Local convergence of generalized Mann iteration
The article deals with generalized Mann iterations \[ x_{n+1} = (I - D_n)x_n + D_nT(x_n), \qquad n = 0,1,2,\ldots,\eqno(1) \] for the approximative construction of fixed points of a nonlinear operator \(T:\;C \to H\), where \(H\) is a real Hilbert space, \(C\) an open subset of \(H\), \(\{D_n\} \subset L(H)\) is a generalized control sequence. Let \(\text{Fix}\, T \neq \emptyset\). The main result is the following: the sequence (1), with starting point in the ball \(B_r \subset C\), remains in \(B_r\) and converges weakly to a fixed point of \(T\), provided that: {\parindent=0.7cm \begin{itemize}\item[(i)] \(T\) is demiclosed at zero on \(B_r \subset C\); \item[(ii)] \(D_x\) is invertible and \(\|D_x^{-1}\| \leq M\) (\(x \in B_r\)); \item[(iii)] \(T\) is generalized demicontractive: \[ \langle D_x(x - T(x)),x - p \rangle \geq \lambda \|D_x(x - T(x))\|^2, \qquad x \in B_r,\quad p \in \operatorname{Fix}T; \] \end{itemize}} with \(\lambda > 0.5\) (\(D_x\) is the control linear mapping, \(D:\;C \to L(C)\)); \noindent moreover, the sequence (1) converges strongly to the unique fixed point \(p\) of \(T\) if, in addition, \(T\) is quasi-expansive on \(B_r\): \[ \|Tx - p\| \geq \frac{1 - \rho}{\rho} k\|x - p\|, \qquad x \in B_r,\quad p \in \operatorname{Fix}T. \] At the end of the article, some numerical experiments are presented. There are some vague places in the article. In particular, the relations between \(D_n\) and \(D_x\) is not explained; the definition of the constant \(\rho\) is incomprehensible.
scientific article; zbMATH DE number 6825425
scientific article