# 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)
An implicit iteration process for nonexpansive semigroups. (English) Zbl 1225.47093
Summary: Let $C$ be a closed convex subset of a Banach space $E$. Let $\left\{T\left(t\right):t\ge 0\right\}$ be a strongly continuous semigroup of nonexpansive mappings on $C$ such that ${\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)\ne \varnothing$. Let $\left\{{\alpha }_{n}\right\}$ and $\left\{{t}_{n}\right\}$ be sequences of real numbers satisfying appropriate conditions, then for arbitrary ${x}_{0}\in C$, the Mann type implicit iteration process $\left\{{x}_{n}\right\}$ given by ${x}_{n}={\alpha }_{n}{x}_{n-1}+\left(1-{\alpha }_{n}\right)T\left({t}_{n}\right){x}_{n}$, $n\ge 0$, weakly (strongly) converges to an element of ${\bigcap }_{t\ge 0}F\left(T\left(t\right)\right)$.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H20 Semigroups of nonlinear operators 47H09 Mappings defined by “shrinking” properties
##### References:
 [1] Browder, F. E.: Convergence of appropriates to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601 · doi:10.1007/BF00251595 [2] Browder, F. E.: Nonexpansive nonlinear operators in Banach space, Proc. nat. Acad. sci. USA 54, 1041-1044 (1965) · Zbl 0128.35801 · doi:10.1073/pnas.54.4.1041 [3] Shioji, N.; Takahashi, W.: Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, Nonlinear anal. 34, 87-99 (1998) · Zbl 0935.47039 · doi:10.1016/S0362-546X(97)00682-2 [4] Suzuki, T.: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert space, Proc. amer. Math. soc. 131, 2133-2136 (2003) · Zbl 1031.47038 · doi:10.1090/S0002-9939-02-06844-2 [5] Xu, H. -K.: A strong convergence theorem for contradiction semigroups in Banach spaces, Bull. austral. Math. soc. 72, 371-379 (2005) · Zbl 1095.47016 · doi:10.1017/S000497270003519X [6] Saejung, S.: Strong convergence theorem for nonexpansive semigroups without Bochner integrals, Fixed point theory appl. 2008, 7 papers (2008) · Zbl 1203.47077 · doi:10.1155/2008/745010