Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces. (English) Zbl 1028.47049

Alber, Yakov; Reich, Simeon; Yao, Jen-Chih

Abstr. Appl. Anal. 2003, No. 4, 193-216 (2003).

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 \(x,y\in G\), \(\|Ax-Ay\|\leq\|x-y\|\), (ii) weakly contractive of class \(C_{\psi(t)}\) on \(G\) if there exists a continuous and increasing function \(\psi(t)\) defined on \(\mathbb{R}^+\) such that \(\psi\) is positive on \(\mathbb{R}^+\setminus \{0\}\), \(\psi(0)=0\), \(\lim_{t\to+ \infty}\psi(t) =+\infty\), and for all \(x,y\in G\), \(\|Ax-Ay\|\leq\|x-y\|-\psi(\|x-y\|)\).
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.

Reviewer: Tulsi Das Narang (Amritsar)

MSC:

47J25	Iterative procedures involving nonlinear operators
47H06	Nonlinear accretive operators, dissipative operators, etc.
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10	Fixed-point theorems

Keywords:

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