International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 698529 | 28 pages | https://doi.org/10.5402/2011/698529

Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

Academic Editor: G. Scheuermann
Received01 Apr 2011
Accepted18 May 2011
Published19 Jul 2011

Abstract

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

1. Introduction

Neutral functional differential equations manifest themselves in many fields including biology, mechanics, and economics [1–4]. For example, in population dynamics, since a growing population consumes more (or less) food than a matured one, depending on individual species, this leads to neutral functional equations [1]. These equations also arise in classical β€œcobweb” models in economics where current demand depends on price but supply depends on the previous periodic solutions [2]. The study on neutral functional differential equations is more intricate than ordinary delay differential equations. In recent years, there has been a good amount of work on periodic solutions for neutral differential equations (see [5–12] and the references cited therein). For example, in [5], Wu and Wang discussed the second-order neutral delay differential equation(π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ))ξ…žξ…ž+π‘Ž(𝑑)π‘₯(𝑑)=πœ†π‘(𝑑)𝑓(π‘₯(π‘‘βˆ’πœ(𝑑))).(1.1) By a fixed point theorem, they obtain some existence results of positive periodic solutions for (1.1). Recently, in [6], Cheung et al. considered second-order neutral functional differential equation(π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’πœ(𝑑)))ξ…žξ…ž=π‘Ž(𝑑)π‘₯(𝑑)βˆ’π‘“(𝑑,π‘₯(π‘‘βˆ’πœ(𝑑))).(1.2) By choosing available operators and applying Krasnoselskii's fixed point theorem, they obtained sufficient conditions for the existence of periodic solutions to (1.2).

In general, most of the existing results are concentrated on first-order and second-order neutral functional differential equations, while studies on third-order neutral functional differential equations are rather infrequent, especially on the positive periodic solutions for third-order neutral functional differential equations. In the study of high-order (in particular third-order) differential equations, the naive idea to translate the equation into a first-order differential system by defining π‘₯1=π‘₯, π‘₯2=π‘₯ξ…ž, π‘₯3=π‘₯ξ…žξ…ž,…, works well for showing existence of periodic solutions, however, it does not obviously lead to existence proofs for positive periodic solutions, since the condition π‘₯=π‘₯1β‰₯0 of positivity for the higher order equation is different from the natural positivity condition (π‘₯1,π‘₯2,…)β‰₯0 for the corresponding system. Another approach, which will be used in this paper, is to transform the third-order equation into a corresponding integral equation and to establish the existence of positive periodic solutions based on a fixed point theorem in cones. Following this path one needs an explicit representation of Green's function which is rather intricate to compute.

In this paper, we consider the following third-order neutral functional differential equation:(π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ(𝑑)))ξ…žξ…žξ…ž=βˆ’π‘Ž(𝑑)π‘₯(𝑑)+πœ†π‘(𝑑)𝑓(π‘₯(π‘‘βˆ’πœ(𝑑))).(1.3) Here πœ† is a positive parameter; π‘“βˆˆπΆ(ℝ,[0,∞)), and 𝑓(π‘₯)>0 for π‘₯>0; π‘ŽβˆˆπΆ(ℝ,(0,∞)), π‘βˆˆπΆ(ℝ,(0,∞)), 𝜏,π›ΏβˆˆπΆ1(ℝ,ℝ), π‘Ž(𝑑), 𝑏(𝑑), 𝛿(𝑑), and 𝜏(𝑑) are πœ”-periodic functions.

Notice that here neutral operator (𝐴π‘₯)(𝑑)=π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ(𝑑)) is a natural generalization of the familiar operator (𝐴1π‘₯)(𝑑)=π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ). But 𝐴 possesses a more complicated nonlinearity than 𝐴1. For example, the neutral operators 𝐴1 is homogeneous in the following senses (𝐴1π‘₯)ξ…ž(𝑑)=(𝐴1π‘₯ξ…ž)(𝑑), whereas the neutral operator 𝐴 in general is inhomogeneous. As a consequence many of the new results for differential equations with the neutral operator 𝐴 will not be a direct extension of known theorems for neutral differential equations.

The paper is organized as follows. In Section 2, we first analyze qualitative properties of the generalized neutral operator 𝐴 which will be helpful for further studies of differential equations with this neutral operator; in Section 3, we consider two types of third-order constant coefficient linear differential equations and present their Green's functions and properties for those equation; in Section 4, by an application of the fixed point index theorem we obtain sufficient conditions for the existence, multiplicity and nonexistence of positive periodic solutions to third-order neutral differential equation. We will give an example to illustrate our results; in Section 5, the Lyapunov stability of periodic solutions for the equation will then be established. And an example is also given in this section.

2. Analysis of the Generalized Neutral Operator

Let 𝑋={π‘₯∈𝐢(ℝ,ℝ)∢π‘₯(𝑑+πœ”)=π‘₯(𝑑),π‘‘βˆˆβ„} with norm β€–π‘₯β€–=maxπ‘‘βˆˆ[0,πœ”]|π‘₯(𝑑)|, and let 𝐢+πœ”={π‘₯∈𝐢(ℝ,(0,∞))∢π‘₯(𝑑+πœ”)=π‘₯(𝑑)}, πΆβˆ’πœ”={π‘₯∈𝐢(ℝ,(βˆ’βˆž,0))∢π‘₯(𝑑+πœ”)=π‘₯(𝑑)}. Then (𝑋,β€–β‹…β€–) is a Banach space. A cone 𝐾 in 𝑋 is defined by 𝐾={π‘₯βˆˆπ‘‹βˆΆπ‘₯(𝑑)β‰₯𝛼‖π‘₯β€–, βˆ€π‘‘βˆˆβ„}, where 𝛼 is a fixed positive number with 𝛼<1. Moreover, define operators 𝐴,π΅βˆΆπ‘‹β†’π‘‹ by (𝐴π‘₯)(𝑑)=π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ(𝑑)),(𝐡π‘₯)(𝑑)=𝑐π‘₯(π‘‘βˆ’π›Ώ(𝑑)).(2.1)

Lemma 2.1. If |𝑐|β‰ 1, then the operator 𝐴 has a continuous inverse π΄βˆ’1 on 𝑋, satisfying (1)ξ€·π΄βˆ’1π‘“ξ€ΈβŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩(𝑑)=𝑓(𝑑)+βˆžξ“π‘—=1π‘π‘—π‘“ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺβˆ’,for|𝑐|<1,βˆ€π‘“βˆˆπ‘‹,𝑓(𝑑+𝛿(𝑑))π‘βˆ’βˆžξ“π‘—=11𝑐𝑗+1𝑓𝑠+𝛿(𝑑)+π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ,for|𝑐|>1,βˆ€π‘“βˆˆπ‘‹,(2.2)(2)||ξ€·π΄βˆ’1𝑓||≀(𝑑)‖𝑓‖||||1βˆ’|𝑐|,βˆ€π‘“βˆˆπ‘‹,(2.3)(3)ξ€œπœ”0||ξ€·π΄βˆ’1𝑓||1(𝑑)𝑑𝑑≀||||ξ€œ1βˆ’|𝑐|πœ”0||||𝑓(𝑑)𝑑𝑑,βˆ€π‘“βˆˆπ‘‹.(2.4)

