Existence of Two Positive Periodic Solutions for a Neutral Multi-Delay Logarithmic Population Model with a Periodic Harvesting Rate

Abstract

By using an abstract existence result based on a coincidence degree theory for 𝑘-set contractive mapping, a new result is obtained for the existence of at least two positive periodic solutions for a neutral multidelay logarithmic population model with a periodic harvesting rate. An example is given to illustrate the effectiveness of the result.

1. Introduction

In recent years, many papers have been published on the existence of positive periodic solutions for neutral delay logarithmic population models by using a topological degree theory for 𝑘-set contractive mapping (see, e.g., [1–5]). Recently, Xia [6] obtained some new sufficient conditions for the existence and uniqueness of an almost periodic solution of a multispecies logarithmic population model with feedback controls. However, few papers deal with the existence of multiple positive periodic solutions for neutral multidelay logarithmic population models with harvesting. The main difficulty is hard to obtain a priori bounds on solutions for neutral multi-delay models with harvesting.

In this paper, we consider the following neutral multi-delay logarithmic population model of single-species population growth with a periodic harvesting rate 𝑑𝑦(𝑡)(𝑡)𝑎(𝑡)−𝑑𝑡=𝑦^𝑛_𝑗=1𝑏_𝑗(𝑡)ln𝑦𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)𝑑𝑑𝑡ln𝑦𝑡−𝜏_𝑖(𝑡)−ℎ(𝑡), (1.1) where 𝑎(𝑡),𝑏_𝑗(𝑡),𝜎_𝑗(𝑡)(𝑗=1,2,…,𝑛),𝑐_𝑖(𝑡),𝜏_𝑖(𝑡)(𝑖=1,2,…,𝑚),ℎ(𝑡) are nonnegative continuous 𝑇-periodic functions, and ℎ(𝑡) denotes the harvesting rate. When ℎ(𝑡)≡0, (1.1) was considered by [2–5]. When 𝑛=𝑚=1,ℎ(𝑡)≡0, and 𝜎₁(𝑡),𝜏₁(𝑡) are constants, (1.1) was considered by [7].

The purpose of this paper is to establish the existence of at least two positive periodic solutions for a neutral multi-delay logarithmic population model (1.1) by using a coincidence degree theory for 𝑘-set contractions. Motivated by the work of Chen [8], some novel techniques are employed to find a priori bounds on solutions.

2. Preliminaries

We now briefly state the part of the coincidence degree theory for 𝑘-set contractive mapping developed by Hetzer [9, 10]. For more details, we refer to [11].

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)=inf𝛿>0∶∃afinitenumberofsubsets𝐴_𝑖⊂𝐴,𝐴=_𝑖𝐴_𝑖,diam𝐴_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)=sup𝑟≥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 be 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 then 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 an isomorphism 𝐽∶Im𝑄→Ker𝐿.

Lemma 2.1 ([11, 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 [[11, 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(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 𝐶⁰_𝑇 denote the linear space of real valued continuous 𝑇-periodic functions on 𝑅. The linear space 𝐶⁰_𝑇 is a Banach space with the usual norm for 𝑥∈𝐶⁰_𝑇 given by |𝑥|₀=max_𝑡∈𝑅|𝑥(𝑡)|. Let 𝐶¹_𝑇 denote the linear space of 𝑇-periodic functions with the first-order continuous derivative. 𝐶¹_𝑇 is a Banach space with norm |𝑥|₁=max{|𝑥|₀,|𝑥^′|₀}.

Let 𝑋=𝐶¹_𝑇 and 𝑍=𝐶⁰_𝑇 and let 𝐿∶𝑋→𝑍 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.

Under the transformation of 𝑦(𝑡)=𝑒^𝑥(𝑡), (1.1) can be rewritten as 𝑥^′(𝑡)=𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡−ℎ(𝑡))𝑒^𝑥(𝑡). (3.1) Next define a nonlinear map 𝑁∶𝑋→𝑍 by 𝑁(𝑥)(𝑡)=𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡−ℎ(𝑡))𝑒^𝑥(𝑡). (3.2) Now, if 𝐿𝑥=𝑁𝑥 for some 𝑥∈𝑋, then the problem (3.1) has a 𝑇-periodic solution 𝑥(𝑡).

In the following, we denote 1𝑔=𝑇^𝑇₀𝑔(𝑡)𝑑𝑡,𝑔^𝑙=min_{[]𝑡∈0,𝑇}𝑔(𝑡),𝑔^𝑢=max_{[]𝑡∈0,𝑇}𝑔(𝑡), (3.3) where 𝑔(𝑡) is a continuous nonnegative 𝑇-periodic solution.

