Well-posedness of a problem for a differential equation given values of a function and its derivatives at several points (Q1358058)

The author considers the following abstract differential equation set in a separable Hilbert space \(H\): Find \(U:[0,T]\to H\) such that \[ {d^nU\over dt^n}+ A_1 {d^{n-1}U\over dt^{n-1}}+\cdots+ A_nU=0,\tag{1} \] \[ U(t_1)= \Phi^{(1)}_0,\dots,\;U^{(k_1)}(t_1)= \Phi^{(1)}_{k_1},\dots,\;U(t_l)= \Phi^{(l)}_0,\dots,\;U^{(k_l)}(t_l)= \Phi^{(l)}_{k_l},\tag{2} \] \[ (0= t_1< t_2<\cdots< t_l= T;\;k_1+1+ k_2+1\cdots+ k_l+1=n). \] Here the coefficients \(A_i\) are all linear operators that are functions of some selfadjoint operator \(A\) on \(H\) and the data points \(\Phi^{(q)}_p\) are known elements of \(H\). Under suitable conditions, the problem (1), (2) is shown to be uniquely solvable and to depend continuously on the data \(\Phi^{(q)}_p\).