Urgent: Existence and Uniqueness theorem

  • Thread starter MT20
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Greating my friends,

I have just returned home today from heart surgery.

I still feeling the effects of the operation, because I'm affried the hospital send me home a bit to early. But I have to have these questions finished before tomorrow.

So therefore I would very much appriciate if somebody could help me answer these questions?

(a)

(I use the triangle inequality in (1) and (3)?)

Let I be a open interval and f: I \rightarrow \mathbb{R}^n be a continious function.

Let || \cdot || be a given norm on \mathbb{R}^n.

Show the following:

1) If there exists a C>0 then ||x|| \leq C ||C||_1; x \in \mathbb{R}^n, ||x|| _1 = \sum _{j=1} ^n |x_j|

2) The mapping I \ni t \rightarrow ||f(t)|| \in \mathbb{R} is continious on.

3) for all t1,t2 \in \ || \int_{t_1} ^{t_2} f(t) dt|| \leq | \int_{t_1} ^{t_2} ||f(t)||dt|


(b)

Looking at the system(*) of equations,

x1' = (a-bx2)* x1
x2' = (cx1 -d)*x2

open the open Quadrant K; here a,b,c and d er positive constants.

I need to show that the system can be integrated. Which means I need to show that there exist a C^1 -function, with the properties F:U \rightarrow \mathbb{R}, where U \subseteq K is open and close in K(close means that for every point in K, is a limit point for q sequence, whos elements belongs to K). Such that gradient F \notequal 0 for alle x \in U and such that F is constant for trajectories of system.

Finally I need to conclude that every max trajectory is contained in a compact subset of K, and that the system (*) field X: R \rightarrow \mathbb{R}^2.

Sincerley Yours
Maria Thomson 20

p.s. I'm so sorry for asking so much, but has been such a difficult couple of days for me.
 

Answers and Replies

  • #2
AKG
Science Advisor
Homework Helper
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I'm just going to copy and past your post, adding in TeX tags so I can read your questions:

Greating my friends,

I have just returned home today from heart surgery.

I still feeling the effects of the operation, because I'm affried the hospital send me home a bit to early. But I have to have these questions finished before tomorrow.

So therefore I would very much appriciate if somebody could help me answer these questions?

(a)

(I use the triangle inequality in (1) and (3)?)

Let I be a open interval and [itex]f: I \rightarrow \mathbb{R}^n[/itex] be a continious function.

Let [itex]|| \cdot ||[/itex] be a given norm on [itex]\mathbb{R}^n.[/itex]

Show the following:

1) If there exists a C>0 then [itex]||x|| \leq C ||C||_1; x \in \mathbb{R}^n, ||x|| _1 = \sum _{j=1} ^n |x_j|[/itex]

2) The mapping [itex]I \ni t \rightarrow ||f(t)|| \in \mathbb{R}[/itex] is continious on.

3) for all [itex]t_1,t_2 \in \ || \int_{t_1} ^{t_2} f(t) dt|| \leq | \int_{t_1} ^{t_2} ||f(t)||dt|[/itex]


(b)

Looking at the system(*) of equations,

x1' = (a-bx2)* x1
x2' = (cx1 -d)*x2

open the open Quadrant K; here a,b,c and d er positive constants.

I need to show that the system can be integrated. Which means I need to show that there exist a [itex]C^1[/itex] -function, with the properties [itex]F:U \rightarrow \mathbb{R}[/itex], where [itex]U \subseteq K[/itex] is open and close in K(close means that for every point in K, is a limit point for q sequence, whos elements belongs to K). Such that [itex]\nabla F \notequal 0[/itex] for alle [itex]x \in U[/itex] and such that F is constant for trajectories of system.

Finally I need to conclude that every max trajectory is contained in a compact subset of K, and that the system (*) field [itex]X: R \rightarrow \mathbb{R}^2[/itex].

Sincerley Yours
Maria Thomson 20

p.s. I'm so sorry for asking so much, but has been such a difficult couple of days for me.
 
  • #3
2
0
Hi

I'm sorry, but I have these answered within the next 4 hours, so if there somebody here who can help me I would be very greatful.

Sincerely Yours and God bless

MT
 

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