Positive Periodic Solution for Second-Order Singular Semipositone Differential Equations

Abstract

We study the existence of a positive periodic solution for second-order singular semipositone differential equation by a nonlinear alternative principle of Leray-Schauder. Truncation plays an important role in the analysis of the uniform positive lower bound for all the solutions of the equation. Recent results in the literature (Chu et al., 2010) are generalized.

1. Introduction

In this paper, we study the existence of positive 𝑇-periodic solutions for the following singular semipositone differential equation: 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=𝑓𝑡,𝑥,𝑥^′, (1) where ℎ,𝑎∈𝐶(𝑅/𝑇𝑍,𝑅) and the nonlinearity 𝑓∈𝐶((𝑅/𝑇𝑍)×(0,+∞)×𝑅,𝑅) satisfies 𝑓(𝑡,𝑥,𝑥^′)≥−𝑀 for some 𝑀>0. In particular, the nonlinearity may have a repulsive singularity at 𝑥=0, which means that lim_𝑥→0+𝑓𝑡,𝑥,𝑦=+∞,uniformlyin𝑡,𝑦∈𝑅². (2) Electrostatic or gravitational forces are the most important examples of singular interactions.

During the last two decades, the study of the existence of periodic solutions for singular differential equations has attracted the attention of many researchers [1–4]. Some strong force conditions introduced by Gordon [5] are standard in the related earlier works [6, 7]. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent [2, 8, 9], but has also attracted many researchers. Some classical tools have been used to study singular differential equations in the literature, including the method of upper and lower solutions [10], degree theory [11], some fixed point theorem in cones for completely continuous operators [12], Schauder’s fixed point theorem [8, 9, 13], and a nonlinear Leray-Schauder alternative principle [2, 3, 14, 15].

However the singular differential equations, in which there is the damping term, that is, the nonlinearity is dependent on the derivative, has not attracted much attention in the literature. Several existence results can be found in [14, 16, 17].

The aim of this paper is to further show that the nonlinear Leray-Schauder alternative principle can be applied to (1) in the semipositone cases, that is, 𝑓(𝑡,𝑥,𝑥^′)≥−𝑀 for some 𝑀>0.

The remainder of the paper is organized as follows. In Section 2, we state some known results. In Section 3, the main results of this paper are stated and proved. To illustrate our result, we select the following system: 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=1+|𝑥|^𝛾𝑥^−𝛼+𝜇𝑥^𝛽+𝑒(𝑡), (3) where 𝛼>1,  𝛽>0,  1>𝛾≥0,  𝜇>0 is a positive parameter, 𝑒(𝑡) is a 𝑇-periodic function.

In this paper, let us fix some notations to be used in the following: given 𝜑∈𝐿¹[0,𝑇], we write 𝜑≻0 if 𝜑≥0 for almost everywhere 𝑡∈[0,𝑇] and it is positive in a set of positive measure. The usual 𝐿^𝑝-norm is denoted by ‖⋅‖_𝑝. 𝑝^∗ and 𝑝_∗ the essential supremum and infinum of a given function 𝑝∈𝐿¹[0,𝑇], if they exist.

2. Preliminaries

We say that 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=0, (4) associated to the periodic boundary conditions 𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇), (5) is nonresonant when its unique solutions is the trival one. When (4)-(5) is nonresonant, as a consequence of Fredholm’s alternative, the nonhomogeneous equation 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=𝑙(𝑡) (6) admits a unique 𝑇-periodic solution, which can be written as 𝑥(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑙(𝑠)𝑑𝑠, (7) where 𝐺(𝑡,𝑠) is the Green’s function of problem (4)-(5). Throughout this paper, we assume that the following standing hypothesis is satisfied.(A) The Green function 𝐺(𝑡,𝑠), associated with (4)-(5), is positive for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇].

In other words, the strict antimaximum principle holds for (4)-(5).

Definition 1. We say that (4) admits the antimaximum principle if (6) has a unique 𝑇-periodic solution for any 𝑙∈ℂ(ℝ/𝑇ℤ) and the unique 𝑇-periodic solution 𝑥_𝑙(𝑡)>0 for all 𝑡 if 𝑙≻0.

