Two Positive Periodic Solutions for a Neutral Delay Model of Single-Species Population Growth with Harvesting

Abstract

By coincidence degree theory for k-set-contractive mapping, this paper establishes a new criterion for the existence of at least two positive periodic solutions for a neutral delay model of single-species population growth with harvesting. An example is given to illustrate the effectiveness of the result.

1. Introduction

In 1993, Kuang [1] proposed the following open problem (Open Problem 9.2): obtain sufficient conditions for the existence of positive periodic solutions for 𝑁^(𝑡)=𝑁(𝑡)𝑎(𝑡)−𝛽(𝑡)𝑁(𝑡)−𝑏(𝑡)𝑁(𝑡−𝜏(𝑡))−𝑐(𝑡)𝑁^(𝑡−𝜏(𝑡)),(1.1) where all parameters are nonnegative continuous 𝑇-periodic functions. Fang and Li [2] gave an answer to the above open problem. In recent years, many papers have been published on the existence of multiple positive periodic solutions for some population systems with periodic harvesting terms by using Mawhin's coincidence degree theory (see, e.g., [3–7]). However, to our knowledge, few papers deal with the existence of multiple positive periodic solutions for neutral delay population models with harvesting. The main difficulty is that Mawhin's coincidence degree theory is generally not available to neutral delay models. Moreover, it is also hard to obtain a priori bounds on solutions for neutral delay models.

In this paper, we consider the following neutral delay model of single-species population growth with harvesting 𝑁^(𝑡)=𝑁(𝑡)𝑎(𝑡)−𝛽(𝑡)𝑁(𝑡)−𝑏(𝑡)𝑁(𝑡−𝜏(𝑡))−𝑐(𝑡)𝑁^(𝑡−𝜏(𝑡))−ℎ(𝑡),(1.2) where 𝑎(𝑡),𝛽(𝑡),𝑏(𝑡),𝜏(𝑡),𝑐(𝑡),andℎ(𝑡) are nonnegative continuous 𝑇-periodic functions, and ℎ(𝑡) denotes the harvesting rate.

The purpose of this paper is to establish the existence of at least two positive periodic solutions for neutral delay model (1.2). To show the existence of solutions to the considered problems, we will use the coincidence degree theory for 𝑘-set contractions [8–10] and a priori bounds on solutions.

2. Preliminaries

We now briefly state the part of coincidence degree theory for 𝑘-set-contractive mapping (see [8–10]).

Let 𝑍 be a Banach space. For a bounded subset 𝐴⊂𝑍, let Γ_𝑍(𝐴) denote the (Kuratowski) measure of noncompactness defined by Γ_𝑍(𝐴)=inf𝛿>0∶∃afinitenumberofsubsets𝐴_𝑖⊂𝐴,𝐴=_𝑖𝐴_𝑖,diam𝐴_𝑖≤𝛿.(2.1) Here, diam (𝐴_𝑖) denotes the maximum distance between the points in the set 𝐴_𝑖.

Let 𝑋 and 𝑍 be Banach spaces with norms ‖⋅‖_𝑥 and ‖⋅‖_𝑧 respectively, and Ω a bounded open subset of 𝑋· A continuous and bounded mapping 𝑁∶Ω→𝑍 is called 𝑘-set-contractive if, for any bounded 𝐴⊂Ω, we have Γ_𝑍(𝑁(𝐴))≤𝑘Γ_𝑋(𝐴).(2.2)

Also, for a continuous and bounded map 𝑇∶𝑋→𝑌, we define 𝑙(𝑇)=sup𝑟≥0∶∀boundedsubset𝐴⊂𝑋,𝑟Γ_𝑋(𝐴)≤Γ_𝑌(𝑇(𝐴)).(2.3)

