Multiple Periodic Solutions of Delayed Predator-Prey Systems with Type IV Functional Responses on Time Scales

Abstract

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the existence of multiple positive periodic solutions of delayed predator-prey systems with type IV functional responses on time scales. Our results not only unify the existing ones but also widen the range of applications.

1. Introduction

As was pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. At first sight, these problems may appear to be simple mathematically. However, in fact, they are often very challenging and complicated. Also, Zhen and Ma [2] argued that the environmental fluctuation is important in an ecosystem, and more realistic models require the inclusion of the effect of environmental changes, especially environmental parameters which are time dependent and periodically changing (e.g., seasonal changes, food supplies, etc.). Hence, just as pointed out by Freedman and Wu [3] and Kuang [4], it would be of great interest and importance to study the existence of periodic solutions for systems with periodic delay. Much progress has been made in this direction (see, e.g., [5–8] and the references cited therein).

In 1959, in order to describe behavior of different kinds of species, Holling [9] proposed three types of functional response functions. However, some authors [10] have also described a type IV functional response that is humped and that declines at high prey densities. This decline may occur due to prey group defense or prey toxicity. Recently, Chen [11] has studied the following periodic predator-prey system with a type IV functional response: 𝑥̇(𝑡)=𝑥(𝑡)𝑏₁(𝑡)−𝑎₁(𝑡)𝑥𝑡−𝜏₁(𝑡)−𝑐(𝑡)𝑦(𝑡−𝜎(𝑡))𝑥²(𝑡)(𝑡),(𝑡)(𝑡/𝑛+𝑥+𝑎𝑦̇=𝑦)−𝑏₂(𝑡)+𝑎₂(𝑡)𝑥𝑡−𝜏₂(𝑡)𝑥²𝑡−𝜏₂(𝑡)/𝑛+𝑥𝑡−𝜏₂(𝑡),+𝑎 (1.1) where 𝑐,𝜎,𝑎_𝑗,𝑏_𝑗, and 𝜏_𝑗(𝑗=1,2) are continuous 𝜔-periodic functions with 𝑐(𝑡)≥0,𝜎(𝑡)≥0,  𝑎_𝑗(𝑡)≥0, and 𝜏_𝑗≥0,∫^𝜔₀𝑐(𝑡)𝑑𝑡>0, and ∫^𝜔₀𝑏_𝑗(𝑡)𝑑𝑡>0,𝑛 and 𝑎 are positive constants. The growth functions 𝑏_𝑗 may change sign, since the environment fluctuates randomly. Under bad conditions, 𝑏_𝑗 may be negative.

Considering that discrete time models governed by difference equations are more appropriate than continuous ones when the populations have nonoverlapping generations, Zhang et al. [12] studied the following discrete time predator-prey system: 𝑥(𝑘+1)=𝑥(𝑘)exp𝑏₁(𝑘)−𝑎₁(𝑘)𝑥𝑘−𝜏₁(𝑘)−𝑐(𝑘)𝑦(𝑘−𝜎(𝑘))𝑥²(𝑘)(𝑘),𝑦()(𝑘)/𝑛+𝑥+𝑎𝑘+1=𝑦exp−𝑏₂(𝑘)+𝑎₂(𝑘)𝑥𝑘−𝜏₂(𝑘)𝑥²𝑘−𝜏₂(𝑘)/𝑛+𝑥𝑘−𝜏₂(𝑘),+𝑎 (1.2) where, for 𝑖=1,2,𝑏_𝑖∶ℤ→ℝ,𝑐,𝑎_𝑖∶ℤ→ℝ⁺,𝜏_𝑖,𝜎∶ℤ→ℤ⁺ are all 𝜔-periodic.

