On the rates of asymptotic regularity for some unbounded trajectories
The author considers a nonexpansive self-mapping \(T:C\mapsto C\) of a closed convex subset \(C\) of some Banach space. Let \(T_\lambda=\lambda T+(1-\lambda)I\) be the averaged mapping where \(0<\lambda<1\), \(I\) is the identity operator. Then the iterative sequence \(x_n=T_\lambda^n x_0\) is exactly the Euler approximation to the initial value problem \[ {du\over dt}=-(I-T)u, \quad u(0)=x_0\in C \] with stepsize \(\lambda\). The facts that \(\|x_n-Tx_n\|=O(n^{-1/2})\) as \(n\to\infty\) and \(\|u'(t)\|=O(t^{-1/2})\) as \(t\to\infty\) are known for the bounded \(C\). The author gives a generalization of the second result for the case when the set \(C\) may be unbounded. Namely he assumes that \(\|u(t)\|=O(t^\alpha)\), \(0\leq\alpha\leq 1\) and proves that \(\|u'(t)\|=O(t^{-\beta})\) as \(t\to\infty\), where \(\alpha+2\beta=1\). The corresponding estimate is obtained with universal constant depending only on \(\alpha\).
scientific article; zbMATH DE number 1417613