# 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 multi-valued contractions in complete metric spaces. (English) Zbl 1213.54063
Let $(X,d)$ be a metric space and $\text{Cl}(X)$ the collection of nonempty closed sets of $X$. The main results of this paper are two fixed point theorems for multivalued functions. The following theorem is a generalization of Theorem 1 of {\it N. Mizoguchi} and {\it W. Takahashi} [J. Math. Anal. Appl. 141, No. 1, 177--188 (1989; Zbl 0688.54028)]. Theorem A. Let $(X,d)$ be a complete metric space and $T: X\to \text{Cl}(X)$ be a mapping of $X$ into itself. If there exists a function $\varphi: [0,\infty)\to (0,1)$ satisfying $\limsup\varphi*r(< 1$ for each $t\in [0,\infty)$ and $r\to t+$ such that for any $x\in X$ there exists $y\in T(x)$ satisfying the following two conditions: $d(x,y)\le (2-\varphi(d(x,y))) D(x,Tx)$ and $D(y,T(y))\le \varphi(d(x,y)) d(x,y)$, then $T$ has a fixed point in $X$ provided $f(x)= D(x,T(x))$ is lower-semicontinuous. The following theorem is a generalization of Theorem 1 [op. cit.], of Theorem 2 of {\it Y. Feng} and {\it S. Liu} [J. Math. Anal. Appl. 317, No. 1, 103--112 (2006; Zbl 1094.47049)] and {\it D. Klim} and {\it D. Wardowski} [J. Math. Anal. Appl. 334, No. 1, 132--139 (2007; Zbl 1133.54025)]. Theorem B. Let $(X,d)$ be a complete metric space and $T: X\to \text{Cl}(X)$ be a mapping of $X$ into itself. If there exists a function $\varphi: [0,\infty)\to (0,1)$ and a nondecreasing function $b: [0,\infty)\to [b,1)$, $b> 0$, such that $\varphi(t)< b(t)$ and $\limsup_{t\to r+}\,\varphi(t)< \limsup_{t\to r+} b(t)$ for all $t\in [0,\infty)$, and for any $x\in X$ there exists $y\in T(x)$ satisfying the following conditions: $b(d(x,y)) d(x,y)\le D(x,Tx)$ and $D(y,T(y))\le \varphi(d(x, y))$ $d(x,y)$, then $T$ has a fixed point in $X$ provided $f(x)= D(x,T(x))$ is lower-semicontinuous. In the last part of the paper the author constructs two examples which show that the results from this paper are genuine generalizations of the results of Mizaguchi and Takahasi, Feng and Liu and Klim and Wardowski.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54C60 Set-valued maps (general topology)
Full Text:
##### References:
 [1] &cacute, L. B.; Irić; Ume, J. S.: Some common fixed point theorems for weakly compatible mappings, J. math. Anal. appl. 314, No. 2, 488-499 (2006) · Zbl 1086.54027 [2] &cacute, L. B.; Irić: Common fixed point theorems for multi-valued mappings, Demonstratio math. 39, No. 2, 419-428 (2006) [3] &cacute, L. B.; Irić: Fixed point theorems for set-valued non-self mappings, Math. balkanica 20, No. 2, 207-217 (2006) [4] Daffer, P. Z.; Kaneko, H.: Fixed points of generalized contractive multi-valued mappings, J. math. Anal. appl. 192, 655-666 (1995) · Zbl 0835.54028 · doi:10.1006/jmaa.1995.1194 [5] Feng, Y.; Liu, S.: Fixed point theorems for multi-valued contractive mappings and multi-valued caristi type mappings, J. math. Anal. appl. 317, 103-112 (2006) · Zbl 1094.47049 · doi:10.1016/j.jmaa.2005.12.004 [6] Klim, D.; Wardowski, D.: Fixed point theorems for set-valued contractions in complete metric spaces, J. math. Anal. appl. 334, 132-139 (2007) · Zbl 1133.54025 · doi:10.1016/j.jmaa.2006.12.012 [7] Mizoguchi, N.; Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces, J. math. Anal. appl. 141, 177-188 (1989) · Zbl 0688.54028 · doi:10.1016/0022-247X(89)90214-X [8] Jr., S. B. Nadler: Multi-valued contraction mappings, Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002 [9] Naidu, S. V. R.: Fixed-point theorems for a broad class of multimaps, Nonlinear anal. 52, 961-969 (2003) · Zbl 1029.54049 · doi:10.1016/S0362-546X(02)00146-3 [10] Reich, S.: Fixed points of contractive functions, Boll. unione mat. Ital. 5, 26-42 (1972) · Zbl 0249.54026 [11] Reich, S.: Some fixed point problems, Atti acad. Naz. lincei 57, 194-198 (1974) · Zbl 0329.47019 [12] Zhong, C. K.; Zhu, J.; Zhao, P. H.: An extension of multi-valued contraction mappings and fixed points, Proc. amer. Math. soc. 128, 2439-2444 (2000) · Zbl 0948.47058 · doi:10.1090/S0002-9939-99-05318-6