Page 61 - Textos de Matemática Vol. 36

2.5. Global existence results 51 As far as ∂Λ(σ,t) is continuous with respect to the first argument, then it is
∂s
bounded on any finite closed interval and we conclude that λ(t) is Lipschitzian (and absolutely continuous) on this interval.
To derive equality (2.45) at any point t0 of differentiability of λ(t), let us choose h > 0 and first estimate the difference
λ(t0 +h)−λ(t0)−x∗(t0)K˙(t0)x(t0)h=λ(t0 +h)−λ(t0)− ∂Λ(t0,t0)h. ∂s
Choosing t′ = t0, t′′ = t0 +h and utilizing the upper estimate of (2.46) we obtain ∂Λ t0+h ∂Λ ∂Λ
(∂s(σ,t0))− ∂s(t0,t0))dσ.
As long as σ → ∂Λ(σ,t0) is continuous, then the latter integral is o(h), as
λ(t0 +h)−λ(t0)− ∂s(t0,t0)h≤ ∂s
t0
h → +0 and we conclude dλ (t0) ≤ ∂Λ (t0, t0). Let us now consider the difference dt ∂s
λ(t0) − λ(t0 − h) − ∂Λ (t0, t0)h. Choosing t′ = t0 − h, t′′ = t0 and utilizing the ∂s
lower estimate of (2.46) we conclude
t0 ∂Λ ∂Λ ∂Λ
( ∂s (σ, t0) − ∂s (t0, t0))dσ ≤ λ(t0) − λ(t0 − h) − ∂s (t0, t0)h. t0 −h
From this estimate we derive dλ (t0) ≥ ∂Λ (t0, t0), arriving at the equality (2.45). dt ∂s

Notice that in general (see [Baum85]) there is needed analyticity of K(t) to
conclude differentiability (and clearly analyticity) of the eigenvalues and eigen-
projections. In that case (2.45) could be derived by differentiating
λ = x∗ (t)K (t)x(t), observing that d (x∗ (t)x(t)) = 2x∗ (t)x˙ (t) = 0. This can dt
become false even if K(t) is infinitely differentiable. (See for instance an exam- ple in [Kato66], p.111).
This theorem will be used now to obtain several results on invariant do- mains for solutions of Riccati differential equations, which, indeed, yield global existence for such solutions starting in such an invariant domain. First results on global existence for solutions of non-symmetric RDE were obtained by Redheffer [Redh59] and Reid [Reid60]. Redheffer and Volkmann (see [Redh75], [ReVo96] and papers of these authors cited therein) obtained, as far as we know for the first time, conditions for the existence of an invariant ball or invariant cone for operator differential equations, which include RDE as a special case; [Walc01] contains an alternative approach and some applications.