A note on Cauchy-Lipschitz-Picard theorem

Abstract

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.

Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term

“…The proof below is enough for the infinite dimensional setting in Lemma 1 because for the drift function f satisfying the Lipschitz condition in the variable y, the Cauchy-Picard theorem, see [38], transmits the result from the finite dimensional case to an infinite dimensional case. Yet, the drift f fulfills the Lipschitz condition because of (11) and its affine structure.…”

Optimal version of the Picard-Lindelöf theorem

“…The usual proof of the Picard-Lindelöf theorem uses contraction on a suitably defined Banach space. For extensions of the Picard-Lindelöf theorem using contractions we refer the reader to [2], [3] and [4]. A different generalization was given in [6].…”