In a recent paper, Kisynski
studied the solutions of the abstract Cauchy problem 𝜖x⋅⋅(t) + x⋅(t) + Ax(t) = 0,
x(0) = x0 and x⋅(0) = x1 where 0 ≦ t ≦ T,𝜖 > 0 is small parameter and A is a
nonnegative self-adjoint operator in a Hilbert space H. With the aid of the functional
calculus of the operator A, he has showed that as 𝜖 → 0 the solution of
this problem converges to the solution of the unperturbed Cauchy problem
x⋅(t) + Ax(t) = 0, x(0) = x0. Smoller has proved the same result for equation of
higher order.
The purpose of this paper is to study the solution of a similar problem and
allowing the operator A to depend on t.
