Attractivity of the origin for the equation ẍ + f(t, x, ẋ) ¦ ẋ ¦α ẋ + g(x) = 0

Asymptotic stability for functional differential equations

“…We consider a system of functional differential equations with finite delay written as (1) x'(t) = f(t, st), Standard existence theory shows that if r E CH and to > 0, then there is at least one continuous solution x(t, to, c)) on [t0,t0 + a) satisfying (1) for t > > to, st(to, r = r and a some positive constant; if there is a closed subset B C CH such that the solution remains in B, then a = oc. Also, I " I will denote the norm in R m with lal = max:<~_<~ Ixi[.…”

Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment

“…For example, if h(t) = sin 2 t, then the zero solution of (17) is uniformly asymptotically stable. On the other hand, if h(t) = 1/(1 + t) or h(t) = sin 2 t/(1 + t), then the zero solution of (17) is asymptotically stable, but it is not uniformly asymptotically stable (for details, see [2,13,28]). If the zero solution of a linear system is uniformly asymptotically stable, then the zero solution of the corresponding quasi-linear system is also uniformly asymptotically stable.…”

Uniform asymptotic stability of time-varying damped harmonic oscillators

“…Hence, condition (A) is not satisfied. However, by means of Ballieu and Peiffer's result [3,Corollary 7], we can verify that the equilibrium of (1) is asymptotically stable in this example.…”

“…Hence, we need only show that each solution of (1) and its derivative tend to zero as t → ∞ in order to prove that the equilibrium is asymptotically stable. Many efforts have been made to find sufficient (also necessary and sufficient) conditions which guarantee that the equilibrium of (1) is asymptotically stable (for example, see [1,3,4,[9][10][11][12][13]16,18,19,21,[24][25][26][27]31]). Among them, we should mention especially the criterion given by Smith [ Then the following result holds.…”