Under hypothesis (A), we denote 𝐴=min_{0≤𝑠,𝑡≤𝑇}𝐺(𝑡,𝑠),𝐵=max_{0≤𝑠,𝑡≤𝑇}𝐺()𝐴𝑡,𝑠,𝜄=𝐵. (8) Thus 𝐵>𝐴>0 and 0<𝜄<1. We also use 𝑤(𝑡) to denote the unique periodic solution of (6) with 𝑙(𝑡)=1 under condition (5), that is, 𝑤(𝑡)=(ℒ1)(𝑡). In particular, 𝑇𝐴≤𝑤(𝑡)≤𝑇𝐵.

With the help of [18, 19], the authors give a sufficient condition to ensure that (4) admits the antimaximum principle in [14]. In order to state this result, let us define the functions 𝜎(ℎ)(𝑡)=exp^𝑡₀ℎ(𝑠),𝜎𝑑𝑠₁(ℎ)(𝑡)=𝜎(ℎ)(𝑇)^𝑡₀𝜎ℎ(𝑠)𝑑𝑠+^𝑇_𝑡𝜎(ℎ)(𝑠)𝑑𝑠. (9)

Lemma 2 (see [14, Corollary 2.6]). Assume that 𝑎≢0 and the following two inequalities are satisfied: ^𝑇₀𝑎(𝑠)𝜎(ℎ)(𝑠)𝜎₁(−ℎ)(𝑠)𝑑𝑠≥0,sup_{0≤𝑡≤𝑇𝑡}^𝑡+𝑇𝜎(−ℎ)(𝑠)𝑑𝑠_𝑡^𝑡+𝑇[𝑎(𝑠)]₊𝜎(ℎ)(𝑠)𝑑𝑠≤4, (10) where [𝑎(𝑠)]₊=max{𝑎(𝑠),0}. Then the Green’s function 𝐺(𝑡,𝑠), associated with (5), is positive for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇].

Next, recall a well-known nonlinear alternative principle of Leray-Schauder, which can be found in [20] and has been used by Meehan and O’Regan in [4].

Lemma 3. Assume Ω is an open subset of a convex set 𝐾 in a normed linear space 𝑋 and 𝑝∈Ω. Let 𝑇∶Ω→𝐾 be a compact and continuous map. Then one of the following two conclusions holds:(I)𝑇 has at least one fixed point in Ω.(II)There exists 𝑥∈𝜕Ω and 0<𝜆<1 such that 𝑥=𝜆𝑇𝑥+(1−𝜆)𝑝.

In applications below, we take 𝐾=𝐶¹_𝑇={𝑥∶𝑥,𝑥^′∈𝐶(𝑅/𝑇𝑍,𝑅)}⊂𝑋 with the norm ‖𝑥‖=max_{𝑡∈[0,𝑇]}|𝑥(𝑡)| and define Ω={𝑥∈𝐶¹_𝑇∶‖𝑥‖<𝑟}.

3. Main Results

In this section, we prove a new existence result of (1).

Theorem 4. Suppose that (4) satisfies (A) and 𝑎(𝑡)≻0. (11) Furthermore, assume that there exist three constants 𝑀,𝑅₀,𝑟>𝑀𝑤^∗/𝜄 such that:(H₁)𝐹(𝑡,𝑥,𝑦)=𝑓(𝑡,𝑥,𝑦)+𝑀≥0 for all (𝑡,𝑥,𝑦)∈[0,𝑇]×(0,𝑟]×𝑅.(H₂)𝑓(𝑡,𝑥,𝑦)≥𝑔₀(𝑥) for (𝑡,𝑥,𝑦)∈[0,𝑇]×(0,𝑅₀]×𝑅, where the nonincreasing continuous function 𝑔₀(𝑥)>0 satisfies lim_𝑥→0⁺𝑔₀(𝑥)=+∞ and lim_𝑥→0⁺∫^𝑅0_𝑥𝑔₀(𝑢)𝑑𝑢=+∞.(H₃)0≤𝐹(𝑡,𝑥,𝑦)≤(𝑔(𝑥)+ℎ(𝑥))𝜚(|𝑦|),  for  all  (𝑡,𝑥,𝑦)∈[0,𝑇]×(0,𝑟]×𝑅, where 𝑔(⋅)>0 is nonincreasing in (0,𝑟] and ℎ(⋅)/𝑔(⋅)≥0,  𝜚(⋅)≥0 are nondecreasing in (0,𝑟]. (H₄)𝑟𝑔(𝜄𝑟−𝑀𝑤^∗)1+ℎ(𝑟)/𝑔(𝑟)𝜚((𝑟+𝑀)𝐿)>𝑤^∗, (12) where 𝐿=2∫^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑑𝑡min_{0≤𝑡≤𝑇}𝜎(ℎ𝑡).)( (13) Then (1) has at least one positive periodic solution 𝑢(𝑡) with 0<‖𝑢+𝑀𝑤‖≤𝑟.

