Positive Periodic Solutions for Second-Order Ordinary Differential Equations with Derivative Terms and Singularity in Nonlinearities

Abstract

The existence results of positive 𝜔-periodic solutions are obtained for the second-order ordinary differential equation 𝑢′′(𝑡)=𝑓(𝑡,𝑢(𝑡),𝑢^′(𝑡)),𝑡∈ℝ where, 𝑓∶ℝ×(0,∞)×ℝ→ℝ is a continuous function, which is 𝜔-periodic in 𝑡 and 𝑓(𝑡,𝑢,𝑣) may be singular at 𝑢=0. The discussion is based on the fixed point index theory in cones.

1. Introduction

In this paper, we discuss the existence of positive 𝜔-periodic solutions of the second-order ordinary differential equation with first-order derivative term in the nonlinearity 𝑢^′′(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡∈ℝ, (1.1) where the nonlinearity 𝑓∶ℝ×(0,∞)×ℝ→ℝ is a continuous function, which is 𝜔-periodic in t and 𝑓(𝑡,𝑢,𝑣) may be singular at 𝑢=0.

The existence problems of periodic solutions for nonlinear second-order ordinary differential equations have attracted many authors’ attention and concern, and most works are on the special equation 𝑢^′′(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈ℝ, (1.2) that does not contain explicitly first-order derivative term in nonlinearity. Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (1.2). These theorems and methods are mainly the upper and lower solutions method and monotone iterative technique [1–4], the continuation method of topological degree [5–7], variational method and critical point theory, [8–10] and so forth.

In recent years, the fixed point theorems of cone mapping, especially the fixed point theorem of Krasnoselskii’s cone expansion or compression type, have been extensively applied to two-point boundary value problems of second-order ordinary differential equations, and some results of existence and multiplicity of positive solutions have been obtained, see [11–15]. Lately, the authors of [16–18] have also applied the Krasnoselskii’s fixed point theorem to periodic problems of second-order nonlinear ordinary differential equations, and obtained existence results of positive periodic solutions. In these works, the new discovered positivity of Green function of the corresponding linear second-order periodic boundary value problems plays an important role. The positivity guarantees that the integral operators of the second-order periodic problems are cone-preserving in the cone 𝐾₀={𝑢∈𝐶[0,𝜔]∣𝑢(𝑡)≥𝜎‖𝑢‖,𝑡∈[0,𝜔]} (1.3) in the Banach space 𝐶[0,𝜔], where 𝜎>0 is a constant. Hence the fixed point theorems of cone mapping can be applied to the second-order periodic problems. For more precise results using the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of second-order ordinary differential equation, see [19–22]. However, all of these works are on the special second-order equation (1.2), and few people consider the existence of the positive periodic solutions for the general second-order equation (1.1) that explicitly contains the first order derivative term.

The purpose of this paper is to extend the results of [16–22] to the general second-order equation (1.1). We will use the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of (1.1). For the periodic problem of (1.1), since the corresponding integral operator has no definition on the cone 𝐾₀ in 𝐶[0,𝜔], the argument methods used in [16–22] are not applicable. We will use a completely different method to treat (1.1). Our main results will be given in Section 3. Some preliminaries to discuss (1.1) are presented in Section 2.

2. Preliminaries

Let 𝐶_𝜔(ℝ)denote the Banach space of all continuous 𝜔-periodic function 𝑢(𝑡) with norm ‖𝑢‖_𝐶=max_{0≤𝑡≤𝜔}|𝑢(𝑡)|. Let 𝐶¹_𝜔(ℝ) be the Banach space of all continuous differentiable 𝜔-periodic function 𝑢(𝑡) with the norm ‖𝑢‖_𝐶¹=‖𝑢‖_𝐶+𝑢^′_𝐶. (2.1) Generally, 𝐶^𝑛_𝜔(ℝ) denotes the 𝑛th-order continuous differentiable 𝜔-periodic function space for 𝑛∈ℕ. Let 𝐶⁺_𝜔(ℝ) be the cone of all nonnegative functions in 𝐶_𝜔(ℝ).

Let 𝑀∈(0,𝜋²/𝜔²) be a constant. For ℎ∈𝐶_𝜔(ℝ), we consider the linear second-order differential equation 𝑢^′′(𝑡)+𝑀𝑢(𝑡)=ℎ(𝑡),𝑡∈ℝ. (2.2) The 𝜔-periodic solutions of (2.2) are closely related with the linear second-order boundary value problem 𝑢^′′(𝑡)(𝑡)𝑢(0)(𝜔)+𝑀𝑢=0,0≤𝑡≤𝜔,−𝑢=0,𝑢^′(0)−𝑢^′(𝜔)=1, (2.3) see [19]. It is easy to see that Problem (2.3) has a unique solution, which is explicitly given by 𝑈(𝑡)=cos𝛽(𝑡−𝜔/2)2𝛽sin𝛽𝜔/2,0≤𝑡≤𝜔, (2.4) where 𝛽=√𝑀. We have the following Lemma.

