Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0760.47029
Takahashi, Wataru
Existence theorems generalizing fixed point theorems for multivalued mappings. (English)
[A] 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.]\par 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 $z\in X$.\par 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 $z\in X$.\par 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)].\par 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)].\par 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)].
[T.Kubiak (Poznań)]

MSC 2000:: *47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
54H25 Fixed-point theorems in topological spaces

Keywords: Ekeland's theorem; minimization theorems; Caristi-Kirk theorem; set- valued contraction principle

Citations: Zbl 0731.00015; Zbl 0305.47029; Zbl 0441.49011; Zbl 0187.450; Zbl 0047.351; Zbl 0185.395; Zbl 0298.46006

Cited in: Zbl 1151.54353 Zbl 1120.47044 Zbl 1113.47044 Zbl 1103.54023 Zbl 1089.49004 Zbl 0893.47036

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]