Let 𝐿∶dom𝐿⊂𝑋→𝑍 be 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, then there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that 𝐼𝑚𝑃=Ker𝐿,𝐼𝑚𝐿=Ker𝑄=𝐼𝑚(𝐼−𝑄). If we define 𝐿_𝑃∶dom𝐿∩Ker𝑃→𝐼𝑚𝐿 as the restriction 𝐿|_{dom𝐿∩Ker𝑃} of 𝐿 to dom𝐿∩Ker𝑃, then 𝐿_𝑃 is invertible. We denote the inverse of that map by 𝐾_𝑃. If Ω is an open bounded subset of 𝑋, the mapping 𝑁 will be called 𝐿-𝑘-set-contractive on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is 𝑘-set contractive. Since 𝐼𝑚𝑄 is isomorphic to Ker𝐿, there exists isomorphism 𝐽∶𝐼𝑚𝑄→Ker𝐿.

Lemma 2.1 (see[8], Proposition XI.2.). Let 𝐿 be a closed Fredholm mapping of index zero, and let 𝑁∶Ω→𝑍 be 𝑘′-set contractive with 0≤𝑘′<𝑙(𝐿).(2.4) Then 𝑁∶Ω→𝑍 is a 𝐿-𝑘-set contraction with constant 𝑘=𝑘′/𝑙(𝐿)<1.

The following lemma (see [8], page 213) will play a key role in this paper.

Lemma 2.2. Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁∶Ω→𝑍 be 𝐿-𝑘-set contractive on Ω,𝑘<1. Suppose that(i)𝐿𝑥≠𝜆𝑁𝑥 for every 𝑥∈dom𝐿∩𝜕Ω and every 𝜆∈(0,1);(ii)𝑄𝑁𝑥≠0 for every 𝑥∈𝜕Ω∩Ker𝐿;(iii)Brouwer degree deg_𝐵(𝐽𝑄𝑁,Ω∩Ker𝐿,0)≠0.Then 𝐿𝑥=𝑁𝑥 has at least one solution in dom𝐿∩Ω.

3. Main Result

Let 𝐶^0𝑇 denote the linear space of real-valued continuous 𝑇-periodic functions on 𝑅. The linear space 𝐶^0𝑇 is a Banach space with the usual norm for 𝑥∈𝐶^0𝑇 given by |𝑥|₀=max_𝑡∈𝑅|𝑥(𝑡)|. Let 𝐶^1𝑇 denote the linear space of 𝑇-periodic functions with the first-order continuous derivative. 𝐶^1𝑇 is a Banach space with norm |𝑥|₁=max{|𝑥|₀,|𝑥^|₀}.

Let 𝑋=𝐶^1𝑇 and 𝑌=𝐶^0𝑇, and let 𝐿∶𝑋→𝑌 be given by 𝐿𝑥=𝑑𝑥/𝑑𝑡. Since |𝐿𝑥|₀=|𝑥^|₀≤|𝑥|₁, we see that 𝐿 is a bounded (with bound =1) linear map.

Since we are concerning with positive solutions of (1.2), we make the change of variables as follows: 𝑁(𝑡)=𝑒^𝑥(𝑡).(3.1) Then (1.2) is rewritten as 𝑥^(𝑡)=𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥(𝑡)−𝑏(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}−𝑐(𝑡)𝑥^(𝑡−𝜏(𝑡))𝑒^{𝑥(𝑡−𝜏(𝑡))}−ℎ(𝑡)𝑒^𝑥(𝑡).(3.2) Next define a nonlinear map 𝑁∶𝑋→𝑌 by 𝑁(𝑥)(𝑡)=𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥(𝑡)−𝑏(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}−𝑐(𝑡)𝑥^(𝑡−𝜏(𝑡))𝑒^{𝑥(𝑡−𝜏(𝑡))}−ℎ(𝑡)𝑒^𝑥(𝑡).(3.3) Now, if 𝐿𝑥=𝑁𝑥 for some 𝑥∈𝑋, then the problem (3.2) has a periodic solution 𝑥(𝑡).