Lemma 2.1. Let 𝑀∈(0,𝜋²/𝜔²). Then for every ℎ∈𝐶_𝜔(ℝ), the linear equation (2.2) has a unique 𝜔-periodic solution 𝑢(𝑡), which is given by 𝑢(𝑡)=^𝑡_𝑡−𝜔𝑈(𝑡−𝑠)ℎ(𝑠)𝑑𝑠∶=𝑆ℎ(𝑡),𝑡∈ℝ. (2.5) Moreover, 𝑆∶𝐶_𝜔(ℝ)→𝐶¹_𝜔(ℝ) is a completely continuous linear operator.

Proof. Taking the derivative in (2.5) and using the boundary condition of 𝑈(𝑡), we obtain that 𝑢^′′(𝑡)=𝑈^′(0)−𝑈^′(𝜔)ℎ(𝑡)+^𝑡_𝑡−𝜔𝑈^′′()ℎ(𝑠)(𝑡)𝑡−𝑠𝑑𝑠=ℎ−𝑀^𝑡_𝑡−𝜔𝑈()ℎ(𝑠)(𝑡)(𝑡).𝑡−𝑠𝑑𝑠=ℎ−𝑀𝑢 (2.6) Therefore, 𝑢(𝑡) satisfies (2.2). Let 𝜏=𝑠+𝜔; it follows from (2.5) that 𝑢(𝑡)=_𝑡^𝑡+𝜔𝑈()ℎ()=𝑡+𝜔−𝜏𝜏−𝜔𝑑𝜏_{𝑡𝑡+𝜔}𝑈(𝑡+𝜔−𝜏)ℎ(𝜏)𝑑𝜏=𝑢(𝑡+𝜔). (2.7) Hence, 𝑢(𝑡) is an 𝜔-periodic solution of (2.2). From the maximum principle for second-order periodic boundary value problems [4], it is easy to see that 𝑢(𝑡) is the unique 𝜔-periodic solution of (2.2).
From (2.5) and (2.6), we easily see that 𝑆∶𝐶_𝜔(ℝ)→𝐶²_𝜔(ℝ) is a linear bounded operator. By the compactness of the embedding 𝐶²_𝜔(ℝ)↪𝐶¹_𝜔(ℝ), 𝑆∶𝐶_𝜔(ℝ)→𝐶¹_𝜔(ℝ) is a completely continuous operator.

Since 𝑈(𝑡)>0 for every 𝑡∈[0,𝜔], by (2.5), if ℎ∈𝐶⁺_𝜔(ℝ)and ℎ(𝑡)≢0, then the 𝜔-periodic solution of (2.2) 𝑢(𝑡)>0 for every 𝑡∈ℝ, and we term it the positive 𝜔-periodic solution. Let 𝑈=max_{0≤𝑡≤𝜔}𝑈(𝑡)=12𝛽sin𝛽𝜔/2,𝑈=min_{0≤𝑡≤𝜔}𝑈(𝑡)=cos𝛽𝜔/2,2𝛽sin𝛽𝜔/2𝑈₁=max_{0≤𝑡≤𝜔}𝑈^′(𝑡)=max_{0≤𝑡≤𝜔}sin𝛽(𝑡−𝜔/2)=12sin𝛽𝜔/22,𝑈𝜎=𝑈=cos𝛽𝜔2,𝐶₀=𝑈₁𝑈=𝛽tan𝛽𝜔2. (2.8) Define the cone 𝐾 in 𝐶¹_𝜔(ℝ) by 𝐾=𝑢∈𝐶¹_𝜔(ℝ)∣𝑢(𝑡)≥𝜎‖𝑢‖_𝐶,𝑢^′(𝑡)≤𝐶₀|𝑢(𝑡)|,𝑡∈ℝ. (2.9) We have the following Lemma.

Lemma 2.2. Let 𝑀∈(0,𝜋²/𝜔²). Then for every ℎ∈𝐶⁺_𝜔(ℝ), the positive 𝜔-periodic solution of (2.2) 𝑢=𝑆ℎ∈𝐾. Namely, 𝑆(𝐶⁺_𝜔(ℝ))⊂𝐾.

Proof. Let ℎ∈𝐶⁺_𝜔(ℝ), 𝑢=𝑆ℎ. For every 𝑡∈ℝ, from (2.5) it follows that 𝑢(𝑡)=^𝑡_𝑡−𝜔𝑈(𝑡−𝑠)ℎ(𝑠)𝑑𝑠≤𝑈^𝑡_𝑡−𝜔ℎ(𝑠)𝑑𝑠=𝑈^𝜔₀ℎ(𝑠)𝑑𝑠, (2.10) and therefore, ‖𝑢‖_𝐶≤𝑈^𝜔₀ℎ(𝑠)𝑑𝑠. (2.11) Using (2.5), we obtain that 𝑢(𝑡)=^𝑡_𝑡−𝜔𝑈(𝑡−𝑠)ℎ(𝑠)𝑑𝑠≥𝑈^𝑡_𝑡−𝜔ℎ(𝑠)𝑑𝑠=𝑈^𝜔₀ℎ(𝑠)𝑑𝑠≥𝜎‖𝑢‖_𝐶. (2.12) For every 𝑡∈ℝ, since 𝑢^′(𝑡)=^𝑡_𝑡−𝜔𝑈^′(𝑡−𝑠)ℎ(𝑠)𝑑𝑠, (2.13) we have 𝑢^′(𝑡)≤^𝑡_𝑡−𝜔𝑈^′(𝑡−𝑠)ℎ(𝑠)𝑑𝑠≤𝑈₁^𝑡_𝑡−𝜔ℎ(𝑠)=𝑑𝑠𝑈₁^𝜔₀ℎ(𝑠)𝑑𝑠=𝐶₀𝑈^𝜔₀ℎ(𝑠)𝑑𝑠≤𝐶₀𝑢(𝑡). (2.14) Hence, 𝑢∈𝐾.

