On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

Abstract

The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays 𝑢′′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡),𝑢(𝑡−𝜏₁(𝑡)),…,𝑢(𝑡−𝜏_𝑛(𝑡))), where 𝑎(𝑡)∈𝐶(ℝ) is a positive ω-periodic function, 𝑓∶ℝ×[0,+∞)^𝑛+1→[0,+∞) is a continuous function which is ω-periodic in t, and 𝜏₁(𝑡),…,𝜏_𝑛(𝑡)∈𝐶(ℝ,[0,+∞)) are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

1. Introduction

In this paper, we deal with the existence of positive periodic solution of the second-order functional differential equation with multiple delays 𝑢^′′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢𝑡−𝜏₁(𝑡),…,𝑢𝑡−𝜏_𝑛(𝑡),𝑡∈ℝ, (1.1) where 𝑎(𝑡)∈𝐶(ℝ) is a positive 𝜔-periodic function, 𝑓∶ℝ×[0,+∞)^𝑛+1→[0,+∞) is a continuous function which is 𝜔-periodic in 𝑡, and 𝜏₁(𝑡),…,𝜏_𝑛(𝑡)∈𝐶(ℝ,[0,+∞)) are 𝜔-periodic functions 𝜔>0 is a constant.

In recent years, the existence of periodic solutions for second-order functional differential equations has been researched by many authors see [1–8] and references therein. In some practice models, only positive periodic solutions are significant. In [4–8], the authors obtained the existence of positive periodic solutions for some second-order functional differential equations by using fixed-point theorems of cone mapping. Especially in [5], Wu considered the second-order functional differential equation 𝑢^′′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝜆𝑓𝑡,𝑢𝑡−𝜏₁(𝑡),…,𝑢𝑡−𝜏_𝑛(𝑡),𝑡∈ℝ, (1.2) and obtained the existence results of positive periodic solutions by using the Krasnoselskii fixed-point theorem of cone mapping when the coefficient 𝑎(𝑡) satisfies the condition that 0<𝑎(𝑡)<𝜋²/𝜔² for every 𝑡∈ℝ. And in [8], Li obtained the existence results of positive 𝜔-periodic solutions for the second-order differential equation with constant delays −𝑢^′′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝑓𝑡,𝑢𝑡−𝜏₁,…,𝑢𝑡−𝜏_𝑛,𝑡∈ℝ, (1.3) by employing the fixed-point index theory in cones. For the second-order differential equations without delay, the existence of positive periodic solutions has been discussed by more authors, see [9–14].

Motivated by the paper mentioned above, we research the existence of positive periodic solutions of (1.1). We aim to obtain the essential conditions on the existence of positive periodic solution of (1.1) by constructing a special cone and applying the fixed-point index theory in cones.

In this paper, we assume the following conditions:(H1) 𝑎∈𝐶(ℝ,(0,+∞)) is 𝜔-periodic function and there exists a constant 1≤𝑝≤+∞ such that ‖𝑎‖_𝑝≤𝐾2𝑝^∗, (1.4) where ‖𝑎‖_𝑝 is the 𝑝-norm of 𝑎 in 𝐿^𝑝[0,𝜔], 𝑝^∗ is the conjugate exponent of 𝑝 defined by (1/𝑝)+(1/𝑝^∗)=1, and the function 𝐾(𝑞) is defined by 𝐾𝑞=2𝜋𝑞𝜔^1+2/𝑞22+𝑞^1−2/𝑞Γ1/𝑞Γ1/2+1/𝑞²4,if1≤𝑞<+∞,𝜔,if𝑞=+∞ (1.5) in which Γ is the Gamma function.(H2) 𝑓∈𝐶(ℝ×[0,+∞)^𝑛+1,[0,+∞)) and 𝑓(𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛) is 𝜔-periodic in 𝑡.(H3) 𝜏₁(𝑡),…,𝜏_𝑛(𝑡)∈𝐶(ℝ,[0,∞)) are 𝜔-periodic functions.

In Assumption (H1), if 𝑝=+∞, since 𝐾(2)=𝜋²/𝜔², then (1.4) implies that 𝑎 satisfies the condition (𝑡)≤𝜋0<𝑎²𝜔²,𝑡∈[0,𝜔]. (1.6) This condition includes the case discussed in [5].

