International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

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

Jingli Ren, Zhibo Cheng, Yueli Chen, "Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation", International Scholarly Research Notices, vol. 2011, Article ID 698529, 28 pages, 2011. 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 [1โ€“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 โ€œcobwebโ€ 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 [5โ€“12] and the references cited therein). For example, in [5], Wu and Wang discussed the second-order neutral delay differential equation(๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ))๎…ž๎…ž+๐‘Ž(๐‘ก)๐‘ฅ(๐‘ก)=๐œ†๐‘(๐‘ก)๐‘“(๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))).(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(๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก)))๎…ž๎…ž=๐‘Ž(๐‘ก)๐‘ฅ(๐‘ก)โˆ’๐‘“(๐‘ก,๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))).(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 ๐‘ฅ1=๐‘ฅ, ๐‘ฅ2=๐‘ฅ๎…ž, ๐‘ฅ3=๐‘ฅ๎…ž๎…ž,โ€ฆ, works well for showing existence of periodic solutions, however, it does not obviously lead to existence proofs for positive periodic solutions, since the condition ๐‘ฅ=๐‘ฅ1โ‰ฅ0 of positivity for the higher order equation is different from the natural positivity condition (๐‘ฅ1,๐‘ฅ2,โ€ฆ)โ‰ฅ0 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:(๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก)))๎…ž๎…ž๎…ž=โˆ’๐‘Ž(๐‘ก)๐‘ฅ(๐‘ก)+๐œ†๐‘(๐‘ก)๐‘“(๐‘ฅ(๐‘กโˆ’๐œ(๐‘ก))).(1.3) Here ๐œ† is a positive parameter; ๐‘“โˆˆ๐ถ(โ„,[0,โˆž)), and ๐‘“(๐‘ฅ)>0 for ๐‘ฅ>0; ๐‘Žโˆˆ๐ถ(โ„,(0,โˆž)), ๐‘โˆˆ๐ถ(โ„,(0,โˆž)), ๐œ,๐›ฟโˆˆ๐ถ1(โ„,โ„), ๐‘Ž(๐‘ก), ๐‘(๐‘ก), ๐›ฟ(๐‘ก), and ๐œ(๐‘ก) are ๐œ”-periodic functions.

Notice that here neutral operator (๐ด๐‘ฅ)(๐‘ก)=๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก)) is a natural generalization of the familiar operator (๐ด1๐‘ฅ)(๐‘ก)=๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ). But ๐ด possesses a more complicated nonlinearity than ๐ด1. For example, the neutral operators ๐ด1 is homogeneous in the following senses (๐ด1๐‘ฅ)๎…ž(๐‘ก)=(๐ด1๐‘ฅ๎…ž)(๐‘ก), whereas the neutral operator ๐ด in general is inhomogeneous. As a consequence many of the new results for differential equations with the neutral operator ๐ด 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 ๐ด 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 ๐‘‹={๐‘ฅโˆˆ๐ถ(โ„,โ„)โˆถ๐‘ฅ(๐‘ก+๐œ”)=๐‘ฅ(๐‘ก),๐‘กโˆˆโ„} with norm โ€–๐‘ฅโ€–=max๐‘กโˆˆ[0,๐œ”]|๐‘ฅ(๐‘ก)|, and let ๐ถ+๐œ”={๐‘ฅโˆˆ๐ถ(โ„,(0,โˆž))โˆถ๐‘ฅ(๐‘ก+๐œ”)=๐‘ฅ(๐‘ก)}, ๐ถโˆ’๐œ”={๐‘ฅโˆˆ๐ถ(โ„,(โˆ’โˆž,0))โˆถ๐‘ฅ(๐‘ก+๐œ”)=๐‘ฅ(๐‘ก)}. Then (๐‘‹,โ€–โ‹…โ€–) is a Banach space. A cone ๐พ in ๐‘‹ is defined by ๐พ={๐‘ฅโˆˆ๐‘‹โˆถ๐‘ฅ(๐‘ก)โ‰ฅ๐›ผโ€–๐‘ฅโ€–, โˆ€๐‘กโˆˆโ„}, where ๐›ผ is a fixed positive number with ๐›ผ<1. Moreover, define operators ๐ด,๐ตโˆถ๐‘‹โ†’๐‘‹ by (๐ด๐‘ฅ)(๐‘ก)=๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก)),(๐ต๐‘ฅ)(๐‘ก)=๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก)).(2.1)

Lemma 2.1. If |๐‘|โ‰ 1, then the operator ๐ด has a continuous inverse ๐ดโˆ’1 on ๐‘‹, satisfying (1)๎€ท๐ดโˆ’1๐‘“๎€ธโŽงโŽชโŽชโŽจโŽชโŽชโŽฉ(๐‘ก)=๐‘“(๐‘ก)+โˆž๎“๐‘—=1๐‘๐‘—๐‘“๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒชโˆ’,for|๐‘|<1,โˆ€๐‘“โˆˆ๐‘‹,๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘โˆ’โˆž๎“๐‘—=11๐‘๐‘—+1๐‘“๎ƒฉ๐‘ +๐›ฟ(๐‘ก)+๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช,for|๐‘|>1,โˆ€๐‘“โˆˆ๐‘‹,(2.2)(2)||๎€ท๐ดโˆ’1๐‘“๎€ธ||โ‰ค(๐‘ก)โ€–๐‘“โ€–||||1โˆ’|๐‘|,โˆ€๐‘“โˆˆ๐‘‹,(2.3)(3)๎€œ๐œ”0||๎€ท๐ดโˆ’1๐‘“๎€ธ||1(๐‘ก)๐‘‘๐‘กโ‰ค||||๎€œ1โˆ’|๐‘|๐œ”0||||๐‘“(๐‘ก)๐‘‘๐‘ก,โˆ€๐‘“โˆˆ๐‘‹.(2.4)

