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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
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