Capitolul 1 Ecuatii diferentiale de ordinul ˆıntâi rezolvabile prin ...

Proprietǎt¸i calitative ale solut¸iilor 123
||X(t; t0, X 0 , µ) − X(t; t0, X 0 , µ 0 )|| ≤
≤
≤
≤
t
t0
t0
t0
t
t
≤ K ·
Contradict¸ie.
Prin urmare:
||F(τ, X(τ; t0, X 0 , µ), µ)−F(τ, X(τ; t0, X 0 , µ 0 ), µ 0 )||dτ ≤
||F(τ, X(τ; t0, X 0 , µ), µ)−F(τ, X(τ; t0, X 0 , µ 0 ), µ)||dτ ≤
||F(τ, X(τ; t0, X 0 , µ 0 ), µ)−F(τ, X(τ; t0, X 0 , µ 0 ), µ 0 )||dτ ≤
t
||X(τ; t0X 0 , µ)−X(τ; t0, X 0 , µ 0 )||dτ+ε·2 −1 ·e −K·h ≤
t0
≤ ε·2 −1 ·e −K·h ·e K(t−t0) ε
<
2 .
||X(t; t0, X 0 , µ) − X(t; t0, X 0 , µ 0 )|| < ε
2 , (∀)t ∈ I∗ ∩ Iµ.
Vom arǎta în continuare cǎ α = inf Iµ ≤ T1 ¸si β = sup Iµ ≥ T2. Rat¸ionǎm
prin reducere la absurd presupunând de exemplu β < T2.
Pentru orice t ∈ [t0, β) are loc inegalitatea:
¸si pentru t ′ , t ′′ ∈ [t0, β] avem:
cu
||X(t; t0, X 0 , µ) − X(t; t0, X 0 , µ 0 )|| < ε
2
||X(t ′ ; t0, X 0 , µ) − X(t ′′ ; t0, X 0 , µ)|| < M · |t ′ − t ′′ |
M = sup ||F(t, X, µ)||
∆×S