Now we consider the nonlinear equation (1.1). Hereafter, we assume that the nonlinearity 𝑓 satisfies the following condition. (F0) There exists 𝑀∈(0,𝜋²/𝜔²) such that 𝑓𝑡,𝑥,𝑦+𝑀𝑥≥0,𝑥>0,𝑡,𝑦∈ℝ. (2.15)Let 𝑓₁(𝑡,𝑥,𝑦)=𝑓(𝑡,𝑥,𝑦)+𝑀𝑥, then 𝑓₁(𝑡,𝑥,𝑦)≥0 for 𝑥>0,  𝑡,𝑦∈ℝ, and (1.1) is rewritten to 𝑢^′′(𝑡)+𝑀𝑢(𝑡)=𝑓₁𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡∈ℝ. (2.16) For 𝑢∈𝐾, if 𝑢≠0, then ‖𝑢‖_𝐶>0 and by the definition of 𝐾, 𝑢(𝑡)≥𝜎‖𝑢‖_𝐶>0 for every 𝑡∈ℝ. Hence 𝐹(𝑢)(𝑡)∶=𝑓₁𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡∈ℝ (2.17) is well defined, and we can define the integral operator 𝐴∶𝐾⧵{0}→𝐶¹_𝜔(ℝ) by 𝐴𝑢(𝑡)=^𝑡_𝑡−𝜔𝑈(𝑡−𝑠)𝑓₁𝑠,𝑢(𝑠),𝑢^′(𝑠)𝑑𝑠=(𝑆∘𝐹)(𝑡). (2.18) By the definition of operator 𝑆, the positive 𝜔-periodic solution of (1.1) is equivalent to the nontrivial fixed point of 𝐴. From Assumption (F0), Lemmas 2.1 and 2.2, we easily see the following Lemma.

Lemma 2.3. 𝐴(𝐾⧵{0})⊂𝐾, and 𝐴∶𝐾⧵{0}→𝐾 is completely continuous.

We will find the nonzero fixed point of 𝐴 by using the fixed point index theory in cones. Since the singularity of 𝑓 at 𝑥=0 implies that 𝐴 has no definition at 𝑢=0, the fixed point index theory in the cone 𝐾 cannot be directly applied to 𝐴. We need to make some Preliminaries.

We recall some concepts and conclusions on the fixed point index in [23, 24]. Let 𝐸 be a Banach space and 𝐾⊂𝐸 a closed convex cone in 𝐸. Assume Ω is a bounded open subset of 𝐸 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 a 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 are needed in our argument.

Lemma 2.4 (see [24]). Let Ω be a bounded open subset of 𝐸 with 𝜃∈Ω and 𝐴∶𝐾∩Ω→𝐾 a completely continuous mapping. If 𝜆𝐴𝑢≠𝑢 for every 𝑢∈𝐾∩𝜕Ω and 0<𝜆≤1, then 𝑖(𝐴,𝐾∩Ω,𝐾)=1.

Lemma 2.5 (see [24]). Let Ω be a bounded open subset of 𝐸 and 𝐴∶𝐾∩Ω→𝐾 a completely continuous mapping. If there exists an 𝑒∈𝐾⧵{𝜃} such that 𝑢−𝐴𝑢≠𝜏𝑒 for every 𝑢∈𝐾∩𝜕Ω and 𝜏≥0, then 𝑖(𝐴,𝐾∩Ω,𝐾)=0.

We use Lemmas 2.4 and 2.5 to show the following fixed-point theorem in cones which is applicable to the operator 𝐴 defined by (2.18).

Theorem 2.6. Let 𝐸 be a Banach space and 𝐾⊂𝐸 a closed convex cone. Assume Ω₁ and Ω₂ are bounded open subsets of 𝐸 with 𝜃∈Ω₁, Ω₁⊂Ω₂. Let 𝐴∶𝐾∩(Ω₂⧵Ω₁)→𝐾 be a completely continuous mapping. If 𝐴 satisfies the following conditions:(1) 𝜆𝐴𝑢≠𝑢 for 𝑢∈𝐾∩𝜕Ω₁, 0<𝜆≤1; (2) there exists 𝑒∈𝐾⧵{𝜃} such that 𝑢−𝐴𝑢≠𝜏𝑒 for 𝑢∈𝐾∩𝜕Ω₂, 𝜏≥0, or the following conditions:(3) there exists 𝑒∈𝐾⧵{𝜃} such that 𝑢−𝐴𝑢≠𝜏𝑒 for 𝑢∈𝐾∩𝜕Ω₁, 𝜏≥0; (4) 𝜆𝐴𝑢≠𝑢 for 𝑢∈𝐾∩𝜕Ω₂, 0<𝜆≤1, then 𝐴 has a fixed-point in 𝐾∩(Ω₂⧵Ω₁).

