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An iterative process for nonlinear Lipschitzian and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces. (English) Zbl 0801.47040

Summary: Suppose $X$ is an $s$-uniformly smooth Banach space $\left(s>1\right)$. Let $T:X\to X$ be a Lipschitzian and strongly accretive map with constant $k\in \left(0,1\right)$ and Lipschitz constant $L$. Define $S$: $X\to X$ by $Sx=f-Tx+x$. For arbitrary ${x}_{0}\in X$, the sequence ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ is defined by

${x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}S{y}_{n},\phantom{\rule{1.em}{0ex}}{y}_{n}=\left(1-{\beta }_{n}\right){x}_{n}+{\beta }_{n}S{x}_{n},\phantom{\rule{4pt}{0ex}}n\ge 0,$

where ${\left\{{\alpha }_{n}\right\}}_{n=0}^{\infty }$, ${\left\{{\beta }_{n}\right\}}_{n=0}^{\infty }$ are two real sequences satisfying:

(i) $0\le {\alpha }_{n}^{p-1}\le {2}^{-1}s\left(k+k{\beta }_{n}-{L}^{2}{\beta }_{n}\right)$ ${\left(w+h\right)}^{-1}$ for each $n$,

(ii) $0\le {\beta }_{n}^{p-1}\le min\left\{k/{L}^{2},\phantom{\rule{4pt}{0ex}}sk/\left(w+h\right)\right\}$ for each $n$,

(iii) ${\sum }_{n}{\alpha }_{n}=\infty$,

where $w=b{\left(1+L\right)}^{s}$ and $b$ is the constant appearing in a characteristic inequality of $X$, $h=max\left\{1,s\left(s-1\right)/2\right\}$, $p=min\left\{2,s\right\}$. Then ${\left\{{x}_{n}\right\}}_{n=1}^{\infty }$ converges strongly to the unique solution of $Tx=f$. Moreover, if $p=2$, ${\alpha }_{n}={2}^{-1}s\left(k+k\beta -{L}^{2}\beta \right)$ ${\left(w+h\right)}^{-1}$, and ${\beta }_{n}=\beta$ for each $n$ and some $0\le \beta \le min\left\{k/{L}^{2},\phantom{\rule{4pt}{0ex}}sk/\left(w+h\right)\right\}$, then

$\parallel {x}_{n+1}-q\parallel \le {\rho }^{n/s}\parallel {x}_{1}-q\parallel ,$

where $q$ denotes the solution of $Tx=f$ and

$\rho =\left(1-{4}^{-1}{s}^{2}{\left(k+k\beta -{L}^{2}\beta \right)}^{2}{\left(w+h\right)}^{-1}\right)\in \left(0,1\right)·$

A related result deals with the iterative approximation of Lipschitz strongly pseudocontractive maps in $X$. Suppose $X$ is an $m$- uniformly convex Banach space $\left(m>1\right)$ and $c$ is the constant appearing in a characteristic inequality of $X$, two similar results are showed in the cases of $L$ satisfying $\left(1-{c}^{2}\right){\left(1+L\right)}^{m}<1+c-cm\left(1-k\right)$ or $\left(1-{c}^{2}\right){L}^{m}<1+c-cm\left(1-s\right)$.

##### MSC:
 47H06 Accretive operators, dissipative operators, etc. (nonlinear) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations)
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