Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th Semester). For me, besides different ways to solve such particular equations (linear or not, constant coefficients or not , order 1, 2 , ... ), the Theorem of Existence and Uniqueness of solutions (Cauchy problem) must be taught to the students (Otherways the course would be empty : I told that to my students !).For this, a Lipschitz condition is needed and, hence, a Mean Value Theorem (MVT) for vector valued functions is useful when differentiability is satisfied . If a function f has a differential Df , Lipschitzity is equivalent to the boundedness of Df (which is easy to check for students when dealing with ordinary differential equations). One shows this equivalence using some MVT. The best statement I found for a such MVT is by Robert M. McLeold (Proc.Edin.Math.Soc. 14(19645),197209). Now my question : is there any nicer formula for the Mean Value Theorem for vector valued functions than that by Robert M. McLeold ? Secondary question : is there any better book than Pontriagin's (Ordinary differential equations ) for my teaching (2nd university year, 4th semester) ? Of course, Arnold's one, I know !

$\begingroup$ A function $f:\mathbb{R}^n\to\mathbb{R}^m$ is decribed by its components $f_1(x_1,\dotsc,x_n),\dotsc, f_m(x_1,\dotsc,x_n)$. The function $f$ is Lipschitz iff each of the components is such. $\endgroup$ – Liviu Nicolaescu Mar 14 '14 at 20:26

$\begingroup$ I know that . My question is about the Mean Value Theorem for vector value functions f : (a,b)→R^m. $\endgroup$ – AlAmrani Mar 14 '14 at 20:39

$\begingroup$ I'm a little lost here. I'm assuming that you want to prove existence and uniqueness of a solution $u$ to a system $u' = F(t, u)$, $u(0) = u_0$. Integrate both sides to obtain an integral operator for which a fixed point is the desired solution. The appropriate Lipschitz condition on $F$ implies that the integral operator is Lipschitz in the space of continuous functions using the sup norm, where the Lipschitz constant goes to zero as the time interval goes to zero. So for small time, the integral operator is a contraction mapping. I don't see where a mean value theorem is needed. $\endgroup$ – Deane Yang Mar 14 '14 at 22:44

$\begingroup$ @ Deane: I assumed he needed the mean value theorem to prove the Lipschitz condition. @ AlAmrani. A function $f:(a,b) \to \mathbb{R}^m$ is Lipschitz iff its components are. You only need the mean value for scalar valued functions of one variable. $\endgroup$ – Liviu Nicolaescu Mar 14 '14 at 23:08

$\begingroup$ @Dean Yang , @ Liviu Nicolaescu : to check that F(t,u) is Lipschitz w.r.t. u, the commune hypothesis is that F has a bounded partial derivative w.r.t. to u . $\endgroup$ – AlAmrani Mar 15 '14 at 10:51
(This is an answer to my secondary question above.)
A very good reference I came to the use of which in my teaching is : Ordinary Differential Equations by Wolfgang Walter , translated by Russel Thompson (Graduate Texts in Math., Springer 1998). That is a living and lively teaching book. It contains many explicit examples, and exercises (with solutions or hints). Notions, results, methods to solve equations, are well introduced, before giving proofs . (Birth and death dates, full names , nationalities ... of classical mathematicians are given.)