Results 1 to 5 of 5

Math Help - Existence and uniqueness of global solutions

  1. #1
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    Posts
    481
    Thanks
    2

    Exclamation [SOLVED] Existence and uniqueness of global solutions

    Dear friends, I need some help with the existence and/or uniqueness of global solutions to first-order linear differential equations.
    For instance, let x_{0},t_{0}\in\mathbb{R} and A,B\in C([t_{0},\infty),\mathbb{R}) and consider the following differential equation
    <br />
\begin{cases}<br />
x^{\prime}(t)=A(t)x(t)+B(t),& t\geq t_{0}\\<br />
x(t_{0})=x_{0}.&<br />
\end{cases}<br />
    Which theorem ensures existence of global solutions to this initial value problem?

    I am actually wondering to find this result for delay differential equations
    <br />
\begin{cases}<br />
x^{\prime}(t)=A(t)x(\alpha(t))+B(t),& t\geq t_{0}\\<br />
x(t)=\varphi(t)&, t_{-1}\leq t\leq t_{0},<br />
\end{cases}<br />
    where t_{-1}:=\min\{\alpha(t):t\geq t_{0}\}, \varphi\in C([t_{-1},t_{0}],\mathbb{T}), \alpha(t)\leq t for all t\geq t_{0} and \lim\nolimits_{t\to\infty}\alpha(t)=\infty.


    Thanks for your help.
    Last edited by bkarpuz; September 19th 2009 at 03:27 AM. Reason: delay equation is added.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hi

    I know nothing of delay differential equation, but there is a theorem that ensures the existence and uniqueness of a solution : Picard?Lindelöf theorem - Wikipedia, the free encyclopedia

    But it requires some conditions.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    Posts
    481
    Thanks
    2
    Quote Originally Posted by Moo View Post
    Hi

    I know nothing of delay differential equation, but there is a theorem that ensures the existence and uniqueness of a solution : Picard?Lindelöf theorem - Wikipedia, the free encyclopedia

    But it requires some conditions.
    TY Lady Moo but I already have them read.
    For delay equations if we have a minimal delay, i.e., \alpha(t) is bounded away from t by a positive constant, then the Global E&U follows very easily by the method of steps.
    But the problem is when the delay is not minimal.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    Posts
    481
    Thanks
    2
    I think I have solved the problem.
    I will soon sketch the proof for the ones who wonder to know about it.

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    Posts
    481
    Thanks
    2

    Thumbs up Proved.

    Quote Originally Posted by bkarpuz View Post
    Dear friends, I need some help with the existence and/or uniqueness of global solutions to first-order linear differential equations.
    For instance, let x_{0},t_{0}\in\mathbb{R} and A,B\in C([t_{0},\infty),\mathbb{R}) and consider the following differential equation
    <br />
\begin{cases}<br />
x^{\prime}(t)=A(t)x(t)+B(t),& t\geq t_{0}\\<br />
x(t_{0})=x_{0}.&<br />
\end{cases}<br />
    Which theorem ensures existence of global solutions to this initial value problem?
    ...
    Consider the following initial value problem
    \begin{cases}<br />
x^{\prime}(t)=f(t,x(\tau(t))),&t\in[b,c]\\<br />
x(t)=\varphi(t),&t\in[a,b],<br />
\end{cases}\quad(1)
    where f:[b,c]\times\mathbb{R}\to\mathbb{R}, \varphi\in C([a,b],\mathbb{R}) and \tau\in C([b,c],[a,c]) satisfies \tau(t)\leq t for all t\in[b,c].
    Set I(\varphi,\varepsilon):=\{x\in\mathbb{R}:x\in B(\varphi(t),\varepsilon)\ \text{for}\ t\in[a,b]\}, where B(x_{0},\varepsilon):=\{x\in\mathbb{R}:|x-x_{0}|\leq\varepsilon\}.

    Theorem 1. Let f:[b,c]\times I(\varphi,\varepsilon)\to\mathbb{R} for some \varepsilon>0. Assume that there exist M>0 and L>0 such that |f(t,x)|\leq M for all (t,x)\in[b,c]\times I(\varphi,\varepsilon) and |f(t,u)-f(t,v)|\leq L|u-v| for all (t,u),(t,v)\in[b,c]\times I(\varphi,\varepsilon). Then (1) has a unique solution on [a,b+\delta], where \delta:=\min\{c-b,\varepsilon/M\}.

    Proof. The proof can be given by following exactly the same arguments in the proof of Picard-Lidelof theorem, and thus omitted here. \rule{0.2cm}{0.2cm}

    Now, consider the following initial value problem
    \begin{cases}<br />
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[b,c]\\<br />
x(t)=\varphi(t),&t\in[a,b],<br />
\end{cases}\quad(2)
    where p,q\in C([b,c],\mathbb{R}), and the other arguments are same to that of (1).

