On existence of analytic solutions of (k+1)-times integrable Cauchy problem
by
Volodymyr M. Gorbachuk
National Technical University of Ukraine (Kyiv Polytechnic Institute)

Let A be a closed linear operator on a Banach space B. We say that a vector x in \cap _{n in N0}D(Aⁿ),  N₀ = {0, 1, 2, ...}, is of finite order if for sufficiently large n in N₀ there exists \gamma in R¹ such that

||Anx|| <= nn\gamma.

The greatest lower bound \beta = \beta(x) of such \gamma is called the order of x. By the type of a vector x of order \beta we mean the number

\alpha = \alpha(x) = inf {\delta > 0: ||Anx|| <= \deltannn\beta,  n in N0  is sufficiently large}.

Let 0 < \tau <= \infty. Consider the following Cauchy problem:

Ck+1(\tau)     ì ï ï í ï ï î y in C([0, \tau), D(A)) \cap C1([0, \tau), B), y'(t) = Ay(t) + \fractkk!x,     t in [0, \tau), y(0) = 0

(k in N₀ is fixed). The problem of finding a solution y(t) of C_k+1(\tau) is called k+1-times integrated Cauchy problem [1]. The problem C_k+1(\tau) is well posed if for any x in B there exists a unique solution of this problem.

We discuss the conditions on a vector x under which the problem C_k+1(\tau) has a solution analytic in a neighborhood of zero, and if this is the case, when such a solution can be extended to an entire vector-function of finite order and type. We show that the following assertion holds.

Theorem. The Cauchy problem C_k+1(\tau) has a solution analytic in a neighborhood of zero if and only if \beta(x) <= 1. In order that the solution admit an extension to an entire vector-function, it is necessary and sufficient that one of the below conditions be satisfied:

(i)  \beta(x) < 1;

(ii)  \beta(x) = 1,  \alpha(x) = 0.

The following relationship between the order \rho and type \sigma of an entire solution y(t) of C_k+1(\tau) and the order \beta and type \alpha of the vector x holds:

\rho = \frac11 - \beta,     \sigma = \frac(\alphae)\rho\rhoe.

It is not hard to give an example of the operator A such that for any \tau the problem C_k+1(\tau) is well posed, but there is no nontrivial solution analytic in a neighborhood of zero. We prove that in the case where A is a normal operator on a Hilbert space, the problem C_k+1(\tau) has solutions in the class of entire vector-functions of order \rho <= 1, and the set of such solutions is dense in the set of all solutions. If A is the generator of an analytic semigroup of bounded linear operators on B, then the set of all entire solutions of the problem C_k+1(\tau) is also dense in the set of all its solutions.

#1 W. Arendt, O. El-Mennaoui, and V. Keyantuo, Local integrated semigroups: evolution with jumps og regularity, J. Math. Anal. Appl. 186 (1994), 572-595.

Date received: May 31, 2000