AN EXPRESSION FOR THE SOLUTION OF A DIFFERENTIAL EQUATION IN TERMS OF ITERATES OF DIFFERENTIAL OPERATORS

Abstract

We obtain theorems on the expression of A−1 in terms of iterates of the operator A, which is the reproducing operator of a 1-parameter group of linear transformations of a Banach space, and whose spectrum does not surround 0. These results are applied to first order differential equations with analytic coefficients and right-hand sides (symmetric first order systems on a compact manifold without boundary), and to second order elliptic equations (equations with a real principal part on a manifold without boundary, selfadjoint equations degenerate on the boundary of the domain, and the Dirichlet problem for a selfadjoint equation in a domain with an analytic boundary). We obtain formulas expressing the value of the solution at a point in terms of the derivatives of the coefficients and the right-hand side at this point. Bibliography: 9 titles.