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An iterative process for nonlinear Lipschitzian strongly accretive mappings in $L\sb p$ 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 $X=L\sb p$ (or $l\sb p)$, $p\ge 2$. 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 $\{C\sb n\}\sp{\infty}\sb{n=1}$ be a real sequence satisfying: (i) $0<C\sb n\le k[(p-1)L\sp 2+2k-1]\sp{-1}$ for each n, (ii) $\sum\sp{\infty}\sb{n=0}C\sb n=\infty.$ Then, for arbitrary $x\sb 0\in X$, the sequence $$ x\sb{n+1}=(1-C\sb n)x\sb n+C\sb nSx\sb n,\ n\ge 0, $$ converges strongly to the unique solution of $Tx=f$. Moreover, if $C\sb n=k[(p-1)L\sp 2+2k-1]\sp{-1}$ for each n, $then$ $\Vert x\sb{n+1}-q\Vert \le \rho\sp{n/2}\Vert x\sb 1-q\Vert$, where q denotes the solution of $Tx=f$ and $$ \rho =(1-k[(p-1)L\sp 2+2k-1]\sp{- 1})\in (0,1).$$ A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in X.”
Reviewer: I.Marek (Praha)

65J15Equations with nonlinear operators (numerical methods)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47J05Equations involving nonlinear operators (general)
47J25Iterative procedures (nonlinear operator equations)
Full Text: DOI
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