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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 $\varphi$ 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 $y\ne x$. 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 $F: X\times X\to\bbfR$ 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 $a\in\bbfR$, then $F(z,z)=M(z)$ for some $z\in X$. From Theorem 1 the author deduces the Caristi-Kirk theorem [{\it J. Caristi}, Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)], the {\it I. Ekeland’s} theorem [Bull. Am. Math. Soc. 1, 443-474 (1979; Zbl 0441.49011)], and the set-valued contraction principle of {\it S. B. Nadler} [Pac. J. Math. 30, 475-488 (1969; Zbl 0187.450)]. From Theorem 4, whose proof essentially uses a result of {\it K. Fan} [Proc. Nat. Acad. Sci. USA 38, 121-126 (1952; Zbl 0047.351)], the author deduces two minimization theorems from {\it 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 {\it A. Brøndsted} [Pac. J. Math. 55, 335-341 (1974; Zbl 0298.46006)].

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