概率度量空间的基本理论及应用(Ⅰ)<sup>*</sup>

概率度量空间的基本理论及应用(Ⅰ)^*

基金项目: * 中国科学院科学基金

摘要: 本文系统地研究概率度量空间的基本理论和应用,讨论了概率度量空间的拓扑结构和性质;给出了概率度量空间,Menger概率度量空间以及概率线性赋范空间可度量化的条件及其度量函数的形式:得出了概率度量空间集合的各种概率有界性的表征等.作为这些结果的应用,我们讨论了概率线性赋范空间中线性算子的理论及概率度量空间中不动点的存在性问题.

Abstract: This paper is devoted to the study of the basic theory and applications of probabilistic metric spaces (PM-space). In this paper the topological structure and properties for PM-space are considered. The conditions of metrization and the form of metric functions for PM-spaces. Menger PM-spaces and probabilistic normed linear spaces (PN-space) are given and the characterizations of various probabilistically bounded sets are presented. As applications we utilize these results obtained in this paper to study the linear operator theory and fixed point theory on PM-spaces.

[1]	张石生,概率度量空间中映象的不动点定理及应用,中国科学,A辑,6(1983),495-504.
[2]	张石生,《不动点理论及应用》,重庆出版社(1984).
[3]	Zhang Shi-sheng, On the theory of probabilistic metric spaces with applications, Acta Math. Sinica, New Series, 1,4 (1985), 366-377.
[4]	Zhang Shi-sheng, The metrization of probabilistic metric spaces with applications, Zbornik Radova Prirodno-Matematickog Fakulteta u Novom Sadu, Serijaza Matematiku, 15, 1 (1985), 107-117.
[5]	Hicks, T.L. and P.L. Sharma, Probabilistic metric structures: Topological classification, Ibid, 14, 1 (1984), 35-42.
[6]	Hicks, T.L. and P.L. Sharma, Random normed structures, Ibid, 14, 1 (1984), 43-50.
[7]	Hadzic, O., Some fixed point theorems in probabilistic metric spaces, Ibid, 15, 1 (1985), 23-36.
[8]	Radu, V., On some fixed point theorems in probabilistic metric spaces, Seminarul de Teoria Probabilitatilor si Applicatii, 74 (1985), 1-10.
[9]	Constantin, Gh., On some classes of contraction mappings in Menger spaces, Ibid, 76 (1981), 1-10.
[10]	Schweizer, B., A. Sklar and E. Thorp, The metrization of statistical metric spaces, Pacific J. Math., 10 (1960), 673-675.
[11]	Schweizer, B. and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, Amsterdam, Oxford (1983).
[12]	Engelking, R., General Topology, Warszawa (1977).
[13]	Nadler, S.B., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), 475-487.