Proof. By Dugundji’s extension theorem, the operator 𝐴∶𝐾∩(Ω₂⧵Ω₁)→𝐾 can be extended into a completely continuous operator from 𝐾∩Ω₂ to 𝐾, says 𝐴∶𝐾∩Ω₂→𝐾.
If 𝐴 satisfies conditions (1) and (2) of Theorem 2.6, then 𝐴 also satisfies them. By Lemmas 2.4 and 2.5, respectively, we have 𝑖𝐴,𝐾∩Ω₁,𝐾=1,𝑖𝐴,𝐾∩Ω₂,𝐾=0. (2.19) By the additivity of the fixed point index, we have 𝑖𝐴,𝐾∩Ω₂⧵Ω₁,𝐾=𝑖𝐴,𝐾∩Ω₂,𝐾−𝑖𝐴,𝐾∩Ω₁,𝐾=−1. (2.20) Hence 𝐴 has a fixed-point in 𝐾∩(Ω₂⧵Ω₁). Since 𝐴 is an extension of 𝐴, it follows that 𝐴 has a fixed-point in 𝐾∩(Ω₂⧵Ω₁).
If 𝐴 satisfies conditions (3) and (4) of Theorem 2.6, with a similar count, we obtain that 𝑖𝐴,𝐾∩Ω₂⧵Ω₁,𝐾=1. (2.21) This means that 𝐴 has a fixed-point in 𝐾∩(Ω₂⧵Ω₁). Hence, 𝐴 has a fixed-point in 𝐾∩(Ω₂⧵Ω₁).

Theorem 2.6 is an improvement of the fixed point theorem of Krasnoselskii’s cone expansion or compression. We will use it to discuss the existence of positive 𝜔-periodic solutions of (1.1) in the next section.

3. Main Results

We consider the the existence of positive 𝜔-periodic solutions of (1.1). Let 𝑓∈𝐶(ℝ×(0,∞)×ℝ) satisfy Assumption (F0) and 𝑓(𝑡,𝑥,𝑦) be 𝜔-periodic in 𝑡. Let 𝐶₀ be the constant defined by (2.8) and 𝐼=[0,𝜔]. To be convenient, we introduce the notations 𝑓₀=liminf_𝑥→0+min_{𝑦≤𝐶0|𝑥|,𝑡∈𝐼}𝑓𝑡,𝑥,𝑦𝑥,𝑓⁰=limsup_𝑥→0+max_{𝑦≤𝐶0|𝑥|,𝑡∈𝐼}𝑓𝑡,𝑥,𝑦𝑥,𝑓_∞=liminf_𝑥→+∞min_{𝑦≤𝐶0|𝑥|,𝑡∈𝐼}𝑓𝑡,𝑥,𝑦𝑥,𝑓^∞=limsup_𝑥→+∞max_{𝑦≤𝐶0|𝑥|,𝑡∈𝐼}𝑓𝑡,𝑥,𝑦𝑥. (3.1) Our main results are as follows.

Theorem 3.1. Let 𝑓∶ℝ×(0,∞)×ℝ→ℝ be continuous and 𝑓(𝑡,𝑥,𝑦) be 𝜔-periodic in t. If 𝑓 satisfies Assumption (F0) and the condition (F1)𝑓⁰<0,𝑓_∞>0, then (1.1) has at least one positive 𝜔-periodic solution.

Theorem 3.2. Let 𝑓∶ℝ×(0,∞)×ℝ→ℝ be continuous and 𝑓(𝑡,𝑥,𝑦) be 𝜔-periodic in t. If 𝑓 satisfies Assumption (F0) and the conditions (F2)𝑓₀>0,𝑓^∞<0, then (1.1) has at least one positive 𝜔-periodic solution.

Noting that 0 is an eigenvalue of the associated linear eigenvalue problems of (1.1) with periodic boundary condition, if one inequality concerning comparison with 0 in (F1) or (F2) of Theorem 3.1 or Theorem 3.2 is not true, the existence of periodic solution to (1.1) cannot be guaranteed. Hence, the 0 is the optimal value in condition (F1) and (F2).

In Theorem 3.1, the condition (F1) allows 𝑓(𝑡,𝑥,𝑦) to have superlinear growth on 𝑥 and 𝑦. For example, 𝑓𝑡,𝑥,𝑦=𝑥²+𝑦²−14𝜋²𝜔²2+sin𝜋𝑡𝜔𝑥 (3.2) satisfies (F0) with 𝑀=(3/4)(𝜋²/𝜔²) and (F1) with 𝑓⁰=−(1/4)(𝜋²/𝜔²) and 𝑓_∞=+∞.

