I want to show that exists an unique solution for the Lotka-Volterra problem for $t \in [0, T[$. With the Cauchy-Lipschitz theorem I could only show that exists an interval of unique solution, but I don't know how to argument that $0$ is in this interval and the interval is positive.
$$\left\{ \begin{array}{c} \\x'(t) = x(t)(a-by(t)) \\y'(t) = y(t)(-c+dx(t)) \\x(0) = x_0 > 0 \\y(0) = y_0 >0 \end{array}\right.$$
as $a, b, c, d > 0$
Can anyone help?
The problem you have written down is an Initial Value Problem, with an initial condition at t=0. Now if you apply the Picard–Lindelöf theorem (or Cauchy-Lipschitz), you automatically get the existence of a unique solution in an open interval $(-\varepsilon,\varepsilon)$ around 0. Hence, 0 is certainly in the interval of existence, as well as a bit of positive time.