Proof. We have the following cases. Case 1 (|๐‘|<1). Let ๐‘กโˆ’๐›ฟ(๐‘ก)=๐‘  and ๐ท๐‘—โˆ‘=๐‘ โˆ’๐‘—โˆ’1๐‘–=1๐›ฟ(๐ท๐‘–), ๐‘—=1,2,โ€ฆ. Therefore ๐ต๐‘—๐‘ฅ(๐‘ก)=๐‘๐‘—๐‘ฅ๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช,โˆž๎“๐‘—=0๎€ท๐ต๐‘—๐‘“๎€ธ(๐‘ก)=๐‘“(๐‘ก)+โˆž๎“๐‘—=1๐‘๐‘—๐‘“๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช.(2.5) Since ๐ด=๐ผโˆ’๐ต, we get from โ€–๐ตโ€–โ‰ค|๐‘|<1 that ๐ด has a continuous inverse ๐ดโˆ’1โˆถ๐‘‹โ†’๐‘‹ with ๐ดโˆ’1=(๐ผโˆ’๐ต)โˆ’1=๐ผ+โˆž๎“๐‘—=1๐ต๐‘—=โˆž๎“๐‘—=0๐ต๐‘—.(2.6) Here ๐ต0=๐ผ. Then ๎€ท๐ดโˆ’1๎€ธ=๐‘“(๐‘ก)โˆž๎“๐‘—=0๎€บ๐ต๐‘—๐‘“๎€ป(๐‘ก)=โˆž๎“๐‘—=0๐‘๐‘—๐‘“๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช,(2.7) and consequently ||๎€ท๐ดโˆ’1๐‘“๎€ธ||=|||||(๐‘ก)โˆž๎“๐‘—=0๎€บ๐ต๐‘—๐‘“๎€ป|||||=|||||(๐‘ก)โˆž๎“๐‘—=0๐‘๐‘—๐‘“๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช|||||โ‰คโ€–๐‘“โ€–1โˆ’|๐‘|.(2.8) Moreover, ๎€œ๐œ”0||๎€ท๐ดโˆ’1๐‘“๎€ธ||๎€œ(๐‘ก)๐‘‘๐‘ก=๐œ”0|||||โˆž๎“๐‘—=0๎€ท๐ต๐‘—๐‘“๎€ธ|||||โ‰ค(๐‘ก)๐‘‘๐‘กโˆž๎“๐‘—=0๎€œ๐œ”0||๎€ท๐ต๐‘—๐‘“๎€ธ(||=๐‘ก)๐‘‘๐‘กโˆž๎“๐‘—=0๎€œ๐œ”0|||||๐‘๐‘—๐‘“๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช|||||โ‰ค1๐‘‘๐‘ก๎€œ1โˆ’|๐‘|๐œ”0||||๐‘“(๐‘ก)๐‘‘๐‘ก.(2.9)Case 2 (|๐‘|>1). Let 1๐ธโˆถ๐‘‹โŸถ๐‘‹,(๐ธ๐‘ฅ)(๐‘ก)=๐‘ฅ(๐‘ก)โˆ’๐‘๐ต๐‘ฅ(๐‘ก+๐›ฟ(๐‘ก)),1๎€ท๐ตโˆถ๐‘‹โŸถ๐‘‹,1๐‘ฅ๎€ธ1(๐‘ก)=๐‘๐‘ฅ(๐‘ก+๐›ฟ(๐‘ก)).(2.10) By definition of the linear operator ๐ต1, we have ๎€ท๐ต๐‘—1๐‘“๎€ธ1(๐‘ก)=๐‘๐‘—๐‘“๎ƒฉ๐‘ +๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช.(2.11) Here ๐ท๐‘– is defined as in Case 1. Summing over ๐‘— yields โˆž๎“๐‘—=0๎€ท๐ต๐‘—1๐‘“๎€ธ(๐‘ก)=๐‘“(๐‘ก)+โˆž๎“๐‘—=11๐‘๐‘—๐‘“๎ƒฉ๐‘ +๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช.(2.12) Since โ€–๐ต1โ€–<1, we obtain that the operator ๐ธ has a bounded inverse ๐ธโˆ’1, ๐ธโˆ’1โˆถ๐‘‹โŸถ๐‘‹,๐ธโˆ’1=๎€ท๐ผโˆ’๐ต1๎€ธโˆ’1=๐ผ+โˆž๎“๐‘—=1๐ต๐‘—1,(2.13) and for all ๐‘“โˆˆ๐‘‹ we get ๎€ท๐ธโˆ’1๐‘“๎€ธ(๐‘ก)=๐‘“(๐‘ก)+โˆž๎“๐‘—=1๎€ท๐ต๐‘—1๐‘“๎€ธ(๐‘ก).(2.14) On the other hand, from (๐ด๐‘ฅ)(๐‘ก)=๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก)), we have ๎‚ƒ1(๐ด๐‘ฅ)(๐‘ก)=๐‘ฅ(๐‘ก)โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก))=โˆ’๐‘๐‘ฅ(๐‘กโˆ’๐›ฟ(๐‘ก))โˆ’๐‘๎‚„,๐‘ฅ(๐‘ก)(2.15) that is, (๐ด๐‘ฅ)(๐‘ก)=โˆ’๐‘(๐ธ๐‘ฅ)(๐‘กโˆ’๐›ฟ(๐‘ก)).(2.16) Let ๐‘“โˆˆ๐‘‹ be arbitrary. We are looking for ๐‘ฅ such that (๐ด๐‘ฅ)(๐‘ก)=๐‘“(๐‘ก),(2.17) that is โˆ’๐‘(๐ธ๐‘ฅ)(๐‘กโˆ’๐›ฟ(๐‘ก))=๐‘“(๐‘ก).(2.18) Therefore (๐ธ๐‘ฅ)(๐‘ก)=โˆ’๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘=โˆถ๐‘“1(๐‘ก),(2.19) and hence ๎€ท๐ธ๐‘ฅ(๐‘ก)=โˆ’1๐‘“1๎€ธ(๐‘ก)=๐‘“1(๐‘ก)+โˆž๎“๐‘—=1๎€ท๐ต๐‘—1๐‘“1๎€ธ(๐‘ก)=โˆ’๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘โˆ’โˆž๎“๐‘—=1๐ต๐‘—1๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘,(2.20) proving that ๐ดโˆ’1 exists and satisfies ๎€บ๐ดโˆ’1๐‘“๎€ป(๐‘ก)=โˆ’๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘โˆ’โˆž๎“๐‘—=1๐ต๐‘—1๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘=โˆ’๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘โˆ’โˆž๎“๐‘—=11๐‘๐‘—+1๐‘“๎ƒฉ๐‘ +๐›ฟ(๐‘ก)+๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช,||๎€บ๐ดโˆ’1๐‘“๎€ป(||=|||||โˆ’๐‘ก)๐‘“(๐‘ก+๐›ฟ(๐‘ก))๐‘โˆ’โˆž๎“๐‘—=11๐‘๐‘—+1๐‘“๎ƒฉ๐‘ +๐›ฟ(๐‘ก)+๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช|||||โ‰คโ€–๐‘“โ€–.|๐‘|โˆ’1(2.21) Statements (1) and (2) are proved. From the above proof, (3) can easily be deduced.

Lemma 2.2. If ๐‘<0 and |๐‘|<๐›ผ, we have for ๐‘ฆโˆˆ๐พ that ๐›ผโˆ’|๐‘|1โˆ’๐‘2โ€–๎€ท๐ด๐‘ฆโ€–โ‰คโˆ’1๐‘ฆ๎€ธ1(๐‘ก)โ‰คโ€–1โˆ’|๐‘|๐‘ฆโ€–.(2.22)

Proof. Since ๐‘<0 and |๐‘|<๐›ผ<1, by Lemma 2.1, one has for ๐‘ฆโˆˆ๐พ that ๎€ท๐ดโˆ’1๐‘ฆ๎€ธ(๐‘ก)=๐‘ฆ(๐‘ก)+โˆž๎“๐‘—=1๐‘๐‘—๐‘ฆ๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช๎“=๐‘ฆ(๐‘ก)+๐‘—โ‰ฅ1even๐‘๐‘—๐‘ฆ๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒชโˆ’๎“๐‘—โ‰ฅ1odd|๐‘|๐‘—๐‘ฆ๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช๎“โ‰ฅ๐›ผโ€–๐‘ฆโ€–+๐›ผ๐‘—โ‰ฅ1even๐‘๐‘—๎“โ€–๐‘ฆโ€–โˆ’โ€–๐‘ฆโ€–๐‘—โ‰ฅ1odd|๐‘|๐‘—=๐›ผ1โˆ’๐‘2โ€–๐‘ฆโ€–โˆ’|๐‘|1โˆ’๐‘2=โ€–๐‘ฆโ€–๐›ผโˆ’|๐‘|1โˆ’๐‘2โ€–๐‘ฆโ€–.(2.23)

Lemma 2.3. If ๐‘>0 and ๐‘<1, then for ๐‘ฆโˆˆ๐พ, one has ๐›ผ๎€ท๐ด1โˆ’๐‘โ€–๐‘ฆโ€–โ‰คโˆ’1๐‘ฆ๎€ธ1(๐‘ก)โ‰ค1โˆ’๐‘โ€–๐‘ฆโ€–.(2.24)

Proof. Since ๐‘>0, ๐‘<1, and ๐›ผ<1, by Lemma 2.1, we have for ๐‘ฆโˆˆ๐พ that ๎€ท๐ดโˆ’1๐‘ฆ๎€ธ๎“(๐‘ก)=๐‘ฆ(๐‘ก)+๐‘—โ‰ฅ1๐‘๐‘—๐‘ฆ๎ƒฉ๐‘ โˆ’๐‘—โˆ’1๎“๐‘–=1๐›ฟ๎€ท๐ท๐‘–๎€ธ๎ƒช๎“โ‰ฅ๐›ผโ€–๐‘ฆโ€–+๐›ผโ€–๐‘ฆโ€–๐‘—โ‰ฅ1๐‘๐‘—=๐›ผ1โˆ’๐‘โ€–๐‘ฆโ€–.(2.25)

3. Green's Functions

Theorem 3.1. For ๐œŒ>0 and โ„Žโˆˆ๐‘‹, the equation ๐‘ข๎…ž๎…ž๎…žโˆ’๐œŒ3๐‘ข=โ„Ž(๐‘ก),๐‘ข(0)=๐‘ข(๐œ”),๐‘ข๎…ž(0)=๐‘ข๎…ž(๐œ”),๐‘ข๎…ž๎…ž(0)=๐‘ข๎…ž๎…ž(๐œ”)(3.1) has a unique solution which is of the form ๎€œ๐‘ข(๐‘ก)=๐œ”0๐บ1(๐‘ก,๐‘ )(โˆ’โ„Ž(๐‘ ))d๐‘ ,(3.2) where ๐บ1โŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๎‚ƒโˆš(๐‘ก,๐‘ )=2exp((1/2)๐œŒ(๐‘ โˆ’๐‘ก))sin๎‚€๎‚€๎‚๎‚๎‚„3/2๐œŒ(๐‘กโˆ’๐‘ )+๐œ‹/6โˆ’๐’ฒ3๐œŒ2๎‚€โˆš1+exp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚€๎‚€๎‚+3/2๐œŒ๐œ”๎‚๎‚exp(๐œŒ(๐‘กโˆ’๐‘ ))3๐œŒ2๎‚ƒโˆš(exp(๐œŒ๐œ”)โˆ’1),0โ‰ค๐‘ โ‰ค๐‘กโ‰ค๐œ”,2exp((1/2)๐œŒ(๐‘ โˆ’๐‘กโˆ’๐œ”))sin๎‚€๎‚€๎‚๎‚๎‚„3/2๐œŒ(๐‘กโˆ’๐‘ +๐œ”)+๐œ‹/6โˆ’๐’ด3๐œŒ2๎‚€โˆš1+exp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚€๎‚€๎‚+3/2๐œŒ๐œ”๎‚๎‚exp(๐œŒ(๐‘ก+๐œ”โˆ’๐‘ ))3๐œŒ2(exp(๐œŒ๐œ”)โˆ’1),0โ‰ค๐‘กโ‰ค๐‘ โ‰ค๐œ”,(3.3) where ๐’ฒ denotes โˆšexp(โˆ’(1/2)๐œŒ๐œ”)sin((3/2)๐œŒ(๐‘กโˆ’๐‘ โˆ’๐œ”)+๐œ‹/6)and๐’ด denotes โˆšexp(โˆ’(1/2)๐œŒ๐œ”)sin((3/2)๐œŒ(๐‘กโˆ’๐‘ )+๐œ‹/6).

