# 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)
Strong convergence of modified Mann iterations. (English) Zbl 1091.47055
Let $X$ be a real Banach space with a norm $\parallel ·\parallel$ and let $C$ be a nonempty, closed and convex subset of $X$. A mapping $T:C\to C$ is nonexpansive provided that $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in C$. Assume that $T$ has at least one fixed point in $C$. The authors consider the following iteration sequence $\left\{{x}_{n}\right\}$ for $T:{x}_{0}=x\in C$, ${y}_{n}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right)T{x}_{n}$, ${x}_{n+1}={\beta }_{n}u+\left(1-{\beta }_{n}\right){y}_{n}$, where $u$ is an arbitrary fixed element in $C$ and $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$ are two sequences in the interval $\left(0,1\right)$ converging to 0 and such that $\sum {\alpha }_{n}=\sum {\beta }_{n}=\infty$. Moreover, $\sum |{\alpha }_{n+1}-{\alpha }_{n}|<\infty$, $\sum |{\beta }_{n+1}-{\beta }_{n}|<\infty$. Under the assumption that $X$ is uniformly smooth, it is shown that the sequence $\left\{{x}_{n}\right\}$ converges strongly to a fixed point of $T$. An analogous result is proved for accretive operators.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 46B25 Classical Banach spaces in the general theory of normed spaces 47H06 Accretive operators, dissipative operators, etc. (nonlinear)