# 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)
Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces. (English) Zbl 1181.47074
The author considers a uniformly convex Banach space $E$ satisfying the Opial condition, a closed, convex subset $C$ of $E$, and a Lipschitzian pseudo-contractive semigroup $\text{T}:=\left\{T\left(t\right):t\ge 0\right\}:C\to C$ such that $\text{F}:={\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne \varnothing$, where $F\left(T\left(t\right)\right)$ denotes the set of fixed points of $T\left(t\right)$. Let ${\left({\alpha }_{n}\right)}_{n}$ be a sequence in $\left(0,1\right)$ and ${\left({t}_{n}\right)}_{n}$ a sequence in $\left(0,\infty \right)$ satisfying (i) ${lim sup}_{n}{\alpha }_{n}<1$ and (ii) ${lim}_{n}{t}_{n}={lim}_{n}\frac{{\alpha }_{n}}{{t}_{n}}=0$. Under these circumstances, he shows that the sequence ${x}_{0}\in C$, ${x}_{n}={\alpha }_{n}{x}_{n-1}+\left(1-{\alpha }_{n}\right)T\left({t}_{n}\right){x}_{n},\phantom{\rule{4pt}{0ex}}n\ge 1$, is weakly convergent to a common fixed point of the semigroup.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H20 Semigroups of nonlinear operators
##### References:
 [1] Browder, F. E. Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Sympos. Pure Math. 18(2) (1976) [2] Kim, T. H. and Xu, H. K. Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 64(5), 1140–1152 (2006) · Zbl 1090.47059 · doi:10.1016/j.na.2005.05.059 [3] Xu, H. K. Strong asymptotic behavior of almost-robits of nonlinear semigroups. Nonlinear Anal. 46(1), 135–151 (2001) · Zbl 0993.47038 · doi:10.1016/S0362-546X(99)00453-8 [4] Deimling, K. Zeros of accretive operators. Manuscripta Math. 13(4), 365–374 (1974) · Zbl 0288.47047 · doi:10.1007/BF01171148 [5] Chang, S. S., Cho, Y. J., and Zhou, H. Y. Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publishers, New York (2002) [6] Xu, H. K. and Ori, R. G. An implicit iteration process for nonexpansive mappings. Numer. Funct. Anal. Optim. 22, 767–773 (2001) · Zbl 0999.47043 · doi:10.1081/NFA-100105317 [7] Osilike, M. O. Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 294(1), 73–81 (2004) · Zbl 1045.47056 · doi:10.1016/j.jmaa.2004.01.038 [8] Chen, R. D., Song, Y. S., and Zhai, H. Y. Convergence theorems for implicit iteration press for a finite family of continuous pseudocontractive mappings. J. Math. Anal. Appl. 314(2), 701–709 (2006) · Zbl 1086.47046 · doi:10.1016/j.jmaa.2005.04.018 [9] Zhou, H. Y. Convergence theorems of common fixed points for a finite family of Lipschitzian pseudocontractions in Banach spaces. Nonlinear Anal. 68(10), 2977–2983 (2008) · Zbl 1145.47055 · doi:10.1016/j.na.2007.02.041 [10] Bruck, R. E. A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math., 32(1), 107–116 (1979) · Zbl 0423.47024 · doi:10.1007/BF02764907