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