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Suppose that:$f: \mathbb{R} \rightarrow \mathbb{R}$ continuous and for some $x_0]in \mathbb{R}$ there exist two distinct solutions to the problem $x'=f(x), x(0)=x_0$ for $0\le t \le 1$. Prove that there exist infinitely many solutions to this problem valid in an interval of the form $[0,a]$ for some $a>0$. I have tried to use Picard iteration but I don't know how to go to further. Could you give me a hint?

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    $\begingroup$ Hint: what is $f(x_0)$? $\endgroup$ Sep 16, 2016 at 10:46

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We know that $f(x_0)=0$, otherwise the method of separation of variables would have guaranteed the unicity of solution of the Cauchy problem $$x'=f(x),\quad x(0)=x_0.$$

Therefore, one of the solutions of that system is non-constant; we will call it $\bar x(t)$. Also, a constant $x_0$ is also a solution of that system.

Now for $s>0$ take a function $$x_s(t) = \begin{cases} x_0, & t\in[0,s]\\\bar x(t-s),&t>s.\end{cases}$$

This is obviously a solution of the initial system, and for different $s$ the functions $x_s$ are distinct, hence we have an infinite family of solutions.

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  • $\begingroup$ Thank you very much for your help. $\endgroup$
    – user52523
    Sep 16, 2016 at 11:20

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