In Theorem 3.2, the condition (F2) allows that 𝑓(𝑡,𝑥,𝑦) has singularity at 𝑥=0. For example, 𝑓𝑡,𝑥,𝑦=𝑥𝑦²+2+sin(𝜋𝑡/𝜔)𝑥³−𝜋²2𝜔²𝑥 (3.3) satisfies (F0) with 𝑀=𝜋²/2𝜔², and (F2) with 𝑓₀=+∞ and 𝑓^∞=−𝜋²/2𝜔². The existence of periodic solutions for singular ordinary differential equations has been studied by several authors, see [20, 25, 26]. But the equations considered by these authors do not contain derivative term 𝑢^′(𝑡).

Proof of Theorem 3.1. Choose the working space 𝐸=𝐶¹_𝜔(ℝ). Let 𝐾⊂𝐶¹_𝜔(ℝ) be the closed convex cone in 𝐶¹_𝜔(ℝ) defined by (2.9) and 𝐴∶𝐾⧵{0}→𝐾 the operator defined by (2.18). 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∈𝐶¹_𝜔(ℝ)∣‖𝑢‖_𝐶¹<𝑟,Ω₂=𝑢∈𝐶¹_𝜔(ℝ)∣‖𝑢‖_𝐶¹<𝑅. (3.4) We show that the operator 𝐴 has a fixed point in 𝐾∩(Ω₂⧵Ω₁) by Theorem 2.6 when 𝑟 is small enough and 𝑅 large enough.
By 𝑓⁰<0 and the definition of 𝑓⁰, there exist 𝜀∈(0,𝑀) and 𝛿>0, such that 𝑓𝑡,𝑥,𝑦≤−𝜀𝑥,𝑡∈[0,𝜔],𝑦≤𝐶₀,0<𝑥≤𝛿. (3.5) Let 𝑟∈(0,𝛿). We now prove that 𝐴 satisfies the Condition (1) of Theorem 2.6, namely, 𝜆𝐴𝑢≠𝑢 for every 𝑢∈𝐾∩𝜕Ω₁ and 0<𝜆≤1. In fact, if there exist 𝑢₀∈𝐾∩𝜕Ω₁ and 0<𝜆₀≤1 such that 𝜆₀𝐴𝑢₀=𝑢₀, then by definition of 𝐴 and Lemma 2.1, 𝑢₀∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝑢₀^′′(𝑡)+𝑀𝑢₀(𝑡)=𝜆₀𝑓₁𝑡,𝑢₀(𝑡),𝑢^′₀(𝑡),𝑡∈ℝ. (3.6) Since 𝑢₀∈𝐾∩𝜕Ω₁, by the definitions of 𝐾 and Ω₁, we have 𝑢^′₀(𝑡)≤𝐶₀𝑢₀(𝑡),0<𝜎𝑢_0𝐶≤𝑢₀(𝑡)≤𝑢_0𝐶1=𝑟<𝛿,𝑡∈ℝ. (3.7) Hence from (3.5) it follows that 𝑓𝑡,𝑢₀(𝑡),𝑢^′₀(𝑡)≤−𝜀𝑢₀(𝑡),𝑡∈ℝ. (3.8) By this, (3.6), and the definition of 𝑓₁ we have 𝑢₀^′′(𝑡)+𝑀𝑢₀(𝑡)≤𝜆₀𝑀𝑢₀(𝑡)−𝜀𝑢₀(𝑡)≤(𝑀−𝜀)𝑢₀(𝑡),𝑡∈ℝ. (3.9) Integrating both sides of this inequality from 0 to 𝜔 and using the periodicity of 𝑢₀, we obtain that 𝑀^𝜔₀𝑢₀(𝑡)𝑑𝑡≤(𝑀−𝜀)^𝜔₀𝑢₀(𝑡)𝑑𝑡. (3.10) Since ∫^𝜔₀𝑢₀(𝑡)𝑑𝑡≥𝜔𝜎‖𝑢₀‖_𝐶>0, it follows that 𝑀≤𝑀−𝜀, which is a contradiction. Hence the Condition (1) of Theorem 2.6 holds.
