Equações diferenciais

Prévia do material em texto

606 Chapter 19. Ordinary Di�erential Equations
19.3 Carathéodory’s Type ODE
19.3.1 Main definitions
The di�erential equation
{˙ (w) = i (w> { (w)) > w � w0 (19.107)
in the regular case (with continuous right-hand side in both variables)
is known to be equivalent to the integral equation
{ (w) = { (w0) +
wZ
v=w0
i (v> { (v)) gv (19.108)
Definition 19.7 If the function i (w> {) is discontinuous in w and
continuous in { 5 Rq, then the functions { (w), satisfying the inte-
gral equation (19.108) where the integral is understood in the Lebesgue
sense, is called solutions ODE (19.107).
The material presented bellow follows (Filippov 1988).
Let us define more exactly the conditions which the function i (w> {)
should satisfy.
Condition 19.1 (Carathéodory’s conditions) Let in the domain
D of the (w> {)-space the following conditions be fulfilled:
1) the function i (w> {) be defined and continuous in { for almost
all w;
2) the function i (w> {) be measurable (see (15.97)) in w for each {;
3)
ki (w> {)k � p (w) (19.109)
where the function p (w) is summable (integrable in the Lebesgue
sense) on each finite interval (if w is unbounded in the domain
D).
19.3. Carathéodory’s Type ODE 607
Definition 19.8
a) The equation (19.107), where the function i (w> {) satisfies the con-
ditions 19.1, is called the Carathéodory’s type ODE.
b) A function { (w), defined on an open or closed interval o, is called
a solution of the Carathéodory’s type ODE if
- it is absolutely continuous on each interval [�> �] 5 o;
- it satisfies almost everywhere this equation or, which under the
conditions 19.1 is the same thing, satisfies the integral equation
(19.108).
19.3.2 Existence and uniqueness theorems
Theorem 19.11 ((Filippov 1988)) For w 5 [w0> w0 + d] and
{ : k{� {0k � e let the function i (w> {) satisfies the Carathéodory’s
conditions 19.1. Then on a closed interval [w0> w0 + g] there exists a
solution of the Cauchy’s problem
{˙ (w) = i (w> { (w)) > { (w0) = {0 (19.110)
In this case one can take an arbitrary number g such that
0 ? g � d> * (w0 + g) � e where * (w) :=
wZ
v=w0
p (v) gv (19.111)
(p (w) is from (19.109)).
Proof. For integer n � 1 define k := g@n, and on the intervals
[w0 + lk> w0 + (l+ 1)k] (l = 1> 2> ===> n) construct iteratively an approx-
imate solution {n (w) as
{n (w) := {0 +
wZ
v=w0
i (v> {n31 (v)) gv (w0 ? w � w0 + g) (19.112)
(for any initial approximation {0 (v) > for example, {0 (v) = const).
Remember that if i (w> {) satisfies the Carathéodory’s conditions 19.1