Proof. It is easy to check that the associated homogeneous equation of (3.1) has the solution ๐‘ฃ(๐‘ก)=๐‘1exp(๐œŒ๐‘ก)+exp(โˆ’๐œŒ๐‘ก/2)(๐‘2โˆšcos(3๐œŒ/2)๐‘ก+๐‘3โˆšsin(3๐œŒ/2)๐‘ก). The only periodic solution of the associated homogeneous equation of (3.1) is the trivial solution, that is, ๐‘1,๐‘2,๐‘3=0. This follows by assuming that ๐‘ฃ(๐‘ก) is periodic; we immediately get that ๐‘1=0 and by assuming that ๐‘22+๐‘23>0 and choosing ๐œ‘ such that sin๐œ‘=๐‘2/๎”๐‘22+๐‘23,โ€‰โ€‰cos๐œ‘=๐‘3/๎”๐‘22+๐‘23, we get ๐‘ฃ(๐‘ก)๎”๐‘22+๐‘23๎‚€โˆ’=exp๐œŒ๐‘ก2๎‚๎ƒฉโˆšsin๐œ‘cos3๐œŒ2โˆš๐‘ก+cos๐œ‘sin3๐œŒ2๐‘ก๎ƒช๎‚€โˆ’=exp๐œŒ๐‘ก2๎‚๎ƒฉโˆšsin๐œ‘+3๐œŒ2๐‘ก๎ƒช(3.4) which for ๐‘กโ†’โˆž contradicts periodicity of ๐‘ฃ, proving that ๐‘2=๐‘3=0.
Applying the method of variation of parameters, we get ๐‘๎…ž1(๐‘ก)=exp(โˆ’๐œŒ๐‘ก)3๐œŒ2โ„Ž๐‘(๐‘ก),๎…ž2๎‚€โˆš(๐‘ก)=๎‚๎‚€โˆš3/3sin๎‚๎‚€โˆš3๐œŒ๐‘ก/2โˆ’(1/3)cos๎‚3๐œŒ๐‘ก/2๐œŒ2๎‚€exp๐œŒ๐‘ก2๎‚๐‘โ„Ž(๐‘ก),๎…ž3๎‚€โˆš(๐‘ก)=โˆ’(1/3)sin๎‚โˆ’๎‚€โˆš3๐œŒ๐‘ก/2๎‚๎‚€โˆš3/3cos๎‚3๐œŒ๐‘ก/2๐œŒ2๎‚€exp๐œŒ๐‘ก2๎‚โ„Ž(๐‘ก),(3.5) and then ๐‘1(๐‘ก)=๐‘1(๎€œ0)+๐‘ก0exp(โˆ’๐œŒ๐‘ )3๐œŒ2๐‘โ„Ž(๐‘ )d๐‘ ,2(๐‘ก)=๐‘2๎€œ(0)+๐‘ก0๎‚€โˆš๎‚๎‚€โˆš3/3sin๎‚โˆ’๎‚€โˆš3๐œŒ๐‘ /2(1/3)cos๎‚3๐œŒ๐‘ /2๐œŒ2๎‚€exp๐œŒ๐‘ 2๎‚๐‘โ„Ž(๐‘ )d๐‘ ,3(๐‘ก)=๐‘3๎€œ(0)+๐‘ก0๎‚€โˆšโˆ’(1/3)sin๎‚โˆ’๎‚€โˆš3๐œŒ๐‘ /2๎‚๎‚€โˆš3/3cos๎‚3๐œŒ๐‘ /2๐œŒ2๎‚€exp๐œŒ๐‘ 2๎‚โ„Ž(๐‘ )d๐‘ ,๐‘ข(๐‘ก)=๐‘1(๎‚€โˆ’๐‘ก)exp(๐œŒ๐‘ก)+exp๐œŒ๐‘ก2๎‚๎ƒฉ๐‘2(โˆš๐‘ก)cos3๐œŒ2๐‘ก+๐‘3(โˆš๐‘ก)sin3๐œŒ2๐‘ก๎ƒช=๐‘1(0)exp(๐œŒ๐‘ก)+๐‘2๎‚€โˆ’exp๐œŒ๐‘ก2๎‚๎ƒฉโˆšcos32๎ƒช๐œŒ๐‘ก+๐‘3๎‚€โˆ’(0)exp๐œŒ๐‘ก2๎‚๎ƒฉโˆšsin32๎ƒช+๎€œ๐œŒ๐‘ก๐‘ก0exp(๐œŒ(๐‘กโˆ’๐‘ ))3๐œŒ2๎€œโ„Ž(๐‘ )d๐‘ +๐‘ก0โˆšsin๎‚€๎‚€๎‚๎‚3/2๐œŒ(๐‘ โˆ’๐‘ก)โˆ’๐œ‹/66๐œŒ2๎‚€๐œŒexp2(๎‚๐‘ โˆ’๐‘ก)โ„Ž(๐‘ )d๐‘ .(3.6) Noting that ๐‘ข(0)=๐‘ข(๐œ”),๐‘ข๎…ž(0)=๐‘ข๎…ž(๐œ”),๐‘ข๎…ž๎…ž(0)=๐‘ข๎…ž๎…ž(๐œ”), we obtain ๐‘1๎€œ(0)=๐œ”0exp(๐œŒ(๐œ”โˆ’๐‘ ))3๐œŒ2๐‘(1โˆ’exp(๐œŒ๐œ”))โ„Ž(๐‘ )d๐‘ ,2๎€œ(0)=๐œ”0๎‚ƒ๎‚€โˆš2exp(๐œŒ(๐‘ โˆ’๐œ”)/2)exp(โˆ’๐œŒ๐œ”/2)sin๐œ‹/6โˆ’๎‚๎‚„3๐œŒ๐‘ /2โˆ’sin๐’Ÿ3๐œŒ2๎‚€๎‚€โˆšexp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚๎‚๐‘3๐œŒ๐œ”/2+1โ„Ž(๐‘ )d๐‘ ,3๎€œ(0)=๐œ”0๎‚ƒ๎‚€โˆš2exp(๐œŒ(๐‘ โˆ’๐œ”)/2)exp(โˆ’๐œŒ๐œ”/2)cos๐œ‹/6โˆ’๎‚๎‚„3๐œŒ๐‘ /2โˆ’cos๐’Ÿ3๐œŒ2๎‚€๎‚€โˆšexp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚๎‚3๐œŒ๐œ”/2+1โ„Ž(๐‘ )d๐‘ ,(3.7) where ๐’Ÿ denotes โˆš(๐œ‹/6โˆ’3๐œŒ(๐‘ โˆ’๐œ”)/2). Therefore ๐‘ข(๐‘ก)=๐‘1๎‚€โˆ’(๐‘ก)exp(๐œŒ๐‘ก)+exp๐œŒ๐‘ก2๎‚๎ƒฉ๐‘2โˆš(๐‘ก)cos3๐œŒ2๐‘ก+๐‘3โˆš(๐‘ก)sin3๐œŒ2๐‘ก๎ƒช=๎€œ๐‘ก0โŽงโŽชโŽจโŽชโŽฉ๎‚ƒโˆš2exp((1/2)๐œŒ(๐‘ โˆ’๐‘ก))sin๎‚€๎‚€๎‚๎‚๎‚„3/2๐œŒ(๐‘กโˆ’๐‘ )+๐œ‹/6โˆ’๐’ฒ3๐œŒ2๎‚€โˆš1+exp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚€๎‚€๎‚+3/2๐œŒ๐œ”๎‚๎‚exp(๐œŒ(๐‘กโˆ’๐‘ ))3๐œŒ2โŽซโŽชโŽฌโŽชโŽญ+๎€œ(1โˆ’exp(๐œŒ๐œ”))โ„Ž(๐‘ )d๐‘ ๐œ”๐‘กโŽงโŽชโŽจโŽชโŽฉ๎‚ƒโˆš2exp((1/2)๐œŒ(๐‘ โˆ’๐‘กโˆ’๐œ”))sin๎‚€๎‚€๎‚๎‚๎‚„3/2๐œŒ(๐‘กโˆ’๐‘ +๐œ”)+๐œ‹/6โˆ’๐’ด3๐œŒ2๎‚€โˆš1+exp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚€๎‚€๎‚+3/2๐œŒ๐œ”๎‚๎‚exp(๐œŒ(๐‘ก+๐œ”โˆ’๐‘ ))3๐œŒ2(โŽซโŽชโŽฌโŽชโŽญ=๎€œ1โˆ’exp(๐œŒ๐œ”))โ„Ž(๐‘ )d๐‘ ๐œ”0๐บ1(๐‘ก,๐‘ )โ„Ž(๐‘ )d๐‘ ,(3.8) where ๐บ1(๐‘ก,๐‘ ) is defined as in (3.3).
By direct calculation, we get the solution ๐‘ข satisfies the periodic boundary value condition of the problem (3.1).