On the other hand, recently, Bohner et al. [13] pointed out that it is unnecessary to explore the existence of periodic solutions of some continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equation on general time scales. The theory of calculus on time scales, which has recently received a lot of attention, was initiated by Hilger in his Ph.D. Thesis in 1988 [14] in order to unify continuous and discrete analysis. Although there has been much research activity concerning the oscillation (nonoscillation) of solutions and periodic solution of differential equation on time scales (or measure chains) (see, e.g., [15–29]), there are few results dealing with multiple periodic solutions of predator-prey systems with time delay.

Motivated by the above work, we consider the following system on time scales 𝕋: 𝑥^Δ(𝑡)=𝑏₁(𝑡)−𝑎₁(𝑡)exp𝑥𝑡−𝜏₁(𝑡)−𝑐(𝑡)exp𝑦(𝑡−𝜏(𝑡)){(𝑡{𝑥(𝑡,𝑦exp2𝑥)}/𝑛+exp)}+𝑎_Δ(𝑡)=−𝑏₂(𝑡)+𝑎₂(𝑡)exp𝑥𝑡−𝜏₂(𝑡)exp2𝑥𝑡−𝜏₂(𝑡)/𝑛+exp𝑥𝑡−𝜏₂(𝑡),+𝑎 (1.3) where, for 𝑖=1,2,𝑏_𝑖∶𝕋→ℝ,𝑐,𝑎_𝑖,𝜏_𝑖,𝜏∶𝕋→ℝ⁺ are 𝜔-periodic functions. 𝑛 and 𝑎 are positive constants.

The main purpose of this paper is to derive a set of easily verifiable sufficient conditions for the existence of multiple positive periodic solutions of (1.3). The method used here will be the coincidence degree theory developed by Gaines and Mawhin [30].

In (1.3), set 𝑧₁(𝑡)=exp{𝑥(𝑡)},𝑧₂(𝑡)=exp{𝑦(𝑡)}. If 𝕋=ℝ, then (1.3) reduces to (1.1). Also, if 𝕋=ℤ, then (1.3) becomes (1.2). Thus, our results also show that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equations on time scales.

The paper is arranged as follows. In Section 2, we present some preliminary results such as the calculus on time scales and the continuation theorem in coincidence degree theory. In Section 3, we prove our main result.

2. Preliminaries

In this section, we give a short introduction to the time scales calculus and recall the continuation theorem from coincidence degree theory.

First, let us present some foundational definitions and results from the calculus on time scales, for proofs and further explanation and results, we refer to the paper by Hilger [14].

Definition 2.1. A time scale is an arbitrary nonempty closed subset 𝕋 of the real numbers ℝ. The set 𝕋 inherits the standard topology of ℝ.

Definition 2.2. For 𝑡∈𝕋, one defines the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡}, (2.1) while the backward jump operator 𝜌∶𝕋→𝕋 is defined by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}. (2.2)
In this definition we put inf∅=sup𝕋 (i.e., 𝜎(𝑡)=𝑡 if 𝕋 has a maximum 𝑡) and sup∅=inf𝕋 (i.e., 𝜌(𝑡)=𝑡 if 𝕋 has a minimum 𝑡), where ∅ denotes the empty set. If 𝜎(𝑡)>𝑡, we say that 𝑡 is right-scattered, while if 𝜌(𝑡)<𝑡 we say that 𝑡 is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if 𝑡<sup𝕋 and 𝜎(𝑡)=𝑡, then 𝑡 is called right-dense, and if 𝑡>inf𝕋 and 𝜌(𝑡)=𝑡, then 𝑡 is called left-dense. Points that are right-dense and left-dense at the same time are called dense.

Definition 2.3. A function 𝑓∶𝕋→ℝ is said to be rd-continuous if it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. The set of rd-continuous functions is denoted by 𝐶_rd=𝐶_rd(𝕋)=𝐶_rd(𝕋,ℝ).

