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An automatic iteration scheme and its application to nonlinear operator equations. (English) Zbl 0541.65042
Treatment of integral equations by numerical methods, Proc. Symp., Durham 1982, 69-78 (1982).
The paper is motivated by the idea that, for the approximation of a solution $$\bar x$$ of a continuous problem, a sequence of discrete problems $$x=T_ nx$$ with solutions $$\bar x_ n$$ frequently are solved. Normally not $$\bar x_ n$$ but an approximation $$y_ n$$ is obtained. If an inequality d(\=x,\=x$${}_ n)\leq C\rho^ n$$ for some metric d and some constants $$c\geq 0$$, $$\rho \in(0,1)$$ is known, the author studies the question of obtaining $$y_ n$$ such that d(\=x,y$${}_ n)\leq C\rho^ n$$. Then the $$y_ n$$ are of the same accuracy as the $$\bar x_ n$$. The basic assumption is that all the $$T_ n$$ are contractions so that the standard error estimate from the contraction mapping theorem is available for approximate solutions of $$T_ nx=x$$. An application to a non-linear integral equation is discussed.
Reviewer: E.Bohl
##### MSC:
 65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx) 65R20 Numerical methods for integral equations 65H10 Numerical computation of solutions to systems of equations