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

Consider the following system of delay differential equations { x 1 ′ ( t ) = − F ( x 1 ( t ) ) + G ( x 2 ( t − r 2 ) ) , x 2 ′ ( t ) = − F ( x 2 ( t ) ) + G ( x 3 ( t − r 3 ) ) , x 3 ′ ( t ) = − F ( x 3 ( t ) ) + G ( x 1 ( t − r 1 ) ) , where r 1 , r 2 and r 3 are positive constants, F , G ∈ C ( R 1 ) , and F is nondecreasing on R 1 . 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.