In the following, we denote that 𝑔=1𝑇^𝑇0𝑔(𝑡)𝑑𝑡,𝑔^𝑙=min_{[]𝑡∈0,𝑇}𝑔(𝑡),𝑔^𝑢=max_{[]𝑡∈0,𝑇}𝑔(𝑡),(3.4) where 𝑔(𝑡) is a continuous nonnegative 𝑇-periodic solution.

From now on, we always assume that(𝐻₁)𝑎(𝑡)∈𝐶(𝑅,(0,+∞)),𝛽(𝑡),𝑏(𝑡)∈𝐶(𝑅,𝑅⁺),𝑐(𝑡),𝜏(𝑡)∈𝐶¹(𝑅,𝑅⁺),𝜏^<1; (𝐻₂)𝑐^0(𝑡)<𝑏(𝑡), where 𝑐₀(𝑡)=𝑐(𝑡)/(1−𝜏^(𝑡));(𝐻₃)𝑎^𝑙>𝑐^𝑢𝑀₀+2√[𝛽^𝑢+𝑏^𝑢]ℎ^𝑢,𝑐^𝑢𝑒^𝑅1<1, where 𝑀₀=𝑎^𝑢𝑒^𝑅1+(𝛽^𝑢+𝑏^𝑢)𝑒^2𝑅1+ℎ^𝑢1−𝑐^𝑢𝑒^𝑅1𝑅₁,=ln𝑎𝛽^𝑙+𝑐^𝑢0𝑎𝑏−𝑐^0^𝑙+2𝑎𝑇.(3.5)

For further convenience, we introduce 6 positive numbers as below 𝑙^±=𝑎^𝑢+𝑐^𝑢𝑀₀±𝑎^𝑢+𝑐^𝑢𝑀₀²−4𝛽^𝑙ℎ^𝑙2𝛽^𝑙𝑥_±,=𝑎±𝑎²−4𝛽+𝑏ℎ𝑢_±2𝛽+𝑏,=𝑎^𝑙−𝑐^𝑢𝑀₀±𝑎^𝑙−𝑐^𝑢𝑀₀²−4𝛽^𝑢+𝑏^𝑢ℎ^𝑢2𝛽^𝑢+𝑏^𝑢.(3.6) Set the following: 𝑔⁻(𝑥,𝑦,𝑧)=𝑦−√𝑦²−4𝑥𝑧𝑔2𝑥=𝑦+√𝑦2₂𝑧−4𝑥𝑧(𝑥>0,𝑦>0,𝑧>0),₊(𝑥,𝑦,𝑧)=𝑦+√𝑦²−4𝑥𝑧2𝑥(𝑥>0,𝑦>0,𝑧>0),(3.7) where 𝑦²>4𝑥𝑧.

By the monotonicity of the functions 𝑔⁻(𝑥,𝑦,𝑧),𝑔⁺(𝑥,𝑦,𝑧) on 𝑥,𝑦,𝑧, it is not difficult to see that 𝑙⁻<𝑥⁻<𝑢⁻<𝑢⁺<𝑥⁺<𝑙⁺.(3.8)

Theorem 3.1. In addition to (𝐻₁), (𝐻₂),   (𝐻₃), assume further that the following condition holds:(𝐻₄)𝑘^∗=𝑐^𝑢max{𝑒^𝑅1,𝑙⁺}<1. Then (1.2) has at least two positive 𝑇-periodic solutions.

Before proving Theorem 3.1, we need the following lemmas.

