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I am trying to prove that when an IVP in ODEs does not enjoy uniqueness, then it has infinitely many different solutions. I know that when the Lipschitz condition is satisfied, then there is a unique solution.

I guess, I am trying to show that if the IVP possesses two solutions, then we can create infinitely many solutions using these two. However, I am slightly lost as where to start the proof.

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1 Answer 1

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This is not something totally trivial.

It is Helmut Kneser's Theorem.

Let the following IVP $$ x'=f(t,x), \quad x(\tau)=\xi,\tag{1} $$ where $x,f,\xi\in\mathbb R^n$. Then the set for any $s\in\mathbb R$, for which there exists a solution of the above in the interval $[\tau,s]$ the set $$ S=\{x(s): x\,\, \text{is a solution of $(1)$}\}, $$ is connected.

In the one-dimensional case, the proof is rather simple. Assume that $\varphi,\psi:[\tau,\sigma]\to\mathbb R$ are solutions of $(1)$ and $\varphi(\sigma)<\eta<\psi(\sigma)$. Then solve "backwards" the IVP $$ x'=f(t,x), \quad x(\sigma)=\eta,\tag{2} $$ which means that you obtain a solution $\zeta$ for $t\le \sigma$. As soon as the graph of $\zeta$ hits the graph of $\varphi$ or $\psi$, say at $t_0\in(\tau,\sigma)$ you have $\zeta(t_0)=\psi(t_0)$, then you can define a new solution $\tilde\zeta$ as $$ \tilde\zeta(t)=\left\{ \begin{array}{lll} \psi(t) & \text{if} & t\in[\tau,t_0], \\ \zeta(t) & \text{if} & t\in[t_0,\sigma]. \end{array} \right. $$ Clearly $\tilde\zeta$ satisfies both $(1)$ and $(2)$, and that's how you obtain a continuum of solutions.

For a more general theorem, see Philip Hartman, Ordinary Differential Equations, page 15.

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  • $\begingroup$ Thanks for the reference. I knew the theorem had a name, but I couldn't recall it. So my question is the other answer I received shows that you can construct additional solutions from the non-uniqueness conditions, but does this imply there is a continuum of solutions? Do I need to go through the proof of connectedness to show that it is indeed a continuum? $\endgroup$
    – user75514
    Commented Jan 30, 2014 at 15:21
  • $\begingroup$ @user75514: See updated answer. $\endgroup$ Commented Jan 30, 2014 at 15:34
  • $\begingroup$ Thanks. That was very helpful. $\endgroup$
    – user75514
    Commented Jan 31, 2014 at 15:33
  • $\begingroup$ There is a late reply. Why the function $\zeta$ is defined in $[\tau ,\sigma]$? Or what's more important, why does its graph reach the line of function $\psi$ or $\phi$? $\endgroup$
    – Limsup
    Commented Oct 1, 2020 at 18:35
  • $\begingroup$ How do we know the new constructed soln is differentiable at $t_0$? Do you need some regularity conditions on $f$? $\endgroup$ Commented Jan 6, 2022 at 9:20

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