The techniques used in this paper are completely different from those in [5]. Our results are more general than those in [5] in two aspects. Firstly, we relax the conditions of the coefficient 𝑎(𝑡) appeared in an equation in [5] and expand the range of its values. Secondly, by constructing a special cone and applying the fixed-point index theory in cones, we obtain the essential conditions on the existence of positive periodic solutions of (1.1). The conditions concern the first eigenvalue of the associated linear periodic boundary problem, which improve and optimize the results in [5]. To our knowledge, there are very few works on the existence of positive periodic solutions for the above functional differential equations under the conditions concerning the first eigenvalue of the corresponding linear equation.

Our main results are presented and proved in Section 3. Some preliminaries to discuss (1.1) are presented in Section 2.

2. Preliminaries

In order to discuss (1.1), we consider the existence of 𝜔-periodic solution of the corresponding linear differential equation 𝑢^′′+𝑎(𝑡)𝑢=ℎ(𝑡),𝑡∈ℝ, (2.1) where ℎ∈𝐶(ℝ) is a 𝜔-periodic function. It is obvious that finding an 𝜔-periodic solution of (2.1) is equivalent to finding a solution of the linear periodic boundary value problem 𝑢^′′(𝑡)(𝑡)[],𝑢(0)(𝜔)+𝑎𝑢=ℎ,𝑡∈0,𝜔=𝑢,𝑢^′(0)=𝑢^′(𝜔). (2.2) In [14], Torres show the following existence resulted.

Lemma 2.1. Assume that (H1 ) holds, then for every ℎ∈𝐶[0,𝜔], the linear periodic boundary problem (2.2) has a unique solution expressed by 𝑢(𝑡)=^𝜔₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠,𝑡∈[0,𝜔], (2.3) where 𝐺(𝑡,𝑠)∈𝐶([0,𝜔]×[0,𝜔]) is the Green function of the linear periodic boundary problem (2.2), which satisfies the positivity: 𝐺(𝑡,𝑠)>0 for every (𝑡,𝑠)∈[0,𝜔]×[0,𝜔].

For the details, see [14, Theorem 2.1 and Corollary 2.3].

For 𝑚∈ℕ, we use 𝐶^𝑚_𝜔(ℝ) to denote the 𝑚th-order continuous differentiable 𝜔-periodic functions space. Let 𝑋=𝐶_𝜔(ℝ) be the Banach space of all continuous 𝜔-periodic functions equipped the norm ‖𝑢‖=max_{0≤𝑡≤𝜔}|𝑢(𝑡)|.

Let 𝐾₀={𝑢∈𝑋∣𝑢(𝑡)≥0,𝑡∈ℝ} (2.4) be the cone of all nonnegative functions in 𝑋. Then 𝑋 is an ordered Banach space by the cone 𝐾₀. 𝐾₀ has a nonempty interior int𝐾₀={𝑢∈𝑋∣𝑢(𝑡)>0,𝑡∈ℝ}. (2.5) Let 𝐺=min_{0≤𝑡,𝑠≤𝜔}𝐺(𝑡,𝑠),𝐺=max_{0≤𝑡,𝑠≤𝜔}𝐺(𝑡,𝑠);𝜎=𝐺/𝐺. (2.6)

Lemma 2.2. Assume that (H1) holds, then for every ℎ∈𝑋, (2.1) has a unique 𝜔-periodic solution 𝑢. Let 𝑇∶ℎ↦𝑢, then 𝑇∶𝑋→𝑋 is a completely continuous linear operator, and when ℎ∈𝐾₀, 𝑇ℎ has the positivity estimate 𝑇ℎ(𝑡)≥𝜎‖𝑇ℎ‖,∀𝑡∈ℝ. (2.7)

