A common fixed point theorem and its applications in abstract convex spaces

Shunyou Xia; Chongyi Zhong; Chunrong Mo; Shunyou Xia; Chongyi Zhong; Chunrong Mo

doi:10.3934/math.2025240

AIMS Mathematics

2025, Volume 10, Issue 3: 5236-5245. doi: 10.3934/math.2025240

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Research article

A common fixed point theorem and its applications in abstract convex spaces

1.
Guizhou Key Laboratory of Artificial Intelligence and Brain-inspired Computing, Guiyang 550018, China
2.
School of Mathematics and Big Data, Guizhou Education University, Guiyang 550018, China
3.
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China

Received: 22 December 2024 Revised: 26 February 2025 Accepted: 04 March 2025 Published: 10 March 2025
MSC : 47H10

First, this paper introduces a family of generalized equi-KKM mappings and the concept of quasi-abstract convexity (concavity) defined by parameter multi-valued mappings in abstract convex spaces that satisfy the $ H_0 $-condition. Then, using the Brouwer fixed-point theorem, we prove a common fixed-point theorem for this family of generalized equi-KKM mappings. Finally, we introduce two classes of generalized abstract equilibrium problems and obtain two existence theorems for their solutions, as an application of the common fixed-point theorem.
- $ H_0 $-condition,
- abstract convex space,
- the family of generalized equi-KKM mappings,
- common fixed points,
- generalized abstract equilibrium problems
Citation: Shunyou Xia, Chongyi Zhong, Chunrong Mo. A common fixed point theorem and its applications in abstract convex spaces[J]. AIMS Mathematics, 2025, 10(3): 5236-5245. doi: 10.3934/math.2025240

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Abstract

First, this paper introduces a family of generalized equi-KKM mappings and the concept of quasi-abstract convexity (concavity) defined by parameter multi-valued mappings in abstract convex spaces that satisfy the $ H_0 $-condition. Then, using the Brouwer fixed-point theorem, we prove a common fixed-point theorem for this family of generalized equi-KKM mappings. Finally, we introduce two classes of generalized abstract equilibrium problems and obtain two existence theorems for their solutions, as an application of the common fixed-point theorem.

References

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