Lemma 3.2 (see [11]). 𝐿 is a Fredholm map of index 0 and satisfies 𝑙(𝐿)≥1.(3.9)

Lemma 3.3. Under the assumptions of Theorem 3.1, let Ω=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑥∈𝑋||||||||||max^{[]𝑡∈0,𝑇}min_{[]𝑡∈0,𝑇}𝑥(𝑡)∈ln(𝑙⁻−𝛿),lnmax𝑒^𝑅1,𝑙⁺+𝛿,max_{[]𝑡∈0,𝑇}𝑥(𝑡)∈ln(𝑙⁻−𝛿),lnmax𝑒^𝑅1,𝑙⁺+𝛿,||𝑥^(𝑡)||<𝑀₁.⎫⎪⎪⎪⎬⎪⎪⎪⎭,(3.10) where 𝑀₁=𝑎^𝑢+(𝛽^𝑢+𝑏^𝑢)𝑒^𝑅1+ℎ^𝑢/𝑙⁻1−𝑐^𝑢𝑒^𝑅1,(3.11) and 0<𝛿<𝑙⁻ such that 𝑘₀=𝑐^𝑢max𝑒^𝑅1,𝑙⁺+𝛿<1.(3.12) Then 𝑁∶Ω→𝑌 is a 𝑘₀-set-contractive map.

Proof. The proof is similar to that of lemma  3.3 in [9], but for the sake of completeness we give the proof here. Let 𝐴⊂Ω be a bounded subset and let 𝜂=Γ_𝑋(𝐴). Then for any 𝜀>0, there is a finite family of subsets {𝐴_𝑖} with 𝐴=⋃_𝑖𝐴_𝑖 and diam₁(𝐴_𝑖)≤𝜂+𝜀.
Set the following: 𝑔𝑡,𝑥,𝑥₁,𝑥₂=𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥−𝑏(𝑡)𝑒^𝑥1−𝑐(𝑡)𝑥₂𝑒^𝑥1−ℎ(𝑡)𝑒^𝑥.(3.13) Now it follows from the fact that 𝑔(𝑡,𝑥,𝑥₁,𝑥₂) is uniformly continuous on any compact subset of 𝑅×𝑅³, and from the fact 𝐴 and 𝐴_𝑖 are precompact in 𝐶^0𝑇 with norm |⋅|₀, that there is a finite family of subsets {𝐴_𝑖𝑗} of 𝐴_𝑖 such that 𝐴_𝑖=⋃_𝑗𝐴_𝑖𝑗 with ||𝑔𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑢^(𝑡−𝜏(𝑡))−𝑔𝑡,𝑢(𝑡),𝑢(𝑡−𝜏(𝑡)),𝑢^(𝑡−𝜏(𝑡))||<𝜀(3.14) for any 𝑥,𝑢∈𝐴_𝑖𝑗. Therefore, for 𝑥,𝑢∈𝐴_𝑖𝑗 we have ||𝑁𝑥−𝑁𝑢||₀=≤ssuupp_{0≤𝑡≤𝑇0≤𝑡≤𝑇}||𝑔𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥^(𝑡−𝜏(𝑡))−𝑔𝑡,𝑢(𝑡),𝑢(𝑡−𝜏(𝑡)),𝑢^(𝑡−𝜏(𝑡))||+sup||_{0≤𝑡≤𝑇}𝑔𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥^(𝑡−𝜏(𝑡))−𝑔𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑢^(𝑡−𝜏(𝑡))||≤𝑐_𝑢||𝑔𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑢^(𝑡−𝜏(𝑡))−𝑔𝑡,𝑢(𝑡),𝑢(𝑡−𝜏(𝑡)),𝑢^(𝑡−𝜏(𝑡))||≤𝑘₀max𝑒^𝑅1,𝑙⁺+𝛿sup_{0≤𝑡≤𝑇}||𝑥^(𝑡−𝜏(𝑡))−𝑢^(𝑡−𝜏(𝑡))||+𝜀≤𝑘₀||𝑥^−𝑢^||₀+𝜀𝜂+𝑘₀+1𝜀.(3.15) That is Γ_𝑌(𝑁(𝐴))≤𝑘₀Γ_𝑋(𝐴). The proof is complete.

