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 theorem for asymptotically nonexpansive mappings. (English) Zbl 0861.47030
In the present paper the authors prove the following Theorem: Let $C$ be a nonempty closed convex subset of a real Hilbert space and $T:C\to C$ be an asymptotically nonexpansive mapping (i.e. for each $n\geq 1$ there exists a real number $k_n\geq 1$ such that $|T^nx- T^ny|\leq k_n|x-y|$, $x,y\in C$ and $\lim_{n\to\infty} k_n=1$). Suppose that the set of fixed points of $T$, $F(T)$ is nonempty. Let $$b_n= {1\over n}\sum^n_{j=1} (1+|1-k_j|+ e^{-j}), \qquad 0<a<1, \quad x_0\in C.$$ Then the mapping $T_n$ on $C$ given by $$T_nx= a_n x_0+ (1-a_n) A_nx\qquad \text{for all }x\in C$$ where $$a_n= {{b_n-1} \over {b_n-1+a}} \qquad \text{and} \qquad A_n= {1\over n}\sum^n_{j=1} T^j,$$ has a unique fixed point $u_n$ in $C$ and the sequence $\{u_n\}$ converges strongly to the element of $F(T)$ which is nearest to $x_0$.

MSC:
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
WorldCat.org
Full Text: DOI
References:
[1] Goebel, K.; Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. am. Math. soc. 35, 171-174 (1972) · Zbl 0256.47045
[2] Ishihara, H.; Takahashi, W.: A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space. Proc. am. Math. soc. 104, 431-436 (1988) · Zbl 0692.47010
[3] Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach space. Archs ration. Mech. analysis 24, 82-90 (1967) · Zbl 0148.13601
[4] Baillon, J.B.: Un théorème de type ergodique pour LES contractions non linéaires dans un espace de Hilbert. CR acad. Sci. Paris 280, 1511-1514 (1975) · Zbl 0307.47006
[5] Hirano, N.; Takahashi, W.: Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces. Kodai math. J. 2, 11-25 (1979) · Zbl 0404.47031
[6] Tan, K.K.; Xu, H.K.: The nonlinear ergodic theorem for asymptotcially nonexpansive mappings in Banach spaces. Proc. am. Math. soc. 114, 399-404 (1992) · Zbl 0781.47045