Definition 2.4. Suppose 𝑓∶𝕋→ℝ is a function, and let 𝑡∈𝕋. Then one defines 𝑓^Δ(𝑡), the delta-derivative of 𝑓 at 𝑡, to be the number (provided it exists) with the property that, given any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)∩𝕋) for some 𝛿>0 such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]≤𝜀|𝜎(𝑡)−𝑠|∀𝑠∈𝑈. (2.3) Thus, 𝑓 is said to be delta-differentiable if its delta-derivative exists. The set of functions 𝑓∶𝕋→ℝ that are delta-differentiable and whose delta-derivative are rd-continuous functions is denoted by 𝐶¹_rd=𝐶¹_rd(𝕋)=𝐶¹_rd(𝕋,ℝ).

Definition 2.5. A function 𝐹∶𝕋→ℝ is called a delta-antiderivative of 𝑓∶𝕋→ℝ provided 𝐹^Δ=𝑓(𝑡), for all 𝑡∈𝕋. Then, one writes ^𝑠_𝑟𝑓(𝑡)Δ𝑡∶=𝐹(𝑠)−𝐹(𝑟)∀𝑠,𝑟∈𝕋. (2.4)

Definition 2.6. One says that a time scale 𝕋 is 𝜔-periodic, if 𝑡∈𝕋 implies 𝑡+𝜔∈𝕋.

Lemma 2.7. Every rd-continuous function has an antiderivative.

Lemma 2.8. If 𝑎,𝑏∈𝕋,𝛼,𝛽∈ℝ, and 𝑓,𝑔∈𝐶_rd(𝕋), then (i)∫^𝑏_𝑎[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+𝛽∫^𝑏_𝑎𝑔(𝑡)Δ𝑡;(ii)if 𝑓(𝑡)≥0 for all 𝑎≤𝑡<𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0;(iii)if |𝑓(𝑡)|≤𝑔(𝑡) on [𝑎,𝑏)∶={𝑡∈𝕋∶𝑎≤𝑡<𝑏}, then |∫^𝑏_𝑎𝑓(𝑡)Δ𝑡|≤∫^𝑏_𝑎𝑔(𝑡)Δ𝑡.

For convenience, one now introduces some notations to be used throughout this paper. Let 𝜅=min{[0,∞)∩𝕋},𝐼_𝜔=[𝜅,𝜅+𝜔]∩𝕋,𝑔^𝑢=sup_𝑡∈𝕋𝑔(𝑡),𝑔^𝑙=inf_𝑡∈𝕋𝑔(𝑡),1𝑔=𝜔_𝐼𝜔𝑔(𝑡)Δ𝑡=_𝜅^𝜅+𝜔𝑔(𝑡)Δ𝑡, (2.5) where 𝑔∈𝐶_rd(𝕋) is an 𝜔-periodic real function.

In order to obtain the existence of positive periodic solutions of (1.3), for the reader’s convenience, we will summarize in the following a few concepts and results from [30] that will be basic for this paper.

Let 𝑋,𝑍 be normed vector spaces, 𝐿∶Dom𝐿⊂𝑋→𝑍 a linear mapping, and 𝑁∶𝑋→𝑍 a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=CodimIm𝐿<+∞ and Im𝐿 is closed in 𝑍. If 𝐿 is a Fredholm mapping of index zero, there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄). It follows that 𝐿∣Dom𝐿∩Ker𝑃∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. We denote the inverse of that map by 𝐾_𝑃. Let Ω be an open bounded subset of 𝑋; the mapping 𝑁 will be called 𝐿-compact on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

Lemma 2.9 (continuation theorem). Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁 be 𝐿-compact on Ω. Suppose(a)for each 𝜆∈(0,1), every solution 𝑥 of 𝐿𝑥=𝜆𝑁𝑥 is such that 𝑥∉𝜕Ω;(b)𝑄𝑁𝑥≠0 for each 𝑥∈𝜕Ω∩Ker𝐿 and deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0. (2.6) Then the equation 𝐿𝑥=𝑁𝑥 has at least one solution lying in Dom𝐿∩Ω.

Now, we give a lemma which will be useful in our following proof. The proofs of the lemmas can be found in [13].

