Product integrals for an ordinary differential equation in a Banach space

David Lowell Lovelady

Vol. 48 (1973), No. 1, 163–168

DOI: 10.2140/pjm.1973.48.163

Abstract

Let Y be a Banach space with norm ∥ ∥, and let R⁺ be the interval [0,∞). Let A be a function on R⁺ having the properties that if t is in R⁺ then A(t) is a function from Y to Y and that the function from R⁺ ×Y to Y described by (t,x) → A(t)[x] is continuous. Suppose there is a continuous real-valued function α on R⁺ such that if t is in R⁺ then A(t) −α(t)I is dissipative. Now it is known that if z is in Y , the differential equation u′(t) = A(t)[u(t)]; u(0) = z has exactly one solution on R⁺. It is shown in this paper that if t is in R⁺ then u(t) = 0∏ ^t exp[(ds)A(s)][z] = 0∏ ^t[I − (ds)A(s)]⁻¹[z], where the exponentials are defined by the solutions of the associated family of autonomous equations.

Mathematical Subject Classification

Primary: 34G05

Milestones

Received: 7 June 1972

Published: 1 September 1973