Asymptotic stability and instability criteria for nonautonomous systems

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The classical criterion of asymptotic stability of the zero solution of equations x′ = f(t, x) is that there exists a function V (t, x), a(∥x∥) ≤ V (t, x) ≤ b(∥x∥) for some a, b ∈ K such that $$ \dot{V} $$ (t, x) ≤ −c(∥x∥) for some c ∈ K. In this paper, we prove that if $$ \mathop {V}\limits^{(m + {1})} $$ (t, x) is bounded on some set [tk − T, tk + T] × BH(tk → +∞ as k → ∞), then the condition that $$ \dot{V} $$ (t, x) ≤ −c(∥x∥) can be weakened and replaced by that $$ \dot{V} $$ (t, x) ≤ 0 and − (− $$ \dot{V} $$ (tk, x)| + − $$ \ddot{V} $$ (tk, x)| + ⋯ + − $$ \mathop {V}\limits^{(m)} $$ (tk, x)|) ≤ −c′(∥x∥) for some c′ ∈ K. Moreover, the author also presents a corresponding instability criterion. [1–10]