Proof. We have the following cases. Case 1 (|𝑐|<1). Let π‘‘βˆ’π›Ώ(𝑑)=𝑠 and π·π‘—βˆ‘=π‘ βˆ’π‘—βˆ’1𝑖=1𝛿(𝐷𝑖), 𝑗=1,2,…. Therefore 𝐡𝑗π‘₯(𝑑)=𝑐𝑗π‘₯ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ,βˆžξ“π‘—=0𝐡𝑗𝑓(𝑑)=𝑓(𝑑)+βˆžξ“π‘—=1π‘π‘—π‘“ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ.(2.5) Since 𝐴=πΌβˆ’π΅, we get from ‖𝐡‖≀|𝑐|<1 that 𝐴 has a continuous inverse π΄βˆ’1βˆΆπ‘‹β†’π‘‹ with π΄βˆ’1=(πΌβˆ’π΅)βˆ’1=𝐼+βˆžξ“π‘—=1𝐡𝑗=βˆžξ“π‘—=0𝐡𝑗.(2.6) Here 𝐡0=𝐼. Then ξ€·π΄βˆ’1ξ€Έ=𝑓(𝑑)βˆžξ“π‘—=0𝐡𝑗𝑓(𝑑)=βˆžξ“π‘—=0π‘π‘—π‘“ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ,(2.7) and consequently ||ξ€·π΄βˆ’1𝑓||=|||||(𝑑)βˆžξ“π‘—=0𝐡𝑗𝑓|||||=|||||(𝑑)βˆžξ“π‘—=0π‘π‘—π‘“ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ|||||≀‖𝑓‖1βˆ’|𝑐|.(2.8) Moreover, ξ€œπœ”0||ξ€·π΄βˆ’1𝑓||ξ€œ(𝑑)𝑑𝑑=πœ”0|||||βˆžξ“π‘—=0𝐡𝑗𝑓|||||≀(𝑑)π‘‘π‘‘βˆžξ“π‘—=0ξ€œπœ”0||𝐡𝑗𝑓(||=𝑑)π‘‘π‘‘βˆžξ“π‘—=0ξ€œπœ”0|||||π‘π‘—π‘“ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ|||||≀1π‘‘π‘‘ξ€œ1βˆ’|𝑐|πœ”0||||𝑓(𝑑)𝑑𝑑.(2.9)Case 2 (|𝑐|>1). Let 1πΈβˆΆπ‘‹βŸΆπ‘‹,(𝐸π‘₯)(𝑑)=π‘₯(𝑑)βˆ’π‘π΅π‘₯(𝑑+𝛿(𝑑)),1ξ€·π΅βˆΆπ‘‹βŸΆπ‘‹,1π‘₯ξ€Έ1(𝑑)=𝑐π‘₯(𝑑+𝛿(𝑑)).(2.10) By definition of the linear operator 𝐡1, we have 𝐡𝑗1𝑓1(𝑑)=𝑐𝑗𝑓𝑠+π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ.(2.11) Here 𝐷𝑖 is defined as in Case 1. Summing over 𝑗 yields βˆžξ“π‘—=0𝐡𝑗1𝑓(𝑑)=𝑓(𝑑)+βˆžξ“π‘—=11𝑐𝑗𝑓𝑠+π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ.(2.12) Since ‖𝐡1β€–<1, we obtain that the operator 𝐸 has a bounded inverse πΈβˆ’1, πΈβˆ’1βˆΆπ‘‹βŸΆπ‘‹,πΈβˆ’1=ξ€·πΌβˆ’π΅1ξ€Έβˆ’1=𝐼+βˆžξ“π‘—=1𝐡𝑗1,(2.13) and for all π‘“βˆˆπ‘‹ we get ξ€·πΈβˆ’1𝑓(𝑑)=𝑓(𝑑)+βˆžξ“π‘—=1𝐡𝑗1𝑓(𝑑).(2.14) On the other hand, from (𝐴π‘₯)(𝑑)=π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ(𝑑)), we have 1(𝐴π‘₯)(𝑑)=π‘₯(𝑑)βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ(𝑑))=βˆ’π‘π‘₯(π‘‘βˆ’π›Ώ(𝑑))βˆ’π‘ξ‚„,π‘₯(𝑑)(2.15) that is, (𝐴π‘₯)(𝑑)=βˆ’π‘(𝐸π‘₯)(π‘‘βˆ’π›Ώ(𝑑)).(2.16) Let π‘“βˆˆπ‘‹ be arbitrary. We are looking for π‘₯ such that (𝐴π‘₯)(𝑑)=𝑓(𝑑),(2.17) that is βˆ’π‘(𝐸π‘₯)(π‘‘βˆ’π›Ώ(𝑑))=𝑓(𝑑).(2.18) Therefore (𝐸π‘₯)(𝑑)=βˆ’π‘“(𝑑+𝛿(𝑑))𝑐=βˆΆπ‘“1(𝑑),(2.19) and hence 𝐸π‘₯(𝑑)=βˆ’1𝑓1ξ€Έ(𝑑)=𝑓1(𝑑)+βˆžξ“π‘—=1𝐡𝑗1𝑓1ξ€Έ(𝑑)=βˆ’π‘“(𝑑+𝛿(𝑑))π‘βˆ’βˆžξ“π‘—=1𝐡𝑗1𝑓(𝑑+𝛿(𝑑))𝑐,(2.20) proving that π΄βˆ’1 exists and satisfies ξ€Ίπ΄βˆ’1𝑓(𝑑)=βˆ’π‘“(𝑑+𝛿(𝑑))π‘βˆ’βˆžξ“π‘—=1𝐡𝑗1𝑓(𝑑+𝛿(𝑑))𝑐=βˆ’π‘“(𝑑+𝛿(𝑑))π‘βˆ’βˆžξ“π‘—=11𝑐𝑗+1𝑓𝑠+𝛿(𝑑)+π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ,||ξ€Ίπ΄βˆ’1𝑓(||=|||||βˆ’π‘‘)𝑓(𝑑+𝛿(𝑑))π‘βˆ’βˆžξ“π‘—=11𝑐𝑗+1𝑓𝑠+𝛿(𝑑)+π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ|||||≀‖𝑓‖.|𝑐|βˆ’1(2.21) Statements (1) and (2) are proved. From the above proof, (3) can easily be deduced.

Lemma 2.2. If 𝑐<0 and |𝑐|<𝛼, we have for π‘¦βˆˆπΎ that π›Όβˆ’|𝑐|1βˆ’π‘2β€–ξ€·π΄π‘¦β€–β‰€βˆ’1𝑦1(𝑑)≀‖1βˆ’|𝑐|𝑦‖.(2.22)

Proof. Since 𝑐<0 and |𝑐|<𝛼<1, by Lemma 2.1, one has for π‘¦βˆˆπΎ that ξ€·π΄βˆ’1𝑦(𝑑)=𝑦(𝑑)+βˆžξ“π‘—=1π‘π‘—π‘¦ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺ=𝑦(𝑑)+𝑗β‰₯1evenπ‘π‘—π‘¦ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺβˆ’ξ“π‘—β‰₯1odd|𝑐|π‘—π‘¦ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺβ‰₯𝛼‖𝑦‖+𝛼𝑗β‰₯1evenπ‘π‘—ξ“β€–π‘¦β€–βˆ’β€–π‘¦β€–π‘—β‰₯1odd|𝑐|𝑗=𝛼1βˆ’π‘2β€–π‘¦β€–βˆ’|𝑐|1βˆ’π‘2=β€–π‘¦β€–π›Όβˆ’|𝑐|1βˆ’π‘2‖𝑦‖.(2.23)

Lemma 2.3. If 𝑐>0 and 𝑐<1, then for π‘¦βˆˆπΎ, one has 𝛼𝐴1βˆ’π‘β€–π‘¦β€–β‰€βˆ’1𝑦1(𝑑)≀1βˆ’π‘β€–π‘¦β€–.(2.24)

Proof. Since 𝑐>0, 𝑐<1, and 𝛼<1, by Lemma 2.1, we have for π‘¦βˆˆπΎ that ξ€·π΄βˆ’1𝑦(𝑑)=𝑦(𝑑)+𝑗β‰₯1π‘π‘—π‘¦ξƒ©π‘ βˆ’π‘—βˆ’1𝑖=1𝛿𝐷𝑖ξƒͺβ‰₯𝛼‖𝑦‖+𝛼‖𝑦‖𝑗β‰₯1𝑐𝑗=𝛼1βˆ’π‘β€–π‘¦β€–.(2.25)

