Continuous dependence for an implicit nonlinear equation
Consider the following problem: (1) \(v'(t)+ w(t)+ C(t)u(t)= f(t)\) for a.e. \(t\in (0,T)\), (2) \(v(t)\in A(t)u(t)\), \(w(t)\in B(t)u(t)\) for a.e. \(t\in (0,T)\), (3) \(v(0)= v_ 0\), where \(A(t)\) and \(B(t)\) are maximal monotone and linearly bounded operators in a real Hilbert space \(V\) and \(A(t)+ B(t)\) is \(V\)-coercive. Moreover, it is assumed that \(A(t)\) is the subdifferential of a continuous convex function on a real Banach space \(W\) and there exists a compact injection of \(V\) in \(W\). Apart from this \(C(t): V\to W^*\) is Lipschitzian. The object of the paper under review is the study of the continuous dependence of solutions of (1)--(3) on \(A\), \(B\), \(C\), \(f\) and \(v_ 0\). The results obtained are applied to control theory.
scientific article; zbMATH DE number 599885