Lemma 3.4. If the assumptions of Theorem 3.1 hold, then every solution 𝑥∈𝑋 of the problem 𝐿𝑥=𝜆𝑁𝑥,𝜆∈(0,1)(3.16) satisfies max_{[]𝑡∈0,𝑇}𝑥(𝑡m)i∈n_{[]𝑡∈0,𝑇}[ln𝑙⁻,ln𝑢⁻]∪ln𝑢⁺,ln𝑙⁺,ma𝑥x_{[]𝑡∈0,𝑇}(𝑡)∈ln𝑙⁻,ln𝑙⁺,||𝑥^(𝑡)||<𝑀₁.(3.17)

Proof. Let 𝐿𝑥=𝜆𝑁𝑥 for 𝑥∈𝑋, that is, 𝑥^(𝑡)=𝜆𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥(𝑡)−𝑏(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}−𝑐(𝑡)𝑥^(𝑡−𝜏(𝑡))𝑒^{𝑥(𝑡−𝜏(𝑡))}−ℎ(𝑡)𝑒^𝑥(𝑡),𝜆∈(0,1).(3.18) Therefore, we have 𝑥^(𝑡)=𝜆𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥(𝑡)−𝑏(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}−𝑐₀(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}^−ℎ(𝑡)𝑒^𝑥(𝑡),𝜆∈(0,1),(3.19) where 𝑐₀(𝑡)=𝑐(𝑡)/(1−𝜏^(𝑡)).
By (3.19), we have 𝑥(𝑡)+𝜆𝑐₀(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}^=𝜆𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥(𝑡)−𝑏(𝑡)−𝑐^0(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}−ℎ(𝑡)𝑒^𝑥(𝑡).(3.20) Integrating this identity leads to ^𝑇0𝛽(𝑡)𝑒^𝑥(𝑡)+𝑏(𝑡)−𝑐^0(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}+ℎ(𝑡)𝑒^𝑥(𝑡)𝑑𝑡=^𝑇0𝑎(𝑡)𝑑𝑡.(3.21) From (3.20),(3.21), we have ^𝑇0|||𝑥(𝑡)+𝜆𝑐₀(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}^≤𝜆_𝑇0|||𝑑𝑡𝑎(𝑡)𝑑𝑡+^𝑇0𝛽(𝑡)𝑒^𝑥(𝑡)+𝑏(𝑡)−𝑐^0(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}+ℎ(𝑡)<2_𝑇0𝑒^𝑥(𝑡)𝑑𝑡𝑎(𝑡)𝑑𝑡=2𝑇𝑎.(3.22) By (3.21), we have ^𝑇0𝑎(𝑡)𝑑𝑡≥^𝑇0𝛽(𝑡)𝑒^𝑥(𝑡)+𝑏(𝑡)−𝑐^0(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}𝑑𝑡.(3.23) It follows that ^𝑇0𝑎(𝑡)𝑑𝑡≥𝑇𝛽^𝑙𝑒^𝑥(𝜉)+𝑇𝑏−𝑐^0^𝑙𝑒^{𝑥(𝜉−𝜏(𝜉))},(3.24) for some 𝜉∈[0,𝑇].
Therefore, we have 𝑥(𝜉)≤ln𝑎𝛽^𝑙,𝑒^{𝑥(𝜉−𝜏(𝜉))}≤𝑎𝑏−𝑐^0^𝑙.(3.25) From (3.22) and (3.25), we see that 𝑥(𝑡)+𝜆𝑐₀(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}≤𝑥(𝜉)+𝜆𝑐₀(𝜉)𝑒^{𝑥(𝜉−𝜏(𝜉))}+^𝑇0|||𝑥(𝑡)+𝜆𝑐₀(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}^<ln|||𝑑𝑡𝑎𝛽^𝑙+𝑐^𝑢0𝑎𝑏−𝑐^0^𝑙+2𝑎𝑇=𝑅₁.(3.26) Hence, we have 𝑥(𝑡)<𝑅₁.