Theorem 3.2. For ๐œŒ>0 and โ„Žโˆˆ๐‘‹, the equation ๐‘ข๎…ž๎…ž๎…ž+๐œŒ3๐‘ข=โ„Ž(๐‘ก),๐‘ข(0)=๐‘ข(๐œ”),๐‘ข๎…ž(0)=๐‘ข๎…ž(๐œ”),๐‘ข๎…ž๎…ž(0)=๐‘ข๎…ž๎…ž(๐œ”)(3.9) has a unique ๐œ”-periodic solution ๎€œ๐‘ข(๐‘ก)=๐œ”0๐บ2(๐‘ก,๐‘ )โ„Ž(๐‘ )d๐‘ ,(3.10) where ๐บ2โŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽฉ๎‚ƒโˆš(๐‘ก,๐‘ )=2exp((1/2)๐œŒ(๐‘กโˆ’๐‘ ))sin๎‚€๎‚€๎‚๎‚๎‚„3/2๐œŒ(๐‘กโˆ’๐‘ )โˆ’๐œ‹/6โˆ’๐’ฐ3๐œŒ2๎‚€โˆš1+exp(๐œŒ๐œ”)โˆ’2exp((1/2)๐œŒ๐œ”)cos๎‚€๎‚€๎‚+3/2๐œŒ๐œ”๎‚๎‚exp(๐œŒ(๐‘ โˆ’๐‘ก))3๐œŒ2๎‚ƒโˆš(1โˆ’exp(โˆ’๐œŒ๐œ”)),0โ‰ค๐‘ โ‰ค๐‘กโ‰ค๐œ”,2exp((1/2)๐œŒ(๐‘ก+๐œ”โˆ’๐‘ ))sin๎‚€๎‚€๎‚๎‚๎‚„3/2๐œŒ(๐‘ก+๐œ”โˆ’๐‘ )โˆ’๐œ‹/6โˆ’๐’ณ3๐œŒ2๎‚€โˆš1+exp(๐œŒ๐œ”)โˆ’2exp((1/2)๐œŒ๐œ”)cos๎‚€๎‚€๎‚+3/2๐œŒ๐œ”๎‚๎‚exp(๐œŒ(๐‘ โˆ’๐‘กโˆ’๐œ”))3๐œŒ2(1โˆ’exp(โˆ’๐œŒ๐œ”)),0โ‰ค๐‘กโ‰ค๐‘ โ‰ค๐œ”.(3.11) where ๐’ฐ denotes โˆšexp((1/2)๐œŒ๐œ”)sin((3/2)๐œŒ(๐‘กโˆ’๐‘ โˆ’๐œ”)โˆ’๐œ‹/6)and๐’ณ denotes โˆšโˆ’exp((1/2)๐œŒ๐œ”)sin((3/2)๐œŒ(๐‘กโˆ’๐‘ )โˆ’๐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)1๐‘™=3๐œŒ2(exp(๐œŒ๐œ”)โˆ’1),๐ฟ=3+2exp(โˆ’๐œŒ๐œ”/2)3๐œŒ2(1โˆ’exp(โˆ’๐œŒ๐œ”/2))2.(3.12)

Theorem 3.3. โˆซ๐œ”0๐บ1(๐‘ก,๐‘ )d๐‘ =1/๐œŒ3, and if โˆš3๐œŒ๐œ”<(4/3)๐œ‹ holds, then 0<๐‘™<๐บ1(๐‘ก,๐‘ )โ‰ค๐ฟ for all ๐‘กโˆˆ[0,๐œ”] and ๐‘ โˆˆ[0,๐œ”].

