| Popis: |
by the method of product integration. We will prove an existence and uniqueness theorem and related results for solutions of (0.2) when A is a (time dependent, possibly non-linear) vector field on a Banach space, satisfying a Lipschitz condition in the “space” variable and an integrability condition in the “time” variable. These results are, of course, not new, and can be proved in the usual way by iteration of Eq. (0.2); however, the use of product integration in the generality presented here will, it is hoped, be of sufficient novelty to justify the present work. To give an idea of our method, we first recall (see [ 1 ] for a discussion of the finite-dimensional case) that if the vector field A satisfies appropriate continuity conditions in the time variable, then the solution x(t) of (0.1) can be obtained by associating with partitions |