Let 𝑠=𝑡−𝜏(𝑡). It follows from (3.19) that 𝑥^(𝑡)=𝜆𝑎(𝑡)−𝛽(𝑡)𝑒^𝑥(𝑡)−𝑏(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}−𝑐(𝑡)𝑑𝑒^𝑥(𝑠)𝑑𝑠−ℎ𝑒(𝑡^𝑥(𝑡)),𝜆∈(0,1).(3.27) Then from (3.27) and the inequality 𝑥(𝑡)<𝑅₁, we obtain that |||𝑒_𝑥(𝑡)^|||≤𝜆𝑎(𝑡)𝑒_𝑥(𝑡)+𝛽(𝑡)𝑒^2𝑥(𝑡)+𝑏(𝑡)𝑒^{𝑥(𝑡)+𝑥(𝑡−𝜏(𝑡))}+𝑐(𝑡)𝑒^𝑥(𝑡)||||𝑑𝑒_𝑥(𝑠)<𝑎_𝑢𝑑𝑠||||+ℎ(𝑡)𝑒^𝑅1+(𝛽^𝑢+𝑏^𝑢)𝑒^2𝑅1+𝑐^𝑢𝑒^𝑅1||||𝑑𝑒_𝑥(𝑠)𝑑𝑠||||+ℎ^𝑢,∀𝑡∈𝑅.(3.28) So that |||𝑒_𝑥(𝑡)^|||<𝑎_𝑢𝑒^𝑅1+𝛽^𝑢+𝑏^𝑢𝑒^2𝑅1+𝑐^𝑢𝑒^𝑅1|||𝑒_𝑥^|||₀+ℎ^𝑢,∀𝑡∈𝑅.(3.29) Since 𝑐^𝑢𝑒^𝑅1<1, we have |||𝑒_𝑥^|||₀<𝑎^𝑢𝑒^𝑅1+(𝛽^𝑢+𝑏^𝑢)𝑒^2𝑅1+ℎ^𝑢1−𝑐^𝑢𝑒^𝑅1=𝑀₀.(3.30) Choose 𝑡_𝑀,𝑡_𝑚∈[0,𝑇], such that 𝑥𝑡_𝑀=max_{[]𝑡∈0,𝑇}𝑥(𝑡),𝑥𝑡_𝑚=min_{[]𝑡∈0,𝑇}𝑥(𝑡).(3.31) Then, it is clear that 𝑥^𝑡_𝑀=0,𝑥^𝑡_𝑚=0.(3.32) From this and (3.27), we obtain that 𝑎𝑡_𝑀𝑒^{𝑥(𝑡𝑀)}=𝛽𝑡_𝑀𝑒^{2𝑥(𝑡𝑀)}+𝑏𝑡_𝑀𝑒^{𝑥(𝑡𝑀)+𝑥(𝑡𝑀−𝜏(𝑡𝑀))}+𝑐𝑡_𝑀𝑒^{𝑥(𝑡𝑀)}𝑑𝑒𝑑𝑠^𝑥(𝑠)_{𝑠=𝑡𝑀−𝜏(𝑡𝑀)}𝑎𝑡+ℎ𝑡_𝑀,(3.33)_𝑚𝑒^{𝑥(𝑡𝑚)}=𝛽𝑡_𝑚𝑒^{2𝑥(𝑡𝑚)}+𝑏𝑡_𝑚𝑒^{𝑥(𝑡𝑚)+𝑥(𝑡𝑚−𝜏(𝑡𝑚))}+𝑐𝑡_𝑚𝑒^{𝑥(𝑡𝑚)}𝑑𝑒𝑑𝑠^𝑥(𝑠)_{𝑠=𝑡𝑚−𝜏(𝑡𝑚)}+ℎ𝑡_𝑚.(3.34) It follows from (3.33) that 𝛽^𝑢+𝑏^𝑢𝑒^{2𝑥(𝑡𝑀)}−𝑎^𝑙−𝑐^𝑢𝑀₀𝑒^{𝑥(𝑡𝑀)}+ℎ^𝑢≥0.(3.35) By (𝐻₃), we have 𝑥𝑡_𝑀≤ln𝑢⁻or𝑥𝑡_𝑀≥ln𝑢⁺.(3.36) It also follows from (3.33) that 𝛽^𝑙𝑒^{2𝑥(𝑡𝑀)}−𝑎^𝑢+𝑐^𝑢𝑀₀𝑒^{𝑥(𝑡𝑀)}+ℎ^𝑙≤0.(3.37) By (𝐻₃), we have ln𝑙⁻≤𝑥𝑡_𝑀≤ln𝑙⁺.(3.38) Similarly, it follows from (3.34) that 𝛽^𝑙𝑒^{2𝑥(𝑡𝑚)}−𝑎^𝑢+𝑐^𝑢𝑀₀𝑒^{𝑥(𝑡𝑚)}+ℎ^𝑙≤0.(3.39) By (𝐻₃), we have ln𝑙⁻≤𝑥𝑡_𝑚≤ln𝑙⁺.(3.40) Hence, it follows from (3.36), (3.38), and (3.40) that 𝑥𝑡_𝑀∈𝑥[𝑡l_𝑚n𝑙⁻,ln𝑢⁻]∪ln𝑢⁺,ln𝑙⁺,∈ln𝑙⁻,ln𝑙⁺.(3.41) From the above inequality and (3.18), we obtain that ||𝑥^(𝑡)||≤𝜆𝑎(𝑡)+𝛽(𝑡)𝑒^𝑥(𝑡)+𝑏(𝑡)𝑒^{𝑥(𝑡−𝜏(𝑡))}+𝑐(𝑡)||𝑥^(𝑡−𝜏(𝑡))||𝑒^{𝑥(𝑡−𝜏(𝑡))}+ℎ(𝑡)<𝑎_𝑢𝑒^𝑥(𝑡)+(𝛽^𝑢+𝑏^𝑢)𝑒^𝑅1+𝑐^𝑢𝑒^𝑅1||𝑥^||₀+ℎ^𝑢𝑒^{𝑥(𝑡𝑚)}.(3.42) So that ||𝑥′||₀<𝑎^𝑢+𝛽^𝑢+𝑏^𝑢𝑒^𝑅1+𝑐^𝑢𝑒^𝑅1||𝑥′||₀+ℎ^𝑢𝑙⁻.(3.43) Since 𝑐^𝑢𝑒^𝑅1<1, we have ||𝑥′||₀<𝑎^𝑢+(𝛽^𝑢+𝑏^𝑢)𝑒^𝑅1+ℎ^𝑢/𝑙⁻1−𝑐^𝑢𝑒^𝑅1=𝑀₁.(3.44) The proof is complete.