Proof. One has the following: ๐ป1(๐‘ก,๐‘ )=exp(๐œŒ(๐‘กโˆ’๐‘ ))3๐œŒ2๎€บ๎€ป,๐ปexp(๐œŒ๐œ”)โˆ’1โˆ—1(๐‘ก,๐‘ )=exp(๐œŒ(๐‘ก+๐œ”โˆ’๐‘ ))3๐œŒ2๎€บ๎€ป,๐ปexp(๐œŒ๐œ”)โˆ’12๎‚ƒโˆš(๐‘ก,๐‘ )=2exp((1/2)๐œŒ(๐‘ โˆ’๐‘ก))sin๎‚€๎‚€๎‚๐œŒ๎‚๎‚„3/2(๐‘กโˆ’๐‘ )+๐œ‹/6โˆ’๐’ฒ3๐œŒ2๎‚€โˆš1+exp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚€๎‚€๎‚,๐ป3/2๐œŒ๐œ”๎‚๎‚โˆ—2๎‚ƒโˆš(๐‘ก,๐‘ )=2exp((1/2)๐œŒ(๐‘ โˆ’๐‘กโˆ’๐œ”))sin๎‚€๎‚€๎‚๐œŒ๎‚๎‚„3/2(๐‘กโˆ’๐‘ +๐œ”)+๐œ‹/6โˆ’๐’ด3๐œŒ2๎‚€โˆš1+exp(โˆ’๐œŒ๐œ”)โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos๎‚€๎‚€๎‚.3/2๐œŒ๐œ”๎‚๎‚(3.13) A direct computation shows that โˆซ๐œ”0๐บ1(๐‘ก,๐‘ )d๐‘ =1/๐œŒ3. It is easy to see that ๐ป1(๐‘ก,๐‘ )>0 for ๐‘ โˆˆ[0,๐‘ก] and ๐ปโˆ—1(๐‘ก,๐‘ )>0 for ๐‘ โˆˆ[๐‘ก,๐œ”] and โˆšexp(โˆ’๐œŒ๐œ”)+1โˆ’2exp(โˆ’๐œŒ๐œ”/2)cos(3๐œŒ๐œ”/2)>[1โˆ’exp(โˆ’๐œŒ๐œ”/2)]2>0.
For convenience, we denote โˆš๐œƒ=(3/2)๐œŒ(๐‘กโˆ’๐‘ )+๐œ‹/6๐‘”1๎ƒฉโˆš(๐‘ก,๐‘ )=sin32๐œ‹๐œŒ(๐‘กโˆ’๐‘ )+6๎ƒช๎‚€โˆ’โˆ’exp๐œŒ๐œ”2๎‚๎ƒฉโˆšsin32๐œ‹๐œŒ(๐‘กโˆ’๐‘ โˆ’๐œ”)+6๎ƒช๎‚€โˆ’=sin(๐œƒ)โˆ’exp๐œŒ๐œ”2๎‚๎ƒฉโˆšsin๐œƒโˆ’32๎ƒช,๐‘”๐œŒ๐œ”โˆ—1๎ƒฉโˆš(๐‘ก,๐‘ )=sin32๐œŒ๐œ‹(๐‘กโˆ’๐‘ +๐œ”)+6๎ƒช๎‚€โˆ’โˆ’exp๐œŒ๐œ”2๎‚๎ƒฉโˆšsin32๐œŒ๐œ‹(๐‘กโˆ’๐‘ )+6๎ƒช๎ƒฉโˆš=sin๐œƒ+32๎ƒช๎‚€โˆ’๐œŒ๐œ”โˆ’exp๐œŒ๐œ”2๎‚sin๐œƒ.(3.14) If ๐‘”1(๐‘ก,๐‘ )>0 and ๐‘”โˆ—1(๐‘ก,๐‘ )>0, then obviously ๐ป2(๐‘ก,๐‘ )>0, ๐ปโˆ—2(๐‘ก,๐‘ )>0, and ๐บ1(๐‘ก,๐‘ )>0.
For 0โ‰ค๐‘ โ‰ค๐‘กโ‰ค๐œ”, Since โˆš3๐œŒ๐œ”<(4/3)๐œ‹, we have๐œ‹6โˆšโ‰ค๐œƒโ‰ค32๐œ‹๐œŒ๐œ”+6<5๐œ‹6,โˆ’๐œ‹2<๐œ‹6โˆ’โˆš32โˆš๐œŒ๐œ”โ‰ค๐œƒโˆ’32๐œ‹๐œŒ๐œ”โ‰ค6.(3.15)(i)For โˆšโˆ’๐œ‹/2<๐œƒโˆ’(3/2)๐œŒ๐œ”โ‰ค0, then sin๐œƒ>0, โˆšsin(๐œƒโˆ’(3/2)๐œŒ๐œ”)<0, we get ๐‘”1(๐‘ก,๐‘ )>0,(ii)For โˆš0<๐œƒโˆ’(3/2)๐œŒ๐œ”โ‰ค๐œ‹/6, we have sin๐œƒ>0, โˆšsin(๐œƒโˆ’(3/2)๐œŒ๐œ”)>0, and โˆš0<34โˆš๐œŒ๐œ”โ‰ค๐œƒโˆ’34๐œ‹๐œŒ๐œ”โ‰ค6+โˆš34๐œ‹๐œŒ๐œ”<2,๐‘”1๎‚€โˆ’(๐‘ก,๐‘ )=sin(๐œƒ)โˆ’exp๐œŒ๐œ”2๎‚๎ƒฉโˆšsin๐œƒโˆ’32๎ƒช๎ƒฉโˆš๐œŒ๐œ”โ‰ฅsin๐œƒโˆ’sin๐œƒโˆ’32๎ƒช๎ƒฉโˆš๐œŒ๐œ”=2cos๐œƒโˆ’34๎ƒช๎ƒฉโˆš๐œŒ๐œ”sin34๎ƒช๐œŒ๐œ”>0.(3.16)
For 0โ‰ค๐‘กโ‰ค๐‘ โ‰ค๐œ”, โˆ’๐œ‹2โˆš<โˆ’32๐œ‹๐œŒ๐œ”+6๐œ‹โ‰ค๐œƒโ‰ค6,๐œ‹6โˆšโ‰ค๐œƒ+32๐œ‹๐œŒ๐œ”โ‰ค6+โˆš325๐œŒ๐œ”<6๐œ‹.(3.17)(i)For โˆ’๐œ‹/2<๐œƒโ‰ค0, we have sin๐œƒ<0, โˆšsin(๐œƒ+(3/2)๐œŒ๐œ”)>0, and then ๐‘”โˆ—1(๐‘ก,๐‘ )>0.(ii)For 0<๐œƒโ‰ค๐œ‹/6, we have sin๐œƒ>0, โˆšsin(๐œƒ+(3/2)๐œŒ๐œ”)>0, and โˆš0<๐œƒ+34๐œ‹๐œŒ๐œ”<2,๐‘”โˆ—1(๎ƒฉโˆš๐‘ก,๐‘ )=sin๐œƒ+32๎ƒช๎‚€โˆ’๐œŒ๐œ”โˆ’exp๐œŒ๐œ”2๎‚๎ƒฉโˆšsin๐œƒโ‰ฅsin๐œƒ+32๎ƒช๎ƒฉโˆš๐œŒ๐œ”โˆ’sin๐œƒ=2cos๐œƒ+34๎ƒช๎ƒฉโˆš๐œŒ๐œ”sin34๎ƒช๐œŒ๐œ”>0.(3.18)
If โˆš3๐œŒ๐œ”<(4/3)๐œŒ๐œ”, we get ๐‘”1(๐‘ก,๐‘ )>0 and ๐‘”โˆ—1(๐‘ก,๐‘ )>0, proving that ๐บ(๐‘ก,๐‘ )>0 for all ๐‘กโˆˆ[0,๐œ”] and ๐‘ โˆˆ[0,๐œ”].
Next we compute a lower and an upper bound for ๐บ1(๐‘ก,๐‘ ) for ๐‘ โˆˆ[0,๐œ”]. We have1๐‘™=3๐œŒ2(โ‰คexp(๐œŒ๐œ”)โˆ’1)exp(๐œŒ(๐‘ก+๐œ”โˆ’๐‘ ))3๐œŒ2(exp(๐œŒ๐œ”)โˆ’1)<๐บ1โ‰ค(๐‘ก,๐‘ )exp(๐œŒ(๐‘ก+๐œ”โˆ’๐‘ ))3๐œŒ2๎€บ๎€ป+๎€บ๎€ปexp(๐œŒ๐œ”)โˆ’1exp(๐œŒ(๐‘ โˆ’๐‘กโˆ’๐œ”)/2)2+2exp(โˆ’๐œŒ๐œ”/2)3๐œŒ2๎‚ƒ๎‚€โˆšexp(โˆ’๐œŒ๐œ”)+1โˆ’2exp(โˆ’๐œŒ๐œ”/2)cosโ‰ค3๐œŒ๐œ”/2๎‚๎‚„exp(๐œŒ๐œ”)3๐œŒ2๎€บ๎€ป+exp(๐œŒ๐œ”)โˆ’12+2exp(โˆ’๐œŒ๐œ”/2)3๐œŒ2๎‚ƒ๎‚€โˆšexp(โˆ’๐œŒ๐œ”)+1โˆ’2exp(โˆ’๐œŒ๐œ”/2)cosโ‰ค13๐œŒ๐œ”/2๎‚๎‚„3๐œŒ2๎€บ๎€ป+1โˆ’exp(โˆ’๐œŒ๐œ”)2+2exp(โˆ’๐œŒ๐œ”/2)3๐œŒ2๎€บ๎€ป1โˆ’exp(โˆ’๐œŒ๐œ”/2)2โ‰ค3+2exp(โˆ’๐œŒ๐œ”/2)3๐œŒ2๎€บ๎€ป1โˆ’exp(โˆ’๐œŒ๐œ”/2)2=๐ฟ.(3.19) The proof is complete.

Similarly, the following dual theorem can be proved.

Theorem 3.4. โˆซ๐œ”0๐บ2(๐‘ก,๐‘ )d๐‘ =1/๐œŒ3, and if โˆš3๐œŒ๐œ”<(4/3)๐œ‹ holds, then 0<๐‘™<๐บ2(๐‘ก,๐‘ )โ‰ค๐ฟ for all [0,๐œ”] and ๐‘ โˆˆ[0,๐œ”].

4. Positive Periodic Solutions for (1.3)

Define the Banach space ๐‘‹ as in Section 2. Denote [][]๐‘€=max{๐‘Ž(๐‘ก)โˆถ๐‘กโˆˆ0,๐œ”},๐‘š=min{๐‘Ž(๐‘ก)โˆถ๐‘กโˆˆ0,๐œ”},๐œŒ3=๐‘€,๐‘˜=๐‘™(๐‘€+๐‘š)+๐ฟ๐‘€,๐‘˜1=โˆš๐‘˜โˆ’๐‘˜2โˆ’4๐ฟ๐‘™๐‘€๐‘š๐‘™[]2๐ฟ๐‘€,๐›ผ=๐‘šโˆ’(๐‘€+๐‘š)|๐‘|.๐ฟ๐‘€(1โˆ’|๐‘|)(4.1) It is easy to see that ๐‘€,๐‘š,๐›ฝ,๐ฟ,๐‘™,๐‘˜,๐‘˜1>0.

Now we consider (1.3). First let ๐‘“0=lim๐‘ฅโ†’0๐‘“(๐‘ฅ)๐‘ฅ,๐‘“โˆž=lim๐‘ฅโ†’โˆž๐‘“(๐‘ฅ)๐‘ฅ,๐‘“0=lim๐‘ฅโ†’0๐‘“(๐‘ฅ)๐‘ฅ,๐‘“โˆž=lim๐‘ฅโ†’โˆž๐‘“(๐‘ฅ)๐‘ฅ,(4.2) and denote ๐‘–0๎‚€:numberof0โ€ฒsin๐‘“0,๐‘“โˆž๎‚,๐‘–0๎‚€๐‘“:numberof0โ€ฒsin0,๐‘“โˆž๎‚,๐‘–โˆž๎‚€:numberofโˆžโ€ฒsin๐‘“0,๐‘“โˆž๎‚,๐‘–โˆž๎‚€๐‘“:numberofโˆžโ€ฒsin0,๐‘“โˆž๎‚.(4.3) It is clear that ๐‘–0,๐‘–0,๐‘–โˆž,๐‘–โˆžโˆˆ{0,1,2}. We will show that (1.3) has ๐‘–0 or ๐‘–โˆž positive ๐‘ค-periodic solutions for sufficiently large or small ๐œ†, respectively.

