zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings. (English) Zbl 0942.47046
The paper concerns some generalizations of convergence of special type of iterations of asymptotically nonexpansive mappings in uniformly convex Banach spaces to a fixed point. A map $T: E\to E$ defined on a subset of a Banach space is said to be asymptotically nonexpansive iff $$\|T^nx- T^nq\|\le k_n\|x-y\|$$ for all $x,y\in E$ and $n\in\bbfN$, where $(k_n)$ is a sequence of real numbers such that $$k_n\ge 1\quad\text{and}\quad \lim_{n\to\infty} k_n= 1.$$ The author considers the convergence of the following iteration process with errors $$x_{n+1}= (1-\alpha_n) x_n+ \alpha_n T^n y_n+ u_n,$$ $$y_n= (1- \beta_n) x_n+ \beta_n T^nx_n+ v_n,$$ where $\{u_n\}$ and $\{v_n\}$ are sequences in $E$ satisfying $$\sum^\infty_{n= 1}\|u_n\|<\infty\quad\text{and} \quad \sum^\infty_{n=1}\|v_n\|< \infty,$$ and $\{\alpha_n\}$ and $\{\beta_n\}$ are sequences of real numbers in $[0,1]$. Under some additional assumptions it has been proved that the sequence of iterations $\{x_n\}$ converges strongly to a fixed point of $T$. The results presented here are some generalizations of the results obtained in 1994 by B. E. Rhoades. Let’s note that some facts (see e.g. Lemma 1, Lemma 6) proved in the paper are very obvious.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
Full Text: DOI
[1] Rhoades, B. E.: Fixed point iterations for certain nonlinear mappings. J. math. Anal. appl. 183, 118-120 (1994) · Zbl 0807.47045
[2] Schu, J.: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. math. Anal. appl. 158, 407-413 (1991) · Zbl 0734.47036
[3] Liu, L. S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. math. Anal. appl. 194, 114-125 (1995) · Zbl 0872.47031
[4] Ishikawa, S.: Fixed points by a new iteration method. Proc. amer. Math. soc. 44, 147-150 (1974) · Zbl 0286.47036
[5] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-510 (1953) · Zbl 0050.11603
[6] Xu, H. K.: Inequalities in Banach spaces with applications. Nonlinear anal. 16, 1127-1138 (1991) · Zbl 0757.46033
[7] Tan, K. K.; Xu, H. K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. math. Anal. appl. 178, 301-308 (1993) · Zbl 0895.47048
[8] Liu, Q. H.: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear analysis 26, 1835-1842 (1996) · Zbl 0861.47047
[9] Goebel, K.; Kirk, W. A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. amer. Math. soc. 35, 171-174 (1972) · Zbl 0256.47045