3. Green's Functions

Theorem 3.1. For 𝜌>0 and β„Žβˆˆπ‘‹, the equation π‘’ξ…žξ…žξ…žβˆ’πœŒ3𝑒=β„Ž(𝑑),𝑒(0)=𝑒(πœ”),π‘’ξ…ž(0)=π‘’ξ…ž(πœ”),π‘’ξ…žξ…ž(0)=π‘’ξ…žξ…ž(πœ”)(3.1) has a unique solution which is of the form ξ€œπ‘’(𝑑)=πœ”0𝐺1(𝑑,𝑠)(βˆ’β„Ž(𝑠))d𝑠,(3.2) where 𝐺1⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©ξ‚ƒβˆš(𝑑,𝑠)=2exp((1/2)𝜌(π‘ βˆ’π‘‘))sin3/2𝜌(π‘‘βˆ’π‘ )+πœ‹/6βˆ’π’²3𝜌2ξ‚€βˆš1+exp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos+3/2πœŒπœ”ξ‚ξ‚exp(𝜌(π‘‘βˆ’π‘ ))3𝜌2ξ‚ƒβˆš(exp(πœŒπœ”)βˆ’1),0β‰€π‘ β‰€π‘‘β‰€πœ”,2exp((1/2)𝜌(π‘ βˆ’π‘‘βˆ’πœ”))sin3/2𝜌(π‘‘βˆ’π‘ +πœ”)+πœ‹/6βˆ’π’΄3𝜌2ξ‚€βˆš1+exp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos+3/2πœŒπœ”ξ‚ξ‚exp(𝜌(𝑑+πœ”βˆ’π‘ ))3𝜌2(exp(πœŒπœ”)βˆ’1),0β‰€π‘‘β‰€π‘ β‰€πœ”,(3.3) where 𝒲 denotes √exp(βˆ’(1/2)πœŒπœ”)sin((3/2)𝜌(π‘‘βˆ’π‘ βˆ’πœ”)+πœ‹/6)and𝒴 denotes √exp(βˆ’(1/2)πœŒπœ”)sin((3/2)𝜌(π‘‘βˆ’π‘ )+πœ‹/6).

Proof. It is easy to check that the associated homogeneous equation of (3.1) has the solution 𝑣(𝑑)=𝑐1exp(πœŒπ‘‘)+exp(βˆ’πœŒπ‘‘/2)(𝑐2√cos(3𝜌/2)𝑑+𝑐3√sin(3𝜌/2)𝑑). The only periodic solution of the associated homogeneous equation of (3.1) is the trivial solution, that is, 𝑐1,𝑐2,𝑐3=0. This follows by assuming that 𝑣(𝑑) is periodic; we immediately get that 𝑐1=0 and by assuming that 𝑐22+𝑐23>0 and choosing πœ‘ such that sinπœ‘=𝑐2/𝑐22+𝑐23,  cosπœ‘=𝑐3/𝑐22+𝑐23, we get 𝑣(𝑑)𝑐22+𝑐23ξ‚€βˆ’=expπœŒπ‘‘2ξ‚ξƒ©βˆšsinπœ‘cos3𝜌2βˆšπ‘‘+cosπœ‘sin3𝜌2𝑑ξƒͺξ‚€βˆ’=expπœŒπ‘‘2ξ‚ξƒ©βˆšsinπœ‘+3𝜌2𝑑ξƒͺ(3.4) which for π‘‘β†’βˆž contradicts periodicity of 𝑣, proving that 𝑐2=𝑐3=0.
Applying the method of variation of parameters, we get π‘ξ…ž1(𝑑)=exp(βˆ’πœŒπ‘‘)3𝜌2β„Žπ‘(𝑑),ξ…ž2ξ‚€βˆš(𝑑)=ξ‚ξ‚€βˆš3/3sinξ‚ξ‚€βˆš3πœŒπ‘‘/2βˆ’(1/3)cos3πœŒπ‘‘/2𝜌2ξ‚€expπœŒπ‘‘2ξ‚π‘β„Ž(𝑑),ξ…ž3ξ‚€βˆš(𝑑)=βˆ’(1/3)sinξ‚βˆ’ξ‚€βˆš3πœŒπ‘‘/2ξ‚ξ‚€βˆš3/3cos3πœŒπ‘‘/2𝜌2ξ‚€expπœŒπ‘‘2ξ‚β„Ž(𝑑),(3.5) and then 𝑐1(𝑑)=𝑐1(ξ€œ0)+𝑑0exp(βˆ’πœŒπ‘ )3𝜌2π‘β„Ž(𝑠)d𝑠,2(𝑑)=𝑐2ξ€œ(0)+𝑑0ξ‚€βˆšξ‚ξ‚€βˆš3/3sinξ‚βˆ’ξ‚€βˆš3πœŒπ‘ /2(1/3)cos3πœŒπ‘ /2𝜌2ξ‚€expπœŒπ‘ 2ξ‚π‘β„Ž(𝑠)d𝑠,3(𝑑)=𝑐3ξ€œ(0)+𝑑0ξ‚€βˆšβˆ’(1/3)sinξ‚βˆ’ξ‚€βˆš3πœŒπ‘ /2ξ‚ξ‚€βˆš3/3cos3πœŒπ‘ /2𝜌2ξ‚€expπœŒπ‘ 2ξ‚β„Ž(𝑠)d𝑠,𝑒(𝑑)=𝑐1(ξ‚€βˆ’π‘‘)exp(πœŒπ‘‘)+expπœŒπ‘‘2𝑐2(βˆšπ‘‘)cos3𝜌2𝑑+𝑐3(βˆšπ‘‘)sin3𝜌2𝑑ξƒͺ=𝑐1(0)exp(πœŒπ‘‘)+𝑐2ξ‚€βˆ’expπœŒπ‘‘2ξ‚ξƒ©βˆšcos32ξƒͺπœŒπ‘‘+𝑐3ξ‚€βˆ’(0)expπœŒπ‘‘2ξ‚ξƒ©βˆšsin32ξƒͺ+ξ€œπœŒπ‘‘π‘‘0exp(𝜌(π‘‘βˆ’π‘ ))3𝜌2ξ€œβ„Ž(𝑠)d𝑠+𝑑0√sin3/2𝜌(π‘ βˆ’π‘‘)βˆ’πœ‹/66𝜌2ξ‚€πœŒexp2(ξ‚π‘ βˆ’π‘‘)β„Ž(𝑠)d𝑠.(3.6) Noting that 𝑒(0)=𝑒(πœ”),π‘’ξ…ž(0)=π‘’ξ…ž(πœ”),π‘’ξ…žξ…ž(0)=π‘’ξ…žξ…ž(πœ”), we obtain 𝑐1ξ€œ(0)=πœ”0exp(𝜌(πœ”βˆ’π‘ ))3𝜌2𝑐(1βˆ’exp(πœŒπœ”))β„Ž(𝑠)d𝑠,2ξ€œ(0)=πœ”0ξ‚ƒξ‚€βˆš2exp(𝜌(π‘ βˆ’πœ”)/2)exp(βˆ’πœŒπœ”/2)sinπœ‹/6βˆ’ξ‚ξ‚„3πœŒπ‘ /2βˆ’sinπ’Ÿ3𝜌2ξ‚€ξ‚€βˆšexp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos𝑐3πœŒπœ”/2+1β„Ž(𝑠)d𝑠,3ξ€œ(0)=πœ”0ξ‚ƒξ‚€βˆš2exp(𝜌(π‘ βˆ’πœ”)/2)exp(βˆ’πœŒπœ”/2)cosπœ‹/6βˆ’ξ‚ξ‚„3πœŒπ‘ /2βˆ’cosπ’Ÿ3𝜌2ξ‚€ξ‚€βˆšexp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos3πœŒπœ”/2+1β„Ž(𝑠)d𝑠,(3.7) where π’Ÿ denotes √(πœ‹/6βˆ’3𝜌(π‘ βˆ’πœ”)/2). Therefore 𝑒(𝑑)=𝑐1ξ‚€βˆ’(𝑑)exp(πœŒπ‘‘)+expπœŒπ‘‘2𝑐2√(𝑑)cos3𝜌2𝑑+𝑐3√(𝑑)sin3𝜌2𝑑ξƒͺ=ξ€œπ‘‘0⎧βŽͺ⎨βŽͺβŽ©ξ‚ƒβˆš2exp((1/2)𝜌(π‘ βˆ’π‘‘))sin3/2𝜌(π‘‘βˆ’π‘ )+πœ‹/6βˆ’π’²3𝜌2ξ‚€βˆš1+exp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos+3/2πœŒπœ”ξ‚ξ‚exp(𝜌(π‘‘βˆ’π‘ ))3𝜌2⎫βŽͺ⎬βŽͺ⎭+ξ€œ(1βˆ’exp(πœŒπœ”))β„Ž(𝑠)dπ‘ πœ”π‘‘βŽ§βŽͺ⎨βŽͺβŽ©ξ‚ƒβˆš2exp((1/2)𝜌(π‘ βˆ’π‘‘βˆ’πœ”))sin3/2𝜌(π‘‘βˆ’π‘ +πœ”)+πœ‹/6βˆ’π’΄3𝜌2ξ‚€βˆš1+exp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos+3/2πœŒπœ”ξ‚ξ‚exp(𝜌(𝑑+πœ”βˆ’π‘ ))3𝜌2(⎫βŽͺ⎬βŽͺ⎭=ξ€œ1βˆ’exp(πœŒπœ”))β„Ž(𝑠)dπ‘ πœ”0𝐺1(𝑑,𝑠)β„Ž(𝑠)d𝑠,(3.8) where 𝐺1(𝑑,𝑠) is defined as in (3.3).
By direct calculation, we get the solution 𝑒 satisfies the periodic boundary value condition of the problem (3.1).