Proof. Let ℎ∈𝑋. By Lemma 2.1, the linear periodic boundary problem (2.2) has a unique solution 𝑢∈𝐶²[0,𝜔] given by (2.3). We extend 𝑢 to a 𝜔-periodic function, which is still denoted by 𝑢, then 𝑢∶=𝑇ℎ∈𝐶²_𝜔(ℝ) is a unique 𝜔-periodic solution of (2.1). By (2.3), 𝑇ℎ(𝑡)=^𝜔₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠,𝑡∈[0,𝜔]. (2.8) From this we see that 𝑇 maps every bounded set in 𝑋 to a bounded equicontinuous set of 𝑋. Hence, by the Ascoli-Arzelà theorem, 𝑇∶𝑋→𝑋 is completely continuous.
Let ℎ∈𝐾₀. For every 𝑡∈[0,𝜔], from (2.8) it follows that 𝑇ℎ(𝑡)=^𝜔₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠≤𝐺^𝜔₀ℎ(𝑠)𝑑𝑠, (2.9) and therefore, ‖𝑇ℎ‖≤𝐺^𝜔₀ℎ(𝑠)𝑑𝑠. (2.10) Using (2.8) and this inequality, we have that 𝑇ℎ(𝑡)=^𝜔₀𝐺(𝑡,𝑠)ℎ(𝑠)𝑑𝑠≥𝐺^𝜔₀ℎ(𝑠)=𝐺𝑑𝑠/𝐺⋅𝐺^𝜔₀ℎ(𝑠)‖‖.𝑑𝑠≥𝜎𝑇ℎ (2.11)
Hence, by the periodicity of 𝑢, (2.7) holds for every 𝑡∈ℝ.

From (2.7) we easily see that 𝑇(𝐾₀)⊂int(𝐾₀); namely, 𝑇∶𝑋→𝑋 is a strongly positive linear operator. By the well-known Krein-Rutman theorem, the spectral radius 𝑟(𝑇)>0 is a simple eigenvalue of 𝑇, and 𝑇 has a corresponding positive eigenfunction 𝜙∈𝐾₀; that is, 𝑇𝜙=𝑟(𝑇)𝜙. (2.12) Since 𝜙 can be replaced by 𝑐𝜙, where 𝑐>0 is a constant, we can choose 𝜙∈𝐾₀ such that ^𝜔₀𝜙(𝑡)𝑑𝑡=1. (2.13) Set 𝜆₁=1/𝑟(𝑇), then 𝜙=𝑇(𝜆₁𝜙). By Lemma 2.2 and the definition of 𝑇, 𝜙∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝜙^′′(𝑡)+𝑎(𝑡)𝜙(𝑡)=𝜆₁𝜙(𝑡),𝑡∈ℝ. (2.14) Thus, 𝜆₁ is the minimum positive real eigenvalue of the linear equation (2.1) under the 𝜔-periodic condition. Summarizing these facts, we have the following lemma.

Lemma 2.3. Assume that (H1) holds, then there exist 𝜙∈𝐾₀∩𝐶²_𝜔(ℝ) such that (2.13) and (2.14) hold.

Let 𝑓∶ℝ×[0,∞)^𝑛+1→[0,∞) satisfy the assumption (H2). For every 𝑢∈𝑋, set 𝐹(𝑢)(𝑡)∶=𝑓𝑡,𝑢(𝑡),𝑢𝑡−𝜏₁,…,𝑢𝑡−𝜏_𝑛,𝑡∈ℝ. (2.15) Then 𝐹∶𝐾₀→𝐾₀ is continuous. Define a mapping 𝐴∶𝐾₀→𝑋 by 𝐴=𝑇∘𝐹. (2.16) By the definition of operator 𝑇, the 𝜔-periodic solution of (1.1) is equivalent to the fixed point of 𝐴. Choose a subcone of 𝐾₀ by 𝐾=𝑢∈𝐾₀∣𝑢(𝑡)≥𝜎‖𝑢‖_𝐶,𝑡∈ℝ. (2.17) By the strong positivity (2.7) of 𝑇 and the definition of 𝐴, we easily obtain the following.

Lemma 2.4. Assume that (H1) holds, then 𝐴(𝐾₀)⊂𝐾 and 𝐴∶𝐾→𝐾 is completely continuous.

Hence, the positive 𝜔-periodic solution of (1.1) is equivalent to the nontrivial fixed point of 𝐴. We will find the nonzero fixed point of 𝐴 by using the fixed-point index theory in cones.

We recall some concepts and conclusions on the fixed-point index in [15, 16]. Let 𝑋 be a Banach space and 𝐾⊂𝑋 a closed convex cone in 𝑋. Assume that Ω is a bounded open subset of 𝑋 with boundary 𝜕Ω, and 𝐾∩Ω≠∅. Let 𝐴∶𝐾∩Ω→𝐾 be a completely continuous mapping. If 𝐴𝑢≠𝑢 for any 𝑢∈𝐾∩𝜕Ω, then the fixed-point index 𝑖(𝐴,𝐾∩Ω,𝐾) has definition. One important fact is that if 𝑖(𝐴,𝐾∩Ω,𝐾)≠0, then 𝐴 has a fixed point in 𝐾∩Ω. The following two lemmas in [16] are needed in our argument.