In the following we discuss (1.3) in two cases, namely, the case where ๐‘<0, and ๐‘>โˆ’min{๐‘˜1,๐‘š/(๐‘€+๐‘š)} (note that ๐‘>โˆ’๐‘š/(๐‘€+๐‘š) implies ๐›ผ>0, ๐‘>โˆ’๐‘˜1 implies |๐‘|<๐›ผ); and the case where ๐‘>0 and ๐‘<min{๐‘š/(๐‘€+๐‘š),(๐ฟ๐‘€โˆ’๐‘™๐‘š)/((๐ฟโˆ’๐‘™)๐‘€โˆ’๐‘™๐‘š)}, (note that ๐‘<๐‘š/(๐‘€+๐‘š) implies ๐›ผ>0, ๐‘<(๐ฟ๐‘€โˆ’๐‘™๐‘š)/((๐ฟโˆ’๐‘™)๐‘€โˆ’๐‘™๐‘š) implies ๐›ผ<1). Obviously, we have |๐‘|<1 which makes Lemma 2.1 applicable for both cases, and also Lemmas 2.2 or 2.3, respectively.

Let ๐พ={๐‘ฅโˆˆ๐‘‹โˆถ๐‘ฅ(๐‘ก)โ‰ฅ๐›ผโ€–๐‘ฅโ€–} denote the cone in ๐‘‹, where ๐›ผ is just as defined above. We also use ๐พ๐‘Ÿ={๐‘ฅโˆˆ๐พโˆถโ€–๐‘ฅโ€–<๐‘Ÿ} and ๐œ•๐พ๐‘Ÿ={๐‘ฅโˆˆ๐พโˆถโ€–๐‘ฅโ€–=๐‘Ÿ}.

Let ๐‘ฆ(๐‘ก)=(๐ด๐‘ฅ)(๐‘ก), then from Lemma 2.1 we have ๐‘ฅ(๐‘ก)=(๐ดโˆ’1๐‘ฆ)(๐‘ก). Hence (1.3) can be transformed into๐‘ฆ๎…ž๎…ž๎…ž๎€ท๐ด(๐‘ก)+๐‘Ž(๐‘ก)โˆ’1๐‘ฆ๎€ธ๐ด(๐‘ก)=๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธ,(๐‘กโˆ’๐œ(๐‘ก))(4.4) which can be further rewritten as๐‘ฆ๎…ž๎…ž๎…ž๐ด(๐‘ก)+๐‘Ž(๐‘ก)๐‘ฆ(๐‘ก)โˆ’๐‘Ž(๐‘ก)๐ป(๐‘ฆ(๐‘ก))=๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธ,(๐‘กโˆ’๐œ(๐‘ก))(4.5) where ๐ป(๐‘ฆ(๐‘ก))=๐‘ฆ(๐‘ก)โˆ’(๐ดโˆ’1๐‘ฆ)(๐‘ก)=โˆ’๐‘(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐›ฟ(๐‘ก)).

Now we discuss the two cases separately.

4.1. Case I: ๐‘<0 and ๐‘>โˆ’min{๐‘˜1,๐‘š/(๐‘€+๐‘š)}

Now we consider๐‘ฆ๎…ž๎…ž๎…ž(๐‘ก)+๐‘Ž(๐‘ก)๐‘ฆ(๐‘ก)โˆ’๐‘Ž(๐‘ก)๐ป(๐‘ฆ(๐‘ก))=โ„Ž(๐‘ก),โ„Žโˆˆ๐ถ+๐œ”,(4.6) and define operators ๐‘‡, ๎๐ปโˆถ๐‘‹โ†’๐‘‹ by (๎€œ๐‘‡โ„Ž)(๐‘ก)=๐‘ก๐‘ก+๐œ”๐บ2(๎‚€๎๎‚(๐‘ก,๐‘ )โ„Ž(๐‘ )d๐‘ ,๐ป๐‘ฆ๐‘ก)=๐‘€โˆ’๐‘Ž(๐‘ก)๐‘ฆ(๐‘ก)+๐‘Ž(๐‘ก)๐ป(๐‘ฆ(๐‘ก)).(4.7) Clearly ๐‘‡, ๎๐ป are completely continuous, (๐‘‡โ„Ž)(๐‘ก)>0 for โ„Ž(๐‘ก)>0 and โ€–๎๐ปโ€–โ‰ค(๐‘€โˆ’๐‘š+๐‘€(|๐‘|/(1โˆ’|๐‘|))). By Theorem 3.2, the solution of (4.6) can be written in the following form:๎‚€๐‘‡๎๎‚๐‘ฆ(๐‘ก)=(๐‘‡โ„Ž)(๐‘ก)+๐ป๐‘ฆ(๐‘ก).(4.8) In view of ๐‘<0 and ๐‘>โˆ’min{๐‘˜1,๐‘š/(๐‘€+๐‘š)}, we haveโ€–โ€–๐‘‡๎๐ปโ€–โ€–โ€–โ€–๎๐ปโ€–โ€–โ‰คโ‰คโ€–๐‘‡โ€–๐‘€โˆ’๐‘š+๐‘š|๐‘|๐‘€(1โˆ’|๐‘|)<1,(4.9) and hence ๎‚€๎๐ป๎‚๐‘ฆ(๐‘ก)=๐ผโˆ’๐‘‡โˆ’1(๐‘‡โ„Ž)(๐‘ก).(4.10) Define an operator ๐‘ƒโˆถ๐‘‹โ†’๐‘‹ by ๎‚€๎๐ป๎‚(๐‘ƒโ„Ž)(๐‘ก)=๐ผโˆ’๐‘‡โˆ’1(๐‘‡โ„Ž)(๐‘ก).(4.11) Obviously, for any โ„Žโˆˆ๐ถ+๐œ”, if (โˆš3/2)๐œŒ๐œ”<๐œ‹ hold, ๐‘ฆ(๐‘ก)=(๐‘ƒโ„Ž)(๐‘ก) is the unique positive ๐œ”-periodic solution of (4.6).

Lemma 4.1. ๐‘ƒ is completely continuous, and (๐‘‡โ„Ž)(๐‘ก)โ‰ค(๐‘ƒโ„Ž)(๐‘ก)โ‰ค๐‘€(1โˆ’|๐‘|)โ€–๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐‘‡โ„Žโ€–,โˆ€โ„Žโˆˆ๐ถ+๐œ”.(4.12)

Proof. By the Neumann expansion of ๐‘ƒ, we have ๎‚€๎๐ป๎‚๐‘ƒ=๐ผโˆ’๐‘‡โˆ’1๐‘‡=๎‚ต๎๎‚€๐‘‡๎๐ป๎‚๐ผ+๐‘‡๐ป+2๎‚€๐‘‡๎๐ป๎‚+โ‹ฏ+๐‘›๎‚ถ๐‘‡๎๎‚€๐‘‡๎๐ป๎‚+โ‹ฏ=๐‘‡+๐‘‡๐ป๐‘‡+2๎‚€๐‘‡๎๐ป๎‚๐‘‡+โ‹ฏ+๐‘›๐‘‡+โ‹ฏ.(4.13) Since ๐‘‡ and ๎๐ป are completely continuous, so is ๐‘ƒ. Moreover, by (4.13) and recalling that ๎โ€–๐‘‡๐ปโ€–โ‰ค(๐‘€โˆ’๐‘š+๐‘š|๐‘|)/๐‘€(1โˆ’|๐‘|)<1, we get (๐‘‡โ„Ž)(๐‘ก)โ‰ค(๐‘ƒโ„Ž)(๐‘ก)โ‰ค๐‘€(1โˆ’|๐‘|)โ€–๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐‘‡โ„Žโ€–.(4.14)

Define an operator ๐‘„โˆถ๐‘‹โ†’๐‘‹ by๎€ท๐ด๐‘„๐‘ฆ(๐‘ก)=๐‘ƒ๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ.(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธ(4.15)

Lemma 4.2. One has that ๐‘„(๐พ)โŠ‚๐พ.

