Boundedness and persistence of delay differential equations with mixed nonlinearity

Abstract

For a nonlinear equation with several variable delays $$\dot{x}(t)=\sum _{k=1}^m{f}_k(t,x(h_1(t)),\dots ,x(h_l(t)))-g(t,x(t)),$$ where the functions $f_k$ increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on $[0,\infty )$, as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide.

24 pages, published in Applied Mathematics and Computation, 2016