A remark on solving large systems of equations in function spaces. (English) Zbl 0759.65033

The authors consider a system \(x_ i=T_ i(x_ i)\), \(i=1,\dots,N\), where \(T_ i\), \(i=1,\dots,N\) is supposed to be a splitting of a nonlinear operator \(T\) mapping a complete metric space \((X,d)\) into itself. \(T_ i\) operates on a complete metric space \((X_ i,d_ i)\) with values in \((X,d)\). The above system is supposed to be equivalent to the fixed point equation \(y=T(y)\).
Assuming \(d_ i(T_ i(x),T_ i(\tilde x))\leq \sum^ N_{j=1}k_{ij}d_ j(x_ j,\tilde x_ j)\), the authors are interested in the convergence of the iterative procedure \(x^ k=T(x^{k-1})\), \(k=1,2,\dots\) and they prove the following result:
Let \(K\) be the real matrix with the entries \(k_{ij}\). If the spectral radius \(\rho(K)<1\), then the authors construct a metric \(\hat d\) on \(X\) such that \(T\) is a uniform contraction on the whole space \((X,\hat d)\).
An application to a system consisting of an initial value problem and a system of algebraic equations closes the paper.
Reviewer: E.Bohl (Konstanz)


65J15 Numerical solutions to equations with nonlinear operators
65H10 Numerical computation of solutions to systems of equations
47J25 Iterative procedures involving nonlinear operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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