Permanence in a class of delay differential equations with mixed monotonicity

Abstract

In this paper we consider a class of delay differential equations of the form $\dot{x}(t)=\alpha (t) h(x(t-\tau), x(t-\sigma))-\beta(t)f(x(t))$, where $h$ is a mixed monotone function. Sufficient conditions are presented for the permanence of the positive solutions. Our results give also lower and upper estimates of the limit inferior and the limit superior of the solutions via a special solution of an associated nonlinear system of algebraic equations. The results are generated to a more general class of delay differential equations with mixed monotonicity.

Keywords

• delay differential equations
• mixed monotonicity
• persistence
• permanence