## Minimization theorems and fixed point theorems in generating spaces of quasi-metric family.(English)Zbl 0986.54015

Summary: Following the approaches of O. Kada, T. Suzuki and W. Takahashi [Math. Jap. 44, No. 2, 381-391 (1996; Zbl 0897.54029)], we define a family of weak quasi-metrics in a generating space of quasi-metric family. By using a family of weak quasi-metrics, we prove a Takahashi-type minimization theorem, a generalized Ekeland variational principle and a general Caristi-type fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family. Also, following the approach of J.-P. Aubin [Optima and equilibria, Grad. Texts Math. 140 (1998; Zbl 0930.91001)], we prove another fixed point theorem for set-valued maps in complete generating spaces of quasi-metric family without the assumption of lower semicontinuity. From our results in complete generating spaces of quasi-metric family, we obtain the corresponding theorems for set-valued maps in complete fuzzy metric spaces.

### MSC:

 54A40 Fuzzy topology 54H25 Fixed-point and coincidence theorems (topological aspects)

### Citations:

Zbl 0781.90012; Zbl 0897.54029; Zbl 0930.91001
Full Text:

### References:

 [1] Aubin, J.P., Optima and equilibria, (1993), Springer Berlin [2] Caristi, J., Fixed point theorem for mappings satisfying inwardness conditions, Trans. amer. math. soc., 215, 241-251, (1976) · Zbl 0305.47029 [3] Chang, Shih-sen; Cho, Yeol Je; Kang, Shin Min, Probabilistic metric spaces and nonlinear operator theory, (1994), Sichuan University Press Chengdu, China · Zbl 1080.47054 [4] Shih-sen Chang, Yeol Je Cho, Byung Soo Lee, Jong Soo Jung and Shin Min Kang, Coincidence point theorems and minimization theorems in fuzzy metric spaces, to appear in Fuzzy Sets and Systems. · Zbl 1108.54304 [5] Congxin, W.; Ginxuan, F., Fuzzy generalization of Kolmogroff’s theorem, J. Harbin inst. of technology, No. 1, 1-7, (1984) [6] Congxin, W.; Ming, M., Continuity and boundedness for the mappings between fuzzy normed spaces, J. fuzzy math., 1, 13-24, (1993) · Zbl 0872.46039 [7] Downing, D.; Kirk, W.A., A generalization of Caristi’s theorem with applications to nonlinear mapping theory, Pacific J. math., 69, 339-346, (1977) · Zbl 0357.47036 [8] Ekeland, I., On the variational principle, J. math. anal. appl., 47, 324-353, (1974) · Zbl 0286.49015 [9] Fan, J.X., On the generalizations of Ekeland’s variational principle and Caristi’s fixed point theorem, () [10] He, Pei-jun, The variational principle in fuzzy metric spaces and its applications, Fuzzy sets and systems, 45, 389-394, (1992) · Zbl 0754.54005 [11] Jung, Jong Soo; Cho, Yeol Je; Kim, Jong Kyu, Minimization theorems for fixed point theorems in fuzzy metric spaces and applications, Fuzzy sets and systems, 61, 199-207, (1994) · Zbl 0845.54004 [12] Jung, Jong Soo; Cho, Yeol Je; Kang, Shin Min; Chang, Shih-sen, Coincidence theorems for set-valued mappings and Ekeland variational principle in fuzzy metric spaces, Fuzzy sets and systems, 79, 239-250, (1996) · Zbl 0867.54018 [13] Kada, O.; Suzuki, T.; Takahashi, W., Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. japonica, 44, 381-391, (1996) · Zbl 0897.54029 [14] Kaleva, O.; Seikkala, S., On fuzzy metric spaces, Fuzzy sets and systems, 12, 215-229, (1984) · Zbl 0558.54003 [15] Kim, T.H.; Kim, E.S.; Shin, S.S., Minimization theorems relating to fixed point theorems on complete metric spaces, Math. japonica, 45, 97-102, (1997) · Zbl 0872.47028 [16] Schweizer, B.; Sklar, A., Statistical metric spaces, Pacific J. math., 10, 313-334, (1960) · Zbl 0091.29801 [17] Takahashi, W., Existence theorems generalizing fixed point theorems for multivalued mapping, (), 397-406, Series 252 · Zbl 0760.47029 [18] Ume, J.S., Some existence theorems generating fixed point theorems on complete metric spaces, Math. japonica, 40, 109-114, (1994) · Zbl 0813.47074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.