Proof. For convinence, let us write 𝑍(𝑡)=𝑥(𝑡)−𝑀𝑤(𝑡), 𝑍_𝑛(𝑡)=𝑥_𝑛(𝑡)−𝑀𝑤(𝑡), where 𝑤(𝑡)=(ℒ1)(𝑡). Let 𝐴_ℎ=min_{0≤𝑡≤𝑇}𝜎(ℎ)(𝑡),𝐵_ℎ=max_{0≤𝑡≤𝑇}𝜎(ℎ)(𝑡),𝜄_ℎ=𝐵_ℎ𝐴_ℎ,𝑀_ℎ=(𝑟+𝑀)𝐿⋅max_{0≤𝑡≤𝑇}|ℎ(𝑡)|. (14)(15) First we show that 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=𝐹𝑡,𝑍(𝑡),𝑍^′(𝑡) (16) has a solution 𝑥 satisfying (5), 0<‖𝑥‖≤𝑟 and 𝑍(𝑡)>0 for 𝑡∈[0,𝑇]. If this is true, it is easy to see that 𝑍(𝑡) will be a positive solution of (1)–(5) with 0<‖𝑍+𝑀𝑤‖≤𝑟.
Choose 𝑛₀∈{1,2,…} such that 1/𝑛₀<𝑟, and then let 𝑁₀={𝑛₀,𝑛₀+1+⋯}.
Consider the family of equations 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=𝜆𝐹_𝑛𝑡,𝑍(𝑡),𝑍^′(𝑡)+𝑎(𝑡)𝑛, (17) where 𝜆∈[0,1],  𝑛∈𝑁₀,  𝑥∈𝐵_𝑟={𝑥∶‖𝑥‖<𝑟} and 𝐹_𝑛(𝑡,𝑥,𝑦)=𝐹(𝑡,max{1/𝑛,𝑥},𝑦).
A 𝑇-periodic solution of (17) is just a fixed of the operator equation 𝑥=𝜆𝑇_𝑛(𝑥)+(1−𝜆)𝑝, (18) where 𝑝=1/𝑛 and 𝑇_𝑛 is a completely continuous operator defined by 𝑇_𝑛𝑥(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝐹_𝑛𝑠,𝑍(𝑠),𝑍^′(𝑠)1𝑑𝑠+𝑛, (19) where we have used the fact ^𝑇₀𝐺(𝑡,𝑠)𝑎(𝑠)𝑑𝑠≡1. (20)
We claim that for any 𝑇-periodic solution 𝑥_𝑛(𝑡) of (17) satisfies 𝑥^′_𝑛≤𝐿𝑟. (21) Note that the solution 𝑥_𝑛(𝑡) of (17) is also satisfies the following equivalent equation 𝜎(ℎ)(𝑡)𝑥^′_𝑛^′+𝑎(𝑡)𝜎(ℎ)(𝑡)𝑥_𝑛=𝜎(ℎ)(𝑡)𝜆𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)+𝑎(𝑡)𝑛. (22) Integrating (22) from 0 to 𝑇, we obtain ^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑥_𝑛(𝑡)=𝑑𝑡_𝑇0𝜎(ℎ)(𝑡)𝜆𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡+𝑎(𝑡))𝑛𝑑𝑡. (23)
By the periodic boundary conditions, we have 𝑥^′(𝑡₀)=0 for some 𝑡₀∈[0,𝑇]. Therefore,𝜎(ℎ)(𝑡)𝑥^′_𝑛(𝑡=_𝑡𝑡0)𝜎(ℎ)(𝑠)𝑥^′_𝑛(𝑠)^′=𝑑𝑠_𝑡𝑡0𝜎(ℎ)(𝑠)𝜆𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)+𝑎(𝑠)𝑛−𝑎(𝑠)𝑥_𝑛(𝑠)≤𝑑𝑠_𝑇0𝜎(ℎ)(𝑠)𝜆𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)+𝑎(𝑠)𝑛+𝑎(𝑠)𝑥_𝑛(𝑠)𝑑𝑠=2^𝑇₀𝜎(ℎ)(𝑠)𝑎(𝑠)𝑥_𝑛(𝑠)𝑑𝑠≤2𝑟^𝑇₀𝜎(ℎ)(𝑠)𝑎(𝑠)𝑑𝑠, (24)where we have used the assumption (11) and ‖𝑥_𝑛‖<𝑟. Therefore, min_{0≤𝑡≤𝑇}𝜎(ℎ)(𝑡)𝑥^′_𝑛(𝑡)≤2𝑟^𝑇₀𝜎(ℎ)(𝑠)𝑎(𝑠)𝑑𝑠, (25) which implies that (21) holds. In particular, let 𝜆𝐹_𝑛(𝑡,𝑍(𝑡),𝑍^′(𝑡))+𝑎(𝑡)/𝑛=1 in (17), we have 𝑤^′(𝑡)≤𝐿. (26)
Choose 𝑛₁∈𝑁₀ such that 1/𝑛₁≤𝑅₁, and then let 𝑁₁={𝑛₁,𝑛₁+1,…}. The following lemma holds.

