Permanence in a class of delay differential equations with mixed monotonicity

István Győri,Ferenc Hartung,Nahed Mohamady

doi:10.14232/ejqtde.2018.1.53

Abstract

Permanence in a class of delay differential equations with mixed monotonicity

Highlights

2010 Mathematics Subject Classification: 34K05. In this manuscript we study persistence and permanence of the scalar delay equation x(t) = α(t)h(x(t − τ), x(t − σ)) − β(t) f (x(t)), t ≥ 0, (1.1)
Where we assume that h : R+ × R+ → R+ is a mixed monotone function, i.e., h is monotone increasing in its first argument, and it is monotone decreasing in its second argument
Under natural conditions, Theorem 2.4 can be applied to prove the boundedness of the positive solutions

Summary

Introduction

Delay differential equations with mixed monotone functions in the delay term were studied, e.g., in [3, 4, 12, 16, 17] For such equations, persistence and permanence of solutions of a class of nonlinear differential equations with multiple delays were first studied in [3]. Our manuscript extends the results of [13] for the case when the birth term in the population model contains a mixed monotone function. Theorem 2.4 below presents sufficient conditions implying the permanence of equation (1.1), and we give a method how to find the lower and upper estimate of the limit inferior and limit superior of the solutions We extend this result to a more general scalar equation with multiple delays in Theorem 2.6.

Main results

Applications and examples

Proofs