A note on Cauchy-Lipschitz-Picard theorem
© Feng et al. 2016
Received: 6 July 2016
Accepted: 18 October 2016
Published: 4 November 2016
In this note, we try to generalize the classical Cauchy-Lipschitz-Picard theorem on the global existence and uniqueness for the Cauchy initial value problem of the ordinary differential equation with global Lipschitz condition, and we try to weaken the global Lipschitz condition. We can also get the global existence and uniqueness.
In his famous book , Brezis gave a very sketchy and interesting proof on the classical Cauchy-Lipschitz-Picard theorem.
It is well known that if we only assume the local Lipschitz condition, we can only get the local existence and uniqueness for Cauchy initial value problems.
In this paper, we try to weaken the global Lipschitz condition, but we also want to get the global existence and uniqueness; we have the following theorem.
2 The proof of Theorem 1.2
, Banach contraction mapping principle
Let X be a nonempty complete metric space and let \(T: X\rightarrow X\) be a strict contraction, i.e., there is \(0< k<1\) such that \(d(T(x),T(y))\leq kd(x,y)\), \(\forall x,y\in X\), then S has a unique fixed point \(u=T(u)\).
The following lemma can be regarded as a natural generalization of the Gronwall inequality.
(1) We now show that, for every \(u\in X\), \(\Phi(u)\) also belongs to X.
Furthermore, by Gronwall’s inequality, we can get the uniqueness.
The authors sincerely thank referee for his/her valuable comments. This paper was partially supported by NSFC (No. 11671278 and No. 11426181) and the National Research Foundation for the Doctoral Program of Higher Education of China under Grant No. 20120181110060.
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