I was working on a problem about a sequence of functions, each of which is a solution to a sequence of differential equations, that converges to a function which is supposed to be the limit of the previous sequence I mentioned.

The problem is as follows

Let $R=\lbrack t_0-a,t_0+a\rbrack\times\lbrack x_0-b,x_0+b\rbrack$, Suppose $(f_n)_{n\in\Bbb N}$ is a sequence of functions defined in $R$ which converges uniformly to $f$. This function is Lipschitz with respect to $x$ and continuous with respect to $t$.

Consider the following differential equations $$ \begin{cases} x'(t)=f(t,x)\\ x(t_0)=x_0 \end{cases} \quad \begin{cases} x'(t)=f_n(t,x)\\ x(t_0)=x_n \end{cases} $$

Let $\varphi,\varphi_n$ be solutions to both differential equations respectively defined on a common subinterval $\lbrack c,d\rbrack\subseteq \lbrack t_0-a,t_0+a\rbrack$. Show that if $x_n\xrightarrow[]{}x_0$ then $\varphi_n\to\varphi$ uniformly in $\lbrack c,d\rbrack$.

What I've done is to try and follow the definitions of uniform convergence for a sequence of functions and I can't quite grasp how to get to the convergence of $\varphi_n$. I know that the Lipschitz condition on $f$ guarantees the existence of such solutions to the differential equations.

It would seem that the problem has a bit of tricks behind it's back. I'm not quite sure on how should I work with this problem, any help is very much appreciated.


You are right that there are some tricks. The first (a standard one) is that we write the solutions of the IVPs as solutions of the integral equations $$ \varphi(t) = x_0 + \int\limits_{t_0}^{t} f(s, \varphi(s)) \, ds, \quad \varphi_n(t) = x_n + \int\limits_{t_0}^{t} f_n(s, \varphi_n(s)) \, ds, \quad \forall{t \in [c, d]}. $$ Now we need to group terms in a suitable way: $$ \varphi_n(t) - \varphi(t) = \biggl( \Bigl( x_n - x_0 \Bigr) + \int\limits_{t_0}^{t} (f_n(s, \varphi_n(s)) - f(s, \varphi_n(s)))\, ds \biggr) \\ + \int\limits_{t_0}^{t} (f(s, \varphi_n(s)) - f(s, \varphi(s))) \, ds, $$ which gives $$ \lvert \varphi_n(t) - \varphi(t) \rvert \le \biggl( \lvert x_n - x_0 \rvert + \biggl\lvert \int\limits_{t_0}^{t} \lvert f_n(s, \varphi_n(s)) - f(s, \varphi_n(s)) \rvert \, ds \biggr\rvert \biggr) \\+ \biggl\lvert\int\limits_{t_0}^{t} \lvert f(s, \varphi_n(s)) - f(s, \varphi(s))\rvert \, ds \biggl\rvert. $$ Take $\varepsilon > 0$. For $n$ sufficiently large, $\lvert x_n - x_0 \rvert < \varepsilon$. Similarly, by the uniform convergence of $f$ to $f_n$ on $R$, we have, for sufficiently large $n$, $$ \biggl\lvert \int\limits_{t_0}^{t} \lvert f_n(s, \varphi_n(s)) - f(s, \varphi_n(s)) \rvert \, ds \biggl\rvert < \varepsilon. $$ Consequently, $$ \lvert \varphi_n(t) - \varphi(t) \rvert \le 2 \varepsilon + L \biggl\lvert \int\limits_{t_0}^{t} \lvert \varphi_n(s) - \varphi(s) \rvert \, ds \biggl\rvert, $$ where $L$ is a Lipschitz constant for $f$. We apply (a simple form of) Grönwall's inequality (another trick) to obtain $$ \lvert \varphi_n(t) - \varphi(t) \rvert \le 2 \varepsilon e^{L \lvert t - t_0 \rvert} \qquad \forall{t \in [c, d]}, $$ hence $$ \lvert \varphi_n(t) - \varphi(t) \rvert \le 2 \varepsilon e^{L a} \qquad \forall{t \in [c, d]}, $$ which concludes the proof.

Observe that we do not assume that $f_n$ are Lipschitz. Indeed, we do not need the uniqueness of the solutions to $$ \begin{cases} x'(t)=f_n(t,x)\\ x(t_0)=x_n \end{cases} $$ in our proof.

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