Uniqueness and approximation theorems for a degenerate operator-differential equation (Q1387371)

Of concern is the abstract Cauchy problem of higher order \[ \sum^n_{j= 0} A_ju^{(j)}(t)= 0,\quad t>0,\quad u^{(j)}(0)= u_j,\quad j=0,\dots,n- 1 \] in Banach spaces. The operator \(A_n\) is not necessarily invertible. The authors discuss uniqueness, completeness of eigenfunctions in the solution subspace, and some examples from partial differential equations. The abstract case is examined via estimates for the operators \(R(\lambda)= L^{-1}(\lambda)\), \(R_j(\lambda)= R(\lambda)A_j\), where \(L(\lambda)= \sum^n_{j= 0}\lambda^k A_j\), using Laplace transform techniques and the Phragmen-Lindelöf principle.