Singular Perturbations and a Theorem of Kisyriski*

where t > 0, E > 0 is a small parameter, and A is a nonnegative self-adjoint (not necessarily bounded) operator in H, converge, as E + 0, to the solution of (1.2). While we borrow Kisynski’s idea of using the functional calculus of the operator A in order to construct a solution of (1.3), our approach is different from Kisynski’s in that we do not employ the techniques of the theory of semigroups. Secondly, we shall show that, in general, one cannot expect higher order perturbations of (1.2) to converge to a solution of (1.2). To this end, we shall show the following: If H = R, , the real line, there is no dense subset D C R, for which the solutions of the Cauchy problem d’(t) + x’(t) + Ax(t) = 0, xyo) = xi ) i = 0, 1, 2,3, (1.4)