The Proof of Theorem 3.1
Clearly, 𝑙^±,𝑢^± are independent of 𝜆. Now, let us consider 𝑄𝑁(𝑥) with 𝑥∈𝑅. Note that 𝑄𝑁(𝑥)=𝑎−𝛽+𝑏𝑒^𝑥−ℎ𝑒^𝑥.(3.45) It is easy to see that 𝑄𝑁(𝑥)=0 has two distinct solutions: 𝑢₁̃=ln𝑥⁻,𝑢₂̃=ln𝑥⁺.(3.46) By (3.8), one can take 𝑣⁻,𝑣⁺>0 such that 𝑢⁻<𝑣⁻<𝑣⁺<𝑢⁺.(3.47) Let Ω₁=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑥∈𝑋||||||||||max^{[]𝑡∈0,𝑇}min_{[]𝑡∈0,𝑇}𝑥(𝑡)∈(ln(𝑙⁻−𝛿),ln𝑣⁻),max_{[]𝑡∈0,𝑇}𝑥(𝑡)∈ln(𝑙⁻−𝛿),ln𝑙⁺+𝛿,Ω₂||𝑥^(𝑡)||<𝑀₁.⎫⎪⎪⎪⎬⎪⎪⎪⎭,=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝑥∈𝑋||||||||||||||max^{[]𝑡∈0,𝑇}min_{[]𝑡∈0,𝑇}𝑥(𝑡)∈ln𝑣⁺,ln𝑙⁺+𝛿,max_{[]𝑡∈0,𝑇}𝑥(𝑡)∈ln(𝑙⁻−𝛿),ln𝑙⁺+𝛿,||𝑥^(𝑡)||<𝑀₁.⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭.(3.48) Then Ω₁andΩ₂ are bounded open subsets of 𝑋. Clearly, Ω_𝑖⊂Ω(𝑖=1,2). It follows from Lemma 3.3 that 𝑁∶Ω_𝑖→𝑌 is a 𝑘₀-set-contractive map (𝑖=1,2). Therefore, it follows from Lemmas 2.1 and 3.2 that 𝑁∶Ω_𝑖→𝑍 is 𝐿-𝑘-set contractive on Ω_𝑖(𝑖=1,2) with 𝑘=𝑘₀/𝑙(𝐿)≤𝑘₀<1.
It follows from (3.8) and (3.46) that 𝑢_𝑖̃∈Ω_𝑖(𝑖=1,2). From (3.8), (3.47) and Lemma 3.4, it is easy to see that Ω₁∩Ω₂=∅ and Ω_𝑖 satisfies (i) in Lemma 2.2 for 𝑖=1,2. Moreover, 𝑄𝑁(𝑥)≠0 for 𝑥∈𝜕Ω_𝑖∩Ker𝐿(𝑖=1,2).
A direct computation gives the following: deg𝐽𝑄𝑁,Ω₁∩Ker𝐿,0=1,deg𝐽𝑄𝑁,Ω₂∩Ker𝐿,0=−1.(3.49) Here, 𝐽 is taken as the identity mapping since 𝐼𝑚𝑄=Ker𝐿. So far we have proved that Ω_𝑖 satisfies all the assumptions in Lemma 2.2(𝑖=1,2). Hence, (3.2) has at least two 𝑇-periodic solutions: 𝑥^∗𝑖(𝑡) and 𝑥^∗𝑖∈dom𝐿∩Ω_𝑖(𝑖=1,2). Obviously, 𝑥^∗𝑖(𝑖=1,2) are different. Let 𝑁^∗𝑖(𝑡)=𝑒^{𝑥∗𝑖(𝑡)}(𝑖=1,2). Then 𝑁^∗𝑖(𝑡)(𝑖=1,2) are two different positive 𝑇-periodic solutions of (1.2). The proof is complete.

