Solutions-set first order ODE's without uniqueness

Question

In short: What can we say about the set of all solutions of an ordinary differential equation (ODE) when we there is no uniqueness?

Consider the ODE

$f:D\to \mathbb{R}, \quad D\subseteq \mathbb{R}^2,$

$y'(x) = f(x,y),\quad y(x_0 ) = y_0, \quad (x_0,y_0 )\in D .$

Two fundamental theorems are:

1. If $f\in C(D)$, then there exists a neighbourhood of $x_0$ for which there is a solution $y(x) $ Peano.
2. If $f$ is also Lipschitz in $y$, then there exists a neighbourhood of $x_0$ in which $y(x)$ exist and is a unique solution.Picard Lindelöf.

The natural question is

What can be generally said about the set of all solutions when there is no uniqueness, i.e. $f$ is continuous but not Lipschitz?

Answer 1

1. In most common applications, uniqueness and the Lipschitz condition are violated only at certain points which form a curve which is a "singular solution". The keyword is Clairault Equation The simplest example is $y'=y^{1/3}$, the singular solution is $y=0$. Such situation usually occurs when one considers $F(y',y,t)=0$ where $F$ is analytic. For a modern exposition, see MR1503299 Ritt, J. F. On the singular solutions of algebraic differential equations. Ann. of Math. (2) 37 (1936), no. 3, 552–617.

2. Lipschitz condition in the uniqueness theorem can be slightly relaxed. $$|F(x,t)-F(x',t)|\leq\omega(x-x'),$$ where $$\int_{0+}\frac{ds}{\omega(s)}=\infty.$$ This is due to Osgood. (See Ph. Hartman, Ordinary DE, Birkhauser 1982).

3. A remarkable theorem of Władysław Orlicz says the following. Consider the differential equation $x'=f(x,t)$, where $x$ and $f$ are vector functions. In the space of all continuous vector functions $f(x,t)$ those functions for which the differential equation has at least one non-uniqueness point is of first category. (Convergence in the space is uniform on compact subsets). This shows that in the generic situation you do not need Lipschitz condition for uniqueness. Ref. W. Orlicz, Zur Theorie der Differentialgleichung $y'=f(x,y)$, Bull. de Acad. Polon. des Sciences, Ser A, 1932, 221-228. (Of course the set of Lipschitz functions is also of the first category:-) Same about functions satisfying Osgood's condition above.

4. A theorem of H. Kneser says that if you have non-uniqueness at some point then the set of solutions passing through this point is a continuum (closed connected set). Über die Lösungen eines Systems gewöhnlicher Differentialgleichungen, das der Lipschitzschen Bedingung nicht genügt. (German) JFM 49.0302.03 Berl. Ber. 1923, 171-174 (1923).

5. Scalar autonomous equation $x'=f(x)$ is exceptional: if $f(x_0)\neq 0$ and $f$ is continuous then the solution with $x(0)=x_0$ is unique.

A reference for general discussion of non-uniqueness (and its relevance for science) is V. I. Yudovich, Mathematical models for natural sciences. Lecture course. Moscow, 2004. (In Russian, unfortunately not translated into Latin-alphabet languages, and not reviewed on Mathscinet). In particular, Yudovich asks for an explicit condition of uniqueness which is satisfied by a residual set of continuous functions. No such condition is known. (Residual means that the complement is of the 1-st category).

Answer 2

Check the exercises in chapter existence theorem in Dieudonne Foundations on Modern Analysis you will get a nice overview for an answer to your question. best jorge