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Zbl 1028.47049
Alber, Yakov; Reich, Simeon; Yao, Jen-Chih
Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.
(English)
[J] Abstr. Appl. Anal. 2003, No.4, 193-216 (2003). ISSN 1085-3375; ISSN 1687-0409/e

Let $G$ be a closed convex subset of a Banach space $B$. A mapping $A:G\to B$ is said to be (i) nonexpansive if for all $x,y\in G$, $\|Ax-Ay\|\le\|x-y\|$, (ii) weakly contractive of class $C_{\psi(t)}$ on $G$ if there exists a continuous and increasing function $\psi(t)$ defined on $\bbfR^+$ such that $\psi$ is positive on $\bbfR^+\setminus \{0\}$, $\psi(0)=0$, $\lim_{t\to+ \infty}\psi(t) =+\infty$, and for all $x,y\in G$, $\|Ax-Ay\|\le\|x-y\|-\psi(\|x-y\|)$.\par The authors study descent-like approximation methods and proximal methods of retraction type for solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. They prove strong and weak convergence for weakly contractive and nonexpansive maps, respectively. They also establish the stability of these methods with respect to perturbations of the operators and the constraint sets.
[Tulsi Das Narang (Amritsar)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H06 Accretive operators, etc. (nonlinear)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: weakly contraction maps; descent-like approximation methods; proximal methods; convergence; nonexpansive maps; stability

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