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Existence theorems generalizing fixed point theorems for multivalued mappings. (English) Zbl 0760.47029
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 ϕ be a bounded from below l.s.c. function on a complete metric space (X,d). Given xX with ϕ(x)>infϕ(X), let d(x,y)ϕ(x)-ϕ(y) for some yx. Then ϕ(z)=infϕ(X) for some zX.

Theorem 4. Let X be a compact convex subset of a locally convex t.v.s. If F:X×X is u.s.c., M(·)=sup{F(·,y): yX} is l.s.c., and {yX: F(x,y)a} is convex for all xX and a, then F(z,z)=M(z) for some zX.

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)].


MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces