Existence for Singular Periodic Problems: A Survey of Recent Results

Abstract

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

1. Introduction

During the last two decades, singular differential equations have attracted many researchers [1–11] because such equations describe many problems in the applied sciences, such as the Brillouin focusing system [12–14], nonlinear elasticity [15], and gravitational forces [3]. Besides these important applications, it has been found that a particular case of singular equations, the Ermakov-Pinney equation, plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations [16–18].

In the literature, two different approaches have been used to establish the existence results for singular equations. The first one is the variational approach [3, 4, 6, 19, 20] and the second one is topological methods [1, 10, 21–28]. In our opinion, the first important result was proved in the pioneering paper of Lazer and Solimini [29]. They proved that a necessary and sufficient condition for the existence of a positive periodic solution for 𝑥^′′=1𝑥^𝜆+𝑒(𝑡) (1) is that the mean value of 𝑒 is negative; that is, 𝑒<0, here 𝜆≥1, which corresponds to a strong force condition, according to a terminology first introduced by Gordon [30]. Moreover, if 0<𝜆<1, which corresponds to a weak force condition, they found examples of functions 𝑒 with negative mean values and yet no periodic solutions exist. Therefore, there is an essential difference between a strong singularity and a weak singularity. Since the work of Lazer and Solimini, the strong force condition became standard in related work, see, for instance, [8, 15, 18, 27, 28]. 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, but it has also attracted many researchers [31–39]. In [39], for the first time in this topic, Torres et al. proved an existence result which is valid for a weak singularity, whereas the validity of such results under a strong force assumption remains as an open problem, which was partially solved in [32].

The main aim of this survey is to present some recent existence results for singular differential equations. In particular, we will consider the scalar singular equations, singular damped equations, singular impulsive equations, and singular differential systems. We will also include some examples to illustrate the results presented.

The rest of this paper is organized as follows. In Section 2, we will state some important results for the second-order scalar singular differential equations. Singular damped equations will be considered in Section 3. In Section 4, singular impulsive differential equations will be studied. Finally in Section 5, we will focus on the singular differential systems. Sections 2 and 3 are mainly written by the first author. Section 4 is mainly written by the second author, and Section 5 is mainly completed by the third author. All the results presented in Sections 3–5 shed some lights on the differences between a strong singularity and a weak singularity.

Finally in this section, we must note that besides the results presented in this survey, many interesting and important results on singular differential equations have been obtained by other researchers, see, for example, [9, 40–45] and the references cited therein.

In this paper, we denote the essential supremum and infimum of 𝑝 by 𝑝^∗ and 𝑝_∗, respectively, for a given function 𝑝∈𝐿¹[0,𝑇] essentially bounded.

2. Second-Order Scalar Singular Equations

In this section, we recall some results for second-order singular differential equations 𝑥^′′+𝑎(𝑡)𝑥=𝑓(𝑡,𝑥)+𝑒(𝑡), (2) here 𝑎(𝑡),𝑒(𝑡) are continuous, 𝑇-periodic functions. The nonlinearity 𝑓(𝑡,𝑥) is continuous in (𝑡,𝑥) and 𝑇-periodic in 𝑡 and has a singularity at 𝑥=0.

First we need to present some preliminary results on the linear equation 𝑥^′′+𝑎(𝑡)𝑥=𝑝(𝑡) (3) with periodic boundary conditions 𝑥(0)=𝑥(𝑇),𝑥^′(0)=𝑥^′(𝑇). (4) We assume the following:(A) the Green function 𝐺(𝑡,𝑠), associated with (3)-(4), is positive for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇], or(B) the Green function 𝐺(𝑡,𝑠), associated with (3)-(4), is nonnegative for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇].

When 𝑎(𝑡)=𝑘², condition (A) is equivalent to 0<𝑘²<𝜆₁=(𝜋/𝑇)² and condition (B) is equivalent to 0<𝑘²≤𝜆₁. In this case, we have 𝐺()=()()𝑡,𝑠sin𝑘𝑡−𝑠+sin𝑘𝑇−𝑡+𝑠()()()2𝑘1−cos𝑘𝑇,0≤𝑠≤𝑡≤𝑇,sin𝑘𝑠−𝑡+sin𝑘𝑇−𝑠+𝑡2𝑘(1−cos𝑘𝑇),0≤𝑡≤𝑠≤𝑇. (5)

For a nonconstant function 𝑎(𝑡), there is an 𝐿^𝑝-criterion proved in [46], which is given in Lemma 1 for the sake of completeness. Let 𝐊(𝑞) denote the best Sobolev constant in the following inequality: 𝐶‖𝑢‖²_𝑞≤‖𝑢^′‖²₂,∀𝑢∈𝐻¹₀(0,𝑇). (6) The explicit formula for 𝐊(𝑞) is 𝐊𝑞=2𝜋𝑞𝑇^1+2/𝑞22+𝑞^1−2/𝑞Γ(1/𝑞)Γ(1/2+1/𝑞)²4if1≤𝑞<∞,𝑇if𝑞=∞, (7) where Γ is the gamma function, see [47, 48].

Lemma 1 (see [46, Corollary 2.3]). Assume that 𝑎(𝑡)≻0 and 𝑎∈𝐿^𝑝[0,𝑇] for some 1≤𝑝≤∞. If ‖𝑎‖_𝑝<𝐊2𝑝^∼, (8) then the condition (A) holds. Moreover, condition (B) holds if ‖𝑎‖_𝑝≤𝐊2𝑝^∼. (9)

When the hypothesis (A) is satisfied, we denote 𝑚=min_{0≤𝑠,𝑡≤𝑇}𝐺(𝑡,𝑠),𝑀=max_{0≤𝑠,𝑡≤𝑇}𝐺()𝑚𝑡,𝑠,𝜎=𝑀. (10) Obviously, 𝑀>𝑚>0 and 0<𝜎<1.

The first existence result deals with the case of a strong singularity and the proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in [49] or [50, pages 120–130].

Lemma 2. 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.

Theorem 3 (see [37, Theorem 4.1]). Suppose that 𝑎(𝑡) satisfies (A) and 𝑓(𝑡,𝑥) satisfies the following.(H₁) There exists a nonincreasing positive continuous function 𝑔₀(𝑥) on (0,∞) and a constant 𝑅₀>0 such that 𝑓(𝑡,𝑥)≥𝑔₀(𝑥) for (𝑡,𝑥)∈[0,𝑇]×(0,𝑅₀], where 𝑔₀(𝑥) satisfies lim_𝑥→0+𝑔₀(𝑥)=+∞,lim_𝑥→0+^𝑅0_𝑥𝑔₀(𝑢)𝑑𝑢=+∞. (11)(H₂) There exist continuous, nonnegative functions 𝑔(𝑥) and ℎ(𝑥) such that 0≤𝑓(𝑡,𝑥)≤𝑔(𝑥)+ℎ(𝑥)∀(𝑡,𝑥)∈[0,𝑇]×(0,∞), (12) 𝑔(𝑥)>0 is nonincreasing and ℎ(𝑥)/𝑔(𝑥) is nondecreasing in 𝑥∈(0,∞). (H₃) There exists a positive number 𝑟 such that 𝜎𝑟+𝛾_∗>0 and 𝑟𝑔𝜎𝑟+𝛾_∗1+ℎ𝑟+𝛾^∗/𝑔𝑟+𝛾^∗>𝜔^∗, (13) here 𝛾(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑒(𝑠)𝑑𝑠,𝜔(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑑𝑠. (14)Then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), (2) has at least one positive periodic solution 𝑥 with 𝑥(𝑡)>𝛾(𝑡) for all 𝑡 and 0<‖𝑥−𝛾‖<𝑟.

Note that the study in [37, Theorem 4.1] is slightly different from the above presentation. However, the proof of the above theorem follows from that of [37, Theorem 4.1] with some minor necessary changes. Condition (H₁) corresponds to the classical strong force condition, which was first introduced by Gordon in [30]. In fact, condition (H₁) is only used when we try to obtain a prior lower bound. In Theorem 4, we will show that, for the case 𝛾_∗≥0, we can remove the strong force condition (H₁) and replace it by one weak force condition.

Theorem 4 (see [33, Theorem 3.1]). Assume that (A) and (H₂)-(H₃) are satisfied. Suppose further the following condition.(H₄) For each constant 𝐿>0, there exists a continuous function 𝜙_𝐿≻0 such that 𝑓(𝑡,𝑥)≥𝜙_𝐿(𝑡) for all (𝑡,𝑥)∈[0,𝑇]×(0,𝐿].
Then for each 𝑒(𝑡) with 𝛾_∗≥0, (2) has at least one positive periodic solution 𝑥 with 𝑥(𝑡)>𝛾(𝑡) for all 𝑡 and 0<‖𝑥−𝛾‖<𝑟.

For the superlinear case, we can establish the multiplicity result. The proof is based on a well-known fixed point theorem in cones, which can be found in [51]. Let 𝐾 be a cone in 𝑋 and 𝐷 is a subset of 𝑋, we write 𝐷_𝐾=𝐷∩𝐾 and 𝜕_𝐾𝐷=(𝜕𝐷)∩𝐾.

Theorem 5 (see [51]). Let 𝑋 be a Banach space and 𝐾 a cone in 𝑋. Assume Ω¹,Ω² are open bounded subsets of X with Ω¹_𝐾≠∅,Ω¹_𝐾⊂Ω²_𝐾. Let 𝑇∶Ω²_𝐾→𝐾 (15) be a completely continuous operator such that(a)‖𝑇𝑥‖≤‖𝑥‖ for 𝑥∈𝜕_𝐾Ω¹,(b)there exists 𝜐∈𝐾⧵{0} such that 𝑥≠𝑇𝑥+𝜆𝜐𝑓𝑜𝑟𝑎𝑙𝑙𝑥∈𝜕_𝐾Ω² and all 𝜆>0.
Then 𝑇 has a fixed point in Ω²_𝐾⧵Ω¹_𝐾.

Theorem 6 (see [33, Theorem 3.2]). Suppose that 𝑎(𝑡) satisfies (A) and 𝑓(𝑡,𝑥) satisfies (H₂)-(H₃). Furthermore, assume the following conditions.(H₅) There exist continuous, nonnegative functions 𝑔₁(𝑥),ℎ₁(𝑥) such that 𝑓(𝑡,𝑥)≥𝑔₁(𝑥)+ℎ₁(𝑥),∀(𝑡,𝑥)∈[0,𝑇]×(0,∞), (16) 𝑔₁(𝑥)>0 is nonincreasing and ℎ₁(𝑥)/𝑔₁(𝑥) is non-decreasing in 𝑥.(H₆) There exists 𝑅>0 with 𝜎𝑅>𝑟 such that 𝜎𝑅𝑔₁𝑅+𝛾^∗1+ℎ₁𝜎𝑅+𝛾_∗/𝑔₁𝜎𝑅+𝛾_∗≤𝜔_∗. (17)Then (2) has one positive periodic solution 𝑥^∼ with 𝑟<‖𝑥^∼−𝛾‖≤𝑅.

Combined Theorems 3 and 4 with Theorem 6, we can get the following two multiplicity results.

Theorem 7. Suppose that 𝑎(𝑡) satisfies (A) and 𝑓(𝑡,𝑥) satisfies (H₁)–(H₃) and (H₅)-(H₆). Then (2) has two different positive periodic solutions 𝑥 and 𝑥^∼ with 0<‖𝑥−𝛾‖<𝑟<‖𝑥^∼−𝛾‖≤𝑅.

Theorem 8. Suppose that 𝑎(𝑡) satisfies (A) and 𝑓(𝑡,𝑥) satisfies (H₂)–(H₆). Then (2) has two different positive periodic solutions 𝑥 and 𝑥^∼ with 0<‖𝑥−𝛾‖<𝑟<‖𝑥^∼−𝛾‖≤𝑅.

To illustrate our results, we have selected the following singular equation: 𝑥^′′+𝑎(𝑡)𝑥=𝑥^−𝛼+𝜇𝑥^𝛽+𝑒(𝑡), (18) here 𝑎,𝑒∈ℂ[0,𝑇], 𝛼,𝛽>0, and 𝜇∈ℝ is a given parameter. The corresponding results are also valid for the general case 𝑥^′′(𝑡)𝑏(𝑡)+𝑎𝑥=𝑥^𝛼+𝜇𝑐(𝑡)𝑥^𝛽+𝑒(𝑡), (19) with 𝑏,𝑐∈ℂ[0,𝑇].

Corollary 9. Assume that 𝑎(𝑡) satisfies (A) and 𝛼>0,𝛽≥0,𝜇>0. Then one has the following results.(i)If 𝛼≥1,𝛽<1, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), (18) has at least one positive periodic solution for all 𝜇>0.(ii)If 𝛼≥1,𝛽≥1, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), (18) has at least one positive periodic solution for each 0<𝜇<𝜇₁; here 𝜇₁ is some positive constant.(iii)If 𝛼≥1,𝛽>1, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), (18) has at least two positive periodic solutions for each 0<𝜇<𝜇₁.(iv)If 𝛼>0,𝛽<1, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), with 𝛾_∗≥0, (18) has at least one positive periodic solution for all 𝜇>0.(v)If 𝛼>0,𝛽≥1, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), with 𝛾_∗≥0, (18) has at least one positive periodic solution for each 0<𝜇<𝜇₁.(vi)If 𝛼>0,𝛽>1, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), with 𝛾_∗≥0, (18) has at least two positive periodic solutions for each 0<𝜇<𝜇₁.

All the above results require that the linear equation satisfies (A), which cannot cover the critical case. The next few results deal with the case when the condition (B) is satisfied and the proof is based on Schauder's fixed point theorem.

Theorem 10 (see [31, Theorem 3.1]). Assume that conditions (B) and (H₂) and (H₄) are satisfied. Furthermore, suppose that(H₇) there exists a positive constant 𝑅>0 such that 𝑅>Φ_∗,Φ_∗+𝛾_∗>0 and 𝑅≥𝑔Φ_∗+𝛾_∗ℎ1+𝑅+𝛾^∗𝑔𝑅+𝛾^∗𝜔^∗, (20) here Φ_∗=min_𝑡Φ(𝑡),Φ(𝑡)=∫^𝑇₀𝐺(𝑡,𝑠)𝜙_𝑅+𝛾∗(𝑠)𝑑𝑠.
Then (2) has at least one positive 𝑇-periodic solution.

As an application of Theorem 10, we consider the case 𝛾_∗=0. Corollary 11 is a direct result of Theorem 10.

Corollary 11 (see [31, Corollary 3.2]). Assume that conditions (B) and (H₂) and (H₄) are satisfied. Furthermore, assume that (H₈) there exists a positive constant 𝑅>0 such that 𝑅>Φ_∗ and 𝑅≥𝑔Φ_∗ℎ1+𝑅+𝛾^∗𝑔𝑅+𝛾^∗𝜔^∗. (21) If 𝛾_∗=0, then (2) has at least one positive 𝑇-periodic solution.

Corollary 12 (see [31, Example 3.5]). Suppose that 𝑎 satisfies (B) and 0<𝛼<1, 𝛽≥0, then for each 𝑒(𝑡)∈ℂ(ℝ/𝑇ℤ,ℝ), with 𝛾_∗=0, one has the following:(i)if 𝛼+𝛽<1−𝛼², then (18) has at least one positive periodic solution for each 𝜇≥0,(ii)if 𝛼+𝛽≥1−𝛼², then (18) has at least one positive 𝑇-periodic solution for each 0≤𝜇<𝜇₂, where 𝜇₂ is some positive constant.

The next results explore the case when 𝛾_∗>0.

Theorem 13 (see [31, Theorem 3.6]). Suppose that 𝑎(𝑡) satisfies (B) and 𝑓(𝑡,𝑥) satisfies condition (H₂). Furthermore, assume that(H₉) there exists 𝑅>𝛾^∗ such that 𝑔𝛾_∗ℎ1+𝑅+𝛾^∗𝑔𝑅+𝛾^∗𝜔^∗≤𝑅. (22) If 𝛾_∗>0, then (2) has at least one positive 𝑇-periodic solution.

Corollary 14 (see [31, Example 3.8]). Suppose that 𝑎(𝑡) satisfies (B) and 𝛼,𝛽≥0, then for each 𝑒∈ℂ(ℝ/𝑇ℤ,ℝ), with 𝛾_∗>0, one has the following:(i)if𝛼+𝛽<1, then (18) has at least one positive 𝑇-periodic solution for each 𝜇≥0,(ii)if𝛼+𝛽≥1, then (18) has at least one positive 𝑇-periodic solution for each 0≤𝜇<𝜇₃, where 𝜇₃ is some positive constant.

3. Singular Damped Equations

In this section, we recall some results on second-order singular damped differential equations 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=𝑓𝑡,𝑥,𝑥^′, (23) where ℎ,𝑎∈ℂ(ℝ/𝑇ℤ,ℝ) and the nonlinearity 𝑓∈ℂ((ℝ/𝑇ℤ)×(0,∞)×ℝ,ℝ). In particular, the nonlinearity may have a repulsive singularity at 𝑥=0, which means that lim_𝑥→0+𝑓𝑡,𝑥,𝑦=+∞,uniformlyin𝑡,𝑦∈ℝ². (24) First we recall some results on the linear damped equation 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=0, (25) associated to periodic boundary conditions (4). As in the last section, we say that (25)-(4) is nonresonant when its unique 𝑇-periodic solution is the trivial one. When (25)-(4) is nonresonant, as a consequence of Fredholm's alternative, the nonhomogeneous equation 𝑥^′′+ℎ(𝑡)𝑥^′+𝑎(𝑡)𝑥=𝑙(𝑡) (26) admits a unique 𝑇-periodic solution which can be written as 𝑥(𝑡)=^𝑇₀𝐺₂(𝑡,𝑠)𝑙(𝑠)𝑑𝑠, (27) where 𝐺₂(𝑡,𝑠) is the Green's function of problem (25)-(4). We also assume that the following standing hypothesis is satisfied.(C) The Green's function 𝐺₂(𝑡,𝑠), associated with (25)-(4), is positive for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇].

To guarantee that (C) is satisfied, we require the antimaximum principle for (25)-(4) proved by Hakl and Torres in [52]. To do this, let us define the functions 𝜎(ℎ)(𝑡)=exp^𝑡₀ℎ(𝑠),𝜎𝑑𝑠₁(ℎ)(𝑡)=𝜎(ℎ)(𝑇)^𝑡₀𝜎(ℎ)(𝑠)𝑑𝑠+^𝑇_𝑡𝜎(ℎ)(𝑠)𝑑𝑠. (28)

Lemma 15 (see [52, Theorem 2.2]). 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, (29) where [𝑎(𝑠)]₊=max{𝑎(𝑠),0}. Then (C) holds.

For the special case ∫^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑑𝑡>0 and ℎ∈ℂ^∼(ℝ/𝑇ℤ)∶={ℎ∈ℂ(ℝ/𝑇ℤ)∶ℎ=0}, one criterion has been developed by Cabada and Cid in [40].

Theorem 16 (see [40, Theorem 5.1]). Assume that ℎ∈ℂ^∼(ℝ/𝑇ℤ) and ∫^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑑𝑡>0. Suppose further that there exists 1≤𝑝≤∞ such that (𝐵(𝑇))^1+1/𝑞𝒜_+𝑝,𝑇<𝐌²2𝑞, (30) where 𝐵(𝑇)=^𝑇₀𝜎(𝑡)𝒜−ℎ)(𝑑𝑡,₊(𝑡)=𝑎₊(𝑡)(𝜎(ℎ)(𝑡))^2−1/𝑝. (31) Then (𝐶) holds.

Theorem 17 (see [35, Theorem 3.2]). Suppose that (25) satisfies (C) and ^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑑𝑡>0. (32) Furthermore, assume that there exists a constant 𝑟>0 such that(G₁) there exists a continuous function 𝜙_𝑟≻0 such that 𝑓(𝑡,𝑥,𝑦)≥𝜙_𝑟(𝑡) for all (𝑡,𝑥,𝑦)∈[0,𝑇]×(0,𝑟]×(−∞,∞),(G₂) there exist continuous, nonnegative functions 𝑔(⋅), ℎ(⋅), and 𝜚(⋅) such that ≤𝑔(𝑥)(𝑥)𝜚𝑦,∀∈[]×(]0≤𝑓𝑡,𝑥,𝑦+ℎ𝑡,𝑥,𝑦0,𝑇0,𝑟×ℝ, (33) where 𝑔(⋅)>0 is nonincreasing, ℎ(⋅)/𝑔(⋅) is non-decreasing in (0,𝑟], and 𝜚(⋅) is non-decreasing in (0,∞),(G₃) the following inequality holds: 𝑟𝑔(𝜄𝑟)1+ℎ(𝑟)/𝑔(𝑟)𝜚(𝐿𝑟)>𝜔^∗, (34) where 𝜔(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑑𝑠,𝐿=2∫^𝑇₀𝑎(𝑡)𝜎(ℎ)(𝑡)𝑑𝑡min_{0≤𝑡≤𝑇}𝜎(ℎ𝑡),𝑚)(𝜄=𝑀,𝑚=min_{0≤𝑠,𝑡≤𝑇}𝐺(𝑡,𝑠),𝑀=max_{0≤𝑠,𝑡≤𝑇}𝐺(𝑡,𝑠), (35)
then (23) has at least one positive 𝑇-periodic solution 𝑥 with 0<‖𝑥‖≤𝑟.

Corollary 18 (see [35, Corollary 3.3]). Let the nonlinearity in (23) be 𝑓𝑡,𝑥,𝑦=1+𝑦^𝛾𝑥^−𝛼+𝜇𝑥^𝛽, (36) where 𝛼>0, 𝛽,𝛾≥0, 𝜇>0 is a positive parameter.(i)If 𝛽+𝛾<1, then (23) has at least one positive periodic solution for each 𝜇>0.(ii)If 𝛽+𝛾≥1, then (23) has at least one positive periodic solution for each 0<𝜇<𝜇^∗₁, where 𝜇^∗₁ is some positive constant.

Corollary 19 (see [35, Corollary 3.4]). Let the nonlinearity in (23) be 𝑓𝑡,𝑥,𝑦=1+𝑦^𝛾1𝑥^𝛼−𝜇𝑥^𝛽, (37) where 𝛼>𝛽>0,𝛾≥0 with 𝛾<𝛼+1, 𝜇>0 is a positive parameter. Then there exists a positive constant 𝜇^∗₂ such that (23) has at least one positive 𝑇-periodic solution for each 0≤𝜇<𝜇^∗₂.

Corollary 19 is interesting because the singularity on the right-hand side combines attractive and repulsive effects. The analysis of such differential equations with mixed singularities is at this moment very incomplete, and few references can be cited [22, 44]. Therefore, the results in Corollary 19 can be regarded as one contribution to the literature trying to fill partially this gap in the study of singularities of mixed type.

As in the last section, if we assume that the linear equation (25)-(4) has a nonnegative Green's function, we can also get some results based on Schauder's fixed point theorem, and the results can cover the critical case.

4. Singular Impulsive Differential Equations

In this section, we will study the existence of periodic solutions for some singular differential equations with impulsive effects by using variational methods.

Firstly, we consider the following second-order nonautonomous singular problem: 𝑢^′′−𝑏(𝑡)𝑢^𝛼(𝑡)(),𝑢(0)(𝑇)=𝑒,a.e.𝑡∈0,𝑇−𝑢=𝑢^′(0)−𝑢^′(𝑇)=0, (38) under the impulse conditions Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗,𝑗=1,2,…,𝑝−1, (39) where 𝑡_𝑗,𝑗=1,2,…,𝑝−1 are the instants where the impulses occur and 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑝−1<𝑡_𝑝=𝑇, 𝐼_𝑗∶ℝ→ℝ(𝑗=1,2,…,𝑝−1) are continuous.

Our result is presented as follows.

Theorem 20 (see [19, Theorem 1.1]). Assume that 𝛼>1 and the following conditions hold.(𝑆₁)𝑏∈ℂ¹([0,𝑇],(0,∞)) is 𝑇-periodic and 𝑏^′(𝑡)≥0 for all 𝑡∈[0,𝑇].(𝑆₂)𝑒∈𝐿²([0,𝑇],ℝ) is 𝑇-periodic and ∫^𝑇₀𝑒(𝑡)𝑑𝑡<0.(𝑆₃) There exist two constants 𝑚,𝑀 such that for any 𝑡∈ℝ, 𝑚≤𝐼_𝑗(𝑡)≤𝑀,𝑗=1,2,…,𝑝−1, (40)where 𝑚<0 and 0≤𝑀<(−1/(𝑝−1))∫^𝑇₀𝑒(𝑡)𝑑𝑡.(𝑆₄) For any 𝑡∈ℝ, ^𝑡₀𝐼_𝑗(𝑠)𝑑𝑠≥0,𝑗=1,2,…,𝑝−1. (41)Then problem (38)-(39) has at least one solution.

Remark 21. In fact, it is not difficult to find some functions 𝐼_𝑗 satisfying (𝑆₃) and (𝑆₄). For example, 𝐼_𝑗(𝑡)=sin𝑡,𝑡∈ℝ. (42)

Let 𝐻¹_𝑇=[]𝑢(0)(𝑇)𝑢∶0,𝑇→ℝ∣𝑢isabsolutelycontinuous,=𝑢and𝑢^′∈𝐿²([0,𝑇],ℝ), (43) with the inner product (𝑢,𝑣)=^𝑇₀𝑢(𝑡)𝑣(𝑡)𝑑𝑡+^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡,∀𝑢,𝑣∈𝐻¹_𝑇. (44) The corresponding norm is defined by ‖𝑢‖_𝐻¹_T=^𝑇₀|𝑢(𝑡)|²𝑑𝑡+^𝑇₀|𝑢^′(𝑡)|²𝑑𝑡^1/2,∀𝑢∈𝐻¹_𝑇. (45) Then 𝐻¹_𝑇 is a Banach space (in fact it is a Hilbert space).

If 𝑢∈𝐻¹_𝑇, then 𝑢 is absolutely continuous and 𝑢^′∈𝐿²([0,𝑇],ℝ). In this case, Δ𝑢^′(𝑡)=𝑢^′(𝑡⁺)−𝑢^′(𝑡⁻)=0 is not necessarily valid for every 𝑡∈(0,𝑇) and the derivative 𝑢^′ may exist some discontinuities. It may lead to impulse effects.

Following the ideas of [53], take 𝑣∈𝐻¹_𝑇 and multiply the two sides of the equality −𝑢^′′+𝑏(𝑡)𝑢^𝛼+𝑒(𝑡)=0 (46) by 𝑣 and integrate from 0 to 𝑇, so we have ^𝑇₀−𝑢^′′+𝑏(𝑡)𝑢^𝛼+𝑒(𝑡)𝑣𝑑𝑡=0. (47) Note that since 𝑢^′(0)−𝑢^′(𝑇)=0, one has ^𝑇₀𝑢^′′(𝑡)𝑣(𝑡)=𝑑𝑡_{𝑝−1𝑗=0𝑡𝑗+1𝑡𝑗}𝑢^′′(𝑡)𝑣(𝑡)=𝑑𝑡_{𝑝−1𝑗=0}𝑢^′𝑡⁻_j+1𝑣𝑡⁻_𝑗+1−𝑢^′𝑡⁺_𝑗𝑣𝑡⁺_𝑗−^𝑝−1_𝑗=0^𝑡𝑗+1_𝑡𝑗𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡=𝑢^′(𝑇)𝑣(𝑇)−𝑢^′(0)𝑣(0)−^𝑝−1_𝑗=1Δ𝑢^′𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡=−^𝑝−1_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡. (48) Combining with (47), we get ^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑝−1_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗+^𝑇₀𝑏(𝑡)𝑢^𝛼𝑣(𝑡)𝑑𝑡+^𝑇₀𝑒(𝑡)𝑣(𝑡)𝑑𝑡=0. (49) As a result, we introduce the following concept of a weak solution for problem (38)-(39).

Definition 22. One says that a function 𝑢∈𝐻¹_𝑇 is a weak solution of problem (38)-(39) if ^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑝−1_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗+^𝑇₀𝑏(𝑡)𝑢^𝛼𝑣(𝑡)𝑑𝑡+^𝑇₀𝑒(𝑡)𝑣(𝑡)𝑑𝑡=0 (50) holds for any 𝑣∈𝐻¹_𝑇.

Define the functional Φ∶𝐻¹_𝑇→ℝ by Φ(𝑢)1∶=2^𝑇₀𝑢^′(𝑡)²𝑑𝑡+^𝑝−1_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)+𝑑𝑠_𝑇0𝑏(𝑡)₁^𝑢(𝑡)1𝑠^𝛼𝑑𝑠𝑑𝑡+^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡, (51) for every 𝑢∈𝐻¹_𝑇. Clearly, Φ_𝜆 is well defined on 𝐻¹_𝑇, continuously Gáteaux differentiable functional whose Gáteaux derivative is the functional Φ^′_𝜆(𝑢), given by Φ^′_𝜆(𝑢)𝑣=^𝑇₀𝑢^′(𝑡)𝑣^′(𝑡)𝑑𝑡+^𝑝−1_𝑗=1𝐼_𝑗𝑢𝑡_𝑗𝑣𝑡_𝑗−^𝑇₀𝑏(𝑡)𝑢^𝛼𝑣(𝑡)𝑑𝑡+^𝑇₀𝑒(𝑡)𝑣(𝑡)𝑑𝑡, (52) for any 𝑣∈𝐻¹_𝑇. Moreover, it is easy to verify that Φ_𝜆 is weakly lower semicontinuous. Indeed, if {𝑢_𝑛}⊂𝐻¹_𝑇,𝑢∈𝐻¹_𝑇, and 𝑢_𝑛⇀𝑢, then {𝑢_𝑛} converges uniformly to 𝑢 on [0,𝑇] and 𝑢_𝑛→𝑢 on 𝐿²([0,𝑇]), and combining the fact that liminf_𝑛→∞‖𝑢_𝑛‖_𝐻¹_𝑇≥‖𝑢‖_𝐻¹_𝑇, one has liminf_𝑛→∞Φ_𝜆𝑢_𝑛=liminf_𝑛→∞12𝑢_𝑛²_𝐻1𝑇−12^𝑇₀𝑢_𝑛(𝑡)²+𝑑𝑡_{𝑝−1𝑗=1𝑢𝑛(𝑡𝑗)0}𝐼_𝑗(𝑠)−𝑑𝑠_𝑇0𝑏(𝑡)^𝑢𝑛₁^(𝑡)1𝑠^𝛼+𝑑𝑠𝑑𝑡_𝑇0𝑒(𝑡)𝑢_𝑛(𝑡)≥1𝑑𝑡2^𝑇₀𝑢^′(𝑡)²𝑑𝑡+^𝑝−1_𝑗=1^{𝑢(𝑡𝑗)}₀𝐼_𝑗(𝑠)−𝑑𝑠_𝑇0𝑏(𝑡)₁^𝑢(𝑡)1𝑠^𝛼𝑑𝑠𝑑𝑡+^𝑇₀𝑒(𝑡)𝑢(𝑡)𝑑𝑡=Φ_𝜆(𝑢). (53) By the standard discussion, the critical points of Φ_𝜆 are the weak solutions of problem (38)-(39), see [53, 54].

The following version of the mountain pass theorem will be used in our argument.

Theorem 23 (see [55, Theorem 4.10]). Let 𝑋 be a Banach space and let 𝜑∈𝐶¹(𝑋,ℝ). Assume that there exist 𝑥₀,𝑥₁∈𝑋 and an open neighborhood Ω of 𝑥₀ such that 𝑥₁∈𝑋⧵Ω and max𝜑𝑥₀,𝜑𝑥₁<inf_{𝑥∈𝜕Ω}𝜑(𝑥). (54) Let Γ=ℎ∈𝐶([0,1],𝑋)∶ℎ(0)=𝑥₀,ℎ(1)=𝑥₁,𝑐=inf_ℎ∈Γmax_𝑠∈[0,1]𝜑(ℎ(𝑠)). (55) If 𝜑 satisfies the (PS)-condition, that is, a sequence {𝑢_𝑛} in 𝑋 satisfying 𝜑(𝑢_𝑛) is bounded and 𝜑^′(𝑢_𝑛)→0 as 𝑛→∞ has a convergent subsequence, then 𝑐 is a critical value of 𝜑 and 𝑐>max{𝜑(𝑥₀),𝜑(𝑥₁)}.

Next we consider 𝑇-periodic solution for another impulsive singular problem: 𝑢^′′(𝑡)−1𝑢^𝛼(𝑡)=𝑒(𝑡), (56) under impulsive conditions Δ𝑢^′𝑡_𝑗=𝐼_𝑗𝑢𝑡_𝑗,𝑗=1,2,…,𝑝−1, (57) where 𝛼≥1, 𝑒∈𝐿¹([0,𝑇],ℝ) is 𝑇-periodic, Δ𝑢^′(𝑡_𝑗)=𝑢^′(𝑡⁺_𝑗)−𝑢^′(𝑡⁻_𝑗) with 𝑢^′(𝑡^±_𝑗)=lim_𝑡→𝑡^±_𝑗𝑢^′(𝑡); 𝑡_𝑗,𝑗=1,2,…,𝑝−1 are the instants where the impulses occur, and 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑝−1<𝑡_𝑝=𝑇,𝑡_𝑗+𝑝=𝑡_𝑗+𝑇;𝐼_𝑗∶ℝ→ℝ(𝑗=1,2,…,𝑝−1) are continuous and 𝐼_𝑗+𝑝≡𝐼_𝑗.

In 1987, Lazer and Solimini [29] proved a famous result as follows.

Theorem 24 (see [29]). Assume that 𝑒∈𝐿¹([0,𝑇],ℝ) is 𝑇-periodic. Then problem (56) has a positive 𝑇-periodic weak solution if and only if ∫^𝑇₀𝑒(𝑡)𝑑𝑡<0.

From Theorem 24, if ∫^𝑇₀𝑒(𝑡)𝑑𝑡≥0, then problem (52) does not have a positive 𝑇-periodic weak solution. However, if the impulses happen, for this singular problem may exist a positive 𝑇-periodic weak solution. Inspired by the above facts, our aim is to reveal a new existence result on positive 𝑇-periodic solution for singular problem (56) when impulsive effects are considered, that is, problem (56)-(57). Indeed, this periodic solution is generated by impulses. Here, we say a solution is generated by impulses if this solution is nontrivial when 𝐼_𝑗≢0 for some 1<𝑗<𝑝−1, but it is trivial when 𝐼_𝑗≡0 for all 1<𝑗<𝑝−1. For example, if problem (56)-(57) does not possess positive periodic solution when 𝐼_𝑗≡0 for all 1<𝑗<𝑝−1, then a positive periodic solution 𝑢 of problem (56)-(57) with 𝐼_𝑗≢0 for some 1<𝑗<𝑝−1 is called a positive periodic solution generated by impulses.

Our result is presented as follows.

Theorem 25 (see [35, Theorem 1.2]). Assume the following:(𝑆₁)𝑒∈𝐿¹([0,𝑇],ℝ) is 𝑇-periodic and ∫^𝑇₀𝑒(𝑡)𝑑𝑡≥0;(𝑆₂) there exist two constants 𝑚,𝑀 such that for any 𝑠∈ℝ, 𝑚≤𝐼_𝑗(𝑠)≤𝑀,𝑗=1,2,…,𝑝−1, (58)where 𝑚≤𝑀<(−1/(𝑝−1))∫^𝑇₀𝑒(𝑡)𝑑𝑡≤0.
Then problem (56)-(57) has at least a positive 𝑇-periodic solution.

5. Singular Differential Systems

In this section, we will consider the system of Hill's equations 𝑢_𝑖^′′(𝑡)+𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)=𝐹_𝑖𝑡,𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡),1≤𝑖≤𝑛. (59) Here, 𝑎_𝑖 and 𝐹_𝑖 are 𝑇-periodic in the variable 𝑡, 𝑎_𝑖∈𝐿¹[0,𝑇], and the nonlinearities 𝐹_𝑖(𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛) can be singular at 𝑥_𝑗=0 where 𝑗∈{1,2,…,𝑛}.

Throughout, let 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛). We are interested in establishing the existence of continuous 𝑇-periodic solutions 𝑢 of the system (59), that is, 𝑢∈(𝐶(ℝ))^𝑛 and 𝑢(𝑡)=𝑢(𝑡+𝑇) for all 𝑡∈ℝ. Moreover, we are concerned with constant-sign solutions 𝑢, by which we mean 𝜃_𝑖𝑢_𝑖(𝑡)≥0 for all 𝑡∈ℝ and 1≤𝑖≤𝑛, where 𝜃_𝑖∈{1,−1} is fixed. Note that positive solution, the usual consideration in the literature, is a special case of constant-sign solution when 𝜃_𝑖=1 for 1≤𝑖≤𝑛.

We will employ the Schauder's fixed point theorem to establish the existence of solutions. Indeed, in Section 5.1 we will first tackle a particular case of (59) when 𝐹_𝑖(𝑡,𝑢(𝑡))=𝜕₂ℎ_𝑖1𝑡,2|𝑢(𝑡)|²𝑢_𝑖(𝑡)+𝑓_𝑖(𝑡). (60) Here, 𝜕₂ℎ_𝑖 is the partial derivative of ℎ_𝑖 with respect to the second variable, and |⋅| is a norm in ℝ^𝑛. The particular case (60) occurs in the problem [36] 𝑢̈(𝑡)+∇_𝑢𝑃(𝑡,𝑢(𝑡))=𝑓(𝑡), (61) where the potential 𝑃()=1𝑡,𝑢2𝑎(𝑡)|𝑢|²1−ℎ𝑡,2|𝑢|², (62) and ℎ presents a singularity of the repulsive type, that is, lim_|𝑥|→0ℎ(𝑡,𝑥)=∞ uniformly in 𝑡. The general problem (59) will be investigated in Section 5.2; here the singularities are not necessarily generated by a potential as in the case of (60). To illustrate our results, several examples will be presented.

In [45], the authors use a nonlinear alternative of the Leray-Schauder type and a fixed point theorem in cones to establish the existence of two positive periodic solutions for the system 𝑢̈(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝐺(𝑢(𝑡)), (63) where 𝐺 can be expressed as a sum of two positive functions satisfying certain monotone conditions. Therefore, the results in [45] are not applicable to (59) with 𝐹_𝑖 as in (60). In [45] it is also shown that the system 𝑢₁^′′(𝑡)+𝑎₁(𝑡)𝑢₁(𝑡)=𝑢²₁+𝑢²₂^−𝛽+𝜈𝑢²₁+𝑢²₂^𝛾,𝑢₂^′′(𝑡)+𝑎₂(𝑡)𝑢₂(𝑡)=𝑢²₁+𝑢²₂^−𝛽+𝜈𝑢²₁+𝑢²₂^𝛾 (64) has a solution when 𝛽>0,𝛾∈[0,1), and 𝜈>0. We will generalize the system (64) in Examples 46–48 to allow 𝜈 to be zero or negative. The improvement is possible probably due to the fact that we do not need to make a technical truncation to get compactness when we employ the Schauder fixed point theorem as compared to when the Leray-Schauder alternative is used. In fact, the set that we work on excludes the singularities. The results presented in this section not only generalize the papers [36, 39, 45] to systems and existence of constant-sign solutions, but also improve and/or complement the results in these earlier work as well as other research papers [56–60]. This section is based on the work in [61].

5.1. Existence Results for (60)

In this section we will consider the system of Hill's equations 𝑢_𝑖^′′(𝑡)+𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)=𝜕₂ℎ_𝑖1𝑡,2|𝑢(𝑡)|²𝑢_𝑖(𝑡)+𝑓_𝑖(𝑡),1≤𝑖≤𝑛. (65) Here, 𝜕₂ℎ_𝑖(𝑡,𝑠)≡(𝜕/𝜕𝑠)ℎ_𝑖(𝑡,𝑠) and |⋅| is a norm in ℝ^𝑛. Moreover, 𝑎_𝑖(𝑡), 𝜕₂ℎ_𝑖(𝑡,𝑠), and 𝑓_𝑖(𝑡) are 𝑇-periodic in 𝑡, 𝑎_𝑖∈𝐿¹[0,𝑇],𝑓_𝑖∈𝐿¹[0,𝑇], and 𝜕₂ℎ_𝑖(𝑡,𝑠) can be singular at 𝑠=0.

To seek a 𝑇-periodic solution 𝑢^𝑇=(𝑢^𝑇₁,𝑢^𝑇₂,…,𝑢^𝑇_𝑛) of the system (65), we first obtain a solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛) of the following system of boundary value problems: 𝑢_𝑖^′′(𝑡)+𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)=𝜕₂ℎ_𝑖1𝑡,2|𝑢(𝑡)|²𝑢_𝑖(𝑡)+𝑓_𝑖(𝑡)[],𝑢,𝑡∈0,𝑇_𝑖(0)=𝑢_𝑖(𝑇),𝑢^′_𝑖(0)=𝑢^′_𝑖(𝑇),1≤𝑖≤𝑛. (66) Then, set 𝑢^𝑇(𝑡)=𝑢^∗(𝑡−𝑚𝑇),𝑡∈[𝑚𝑇,(𝑚+1)𝑇],𝑚∈ℤ. (67)

Our main tool is Schauder's fixed point theorem, which is stated below for completeness.

Theorem 26 (see [62]). Let Ω be a convex subset of a Banach space 𝐵 and 𝑆∶Ω→Ω a continuous and compact map. Then 𝑆 has a fixed point.

To begin, let 𝑔_𝑖 be Green's function of the boundary value problem 𝑥^′′(𝑡)+𝑎_𝑖(𝑡)𝑥(𝑡)[],𝑥(0)(𝑇)=0,𝑡∈0,𝑇=𝑥,𝑥^′(0)=𝑥^′(𝑇). (68) Throughout, we will assume that the functions 𝑎_𝑖∈𝐿¹[0,𝑇] are such that (C1) the Hill's equation 𝑥^′′(𝑡)+𝑎_𝑖(𝑡)𝑥(𝑡)=0 is nonresonant (i.e., the unique periodic solution is the trivial solution), and 𝑔_𝑖(𝑡,𝑠)≥0 for all (𝑡,𝑠)∈[0,𝑇]×[0,𝑇].

Note that Torres [46] has a result on 𝑎_𝑖(𝑡) that ensures that condition (C1) is satisfied. In fact, if 𝑎_𝑖(𝑡)=𝑘², then (C1) holds if 𝑘∈(0,𝜋/𝑇]; if 𝑎_𝑖(𝑡) is not a constant, then (C1) is valid if the 𝐿_𝑝 norm of 𝑎_𝑖(𝑡) is bounded above by some specific constant.

Let 𝜃_𝑖∈{1,−1},1≤𝑖≤𝑛 be fixed. Define 𝜙_𝑖(𝑡)=^𝑇₀𝑔_𝑖(𝑡,𝑠)𝜃_𝑖𝑓_𝑖(𝑠)𝑑𝑠,𝑡∈[0,𝑇],1≤𝑖≤𝑛. (69) We also let 𝜙_𝑖^min=min_{[]𝑡∈0,𝑇}𝜙_𝑖(𝑡),𝜙_𝑖^max=max_{[]𝑡∈0,𝑇}𝜙_𝑖(𝑡). (70)

We now present our main result which tackles (65) when the norm |⋅| in ℝ^𝑛 is the 𝑙_𝑝 norm or the 𝑙_∞ norm.

Theorem 27. Assume that the following conditions hold for each 1≤𝑖≤𝑛∶(𝐶1), (C2)𝜙_𝑖^min>0; (C3) let 𝐻_𝑖(𝑡,𝑠)=(𝜕/𝜕𝑠)ℎ_𝑖(𝑡,𝑠); for any numbers 𝑏,𝑏^′ with 𝑏^′≥𝑏>0, the function 𝐻_𝑖∶[0,T]×[𝑏,𝑏^′]→ℝ is an 𝐿¹-Carathéodory function, that is, (i) the map 𝑠↦𝐻_𝑖(𝑡,𝑠) is continuous for almost all 𝑡∈[0,𝑇],(ii) the map 𝑡↦𝐻_𝑖(𝑡,𝑠) is measurable for all 𝑠∈[𝑏,𝑏^′],(iii) for any 𝑟>0, there exists 𝜇_𝑟,𝑖∈𝐿¹[0,𝑇] such that |𝑠|≤𝑟(𝑠∈[𝑏,𝑏^′]) implies |𝐻_𝑖(𝑡,𝑠)|≤𝜇_𝑟,𝑖(𝑡) for almost all 𝑡∈[0,𝑇];(C4)(𝜕/𝜕𝑠)ℎ_𝑖(𝑡,𝑠)≥0 for 𝑡∈[0,𝑇] and 𝑠>0;(C5) there exist 𝑐_𝑖>0 and 𝛼_𝑖>0 such that 𝜕ℎ𝜕𝑠^𝑖(𝑡,𝑠)≤𝑐_𝑖𝑠^−𝛼𝑖,𝑡∈[0,𝑇],𝑠>0; (71)(C6)the norm |⋅| is the 𝑙_𝑝 norm where 1≤𝑝≤∞ is fixed, and ^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠<𝐴^𝑝_𝑖𝑐_𝑖2^𝛼𝑖−1,𝑡∈[0,𝑇], (72) where 𝐴^𝑝_𝑖=^𝑛_𝑘∑₌₁𝜙_𝑘^{min𝑝2𝛼𝑖/𝑝},1≤𝑝<∞,max_{1≤𝑘≤𝑛}𝜙_𝑘^min2𝛼𝑖,𝑝=∞. (73)Then, (65) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝜙_𝑖^min≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅_𝑖,𝑡∈ℝ,1≤𝑖≤𝑛, (74) where 𝑅_𝑖≥𝜙_𝑖^min,𝑅_𝑖≥𝜙_𝑖^max𝑐1−^𝑖2^𝛼𝑖𝐴^𝑝_𝑖max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠⁻¹,1≤𝑖≤𝑛. (75)

Theorem 27 is proved using Theorem 26; in fact we will seek a constant-sign solution of (66) in (𝐶[0,𝑇])^𝑛 and then extend it to a 𝑇-periodic constant-sign solution of (65) as in (67). Here, let Ω be the closed convex set given by Ω=𝑢∈(𝐶[0,𝑇])^𝑛∣𝜙_𝑖^min≤𝜃_𝑖𝑢_𝑖(𝑡)≤𝑅_𝑖,𝑡∈[0,𝑇],1≤𝑖≤𝑛∣𝜙_𝑖^min, (76) where 𝑅_𝑖(≥𝜙_𝑖^min>0) is chosen as in (75), and define the operator 𝑆∶Ω→(𝐶[0,𝑇])^𝑛 as 𝑆𝑢(𝑡)=𝑆₁𝑢(𝑡),𝑆₂𝑢(𝑡),…,𝑆_𝑛𝑢(𝑡),𝑡∈[0,𝑇], (77) where 𝑆_𝑖𝑢(𝑡)=^𝑇₀𝑔_𝑖(𝑡,𝑠)𝜕₂ℎ_𝑖1𝑠,2|𝑢(𝑠)|²𝑢_𝑖(𝑠)+𝑓_𝑖(𝑠[])𝑑𝑠,𝑡∈0,𝑇,1≤𝑖≤𝑛. (78) Clearly, a fixed point of 𝑆𝑢=𝑢 is a solution of (66). We can show that 𝑆(Ω)⊆Ω; that is, 𝑆_𝑖(Ω)⊆Ω for each 1≤𝑖≤𝑛. Further, we can prove that 𝑆∶Ω→Ω is continuous and compact; that is, 𝑆_𝑖𝑢 is bounded and is equicontinuous for any 𝑢∈Ω and 1≤𝑖≤𝑛. By Theorem 26, the system (66) has a constant-sign solution 𝑢^∗∈Ω. Now, a 𝑇-periodic constant-sign solution 𝑢^𝑇 of (65) can be obtained as in (67).

Remark 28. The constants 𝑐_𝑖 that appear in (C5) determine the upper bounds 𝑅_𝑖 of the solution 𝑢^𝑇_𝑖,1≤𝑖≤𝑛. Noting (75), we see that a smaller (bigger) 𝑐_𝑖 gives a smaller (bigger) 𝑅_𝑖, and hence a smaller (bigger) set Ω where the solution lies.

In the next result, we will relax the condition (C6). The tradeoff is the upper bounds 𝑅_𝑖 of the solution that may be bigger than those in (75). Also the bounds 𝑅_𝑖 do not depend on 𝑝 (𝑝 as in 𝑙_𝑝 norm) and so the information of 𝑝 is not utilized. This result is obtained by following the main arguments in the derivation of Theorem 27 but modify the proof of 𝜃_𝑖𝑆_𝑖𝑢(𝑡)≤𝑅_𝑖,𝑡∈[0,𝑇].

Theorem 29. Assume that (C1)–(C5) hold for each 1≤𝑖≤𝑛. The norm |⋅| is the 𝑙_𝑝 norm where 1≤𝑝≤∞ is fixed. Then (65) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝜙_𝑖^min≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅_𝑖,𝑡∈ℝ,1≤𝑖≤𝑛, (79) where, for 1≤𝑖≤𝑛 we have 𝑅_𝑖≥𝜙_𝑖^min, 𝑅^2𝛼𝑖_𝑖>𝑐_𝑖2^𝛼𝑖max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠,if𝛼_𝑖∈10,2,𝑅_𝑖1−𝑐_𝑖2^𝛼𝑖𝑅^−2𝛼𝑖_𝑖max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠≥𝜙_𝑖^max,if𝛼_𝑖∈10,2,𝑅_𝑖≥𝑐_𝑖2^𝛼𝑖𝜙_𝑖^{min1−2𝛼𝑖}max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠+𝜙_𝑖^max,if𝛼_𝑖≥12. (80)(81)(82)

Remark 30. A similar remark as Remark 28 also holds for Theorem 29. Moreover, we note that the upper bounds 𝑅_𝑖 that fulfill (80)–(82) are independent of 𝑝, thus the information of |⋅| being a particular 𝑙_𝑝 norm is not used. On the other hand, in Theorem 27, the upper bounds 𝑅_𝑖 that satisfy (75) depend on 𝑝. The sharpness of the bounds in both theorems cannot be compared in general; however, we will give an example at the end of this section to illustrate the results.

In the next result, we will relax the condition (C2). Here, we allow 𝜙_𝑖(𝑡)≤0 for some 𝑖∈{1,2,…,𝑛} and some 𝑡∈[0,𝑇].

Theorem 31. Suppose that(C7) there exists 𝑗∈{1,2,…,𝑛} such that 𝜙_𝑗^min>0.
Let 𝐽={𝑗∈{1,2,…,𝑛}|𝜙_𝑗^min>0} and let 𝐽^′={1,2,…,𝑛}⧵𝐽. Assume that the following conditions hold for each 1≤𝑖≤𝑛∶ (C1), (C3), (C4), and (C8)there exist 𝑐_𝑖>0 such that 𝜕ℎ𝜕𝑠^𝑖(𝑡,𝑠)≤𝑐_𝑖𝑠^−𝛼𝑖,𝑡∈[0,𝑇],𝑠>0, (83) where 𝛼_𝑗>0 for 𝑗∈𝐽 and 𝛼_𝑘∈(0,1/2) for 𝑘∈𝐽^′.
Further, let the following hold for each 𝑗∈𝐽∶(C9) the norm |⋅| is the 𝑙_𝑝 norm where 1≤𝑝≤∞ is fixed, and ^𝑇₀𝑔_𝑗(𝑡,𝑠)𝑑𝑠<𝐴^𝑝_𝑗𝑐_𝑗2^𝛼𝑗−1,𝑡∈[0,𝑇], (84) where 𝐴^𝑝_𝑗=_ℓ∑_∈𝐽𝜙_ℓ^{min𝑝2𝛼𝑗/𝑝},1≤𝑝<∞,max_ℓ∈𝐽𝜙_ℓ^min2𝛼𝑗,𝑝=∞. (85)Then, (65) has a 𝑇-periodic solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝜙_𝑗^min≤𝜃_𝑗𝑢^𝑇_𝑗(𝑡)≤𝑅_𝑗𝑢,𝑡∈ℝ,𝑗∈𝐽,_𝑇𝑘(𝑡)≤𝑅_𝑘,𝑡∈ℝ,𝑘∈𝐽^′, (86) where 𝑅_𝑗≥𝜙_𝑗^min,𝑅_𝑗≥𝜙_𝑗^max1−𝑐_𝑗2^𝛼𝑗𝐴^𝑝_𝑗⁻¹max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑗(𝑡,𝑠)𝑑𝑠⁻¹𝑅,𝑗∈𝐽,_{2𝛼𝑘𝑘}>𝑐_𝑘2^𝛼𝑘max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑑𝑠,𝑘∈𝐽^′,𝑅_𝑘1−𝑐_𝑘2^𝛼𝑘𝑅^−2𝛼𝑘_𝑘max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑑𝑠≥max_{[]𝑡∈0,𝑇}𝜙_𝑘(𝑡),𝑘∈𝐽^′. (87)(88)(89)

To derive Theorem 31, we let the closed convex set Ω^∗ be Ω^∗=𝑢∈(𝐶[0,𝑇])^𝑛∣𝜙_𝑗^min≤𝜃_𝑗𝑢_𝑗(𝑡)≤𝑅_𝑗[],𝑢,𝑡∈0,𝑇𝑗∈𝐽;_𝑘(𝑡)≤𝑅_𝑘,𝑡∈[0,𝑇],𝑘∈𝐽^′, (90) where 𝑅_𝑗(≥𝜙_𝑗^min>0) and 𝑅_𝑘 are chosen as in (87)–(89). Next, we define the operator 𝑆∶Ω^∗→(𝐶[0,𝑇])^𝑛 as in (78) and show that Theorem 26 is applicable.

Remark 32. From the conclusion of Theorem 29, we see that the solution 𝑢^𝑇 is “partially” of constant sign, in the sense that 𝜃_𝑗𝑢^𝑇_𝑗(𝑡)≥0 for 𝑗∈𝐽, but may not be so for 𝑗∈𝐽^′. Further, the constants 𝑐_𝑖 that appear in (C8) determine the upper bounds 𝑅_𝑖 of the solution 𝑢^𝑇_𝑖,1≤𝑖≤𝑛. From (87) and (88), we see that a smaller (bigger) 𝑐_𝑖 gives a smaller (bigger) 𝑅_𝑖, and hence a smaller (bigger) set Ω^∗ where the solution lies.

Using similar arguments as in the derivation of Theorems 31 and 29 (in getting 𝑆_𝑗𝑢∈Ω^∗ for 𝑗∈𝐽 and 𝑢∈Ω^∗), we obtain the following result.

Theorem 33. Suppose that (C7) hold. Let 𝐽={𝑗∈{1,2,…,𝑛}|𝜙_𝑗^min>0} and let 𝐽^′={1,2,…,𝑛}⧵𝐽. Assume the following conditions hold for each 1≤𝑖≤𝑛∶ (C1), (C3), (C4), and (C8). Then, (65) has a 𝑇-periodic solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝜙_𝑗^min≤𝜃_𝑗𝑢^𝑇_𝑗(𝑡)≤𝑅_𝑗𝑢,𝑡∈ℝ,𝑗∈𝐽,_𝑇𝑘(𝑡)≤𝑅_𝑘,𝑡∈ℝ,𝑘∈𝐽^′, (91) where 𝑅_𝑗≥𝜙_𝑗^min𝑅,𝑗∈𝐽,_{2𝛼𝑗𝑗}>𝑐_𝑗2^𝛼𝑗max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑗(𝑡,𝑠)𝑑𝑠,if𝛼_𝑗∈10,2𝑅,𝑗∈𝐽,_𝑗1−𝑐_𝑗2^𝛼𝑗𝑅^−2𝛼𝑗_𝑗max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑗(𝑡,𝑠)𝑑𝑠≥𝜙_𝑗^max,if𝛼_𝑗∈10,2𝑅,𝑗∈𝐽,_𝑗≥𝑐_𝑗2^𝛼𝑗𝜙_𝑗^{min1−2𝛼𝑗}max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑗(𝑡,𝑠)𝑑𝑠+𝜙_𝑗^max,if𝛼_𝑗≥12𝑅,𝑗∈𝐽,_{2𝛼𝑘𝑘}>𝑐_𝑘2^𝛼𝑘max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑑𝑠,𝑘∈𝐽^′,𝑅_𝑘1−𝑐_𝑘2^𝛼𝑘𝑅^−2𝛼𝑘_𝑘max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑑𝑠≥max_{[]𝑡∈0,𝑇}𝜙_𝑘(𝑡),𝑘∈𝐽^′. (92)(93)

Remark 34. A similar remark as Remark 32 holds for Theorem 33. Also, we observe once again that the upper bounds 𝑅_𝑗 that fulfill (92) are independent of 𝑝, thus the information of |⋅| being a particular 𝑙_𝑝 norm is not used. On the other hand, in Theorem 31, the upper bounds 𝑅_𝑗 that satisfy (87) depend on 𝑝.