On the other hand, since 𝑓_∞>0, by the definition of 𝑓_∞, there exist 𝜀₁>0 and 𝐻>0 such that 𝑓𝑡,𝑥,𝑦≥𝜀₁𝑥,𝑡∈[0,𝜔],𝑦≤𝐶₀𝑥,𝑥≥𝐻. (3.11) Define a function 𝑔∶(0,∞)→ℝ⁺ by 𝑔(𝑥)𝑓=max𝑡,𝑥,𝑦+𝑀𝑥𝑥∣𝑡∈[0,𝜔],𝑦≤𝑥. (3.12) Then 𝑔∶(0,∞)→ℝ⁺ is continuous. By (3.5) and Assumption (F0), 0≤𝑔(𝑥)≤𝜀+𝑀,0<𝑥≤𝛿. (3.13) This implies that 𝐶₁∶=sup𝑥𝑔(𝑥)∣0<𝑥≤𝐻<+∞. (3.14) Hence for every 𝑡∈[0,𝜔], 0<𝑥≤𝐻, and |𝑦|≤𝐶₀𝑥, we have 𝑓𝑡,𝑥,𝑦−𝜀₁𝑥≤𝑓𝑡,𝑥,𝑦+𝑀𝑥+𝑀+𝜀₁𝑥=𝑓𝑡,𝑥,𝑦+𝑀𝑥+𝑀+𝜀₁𝑥≤𝑥𝑔(𝑥)+𝑀+𝜀₁𝑥≤𝐶₁+𝑀+𝜀₁𝐻∶=𝐶₂. (3.15) Combining this with (3.11), it follows that 𝑓𝑡,𝑥,𝑦≥𝜀₁𝑥−𝐶₂,𝑡∈[0,𝜔],𝑦≤𝐶₀𝑥,𝑥>0. (3.16) Choose 𝑒(𝑡)≡1. Clearly,𝑒∈𝐾⧵{𝜃}. We show that 𝐴 satisfies the Condition (2) of Theorem 2.6 if 𝑅 is large enough, namely, 𝑢−𝐴𝑢≠𝜏𝑒 for every 𝑢∈𝐾∩𝜕Ω₂ and 𝜏≥0. In fact, if there exist 𝑢₁∈𝐾∩𝜕Ω₂ and 𝜏₁≥0 such that 𝑢₁−𝐴𝑢₁=𝜏₁𝑒, since 𝑢₁−𝜏₁𝑒=𝐴𝑢₁, by definition of 𝐴 and Lemma 2.1, 𝑢₁∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝑢₁^′′(𝑡)+𝑀𝑢₁(𝑡)−𝜏₁=𝑓₁𝑡,𝑢₁(𝑡),𝑢^′₁(𝑡),𝑡∈ℝ. (3.17) From (3.17) and (3.16), it follows that 𝑢₁^′′(𝑡)=𝑓𝑡,𝑢₁(𝑡),𝑢^′₁(𝑡)+𝑀𝜏₁≥𝑓𝑡,𝑢₁(𝑡),𝑢^′₁(𝑡)≥𝜀₁𝑢₁(𝑡)−𝐶₂,𝑡∈ℝ. (3.18) Integrating this inequality on [0,𝜔] and using the periodicity of 𝑢₁, we get that ^𝜔₀𝑢₁(𝑡)𝐶𝑑𝑡≤²𝜀₁. (3.19) Since 𝑢₁∈𝐾∩𝜕Ω₂, by the definition of 𝐾, we have 𝑢₁(𝑡)≥𝜎𝑢_1𝐶,𝑢^′₁(𝑡)≤𝐶₀𝑢₁(𝑡),𝑡∈ℝ. (3.20) By the first inequality of (3.20), we have ^𝜔₀𝑢₁(𝑡)𝑑𝑡≥𝜔𝜎𝑢_1𝐶. (3.21) From this and (3.19), it follows that 𝑢_1𝐶≤1𝜔𝜎^𝜔₀𝑢₁(𝑡)𝐶𝑑𝑡≤²𝜔𝜎𝜀₁. (3.22) By this and the second inequality of (3.20), we have 𝑢_1𝐶1=𝑢_1𝐶+𝑢^′_1𝐶≤𝑢_1𝐶+𝐶₀𝑢_1𝐶≤1+𝐶₀𝐶₂𝜔𝜎𝜀₁∶=𝑅. (3.23) Therefore, choose 𝑅>max{𝑅,𝛿}, then 𝐴 satisfies the Condition (2) of Theorem 2.6.
Now by the first part of Theorem 2.6, 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⊂𝐶¹_𝜔(ℝ) be defined by (3.4). We use Theorem 2.6 to prove that the operator 𝐴 has a fixed point in 𝐾∩(Ω₂⧵Ω₁) if 𝑟 is small enough and 𝑅 large enough.
By 𝑓₀>0 and the definition of 𝑓₀, there exist 𝜀>0 and 𝛿>0, such that 𝑓𝑡,𝑥,𝑦≥𝜀𝑥,𝑡∈[0,𝜔],𝑦≤𝐶₀𝑥,0<𝑥≤𝛿. (3.24) Let 𝑟∈(0,𝛿) and 𝑒(𝑡)≡1. We prove that 𝐴 satisfies the Condition (3) of Theorem 2.6, namely, 𝑢−𝐴𝑢≠𝜏𝑒 for every 𝑢∈𝐾∩𝜕Ω₁ and 𝜏≥0. In fact, if there exist 𝑢₀∈𝐾∩𝜕Ω₁ and 𝜏₀≥0 such that 𝑢₀−𝐴𝑢₀=𝜏₀𝑒, since 𝑢₀−𝜏₀𝑒=𝐴𝑢₀, by definition of 𝐴 and Lemma 