Theorem 3.2. For 𝜌>0 and β„Žβˆˆπ‘‹, the equation π‘’ξ…žξ…žξ…ž+𝜌3𝑒=β„Ž(𝑑),𝑒(0)=𝑒(πœ”),π‘’ξ…ž(0)=π‘’ξ…ž(πœ”),π‘’ξ…žξ…ž(0)=π‘’ξ…žξ…ž(πœ”)(3.9) has a unique πœ”-periodic solution ξ€œπ‘’(𝑑)=πœ”0𝐺2(𝑑,𝑠)β„Ž(𝑠)d𝑠,(3.10) where 𝐺2⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©ξ‚ƒβˆš(𝑑,𝑠)=2exp((1/2)𝜌(π‘‘βˆ’π‘ ))sin3/2𝜌(π‘‘βˆ’π‘ )βˆ’πœ‹/6βˆ’π’°3𝜌2ξ‚€βˆš1+exp(πœŒπœ”)βˆ’2exp((1/2)πœŒπœ”)cos+3/2πœŒπœ”ξ‚ξ‚exp(𝜌(π‘ βˆ’π‘‘))3𝜌2ξ‚ƒβˆš(1βˆ’exp(βˆ’πœŒπœ”)),0β‰€π‘ β‰€π‘‘β‰€πœ”,2exp((1/2)𝜌(𝑑+πœ”βˆ’π‘ ))sin3/2𝜌(𝑑+πœ”βˆ’π‘ )βˆ’πœ‹/6βˆ’π’³3𝜌2ξ‚€βˆš1+exp(πœŒπœ”)βˆ’2exp((1/2)πœŒπœ”)cos+3/2πœŒπœ”ξ‚ξ‚exp(𝜌(π‘ βˆ’π‘‘βˆ’πœ”))3𝜌2(1βˆ’exp(βˆ’πœŒπœ”)),0β‰€π‘‘β‰€π‘ β‰€πœ”.(3.11) where 𝒰 denotes √exp((1/2)πœŒπœ”)sin((3/2)𝜌(π‘‘βˆ’π‘ βˆ’πœ”)βˆ’πœ‹/6)and𝒳 denotes βˆšβˆ’exp((1/2)πœŒπœ”)sin((3/2)𝜌(π‘‘βˆ’π‘ )βˆ’πœ‹/6).

Proof. It is similar to the proof of Theorem 3.1 and can therefore be omitted.

Now we present the properties of Green's functions for (3.1) and (3.9)1𝑙=3𝜌2(exp(πœŒπœ”)βˆ’1),𝐿=3+2exp(βˆ’πœŒπœ”/2)3𝜌2(1βˆ’exp(βˆ’πœŒπœ”/2))2.(3.12)

Theorem 3.3. βˆ«πœ”0𝐺1(𝑑,𝑠)d𝑠=1/𝜌3, and if √3πœŒπœ”<(4/3)πœ‹ holds, then 0<𝑙<𝐺1(𝑑,𝑠)≀𝐿 for all π‘‘βˆˆ[0,πœ”] and π‘ βˆˆ[0,πœ”].

