Fixed point theory and applications, Proc. Int. Conf., Marseille- Luminy/Fr. 1989, Pitman Res. Notes Math. Ser. 252, 397-406 (1991).
[For the entire collection see Zbl 0731.00015.]
The paper includes two minimization theorems: Theorem 1. Let be a bounded from below l.s.c. function on a complete metric space . Given with , let for some . Then for some .
Theorem 4. Let be a compact convex subset of a locally convex t.v.s. If is u.s.c., : is l.s.c., and : is convex for all and , then for some .
From Theorem 1 the author deduces the Caristi-Kirk theorem [J. Caristi, Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)], the I. Ekeland’s theorem [Bull. Am. Math. Soc. 1, 443-474 (1979; Zbl 0441.49011)], and the set-valued contraction principle of S. B. Nadler [Pac. J. Math. 30, 475-488 (1969; Zbl 0187.450)].
From Theorem 4, whose proof essentially uses a result of K. Fan [Proc. Nat. Acad. Sci. USA 38, 121-126 (1952; Zbl 0047.351)], the author deduces two minimization theorems from K. Fan [Math. Z. 112, 234- 240 (1969; Zbl 0185.395)].
Reviewer’s remark. Theorem 1 is clearly equivalent to the Caristi-Kirk theorem. It is also an obvious consequence of a result of A. Brøndsted [Pac. J. Math. 55, 335-341 (1974; Zbl 0298.46006)].