A certain ill-posed Cauchy problem for first-order and second-order differential equations with operator coefficients (Q1920784)

This article deals with the Cauchy problem for the ordinary differential equation \(Bu_t'=Lu\), \(Bu''_{tt}=Lu\) in a Hilbert space \(H\) with a selfadjoint and positively definite operator \(L\) with a dense domain and a selfadjoint isomorphism \(U\) of \(H\). The basic results are some ``energy'' inequalities for generalized solutions of the problem and a natural method of constructing approximate solutions to given exact solutions; some error estimates are also offered.