Unformatted text preview: system. De nition 1.2.2. The IVP (1.1.2) is well-posed if there exists positive constants k and
^ such that, for any ^, the perturbed IVP z = f (t z) + (t) z(0) = y0 + 0 0 (1.2.5) satis es
jy (t) ; z (t)j k (1.2.6) whenever j 0 j < and j (t)j < for t 2 0 T ]. Again, a Lipschitz condition ensures that we are dealing with a well-posed IVP. Theorem 1.2.2. If f (t y) satis es a Lipschitz condition on
f(t y) j 0 t T ;1 < y < 1g then y = f (t y ) is well posed on 0 T ] with respect to all initial data.
0 Proof. Let (t) = z(t) ; y(t)
and subtract (1.1.2) from (1.2.5) to obtain
0 (t) = f (t z) ; f (t y) + (t) (0) = 0 : Taking an absolute value and using the Lipschitz condition (1.2.4)
j 0 (t)j Lj (t)j + j (t)j
10 j (0)j = 0 : We easily see that j (t)j j (t)j provided that j (t)j exists and it is fairly easy to show
that j (t)j exists. Additionally, by assumption,
0 0 0 0 max j (t)j <
tT mtax j 0 j <
T 0 0 so
(t)j j 0 Lj (t)j + Multiply by the integrating factor e
(e Lt j ; Lt ; j (0)j < : j (0)j < : to obtain (t)j) e 0 Lt ; Integrating L (L + 1)eLt ; 1] :
Since the denominator is smallest when t = 0
j (t)j jz (t) ; y (t)j L2 = k 8t 2 0 T] thus, y = f (t y) is well posed on 0 T ].
0 The notion of a well-posed system is related to the more common notion of stability
as indicated by the following de n...
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- Spring '14
- Boundary value problem, lipschitz condition, IVPs, di erential equations