Lemma 2.5. Let Ω be a bounded open subset of 𝑋 with 𝜃∈Ω, and 𝐴∶𝐾∩Ω→𝐾 a completely continuous mapping. If 𝜇𝐴𝑢≠𝑢 for every 𝑢∈𝐾∩𝜕Ω and 0<𝜇≤1, then 𝑖(𝐴,𝐾∩Ω,𝐾)=1.

Lemma 2.6. Let Ω be a bounded open subset of 𝑋 and 𝐴∶𝐾∩Ω→𝐾 a completely continuous mapping. If there exists an 𝑒∈𝐾⧵{𝜃} such that 𝑢−𝐴𝑢≠𝜇𝑒 for every 𝑢∈𝐾∩𝜕Ω and 𝜇≥0, then 𝑖(𝐴,𝐾∩Ω,𝐾)=0.

3. Main Results

We consider the existence of positive 𝜔-periodic solutions of (1.1). Assume that 𝑓∶ℝ×[0,∞)^𝑛+1→[0,∞) satisfy (H2). To be convenient, we introduce the notations 𝑓₀=liminf_𝑥→0+min_{[]𝑡∈0,𝜔}𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛𝑥,𝑓⁰=limsup_𝑥→0+max_{[]𝑡∈0,𝜔}𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛𝑥,𝑓_∞=liminf_𝑥→+∞min_{𝑡∈[0,𝜔]}𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛𝑥,𝑓^∞=limsup_𝑥→+∞max_{[]𝑡∈0,𝜔}𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛𝑥, (3.1) where 𝑥=max{𝑥₀,𝑥₁,…,𝑥_𝑛} and 𝑥=min{𝑥₀,𝑥₁,…,𝑥_𝑛}. Our main results are as follows.

Theorem 3.1. Suppose that (H1)–(H3) hold. If 𝑓 satisfies the condition (H4) 𝑓⁰<𝜆₁<𝑓_∞,then (1.1) has at least one positive 𝜔-periodic solution.

Theorem 3.2. Suppose that (H1)–(H3) hold. If 𝑓 satisfies the condition(H5) 𝑓^∞<𝜆₁<𝑓₀,then (1.1) has at least one positive 𝜔-periodic solution.

In Theorem 3.1, the condition (H4) allows 𝑓(𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛) to be superlinear growth on 𝑥₀, 𝑥₁,…,𝑥_𝑛. For example, 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛=𝑏₀(𝑡)𝑥²₀+𝑏₁(𝑡)𝑥²₁+⋯+𝑏_𝑛(𝑡)𝑥²_𝑛 (3.2) satisfies (H4) with 𝑓⁰=0 and 𝑓_∞=+∞, where 𝑏₀,𝑏₁,…,𝑏_𝑛∈𝐶_𝜔(ℝ) are positive 𝜔-periodic functions.

In Theorem 3.2, the condition (H5) allows 𝑓(𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛) to be sublinear growth on 𝑥₀, 𝑥₁,…,𝑥_𝑛. For example, 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛=𝑐₀(𝑡)𝑥₀+𝑐₁(𝑡)𝑥₁+⋯+𝑐_𝑛(𝑡)𝑥_𝑛 (3.3) satisfies (H5) with 𝑓₀=+∞ and 𝑓^∞=0, where 𝑐₀,𝑐₁,…,𝑐_𝑛∈𝐶_𝜔(ℝ) are positive 𝜔-periodic solution.

Applying Theorems 3.1 and 3.2 to (1.2), we have the following.

Corollary 3.3. Suppose that (H1)–(H3) hold. If the parameter 𝜆 satisfies one of the following conditions (1)𝜆₁/𝑓_∞<𝜆<𝜆₁/𝑓⁰, (2)𝜆₁/𝑓₀<𝜆<𝜆₁/𝑓^∞, then (1.2) has at least one positive 𝜔-periodic solution.

This result improves and extends [5, Theorem 1.3].

