[For the entire collection see Zbl 0538.00034.]
For \(\epsilon\geq 0\), let \(L_{\epsilon}\) be linear operators between Banach spaces X and Y such that \(\dim N(L_{\epsilon})\) is a finite constant for \(\epsilon >0\) and changes at \(\epsilon =0\). If \(L_{\epsilon}\to L_ 0,\) \(f_{\epsilon}\to f_ 0\), solutions of \(L_{\epsilon}x=f_{\epsilon}\) need not converge as \(\epsilon\) \(\to 0\). In this paper, conditions are given under which convergence for \(\epsilon\) \(\to 0\) takes place. The results deals with two situations:
(1) Equations of the \(2^{nd}\) kind: \(L_{\epsilon}\) Fredholm, \(f_{\epsilon}\in R(L_{\epsilon})\), \(X=N(L_ 0)\dot +R(L_ 0).\) These results generalize those of H. W. Engl and R. Kress [Math. Methods Appl. Sci. 3, 249-274 (1981; Zbl 0476.47011)]; an extension to nonlinear problems can be found in H. Engl [Lect. Notes Math. 1107, 82-101 (1984)].
(2) Equations of the \(1^{st}\) kind: \(L_{\epsilon}\) compact. Since then the equations are ill-posed, the proof uses Tikhonov-regularization. As a by-product, a result about speed of convergence of Tikhonov regularization is obtained.