# 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)
A strong convergence theorem for relatively nonexpansive mappings in a Banach space. (English) Zbl 1071.47063

The present paper is concerned with the problem of finding a fixed point of a relatively nonexpansive mapping defined on a closed convex subset of a Banach space. To this end, the authors investigate under what conditions a sequence constructed by the so-called “hybrid method in mathematical programming” converges strongly to a fixed point of the mapping. The main result (Theorem 3.1) goes as follows:

Theorem. Let $E$ be a uniformly convex and uniformly smooth Banach space, let $C$ be a nonempty closed convex subset of $E$, let $T$ be a relatively nonexpansive mapping from $C$ into itself, and let $\left\{\alpha \left(n\right)\right\}$ be a sequence of real numbers such that $0\le \alpha \left(n\right)\le 1$ and $lim sup\alpha \left(n\right)<1$ when $n\to \infty$. When the set $F\left(T\right)$ of the fixed points of $T$ is nonempty, then the sequence $x\left(n\right)$ constructed by the hybrid method converges strongly to the point that is the generalised projection from $C$ onto $F\left(T\right)$.

As special cases, they obtain analogous strong convergence results for a nonexpansive mapping on a Hilbert space (using the metric projection) and for a maximal monotone operator on a Banach space (using the generalised projection).

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H05 Monotone operators (with respect to duality) and generalizations