A regularized dynamical system method for nonlinear ill-posed Hammerstein type operator equations (Q2895088)

The authors consider nonlinear ill-posed Hammerstein type operator equations \(KF(x)=y\), where \(F:D(F)\subset X \to Z\) is nonlinear and \(K: Z \to Y\) is a bounded linear operator, and \(X\), \(Y\), \(Z\) are Hilbert spaces. Instead of \(y \in Y\) an approximation \(y^{\delta}\) is available, \(\| y^{\delta}-y\| \leq \delta\). The aim of the paper is to analyze the following solution method for the original equation: \(z_{\alpha}^{\delta}=(K^*K+\alpha I)^{-1} K^* y^{\delta}\), \(x^{\prime}(t)=-(F^{\prime}(x_0)+\varepsilon(t)I)^{-1} (F(x)-z_{\alpha}^{\delta})\), \(x(0)=x_0\); \(\alpha=\alpha(\delta)>0\), \(0\leq \varepsilon(t) \leq K\).