From now on, we always assume that(𝐻₁)𝑎(𝑡),𝑏_𝑗(𝑡)∈𝐶(𝑅,(0,+∞)),𝜎_𝑗(𝑡),𝑐_𝑖(𝑡),𝜏_𝑖(𝑡)∈𝐶¹(𝑅,(0,+∞)),forall𝑗∈{1,2,…,𝑛},  for all  𝑖∈{1,2,…,𝑚}.(𝐻₂)𝜎^′_𝑗(𝑡)<1,𝜏^′_𝑖(𝑡)<1,forall𝑡∈𝑅, and Γ(𝑡)=^𝑛_𝑗=1𝑏_𝑗𝜇_𝑗(𝑡)1−𝜎^′_𝑗𝜇_𝑗(𝑡)−^𝑚_𝑖=1𝑐^′_𝑖𝛾_𝑖(𝑡)1−𝜏^′_𝑖𝛾_𝑖(𝑡)>0, (3.4)where 𝜇_𝑗(𝑡) is the inverse function of 𝑡−𝜎_𝑗(𝑡),𝛾_𝑖(𝑡) is the inverse function of 𝑡−𝜏_𝑖(𝑡),forall𝑗∈{1,2,…,𝑛},forall𝑖∈{1,2,…,𝑚}.(𝐻₃)  Let ℎ1+ln^𝑙Γ^𝑢<𝑎^𝑢Γ^𝑙≤ℎ^𝑙Γ^𝑢<1. (3.5)(𝐻₄)  Let 𝑇Γ+^𝑛_𝑗=1𝑏_𝑗+^𝑚_𝑖=1𝑐^𝑢_𝑖ℎ<1,max1+ln^𝑢∑^𝑛_𝑗=1𝑏^𝑙_𝑗<𝑎,0^𝑙−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑢_𝑗<𝑅₀, (3.6)where 𝑀₀=𝑎^𝑢+∑^𝑛_𝑗=1𝑏^𝑢_𝑗𝑅₀+ℎ^𝑢𝑒^𝑅01−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢,𝑅₀ℎ=𝑞+𝑝,𝑞=2ln^𝑙Γ^𝑢,𝑝=2𝑇𝑎+Γ+∑^𝑛_𝑗=1𝑏_𝑗𝑇𝑞1−𝑇Γ+∑^𝑛_𝑗=1𝑏_𝑗−∑^𝑚_𝑖=1𝑐^𝑢_𝑖. (3.7) We first give some technological lemmas.

Set 𝑓(𝑥)=𝑑−𝑥−𝑟𝑒^−𝑥,𝑥∈(−∞,+∞). (3.8)

Lemma 3.1. Assume that 𝑑,𝑟 are positive constants such that 1+ln𝑟<𝑑. (3.9) Then there exist 𝑥⁻_∗,𝑥⁺_∗ such that 𝑓𝑥⁻_∗=𝑓𝑥⁺_∗𝑥=0,_−∗<ln𝑟<𝑥⁺_∗𝑓(𝑥)𝑥<𝑑,>0for𝑥∈⁻_∗,𝑥⁺_∗;𝑓(𝑥)<0for𝑥∈−∞,𝑥⁻_∗∪𝑥⁺_∗,+∞. (3.10) If one assumes further that 𝑑≤𝑟<1, (3.11) then the following inequalities also hold: 2ln𝑟<𝑥⁻_∗<ln𝑟<𝑥⁺_∗≤0. (3.12)

Proof. Clearly, 𝑓^′(𝑥)=𝑟/𝑒^𝑥−1=0 if and only if 𝑥=ln𝑟. Therefore, noticing that lim_𝑥→±∞𝑓(𝑥)=−∞, we have sup_{()𝑥∈−∞,+∞}𝑓(𝑥)=𝑑−ln𝑟−1>0. (3.13) Set 𝑔(𝑟)1=2ln𝑟+𝑟−𝑟. (3.14) Since 𝑔^′(𝑟)=2𝑟−1𝑟²()−1=−1−𝑟²𝑟²<0for0<𝑟<1, (3.15)𝑔(𝑟) is monotonically decreasing on (0,1].
Therefore, we have 𝑔(𝑟)>𝑔(1)=0, (3.16) that is, 12ln𝑟+𝑟>𝑟, (3.17) which implies 𝑓()12ln𝑟=𝑑−2ln𝑟−𝑟<𝑑−𝑟≤0. (3.18) Again, noticing that 𝑓(0)=𝑑−𝑟≤0, (3.19) by the monotonicity of the function 𝑓(𝑥) on the interval (−∞,ln𝑟) and (ln𝑟,+∞), it is easy to see that the assertion holds.

