I have a Cauchy problem for the differential equation \begin{equation} y' = f(t, y), \end{equation} with initial condition $y(0) = y^0$; here, $y$ and $f$ are two-dimensional vector-functions. The function $f$ is smooth with respect to $(t, y) \in \mathbb R \times \mathbb R^2$. So, in accordance with classical results, the solution to this problem exists and is unique in some vicinity of $t = 0$.
I also have auxiliary Cauchy problems for differential equations \begin{equation} u' = U(u), \end{equation} and \begin{equation} u' = -U(u), \end{equation} with initial conditions $u(0) = u^0$ and $u(0) = u_0$, respectively; here, $u$ and $U$ are scalar (one-dimensional) functions and $U(u)$ is continuous with respect to $u \in \mathbb R$. Besides this, $U(u) \geqslant 0$ for $u \in \mathbb R$ and it is true that $$ |y'(t)| = |f(t, y)| \leqslant U(|y(t)|) $$ (that's why these Cauchy problems are called "auxiliary" for the original one).
Let $u^0 (t)$ and $u_0 (t)$ be solutions of the auxiliary Cauchy problems; it is supposed that they both exist on the segment $[0, \theta]$.
Finally, the following estimate \begin{equation} u_0 (t) \leqslant |y(t)| \leqslant u^0(t), \end{equation} is given, where $y(t)$ is the solution to the first (two-dimensional) Cauchy problem and $|\cdot|$ is the euclidean norm, i.e. $|y(t)| = \sqrt{y_1^2 + y_2^2}$.
So, I am interested if there is any theorem guaranteeing that my original problem has a solution defined on the same segment where $u^0 (t)$ and $u_0 (t)$ exist (that is $[0, \theta]$) and that on this segment the estimate mentioned above is true.
