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The theory and applications of generalized H-space. (English) Zbl 0764.54022

Summary: This paper brings forward the concept of generalized \(H\)-spaces which extends the concepts of \(H\)-spaces and almost probabilistic metric spaces. In this paper, the uniformity and properties for generalized \(H\)- spaces are considered. The conditions of metrization and the form of metric functions for generalized \(H\)-spaces, \(H\)-spaces and Menger PM- spaces are given and the characteristics of completeness and compactness for generalized \(H\)-spaces are presented. The results of this paper generalize and unify some recent results of T. L. Hicks and P. L. Sharma [Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 14, No. 1, 35-42 (1984; Zbl 0587.54049)], Zhou Zhongqun [Acta Math. Sin. 29, No. 4, 569-572 (1986), see also the review of Pt. II, ibid. 33, No. 5, 641-645 (1990; Zbl 0712.54019)], B. Schweizer and A. Sklar [Pac. J. Math. 10, 313-334 (1960; Zbl 0091.298) and with E. Thorp, ibid., 673-675 (1960; Zbl 0096.332)].

MSC:

54E70 Probabilistic metric spaces
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References:

[1] Hicks T. L., and P. L. Sharm, Probabilistic metric structures: topological classification,Zbornik Radova Prirodno-Matematičkog Fakulteta u Novom Sadu, Serija za Matematiku,14 (1984), 35–42. · Zbl 0587.54049
[2] Zhou Zhong-qun,Acta. Math. Sinica,29, #4, (1986), 569–572. (in Chinese).
[3] Chang Shin-sen,Fixed Point Theorem and Application, Chongqing Publishing House, Chongqing (1984). (in Chinese).
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[12] Chang Shin-sen, The metrization of probabilistic metric spaces with applications,Zbornik Radova Prirodno-Matematičkog Fakulteta u Novom Sadu, Serija za Matematiku,15, #1 (1985), 107–117.
[13] Chang Shin-sen and Che Su-bing, The metrization of probabilistic metric space and fixed point theorem,Chinese Quarterly Journal of Mathematics, 2 (1987), 54–60. (in Chinese).
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