Set 𝐹(𝑥)=𝑎^𝑙−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑢_𝑗ℎ−𝑥−^𝑢∑^𝑛_𝑗=1𝑏^𝑙_𝑗𝑒^−𝑥,𝐺(𝑥)=𝑎∑^𝑛_𝑗=1𝑏_𝑗−𝑥−ℎ∑^𝑛_𝑗=1𝑏_𝑗𝑒^−𝑥,𝐻(𝑥)=𝑎^𝑢+∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑙_𝑗ℎ−𝑥−^𝑙∑^𝑛_𝑗=1𝑏^𝑢_𝑗𝑒^−𝑥. (3.20)

Lemma 3.2. Assume that (𝐻₁),(𝐻₂),  and  (𝐻₄) hold. Then the following assertions hold.
(1) There exist 𝑢⁻,𝑢⁺ such that 𝐹(𝑢⁻)=𝐹𝑢⁺𝑢=0,₋ℎ<ln^𝑢∑^𝑛_𝑗=1𝑏^𝑙_𝑗<𝑢⁺<𝑎^𝑙−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑢_𝑗,𝐹(𝑥)>0for𝑥∈𝑢⁻,𝑢⁺;𝐹(𝑥)<0for𝑥∈(−∞,𝑢⁻)∪𝑢⁺,+∞. (3.21)
(2) There exist 𝑥⁻,𝑥⁺ such that 𝐺(𝑥⁻)=𝐺𝑥⁺𝑥=0,₋<lnℎ∑^𝑛_𝑗=1𝑏_𝑗<𝑥⁺<𝑎∑^𝑛_𝑗=1𝑏_𝑗,𝐺(𝑥)>0for𝑥∈𝑥⁻,𝑥⁺;𝐺(𝑥)<0for𝑥∈(−∞,𝑥⁻)∪𝑥⁺,+∞. (3.22)
(3) There exist 𝑙⁻,𝑙⁺ such that 𝐻(𝑙⁻)=𝐻𝑙⁺𝑙=0,₋ℎ<ln^𝑙∑^𝑛_𝑗=1𝑏^𝑢_𝑗<𝑙⁺<𝑎^𝑢+∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑙_𝑗,𝐻(𝑥)>0for𝑥∈𝑙⁻,𝑙⁺;𝐻(𝑥)<0for𝑥∈(−∞,𝑙⁻)∪𝑙⁺,+∞. (3.23)
(4) 𝑙⁻<𝑥⁻<𝑢⁻<𝑢⁺<𝑥⁺<𝑙⁺. (3.24)

Proof. Noticing that 𝑎^𝑙−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑢_𝑗<𝑎∑^𝑛_𝑗=1𝑏_𝑗<𝑎^𝑢+∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑙_𝑗,ℎ^𝑙∑^𝑛_𝑗=1𝑏^𝑢_𝑗≤ℎ∑^𝑛_𝑗=1𝑏_𝑗≤ℎ^𝑢∑^𝑛_𝑗=1𝑏^𝑙_𝑗, (3.25) we have 𝐹(𝑥)<𝐺(𝑥)<𝐻(𝑥). (3.26) It follows from (𝐻₁),(𝐻₂),and(𝐻₄), (3.25) that ℎ1+ln^𝑢∑^𝑛_𝑗=1𝑏^𝑙_𝑗<𝑎^𝑙−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑢_𝑗,1+lnℎ∑^𝑛_𝑗=1𝑏_𝑗<𝑎∑^𝑛_𝑗=1𝑏_𝑗,ℎ1+ln^𝑙∑^𝑛_𝑗=1𝑏^𝑢_𝑗<𝑎^𝑢+∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑙_𝑗. (3.27) Therefore, by Lemma 3.1, the assertions (1)–(3) hold. Furthermore, by (3.26) and the assertions (1)–(3), the assertion (4) also holds.