    Corollary 1. (2) admits a unique solution on [a,c].

    Proof. Let f(t,u):=p(t)u+q(t) for (t,u)\in[b,c]\times\mathbb{R}, and M_{1},M_{2}>0 satisfy |p(t)|\leq M_{1} and |q(t)|\leq M_{2} for all t\in[b,c] (since p,q are continuous, we may always find such constants). The Lipschitz condition holds on [\xi_{0},\xi_{1}]\times\mathbb{R} with the Lipschitz constant M_{1}>0. Let b=\xi_{0}<\xi_{1}<\cdots<\xi_{k_{0}}=c satisfy \xi_{k}-\xi_{k-1}\leq1/(2M_{1}) for k=1,2,\ldots,k_{0}. For convenience in the notation define x_{0}:=\varphi and N_{0}:=\max\nolimits_{t\in[a,\xi_{0}]}\{|x_{0}(t)|\}. We may pick \varepsilon_{0}>0 such that \varepsilon_{0}/(M_{1}(\varepsilon_{0}+N_{0})+M_{2})\geq1/(2M_{1}), we see that |f(t,u)|\leq M_{1}(\varepsilon_{0}+N_{0})+M_{2} for all (t,u)\in[\xi_{0},\xi_{1}]\times B(0,\varepsilon_{0}+N_{0}). Applying Theorem 1, we see that
    \begin{cases}<br />
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[\xi_{0},\xi_{1}]\\<br />
x(t)=x_{0}(t),&t\in[a,\xi_{0}]<br />
\end{cases}
    admits a unique solution x_{1} on [a,\xi_{1}] since \min\{\xi_{1}-\xi_{0},\varepsilon_{0}/(M_{1}(\varepsilon_{0}+N_{0})+M_{2})\}\geq\min\{\x  i_{1}-\xi_{0},1/(2M_{1})\}. Next, let N_{1}:=\max\nolimits_{t\in[a,\xi_{1}]}\{|x_{1}(t)|\}. We may find \varepsilon_{1}>0 such that \varepsilon_{1}/(M_{1}(\varepsilon_{1}+N_{1})+M_{2})\geq1/(2M_{1}). And we have |f(t,u)|\leq M_{1}(\varepsilon_{1}+N_{1})+M_{2} for all (t,u)\in[\xi_{1},\xi_{2}]\times B(0,\varepsilon_{1}+N_{1}). Applying Theorem 1, we see that
    \begin{cases}<br />
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[\xi_{1},\xi_{2}]\\<br />
x(t)=x_{1}(t),&t\in[a,\xi_{1}]<br />
\end{cases}
    admits a unique solution x_{2} on [a,\xi_{2}]. Repeating in this manner, we obtain the unique solution x_{k_{0}} of (2) on [a,c]. \rule{0.2cm}{0.2cm}

    Remark 1
    . If we need to obtain the unique global solution to
    \begin{cases}<br />
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[b,\infty)\\<br />
x(t)=\varphi(t),&t\in[a,b],<br />
\end{cases}\quad(3)
    where in addition \lim\nolimits_{t\to\infty}\tau(t)=\infty is assumed to hold, we may pick an increasing divergent sequence \{\xi_{k}\}_{k\in\mathbb{N}}\subset[b,\infty) with the convenience \xi_{-1}:=a and \xi_{0}:=b such that \xi_{k-1}\leq\min\nolimits_{t\in[\xi_{k},\infty)}\{\tau(t)\} for all k\in\mathbb{N}, and apply Corollary 1 successively to obtain the unique solution x_{k} on each of the intervals [\xi_{k-1},\xi_{k}] for k\in\mathbb{N} by assuming the solution x_{k-1} obtained in the previous step as the initial function on the current interval with the convenience x_{0}:=\varphi. Then, letting x(t)=x_{k}(t) for t\in[\xi_{k-1},\xi_{k}] for k\in\mathbb{N}, we obtain the unique global solution to (3).

    Appendix. It is clear that the function t/(at+b)\to1/a\geq1/(2a) as t\to\infty for any fixed a,b>0.

    Remark. The proof is very simple in the case \inf\nolimits_{t\in[b,\infty)}\{t-\tau(t)\}>0.

    proof by bkarpuz
    Last edited by bkarpuz; September 20th 2010 at 12:51 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Existence and Uniqueness
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 11th 2011, 12:17 PM
  2. existence and uniqueness of nth-order IVP solutions
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: September 8th 2010, 02:51 PM
  3. global existence and uniqueness
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: April 25th 2009, 01:27 AM
  4. Existence and Uniqueness Th.
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: April 2nd 2009, 03:37 AM
  5. Existence and Uniqueness
    Posted in the Calculus Forum
    Replies: 4
    Last Post: October 16th 2007, 10:44 PM

Search Tags


/mathhelpforum @mathhelpforum