Boundedness and persistence of delay differential equations with mixed nonlinearity

TL;DR

This paper investigates the existence, boundedness, and persistence of solutions for nonlinear delay differential equations with mixed monotonicity and variable delays, including examples like the Mackey-Glass equation with non-monotone feedback.

Contribution

It provides new conditions for positive solutions, boundedness, and unboundedness in delay differential equations with mixed nonlinearities and variable delays.

Findings

Conditions for positive solution existence

Criteria for boundedness and persistence

Examples showing unbounded solutions with non-monotone feedback

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.