A generalization of the three-dimensional Bernfeld-Haddock conjecture and its proof (Q732156)

This paper deals with 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, and \(F\) is nondecreasing on \(\mathbb R^1\). Such a system is related to a three-dimensional generalization of the Bernfeld-Haddock conjecture. By means of the monotone technique, the authors find that each bounded solution of the systems tends to a constant vector under some desirable conditions.