Proof. One has the following: 𝐻1(𝑑,𝑠)=exp(𝜌(π‘‘βˆ’π‘ ))3𝜌2ξ€Ίξ€»,𝐻exp(πœŒπœ”)βˆ’1βˆ—1(𝑑,𝑠)=exp(𝜌(𝑑+πœ”βˆ’π‘ ))3𝜌2ξ€Ίξ€»,𝐻exp(πœŒπœ”)βˆ’12ξ‚ƒβˆš(𝑑,𝑠)=2exp((1/2)𝜌(π‘ βˆ’π‘‘))sinξ‚€ξ‚€ξ‚πœŒξ‚ξ‚„3/2(π‘‘βˆ’π‘ )+πœ‹/6βˆ’π’²3𝜌2ξ‚€βˆš1+exp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos,𝐻3/2πœŒπœ”ξ‚ξ‚βˆ—2ξ‚ƒβˆš(𝑑,𝑠)=2exp((1/2)𝜌(π‘ βˆ’π‘‘βˆ’πœ”))sinξ‚€ξ‚€ξ‚πœŒξ‚ξ‚„3/2(π‘‘βˆ’π‘ +πœ”)+πœ‹/6βˆ’π’΄3𝜌2ξ‚€βˆš1+exp(βˆ’πœŒπœ”)βˆ’2exp(βˆ’πœŒπœ”/2)cos.3/2πœŒπœ”ξ‚ξ‚(3.13) A direct computation shows that βˆ«πœ”0𝐺1(𝑑,𝑠)d𝑠=1/𝜌3. It is easy to see that 𝐻1(𝑑,𝑠)>0 for π‘ βˆˆ[0,𝑑] and π»βˆ—1(𝑑,𝑠)>0 for π‘ βˆˆ[𝑑,πœ”] and √exp(βˆ’πœŒπœ”)+1βˆ’2exp(βˆ’πœŒπœ”/2)cos(3πœŒπœ”/2)>[1βˆ’exp(βˆ’πœŒπœ”/2)]2>0.
For convenience, we denote βˆšπœƒ=(3/2)𝜌(π‘‘βˆ’π‘ )+πœ‹/6𝑔1ξƒ©βˆš(𝑑,𝑠)=sin32πœ‹πœŒ(π‘‘βˆ’π‘ )+6ξƒͺξ‚€βˆ’βˆ’expπœŒπœ”2ξ‚ξƒ©βˆšsin32πœ‹πœŒ(π‘‘βˆ’π‘ βˆ’πœ”)+6ξƒͺξ‚€βˆ’=sin(πœƒ)βˆ’expπœŒπœ”2ξ‚ξƒ©βˆšsinπœƒβˆ’32ξƒͺ,π‘”πœŒπœ”βˆ—1ξƒ©βˆš(𝑑,𝑠)=sin32πœŒπœ‹(π‘‘βˆ’π‘ +πœ”)+6ξƒͺξ‚€βˆ’βˆ’expπœŒπœ”2ξ‚ξƒ©βˆšsin32πœŒπœ‹(π‘‘βˆ’π‘ )+6ξƒͺξƒ©βˆš=sinπœƒ+32ξƒͺξ‚€βˆ’πœŒπœ”βˆ’expπœŒπœ”2sinπœƒ.(3.14) If 𝑔1(𝑑,𝑠)>0 and π‘”βˆ—1(𝑑,𝑠)>0, then obviously 𝐻2(𝑑,𝑠)>0, π»βˆ—2(𝑑,𝑠)>0, and 𝐺1(𝑑,𝑠)>0.
For 0β‰€π‘ β‰€π‘‘β‰€πœ”, Since √3πœŒπœ”<(4/3)πœ‹, we haveπœ‹6βˆšβ‰€πœƒβ‰€32πœ‹πœŒπœ”+6<5πœ‹6,βˆ’πœ‹2<πœ‹6βˆ’βˆš32βˆšπœŒπœ”β‰€πœƒβˆ’32πœ‹πœŒπœ”β‰€6.(3.15)(i)For βˆšβˆ’πœ‹/2<πœƒβˆ’(3/2)πœŒπœ”β‰€0, then sinπœƒ>0, √sin(πœƒβˆ’(3/2)πœŒπœ”)<0, we get 𝑔1(𝑑,𝑠)>0,(ii)For √0<πœƒβˆ’(3/2)πœŒπœ”β‰€πœ‹/6, we have sinπœƒ>0, √sin(πœƒβˆ’(3/2)πœŒπœ”)>0, and √0<34βˆšπœŒπœ”β‰€πœƒβˆ’34πœ‹πœŒπœ”β‰€6+√34πœ‹πœŒπœ”<2,𝑔1ξ‚€βˆ’(𝑑,𝑠)=sin(πœƒ)βˆ’expπœŒπœ”2ξ‚ξƒ©βˆšsinπœƒβˆ’32ξƒͺξƒ©βˆšπœŒπœ”β‰₯sinπœƒβˆ’sinπœƒβˆ’32ξƒͺξƒ©βˆšπœŒπœ”=2cosπœƒβˆ’34ξƒͺξƒ©βˆšπœŒπœ”sin34ξƒͺπœŒπœ”>0.(3.16)
For 0β‰€π‘‘β‰€π‘ β‰€πœ”, βˆ’πœ‹2√<βˆ’32πœ‹πœŒπœ”+6πœ‹β‰€πœƒβ‰€6,πœ‹6βˆšβ‰€πœƒ+32πœ‹πœŒπœ”β‰€6+√325πœŒπœ”<6πœ‹.(3.17)(i)For βˆ’πœ‹/2<πœƒβ‰€0, we have sinπœƒ<0, √sin(πœƒ+(3/2)πœŒπœ”)>0, and then π‘”βˆ—1(𝑑,𝑠)>0.(ii)For 0<πœƒβ‰€πœ‹/6, we have sinπœƒ>0, √sin(πœƒ+(3/2)πœŒπœ”)>0, and √0<πœƒ+34πœ‹πœŒπœ”<2,π‘”βˆ—1(ξƒ©βˆšπ‘‘,𝑠)=sinπœƒ+32ξƒͺξ‚€βˆ’πœŒπœ”βˆ’expπœŒπœ”2ξ‚ξƒ©βˆšsinπœƒβ‰₯sinπœƒ+32ξƒͺξƒ©βˆšπœŒπœ”βˆ’sinπœƒ=2cosπœƒ+34ξƒͺξƒ©βˆšπœŒπœ”sin34ξƒͺπœŒπœ”>0.(3.18)
If √3πœŒπœ”<(4/3)πœŒπœ”, we get 𝑔1(𝑑,𝑠)>0 and π‘”βˆ—1(𝑑,𝑠)>0, proving that 𝐺(𝑑,𝑠)>0 for all π‘‘βˆˆ[0,πœ”] and π‘ βˆˆ[0,πœ”].
Next we compute a lower and an upper bound for 𝐺1(𝑑,𝑠) for π‘ βˆˆ[0,πœ”]. We have1𝑙=3𝜌2(≀exp(πœŒπœ”)βˆ’1)exp(𝜌(𝑑+πœ”βˆ’π‘ ))3𝜌2(exp(πœŒπœ”)βˆ’1)<𝐺1≀(𝑑,𝑠)exp(𝜌(𝑑+πœ”βˆ’π‘ ))3𝜌2ξ€Ίξ€»+ξ€Ίξ€»exp(πœŒπœ”)βˆ’1exp(𝜌(π‘ βˆ’π‘‘βˆ’πœ”)/2)2+2exp(βˆ’πœŒπœ”/2)3𝜌2ξ‚ƒξ‚€βˆšexp(βˆ’πœŒπœ”)+1βˆ’2exp(βˆ’πœŒπœ”/2)cos≀3πœŒπœ”/2exp(πœŒπœ”)3𝜌2ξ€Ίξ€»+exp(πœŒπœ”)βˆ’12+2exp(βˆ’πœŒπœ”/2)3𝜌2ξ‚ƒξ‚€βˆšexp(βˆ’πœŒπœ”)+1βˆ’2exp(βˆ’πœŒπœ”/2)cos≀13πœŒπœ”/23𝜌2ξ€Ίξ€»+1βˆ’exp(βˆ’πœŒπœ”)2+2exp(βˆ’πœŒπœ”/2)3𝜌2ξ€Ίξ€»1βˆ’exp(βˆ’πœŒπœ”/2)2≀3+2exp(βˆ’πœŒπœ”/2)3𝜌2ξ€Ίξ€»1βˆ’exp(βˆ’πœŒπœ”/2)2=𝐿.(3.19) The proof is complete.

Similarly, the following dual theorem can be proved.

Theorem 3.4. βˆ«πœ”0𝐺2(𝑑,𝑠)d𝑠=1/𝜌3, and if √3πœŒπœ”<(4/3)πœ‹ holds, then 0<𝑙<𝐺2(𝑑,𝑠)≀𝐿 for all [0,πœ”] and π‘ βˆˆ[0,πœ”].

4. Positive Periodic Solutions for (1.3)

Define the Banach space 𝑋 as in Section 2. Denote [][]𝑀=max{π‘Ž(𝑑)βˆΆπ‘‘βˆˆ0,πœ”},π‘š=min{π‘Ž(𝑑)βˆΆπ‘‘βˆˆ0,πœ”},𝜌3=𝑀,π‘˜=𝑙(𝑀+π‘š)+𝐿𝑀,π‘˜1=βˆšπ‘˜βˆ’π‘˜2βˆ’4πΏπ‘™π‘€π‘šπ‘™[]2𝐿𝑀,𝛼=π‘šβˆ’(𝑀+π‘š)|𝑐|.𝐿𝑀(1βˆ’|𝑐|)(4.1) It is easy to see that 𝑀,π‘š,𝛽,𝐿,𝑙,π‘˜,π‘˜1>0.

Now we consider (1.3). First let 𝑓0=limπ‘₯β†’0𝑓(π‘₯)π‘₯,π‘“βˆž=limπ‘₯β†’βˆžπ‘“(π‘₯)π‘₯,𝑓0=limπ‘₯β†’0𝑓(π‘₯)π‘₯,π‘“βˆž=limπ‘₯β†’βˆžπ‘“(π‘₯)π‘₯,(4.2) and denote 𝑖0ξ‚€:numberof0β€²sin𝑓0,π‘“βˆžξ‚,𝑖0𝑓:numberof0β€²sin0,π‘“βˆžξ‚,π‘–βˆžξ‚€:numberofβˆžβ€²sin𝑓0,π‘“βˆžξ‚,π‘–βˆžξ‚€π‘“:numberofβˆžβ€²sin0,π‘“βˆžξ‚.(4.3) It is clear that 𝑖0,𝑖0,π‘–βˆž,π‘–βˆžβˆˆ{0,1,2}. We will show that (1.3) has 𝑖0 or π‘–βˆž positive 𝑀-periodic solutions for sufficiently large or small πœ†, respectively.

In the following we discuss (1.3) in two cases, namely, the case where 𝑐<0, and 𝑐>βˆ’min{π‘˜1,π‘š/(𝑀+π‘š)} (note that 𝑐>βˆ’π‘š/(𝑀+π‘š) implies 𝛼>0, 𝑐>βˆ’π‘˜1 implies |𝑐|<𝛼); and the case where 𝑐>0 and 𝑐<min{π‘š/(𝑀+π‘š),(πΏπ‘€βˆ’π‘™π‘š)/((πΏβˆ’π‘™)π‘€βˆ’π‘™π‘š)}, (note that 𝑐<π‘š/(𝑀+π‘š) implies 𝛼>0, 𝑐<(πΏπ‘€βˆ’π‘™π‘š)/((πΏβˆ’π‘™)π‘€βˆ’π‘™π‘š) implies 𝛼<1). Obviously, we have |𝑐|<1 which makes Lemma 2.1 applicable for both cases, and also Lemmas 2.2 or 2.3, respectively.