Lemma 3.3 (see [12]). 𝐿 is a Fredholm map of index 0 and satisfies 𝑙(𝐿)≥1. (3.28)

Lemma 3.4 (see [2]). Suppose 𝜎∈𝐶¹_𝑇 and 𝜎^′(𝑡)<1,  for all  𝑡∈[0,𝑇]. Then the function 𝑡−𝜎(𝑡) has an inverse function 𝜇(𝑡) satisfying 𝜇∈𝐶(𝑅,𝑅) with 𝜇(𝑎+𝑇)=𝜇(𝑎)+𝑇.

Lemma 3.5. Assume that (𝐻₁)–(𝐻₄) hold. Let 𝑘₀=∑^𝑚_𝑖=1𝑐^𝑢_𝑖(1−𝜏^′_𝑖)^𝑢, and Ω=𝑥∈𝑋max_{𝑡∈[0,𝑇]}𝑥(𝑡)∈𝑙⁻−𝛿,𝑅₀+𝛿,min_{𝑡∈[0,𝑇]}𝑥(𝑡)∈𝑙⁻−𝛿,𝑙⁺+𝛿,max_{𝑡∈[0,𝑇]}𝑥^′(𝑡)<𝑀₀., (3.29) where 0<𝛿<𝑙⁻. Then 𝑁∶Ω→𝑍 is a 𝑘₀-set-contractive map.

Proof. The proof is similar to that of Lemma  3.3 in [12], so we omit it.

Theorem 3.6. Assume that (𝐻₁)–(𝐻₄) hold. Then (1.1) has at least two positive 𝑇-periodic solutions.