Example 3.5. Take the following: 𝜏(𝑡)=1+0𝑐.5(𝑡s)i=n𝑡𝜖,(1−𝑎0(𝑡.5)=co2s𝑡−),sin𝑡,ℎ(𝑡)𝛽=(𝑡𝜖)(=2+𝑏(s𝑡i)n=𝑡)1,+0.5sin𝑡,(3.50) where the constant 𝜖>0.
Clearly, we have 𝑎^𝑙=1,𝛽^𝑙=0.5,𝛽^𝑢=𝑏^𝑢=1.5,𝑐^𝑢=1.5𝜖,ℎ^𝑢=3𝜖,𝑐₀(𝑡)=𝜖.(3.51) Therefore, we have 𝑐^0(𝑡)<𝑏(𝑡). Moreover, it is easy to see that 𝑅₁,𝑀₀,and𝑙⁺ are bounded with respect to 𝜖. Hence, for some sufficiently small 𝜖>0, we have 𝑎^𝑙>𝑐^𝑢𝑀₀+2𝛽^𝑢+𝑏^𝑢ℎ^𝑢,𝑘^∗=𝑐^𝑢max𝑒^𝑅1,𝑙⁺=1.5𝜖max𝑒^𝑅1,𝑙⁺<1.(3.52) In this case, all necessary conditions of Theorem 3.1 hold. By Theorem 3.1, (1.2) has at least two positive 2𝜋-periodic solutions.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (Grant no. 10971085).