Let 𝐾={π‘₯βˆˆπ‘‹βˆΆπ‘₯(𝑑)β‰₯𝛼‖π‘₯β€–} denote the cone in 𝑋, where 𝛼 is just as defined above. We also use πΎπ‘Ÿ={π‘₯βˆˆπΎβˆΆβ€–π‘₯β€–<π‘Ÿ} and πœ•πΎπ‘Ÿ={π‘₯βˆˆπΎβˆΆβ€–π‘₯β€–=π‘Ÿ}.

Let 𝑦(𝑑)=(𝐴π‘₯)(𝑑), then from Lemma 2.1 we have π‘₯(𝑑)=(π΄βˆ’1𝑦)(𝑑). Hence (1.3) can be transformed intoπ‘¦ξ…žξ…žξ…žξ€·π΄(𝑑)+π‘Ž(𝑑)βˆ’1𝑦𝐴(𝑑)=πœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦,(π‘‘βˆ’πœ(𝑑))(4.4) which can be further rewritten asπ‘¦ξ…žξ…žξ…žπ΄(𝑑)+π‘Ž(𝑑)𝑦(𝑑)βˆ’π‘Ž(𝑑)𝐻(𝑦(𝑑))=πœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦,(π‘‘βˆ’πœ(𝑑))(4.5) where 𝐻(𝑦(𝑑))=𝑦(𝑑)βˆ’(π΄βˆ’1𝑦)(𝑑)=βˆ’π‘(π΄βˆ’1𝑦)(π‘‘βˆ’π›Ώ(𝑑)).

Now we discuss the two cases separately.

4.1. Case I: 𝑐<0 and 𝑐>βˆ’min{π‘˜1,π‘š/(𝑀+π‘š)}

Now we considerπ‘¦ξ…žξ…žξ…ž(𝑑)+π‘Ž(𝑑)𝑦(𝑑)βˆ’π‘Ž(𝑑)𝐻(𝑦(𝑑))=β„Ž(𝑑),β„ŽβˆˆπΆ+πœ”,(4.6) and define operators 𝑇, ξπ»βˆΆπ‘‹β†’π‘‹ by (ξ€œπ‘‡β„Ž)(𝑑)=𝑑𝑑+πœ”πΊ2((𝑑,𝑠)β„Ž(𝑠)d𝑠,𝐻𝑦𝑑)=π‘€βˆ’π‘Ž(𝑑)𝑦(𝑑)+π‘Ž(𝑑)𝐻(𝑦(𝑑)).(4.7) Clearly 𝑇, 𝐻 are completely continuous, (π‘‡β„Ž)(𝑑)>0 for β„Ž(𝑑)>0 and ‖𝐻‖≀(π‘€βˆ’π‘š+𝑀(|𝑐|/(1βˆ’|𝑐|))). By Theorem 3.2, the solution of (4.6) can be written in the following form:𝑇𝑦(𝑑)=(π‘‡β„Ž)(𝑑)+𝐻𝑦(𝑑).(4.8) In view of 𝑐<0 and 𝑐>βˆ’min{π‘˜1,π‘š/(𝑀+π‘š)}, we haveβ€–β€–π‘‡ξπ»β€–β€–β€–β€–ξπ»β€–β€–β‰€β‰€β€–π‘‡β€–π‘€βˆ’π‘š+π‘š|𝑐|𝑀(1βˆ’|𝑐|)<1,(4.9) and hence 𝐻𝑦(𝑑)=πΌβˆ’π‘‡βˆ’1(π‘‡β„Ž)(𝑑).(4.10) Define an operator π‘ƒβˆΆπ‘‹β†’π‘‹ by 𝐻(π‘ƒβ„Ž)(𝑑)=πΌβˆ’π‘‡βˆ’1(π‘‡β„Ž)(𝑑).(4.11) Obviously, for any β„ŽβˆˆπΆ+πœ”, if (√3/2)πœŒπœ”<πœ‹ hold, 𝑦(𝑑)=(π‘ƒβ„Ž)(𝑑) is the unique positive πœ”-periodic solution of (4.6).

Lemma 4.1. 𝑃 is completely continuous, and (π‘‡β„Ž)(𝑑)≀(π‘ƒβ„Ž)(𝑑)≀𝑀(1βˆ’|𝑐|)β€–π‘šβˆ’(𝑀+π‘š)|𝑐|π‘‡β„Žβ€–,βˆ€β„ŽβˆˆπΆ+πœ”.(4.12)

Proof. By the Neumann expansion of 𝑃, we have 𝐻𝑃=πΌβˆ’π‘‡βˆ’1𝑇=𝑇𝐻𝐼+𝑇𝐻+2𝑇𝐻+β‹―+𝑛𝑇𝑇𝐻+β‹―=𝑇+𝑇𝐻𝑇+2𝑇𝐻𝑇+β‹―+𝑛𝑇+β‹―.(4.13) Since 𝑇 and 𝐻 are completely continuous, so is 𝑃. Moreover, by (4.13) and recalling that ‖𝑇𝐻‖≀(π‘€βˆ’π‘š+π‘š|𝑐|)/𝑀(1βˆ’|𝑐|)<1, we get (π‘‡β„Ž)(𝑑)≀(π‘ƒβ„Ž)(𝑑)≀𝑀(1βˆ’|𝑐|)β€–π‘šβˆ’(𝑀+π‘š)|𝑐|π‘‡β„Žβ€–.(4.14)

Define an operator π‘„βˆΆπ‘‹β†’π‘‹ by𝐴𝑄𝑦(𝑑)=π‘ƒπœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦.(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ(4.15)

Lemma 4.2. One has that 𝑄(𝐾)βŠ‚πΎ.

Proof. From the definition of 𝑄, it is easy to verify that 𝑄𝑦(𝑑+πœ”)=𝑄𝑦(𝑑). For π‘¦βˆˆπΎ, we have from Lemma 4.1 that 𝐴𝑄𝑦(𝑑)=π‘ƒπœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦𝐴(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έβ‰₯π‘‡πœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1π‘¦ξ€Έξ€œ(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ=πœ†π‘‘π‘‘+πœ”πΊ2𝐴(𝑑,𝑠)𝑏(𝑠)π‘“ξ€Ίξ€·βˆ’1π‘¦ξ€Έξ€»ξ€œ(π‘ βˆ’πœ(𝑠))d𝑠β‰₯πœ†π‘™πœ”0𝐴𝑏(𝑠)π‘“ξ€Ίξ€·βˆ’1𝑦(π‘ βˆ’πœ(𝑠))d𝑠.(4.16) On the other hand, 𝐴𝑄𝑦(𝑑)=π‘ƒπœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦≀(π‘‘βˆ’πœ(𝑑))𝑀(1βˆ’|𝑐|)β€–β€–π‘‡ξ€·π΄π‘šβˆ’(𝑀+π‘š)|𝑐|πœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦‖‖(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ=πœ†π‘€(1βˆ’|𝑐|)π‘šβˆ’(𝑀+π‘š)|𝑐|max[]π‘‘βˆˆ0,πœ”ξ€œπ‘‘π‘‘+πœ”πΊ2𝐴(𝑑,𝑠)𝑏(𝑠)π‘“ξ€·ξ€·βˆ’1𝑦(π‘ βˆ’πœ(𝑠))dπ‘ β‰€πœ†π‘€(1βˆ’|𝑐|)πΏξ€œπ‘šβˆ’(𝑀+π‘š)|𝑐|πœ”0𝐴𝑏(𝑠)π‘“ξ€·ξ€·βˆ’1𝑦(π‘ βˆ’πœ(𝑠))d𝑠.(4.17) Therefore 𝑙[]𝑄𝑦(𝑑)β‰₯π‘šβˆ’(𝑀+π‘š)|𝑐|‖𝐿𝑀(1βˆ’|𝑐|)𝑄𝑦‖=𝛼‖𝑄𝑦‖,(4.18) that is, 𝑄(𝐾)βŠ‚πΎ.

