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The rate of asymptotic regularity is \(O(1/n)\). (English) Zbl 0865.47038

Kartsatos, Athanassios G. (ed.), Theory and applications of nonlinear operators of accretive and monotone type. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 178, 51-81 (1996).
Let \(E\) be a real normed space and \(T:E\supset D(T)\to E\) be a nonexpansive mapping with domain \(D(T)\) and range \(E\) (i.e. \(|Tx-Ty|\leq|x-y|\) for all \(x,y\in D(T)\)). By an averaged mapping we mean a mapping of the form \(T_\lambda= \lambda T+(1-\lambda)I\), where \(\lambda\in(0,1)\) and \(I\) is the identity operator. In the present paper the authors introduce a new “symmetric method” for estimating the asymptotics of \(|T_\lambda^{m+1}x- T^m_\lambda x|\) as \(m\to+\infty\).
The main result of this paper is the following: Put \(R=T_{1/2}\). If \(|x-TR^kx|\leq 1\) for all \(0\leq k<m\), then \[ |R^mx-R^{m-1}x|\leq 4^{-m}{{2m}\choose m}<{1\over {\sqrt{\pi m}}}. \] More general, for any \(0<\lambda<1\), if \(|x-TT^k_\lambda x|\leq 1\) for \(0\leq k\leq m\), then \[ |T^m_\lambda x-TT^m_\lambda x|\leq {2\over\pi}\int^1_0 (1-t)^{1/2} t^{-1/2}(1-4t\lambda (1-\lambda))^m dt<{1\over {\sqrt{\lambda(1-\lambda)\pi m}}}. \] Finally, if \(T\) is the resolvent \((I+A)^{-1}\) of an accretive operator \(A\) and \(|x-T^kx|\leq 1\) for \(0\leq k\leq m+1\), then \[ |T^{m+1}x-T^mx|\leq 4^{-m} {{2m}\choose m}<{1\over {\sqrt{\pi m}}}. \] Some unsolved problems associated with iterating averaged mappings and firmly nonexpansive mappings are given.
For the entire collection see [Zbl 0840.00034].

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H06 Nonlinear accretive operators, dissipative operators, etc.
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