We will now present an example that illustrates Theorems 27 and 29.

Example 35. Consider (65) when 𝑇=2𝜋,𝑛=2,𝑎₁(𝑡)=𝑎₂(𝑡)=14,𝑓₁(𝑡)=1,𝑓₂(𝑡)=12,ℎ₁()=()𝑡,𝑠ln𝑠+1||,ℎsin𝑡+1₂()=()𝑡,𝑠ln𝑠+13|),|⋅|(|cos𝑡+1=𝑙_𝑝norm1≤𝑝≤∞. (94)

Fix 𝜃_𝑖=1,1≤𝑖≤𝑛, that is, we are seeking positive solutions. The corresponding Green's function has the explicit expression [36] 𝑔₁(𝑡,𝑠)=𝑔₂()=1𝑡,𝑠cos2()1𝑡−𝑠−𝜋,0≤𝑠≤𝑡≤2𝜋,cos2(𝑠−𝑡−𝜋),0≤𝑡≤𝑠≤2𝜋. (95) Condition (C1) is satisfied. By direct computation, we get 𝜙₁(𝑡)=4 and 𝜙₂(𝑡)=2 for 𝑡∈[0,2𝜋]. Thus, (C2) is fulfilled with 𝜙₁^min=𝜙₁^max=4,𝜙₂^min=𝜙₂^max=2. (96) Moreover, we have 𝜕ℎ𝜕𝑠¹()=1𝑡,𝑠||1sin𝑡+1≤1𝑠+1≤1𝑠+1𝑠,𝜕ℎ𝜕𝑠²()=1𝑡,𝑠3|)1(|cos𝑡+1≤1𝑠+13()≤1𝑠+13𝑠 (97) and so it is clear that (C4) and (C5) are satisfied with 𝛼₁=1,𝑐₁=1,𝛼₂=1,𝑐₂=13. (98) Finally, we compute 𝐴^𝑝₁=A^𝑝₂=(4^𝑝+2^𝑝)^2/𝑝𝐴,1≤𝑝<∞,_∞1=𝐴^∞₂=16. (99) Since ∫₀^2𝜋𝑔_𝑖(𝑡,𝑠)𝑑𝑠=4 for 𝑡∈[0,2𝜋] and 𝑖=1,2, we check that (C6) holds for all 1≤𝑝≤∞.

All the conditions of Theorem 27 are satisfied, thus we conclude that the problem (65) with (94) has a positive 2𝜋-periodic solution 𝑢=(𝑢₁,𝑢₂) such that 𝜙_𝑖^min≤𝑢_𝑖(𝑡)≤𝑅_𝑖,𝑡∈ℝ,𝑖=1,2, (100) where (from (75)) 𝑅_𝑖≥𝜙_𝑖^max1−8𝑐_𝑖𝐴^𝑝_𝑖⁻¹≡𝐿^𝑝_𝑖,1≤𝑝≤∞,𝑖=1,2. (101)

We can also apply Theorem 29 to conclude that the problem (65) with (94) has a positive 2𝜋-periodic solution 𝑢=(𝑢₁,𝑢₂) satisfying (100) and (from (82)) 𝑅_𝑖≥8𝑐_𝑖𝜙_𝑖^min−1+𝜙_𝑖^max≡𝑀_𝑖,𝑖=1,2. (102)

As mentioned in Remark 30, in general we cannot compare 𝐿^𝑝_𝑖 and 𝑀_𝑖. In fact, a direct calculation gives 𝐿𝑝=1∶₁₁=5.14<𝑀₁=6,𝐿¹₂=2.16<𝑀₂𝐿=3.33,𝑝=2∶₂₁=6.67>𝑀₁=6,𝐿²₂=2.31<𝑀₂𝐿=3.33,𝑝=∞∶_∞1=8>𝑀₁=6,𝐿^∞₂=2.4<𝑀₂=3.33. (103)

5.2. Existence Results for (59)

In this section we will consider the general system of Hill's equations 𝑢_𝑖^′′(𝑡)+𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)=𝐹_𝑖(𝑡,𝑢(𝑡)),1≤𝑖≤𝑛. (104) Here, 𝑎_𝑖 and 𝐹_𝑖 are 𝑇-periodic in the variable 𝑡, 𝑎_𝑖∈𝐿¹[0,𝑇], and the nonlinearities 𝐹_𝑖(𝑡,𝑥₁,𝑥₂,…,𝑥_𝑛) can be singular at 𝑥_𝑗=0 where 𝑗∈{1,2,…,𝑛}.

Once again, to obtain a 𝑇-periodic solution 𝑢^𝑇=(𝑢^𝑇₁,𝑢^𝑇₂,…,𝑢^𝑇_𝑛) of the system (104), we first seek a solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛) of the following system of boundary value problems: 𝑢_𝑖^′′(𝑡)+𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)=𝐹_𝑖((𝑡[],𝑢𝑡,𝑢)),𝑡∈0,𝑇_𝑖(0)=𝑢_𝑖(𝑇),𝑢^′_𝑖(0)=𝑢^′_𝑖(𝑇),1≤𝑖≤𝑛. (105) The periodic solution is then given by 𝑢^𝑇(𝑡)=𝑢^∗(𝑡−𝑚𝑇),𝑡∈[𝑚𝑇,(𝑚+1)𝑇],𝑚∈ℤ. (106)

With 𝑔_𝑖 being the Green's function of the boundary value problem (68), throughout we will assume that (C1) is satisfied. Moreover, for fixed 𝜃_𝑖∈{−1,1} and 𝑇-periodic functions 𝑞_𝑖∈𝐿¹[0,𝑇],1≤𝑖≤𝑛, we define 𝜂_𝑖(𝑡)=^𝑇₀𝑔_𝑖(𝑡,𝑠)𝜃_𝑖𝑞_𝑖(𝑠)𝑑𝑠,𝑡∈[0,𝑇],1≤𝑖≤𝑛 (107) and also 𝜂_𝑖^min=min_{[]𝑡∈0,𝑇}𝜂_𝑖(𝑡),𝜂_𝑖^max=max_{𝑡∈[0,𝑇]}𝜂_𝑖(𝑡). (108) For 𝑏≥𝑏^′≥0 and 1≤𝑖≤𝑛, we denote the interval [𝑏,𝑏^′]_𝑖=𝑏,𝑏^′,if𝜃_𝑖=1,−𝑏^′,−𝑏,if𝜃_𝑖=−1. (109) A similar definition is valid for (𝑏,𝑏^′)_𝑖.

Using Schauder's fixed point theorem, we will establish existence results for the system (104).

Theorem 36. Assume the following conditions hold for each 1≤𝑖≤𝑛∶ (C1);(C10) for any numbers 𝑏_𝑗,𝑏^′_𝑗,1≤𝑗≤𝑛 with 𝑏^′_𝑗≥𝑏_𝑗>0, the function 𝐹_𝑖∶[0,𝑇]×∏^𝑛_𝑗=1[𝑏_𝑗,𝑏^′_𝑗]_𝑗→ℝ is a 𝐿¹-Carathéodory function, that is, (i)the map 𝑢↦𝐹_𝑖(𝑡,𝑢) is continuous for almost all 𝑡∈[0,𝑇],(ii)the map 𝑡↦𝐹_𝑖(𝑡,𝑢) is measurable for all 𝑢∈∏^𝑛_𝑗=1[𝑏_𝑗,𝑏^′_𝑗]_𝑗,(iii)for any 𝑟>0, there exists 𝜇_𝑟,𝑖∈𝐿¹[0,𝑇] such that |𝑢|≤𝑟(𝑢∈∏^𝑛_𝑗=1[𝑏_𝑗,𝑏^′_𝑗]_𝑗) implies |𝐹_𝑖(𝑡,𝑢)|≤𝜇_𝑟,𝑖(𝑡) for almost all 𝑡∈[0,𝑇];(C11)there exist 𝛽_𝑖>0,𝛾_𝑖∈[0,1), and 𝑇-periodic functions 𝑤_𝑖,𝑞_𝑖 with 𝑤_𝑖∈𝐿¹[0,𝑇],q_𝑖∈𝐿¹[0,𝑇] and 𝑤_𝑖(𝑡)>0 for a.e.𝑡∈[0,𝑇] such that 𝜃_𝑖𝑞_𝑖(𝑡)|𝑢|^𝛾𝑖≤𝜃_𝑖𝐹_𝑖(𝑡,𝑢)≤𝜃_𝑖𝑞_𝑖(𝑡)|𝑢|^𝛾𝑖+𝑤_𝑖(𝑡)|𝑢|^−𝛽𝑖,𝑡∈[0,𝑇],𝑢∈^𝑛_𝑘=1(0,∞)_𝑘 (110) (here |⋅| is the 𝑙_𝑝 norm where 1≤𝑝≤∞ is fixed);(C12)𝜂_𝑖^min>0.
Then, (104) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝑟_𝑖≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅_𝑖,𝑡∈ℝ,1≤𝑖≤𝑛, (111) where, for 1≤𝑖≤𝑛 one has 0<𝑟_𝑖≤𝑅_𝑖,𝑟_𝑖≤𝜂_𝑖^{min1/(1−𝛾𝑖)},𝑅_𝑖≥𝜂_𝑖^max|𝑅|^𝛾𝑖_𝑝+|𝑟|^−𝛽𝑖_𝑝max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑤_𝑖(𝑠)𝑑𝑠, (112)(113) (here |𝑅|_𝑝 is the 𝑙_𝑝 norm of (𝑅₁,𝑅₂,…,𝑅_𝑛), likewise |𝑟|_𝑝 is the 𝑙_𝑝 norm of (𝑟₁,𝑟₂,…,𝑟_𝑛)).

