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

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:

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

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

1. $$|f(t,x)| \leq M(t)(1+|x|)$$, $$M(t)$$ 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!

• I’m voting to close this question because I would like to ask the same question on Math-Overflow, since I am not receiving answers here. When I posted the question here I wasn't aware of the existence of Math-Overflow, better suited for research related questions Mar 22, 2021 at 22:56
• Hi, I'm putting a bounty for your question. I hope to retract your closure. Mar 30, 2021 at 19:38
• Also remember that, in order to cross posting a question to MathOverflow there's no need to remove a question from Math.SE. It is sufficient to wait a reasonable amount of time and then precisely state in the text that you are cross posting since you've not received any satisfactory answer. Mar 31, 2021 at 7:35
• Thank you @Sebastiano! And thank you @Daniele, I didn't know that. Mar 31, 2021 at 9:30
• Don't worry: it is a pleasure for me. :-)) @DanieleTampieri best regards to you and to Daniele. Mar 31, 2021 at 10:56

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, approaches the problem with the (now) standard real analysis methods such as Lebesgue integration, the Baire category theorem and the like (see reference , 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 , who extends the results of Reid to systems of ODE instead of single equations (see also the historical survey in , 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 , 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  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 , though also Schwabik reviews it (, 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 .

Final note. Schwabik reviews also another approach (, 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  and  and briefly described at point 3 above. However, is it possible to check for some references detailing it by looking in Schwabik's monograph . Well, my two cents.

References

 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.

 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.

 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.

 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.

 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.

 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.

• I thank you from the bottom of my heart for providing an answer. I will keep Daniele in mind your best efforts in the future as well. My best regards. Apr 6, 2021 at 15:17
• @Sebastiano thank you. Despite not being my field of expertise, I decided to try answering this question since I found it interesting per se, as ODEs with discontinuous vector fields are very common in applications of mathematics, and several upvotes testify I am not the only one who thinks so. I am aware that several questions are not answered here in Math.SE as well as in the MathOverflow, since there are many members who are skilled mathematicians but definitely not experts on each specific topic. Apr 6, 2021 at 17:30
• I really thank you again because regardless of your field of experience it has been like a gesture of humility and love for the others. I am not as good in English as you are and for the four and a half years that I have been registered on TeX.SE almost all the users are conscious of my handicap. (I use Deepl translate :-( ). My best and sincere regards from the Sicily. Apr 6, 2021 at 19:33