Proof of Theorem 3.1. Let 𝐾⊂𝑋 be the cone defined by (2.17) and 𝐴∶𝐾→𝐾 the operator defined by (2.16). Then the positive 𝜔-periodic solution of (1.1) is equivalent to the nontrivial fixed point of 𝐴. Let 0<𝑟<𝑅<+∞ and set Ω₁={𝑢∈𝑋∣‖𝑢‖<𝑟},Ω₂={𝑢∈𝑋∣‖𝑢‖<𝑅}. (3.4) We show that the operator 𝐴 has a fixed point in 𝐾∩(Ω₂⧵Ω₁) when 𝑟 is small enough and 𝑅 large enough.
Since 𝑓⁰<𝜆₁, by the definition of 𝑓⁰, there exist 𝜀∈(0,𝜆₁) and 𝛿>0, such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛≤𝜆₁−𝜀𝑥,𝑡∈[0,𝜔],𝑥∈(0,𝛿), (3.5) where 𝑥=max{𝑥₀,𝑥₁,…,𝑥_𝑛} and 𝑥=min{𝑥₀,𝑥₁,…,𝑥_𝑛}. Choosing 𝑟∈(0,𝛿), we prove that 𝐴 satisfies the condition of Lemma 2.5 in 𝐾∩𝜕Ω₁; namely, 𝜇𝐴𝑢≠𝑢 for every 𝑢∈𝐾∩𝜕Ω₁ and 0<𝜇≤1. In fact, if there exist 𝑢₀∈𝐾∩𝜕Ω₁ and 0<𝜇₀≤1 such that 𝜇₀𝐴𝑢₀=𝑢₀ and since 𝑢₀=𝑇(𝜇₀𝐹(𝑢₀)), by definition of 𝑇 and Lemma 2.2, 𝑢₀∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)=𝜇₀𝐹𝑢₀(𝑡),𝑡∈ℝ, (3.6) where 𝐹(𝑢) is defined by (2.15). Since 𝑢₀∈𝐾∩𝜕Ω₁, by the definitions of 𝐾 and Ω₁, we have 0<𝜎𝑢₀≤𝑢₀(𝜏)≤𝑢₀=𝑟<𝛿,∀𝜏∈ℝ. (3.7) This implies that 0<max𝑢₀(𝑡),𝑢₀𝑡−𝜏₁(𝑡),…,𝑢₀𝑡−𝜏_𝑛(𝑡)<𝛿,𝑡∈ℝ. (3.8) From this and (3.5), it follows that 𝐹𝑢₀(𝑡)=𝑓𝑡,𝑢₀(𝑡),𝑢₀𝑡−𝜏₁(𝑡),…,𝑢₀𝑡−𝜏_𝑛(𝑡)≤𝜆₁−𝜀⋅min𝑢₀(𝑡),𝑢₀𝑡−𝜏₁(𝑡),…,𝑢₀𝑡−𝜏_𝑛(𝑡)≤𝜆₁−𝜀𝑢₀(𝑡),𝑡∈ℝ. (3.9) By this inequality and (3.6), we have 𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)=𝜇₀𝐹𝑢₀(𝑡)≤𝜆₁−𝜀𝑢₀(𝑡),𝑡∈ℝ. (3.10) Let 𝜙∈𝐾∩𝐶²_𝜔(ℝ) be the function given in Lemma 2.4. Multiplying the inequality (3.10) by 𝜙(𝑡) and integrating on [0,𝜔], we have ^𝜔₀𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)𝜙(𝑡)𝑑𝑡≤𝜆₁−𝜀^𝜔₀𝑢₀(𝑡)𝜙(𝑡)𝑑𝑡. (3.11) For the left side of the above inequality using integration by parts, then using the periodicity of 𝑢₀ and 𝜙 and (2.14), we have ^𝜔₀𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)𝜙(𝑡)𝑑𝑡=^𝜔₀𝑢₀(𝑡)𝜙^′′(𝑡)+𝑎(𝑡)𝜙(𝑡)𝑑𝑡=𝜆₁^𝜔₀𝑢₀(𝑥)𝜙(𝑡)𝑑𝑡. (3.12) Consequently, we obtain that 𝜆₁^𝜔₀𝑢₀(𝑥)𝜙(𝑡)𝑑𝑡≤𝜆₁−𝜀^𝜔₀𝑢₀(𝑥)𝜙(𝑡)𝑑𝑡. (3.13) Since 𝑢₀∈𝐾∩𝜕Ω₁, by the definition of 𝐾 and (2.13), ^𝜔₀𝑢₀(𝑥)𝜙(𝑡)𝑑𝑡≥𝜎𝑢₀^𝜔₀𝜙(𝑡)𝑑𝑡=𝜎𝑢₀>0. (3.14) From this and (3.13), we conclude that 𝜆₁≤𝜆₁−𝜀, which is a contradiction. Hence, 𝐴 satisfies the condition of Lemma 2.5 in 𝐾∩𝜕Ω₁. By Lemma 2.5, we have 𝑖𝐴,𝐾∩Ω₁,𝐾=1. (3.15)
