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).
The paper includes two minimization theorems: Theorem 1. Let \(\varphi\) be a bounded from below l.s.c. function on a complete metric space \((X,d)\). Given \(x\in X\) with \(\varphi(x)>\inf \varphi(X)\), let \(d(x,y)\leq\varphi(x)-\varphi(y)\) for some \(y\neq x\). Then \(\varphi(z)=\inf \varphi(X)\) for some \(z\in X\).
Theorem 4. Let \(X\) be a compact convex subset of a locally convex t.v.s. If \(F: X\times X\to\mathbb{R}\) is u.s.c., \(M(\cdot)=\sup\{F(\cdot,y)\): \(y\in X\}\) is l.s.c., and \(\{y\in X\): \(F(x,y)\geq a\}\) is convex for all \(x\in X\) and \(a\in\mathbb{R}\), then \(F(z,z)=M(z)\) for some \(z\in X\).
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 0187.45002)].
From Theorem 4, whose proof essentially uses a result of K. Fan [Proc. Natl. Acad. Sci. USA 38, 121–126 (1952; Zbl 0047.35103)], the author deduces two minimization theorems from K. Fan [Math. Z. 112, 234–240 (1969; Zbl 0185.39503)].
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)].
[For the entire collection see Zbl 0731.00015.]


47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)