ON SOLVABILITY AND WELL-POSEDNESS OF (N + 1)-TIMES INTEGRATED CAUCHY PROBLEM

Abstract

For a closed operator $A$ in a Banach space $X$, the $(n+1)$-times integrated Cauchy problem $C_{n+1}[\tau], \ 0 < \tau < \infty,$ of finding a solution $v(t)$ of the problem $v'(t) = Av(t) + \frac{t^{n}}{n!}x, v(0) = 0, (t \in [0, \tau], x \in X)$ is considered. In the case where the operator $A$ is normal in a Hilbert space, all its solutions are described. The necessary and sufficient conditions on the spectrum of $A$ under which this problem is well-posed are established.