Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays

1 Introduction

Consider the first-order nonlinear equation

where t ∈ R . Let ω > 0 and μ ∈ [ 0 , 1 ] .

We assume that

(H1) a : R → [ 0 , 1 ] and b i : R → [ 0 , ∞ ) ( 1 ≤ i ≤ n ) are ω -periodic continuous functions satisfying a ( t ) = a ( t + ω ) and b i ( t ) = b i ( t + ω ) for t ∈ R ;

(H2) τ i : R → [ 0 , ∞ ) ( 1 ≤ i ≤ n ) and H i : R → [ 0 , ∞ ) ( 1 ≤ i ≤ n ) are ω -periodic continuous functions satisfying τ i ( t ) = τ i ( t + ω ) and H i ( t ) = H i ( t + ω ) for t ∈ R ;

(H3) f : [ 0 , ∞ ) → [ 0 , ∞ ) is a Lipschitz continuous function, i.e., there is a non-negative constant L , for any x , y , the following inequality holds:

Equation (1.1) is a delay differential equation. Delay differential equations are mainly used to describe dynamic systems that rely on current and past historical states. The time-delay phenomenon is widely used in the fields of population dynamics [1,2, 3,4] and infectious diseases [5,6]. Scholars have shown that delay differential equations can more accurately reflect the changing laws of objective things than differential equations without time delay. In the past few decades, delay differential equations have received the attention of many scholars and achieved certain results.

For equation (1.1), when the nonlinear term f ( x ) has different expressions, model (1.1) also has different application backgrounds. For example, when the nonlinear term f ( N ) = N e − α N , then equation (1.1) will degenerate into the following Nicholson’s blowflies model [7] through certain assumptions:

where N ( t ) is the size of the population at time t , B is the maximum per capita daily egg production rate, δ is per capita daily adult death rate, τ denotes the approximate time of the life cycle, and 1 γ is the size at which the population reproduces at its maximum rate. Nicholson’s blowflies model has received wide attention due to its extensive practical significance, and its theoretical achievements have made remarkable progress in the past few decades, see [8,9,10, 11,12]. For example, Li and Du [13] studied the existence of positive periodic solutions of the generalized Nicholson’s blowflies model,

where m is a positive integer, p i , δ , q i ∈ C ( R + , ( 0 , ∞ ) ) , and τ i ∈ C ( R + , R + ) are T -periodic functions for i = 1 , 2 , 3 , … , m .

When the nonlinear term f ( x ) = x m 1 + x l , then model (1.1) represents the hematopoiesis model. The term ∑ i = 1 n b i ( t ) x m ( t − τ i ( t ) ) 1 + x l ( t − τ i ( t ) ) represents the current production (density) of blood cells that is affected by the past blood cell count (density). The hematopoiesis model was proposed by Mackey and Glass in 1977 [14]. Since then, hematopoiesis models have been studied by many scholars [15,16,17, 18,19]. For example, Liu et al. [20] used the fixed point theorem to study the existence of the positive periodic solution of the following hematopoiesis model and gave the necessary conditions for the existence of a unique positive periodic solution,

where m > 0 is a constant, a , b i , τ i are positive continuous ω -periodic functions.

In recent years, many authors have studied the existence of positive ω -periodic solutions for periodic ecological models with multiple delays. See [21,22, 23,24] and references therein. However, in this article, we will study the existence of positive periodic solutions of the first-order nonlinear differential equation with multiple time-varying delays. When nonlinear term f takes some special forms, problem (1.1) can degrade into some well-known models. Therefore, it is more universal than literature [7,13,20]. First, we give the existence condition of positive periodic solutions of equation (1.1), then give the uniqueness condition of positive periodic solution by using the contraction mapping principle, and finally give the oscillation condition of periodic solutions.

Here, we introduce the notations required to describe our main results. For each i = 1 , 2 , … , n , let

Define

Let

where

In this article, we also assume that (H4) holds:

(H4) β ω ∑ i = 1 n b ¯ i > β ρ ω μ ∑ i = 1 n H ̲ i + 1 .

For simplicity, we refer to points belonging to Δ (resp., ∇ ) as Δ -points (resp., ∇ -points) of (1.1). Note that there are no positive real numbers belonging to both Δ and ∇ .

2 Auxiliary lemmas and preparations

For convenience, in this section, we would like to introduce some notations, definitions, lemmas, and assumptions which are used in what follows.

The following is the well-known Krasnoselskii’s fixed point theorem in a cone.

Let

be the Banach space of ω -periodic continuous functions equipped with the norm

Define a subset in M by

It is easy to see that P is a cone in M .

Define a functional G : R × R → [ 0 , ∞ ) by

It is clear that G is strictly increasing with respect to the second component s ∈ [ t , t + ω ] . Also, we see that

and

Hence, when s ∈ [ t , t + ω ] and t ∈ R , we have

Moreover, G has the relationship

Define an operator K : M → M by:

The operator K has the following properties.

3 Simple conditions to ensure the existence of ∇ -point and Δ -point

We can prove that (1.1) has sufficient conditions for ∇ -point and Δ -point. For ∇ -point, we can choose ε such that

If function f satisfies

then we can find a sufficiently small number x ∗ > 0 such that f ( x ) < ε x for 0 ≤ x ≤ x ∗ . Since f is continuous on [ 0 , ∞ ) , we can see that f ( x ∗ ) < ε x ∗ . Hence, from (3.1), we have

Hence, we see that x ∗ ∈ ∇ .

For Δ -point, we can choose G such that

If for given G , there exists M > 0 , so that when x > M = ρ x ∗ , there is f ( x ) > G x . Then, we have

Hence, we see that x ∗ ∈ Δ .

4 Main results