Strongly P-Positive Operators and Explicit Representations of the Solutions of Initial Value Problems for Second-Order Differential Equations in Banach Space

The initial value problems for two second-order differential equations with an unbounded operator coefficient A in a Banach space are considered. Using a linear fractional transform the Cayley transform.of the operator A we give explicit formulas for the solution of these problems if the spectrum of A is situated inside of a parabola. These formulas also provide the algorithmic representations of the operator cosine-function and of the operator Bessel-function with the generator A being a basis for approximate solutions for which error estimates are given. One of the important properties of our approach is the following: the accuracy of the approximate solutions depends automatically on the regularity of the initial data.