Lemma 2.10. Let 𝑡₁,𝑡₂∈𝐼_𝜔 and 𝑡∈𝕋. If 𝑓∶𝕋→ℝ is 𝜔-periodic, then 𝑓(𝑡)≤𝑓𝑡₁+_𝐼𝜔𝑓^ΔΔ𝑠,𝑓(𝑡)≥𝑓𝑡₂−_𝐼𝜔𝑓^ΔΔ𝑠. (2.7)

3. Existence of Periodic Solutions

The goal of this section is to establish sufficient conditions on the existence of periodic solution for system (1.3), where, for 𝑖=1,2,𝑏_𝑖,𝑐,𝑎_𝑖,𝜏_𝑖,𝜏 are rd-continuous functions. Firstly, we always assume that𝑎₂>𝑏₂1+2𝑎𝑛exp𝑏₁+𝑏₁𝜔. (𝐻₁) For further convenience, we define the following six positive numbers: 𝑙_±=𝑛𝑎₂exp𝑏₁+𝑏₁𝜔−𝑏₂±𝑛²𝑎₂exp𝑏₁+𝑏₁𝜔−𝑏₂²−4𝑛𝑎𝑏₂²2𝑏₂,𝑢_±=𝑛𝑎₂−𝑏₂exp𝑏₁+𝑏₁𝜔±𝑛²𝑎₂−𝑏₂exp𝑏₁+𝑏₁𝜔²−4𝑛𝑎𝑏₂²exp2𝑏₁+𝑏₁𝜔2𝑏₂exp𝑏₁+𝑏₁𝜔𝑣_±=𝑛𝑎₂−𝑏₂±𝑛²𝑎₂−𝑏₂²−4𝑛𝑎𝑏₂²2𝑏₂., (3.1) It is easy to show that 𝑙₋<𝑣₋<𝑢₋<𝑢₊<𝑣₊<𝑙₊. (3.2) We now come to the main result of this paper.

Theorem 3.1. In addition to (𝐻₁), assume further that 𝑎₁𝑙₊exp𝑏₁+𝑏₁𝜔<𝑏₁ (𝐻₂) holds and the system (1.3) has at least two 𝜔-periodic solutions.