Proof. From the definition of ๐‘„, it is easy to verify that ๐‘„๐‘ฆ(๐‘ก+๐œ”)=๐‘„๐‘ฆ(๐‘ก). For ๐‘ฆโˆˆ๐พ, we have from Lemma 4.1 that ๎€ท๐ด๐‘„๐‘ฆ(๐‘ก)=๐‘ƒ๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ท๐ด(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธโ‰ฅ๐‘‡๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€œ(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธ=๐œ†๐‘ก๐‘ก+๐œ”๐บ2๐ด(๐‘ก,๐‘ )๐‘(๐‘ )๐‘“๎€บ๎€ทโˆ’1๐‘ฆ๎€ธ๎€ป๎€œ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ โ‰ฅ๐œ†๐‘™๐œ”0๐ด๐‘(๐‘ )๐‘“๎€บ๎€ทโˆ’1๐‘ฆ๎€ธ๎€ป(๐‘ โˆ’๐œ(๐‘ ))d๐‘ .(4.16) On the other hand, ๎€ท๐ด๐‘„๐‘ฆ(๐‘ก)=๐‘ƒ๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธโ‰ค(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธ๐‘€(1โˆ’|๐‘|)โ€–โ€–๐‘‡๎€ท๐ด๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธโ€–โ€–(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธ=๐œ†๐‘€(1โˆ’|๐‘|)๐‘šโˆ’(๐‘€+๐‘š)|๐‘|max[]๐‘กโˆˆ0,๐œ”๎€œ๐‘ก๐‘ก+๐œ”๐บ2๐ด(๐‘ก,๐‘ )๐‘(๐‘ )๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ โ‰ค๐œ†๐‘€(1โˆ’|๐‘|)๐ฟ๎€œ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐œ”0๐ด๐‘(๐‘ )๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ .(4.17) Therefore ๐‘™[]๐‘„๐‘ฆ(๐‘ก)โ‰ฅ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|โ€–๐ฟ๐‘€(1โˆ’|๐‘|)๐‘„๐‘ฆโ€–=๐›ผโ€–๐‘„๐‘ฆโ€–,(4.18) that is, ๐‘„(๐พ)โŠ‚๐พ.

From the continuity of ๐‘ƒ, it is easy to verify that ๐‘„ is completely continuous in ๐‘‹. 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 ๐‘„ in ๐‘‹. Recalling Lemma 4.2, the existence of positive periodic solutions for (4.5) is equivalent to the existence of fixed points of ๐‘„ in ๐พ. Furthermore, if ๐‘„ has a fixed point ๐‘ฆ in ๐พ, it means that (๐ดโˆ’1๐‘ฆ)(๐‘ก) is a positive ๐œ”-periodic solutions of (1.3).

Lemma 4.3. If there exists ๐œ‚>0 such that ๐‘“๐ด๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธโ‰ฅ๎€ท๐ด(๐‘กโˆ’๐œ(๐‘ก))โˆ’1๐‘ฆ๎€ธ[](๐‘กโˆ’๐œ(๐‘ก))๐œ‚,for๐‘กโˆˆ0,๐œ”,๐‘ฆโˆˆ๐พ,(4.19) then โ€–๐‘„๐‘ฆโ€–โ‰ฅ๐œ†๐‘™๐œ‚๐›ผโˆ’|๐‘|1โˆ’๐‘2๎€œ๐œ”0๐‘(๐‘ )d๐‘ โ€–๐‘ฆโ€–,๐‘ฆโˆˆ๐พ.(4.20)

Proof. By Lemma 2.2, Theorem 3.4, and Lemma 4.1, we have for ๐‘ฆโˆˆ๐พ that ๎€ท๐ด๐‘„๐‘ฆ(๐‘ก)=๐‘ƒ๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ท๐ด(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธโ‰ฅ๐‘‡๐œ†๐‘(๐‘ก)๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€œ(๐‘กโˆ’๐œ(๐‘ก))๎€ธ๎€ธ=๐œ†๐‘ก๐‘ก+๐œ”๐บ2๐ด(๐‘ก,๐‘ )๐‘(๐‘ )๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธ๎€œ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ โ‰ฅ๐œ†๐‘™๐œ‚๐œ”0๎€ท๐ด๐‘(๐‘ )โˆ’1๐‘ฆ๎€ธ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ โ‰ฅ๐œ†๐‘™๐œ‚๐›ผโˆ’|๐‘|1โˆ’๐‘2๎€œ๐œ”0๐‘(๐‘ )d๐‘ โ€–๐‘ฆโ€–.(4.21) Hence โ€–๐‘„๐‘ฆโ€–โ‰ฅ๐œ†๐‘™๐œ‚๐›ผโˆ’|๐‘|1โˆ’๐‘2๎€œ๐œ”0๐‘(๐‘ )d๐‘ โ€–๐‘ฆโ€–,๐‘ฆโˆˆ๐พ.(4.22)

Lemma 4.4. If there exists ๐œ€>0 such that ๐‘“๐ด๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธโ‰ค๎€ท๐ด(๐‘กโˆ’๐œ(๐‘ก))โˆ’1๐‘ฆ๎€ธ[](๐‘กโˆ’๐œ(๐‘ก))๐œ€,for๐‘กโˆˆ0,๐œ”,๐‘ฆโˆˆ๐พ,(4.23) then โˆซโ€–๐‘„๐‘ฆโ€–โ‰ค๐œ†๐œ€๐ฟ๐‘€๐œ”0๐‘(๐‘ )d๐‘ ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|โ€–๐‘ฆโ€–,๐‘ฆโˆˆ๐พ.(4.24)

Proof. By Lemma 2.2, Theorem 3.4, and Lemma 4.1, we have โ€–๐‘„๐‘ฆ(๐‘ก)โ€–โ‰ค๐œ†๐‘€(1โˆ’|๐‘|)๐ฟ๎€œ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐œ”0๐ด๐‘(๐‘ )๐‘“๎€ท๎€ทโˆ’1๐‘ฆ๎€ธ๎€ธ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ โ‰ค๐œ†๐‘€(1โˆ’|๐‘|)๎€œ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐ฟ๐œ€๐œ”0๎€ท๐ด๐‘(๐‘ )โˆ’1๐‘ฆ๎€ธโˆซ(๐‘ โˆ’๐œ(๐‘ ))d๐‘ โ‰ค๐œ†๐œ€๐ฟ๐‘€๐œ”0๐‘(๐‘ )d๐‘ ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|โ€–๐‘ฆโ€–.(4.25)

Define ๎‚ป๐‘Ÿ๐น(๐‘Ÿ)=max๐‘“(๐‘ก)โˆถ0โ‰ค๐‘กโ‰ค๎‚ผ,๐‘“1โˆ’|๐‘|1๎‚ป(๐‘Ÿ)=min๐‘“(๐‘ก)โˆถ๐›ผโˆ’|๐‘|1โˆ’๐‘2๐‘Ÿ๐‘Ÿโ‰ค๐‘กโ‰ค๎‚ผ.1โˆ’|๐‘|(4.26)

Lemma 4.5. If ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ, then โ€–๐‘„๐‘ฆโ€–โ‰ฅ๐œ†๐‘™๐‘“1๎€œ(๐‘Ÿ)๐œ”0๐‘(๐‘ )d๐‘ โ€–๐‘ฆโ€–.(4.27)

Proof. By Lemma 2.2, we obtain ((๐›ผโˆ’|๐‘|)/(1โˆ’๐‘2))๐‘Ÿโ‰ค(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก))โ‰ค๐‘Ÿ/(1โˆ’|๐‘|) for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ, which yields ๐‘“((๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)))โ‰ฅ๐‘“1(๐‘Ÿ). The Lemma now follows analog to the proof of Lemma 4.3.

Lemma 4.6. If ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ, then โ€–๐‘„๐‘ฆโ€–โ‰ค๐œ†๐ฟ๐‘€(1โˆ’|๐‘|)๐น(๐‘Ÿ)๎€œ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|๐œ”0๐‘(๐‘ )d๐‘ โ€–๐‘ฆโ€–.(4.28)

Proof. By Lemma 2.2, we can have 0โ‰ค(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก))โ‰ค๐‘Ÿ/(1โˆ’|๐‘|) for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ, which yields ๐‘“((๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)))โ‰ค๐น(๐‘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 ๐‘‹ be a Banach space and ๐พ a cone in ๐‘‹. For ๐‘Ÿ>0, define ๐พ๐‘Ÿ={๐‘ขโˆˆ๐พโˆถโ€–๐‘ขโ€–<๐‘Ÿ}. Assume that ๐‘‡โˆถ๐พ๐‘Ÿโ†’๐พ is completely continuous such that ๐‘‡๐‘ฅโ‰ ๐‘ฅ for ๐‘ฅโˆˆ๐œ•๐พ๐‘Ÿ={๐‘ขโˆˆ๐พโˆถโ€–๐‘ขโ€–=๐‘Ÿ}. (i)If โ€–๐‘‡๐‘ฅโ€–โ‰ฅโ€–๐‘ฅโ€– for ๐‘ฅโˆˆ๐œ•๐พ๐‘Ÿ, then ๐‘–(T,๐พ๐‘Ÿ,๐พ)=0.(ii)If โ€–๐‘‡๐‘ฅโ€–โ‰คโ€–๐‘ฅโ€– for ๐‘ฅโˆˆ๐œ•๐พ๐‘Ÿ, then ๐‘–(๐‘‡,๐พ๐‘Ÿ,๐พ)=1.

