A global stability criterion in nonautonomous delay differential equations

Abstract

Consider the following nonautonomous nonlinear delay differential equation: { d y ( t ) d t = − a ( t ) y ( t ) − ∑ i = 0 m a i ( t ) g i ( y ( τ i ( t ) ) ) , t ⩾ t 0 , y ( t ) = ϕ ( t ) , t ⩽ t 0 , where we assume that there is a strictly monotone increasing function f ( x ) on ( − ∞ , + ∞ ) such that { f ( 0 ) = 0 , 0 < g i ( x ) f ( x ) ⩽ 1 , x ≠ 0 , 0 ⩽ i ⩽ m , and if f ( x ) ≢ x , then lim x → − ∞ f ( x ) or lim x → + ∞ f ( x ) is finite . In this paper, to the above nonautonomous nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f ( x ) which contains two cases f ( x ) = e x − 1 and f ( x ) = x , we give more explicit conditions which are some extension of the “3/2-type criterion.” Applying these to discrete models of nonautonomous delay differential equations, we also obtain new sufficient conditions of the global asymptotic stability of the zero solution.