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Zbl 0606.47067
Chidume, C.E.
The iterative solution of the equation $f\in x+Tx$ for a monotone operator T in $L\sp p$ spaces.
(English)
[J] J. Math. Anal. Appl. 116, 531-537 (1986). ISSN 0022-247X

Suppose T is a multivalued monotone operator (not necessarily continuous) with open domain D(T) in $L\sb p$ $(2\le p<\infty)$, $f\in R(I+T)$ and the equation $f\in x+Tx$ has a solution $q\in D(T)$. Then there exists a neighbourhood $B\subset D(T)$ of q and a real number $r\sb 1>0$ such that for any $r\ge r\sb 1$, for any initial guess $x\sb 1\in B$, and any single-valued section $T\sb 0$ of T, the sequence $\{x\sb n\}\sp{\infty}\sb{n=1}$ generated from $x\sb 1$ by $x\sb{n+1}=(1-C\sb n)x\sb n+C\sb n(f-T\sb 0x\sb n)$ remains in D(T) and converges strongly to q with $\Vert x\sb n-q\Vert =O(n\sp{-})$. Furthermore, for $X=L\sb p(E)$, $\mu (E)<\infty$, $\mu =Lebesgue$ measure and $1<p<2$, suppose T is a single-valued locally Lipschitzian monotone operator with open domain D(T) in X. For $f\in R(I+T)$, a solution of the equation $x+Tx=f$ is obtained as the limit of an iteratively constructed sequence with an explicit error estimate.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H06 Accretive operators, etc. (nonlinear)
65J15 Equations with nonlinear operators (numerical methods)

Keywords: single-valued locally Lipschitzian monotone operator with open domain; iteratively constructed sequence; explicit error estimate

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