Vol. 48, No. 1, 1973

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Product integrals for an ordinary differential equation in a Banach space

David Lowell Lovelady

Vol. 48 (1973), No. 1, 163–168

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
Received: 7 June 1972
Published: 1 September 1973
David Lowell Lovelady