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

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 stability 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.

Uniqueness of fast travelling fronts in reaction–diffusion equations with delay

“…The reader can be referred to Röst & Wu (2007) for the case of unimodal nonlinearities, and for further references. On the other hand, Ivanov et al (2002) provide various conditions that are sufficient to guarantee the exponential stability of the positive steady state. The works mentioned above yield the following.…”

“…Consequently, if g 0 (k)2[0,1), then the positive equilibrium is exponentially stable (e.g. see corollary 3.2 in Ivanov et al (2002)) and globally attracting (e.g. see proposition 3.2 in Röst & Wu (2007)).…”

“…see proposition 3.2 in Röst & Wu (2007)). The second line of condition (v) also ensures the exponential stability of k (see theorem 2.9 in Ivanov et al (2002)) and the global attractivity of k (see corollary 2.3 in Liz et al (2005) which has been intensively studied for the last decade. Equation (1.4) takes into account the spatial distribution of the species, and there is growing interest in understanding the factors that influence the spatial spread of the growing population in that model.…”

Halanay type inequalities on time scales with applications

“…Stability analysis of differential equations using Halanay type inequalities has been studied in [2], [4], and [5]. For stability analysis of difference equations using Halanay inequality one may consult with [6]- [9].…”

A global stability criterion for a family of delayed population models

“…Now, the importance of Proposition 1.1 relies on its sharp nature: for every monotone f * satisfying (D1) or (D2) and for every triple a < 0, δ > 0, h > 0 which does not satisfy (1.3), the equilibrium x(t) = x * loses its asymptotical stability with an appropriate choice of piecewise-continuous periodic delay function τ : R → [0, h]. (See [8,13] for details.) However, the results in [13] do not apply directly to the nonmonotone case of Eq.…”