On the other hand, since 𝑓_∞>𝜆₁, by the definition of 𝑓_∞, there exist 𝜀₁>0 and 𝐻>0 such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛≥𝜆₁+𝜀₁𝑥,𝑡∈[0,𝜔],𝑥>𝐻, (3.16) where 𝑥=max{𝑥₀,𝑥₁,…,𝑥_𝑛} and 𝑥=min{𝑥₀,𝑥₁,…,𝑥_𝑛}. Choose 𝑅>max{𝐻/𝜎,𝛿} and 𝑒(𝑡)≡1. Clearly, 𝑒∈𝐾⧵{𝜃}. We show that 𝐴 satisfies the condition of Lemma 2.6 in 𝐾∩𝜕Ω₂; namely, 𝑢−𝐴𝑢≠𝜇𝜙 for every 𝑢∈𝐾∩𝜕Ω₂ and 𝜇≥0. In fact, if there exist 𝑢₁∈𝐾∩𝜕Ω₂ and 𝜇₁≥0 such that 𝑢₁−𝐴𝑢₁=𝜇₁𝑒, since 𝑢₁−𝜇₁𝑒=𝐴𝑢₁=𝑇(𝐹(𝑢₁)), by the definition of 𝑇 and Lemma 2.2, 𝑢₁∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)−𝜇₁=𝐹𝑢₁(𝑡),𝑡∈ℝ. (3.17) Since 𝑢₁∈𝐾∩𝜕Ω₂, by the definitions of 𝐾 and Ω₂, we have 𝑢₁(𝜏)≥𝜎𝑢₁=𝜎𝑅>𝐻,∀𝜏∈ℝ. (3.18) This means that min𝑢₁(𝑡),𝑢₁𝑡−𝜏₁(𝑡),…,𝑢₁𝑡−𝜏_𝑛(𝑡)>𝐻,𝑡∈ℝ. (3.19) Combining this with (3.16), we have that 𝐹𝑢₁(𝑡)=𝑓𝑡,𝑢₁(𝑡),𝑢₁𝑡−𝜏₁(𝑡),…,𝑢₁𝑡−𝜏_𝑛(𝑡)≥𝜆₁+𝜀₁⋅max𝑢₁(𝑡),𝑢₁𝑡−𝜏₁(𝑡),…,𝑢₁𝑡−𝜏_𝑛(𝑡)≥𝜆₁+𝜀₁𝑢₁(𝑡),𝑡∈ℝ. (3.20) From this inequality and (3.17), it follows that 𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)=𝜇₁𝑎(𝑡)+𝐹𝑢₁(𝑡)≥𝜆₁+𝜀₁𝑢₁(𝑡),𝑡∈ℝ. (3.21) Multiplying this inequality by 𝜙(𝑡) and integrating on [0,𝜔], we have ^𝜔₀𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)𝜙(𝑡)𝑑𝑡≥𝜆₁+𝜀₁^𝜔₀𝑢₁(𝑡)𝜙(𝑡)𝑑𝑡. (3.22) For the left side of the above inequality using integration by parts and (2.14), we have ^𝜔₀𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)𝜙(𝑡)𝑑𝑡=^𝜔₀𝑢₁(𝑡)𝜙^′′(𝑡)+𝑎(𝑡)𝜙(𝑡)𝑑𝑡=𝜆₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡. (3.23) From this and (3.22), it follows that 𝜆₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡≥𝜆₁+𝜀₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡. (3.24) Since 𝑢₁∈𝐾∩𝜕Ω₂, by the definition of 𝐾 and (2.13), we have ^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡≥𝜎𝑢₁^𝜔₀𝜙(𝑡)𝑑𝑡=𝜎𝑢₁>0. (3.25) Hence, from (3.24) it follows that 𝜆₁≥𝜆₁+𝜀₁, which is a contradiction. Therefore, 𝐴 satisfies the condition of Lemma 2.6 in 𝐾∩𝜕Ω₂. By Lemma 2.6, we have 𝑖𝐴,𝐾∩Ω₂,𝐾=0. (3.26)
Now by the additivity of the fixed-point index (3.15), and (3.26), we have 𝑖𝐴,𝐾∩Ω₂⧵Ω₁,𝐾=𝑖𝐴,𝐾∩Ω₂,𝐾−𝑖𝐴,𝐾∩Ω₁,𝐾=−1. (3.27) Hence 𝐴 has a fixed point in 𝐾∩(Ω₂⧵Ω₁), which is a positive 𝜔-periodic solution of (1.1).

