# 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)
Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. (English) Zbl 1168.47047

The purpose of the paper is to study the existence and approximation of fixed points of firmly nonexpansive mappings in Banach spaces. The following definition is used. Let $E$ be a smooth Banach space, $C$ be a nonempty closed convex subset of $E$, and $J$ be the normalized duality mapping from $E$ into ${E}^{*}$. The mapping $T$ is of firmly nonexpansive type if, for all $x,y\in C$,

$〈Tx-Ty,JTx-JTy〉\le 〈Tx-Ty,Jx-Jy〉·$

A fixed point theorem for firmly nonexpansive-type mappings in Banach spaces is obtained. It is shown that every nonexpansive-type mapping which has a fixed point is strongly relatively nonexpansive. A weak convergence theorem is presented. The obtained results are applied to the proximal point algorithm for monotone operators satisfying the range condition in Banach spaces.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H05 Monotone operators (with respect to duality) and generalizations 47H09 Mappings defined by “shrinking” properties 47J25 Iterative procedures (nonlinear operator equations)