Lemma 5. There exists an integer 𝑛₂>𝑛₁ large enough such that, for all 𝑛∈𝑁₂={𝑛₂,𝑛₂+1,…}, 𝑍_𝑛(𝑡)=𝑥_𝑛(𝑡)(𝑡)≥1−𝑀𝑤𝑛. (27)

Proof. The lower bound in (27) is established by using the strong force condition of 𝑓(𝑡,𝑥,𝑦). By condition (H₂), there exists 𝑅₁∈(0,𝑅₀) and a continuous function 𝑔₀(𝑥) such that 𝐹𝑡,𝑥,𝑦−𝑎(𝑡)𝑥≥𝑔₀(𝑥)>max𝑀+𝑀,𝜄_ℎ𝑟‖𝑎‖₁ (28) for all (𝑡,𝑥,𝑦)∈[0,𝑇]×(0,𝑅₁]×𝑅, where 𝑔₀(𝑥) satisfies also the strong force condition like in (H₂).
For 𝑛∈𝑁₁, let 𝛼_𝑛=min_{0≤𝑡≤𝑇}𝑍_𝑛(𝑡),  𝛽_𝑛=max_{0≤𝑡≤𝑇}𝑍_𝑛(𝑡).
If 𝛼_𝑛≥𝑅₁, due to 𝑛∈𝑁₁, (27) holds.
If 𝛼_𝑛<𝑅₁, we claim that, for all 𝑛∈𝑁₁, 𝛽_𝑛>𝑅₁. (29) Otherwise, suppose that 𝛽_𝑛≤𝑅₁ for some 𝑛∈𝑁₁. Then it is easy to verify 𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)>𝜄_ℎ𝑟‖𝑎‖₁. (30) In fact, if 1/𝑛≤𝑍_𝑛(𝑡)≤𝑅₁, we obtain from (28) 𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)=𝐹𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)≥𝑎(𝑡)𝑍_𝑛(𝑡)+𝑔₀𝑍_𝑛(𝑡)≥𝑔₀𝑍_𝑛(𝑡)>𝜄_ℎ𝑟‖𝑎‖₁. (31) and, if 𝑍_𝑛(𝑡)≤1/𝑛, we have 𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)1=𝐹𝑡,𝑛,𝑍^′_𝑛(𝑡)≥𝑎(𝑡)𝑛+𝑔₀1𝑛≥𝑔₀1𝑛>𝜄_ℎ𝑟‖𝑎‖₁. (32) Integrating (22) (with 𝜆=1) from 0 to 𝑇, we deduce that 0=^𝑇₀𝜎(ℎ)(𝑡)𝑥^′_𝑛^′+𝑎(𝑡)𝜎(ℎ)(𝑡)𝑥_𝑛−𝜎(ℎ)(𝑡)𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)+𝑎(𝑡)𝑛=𝑑𝑡_𝑇0𝑎(𝑡)𝜎(ℎ)(𝑡)𝑥_𝑛−𝑑𝑡_𝑇0𝜎(ℎ)(𝑡)𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)−𝑑𝑡_𝑇0𝜎(ℎ𝑡)𝑎(𝑡))(𝑛𝑑𝑡<^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑥_𝑛−𝑑𝑡_𝑇0𝜎(ℎ)(𝑡)𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)𝑑𝑡<0, (33) where estimation (30) and the fact ‖𝑥_𝑛‖<𝑟 are used. This is a contradiction. Hence (29) holds.
Due to 𝛼_𝑛<𝑅₁, that is, 𝛼_𝑛=min_{0≤𝑡≤𝑇}[𝑥_𝑛(𝑡)−𝑀𝑤(𝑡)]=𝑥_𝑛(𝑎_𝑛)−𝑀𝑤(𝑎_𝑛)<𝑅₁ for some 𝑎_𝑛∈[0,𝑇]. By (29), there exists 𝑐_𝑛∈[0,𝑇] (without loss of generality, we assume 𝑎_𝑛<𝑐_𝑛.) such that 𝑥_𝑛(𝑐_𝑛)=𝑀𝑤(𝑐_𝑛)+𝑅₁ and 𝑥_𝑛(𝑡)≤𝑀𝑤(𝑡)+𝑅₁ for 𝑎_𝑛≤𝑡≤𝑐_𝑛.
It can be checked that 𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)>𝑎(𝑡)𝑍_𝑛(𝑡)+𝑀+𝑀_ℎ, (34) where 𝑀_ℎ is defined by (15).