Proof of Theorem 3.2. Let Ω₁,Ω₂⊂𝑋 be defined by (3.4). We prove that the operator 𝐴 defined by (2.16) has a fixed point in 𝐾∩(Ω₂⧵Ω₁) if 𝑟 is small enough and 𝑅 is large enough.
By 𝑓₀>𝜆₁ and the definition of 𝑓₀, there exist 𝜀>0 and 𝛿>0, such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛≥𝜆₁+𝜀𝑥,𝑡∈[0,𝜔],𝑥∈(0,𝛿), (3.28) where 𝑥=max{𝑥₀,𝑥₁,…,𝑥_𝑛}. Let 𝑟∈(0,𝛿) and 𝑒(𝑡)≡1. We prove that 𝐴 satisfies the condition of Lemma 2.6 in 𝐾∩𝜕Ω₁; namely, 𝑢−𝐴𝑢≠𝜇𝑒 for every 𝑢∈𝐾∩𝜕Ω₁ and 𝜇≥0. In fact, if there exist 𝑢₀∈𝐾∩𝜕Ω₁ and 𝜇₀≥0 such that 𝑢₀−𝐴𝑢₀=𝜇₀𝑒 and since 𝑢₀−𝜇₀𝑒=𝐴𝑢₀=𝑇(𝐹(𝑢₀)), by the definition of 𝑇 and Lemma 2.2, 𝑢₀∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)−𝜇₀=𝐹𝑢₀(𝑡),𝑡∈ℝ. (3.29) Since 𝑢₀∈𝐾∩𝜕Ω₁, by the definitions of 𝐾 and Ω₁, 𝑢₀ satisfies (3.7), and hence (3.8) holds. From (3.8) and (3.28), it follows that 𝐹𝑢₀(𝑡)=𝑓𝑡,𝑢₀(𝑡),𝑢₀0−𝜏₁(𝑡),…,𝑢₀𝑡−𝜏_𝑛(𝑡)≥𝜆₁+𝜀⋅max𝑢₀(𝑡),𝑢₀𝑡−𝜏₁(𝑡),…,𝑢₀𝑡−𝜏_𝑛(𝑡)≥𝜆₁+𝜀𝑢₀(𝑡),𝑡∈ℝ. (3.30) By this and (3.29), we obtain that 𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)=𝜇₀𝑎(𝑡)+𝐹𝑢₀(𝑡)≥𝜆₁+𝜀𝑢₀(𝑡),𝑡∈ℝ. (3.31) Multiplying this inequality by 𝜙(𝑡) and integrating on [0,𝜔], we have ^𝜔₀𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)𝜙(𝑡)𝑑𝑡≥𝜆₁+𝜀^𝜔₀𝑢₀(𝑡)𝜙(𝑡)𝑑𝑡. (3.32) For the left side of this inequality using integration by parts and (2.14), we have ^𝜔₀𝑢₀^′′(𝑡)+𝑎(𝑡)𝑢₀(𝑡)𝜙(𝑡)𝑑𝑡=^𝜔₀𝑢₀(𝑡)𝜙^′′(𝑡)+𝑎(𝑡)𝜙(𝑡)𝑑𝑡=𝜆₁^𝜔₀𝑢₀(𝑥)𝜙(𝑡)𝑑𝑡. (3.33) From this and (3.32), it follows that 𝜆₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡≥𝜆₁+𝜀₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡. (3.34) Since 𝑢₀∈𝐾∩𝜕Ω₁, from the definition of 𝐾 and (2.13) it follows that (3.14) holds. By (3.14) and (3.34), we see that 𝜆₁≥𝜆₁+𝜀, which is a contradiction. Hence, 𝐴 satisfies the condition of Lemma 2.6 in 𝐾∩𝜕Ω₁. By Lemma 2.6, we have 𝑖𝐴,𝐾∩Ω₁,𝐾=0. (3.35)
