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An iterative process for nonlinear Lipschitzian strongly accretive mappings in $$L_ p$$ spaces. (English) Zbl 0724.65058
A very accurate description of the content of the paper under review is given in the author’s abstract:
“Suppose $$X=L_ p$$ (or $$l_ p)$$, $$p\geq 2$$. Let $$T: X\to X$$ be a Lipschitzian and strongly accretive map with constant $$k\in (0,1)$$ and Lipschitz constant L. Define $$S: X\to X$$ by $$Sx=f-Tx-x$$. Let $$\{C_ n\}^{\infty}_{n=1}$$ be a real sequence satisfying:
(i) $$0<C_ n\leq k[(p-1)L^ 2+2k-1]^{-1}$$ for each n,
(ii) $$\sum^{\infty}_{n=0}C_ n=\infty.$$
Then, for arbitrary $$x_ 0\in X$$, the sequence $x_{n+1}=(1-C_ n)x_ n+C_ nSx_ n,\;n\geq 0,$ converges strongly to the unique solution of $$Tx=f$$. Moreover, if $$C_ n=k[(p-1)L^ 2+2k-1]^{-1}$$ for each n, $$then$$
$$\| x_{n+1}-q\| \leq \rho^{n/2}\| x_ 1-q\|$$, where q denotes the solution of $$Tx=f$$ and $\rho =(1-k[(p-1)L^ 2+2k-1]^{- 1})\in (0,1).$ A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X.”
Reviewer: I.Marek (Praha)

##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 47H06 Nonlinear accretive operators, dissipative operators, etc. 47J05 Equations involving nonlinear operators (general) 47J25 Iterative procedures involving nonlinear operators
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