# 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 points for mean non-expansive mappings. (English) Zbl 1128.47053
Summary: For a Banach space $X$, {\it J. García--Falset} [Houston J. Math. 20, No. 3, 495--506 (1994; Zbl 0816.47062)] introduced the coefficient $R(X)$ and showed that if $R(X) < 2$, then $X$ has a fixed point. In the present paper, we define a mean nonexpansive mapping $T$ on $X$ in the sense that $\|Tx - Ty\|\leq a\|x - y\| + b\|x - Ty\|$ for any $x, y \in X$, where $a, b \geq 0$, $a + b \leq 1$. We show that if $R{\left(X \right)}<\frac{2}{1 + b}$, then $T$ has a fixed point in $X$.
##### MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 46B20 Geometry and structure of normed linear spaces 47H05 Monotone operators (with respect to duality) and generalizations
Full Text:
##### References:
 [1] Brodskii, M.S., Milman, D.P. On the center of a convex set. Dokl. Akad. Nauk. SSSR, 59:837--840 (1948) [2] Garcia-Falset, J. Stability and fixed points for nonexpansive mappings. Houston Math., 20:495--505 (1994) · Zbl 0816.47062 [3] Garcia-Falset, J. The fixed point property in Banach space with NUS property. J. Math. Anal. Appl., 215:532--542 (1997) · Zbl 0902.47048 · doi:10.1006/jmaa.1997.5657 [4] Goebel, K., Kirk, W.A. Topics in metric fixed point theory. Cambridge, Cambridge University Press, 1990 · Zbl 0708.47031 [5] Huff, R. Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math., 10:743--749 (1980) · Zbl 0505.46011 · doi:10.1216/RMJ-1980-10-4-743 [6] Kirk, W.A. A fixed point theorem for mappings which do not increase distance. Amer. Math. Mon., 72:1004--1006 (1965) · Zbl 0141.32402 · doi:10.2307/2313345 [7] Kirk, W.A., Sims, B. Handbook of metric fixed point theory. Dordrecht, Kluwer Academic Publishers, 2001 · Zbl 0970.54001 [8] Zhang, S. About fixed point theory for mean nonexpansive mapping in Banach spaces. Journal of Sichuan University, 2:67--68 (1975)