From the continuity of 𝑃, it is easy to verify that 𝑄 is completely continuous in 𝑋. Comparing (4.5) to (4.6), it is obvious that the existence of periodic solutions for (4.5) is equivalent to the existence of fixed points for the operator 𝑄 in 𝑋. Recalling Lemma 4.2, the existence of positive periodic solutions for (4.5) is equivalent to the existence of fixed points of 𝑄 in 𝐾. Furthermore, if 𝑄 has a fixed point 𝑦 in 𝐾, it means that (π΄βˆ’1𝑦)(𝑑) is a positive πœ”-periodic solutions of (1.3).

Lemma 4.3. If there exists πœ‚>0 such that π‘“π΄ξ€·ξ€·βˆ’1𝑦β‰₯𝐴(π‘‘βˆ’πœ(𝑑))βˆ’1𝑦[](π‘‘βˆ’πœ(𝑑))πœ‚,forπ‘‘βˆˆ0,πœ”,π‘¦βˆˆπΎ,(4.19) then ‖𝑄𝑦‖β‰₯πœ†π‘™πœ‚π›Όβˆ’|𝑐|1βˆ’π‘2ξ€œπœ”0𝑏(𝑠)d𝑠‖𝑦‖,π‘¦βˆˆπΎ.(4.20)

Proof. By Lemma 2.2, Theorem 3.4, and Lemma 4.1, we have for π‘¦βˆˆπΎ that 𝐴𝑄𝑦(𝑑)=π‘ƒπœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1𝑦𝐴(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έβ‰₯π‘‡πœ†π‘(𝑑)π‘“ξ€·ξ€·βˆ’1π‘¦ξ€Έξ€œ(π‘‘βˆ’πœ(𝑑))ξ€Έξ€Έ=πœ†π‘‘π‘‘+πœ”πΊ2𝐴(𝑑,𝑠)𝑏(𝑠)π‘“ξ€·ξ€·βˆ’1π‘¦ξ€Έξ€Έξ€œ(π‘ βˆ’πœ(𝑠))d𝑠β‰₯πœ†π‘™πœ‚πœ”0𝐴𝑏(𝑠)βˆ’1𝑦(π‘ βˆ’πœ(𝑠))d𝑠β‰₯πœ†π‘™πœ‚π›Όβˆ’|𝑐|1βˆ’π‘2ξ€œπœ”0𝑏(𝑠)d𝑠‖𝑦‖.(4.21) Hence ‖𝑄𝑦‖β‰₯πœ†π‘™πœ‚π›Όβˆ’|𝑐|1βˆ’π‘2ξ€œπœ”0𝑏(𝑠)d𝑠‖𝑦‖,π‘¦βˆˆπΎ.(4.22)

Lemma 4.4. If there exists πœ€>0 such that π‘“π΄ξ€·ξ€·βˆ’1𝑦≀𝐴(π‘‘βˆ’πœ(𝑑))βˆ’1𝑦[](π‘‘βˆ’πœ(𝑑))πœ€,forπ‘‘βˆˆ0,πœ”,π‘¦βˆˆπΎ,(4.23) then βˆ«β€–π‘„π‘¦β€–β‰€πœ†πœ€πΏπ‘€πœ”0𝑏(𝑠)dπ‘ π‘šβˆ’(𝑀+π‘š)|𝑐|‖𝑦‖,π‘¦βˆˆπΎ.(4.24)

Proof. By Lemma 2.2, Theorem 3.4, and Lemma 4.1, we have ‖𝑄𝑦(𝑑)β€–β‰€πœ†π‘€(1βˆ’|𝑐|)πΏξ€œπ‘šβˆ’(𝑀+π‘š)|𝑐|πœ”0𝐴𝑏(𝑠)π‘“ξ€·ξ€·βˆ’1𝑦(π‘ βˆ’πœ(𝑠))dπ‘ β‰€πœ†π‘€(1βˆ’|𝑐|)ξ€œπ‘šβˆ’(𝑀+π‘š)|𝑐|πΏπœ€πœ”0𝐴𝑏(𝑠)βˆ’1π‘¦ξ€Έβˆ«(π‘ βˆ’πœ(𝑠))dπ‘ β‰€πœ†πœ€πΏπ‘€πœ”0𝑏(𝑠)dπ‘ π‘šβˆ’(𝑀+π‘š)|𝑐|‖𝑦‖.(4.25)

Define ξ‚»π‘ŸπΉ(π‘Ÿ)=max𝑓(𝑑)∢0≀𝑑≀,𝑓1βˆ’|𝑐|1ξ‚»(π‘Ÿ)=min𝑓(𝑑)βˆΆπ›Όβˆ’|𝑐|1βˆ’π‘2π‘Ÿπ‘Ÿβ‰€π‘‘β‰€ξ‚Ό.1βˆ’|𝑐|(4.26)

Lemma 4.5. If π‘¦βˆˆπœ•πΎπ‘Ÿ, then ‖𝑄𝑦‖β‰₯πœ†π‘™π‘“1ξ€œ(π‘Ÿ)πœ”0𝑏(𝑠)d𝑠‖𝑦‖.(4.27)

Proof. By Lemma 2.2, we obtain ((π›Όβˆ’|𝑐|)/(1βˆ’π‘2))π‘Ÿβ‰€(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑))β‰€π‘Ÿ/(1βˆ’|𝑐|) for π‘¦βˆˆπœ•πΎπ‘Ÿ, which yields 𝑓((π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)))β‰₯𝑓1(π‘Ÿ). The Lemma now follows analog to the proof of Lemma 4.3.

Lemma 4.6. If π‘¦βˆˆπœ•πΎπ‘Ÿ, then β€–π‘„π‘¦β€–β‰€πœ†πΏπ‘€(1βˆ’|𝑐|)𝐹(π‘Ÿ)ξ€œπ‘šβˆ’(𝑀+π‘š)|𝑐|πœ”0𝑏(𝑠)d𝑠‖𝑦‖.(4.28)

Proof. By Lemma 2.2, we can have 0≀(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑))β‰€π‘Ÿ/(1βˆ’|𝑐|) for π‘¦βˆˆπœ•πΎπ‘Ÿ, which yields 𝑓((π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)))≀𝐹(π‘Ÿ). Similar to the proof of Lemma 4.4, we get the conclusion.

We quote the fixed point theorem on which our results will be based.

Lemma 4.7 (see [13]). Let 𝑋 be a Banach space and 𝐾 a cone in 𝑋. For π‘Ÿ>0, define πΎπ‘Ÿ={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<π‘Ÿ}. Assume that π‘‡βˆΆπΎπ‘Ÿβ†’πΎ is completely continuous such that 𝑇π‘₯β‰ π‘₯ for π‘₯βˆˆπœ•πΎπ‘Ÿ={π‘’βˆˆπΎβˆΆβ€–π‘’β€–=π‘Ÿ}. (i)If ‖𝑇π‘₯β€–β‰₯β€–π‘₯β€– for π‘₯βˆˆπœ•πΎπ‘Ÿ, then 𝑖(T,πΎπ‘Ÿ,𝐾)=0.(ii)If ‖𝑇π‘₯‖≀‖π‘₯β€– for π‘₯βˆˆπœ•πΎπ‘Ÿ, then 𝑖(𝑇,πΎπ‘Ÿ,𝐾)=1.

Now we give our main results on positive periodic solutions for (1.3).

Theorem 4.8. (a) If 𝑖0=1 or 2, then (1.3) has 𝑖0 positive πœ”-periodic solutions for πœ†>1/𝑓1∫(1)π‘™πœ”0𝑏(𝑠)d𝑠>0,
(b) if π‘–βˆž=1 or 2, then (1.3) has π‘–βˆž positive πœ”-periodic solutions for ∫0<πœ†<(π‘šβˆ’(𝑀+π‘š)|𝑐|)/𝐿𝑀(1βˆ’|𝑐|)𝐹(1)πœ”0𝑏(𝑠)d𝑠,
(c) if π‘–βˆž=0 or 𝑖0=0, then (1.3) has no positive πœ”-periodic solutions for sufficiently small or sufficiently large πœ†>0, respectively.