2.1, 𝑢₀∈𝐶²_𝜔(ℝ) satisfies the differential equation 𝑢₀^′′(𝑡)+𝑀𝑢₀(𝑡)−𝜏₀=𝑓₁𝑡,𝑢₀(𝑡),𝑢^′₀(𝑡),𝑡∈ℝ. (3.25) Since 𝑢₀∈𝐾∩𝜕Ω₁, by the definitions of 𝐾 and Ω₁, 𝑢₀ satisfies (3.7). From (3.7), and (3.24) it follows that 𝑓𝑡,𝑢₀(𝑡),𝑢^′₀(𝑡)≥𝜀𝑢₀(𝑡),𝑡∈ℝ. (3.26) By this, (3.25), and the definition of 𝑓₁, we have 𝑢₀^′′(𝑡)+𝑀𝑢₀(𝑡)=𝑓₁𝑡,𝑢₀(𝑡),𝑢₀^′′(𝑡)+𝑀𝜏₀≥(𝑀+𝜀)𝑢₀(𝑡),𝑡∈ℝ. (3.27) Integrating this inequality on [0,𝜔] and using the periodicity of 𝑢₀(𝑡), we obtain that 𝑀^𝜔₀𝑢₀(𝑡)𝑑𝑡≥(𝑀+𝜀)^𝜔₀𝑢₀(𝑡)𝑑𝑡. (3.28) Since ∫^𝜔₀𝑢₀(𝑡)𝑑𝑡≥𝜔𝜎‖𝑢₀‖_𝐶>0, from this inequality it follows that 𝑀≥𝑀+𝜀, which is a contradiction. Hence 𝐴 satisfies the Condition (3) of Theorem 2.6.
Since 𝑓^∞<0, by the definition of 𝑓^∞, there exist 𝜀₁∈(0,𝑀) and 𝐻>0 such that 𝑓𝑡,𝑥,𝑦≤−𝜀₁𝑥,𝑡∈[0,𝜔],𝑦≤𝐶₀𝑥,𝑥≥𝐻. (3.29) Choosing 𝑅>max{(1+𝐶₀)𝐻/𝜎,𝛿}, we show that 𝐴 satisfies the Condition (4) of Theorem 2.6, namely, 𝜆𝐴𝑢≠𝑢 for every 𝑢∈𝐾∩𝜕Ω₂ and 0<𝜆≤1. In fact, if there exist 𝑢₁∈𝐾∩𝜕Ω₂ and 0<𝜆₁≤1 such that 𝜆₁𝐴𝑢₁=𝑢₁, then by the definition of 𝐴 and Lemma 2.1, 𝑢₁∈𝐶²_𝜔(Ω) satisfies the differential equation 𝑢₁^′′(𝑡)+𝑀𝑢₁(𝑡)=𝜆₁𝑓₁𝑡,𝑢₁(𝑡),𝑢^′₁(𝑡),𝑡∈ℝ. (3.30) Since 𝑢₁∈𝐾∩𝜕Ω₂, by the definition of 𝐾, 𝑢₁ satisfies (3.20). By the second inequality of (3.20), we have 𝑢_1𝐶1=𝑢_1𝐶+𝑢^′_1𝐶≤𝑢_1𝐶+𝐶₀𝑢_1𝐶=1+𝐶₀𝑢_1𝐶. (3.31) Consequently, 𝑢_1𝐶≥11+𝐶₀𝑢_1𝐶1. (3.32) By (3.32) and the first inequality of (3.20), we have 𝑢₁(𝑡)≥𝜎𝑢_1𝐶≥𝜎1+𝐶₀𝑢_1𝐶1=𝜎1+𝐶₀𝑅>𝐻,𝑡∈ℝ. (3.33) From this, the second inequality of (3.20) and (3.29), it follows that 𝑓𝑡,𝑢₁(𝑡),𝑢^′₁(𝑡)≤−𝜀₁𝑢₁(𝑡),𝑡∈ℝ. (3.34) By this and (3.30), we have 𝑢₁^′′(𝑡)+𝑀𝑢₁(𝑡)≤𝜆₁𝑀𝑢₁(𝑡)−𝜀₁𝑢₁(𝑡)≤𝑀−𝜀₁𝑢₁(𝑡),𝑡∈ℝ. (3.35) Integrating this inequality on [0,𝜔] and using the periodicity of 𝑢₁(𝑡), we obtain that 𝑀^𝜔₀𝑢₁(𝑡)𝑑𝑡≤𝑀−𝜀₁^𝜔₀𝑢₁(𝑡)𝑑𝑡. (3.36) Since ∫^𝜔₀𝑢₁(𝑡)𝑑𝑡≥𝜔𝜎‖𝑢₁‖_𝐶>0, from this inequality it follows that 𝑀≤𝑀−𝜀₁, which is a contradiction. This means that 𝐴 satisfies the Condition (4) of Theorem 2.6.
By the second part of Theorem 2.6, 𝐴 has a fixed point in 𝐾∩(Ω₂⧵Ω₁), which is a positive 𝜔-periodic solution of (1.1).

