A generalization of the three-dimensional Bernfeld–Haddock conjecture and its proof

Consider the following system of delay differential equations {x1′(t)=−F(x1(t))+G(x2(t−r2)),x2′(t)=−F(x2(t))+G(x3(t−r3)),x3′(t)=−F(x3(t))+G(x1(t−r1)), where r1r1, r2r2 and r3r3 are positive constants, FF, G∈C(R1)G∈C(R1), and FF is nondecreasing on R1R1. These systems have important practical applications and also are a three-dimensional generalization of the Bernfeld–Haddock conjecture. In this paper, by using comparative technique, we obtain the asymptotic behavior of solutions that each bounded solution of the systems tends to a constant vector under a desirable condition.