An expression for the solution of a~differential equation in terms of iterates of differential operators

A. V. Babin

Sbornik. Mathematics, Tome 34 (1978) no. 4, pp. 411-424

Résumé

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.