Proof. Let 𝐿𝑥=𝜆𝑁𝑥 for 𝑥∈𝑋, that is, 𝑥^′(𝑡)=𝜆𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡)−ℎ(𝑡)𝑒^𝑥(𝑡),𝜆∈(0,1). (3.30) Therefore, we have 𝑥^′(𝑡)=𝜆𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)𝑥𝑡−𝜏_𝑖(𝑡)^′−ℎ(𝑡)𝑒^𝑥(𝑡),𝜆∈(0,1). (3.31) By (3.31), we have 𝑥(𝑡)+𝜆^𝑚_𝑖=1𝑐_𝑖(𝑡)𝑥𝑡−𝜏_𝑖(𝑡)^′=𝜆𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)+^𝑚_𝑖=1𝑐^′_𝑖(𝑡)𝑥𝑡−𝜏_𝑖(𝑡)−ℎ(𝑡)𝑒^𝑥(𝑡). (3.32) Integrating this identity leads to ^𝑇₀^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐^′_𝑖(𝑡)𝑥𝑡−𝜏_𝑖(𝑡)+ℎ(𝑡)𝑒^𝑥(𝑡)𝑑𝑡=^𝑇₀𝑎(𝑡)𝑑𝑡. (3.33) By Lemma 3.4, we have 𝜇_𝑗(𝑇+𝑠)=𝑇+𝜇_𝑗(𝑠),𝛾_𝑖(𝑇+𝑠)=𝑇+𝛾_𝑖(𝑠), (3.34) where 𝑡=𝜇_𝑗(𝑠) is the inverse function of 𝑠=𝑡−𝜎_𝑗(𝑡), and 𝑡=𝛾_𝑖(𝑠) is the inverse function of 𝑠=𝑡−𝜏_𝑖(𝑡),forall𝑗∈{1,2,…,𝑛}andforall𝑖∈{1,2,…,𝑚}.
Then ^𝑇₀^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)𝑑𝑡=^𝑛_𝑗=1^{𝑇−𝜎𝑗(𝑇)}_{−𝜎𝑗(0)}𝑏_𝑗𝜇_𝑗(𝑠𝑥(𝑠))1−𝜎^′_𝑗𝜇_𝑗(𝑠)=𝑑𝑠_{𝑛𝑗=1𝑇0}𝑏_𝑗𝜇_𝑗(𝑠)1−𝜎^′_𝑗𝜇_𝑗(𝑠)𝑥(𝑠)𝑑𝑠,_{𝑇0𝑚𝑖=1}𝑐^′_𝑖(𝑡)𝑥𝑡−𝜏_𝑖(𝑡)𝑑𝑡=^𝑚_𝑖=1^{𝑇−𝜏𝑖(𝑇)}_{−𝜏𝑖(0)}𝑐^′_𝑖𝛾_𝑖(𝑠)𝑥(𝑠)1−𝜏^′_𝑖𝛾_𝑖(𝑠)=𝑑𝑠_{𝑚𝑖=1𝑇0}𝑐^′_𝑖𝛾_𝑖(𝑠)1−𝜏^′_𝑖𝛾_𝑖(𝑠)𝑥(𝑠)𝑑𝑠. (3.35) From (3.33)–(3.35), we have ^𝑇₀𝑎(𝑠)−^𝑛_𝑗=1𝑏_𝑗𝜇_𝑗(𝑠)1−𝜎^′_𝑗𝜇_𝑗(𝑠)−^𝑚_𝑖=1𝑐^′_𝑖𝛾_𝑖(𝑠)1−𝜏^′_𝑖𝛾_𝑖(𝑠𝑥(𝑠)−ℎ(𝑠))𝑒^𝑥(𝑠)𝑑𝑠=0, (3.36) which implies 𝑎𝜂𝜂𝑥𝜂−ℎ𝜂−Γ𝑒^𝑥(𝜂)=0, (3.37) for some 𝜂∈[0,𝑇].
Therefore, by (𝐻₂), we have 𝑎^𝑢Γ^𝑙𝜂−ℎ−𝑥^𝑙/Γ^𝑢𝑒^𝑥(𝜂)≥0. (3.38) By (𝐻₃) and Lemma 3.1, we have ℎ2ln^𝑙Γ^𝑢<𝑥𝜂≤0. (3.39) Set ℎ𝑞∶=2ln^𝑙Γ^𝑢, (3.40) then we have |𝑥(𝑡)|≤𝑥𝜂+^𝑇₀𝑥^′(𝑡)𝑑𝑡≤𝑞+^𝑇₀𝑥^′(𝑡)𝑑𝑡, (3.41) which implies |𝑥|₀≤𝑞+^𝑇₀𝑥^′(𝑡)𝑑𝑡. (3.42) It follows from (3.30) that ^𝑇₀𝑥^′(𝑡)𝑑𝑡=𝜆^𝑇₀𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡−ℎ(𝑡))𝑒^𝑥(𝑡)𝑑𝑡≤𝜆^𝑇₀𝑎(𝑡)−^𝑛_𝑗=1𝑏_𝑗(𝑡)𝑥𝑡−𝜎_𝑗(𝑡)−^𝑚_𝑖=1𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡)𝑑𝑡+^𝑇₀ℎ(𝑡)𝑒^𝑥(𝑡)𝑑𝑡. (3.43) By (3.36) and (𝐻₂), we have ^𝑇₀ℎ(𝑡)𝑒^𝑥(𝑡)𝑑𝑡≤^𝑇₀𝑎(𝑡)𝑑𝑡+^𝑇₀Γ(𝑡)𝑑𝑡|𝑥|₀=𝑇𝑎+𝑇Γ|𝑥|₀. (3.44) By this and (3.43), we obtain ^𝑇₀𝑥^′(𝑡)𝑑𝑡≤𝑇2𝑎+Γ|𝑥|₀+^𝑛_𝑗=1𝑏_𝑗|𝑥|₀+^𝑚_𝑖=1^𝑇₀𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡)𝑑𝑡. (3.45) Meanwhile, ^𝑚_𝑖=1^𝑇₀𝑐_𝑖(𝑡)1−𝜏^′_𝑖(𝑡)𝑥^′𝑡−𝜏_𝑖(𝑡)𝑑𝑡=^𝑚_𝑖=1^𝑇₀𝑐_𝑖𝛾_𝑖(𝑠)𝑥^′(𝑠)𝑑𝑠≤^𝑚_𝑖=1𝑐^𝑢_𝑖^𝑇₀𝑥^′(𝑠)𝑑𝑠. (3.46) Substituting (3.42) and (3.46) into (3.45) gives ^𝑇₀𝑥^′(𝑡2)𝑑𝑡≤𝑇𝑎+Γ|𝑥|₀+^𝑛_𝑗=1𝑏_𝑗|𝑥|₀+^𝑚_𝑖=1𝑐^𝑢_𝑖^𝑇₀𝑥^′(𝑠2)𝑑𝑠≤𝑇𝑎+Γ+^𝑛_𝑗=1𝑏_𝑗𝑞+^𝑇₀𝑥^′(𝑡)𝑑𝑡+^𝑚_𝑖=1𝑐^𝑢_𝑖^𝑇₀𝑥^′(𝑠2)𝑑𝑠≤𝑇𝑎+Γ+^𝑛_𝑗=1𝑏_𝑗𝑞+𝑇Γ+^𝑛_𝑗=1𝑏_𝑗+^𝑚_𝑖=1𝑐^𝑢_𝑖^𝑇₀𝑥^′(𝑡)𝑑𝑡. (3.47) Since 𝑇Γ+^𝑛_𝑗=1𝑏_𝑗+^𝑚_𝑖=1𝑐^𝑢_𝑖<1, (3.48) we have ^𝑇₀𝑥^′(𝑡)𝑑𝑡<2𝑇𝑎+Γ+∑^𝑛_𝑗=1𝑏_𝑗𝑇𝑞1−𝑇Γ+∑^𝑛_𝑗=1𝑏_𝑗−∑^𝑚_𝑖=1𝑐^𝑢_𝑖∶=𝑝. (3.49) Then, |𝑥|₀≤𝑞+^𝑇₀𝑥^′(𝑡)𝑑𝑡≤𝑞+𝑝∶=𝑅₀. (3.50) Again from (3.30), we get 𝑥^′₀<𝑎^𝑢+^𝑛_𝑗=1𝑏^𝑢_𝑗|𝑥|₀+^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑥^′₀+ℎ^𝑢𝑒^𝑅0. (3.51) Since ∑^𝑚_𝑖=1𝑐^𝑢_𝑖(1−𝜏^′_𝑖)^𝑢<1, we have 𝑥^′₀<𝑎^𝑢+∑^𝑛_𝑗=1𝑏^𝑢_𝑗𝑅₀+ℎ^𝑢𝑒^𝑅01−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢∶=𝑀₀. (3.52) Choose 𝑡_𝑀,𝑡_𝑚∈[0,𝑇], such that 𝑥𝑡_𝑀=max_{[]𝑡∈0,𝑇}𝑥(𝑡),𝑥𝑡_𝑚=min_{[]𝑡∈0,𝑇}𝑥(𝑡). (3.53) Then, it is clear that 𝑥^′𝑡_𝑀=0,𝑥^′𝑡_𝑚=0. (3.54) From this and (3.30), we obtain that 𝑎𝑡_𝑀=^𝑛_𝑗=1𝑏_𝑗𝑡_𝑀𝑥𝑡_𝑀−𝜎_𝑗𝑡_𝑀+^𝑚_𝑖=1𝑐_𝑖𝑡_𝑀1−𝜏^′_𝑖𝑡_𝑀𝑥^′𝑡_𝑀−𝜏_𝑖𝑡_𝑀+ℎ𝑡_𝑀𝑒^{𝑥(𝑡𝑀)},𝑎𝑡_𝑚=^𝑛_𝑗=1𝑏_𝑗𝑡_𝑚𝑥𝑡_𝑚−𝜎_𝑗𝑡_𝑚+^𝑚_𝑖=1𝑐_𝑖𝑡_𝑚1−𝜏^′_𝑖𝑡_𝑚𝑥^′𝑡_𝑚−𝜏_𝑖𝑡_𝑚+ℎ𝑡_𝑚𝑒^{𝑥(𝑡𝑚)}. (3.55)(3.56) It follows from (3.55) that 𝑎𝑡_𝑀−^𝑚_𝑖=1𝑐_𝑖𝑡_𝑀1−𝜏^′_𝑖𝑡_𝑀𝑀₀−^𝑛_𝑗=1𝑏_𝑗𝑡_𝑀𝑥𝑡_𝑀−ℎ𝑡_𝑀𝑒^{𝑥(𝑡𝑀)}≤0, (3.57) which implies 𝑎^𝑙−∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑢_𝑗−𝑥𝑡_𝑀−ℎ^𝑢∑^𝑛_𝑗=1𝑏^𝑙_𝑗𝑒^{−𝑥(𝑡𝑀)}≤0. (3.58) By the assertion (1) of Lemma 3.2, we have 𝑥𝑡_𝑀≤𝑢⁻or𝑥𝑡_𝑀≥𝑢⁺. (3.59) It follows from (3.56) that 𝑎𝑡_𝑚+^𝑚_𝑖=1𝑐_𝑖𝑡_𝑚1−𝜏^′_𝑖𝑡_𝑚𝑀₀−^𝑛_𝑗=1𝑏_𝑗𝑡_𝑚𝑥𝑡_𝑚−ℎ𝑡_𝑚𝑒^{𝑥(𝑡𝑚)}≥0, (3.60) which