Proof. In order to apply Lemma 2.9 (continuation theorem) to (1.3), we first define 𝑋=𝑍=𝑥,𝑦^𝑇∣𝑥,𝑦∈𝐶_rd,𝑥(𝑡+𝜔)=𝑥(𝑡),𝑦(𝑡+𝜔)=𝑦(𝑡),𝑥,𝑦^𝑇=max_{𝑡∈𝑇𝜔}|𝑥(𝑡)|+max_{𝑡∈𝑇𝜔}𝑦(𝑡) (3.3) for any (𝑥,𝑦)∈𝑋(or𝑍). Then 𝑋, 𝑍 are both Banach spaces when they are endowed with the above norm ‖⋅‖.
For ^𝑥_𝑦∈𝑋, we define 𝑁𝑥𝑦=𝑁₁(𝑡)𝑁₂(𝑡)=𝑏₁(𝑡)−𝑎₁(𝑡)exp𝑥𝑡−𝜏₁(𝑡)−𝑐(𝑡)exp𝑦(𝑡−𝜏(𝑡))exp{2𝑥(𝑡)}/𝑛+exp{𝑥(𝑡)}+𝑎−𝑏₂(𝑡)+𝑎₂(𝑡)exp𝑥𝑡−𝜏₂(𝑡)exp2𝑥𝑡−𝜏₂(𝑡)/𝑛+exp𝑥𝑡−𝜏₂(𝑡,𝐿𝑥𝑦=𝑥)+𝑎_Δ𝑦^Δ𝑥𝑦𝑥𝑦=1,𝑃=𝑄𝜔_𝐼𝜔𝑥(𝑡)1Δ𝑡𝜔_𝐼𝜔𝑦(𝑡)=Δ𝑡𝑥𝑦. (3.4) Then, it follows that Ker𝐿=𝑥,𝑦^𝑇∈𝑋∣𝑥,𝑦^𝑇=ℎ₁,ℎ₂^𝑇∈ℝ²for𝑡∈𝕋,Im𝐿=𝑥,𝑦^𝑇∈𝑋∣_𝐼𝜔𝑥(𝑡)Δ𝑡=0,_𝐼𝜔𝑦(𝑡)Δ𝑡𝑡∈𝕋isclosedin𝑍,dimKer𝐿=2=CodimIm𝐿, (3.5) and 𝑃,𝑄 are continuous projectors such that Im𝑃=Ker𝐿,Ker𝑄=Im𝐿=Im(𝐼−𝑄). (3.6)
Therefore, 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 reads 𝐾_𝑃𝑥𝑦=^𝑡𝜅𝑥(𝑠)1Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑥(𝑠)Δ𝑠Δ𝑡_𝑡𝜅𝑦(𝑠)1Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑦(𝑠)Δ𝑠Δ𝑡. (3.7) Thus, 𝑥𝑦=1𝑄𝑁𝜔_𝜅^𝜅+𝜔𝑏₁(𝑠)−𝑎₁(𝑠)exp𝑥𝑡−𝜏₁(𝑠)−𝑐(𝑠)exp𝑦(𝑠−𝜏(𝑠)){(𝑠{𝑥(𝑠1exp2𝑥)}/𝑛+exp)}+𝑎Δ𝑠𝜔_𝜅^𝜅+𝜔−𝑏₂(𝑠)+𝑎₂(𝑠)exp𝑥𝑡−𝜏₂(𝑠)exp2𝑥𝑠−𝜏₂(𝑠)/𝑛+exp𝑥𝑠−𝜏₂(𝑠),𝐾+𝑎Δ𝑠_𝑃()𝑁𝑥𝑦𝐼−𝑄=^𝑡𝜅𝑁₁(𝑠)1Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑁₁(𝑠)1Δ𝑠Δ𝑡−𝑡−𝜅−𝜔_𝜅^𝜅+𝜔(𝑡−𝜅)Δ𝑡𝑁_1𝑡𝜅𝑁₂(𝑠)1Δ𝑠−𝜔_𝜅^{𝜅+𝜔𝑡}_𝜅𝑁₂(𝑠)1Δ𝑠Δ𝑡−𝑡−𝜅−𝜔_𝜅^𝜅+𝜔(𝑡−𝜅)Δ𝑡𝑁₂. (3.8) Obviously, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. It is not difficult to show that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑍 by using the Arzela-Ascoli theorem. Moreover, 𝑄𝑁(Ω) is clearly bounded. Thus, 𝑁 is 𝐿-compact on Ω with any open bounded set Ω⊂𝑍.
