What is the most general Carathéodory-type global existence theorem?

Question

I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$ $$ \begin{equation} \left\{ \begin{aligned} x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\ x(a) &= x_0 \end{aligned} \right. \end{equation}\label{1}\tag{1} $$ By "global" I mean that the time interval is fixed, i.e. $[a,b]$, but I am not asking the solution to stay in an a priori fixed compact set of $\mathbb{R}^n$ (though the final solution will be absolutely continuous and thus bounded). The setting is that of a possibly discontinuous vector field, described by the Carathéodory conditions, that is

1. $x \mapsto f(t,x)$ is continuous for a.e. $t$
2. $t \mapsto f(t,x)$ is measurable for each $x$
3. $|f(t,x)| \leq m(t)$, $m(t)$ being summable

A classical Carathéodory existence theorem (see e.g. Filippov, "Differential Equations with Discontinuous Right-Hand Side" (1988)) gives a local existence result in a compact set $K \subset \mathbb{R}^n$ under the above Charathéodory conditions.

Another classical Carathéodory theorem gives instead the global existence and uniqueness under a further Lipschitz continuity assumption:

4. $|f(t,x)-f(t,y)| \leq L(t) |x-y|$, $L(t)$ being summable

Finally, I found a global existence theorem (see Theorem II.3.2 on Reid, "Ordinary Differential Equations" (1971)), under the assumption

5. $|f(t,x)| \leq M(t)(1+|x|)$, $M(t)$ being summable

This last result require the vector field to have an at most linear growth in the variable $x$. I was wondering if anyone knows more general results for the existence of a global solutions, which can include also more than linear growth, or if the results I quoted are already the best I can get.

Thank you!

Answer 1

This is definitely a long comment, not an answer: I am not an ODE expert and furthermore the search for references was harder than I expected, even if I was aware that the question had been viewed several times before it was bountied by @Sebastiano. I have found evidence that at least three paths for the generalization of the classical Cauchy problem for a first order ODE or system of ODEs have been taken: I'll briefly describe them below.

1. The "classical" approach. This approach stems from the works of Federico Cafiero and perhaps culminates with the works of Alexey F. Filippov and William Thomas Reid. It deals with the problem by using the (now) standard real analysis methods such as Lebesgue integration, the Baire category theorem and the like (see reference [4], chapter III, §III.5 p. 201). I found only one paper generalizing in the sense of discontinuity in the second side of the ODE(s) in \eqref{1}, and it is the one of Perrson [3], who extends the results of Reid to systems of ODE instead of single equations (see also the historical survey in [4], loc. cit.).

2. The approach trough Perron-Henstock-Kutzweil integration. In this approach, the ODE in \eqref{1} is given a "generalized" meaning by using the generalized integral of Perron-Henstock-Kutzweil. In particular, the classical approach with Carathéodory function vector fields is completely recovered by using this approach (see [5], chapter V, theorem 5.14, p. 141-142): as a matter of fact, this is due because the vector field $f$ in \eqref{1} is allowed to be the sum of a Carathéodory function and a non-absolutely integrable function, while the solution $x(t)$ happens to belong to $BV_\text{loc}$. The vector fields considered by Kurzweil himself and analyzed by similar methods in reference [2] are even more general, allowing for the existence of non-absolutely continuous solutions.

3. The approach through nonlinear theories of generalized functions. This approach is based on the interpretation of the ODE in \eqref{1} in the framewoerk of distribution theory: for example, if $x(t)\in BV_\text{loc}$, then the vector field $f(x,t)$ can have Dirac $\delta(t)$ distributions in its structure. This in turn implies the necessity to multiply distributions in order to solve problem \eqref{1}, and thus it must be considered in the framework of the so called algebras of generalized functions, strictly outside the "classical" Laurent Schwartz, framework. This approach is adopted in reference [6], though also Schwabik reviews it ([5], chapter V, pp. 152-159) in order to show that his approach is able to encompass standard "impulse systems": and in this context, perhaps it would be worth to give a look also at the works of Derr and Knizebulatov [1].

Final note. Schwabik reviews also another approach ([5], chapter V, pp. 146-152), where the objects under study are called "measure differential equations": I have not put it in the above list since these objects, at (my) first sight, seem to fall within the reach of the theory developed in [1] and [6] and briefly described at point 3 above. However, is it possible to check for some references detailing it by looking in Schwabik's monograph [5]. Well, my two cents.

References

[1] Vasilii Ya. Derr and Damir M. Kinzebulatov, "Dynamical generalized functions and the multiplication problem", (English. Russian original) Russian Mathematics 51, No. 5, pp. 32-43 (2007), MR2380839, Zbl 1442.46030.

[2] Jaroslav Kurzweil, Generalized ordinary differential equations. Not absolutely continuous solutions, (English) Series in Real Analysis 11. Hackensack, NJ-Singapore: World Scientific, ISBN 978-981-4324-02-1/hbk; 978-981-4324-03-8/ebook, pp. ix+197 (2012), MR2906899, Zbl 1248.34001.

[3] Jan Persson, "A generalization of Caratheodory’s existence theorems for ordinary differential equations", (English) Journal of Mathematical Analysis and Applications 49, pp. 496-503 (1975), MR0372290, Zbl 0296.34004.

[4] Livio Clemente Piccinini, Guido Stampacchia and Giovanni Vidossich, Ordinary differential equations in $\mathbf R^n$. Problems and methods, Translated from the Italian by A. LoBello (English). Applied Mathematical Sciences 39. Berlin-Heidelberg-New York: Springer-Verlag, pp. XII+385 (1984), ISBN: 0-387-90723-8, MR0740539, Zbl 0535.34001.

[5] Štefan Schwabik, Generalized ordinary differential equations, (English) Series in Real Analysis 5. Singapore: World Scientific, pp. ix+382 (1992), ISBN: 981-02-1225-9, MR1200241, Zbl 0781.34003.

[6] S. T. Zavalishchin and A. N. Sesekin, Dynamic impulse systems: theory and applications, (English) Mathematics and its Applications 394. Dordrecht-Boston-London: Kluwer Academic Publishers, pp. xi +256 (1997), ISBN: 0-7923-4394-8, MR1441079, Zbl 0880.46031.