Vol. 16, No. 2, 1966

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Differentiability of solutions of ordinary differential equations in Hilbert space

Avner Friedman

Vol. 16 (1966), No. 2, 267–271
Abstract

Consider the differential equation

1du-− A(t)u = f(t) (a < t < b)
idt
(1.1)

where u(t), f(t) are elements of a Hilbert space E and A(t) is a closed linear operator in E with a domain D(A) independent of t and dense in E. Denote by Cm(a,b) the set of functions v(t) with values in E which have m strongly continuous derivatives in (a,b). Introducing the norm

       ∑m ∫ b (j)  2   1∕2
|v|m = {    a |v  (t)|dt}
j=0
(1.2)

where |v(t)| is the E-norm of v(t), we denote by Hm(a,b) the completion with respect to the norm (1.2) of the subset of functions in Cm(a,b) whose norm is finite. Set Hm = Hm(−∞,) and denote by H0m the subset of functions in Hm which have compact support. The solutions u(t) of (1.1) are understood in the sense that u(t) H1(a,b) for any a < a< b< b.

Theorem 1. Assume that, for each a < t < b, the resolvent R(λ,A(t)) = (λA(t))1 of A(t) exists for all real λ, |λ|N(t), and that

            C(t)
|R (λ,A (t))| ≦ |λ| if λ real ,|λ| ≧ N (t),
(1.3)

where N(t), C(t) are constants. Assume next that for each s (a,b), A1(s) exists and

A (t)A− 1(s) has m uniformly continuous t-derivatives,
(1.4)

for a < t < b, where m is any integer 1. If u is a solution of (1.1) and if f Hm(a,b), then u Hm+1(a,b) for any a < a< b< b.

Theorem 2. If the assumptions of Theorem 1 hold with m = , if A(t)A1(s) is analytic in t(a < t < b) for each s (a,b), and if f(t) is analytic in (a,b), then u(t) is also analytic in (a,b).

Mathematical Subject Classification
Primary: 34.95
Milestones
Received: 1 May 1964
Published: 1 February 1966
Authors
Avner Friedman