Now we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem (Lemma 2.9). Corresponding to the operator equation 𝐿𝑥=𝜆𝑁𝑥,𝐿𝑦=𝜆𝑁𝑦,𝜆∈(0,1), we have 𝑥^Δ(𝑡)=𝜆𝑏₁(𝑡)−𝑎₁(𝑡)exp𝑥𝑡−𝜏₁(𝑡)−𝑐(𝑡)exp𝑦(𝑡−𝜏(𝑡)){(𝑡{𝑥(𝑡,𝑦exp2𝑥)}/𝑛+exp)}+𝑎_Δ(𝑡)=𝜆−𝑏₂(𝑡)+𝑎₂(𝑡)exp𝑥𝑡−𝜏₂(𝑡)exp2𝑥𝑡−𝜏₂(𝑡)/𝑛+exp𝑥𝑡−𝜏₂(𝑡).+𝑎 (3.9) Suppose that (𝑥(𝑡),𝑦(𝑡))^𝑇∈𝑋 is a solution of system (3.9) for a certain 𝜆∈(0,1). Integrating (3.2) over the set 𝐼_𝜔, we obtain 𝑏₁𝜔=_𝐼𝜔𝑎₁(𝑡)exp𝑥𝑡−𝜏₁(𝑡)+𝑐(𝑡)exp𝑦(𝑡−𝜏(𝑡))exp{2𝑥(𝑡)}/𝑛+exp{𝑥(𝑡)}+𝑎Δ𝑡,𝑏₂𝜔=_𝐼𝜔𝑎₂(𝑡)exp𝑥𝑡−𝜏₂(𝑡)exp2𝑥𝑡−𝜏₂(𝑡)/𝑛+exp𝑥𝑡−𝜏₂(𝑡)+𝑎Δ𝑡. (3.10)(3.11) It follows from (3.9)–(3.11) that _𝐼𝜔𝑥^Δ(𝑡)Δ𝑡<_𝐼𝜔𝑏₁(𝑡)Δ𝑡+_𝐼𝜔𝑎₁(𝑡)exp𝑥𝑡−𝜏₁(𝑡)+𝑐(𝑡)exp𝑦(𝑡−𝜏(𝑡)){(𝑡{𝑥(𝑡=exp2𝑥)}/𝑛+exp)}+𝑎Δ𝑡𝑏₁+𝑏₁𝜔,_𝐼𝜔𝑦^Δ(𝑡)Δ𝑡<_𝐼𝜔𝑏₂(𝑡)Δ𝑡+_𝐼𝜔𝑎₂(𝑡)exp𝑥𝑡−𝜏₂(𝑡)exp2𝑥𝑡−𝜏₂(𝑡)/𝑛+exp𝑥𝑡−𝜏₂(𝑡)=+𝑎Δ𝑡𝑏₂+𝑏₂𝜔. (3.12)(3.13) Note that (𝑥(𝑡),𝑦(𝑡))^𝑇∈𝑋, then there exist 𝜉_𝑖,𝜂_𝑖∈𝐼_𝜔,𝑖=1,2, such that 𝑥𝜉₁=min_{𝑡∈𝐼𝜔}𝑥(𝑡),𝑥𝜂₁=max_{𝑡∈𝐼𝜔}𝑥(𝑡),𝑦𝜉₂=min_{𝑡∈𝐼𝜔}𝑦(𝑡),𝑦𝜂₂=max_{𝑡∈𝐼𝜔}𝑦(𝑡). (3.14) Then, By (3.11) and (3.14), we have 𝑏₂𝜔≤_𝐼𝜔𝑎₂(𝑡)exp𝑥𝜂₁exp2𝑥𝜉₁/𝑛+exp𝑥𝜉₁+𝑎Δ𝑡=𝑎₂𝜔exp𝑥𝜂₁exp2𝑥𝜉₁/𝑛+exp𝑥𝜉₁+𝑎, (3.15) that is, 𝑥𝜂₁≥ln𝑏₂exp2𝑥𝜉₁/𝑛+exp𝑥𝜉₁+𝑎𝑎₂. (3.16) According to (3.12), (3.16), and Lemma 2.10, we derive 𝑥(𝑡)≥𝑥𝜂₁−_𝐼𝜔𝑥^ΔΔ𝑡>ln𝑏₂exp2𝑥𝜉₁/𝑛+exp𝑥𝜉₁+𝑎𝑎₂−𝑏₁+𝑏₁𝜔. (3.17) In particular, we have 𝑥𝜉₁>𝑙𝑛𝑏₂exp2𝑥𝜉₁/𝑛+exp𝑥𝜉₁+𝑎𝑎₂−|𝑏₁|+𝑏₁𝜔 (3.18) or 𝑏₂𝑛exp2𝑥𝜉₁−𝑎₂exp𝑏₁+𝑏₁−𝑏₂exp𝑥𝜉₁+𝑏₂𝑎<0. (3.19) According to (𝐻₁), we have ln𝑙₋<𝑥𝜉₁<ln𝑙₊. (3.20) Similarly, we also can obtain 𝑥𝜂₁<ln𝑢₋or𝑥𝜂₁<ln𝑢₊. (3.21) From (3.12) and (3.20) and Lemma 2.10, one has 𝑥(𝑡)≤𝑥𝜉₁+_𝐼𝜔𝑥^ΔΔ𝑡<𝑙𝑛𝑙₊+𝑏₁+𝑏₁𝜔=^def𝑀₁. (3.22) This, combined with (3.10) and (3.14), gives 𝑏₁𝜔≥𝑐𝜔exp𝑦𝜉₂exp2𝑀₁/𝑛+exp𝑀₁,+𝑎𝑏₁𝜔≤𝑎₁𝜔exp𝑀₁+𝑐𝜔exp𝑦𝜂₂𝑎. (3.23)(3.24) It follows from (3.23) that 𝑦𝜉₂≤ln𝑏₁exp2𝑀₁/𝑛+exp𝑀₁+𝑎𝑐₁. (3.25) This, together with (3.13) and Lemma 2.10, yields 𝑦(𝑡)≤𝑦𝜉₂+_𝐼𝜔𝑦^ΔΔ𝑡<𝑙𝑛𝑏₁exp2𝑀₁/𝑛+exp𝑀₁+𝑎𝑐₁+𝑏₂+𝑏₂𝜔=^def𝑀₂. (3.26) Moreover, because of (𝐻₂), it follows from (3.24) that 𝑦𝜂₂𝑎≥ln𝑏₁−𝑎₁𝑙₊exp𝑏₁+𝑏₁𝜔𝑐. (3.27) This, together with (3.13) and Lemma 2.10 again, yields 𝑦(𝑡)≥𝑦𝜂₂−_𝐼𝜔𝑦^Δ𝑎Δ𝑡≥ln𝑏₁−𝑎₁𝑙₊exp𝑏₁+𝑏₁𝜔𝑐−𝑏₂+𝑏₂𝜔=^def𝑚₂. (3.28) It follows from (3.26) and (3.28) that max_{𝑡∈𝐼𝜔}𝑦(𝑡)<max𝑚₂,𝑀=₂^def𝑀. (3.29) Obviously, ln𝑙_±, ln𝑢_±, 𝑀₁, and 𝑀 are independent of 𝜆.
Now, let us consider 𝑄𝑁𝑧₁ with 𝑧₁=(𝑥,𝑦)^𝑇∈ℝ. Note that 𝑄𝑁𝑧₁=𝑏₁𝜔−𝑎₁𝜔exp{𝑥}−𝑐𝜔exp𝑦{}{𝑥}−exp2𝑥/𝑛+exp+𝑎𝑏₂𝜔+𝑎₂𝜔exp{𝑥}exp{2𝑥}/𝑛+exp{𝑥}+𝑎. (3.30) By virtue of (𝐻₁) and (𝐻₂), we can show that 𝑄𝑁𝑧₁=0 has two distinct solutions 𝑧₁₁=(ln𝑣₋,ln((𝑏₁−𝑎₁𝑣₋)(𝑣²₋/𝑛+𝑣₋+𝑎)/𝑐))^𝑇 and 𝑧₁₂=(ln𝑣₊,ln((𝑏₁−𝑎₁𝑣₊)(𝑣²₋/𝑛+𝑣₊+𝑎)/𝑐))^𝑇. Choose 𝐶>0 such that 𝐶>max𝑏₁−𝑎₁𝑣₋𝑣²₋/𝑛+𝑣₋+𝑎𝑐,𝑏₁−𝑎₁𝑣₊𝑣²₋/𝑛+𝑣₊+𝑎𝑐. (3.31) Let Ω₁=𝑥,𝑦∈𝑋∣𝑥(𝑡)∈ln𝑙₋,ln𝑢₋,max_{𝑡∈𝐼𝜔}𝑦(𝑡,Ω)<𝑀+𝐶₂=𝑥,𝑦∈𝑋∣min_{𝑡∈𝐼𝜔}𝑥(𝑡)∈ln𝑙₋,ln𝑙₊,max_{𝑡∈𝐼𝜔}𝑥(𝑡)∈ln𝑢₊,𝑀₁andmax_{𝑡∈𝐼𝜔}𝑦(𝑡)<𝑀+𝐶. (3.32) Then both Ω₁ and Ω₂ are bounded open subsets of 𝑋. It follows from (3.2) and (3.31) that 𝑧₁₁∈Ω₁ and 𝑧₁₂∈Ω₂. With the help of (3.2), (3.20)–(3.22), (3.29), and (3.31), it is easy to see that Ω₁∩Ω₂=∅, and Ω_𝑖 satisfy the requirement (a) in Lemma 2.9 for 𝑖=1,2. Moreover, 𝑄𝑁𝑧≠0 for 𝑧∈𝜕Ω_𝑖∩Ker𝐿. A direct calculation shows that deg𝐽𝑄𝑁,Ω_𝑖∩Ker𝐿,0=(−1)^𝑖+1≠0. (3.33) Here, 𝐽 can be the identity mapping since Im𝑃=Ker𝐿. So far, we have proved that Ω_𝑖 verifies all the requirements in Lemma 2.9. Hence (1.3) has at least two 𝜔-periodic solutions. This completes the proof.

