Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

Abstract

The following system considered in this paper:This system is referred to as a half-linear system. The coefficient f (t) is assumed to be bounded, but the coefficients e(t), g(t) and h(t) are not necessarily bounded. Sufficient conditions are obtained for global asymptotic stability of the zero solution.Our results can be applied to not only the case that the signs of f (t) and g(t) change like the periodic function but also the case that f (t) and g(t) irregularly have zeros.Some suitable examples are included to illustrate our results.

Asymptotic stability for quasi-linear systems whose linear approximation is not assumed to be uniformly attractive

“…However, system (H L) does not correspond to the quasi-linear system (QL) in any case. Hence, we cannot apply results in [25] to system (QL). In [25], the function ψ(t) is assumed to be non-negative for t ≥ 0.…”

“…Hence, we cannot apply results in [25] to system (QL). In [25], the function ψ(t) is assumed to be non-negative for t ≥ 0. We would like to be able to deal with a possibly signchanging function ψ(t).…”

“…To answer our question mentioned above, we develop an idea of Sugie and Onitsuka [25]. They have considered a system of the form…”

Asymptotic stability of a pendulum with quadratic damping

“…We say that a nonnegative function ψ(t) is said to be weakly integrally positive if ∞ n=1 σn τn ψ(t)dt = ∞ for every pairs of sequences {τ n } and {σ n } satisfying τ n + λ < σ n ≤ τ n+1 ≤ σ n + Λ for some λ > 0 and Λ > 0. The typical example of the weakly integrally positive function is 1/(1 + t) or sin 2 t/(1 + t) (for example, see [13,14,28,30,31]). Even if h(t) has infinitely many isolated zeros, it may be weakly integrally positive.…”

“…As mentioned in Sect. 1, the function sin 2 t/(1 + t) is weakly integrally positive (for the proof, see [31]). Taking the inequality…”