In fact, if 𝑡∈[𝑎_𝑛,𝑐_𝑛] is such that 1/𝑛≤𝑍_𝑛(𝑡)≤𝑅₁, we have 𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)=𝐹𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)≥𝑎(𝑡)𝑍_𝑛(𝑡)+𝑔₀(𝑥)>max𝑀+𝑀,𝜄_ℎ𝑟‖𝛼‖₁≥𝑎(𝑡)𝑍_𝑛(𝑡)+𝑀+𝑀. (35) and, if 𝑡∈[𝑎_𝑛,𝑐_𝑛] is such that 𝑍_𝑛(𝑡)≤1/𝑛, we have 𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡1)=𝐹𝑡,𝑛,𝑍^′_𝑛(𝑡≥𝑎(𝑡))𝑛+𝑔₀1𝑛>𝑎(𝑡)𝑛+𝑀+𝑀_ℎ≥𝑎(𝑡)𝑍_𝑛(𝑡)+𝑀+𝑀_ℎ. (36) So (34) holds.
Using (17) (with 𝜆=1) for 𝑥_𝑛(𝑡) and the estimation (34), we have, for 𝑡∈[𝑎_𝑛,𝑐_𝑛]𝑍_𝑛^′′(𝑡)=−ℎ(𝑡)𝑍^′_𝑛(𝑡)−𝑎(𝑡)𝑍_𝑛(𝑡)−𝑀+𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)+𝑎(𝑡)𝑛>−ℎ(𝑡)𝑍^′_𝑛(𝑡)−𝑎(𝑡)𝑍_𝑛(𝑡)−𝑀+𝑎(𝑡)𝑍_𝑛(𝑡)+𝑀+𝑀_ℎ+𝑎(𝑡)𝑛≥−𝑀_ℎ−𝑎(𝑡)𝑍_𝑛(𝑡)(𝑡)𝑍−𝑀+𝑎_𝑛(𝑡)+𝑀+𝑀_ℎ+𝑎(𝑡)𝑛≥𝑎(𝑡)𝑛≥0. (37) As 𝑍^′_𝑛(𝑎_𝑛)=0, 𝑍^′_𝑛(𝑡)>0 for all 𝑡∈[𝑎_𝑛,𝑐_𝑛], so 𝑍_𝑛(𝑡) is strictly increasing on [𝑎_𝑛,𝑐_𝑛]. We use 𝜉_𝑛 to denote the inverse function of 𝑍_𝑛 restricted to [𝑎_𝑛,𝑐_𝑛].
Suppose that (27) does not hold, that is, for some 𝑛∈𝑁₁, 𝑍_𝑛(𝑡)<1/𝑛<𝑅₁. Then there would exist 𝑏_𝑛∈(𝑎_𝑛,𝑐_𝑛) such that 𝑍_𝑛(𝑏_𝑛)=1/𝑛 and 𝑍_𝑛(𝑡)≤1𝑛for𝑎_𝑛≤𝑡≤𝑏_𝑛,1𝑛≤𝑍_𝑛(𝑡)≤𝑅₁for𝑏_𝑛≤𝑡≤𝑐_𝑛. (38) Multiplying (17) (with 𝜆=1) by 𝑍^′_𝑛(𝑡) and integrating from 𝑏_𝑛 to 𝑐_𝑛, we obtain ^𝑅1_1/𝑛𝐹𝜉_𝑛(𝑍),𝑍,𝑍^′=𝑑𝑍_{𝑐𝑛𝑏𝑛}𝐹𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)𝑍^′_𝑛(𝑡)=𝑑𝑡_{𝑐𝑛𝑏𝑛}𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)𝑍^′_𝑛(𝑡)=𝑑𝑡_{𝑐𝑛𝑏𝑛}𝑥_𝑛^′′(𝑡)+ℎ(𝑡)𝑥^′_𝑛(𝑡)+𝑎(𝑡)𝑥_𝑛(𝑡)−𝑎(𝑡)𝑛𝑍^′_𝑛(𝑡)𝑑𝑡=^𝑐𝑛_𝑏𝑛𝑥_𝑛^′′(𝑡)𝑥^′_𝑛(𝑡)−𝑀𝑤^′(𝑡)𝑑𝑡+^𝑐𝑛_𝑏𝑛ℎ(𝑡)𝑥^′_𝑛(𝑡)𝑍^′_𝑛(𝑡)+𝑑𝑡_{𝑐𝑛𝑏𝑛}𝑎(𝑡)𝑥_𝑛(𝑡)−𝑎(𝑡)𝑛𝑍^′_𝑛(𝑡)𝑑𝑡. (39)
By the facts ‖𝑥_𝑛‖<𝑟,  ‖𝑥^′_𝑛‖≤𝐿𝑟,  ‖𝑤^′‖≤𝑟 and the definition of 𝑍_𝑛(𝑡), we can obtain |𝑍_𝑛(𝑡)|≤𝑟+𝑇𝐵,|𝑍^′_𝑛(𝑡)|≤(𝑟+𝑀)𝐿, together with ‖𝑥_𝑛‖<𝑟, implies that the second term and the third term are bounded. The first term is 𝑥^′_𝑛𝑐_𝑛²−𝑥^′_𝑛𝑏_𝑛²2−𝑀𝑥^′_𝑛𝑐_𝑛𝑤^′𝑐_𝑛−𝑥^′_𝑛𝑏_𝑛𝑤^′𝑏_𝑛+𝑀^𝑐𝑛_𝑏𝑛𝑥^′_𝑛(𝑡)𝑤^′′(𝑡)𝑑𝑡, (40) which is also bounded. As a consequence, there exists a 𝐵₁>0 such that ^𝑅1_1/𝑛𝐹𝜉_𝑛(𝑍),𝑍,𝑍^′𝑑𝑍≤𝐵₁. (41) On the other hand, by (H₂), we can choose 𝑛₂∈𝑁₁ large enough such that ^𝑅1_1/𝑛𝐹𝜉_𝑛(𝑍),𝑍,𝑍^′𝑑𝑍≥^𝑅1_1/𝑛𝑔₀(𝑍)𝑑𝑍>𝐵₁ (42) for all 𝑛∈𝑁₂={𝑛₂,𝑛₂+1,…}. So (27) holds.

