Two conjectures on global asymptotic stability have been recently disproved. The first conjecture, due to Markus-Yamabe (1960), regards continuous and autonomous systems of Differential Equations. The second, due to La Salle (1976), regards autonomous systems of Difference Equations. A counterexample to the first was found in 1997 and a counterexample to the second was discovered in 1998.

However, the story is not that simple. Global asymptotic stability of an equilibrium point can be obtained in the case discussed by Markus-Yamabe when the function that governs the system is continuous, its Gateaux derivative exists except possibly on a linearly countable set S, and the spectrum of the symmetric part of the derivative is strictly contained in the left hand side of the real line.

Similarly, global asymptotic stability can be proved in the case discussed by La Salle when the function is continuous, Gateaux differentiable except possibly on S, and the spectral radius of the matrix obtained by multiplying the Gateaux derivative F’G(x) with its transpose is strictly smaller than 1.

In this talk I shall present the counterexamples and the proof of the two positive outcomes. I shall also show that the set S cannot be uncountable, even in the case when its Lebesgue measure is 0.