Remark 3.2. In (1.3), set 𝑧₁(𝑡)=exp{𝑥(𝑡)},𝑧₂(𝑡)=exp{𝑦(𝑡)}. When 𝕋=ℝ, then (1.3) reduces to (1.1). Also, if 𝕋=ℤ, then (1.3) becomes (1.2). Hence, our result unifies the main results of [11, Theorem  2.2] and [12, Theorem  2.1]. Moreover, our result will also be useful when 𝕋=ℎℤ={ℎ𝑘∣𝑘∈ℤ}, where ℎ>0 and ℎ≠1; however, [11, Theorem  2.2] and [12, Theorem  2.1] are not applicable. In this case, we have 𝑔^𝑢=max_{0≤𝑘≤𝜔/ℎ−1}𝑔(ℎ𝑘),𝑔^𝑙=min_{0≤𝑘≤𝜔/ℎ−1}𝑔(ℎ𝑘),1𝑔=𝜔^𝜔/ℎ−1_𝑘=0𝑔(ℎ𝑘)ℎ,𝑓^Δ(𝑡)=𝑓(𝑡+ℎ)−𝑓(𝑡)ℎ, (3.34) for every 𝜔-periodic sequence of positive real numbers, 𝑔 with 𝜔>1 and 𝑡∈𝕋.

Remark 3.3. According to the above proof, we also can obtain that Theorem 3.1 is true for the following general system: 𝑥^Δ(𝑡)=𝑏₁(𝑡)−𝑎₁(𝑡)exp𝑥𝑡−𝜏₁(𝑡)−𝑐(𝑡)exp𝑦(𝑡−𝜏(𝑡))exp2𝑥𝑡−𝜏₂(𝑡)/𝑛+exp𝑥𝑡−𝜏₂(𝑡),𝑦+𝑎_Δ(𝑡)=−𝑏₂(𝑡)+𝑎₂(𝑡)exp𝑥𝑡−𝜏₃(𝑡)exp2𝑥𝑡−𝜏₄(𝑡)/𝑛+exp𝑥𝑡−𝜏₄(𝑡).+𝑎 (3.35) Moreover, just the same as the authors in [11, 12] point out, Theorem 3.1 will remain valid if some or all terms are replaced by terms with discrete time delays, distributed delays (finite or infinite), state-dependent delays, or deviating arguments. That is to say that time delays of any type or the deviating arguments can have no effect on the existence of positive periodic solutions.

Acknowledgment

The authors are deeply indebted to an anonymous referee for his excellent suggestions, which greatly improve the presentation of this paper.