Furthermore, we can prove 𝑍_𝑛(𝑡) has a uniform positive lower bound 𝛿.

Lemma 6. There exist a constant 𝛿>0 such that, for all 𝑛∈𝑁₂, 𝑍_𝑛(𝑡)≥𝛿. (43)

Proof. Multiplying (17) (with 𝜆=1) by 𝑍^′_𝑛(𝑡) and integrating from 𝑎_𝑛 to 𝑐_𝑛, we obtain ^𝑅1_𝛼𝑛𝐹𝜉_𝑛(𝑍),𝑍,𝑍^′𝑑𝑍=^𝑐𝑛_𝑎𝑛𝐹𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)𝑍^′_𝑛(𝑡)=𝑑𝑡_{𝑐𝑛𝑎𝑛}𝐹_𝑛𝑡,𝑍_𝑛(𝑡),𝑍^′_𝑛(𝑡)𝑍^′_𝑛(𝑡)=𝑑𝑡_{𝑐𝑛𝑎𝑛}𝑥_𝑛^′′(𝑡)+ℎ(𝑡)𝑥^′_𝑛(𝑡)+𝑎(𝑡)𝑥_𝑛(𝑡)−𝑎(𝑡)𝑛𝑍^′_𝑛(𝑡)=𝑑𝑡_{𝑐𝑛𝑎𝑛}𝑥_𝑛^′′(𝑡)𝑥^′_𝑛(𝑡)−𝑀𝑤^′(𝑡)𝑑𝑡+^𝑐𝑛_𝑎𝑛ℎ(𝑡)𝑥^′_𝑛(𝑡)𝑍^′_𝑛(𝑡)+𝑑𝑡_{𝑐𝑛𝑎𝑛}𝑎(𝑡)𝑥_𝑛(𝑡)−𝑎(𝑡)𝑛𝑍^′_𝑛(𝑡)𝑑𝑡. (44)
In the same way as in the proof of (41), one way readily prove that the right-hand side of the above equality is bounded. On the other hand, if 𝑛∈𝑁₂, by (H₂), ^𝑅1_𝛼𝑛𝐹𝜉_𝑛(𝑍),𝑍,𝑍^′≥𝑑𝑍_{𝑅1𝛼𝑛}𝑔₀(𝑍)𝑑𝑍+𝑀𝑅₁−𝛼_𝑛→+∞ (45) if 𝛼_𝑛→0₊. Thus we know that there exists a constant 𝛿>0 such that 𝛼_𝑛≥𝛿. Hence (43) holds.
Next, we will prove (17) has periodic solution 𝑥_𝑛(𝑡).
For 𝜄𝑟>0, we can choose 𝑛₃∈𝑁₂ such that 1/𝑛₃<𝜄𝑟, which together with (H₄) imply 𝑤^∗𝑔𝜄𝑟−𝑀𝑤^∗ℎ(𝑟)1+𝑔(𝑟)𝜚)𝐿)+1((𝑟+𝑀𝑛₃<𝑟. (46) Let 𝑁₃={𝑛₃,𝑛₃+1,…}. For 𝑛∈𝑁₃, consider (17).
Next we claim that any fixed point 𝑥_𝑛 of (18) for any 𝜆∈[0,1] must satisfy ‖𝑥_𝑛‖≠𝑟. So, by using the Leray-Schauder alternative principle, (17) (with 𝜆=1) has a periodic solution 𝑥_𝑛(𝑡). Otherwise, assume that 𝑥_𝑛 is a fixed point 𝑥_𝑛 of (18) for some 𝜆∈[0,1] such that ‖𝑥_𝑛‖=𝑟. Note that 𝑥_𝑛(𝑡)−1𝑛=𝜆^𝑇₀𝐺(𝑡,𝑠)𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)𝑑𝑠≥𝜆𝐴^𝑇₀𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)𝑑𝑠=𝜄𝐵𝜆^𝑇₀𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)𝑑𝑠≥𝜄max_{𝑡∈[0,𝑇]}𝜆^𝑇₀𝐺(𝑡,𝑠)𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)𝑥𝑑𝑠=𝜄_𝑛−1𝑛. (47) For 