Minimization theorems for fixed point theorems in fuzzy metric spaces and applications. (English) Zbl 0845.54004

The authors prove a minimization theorem in fuzzy metric spaces. The theorem goes as follows: If \(X\) and \(Y\) are complete fuzzy metric spaces of a given type, \(f: X\to Y\) is closed and continuous and \(\varphi: f(X)\to (-\infty, \infty]\) is lower semicontinuous and bounded from below then under certain assumptions \(\varphi \circ f\) attains its minimum.
Using the theorem above the authors obtain a variational principle as well as a fixed point theorem for fuzzy metric spaces. Finally the equivalence of these theorems is proved.
The variational principle, the fixed point theorem and their equivalence also appear in M. Stojaković and Z. Ovcin [ibid. 66, 353-356 (1994; Zbl 0842.54020)].
Reviewer: O.Kaleva (Tampere)


54A40 Fuzzy topology


Zbl 0842.54020
Full Text: DOI


[1] Caristi, J., Fixed point theorem for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215, 241-251 (1976) · Zbl 0305.47029
[2] Downing, D.; Kirk, W. A., A generalization of Caristi’s theorem with application to nonlinear mapping theory, Pacific J. Math., 69, 339-346 (1977) · Zbl 0357.47036
[3] Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353 (1974) · Zbl 0286.49015
[4] He, Pei-jun, The variational principle in fuzzy metric spaces and its applications, Fuzzy Sets and Systems, 45, 389-394 (1992) · Zbl 0754.54005
[5] Kaleva, O.; Seikkala, S., On fuzzy metric spaces, Fuzzy Sets and Systems, 12, 215-229 (1984) · Zbl 0558.54003
[6] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. Math., 10, 313-334 (1960) · Zbl 0091.29801
[7] Takahashi, W., Existence theorem generalizing fixed point theorems for multivalued mappings, (Théra, M. A.; Baillon, J.-B., Fixed Point Theory and Applications. Fixed Point Theory and Applications, Pitman Research Notes in Math. Series, 152 (1991)), 397-406 · Zbl 0760.47029
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