Regularization of degenerated equations and inequalities under explicit data parameterization (Q2784192)

Let \(X, Z\) be Banach spaces, \(D \subset X\) a linear manifold and \(Y\) a compact metric space. Denote \(P(y)=\{x \in D: F(x,y)=0 \}\). Let \(\Omega: X \to {\mathbb R}\) be a lower semicontinuous functional with compact level sets. Given \(z \in D\) consider the problem of finding the set \(M(y)=\text{argmin} \{\Omega (x-z): x \in P(y)\}\) and the approximating problem of finding the set \(\{R_{\delta}({\widetilde y}), S_{\delta} \}= \text{argmin} \{\Omega(x-z): x \in P(y), \|y -{\widetilde y} \|\leq \delta \}\). The main result says that if the system \(F(x,y^0)=0, x \in D\) is consistent then \(R_{\delta}({\widetilde y})\) is compact and \(\lim_{\delta \to 0} \sup \{\beta (R_{\delta}({\widetilde y}), M(y^0)): \|y^0- {\widetilde y} \|\leq \delta \}=0\), where \(\beta (A,B)=\sup \{\inf \{\|a-b \|: b \in B \}: a \in A \}\).