𝑛∈𝑁₃, we have 𝑥_𝑛(𝑡)≥𝜄𝑥_𝑛−1𝑛+1𝑛≥𝜄𝑥_𝑛−1𝑛+1𝑛≥𝜄𝑟. (48) By (27) and assumption (H₃), for all 𝑡∈[0,𝑇] and 𝑛∈𝑁₃, we have 𝑥_𝑛(𝑡)=𝜆^𝑇₀𝐺(𝑡,𝑠)𝐹_𝑛𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠1)𝑑𝑠+𝑛=𝜆^𝑇₀𝐺(𝑡,𝑠)𝐹𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)1𝑑𝑠+𝑛≤^𝑇₀𝐺(𝑡,𝑠)𝐹𝑠,𝑍_𝑛(𝑠),𝑍^′_𝑛(𝑠)1𝑑𝑠+𝑛≤^𝑇₀𝐺(𝑡,𝑠)𝑔𝑍_𝑛(𝑠)+ℎ𝑍_𝑛(𝑠)𝜚𝑍^′_𝑛(𝑠+1)𝑑𝑠𝑛≤^𝑇₀𝐺(𝑡,𝑠)𝑔𝑍_𝑛(𝑠)ℎ𝑍1+^𝑛(𝑠)𝑔𝑍_𝑛(𝑠)𝜚𝑍^′_𝑛(𝑠+1)𝑑𝑠𝑛≤^𝑇₀𝐺(𝑡,𝑠)𝑔𝜄𝑟−𝑀𝑤^∗ℎ(𝑟)1+𝑔(𝑟)𝜚)𝐿)+1((𝑟+𝑀𝑑𝑠𝑛≤𝑔𝜄𝑟−𝑀𝑤^∗ℎ(𝑟)1+𝑔(𝑟)𝜚((𝑟+𝑀)𝐿)𝑤^∗+1𝑛₃. (49) Therefore, 𝑟=‖𝑥‖≤𝑔𝜄𝑟−𝑀𝑤^∗ℎ(𝑟)1+𝑔(𝑟)𝜚((𝑟+𝑀)𝐿)𝑤^∗+1𝑛₃. (50) This is a contradiction to the choice of 𝑛₃ and the claim is proved.
The fact ‖𝑥_𝑛‖<𝑟 and ‖𝑥^′_𝑛(𝑡)‖<𝐿𝑟 show that {𝑥_𝑛}_𝑛∈𝑁3 is a bounded and equicontinuous family on [0,𝑇]. Now Arzela-Ascoli Theorem guarantees that {𝑥_𝑛}_𝑛∈𝑁3 has a subsequence {𝑥_𝑛𝑘}_𝑘∈ℕ, converging uniformly on [0,𝑇] to a function 𝑥∈𝐶[0,𝑇]. From the fact ‖𝑥_𝑛‖<𝑟 and 𝑥_𝑛(𝑡)>𝛿, 𝑥 satisfies 𝛿≤𝑥(𝑡)≤𝑟 for all 𝑡. Moreover, {𝑥_𝑛𝑘} satisfies the integral equation 𝑥_𝑛𝑘(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝐹𝑠,𝑍_𝑛𝑘(𝑠),𝑍^′_𝑛𝑘(𝑠)1𝑑𝑠+𝑛_𝑘. (51) Letting 𝑘→∞, we arrive at 𝑥(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝐹𝑠,𝑥(𝑠)−𝑀𝑤(𝑠),𝑥^′(𝑠)−𝑀𝑤^′(𝑠)𝑑𝑠, (52) where the uniform continuity of 𝐹(𝑡,𝑥,𝑦) on [0,𝑇]×[𝛿,𝑟]×[−(𝑟+𝑀)𝐿,(𝑟+𝑀)𝐿] is used. Therefore, 𝑥 is a positive periodic solution of (16) and 𝑍(𝑡)=𝑥(𝑡)−𝑀𝑤(𝑡)≥𝛿. Thus we complete the prove of Theorem 4.

