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DOI: 10.23952/cot.2023.30

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

**Abstract**. Let be a locally Lipschitz map with Denote by the generalized Jacobian of at We show that the following three statements hold true:

(1) Assume that, for any and any the symmetric part of (i.e., the matrix ),is negative definite. Then the map is injective and every solution of the autonomous system goes to $0,$ as $t$ tends to

(2) Assume that, for any and any the spectrum radius of the matrix is less than $latex1.$ Then is the unique fixed point of the map and every orbit of the discrete dynamical system goes to as tends to

(3) Assume that, for any and any the symmetric part of is negative definite. Then, for any there exists a real number such that for every the orbit of the discrete dynamical system converges to as goes to

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 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.