Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces
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.
scientific article; zbMATH DE number 1883745