Math Help Forum - View Single Post - Existence and uniqueness of global solutions

bkarpuz · #1 June 4th, 2009, 08:13 AM

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.