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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.

Ignacio Rojas
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1 Answers1

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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.

user539887
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