Boundedness and persistence of delay differential equations with mixed nonlinearity

For a nonlinear equation with several variable delays x˙(t)=∑k=1mfk(t,x(h1(t)),⋯,x(hl(t)))−g(t,x(t)),where the functions fk increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on [0, ∞), 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.