A generalization of the three-dimensional Bernfeld-Haddock conjecture and its proof. (English) Zbl 1209.34092
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.
\[ x'_1(t)=-F(x_1(t))+G(x_2(t-r_2)), \]
\[ x'_2(t)=-F(x_2(t))+G(x_3(t-r_3)), \]
\[ x'_3(t)=-F(x_3(t))+G(x_1(t-r_1)), \]
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.
Reviewer: Mingshu Peng (Beijing)
MSC:
| 34K25 | Asymptotic theory of functional-differential equations |
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