Abstract
In this paper we first introduce the concept of probabilistic interval space. Under this framework, a new version of KKM theorem is obtained. As application, we utilize this result to study some new minimax theorem, section theorem, matching theorem, coincidence theorem and fixed point theorem in probabilistic metric spaces. The results presented in this paper not only contain the main result of von Neumann[7] as its special case but also extend the corresponding results of [1, 3, 4, 6, 8] to the case of probabilistic metric spaces.
Similar content being viewed by others
References
Zhang Shisheng and Ma Yihai, Generalized KKM theorem on H-space with applications, J. Math. Anal. Appl., 163, (1992), 406–421.
Zhang Shisheng, Basic theory and applications of probabilistic metric spaces (I), (II), Applied Math. and Mech., 9, 2–3 (1988), 123–133, 213–225.
K. Fan, A generalization of Tychonoffs fixed point theorem, Math. Ann., 142 (1961), 303–310.
K. Fan, Some properties of convex sets related to fixed point theorem, Math. Ann., 266 (1984), 519–537.
B. Knaster, B. Kuratowski and S. Mazurkiewicz, Ein beweis des fixpunktsatzes für n-dimensionale simplexe, Fund. Math., 14 (1929), 132–137.
V. Komorink, Minimax theorems for upper semi-continuous functions, Acta Math. Acad. Sci. Hunger, 40 (1982), 159–163.
J.von Neumann, Zur theoric der gesellshaftsphiele, Math. Ann., 100 (1928), 295–320.
S. Park, Generalizations of Ky Fan's matching theorems and their applications, J. Math. Anal. Appl., 141 (1989), 164–176.
B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York Amsterdam, Oxford (1992).
B. Schweizer and A. Sklar, Probabilistic metric spaces, Pacific J. Math., 10 (1960), 313–334.
L. L. Stacho, Minimax theorems beyond topological vector spaces, Acta Sci. Math., 42 (1980), 157–164.
Zhang Shisheng, Yeol Je Cho, Shin Min Kang, Probabilistic Metric Spaces and Nonlinear Operator Theory, Sichuan University Publishing House (1994). (in hinese)
Author information
Authors and Affiliations
Additional information
Project supported by the National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Shisheng, Z., Cho, Y.J. & Ziang, W. New version of KKM theorem in probabilistic metric spaces with applications. Appl Math Mech 17, 1009–1019 (1996). https://doi.org/10.1007/BF00119948
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00119948