Existence and uniqueness of global solutions

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• Jun 4th 2009, 06:13 AM
bkarpuz
[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
$
\begin{cases}
x^{\prime}(t)=A(t)x(t)+B(t),& t\geq t_{0}\\
x(t_{0})=x_{0}.&
\end{cases}
$

Which theorem ensures existence of global solutions to this initial value problem?

I am actually wondering to find this result for delay differential equations
$
\begin{cases}
x^{\prime}(t)=A(t)x(\alpha(t))+B(t),& t\geq t_{0}\\
x(t)=\varphi(t)&, t_{-1}\leq t\leq t_{0},
\end{cases}
$

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.
• Jun 4th 2009, 11:16 AM
Moo
Hi (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.
• Jun 4th 2009, 11:21 AM
bkarpuz
Quote:

Originally Posted by Moo
Hi (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. (Speechless)
• Jun 5th 2009, 12:32 PM
bkarpuz
I think I have solved the problem.
I will soon sketch the proof for the ones who wonder to know about it.

Thanks. (Happy)
• Jun 11th 2009, 10:27 AM
bkarpuz
Proved.
Quote:

Originally Posted by bkarpuz
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
$
\begin{cases}
x^{\prime}(t)=A(t)x(t)+B(t),& t\geq t_{0}\\
x(t_{0})=x_{0}.&
\end{cases}
$

Which theorem ensures existence of global solutions to this initial value problem?
...

Consider the following initial value problem
$\begin{cases}
x^{\prime}(t)=f(t,x(\tau(t))),&t\in[b,c]\\
x(t)=\varphi(t),&t\in[a,b],
\end{cases}\quad(1)$

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
$\begin{cases}
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[b,c]\\
x(t)=\varphi(t),&t\in[a,b],
\end{cases}\quad(2)$

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
$\begin{cases}
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[\xi_{0},\xi_{1}]\\
x(t)=x_{0}(t),&t\in[a,\xi_{0}]
\end{cases}$

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
$\begin{cases}
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[\xi_{1},\xi_{2}]\\
x(t)=x_{1}(t),&t\in[a,\xi_{1}]
\end{cases}$

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
$\begin{cases}
x^{\prime}(t)=p(t)x(\tau(t))+q(t),&t\in[b,\infty)\\
x(t)=\varphi(t),&t\in[a,b],
\end{cases}\quad(3)$

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