Since 𝑓^∞<𝜆₁, by the definition of 𝑓^∞, there exist 𝜀₁∈(0,𝜆₁) and 𝐻>0 such that 𝑓𝑡,𝑥₀,𝑥₁,…,𝑥_𝑛≤𝜆₁−𝜀₁𝑥,𝑡∈[0,𝜔],𝑥>𝐻, (3.36) where 𝑥=min{𝑥₀,𝑥₁,…,𝑥_𝑛}. Choosing 𝑅>max{𝐻/𝜎,𝛿}, we show that 𝐴 satisfies the condition of Lemma 2.5 in 𝐾∩𝜕Ω₂; namely, 𝜇𝐴𝑢≠𝑢 for every 𝑢∈𝐾∩𝜕Ω₂ and 0<𝜇≤1. In fact, if there exist 𝑢₁∈𝐾∩𝜕Ω₂ and 0<𝜇₁≤1 such that 𝜇₁𝐴𝑢₁=𝑢₁, since 𝑢₁=𝑇(𝜇₁𝐹(𝑢₁)), by the definition of 𝑇 and Lemma 2.2, 𝑢₁∈𝐶²_𝜔(Ω) satisfies the differential equation 𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)=𝜇₁𝐹𝑢₁(𝑡),𝑡∈ℝ. (3.37) Since 𝑢₁∈𝐾∩𝜕Ω₂, by the definitions of 𝐾 and Ω₂, 𝑢₁ satisfies (3.18), and hence (3.19) holds. By (3.19) and (3.36), we have 𝐹𝑢₁(𝑡)=𝑓𝑡,𝑢₁(𝑡),𝑢₁𝑡−𝜏₁(𝑡),…,𝑢₁𝑡−𝜏_𝑛(𝑡)≤𝜆₁−𝜀₁⋅min𝑢₁(𝑡),𝑢₁𝑡−𝜏₁(𝑡),…,𝑢₁𝑡−𝜏_𝑛(𝑡)≤𝜆₁−𝜀₁𝑢₁(𝑡),𝑡∈ℝ. (3.38)
From this inequality and (3.37), it follows that 𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)=𝜇₁𝐹𝑢₁(𝑡)≤𝜆₁−𝜀₁𝑢₁(𝑡),𝑡∈ℝ. (3.39) Multiplying this inequality by 𝜙(𝑡) and integrating on [0,𝜔], we have ^𝜔₀𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)𝜙(𝑡)𝑑𝑡≤𝜆₁−𝜀₁^𝜔₀𝑢₁(𝑡)𝜙(𝑡)𝑑𝑡. (3.40) For the left side of this inequality using integration by parts and (2.14), we have ^𝜔₀𝑢₁^′′(𝑡)+𝑎(𝑡)𝑢₁(𝑡)𝜙(𝑡)𝑑𝑡=^𝜔₀𝑢₁(𝑡)𝜙^′′(𝑡)+𝑎(𝑡)𝜙(𝑡)𝑑𝑡=𝜆₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡. (3.41) Consequently, we obtain that 𝜆₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡≤𝜆₁−𝜀₁^𝜔₀𝑢₁(𝑥)𝜙(𝑡)𝑑𝑡. (3.42) Since 𝑢₁∈𝐾∩𝜕Ω₂, by the definition of 𝐾 and (2.13) we see that (3.25) holds. From (3.25) and (3.42), we see that 𝜆₁≤𝜆₁−𝜀₁, which is a contradiction. Hence, 𝐴 satisfies the condition of Lemma 2.5 in 𝐾∩𝜕Ω₂. By Lemma 2.5 we have𝑖𝐴,𝐾∩Ω₂,𝐾=1. (3.43)
Now, from (3.35) and (3.43) it follows that 𝑖𝐴,𝐾∩Ω₂⧵Ω₁,𝐾=𝑖𝐴,𝐾∩Ω₂,𝐾−𝑖𝐴,𝐾∩Ω₁,𝐾=1. (3.44) Hence, 𝐴 has a fixed point in 𝐾∩(Ω₂⧵Ω₁), which is a positive 𝜔-periodic solution of (1.1).

Acknowledgment

This research supported by NNSF of China (11261053 and 11061031).