Example 3.3. Consider the second-order differential equation 𝑢^′′=𝑎₁(𝑡)𝑢+𝑎₂(𝑡)𝑢²+𝑎₃(𝑡)𝑢^′2𝑢,𝑡∈ℝ, (3.37) where 𝑎_𝑖(𝑡)∈𝐶_𝜔(ℝ),𝑖=1,2,3. If −𝜋²/𝜔²<𝑎₁(𝑡)<0 and 𝑎₂(𝑡),𝑎₃(𝑡)>0 for 𝑡∈[0,𝜔], then 𝑓(𝑡,𝑥,𝑦)=𝑎₁(𝑡)𝑥+𝑎₂(𝑡)𝑥²+𝑎₃𝑥𝑦² satisfies the conditions (F0) and (F1). By Theorem 3.1, (3.37) has at least one positive 𝜔-periodic solution.

Example 3.4. Consider the singular differential equation: 𝑢^′′(𝑡)𝑏(𝑡)(𝑡𝑢=𝑎𝑢+𝑢+𝑐)^′2𝑢²,𝑡∈ℝ, (3.38) where 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡)∈𝐶_𝜔(ℝ). If −𝜋²/𝜔²<𝑎(𝑡)<0 and 𝑏(𝑡),𝑐(𝑡)>0 for 𝑡∈[0,𝜔], then 𝑓(𝑡,𝑥,𝑦)=𝑎(𝑡)𝑥+(𝑏(𝑡)𝑥+𝑐(𝑡)𝑦²)/𝑥² satisfies the conditions (F0) and (F2). By Theorem 3.2, the (3.38) has a positive 𝜔-periodic solution.

4. Remarks

Our discussion on the existence of the positive 𝜔-periodic solutions to (1.1) is applicable to the following ordinary differential equation: −𝑢^′′(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢^′(𝑡),𝑡∈ℝ, (4.1) where the nonlinearity 𝑓∶ℝ×(0,∞)×ℝ→ℝ is continuous and 𝑓(𝑡,𝑥,𝑦) is 𝜔-periodic in 𝑡. For (4.1), we need the following assumption.(F0)^∗ There exists 𝑀>0 such that 𝑓𝑡,𝑥,𝑦+𝑀𝑥≥0,𝑥>0,𝑡,𝑦∈ℝ. (4.2) Similarly to Lemma 2.1, we have the following conclusion.

Lemma 4.1. Let 𝑀>0 be a constant. Then for every ℎ∈𝐶_𝜔(ℝ), the linear second order differential equation −𝑢^′′(𝑡)+𝑀𝑢(𝑡)=ℎ(𝑡),𝑡∈ℝ, (4.3) has a unique 𝜔-periodic solution 𝑢(𝑡), which is given by 𝑢(𝑡)=^𝑡_𝑡−𝜔𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠𝑡∈ℝ, (4.4) where 𝑉(𝑡) is the unique solution of the linear second-order boundary value problem −𝑢^′′(𝑡)(𝑡)𝑢(0)(𝜔)+𝑀𝑢=0,0≤𝑡≤𝜔,−𝑢=0,𝑢^′(0)−𝑢^′(𝜔)=−1, (4.5) which is explicitly given by 𝑉(𝑡)=cosh𝛽(𝑡−𝜔/2)2𝛽sinh𝛽𝜔/2,0≤𝑡≤𝜔, (4.6) with 𝛽=√𝑀.

Since 𝑉∶=max_{0≤𝑡≤𝜔}𝑉(𝑡)=cosh𝛽𝜔/22𝛽sinh𝛽𝜔/2,𝑉∶=min_{0≤𝑡≤𝜔}𝑉(𝑡)=1,2𝛽sinh𝛽𝜔/2𝑉₁∶=max_{0≤𝑡≤𝜔}𝑉^′(𝑡)=max_{0≤𝑡≤𝜔}sinh𝛽(𝑡−𝜔/2)=12sinh𝛽𝜔/22, (4.7) we renew to define 𝜎 and 𝐶₀ by 𝑉𝜎=𝑉=1cosh𝛽𝜔/2,𝐶₀=𝑉₁𝑉=𝛽sinh𝛽𝜔2. (4.8)

Now, using the similar arguments to Theorems 3.1 and 3.2, we can obtain the following results.

Theorem 4.2. Let 𝑓∶ℝ×(0,∞)×ℝ→ℝ be continuous and 𝑓(𝑡,𝑥,𝑦) be 𝜔-periodic in t. If 𝑓 satisfies Assumption (F0)* and the condition (F1)𝑓⁰<0,𝑓_∞>0, then (4.1) has at least one positive 𝜔-periodic solution.

Theorem 4.3. Let 𝑓∶ℝ×(0,∞)×ℝ→ℝ be continuous and 𝑓(𝑡,𝑥,𝑦) be 𝜔-periodic in t. If 𝑓 satisfies Assumption (F0)* and the conditions (F2)𝑓₀>0,𝑓^∞<0, then (4.1) has at least one positive 𝜔-periodic solution.

Theorems 4.2 and 4.3 improve and extend some results in References [18, 19, 22].

Acknowledgment

This paper is supported by NNSFs of China (11261053 and 11061031).