Corollary 7. Let the nonlinearity in (1) be 𝑓𝑡,𝑥,𝑦=1+𝑦^𝛾𝑥^−𝛼+𝜇𝑥^𝛽+𝑒(𝑡), (53) where 𝛼>1,  𝛽>0,  1>𝛾≥0,  𝜇>0 is a positive parameter, 𝑒(𝑡) is a 𝑇-periodic function. (i)If 𝛽+𝛾<1, then (1) has at least one positive periodic solution for each 𝜇>0.(ii)If 𝛽+𝛾≥1, then (1) has at least one positive periodic solution for each 0<𝜇<𝜇₁, where 𝜇₁ is some positive constant.

Proof. We will apply Theorem 4 with 𝑀=max_{0≤𝑡≤𝑇}|𝑒(𝑡)| and 𝑔(𝑥)=𝑥^−𝛼,  ℎ(𝑥)=𝜇𝑥^𝛽+2𝑀,  𝜚(𝑦)=1+|𝑦|^𝛾. Then condition (H₁)–(H₃) are satisfied and existence condition (H₄) becomes 𝑟𝜇<𝜄𝑟−𝑀𝑤^∗𝛼−𝑤^∗1+(𝑟+𝑀)^𝛾𝐿^𝛾1+2𝑀𝑟^𝛼𝑤^∗1+(𝑟+𝑀)^𝛾𝐿^𝛾𝑟^𝛼+𝛽. (54) So (1) has at least one positive periodic solution for 0<𝜇<𝜇₁=sup_{𝑟>𝑀𝑤∗/𝜄}𝑟𝜄𝑟−𝑀𝑤^∗𝛼−𝑤^∗1+(𝑟+𝑀)^𝛾𝐿^𝛾×1+2𝑀𝑟^𝛼×𝑤^∗(1+(𝑟+𝑀)^𝛾𝐿^𝛾)𝑟^{𝛼+𝛽−1}. (55) Note that 𝜇₁=∞ if 𝛽+𝛾<1 and 𝜇₁<∞ if 𝛽+𝛾≥1. We have the desired results.

Acknowledgments

The research of X. Xing is supported by the Fund of the Key Disciplines in the General Colleges and Universities of Xin Jiang Uygur Autonomous Region (Grant no. 2012ZDKK13). It is a pleasure for the author to thank Professor J. Chu for his encouragement and helpful suggestions.