implies 𝑎^𝑢+∑^𝑚_𝑖=1𝑐^𝑢_𝑖1−𝜏^′_𝑖^𝑢𝑀₀∑^𝑛_𝑗=1𝑏^𝑙_𝑗−𝑥𝑡_𝑚−ℎ^𝑙∑^𝑛_𝑗=1𝑏^𝑢_𝑗𝑒^{−𝑥(𝑡𝑚)}≥0. (3.61) By the assertion (3) of Lemma 3.2, we have 𝑙⁻≤𝑥𝑡_𝑚≤𝑙⁺. (3.62) Hence, it follows from (3.50), (3.59), and (3.62) that 𝑥𝑡_𝑀∈[𝑙⁻,𝑢⁻]∪𝑢⁺,𝑅₀,𝑥𝑡_𝑚∈𝑙⁻,𝑙⁺. (3.63) Clearly, 𝑙^±,𝑢^± are independent of 𝜆. Now, let us consider 𝑄𝑁(𝑥) with 𝑥∈𝑅. Note that 𝑄𝑁(𝑥)=𝑎−^𝑛_𝑗=1𝑏_𝑗𝑥−ℎ𝑒^𝑥. (3.64) It follows from the assertion (2) of Lemma 3.2 that 𝑄𝑁(𝑥)=0 has two distinct solutions: 𝑢₁=𝑥⁻,𝑢₂=𝑥⁺. (3.65) By the assertion (4) of Lemma 3.2, one can take 𝑣⁻,𝑣⁺>0 such that 𝑢⁻<𝑣⁻<𝑣⁺<𝑢⁺. (3.66) Let Ω₁=𝑥∈𝑋max_{[]𝑡∈0,𝑇}𝑥(𝑡)∈(𝑙⁻−𝛿,𝑣⁻),min_{[]𝑡∈0,𝑇}𝑥(𝑡)∈𝑙⁻−𝛿,𝑙⁺+𝛿,max_{[]𝑡∈0,𝑇}𝑥^′(𝑡)<𝑀₀.,Ω₂=𝑥∈𝑋max_{[]𝑡∈0,𝑇}𝑥(𝑡)∈𝑣⁺,𝑅₀+𝛿,min_{[]𝑡∈0,𝑇}𝑥(𝑡)∈𝑙⁻−𝛿,𝑙⁺+𝛿,max_{[]𝑡∈0,𝑇}𝑥^′(𝑡)<𝑀₀.. (3.67) Then Ω₁,Ω₂ are bounded open subsets of 𝑋. Clearly, Ω_𝑖⊂Ω(𝑖=1,2), where Ω as defined in Lemma 3.5. It follows from Lemma 3.5 that 𝑁∶Ω_𝑖→𝑍 is a 𝑘₀-set-contractive map (𝑖=1,2). Therefore, it follows from Lemmas 2.1 and 3.3 that 𝑁∶Ω_𝑖→𝑍 is 𝐿-𝑘-set contractive on Ω_𝑖(𝑖=1,2) with 𝑘=𝑘₀/𝑙(𝐿)≤𝑘₀<1.
By the assertion (4) of Lemma 3.2, (3.52), (3.63), (3.65), and (3.66), it is easy to verify that Ω_𝑖 satisfies the assumptions (i) and (ii) in Lemma 2.2  (𝑖=1,2). By the assertion (2) of Lemma 3.2, a direct computation gives deg_𝐵𝐽𝑄𝑁,Ω₁∩Ker𝐿,0=sgn−^𝑛_𝑗=1𝑏_𝑗+ℎ𝑒^𝑥−=1,deg_𝐵𝐽𝑄𝑁,Ω₂∩Ker𝐿,0=sgn−^𝑛_𝑗=1𝑏_𝑗+ℎ𝑒^𝑥+=−1. (3.68) 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, (3.1) has at least two 𝑇-periodic solutions 𝑥^∗_𝑖(𝑡) and 𝑥^∗_𝑖∈dom𝐿∩Ω_𝑖(𝑖=1,2). Since Ω₁∩Ω₂=∅, 𝑥^∗_𝑖(𝑖=1,2) are different. Let 𝑦^∗_𝑖(𝑡)=𝑒^{𝑥∗𝑖(𝑡)}(𝑖=1,2). Then 𝑦^∗_𝑖(𝑡)(𝑖=1,2) are two different positive 𝑇-periodic solutions of (1.1). The proof is complete.

