Linear Algebra, Theory And Applications, 2012a

430 APPLICATIONS TO DIFFERENTIAL EQUATIONS
Proof: Let
{∣ ∣ }
∣∣Pk ∣∣
M ≡ max (A) ij for all i, j, k .
Then from Putzer’s formula for Φ (t) and Lemma C.5.1, there exists a constant, C such that
i=1
∣ n−1
∣ ∣∣
∑
∣Φ(t) ij ≤ Ce −δt M.
j=1
k=0
Let the new C be given by nCM.
Next,
⎛
⎞2
n∑ n∑
|Φ(t) x| 2 ≡ ⎝ Φ ij (t) x j
⎠ ≤
≤
⎛
⎞
n∑ n∑
⎝ Ce −δt |x| ⎠
i=1
j=1
This proves (3.30) and completes the proof.
2
⎛
⎞
n∑ n∑
⎝ |Φ ij (t)||x j | ⎠
i=1
j=1
∑ n
= C 2 e −2δt (n |x|) 2 = C 2 e −2δt n 3 |x| 2
Definition C.5.4 Let f : U → R n where U is an open subset of R n such that a ∈ U and
f (a) =0. A point, a where f (a) =0 is called an equilibrium point. Then a is asymptotically
stable if for any ε>0 there exists r>0 such that whenever |x 0 − a| 0 such that if |x 0 − a|