G.M. Lee, T.S. Phạm, A note on global stability of equilibria for locally Lipschitz maps, Vol. 2023 (2023), Article ID 30, pp. 1-7

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DOI: 10.23952/cot.2023.30

Received January 7, 2023; Accepted March 30, 2023; Published June 13, 2023

Abstract. Let $F \colon \mathbb{R}^p \to \mathbb{R}^p$ be a locally Lipschitz map with $F(0) = 0.$ Denote by $\partial F(x)$ the generalized Jacobian of $F$ at $x.$ We show that the following three statements hold true:
(1) Assume that, for any $x \in \mathbb{R}^p$ and any $Z \in \partial F(x),$ the symmetric part of $Z$ (i.e., the matrix $\frac{Z + Z^T}{2}$ ),is negative definite. Then the map $F$ is injective and every solution of the autonomous system $\dot{x}(t) = F(x(t))$ goes to $0,$ as $t$ tends to $+\infty.$
(2) Assume that, for any $x \in \mathbb{R}^p$ and any $Z \in \partial F(x),$ the spectrum radius of the matrix $Z^T Z$ is less than $latex1.$ Then $0$ is the unique fixed point of the map $F$ and every orbit of the discrete dynamical system $x_{n + 1} = F(x_n)$ goes to $0$ as $n$ tends to $+\infty.$
(3) Assume that, for any $x \in \mathbb{R}^p$ and any $Z \in \partial F(x),$ the symmetric part of $Z$ is negative definite. Then, for any $x_0 \in \mathbb{R}^p,$ there exists a real number $h_0 >0$ such that for every $h \in (0, h_0),$ the orbit of the discrete dynamical system $x_{n + 1} = x_n + h F(x_n)$ converges to $0$ as $n$ goes to $+\infty.$
These results strengthen those obtained by Furi, Martelli, and O’Neill in [J. Difference Equ. Appl. 15 (2009) 387-397], which requires further that the map $F$ is Gateaux differentiable except possibly on a linearly countable set.

How to Cite this Article:
G.M. Lee, T.S. Phạm, A note on global stability of equilibria for locally Lipschitz maps, Commun. Optim. Theory 2023 (2023) 30.