Condition of completeness of elementary solutions for degenerate higher order operator differential equations

In a Banach space, the higher order abstract differential equation $\sum\limits_{j=0}^nA_ju^{(j)}(t)=0$ is investigated. The operator $A_n$ in the higher derivative may be degenerate. Conditions of approximation of solutions by linear combinations of elementary solutions have been obtained. Abstract results are applied to partial differential equations.