Example 3.7. Consider the following equation: 𝑑𝑦(𝑡)(𝑡𝑎(𝑡)(𝑡)((𝑡(𝑡)𝑑𝑑𝑡=𝑦)−𝑏ln𝑦𝑡−𝜎))−𝑐((𝑡(𝑡),𝑑𝑡ln𝑦𝑡−𝜏))−ℎ (3.69) where 𝑎(𝑡)=1+14.2𝑒4.2𝑒sin²𝑡(𝑡)=()𝑐(𝑡)=()(𝑡)(𝑡)𝜖,𝑏2.35+0.15sin𝑡𝜖,0.9+0.1cos𝑡𝜖,𝜎≡2𝜋,𝜏≡4𝜋,ℎ=3𝜖+𝜖2𝑒2𝑒sin𝑡, (3.70) and the constant 𝜖>0. Clearly, (𝐻₁) is satisfied.
Let 𝜇(𝑡) be the inverse function of 𝑡−𝜎(𝑡), and 𝛾(𝑡) be the inverse function of 𝑡−𝜏(𝑡). Then we have 𝜇(𝑡)(𝑡)Γ(𝑡)=𝑏𝜇(𝑡=𝑡+2𝜋,𝛾=𝑡+4𝜋,)1−𝜎^′𝜇(𝑡)−𝑐^′𝛾(𝑡)1−𝜏^′𝛾(𝑡)=𝑏(𝑡+2𝜋)−𝑐^′(𝑡+4𝜋)=(2.35+0.25sin𝑡)𝜖. (3.71) Hence, (𝐻₂) is satisfied.
It is easy to see that 𝑇=2𝜋,𝑎^𝑢=𝜖2.1𝑒,𝑎^𝑙=𝜖,4.2𝑒𝜖𝑎=2.8𝑒,𝑏^𝑢=2.5𝜖,𝑏^𝑙=2.2𝜖,Γ𝑏=2.35𝜖,_𝑢=2.6𝜖,Γ^𝑙=2.1𝜖,Γ=2.35𝜖,𝑐^𝑢=𝜖,ℎ^𝑢=2𝜖𝑒,ℎ^𝑙=𝜖𝑒,ℎ=3𝜖.2𝑒 (3.72) Therefore, we have ℎ^𝑙Γ^𝑢=1,𝑎2.6𝑒Γ^𝑢𝑙=1,ℎ4.41𝑒1+ln^𝑙Γ^𝑢=−ln2.6. (3.73) So, ℎ1+ln^𝑙Γ^𝑢<𝑎^𝑢Γ^𝑙<ℎ^𝑙Γ^𝑢<1. (3.74) Hence, (𝐻₃) is satisfied.
Also, it is easy to see that 𝑎^𝑙−𝑐^𝑢1−𝜏^′𝑢𝑀₀𝑏^𝑢=𝜖/(4.2𝑒)−𝜖𝑀₀=12.5𝜖−𝑀10.5𝑒⁰ℎ2.5<1,𝑞=2ln^𝑙Γ^𝑢=2(ln2.6+1)>1,𝑝=2𝑇𝑎+Γ+𝑏𝑇𝑞1−𝑇Γ+𝑏−𝑐^𝑢=𝜋𝜖/(0.7𝑒)+18.8(ln2.6+1)𝜋𝜖,𝑅1−9.4𝜋𝜖−𝜖₀𝑀=𝑞+𝑝,₀=𝑎^𝑢+𝑏^𝑢𝑅₀+ℎ^𝑢𝑒^𝑅01−𝑐^𝑢1−𝜏^′𝑢=𝜖/(2.1𝑒)+2.5𝜖𝑅₀+2𝜖𝑒^𝑅0−1.1−𝜖 (3.75) Therefore, we obtain that 𝑅₀𝑎>𝑞>^𝑙−𝑐^𝑢1−𝜏^′𝑢𝑀₀𝑏^𝑢. (3.76) Noticing that lim_𝜖→0⁺𝑅₀=𝑞, we have lim_𝜖→0⁺𝑀₀=0. Therefore, for some sufficiently small 𝜖>0, the following inequalities hold: ℎmax1+ln^𝑢𝑏^𝑙{}𝑎,0=max−ln1.1,0=0<^𝑙−𝑐^𝑢1−𝜏^′𝑢𝑀₀𝑏^𝑢,𝑇Γ+𝑏+𝑐^𝑢=9.4𝜋𝜖+𝜖<1. (3.77) By (3.76)-(3.77), (𝐻₄) is also satisfied. Therefore, all necessary conditions of Theorem 3.6 are satisfied. By Theorem 3.6, (3.69) has at least two positive 2𝜋-periodic solutions.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (Grant nos. 10971085 and 11061016).