Properties of solutions of equations containing powers of an unbounded operator

The Cauchy problem u"(t) + (k/t)u'(t) = (-1)(m+1)A(m)u(t), t > 0, (1) u(0) = u(0), u'(0) = 0, (2) for the Euler-Poisson-Darboux equation in a Banach space E was considered in [1]. Here k greater than or equal to 0, m is an element of N, u(0) is an element of D (A(m)), and A is a linear closed operator such that the problem u"(t) + (p/t)u'(t) = Au(t), u(0) = u(0), u'(0) = 0, (3) is uniformly well-posed for some p greater than or equal to 0. We denote the set of such operators A by G(p) and the resolving operator of problem (3) (the Bessel operator function) by Y-p(t, A). Thus if A is an element of G(p), then problem (3) has a solution, which is unique and continuously depends oil the initial data; moreover, u(t) = Y-p(t, A)u(0) and \\Y-p (t, A)\\ less than or equal to M-0 exp (omegat), M-0 greater than or equal to 1, omega greater than or equal to 0. (4) A criterion for the well-posedness of problem (3) and the properties of the Bessel operator function Y-p(t, A) were studied in [2]. Note that a solution of the differential equation is defined as a function that is differentiable appropriately many times, belongs to the domain of the corresponding power of A, and makes the differential equation an identity. In the present paper, we establish relationships between solutions of a number of equations containing powers of the unbounded operator A, derive integral representations of solutions of these equations, and indicate some applications; in particular, we study the stabilization issues and present the quasi-inversion method in a Banach space.