On approximate solution of nonlinear operator equations
Let \(A\) be an operator with domain \(D(A)\) and range \(R(A)\) which acts in a Hilbert space \(H\) and, for \(f=f_0\), there is an exact solution \(u_0\) to the operator equation of the first kind \[ Au = f, \] i.e., \(f_0\in R(A)\), but, instead of an exact value \(f_0\) of the right-hand side, only an approximate value \(f_{\delta}\) and an error \(\delta\) are given, \(\| f_{\delta} - f_0\| \leq\delta\). The problem reads: given \(f_{\delta}\), \(\delta >0\), construct an approximate solution to the equation ``close'' to the set \(M_0\) of exact solutions. The author solves this problem and exposes necessary and sufficient conditions on the operator \(A\) for convergence of the approximate solutions obtained by the regularization method.
scientific article; zbMATH DE number 1286347