhang; Rong Li

doi:10.3934/math.2021718

AIMS Mathematics

2021, Volume 6, Issue 11: 12422-12459. doi: 10.3934/math.2021718

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Research article

Collectively fixed point theorems in noncompact abstract convex spaces with applications

School of Business, Jiangsu University of Technology, Changzhou 213001, Jiangsu, China

Received: 29 March 2021 Accepted: 24 August 2021 Published: 30 August 2021
MSC : 47H04, 47H10, 91A10

In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.
- abstract convex space,
- collectively fixed point,
- Nash equilibrium,
- nonempty intersection theorem,
- generalized weak implicit inclusion problem
Citation: Haishu Lu, Kai Zhang, Rong Li. Collectively fixed point theorems in noncompact abstract convex spaces with applications[J]. AIMS Mathematics, 2021, 6(11): 12422-12459. doi: 10.3934/math.2021718

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Abstract

In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.

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