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Urgent: Existence and Uniqueness theorem

  1. Oct 1, 2006 #1
    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.
     
  2. jcsd
  3. Oct 1, 2006 #2

    AKG

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  4. Oct 1, 2006 #3
    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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