Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima

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Abstract
We present an extension of the well-known 3/2-stability criterion by Yorke for two term functional differential equations. We prove the exact nature of the obtained sta- bility region which coincides with the Yorke result in the special case when the decay term is absent. Moreover, we reveal some interesting links existing between the Yorke conditions, Halanay inequalities and differential equations with maxima, all of them essentially involving the maximum functionals. 1. Introduction. It can be observed that several important approaches in the stability theory of delay differential equations of the form (1.1 )x � (t) + ax(t) + bf (t, x t ) = 0 ,t ∈ R ,( x t (s) = x(t + s), s ∈( −h, 0) ), involve the maximum functional maxs∈(−h,0) φ(s) on the space C := C((−h, 0), R) in an essential and subtle way which sometimes is far away from the simple use of the sup-norm relations like