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

Theorem 4.8. (a) If ๐‘–0=1 or 2, then (1.3) has ๐‘–0 positive ๐œ”-periodic solutions for ๐œ†>1/๐‘“1โˆซ(1)๐‘™๐œ”0๐‘(๐‘ )d๐‘ >0,
(b) if ๐‘–โˆž=1 or 2, then (1.3) has ๐‘–โˆž positive ๐œ”-periodic solutions for โˆซ0<๐œ†<(๐‘šโˆ’(๐‘€+๐‘š)|๐‘|)/๐ฟ๐‘€(1โˆ’|๐‘|)๐น(1)๐œ”0๐‘(๐‘ )d๐‘ ,
(c) if ๐‘–โˆž=0 or ๐‘–0=0, then (1.3) has no positive ๐œ”-periodic solutions for sufficiently small or sufficiently large ๐œ†>0, respectively.

Proof. (a) Choose ๐‘Ÿ1=1. Taking ๐œ†0=1/๐‘“1(๐‘Ÿ1โˆซ)๐‘™๐œ”0๐‘(๐‘ )d๐‘ >0, then for all ๐œ†>๐œ†0, we have from Lemma 4.5 that โ€–๐‘„๐‘ฆโ€–>โ€–๐‘ฆโ€–,for๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ1.(4.29)Case 1. If ๐‘“0=0, we can choose 0<๐‘Ÿ2<๐‘Ÿ1, so that ๐‘“(๐‘ข)โ‰ค๐œ€๐‘ข for 0โ‰ค๐‘ขโ‰ค๐‘Ÿ2, where the constant ๐œ€>0 satisfies โˆซ๐œ†๐œ€๐ฟ๐‘€๐œ”0๐‘(๐‘ )d๐‘ ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|<1.(4.30) Letting ๐‘Ÿ2=(1โˆ’|๐‘|)๐‘Ÿ2, we have ๐‘“((๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)))โ‰ค๐œ€(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)) for ๐‘ฆโˆˆ๐พ๐‘Ÿ2. By Lemma 2.2, we have 0โ‰ค(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก))โ‰คโ€–๐‘ฆโ€–/(1โˆ’|๐‘|)โ‰ค๐‘Ÿ2 for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ2. In view of Lemma 4.4 and (4.30), we have for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ2 that โˆซโ€–๐‘„๐‘ฆโ€–โ‰ค๐œ†๐œ€๐ฟ๐‘€๐œ”0๐‘(๐‘ )d๐‘ ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|โ€–๐‘ฆโ€–<โ€–๐‘ฆโ€–.(4.31) It follows from Lemma 4.7 and (4.29) that ๐‘–๎€ท๐‘„,๐พ๐‘Ÿ2๎€ธ๎€ท,๐พ=1,๐‘–๐‘„,๐พ๐‘Ÿ1๎€ธ,๐พ=0.(4.32) thus ๐‘–(๐‘„,๐พ๐‘Ÿ1โงต๐พ๐‘Ÿ2,๐พ)=โˆ’1 and ๐‘„ has a fixed point ๐‘ฆ in ๐พ๐‘Ÿ1โงต๐พ๐‘Ÿ2, which means (๐ดโˆ’1๐‘ฆ)(๐‘ก) is a positive ๐œ”-positive solution of (1.3) for ๐œ†>๐œ†0.Case 2. If ๐‘“โˆž=0, there exists a constant ๎‚๐ป>0 such that ๐‘“(๐‘ข)โ‰ค๐œ€๐‘ข for ๎‚๐ป๐‘ขโ‰ฅ, where the constant ๐œ€>0 satisfies โˆซ๐œ†๐œ€๐ฟ๐‘€๐œ”0๐‘(๐‘ )d๐‘ ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|<1.(4.33) Letting ๐‘Ÿ3=max{2๐‘Ÿ1,๎‚๐ป(1โˆ’๐‘2)/(๐›ผโˆ’|๐‘|)}, we have ๐‘“((๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)))โ‰ค๐œ€(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)) for ๐‘ฆโˆˆ๐พ๐‘Ÿ3. By Lemma 2.2, we have (๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก))โ‰ฅ((๐›ผโˆ’|๐‘|)/(1โˆ’๐‘2๎‚๐ป))โ€–๐‘ฆโ€–โ‰ฅ for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ3. Thus by Lemma 4.4 and (4.33), we have for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ3 that โˆซโ€–๐‘„๐‘ฆโ€–โ‰ค๐œ†๐œ€๐ฟ๐‘€๐œ”0๐‘(๐‘ )d๐‘ ๐‘šโˆ’(๐‘€+๐‘š)|๐‘|โ€–๐‘ฆโ€–<โ€–๐‘ฆโ€–.(4.34) Recalling Lemma 4.7 and (4.29) and that ๐‘–๎€ท๐‘„,๐พ๐‘Ÿ3๎€ธ๎€ท,๐พ=1,๐‘–๐‘„,๐พ๐‘Ÿ1๎€ธ,๐พ=0,(4.35) then ๐‘–(๐‘„,๐พ๐‘Ÿ3โงต๐พ๐‘Ÿ1,๐พ)=1 and ๐‘„ has a fixed point ๐‘ฆ in ๐พ๐‘Ÿ3โงต๐พ๐‘Ÿ1, which means (๐ดโˆ’1๐‘ฆ)(๐‘ก) is a positive ๐œ”-positive solution of (1.3) for ๐œ†>๐œ†0.Case 3. If ๐‘“0=๐‘“โˆž=0, from the above arguments, there exist 0<๐‘Ÿ2<๐‘Ÿ1<๐‘Ÿ3 such that ๐‘„ has a fixed point ๐‘ฆ1(๐‘ก) in ๐พ๐‘Ÿ1โงต๐พ๐‘Ÿ2 and a fixed point ๐‘ฆ2(๐‘ก) in ๐พ๐‘Ÿ3โงต๐พ๐‘Ÿ1. Consequently, (๐ดโˆ’1๐‘ฆ1)(๐‘ก) and (๐ดโˆ’1๐‘ฆ2)(๐‘ก) are two positive ๐œ”-periodic solutions of (1.3) for ๐œ†>๐œ†0.(b)Let ๐‘Ÿ1=1. Taking ๐œ†0=(๐‘šโˆ’(๐‘€+๐‘š)|๐‘|)/๐ฟ๐‘€(1โˆ’|๐‘|)๐น(๐‘Ÿ1)โˆซ๐œ”0๐‘(๐‘ )d๐‘ >0, then by Lemma 4.6 we know if ๐œ†<๐œ†0, then โ€–๐‘„๐‘ฆโ€–<โ€–๐‘ฆโ€–,๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ1.(4.36)Case 1. If ๐‘“0=โˆž, we can choose 0<๐‘Ÿ2<๐‘Ÿ1 so that ๐‘“(๐‘ข)โ‰ฅ๐œ‚๐‘ข for 0โ‰ค๐‘ขโ‰ค๐‘Ÿ2, where the constant ๐œ‚>0 satisfies ๐œ†๐‘™๐œ‚๐›ผโˆ’|๐‘|1โˆ’๐‘2๎€œ๐œ”0๐‘(๐‘ )d๐‘ >1.(4.37) Letting ๐‘Ÿ2=(1โˆ’|๐‘|)๐‘Ÿ2, we have ๐‘“((๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)))โ‰ฅ๐œ‚(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก)) for ๐‘ฆโˆˆ๐พ๐‘Ÿ2. By Lemma 2.2, we have 0โ‰ค(๐ดโˆ’1๐‘ฆ)(๐‘กโˆ’๐œ(๐‘ก))โ‰คโ€–๐‘ฆโ€–/(1โˆ’|๐‘|)โ‰ค๐‘Ÿ2 for ๐‘ฆโˆˆ๐œ•๐พ๐‘Ÿ2. Thus by Lemma 4.3 and (4.37), โ€–๐‘„๐‘ฆโ€–โ‰ฅ๐œ†๐‘™๐œ‚๐›ผโˆ’|๐‘|1โˆ’๐‘2๎€œ๐œ”0