# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Fixed point theorems for multivalued mappings on complete metric spaces. (English) Zbl 0688.54028
The authors give the following “multi-version” of Caristi’s fixed point theorem [{\it J. Caristi}, Trans. Am. Math. Soc. 215, 241-251 (1976; Zbl 0305.47029)]. Let (X,d) be a complete metric space, $\psi$ : $X\to (- \infty,+\infty]$ be a proper, bounded below and lower semicontinuous function and multimap T: $X\to P(X)$ is such that for every $x\in X$, there exists $y\in Tx$ satisfying $$\psi (y)+d(x,y)\le \psi (x).$$ Then T has a fixed point. It is shown that this result is equivalent to the $\epsilon$-variational principle of Ekeland. Then it is used to generalize Nadler’s fixed point theorem and to obtain a common fixed point theorem for a single-valued map and a multimap. Next, some generalizations of Reich’s fixed point theorems for multimaps of contractive type are considered. $\{$ Reviewer’s remark: Another generalization of the Caristi’s theorem on multifunctions was given in the work of {\it J. Madhusudana Rao} [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Ser. 29(77), No.1, 79-80 (1985; Zbl 0561.54041)]$\}$.
Reviewer: V.V.Obukhovskij

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54C60 Set-valued maps (general topology)
Full Text:
##### References:
 [1] Assad, N. A.; Kirk, W. A.: Fixed point theorems for set-valued mappings of contractive type. Pacific J. Math. 43, 553-562 (1972) · Zbl 0239.54032 [2] Bishop, E.; Phelps, R. R.: The support functional of a convex set. Proc. sympos. Pure math., 27-35 (1963) · Zbl 0149.08601 [3] Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. amer. Math. soc. 215, 241-251 (1976) · Zbl 0305.47029 [4] Ekeland, I.: Remarques sur LES problèmes variationnels, I. C. R. Acad. sci. Paris sér. A--B 275, 1057-1059 (1972) · Zbl 0249.49004 [5] Ekeland, I.: Nonconvex minimization problems. Bull. amer. Math. soc. 1, 443-474 (1979) · Zbl 0441.49011 [6] Itoh, S.; Takahashi, W.: Single-valued mappings, multivalued mappings and fixed-point theorems. J. math. Anal. appl. 59, 514-521 (1977) · Zbl 0351.47040 [7] Itoh, S.; Takahashi, W.: The common fixed point theory of singlevalued mappings and multivalued mappings. Pacific J. Math. 79, 493-508 (1978) · Zbl 0371.47042 [8] Jr., S. B. Nadler: Multi-valued contraction mappings. Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002 [9] Reich, S.: Fixed points of contractive functions. Boll. un. Mat. ital. 5, 26-42 (1972) · Zbl 0249.54026 [10] Reich, S.: Some fixed point problems. Atti. acad. Naz. lincei 57, 194-198 (1974) · Zbl 0329.47019 [11] Reich, S.: Approximate selections, best approximations, fixed points, and invariant sets. J. math. Anal. appl. 62, 104-113 (1978) · Zbl 0375.47031