Proof. (a) Choose π‘Ÿ1=1. Taking πœ†0=1/𝑓1(π‘Ÿ1∫)π‘™πœ”0𝑏(𝑠)d𝑠>0, then for all πœ†>πœ†0, we have from Lemma 4.5 that ‖𝑄𝑦‖>‖𝑦‖,forπ‘¦βˆˆπœ•πΎπ‘Ÿ1.(4.29)Case 1. If 𝑓0=0, we can choose 0<π‘Ÿ2<π‘Ÿ1, so that 𝑓(𝑒)β‰€πœ€π‘’ for 0β‰€π‘’β‰€π‘Ÿ2, where the constant πœ€>0 satisfies βˆ«πœ†πœ€πΏπ‘€πœ”0𝑏(𝑠)dπ‘ π‘šβˆ’(𝑀+π‘š)|𝑐|<1.(4.30) Letting π‘Ÿ2=(1βˆ’|𝑐|)π‘Ÿ2, we have 𝑓((π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)))β‰€πœ€(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)) for π‘¦βˆˆπΎπ‘Ÿ2. By Lemma 2.2, we have 0≀(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑))≀‖𝑦‖/(1βˆ’|𝑐|)β‰€π‘Ÿ2 for π‘¦βˆˆπœ•πΎπ‘Ÿ2. In view of Lemma 4.4 and (4.30), we have for π‘¦βˆˆπœ•πΎπ‘Ÿ2 that βˆ«β€–π‘„π‘¦β€–β‰€πœ†πœ€πΏπ‘€πœ”0𝑏(𝑠)dπ‘ π‘šβˆ’(𝑀+π‘š)|𝑐|‖𝑦‖<‖𝑦‖.(4.31) It follows from Lemma 4.7 and (4.29) that 𝑖𝑄,πΎπ‘Ÿ2ξ€Έξ€·,𝐾=1,𝑖𝑄,πΎπ‘Ÿ1ξ€Έ,𝐾=0.(4.32) thus 𝑖(𝑄,πΎπ‘Ÿ1β§΅πΎπ‘Ÿ2,𝐾)=βˆ’1 and 𝑄 has a fixed point 𝑦 in πΎπ‘Ÿ1β§΅πΎπ‘Ÿ2, which means (π΄βˆ’1𝑦)(𝑑) is a positive πœ”-positive solution of (1.3) for πœ†>πœ†0.Case 2. If π‘“βˆž=0, there exists a constant 𝐻>0 such that 𝑓(𝑒)β‰€πœ€π‘’ for 𝐻𝑒β‰₯, where the constant πœ€>0 satisfies βˆ«πœ†πœ€πΏπ‘€πœ”0𝑏(𝑠)dπ‘ π‘šβˆ’(𝑀+π‘š)|𝑐|<1.(4.33) Letting π‘Ÿ3=max{2π‘Ÿ1,𝐻(1βˆ’π‘2)/(π›Όβˆ’|𝑐|)}, we have 𝑓((π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)))β‰€πœ€(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)) for π‘¦βˆˆπΎπ‘Ÿ3. By Lemma 2.2, we have (π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑))β‰₯((π›Όβˆ’|𝑐|)/(1βˆ’π‘2𝐻))‖𝑦‖β‰₯ for π‘¦βˆˆπœ•πΎπ‘Ÿ3. Thus by Lemma 4.4 and (4.33), we have for π‘¦βˆˆπœ•πΎπ‘Ÿ3 that βˆ«β€–π‘„π‘¦β€–β‰€πœ†πœ€πΏπ‘€πœ”0𝑏(𝑠)dπ‘ π‘šβˆ’(𝑀+π‘š)|𝑐|‖𝑦‖<‖𝑦‖.(4.34) Recalling Lemma 4.7 and (4.29) and that 𝑖𝑄,πΎπ‘Ÿ3ξ€Έξ€·,𝐾=1,𝑖𝑄,πΎπ‘Ÿ1ξ€Έ,𝐾=0,(4.35) then 𝑖(𝑄,πΎπ‘Ÿ3β§΅πΎπ‘Ÿ1,𝐾)=1 and 𝑄 has a fixed point 𝑦 in πΎπ‘Ÿ3β§΅πΎπ‘Ÿ1, which means (π΄βˆ’1𝑦)(𝑑) is a positive πœ”-positive solution of (1.3) for πœ†>πœ†0.Case 3. If 𝑓0=π‘“βˆž=0, from the above arguments, there exist 0<π‘Ÿ2<π‘Ÿ1<π‘Ÿ3 such that 𝑄 has a fixed point 𝑦1(𝑑) in πΎπ‘Ÿ1β§΅πΎπ‘Ÿ2 and a fixed point 𝑦2(𝑑) in πΎπ‘Ÿ3β§΅πΎπ‘Ÿ1. Consequently, (π΄βˆ’1𝑦1)(𝑑) and (π΄βˆ’1𝑦2)(𝑑) are two positive πœ”-periodic solutions of (1.3) for πœ†>πœ†0.(b)Let π‘Ÿ1=1. Taking πœ†0=(π‘šβˆ’(𝑀+π‘š)|𝑐|)/𝐿𝑀(1βˆ’|𝑐|)𝐹(π‘Ÿ1)βˆ«πœ”0𝑏(𝑠)d𝑠>0, then by Lemma 4.6 we know if πœ†<πœ†0, then ‖𝑄𝑦‖<‖𝑦‖,π‘¦βˆˆπœ•πΎπ‘Ÿ1.(4.36)Case 1. If 𝑓0=∞, we can choose 0<π‘Ÿ2<π‘Ÿ1 so that 𝑓(𝑒)β‰₯πœ‚π‘’ for 0β‰€π‘’β‰€π‘Ÿ2, where the constant πœ‚>0 satisfies πœ†π‘™πœ‚π›Όβˆ’|𝑐|1βˆ’π‘2ξ€œπœ”0𝑏(𝑠)d𝑠>1.(4.37) Letting π‘Ÿ2=(1βˆ’|𝑐|)π‘Ÿ2, we have 𝑓((π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)))β‰₯πœ‚(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)) for π‘¦βˆˆπΎπ‘Ÿ2. By Lemma 2.2, we have 0≀(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑))≀‖𝑦‖/(1βˆ’|𝑐|)β‰€π‘Ÿ2 for π‘¦βˆˆπœ•πΎπ‘Ÿ2. Thus by Lemma 4.3 and (4.37), ‖𝑄𝑦‖β‰₯πœ†π‘™πœ‚π›Όβˆ’|𝑐|1βˆ’π‘2ξ€œπœ”0𝑏(𝑠)d𝑠‖𝑦‖>‖𝑦‖.(4.38) It follows from Lemma 4.7 and (4.36) that 𝑖𝑄,πΎπ‘Ÿ2ξ€Έξ€·,𝐾=0,𝑖𝑄,πΎπ‘Ÿ1ξ€Έ,𝐾=1,(4.39) which implies 𝑖(𝑄,πΎπ‘Ÿ1β§΅πΎπ‘Ÿ2,𝐾)=1 and 𝑄 has a fixed point 𝑦 in πΎπ‘Ÿ1β§΅πΎπ‘Ÿ2. Therefore (π΄βˆ’1𝑦)(𝑑) is a positive πœ”-periodic solution of (1.3) for 0<πœ†<πœ†0.Case 2. If π‘“βˆž=∞, there exists a constant 𝐻>0 such that 𝑓(𝑒)β‰₯πœ‚π‘’ for 𝐻𝑒β‰₯, where the constant πœ‚>0 satisfies πœ†π‘™πœ‚π›Όβˆ’|𝑐|1βˆ’π‘2ξ€œπœ”0𝑏(𝑠)d𝑠>1.(4.40) Let π‘Ÿ3=max{2π‘Ÿ1,𝐻(1βˆ’π‘2)/(π›Όβˆ’|𝑐|)}, we have 𝑓((π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)))β‰₯πœ‚(π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑)) for π‘¦βˆˆπΎπ‘Ÿ3. By Lemma 2.2, we have (π΄βˆ’1𝑦)(π‘‘βˆ’πœ(𝑑))β‰₯((π›Όβˆ’|𝑐|)/(1βˆ’π‘2