In proving Theorem 36, we actually seek a constant-sign solution of (105) in (𝐶[0,𝑇])^𝑛 and then extend it to a 𝑇-periodic constant-sign solution of (104) as in (106). Let Ω be the closed convex set given by Ω=𝑢∈(𝐶[0,𝑇])^𝑛∣𝑟_𝑖≤𝜃_𝑖𝑢_𝑖(𝑡)≤𝑅_𝑖[],,,𝑡∈0,𝑇1≤𝑖≤𝑛 (114) where 𝑅_𝑖≥𝑟_𝑖>0 are chosen as in (112) and (113), and define the operator 𝑆∶Ω→(𝐶[0,𝑇])^𝑛 as 𝑆𝑢(𝑡)=𝑆₁𝑢(𝑡),𝑆₂𝑢(𝑡),…,𝑆_𝑛𝑢(𝑡),𝑡∈[0,𝑇], (115) where 𝑆_𝑖𝑢(𝑡)=^𝑇₀𝑔_𝑖(𝑡,𝑠)𝐹_𝑖(𝑠,𝑢(𝑠))𝑑𝑠,𝑡∈[0,𝑇],1≤𝑖≤𝑛. (116) Clearly, a fixed point of 𝑆𝑢=𝑢 is a solution of (105). The conditions of Theorem 26 are then shown to be satisfied.

Remark 37. As seen from (112) and (113), the functions 𝑤_𝑖 and 𝑞_𝑖 that appear in (C11) determine the lower and upper bounds of the solution 𝑢^𝑇_𝑖,1≤𝑖≤𝑛.

Theorem 38. Assume that the following conditions hold for each 1≤𝑖≤𝑛∶ (C1), (C10), (C11), and (C12). Then, (104) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝑟≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,1≤𝑖≤𝑛, (117) where 0<𝑟≤𝑅, and for all 1≤𝑖≤𝑛, 𝜂𝑟≤^𝑖min𝑛^{𝛾𝑖/𝑝1/(1−𝛾𝑖)}𝜂,1≤𝑝<∞,_{𝑖min1/(1−𝛾𝑖)}𝑅,𝑝=∞,𝑅≥^𝛾𝑖𝜂_𝑖^max𝑛^{𝛾𝑖/𝑝}+𝑟^−𝛽𝑖𝑛^{−𝛽𝑖/𝑝}max_{[]𝑡∈0,𝑇}∫^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑤_𝑖(𝑠),𝑅𝑑𝑠1≤𝑝<∞,_𝛾𝑖𝜂_𝑖^max+𝑟^−𝛽𝑖max_{𝑡∈[0,𝑇]}∫^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑤_𝑖(𝑠)𝑑𝑠,𝑝=∞. (118)(119)

Theorem 38 is obtained by similar arguments used in the derivation of Theorem 36, with a new Ω defined as Ω=𝑢∈(𝐶[0,𝑇])^𝑛∣𝑟≤𝜃_𝑖𝑢_𝑖(𝑡)[],},≤𝑅,𝑡∈0,𝑇1≤𝑖≤𝑛 (120) where 𝑅≥𝑟>0 are chosen as in (118) and (119).

Remark 39. Remark 37 also holds for Theorem 38. Further, comparing the bounds 𝑟_𝑖, 𝑅_𝑖, 1≤𝑖≤𝑛 in Theorem 36 (see (112), (113)) with the bounds 𝑟,𝑅 in Theorem 38 (see (118), (119)), we note that 𝑟_𝑖 and 𝑅_𝑖 are lower and upper bounds for a particular 𝜃_𝑖𝑢^𝑇_𝑖, whereas 𝑟 and 𝑅 are uniform lower and upper bounds for all 𝜃_𝑖𝑢^𝑇_𝑖,1≤𝑖≤𝑛. However, the computation of 𝑅_𝑖 from (113) is more difficult than calculating 𝑅 from (119).

Our next result tackles the case when 𝜂_𝑖^min=0.

Theorem 40. Assume that the following conditions hold for each 1≤𝑖≤𝑛∶ (C1), (C10), (C13) there exist 𝛽_𝑖∈(0,1),𝛾_𝑖∈[0,1), and 𝑇-periodic functions 𝑤_𝑖,𝑣_𝑖,𝑞_𝑖 with 𝑤_𝑖∈𝐿¹[0,𝑇],𝑣_𝑖∈𝐿¹[0,𝑇],𝑞_𝑖∈𝐿¹[0,𝑇], and 𝑤_𝑖(𝑡),𝑣_𝑖(𝑡)>0 for a.e.𝑡∈[0,𝑇] such that 𝜃_𝑖𝑞_𝑖(𝑡)|𝑢|^𝛾𝑖+𝑣_𝑖(𝑡)|𝑢|^−𝛽𝑖≤𝜃_𝑖𝐹_𝑖(𝑡,𝑢)≤𝜃_𝑖𝑞_𝑖(𝑡)|𝑢|^𝛾𝑖+𝑤_𝑖(𝑡)|𝑢|^−𝛽𝑖,𝑡∈[0,𝑇],𝑢∈^𝑛_𝑘=1(0,∞)_𝑘 (121) (here |⋅| is the 𝑙_𝑝 norm where 1≤𝑝≤∞ is fixed);(C14)𝜂_𝑖^min=0.
Then, (104) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 1𝑅≤𝜃_𝑖𝑢^𝑇_𝑖(t)≤𝑅,𝑡∈ℝ,1≤𝑖≤𝑛, (122) where 𝑅≥1, and for all 1≤𝑖≤𝑛, 𝑛𝑅≥^{𝛽𝑖/𝑝(1−𝛽𝑖)}min_{𝑡∈[0,𝑇]}∫^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑣_𝑖(𝑠)𝑑𝑠^{−1/(1−𝛽𝑖)},1≤𝑝<∞,min_{[]𝑡∈0,𝑇}∫^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑣_𝑖(𝑠)𝑑𝑠^{−1/(1−𝛽𝑖)},𝑅𝑝=∞,𝑅≥^𝛾𝑖𝜂_𝑖^max𝑛^{𝛾𝑖/𝑝}+𝑅^𝛽𝑖𝑛^{−𝛽𝑖/𝑝}max_{[]𝑡∈0,𝑇}∫^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑤_𝑖(𝑠),𝑅𝑑𝑠1≤𝑝<∞,_𝛾𝑖𝜂_𝑖^max+𝑅^𝛽𝑖max_{[]𝑡∈0,𝑇}∫^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑤_𝑖(𝑠)𝑑𝑠,𝑝=∞. (123)

The closed convex set used to get Theorem 40 is given by Ω=𝑢∈(𝐶[0,𝑇])^𝑛∣𝑟≤𝜃_𝑖𝑢_𝑖(𝑡)[],≤𝑅,𝑡∈0,𝑇,1≤𝑖≤𝑛 (124) where 𝑟=1/𝑅 and 𝑅≥1 satisfies (123).

Remark 41. As seen from (123), the functions 𝑤_𝑖,𝑣_𝑖, and 𝑞_𝑖 that appear in (C13) determine the lower and upper bounds of the solution 𝑢^𝑇_𝑖,1≤𝑖≤𝑛.

Finally, the next result tackles the case when 𝜂_𝑖^max<0.

Theorem 42. Assume that the following conditions hold for each 1≤𝑖≤𝑛∶ (C1), (C10), (C15) there exist 𝛽∈(0,1) and 𝑇-periodic functions 𝑤_𝑖,𝑣_𝑖,𝑞_𝑖 with 𝑤_𝑖∈𝐿¹[0,𝑇],𝑣_𝑖∈𝐿¹[0,𝑇],𝑞_𝑖∈𝐿¹[0,𝑇], and 𝑤_𝑖(𝑡),𝑣_𝑖(𝑡)>0 for a.e.𝑡∈[0,𝑇] such that 𝜃_𝑖𝑞_𝑖(𝑡)+𝑣_𝑖(𝑡)|𝑢|^−𝛽≤𝜃_𝑖𝐹_𝑖(𝑡,𝑢)≤𝜃_𝑖𝑞_𝑖(𝑡)+𝑤_𝑖(𝑡)|𝑢|^−𝛽,𝑡∈[0,𝑇],𝑢∈^𝑛_𝑘=1(0,∞)_𝑘 (125) (here |⋅| is the 𝑙_𝑝 norm where 1≤𝑝≤∞ is fixed);(C16)𝜂_𝑖^max<0; (C17)𝜂_𝑖^min≥𝑛^{−𝛽/(1+𝛽)𝑝}𝑊^{−𝛽/(1−𝛽2)}(𝑉𝛽²)^{1/(1−𝛽2)}(1−1/𝛽²) where 𝑊=max_{1≤𝑘≤𝑛}max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑤_𝑘(𝑠)𝑑𝑠,𝑉=min_{1≤𝑘≤𝑛}min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑣_𝑘(𝑠)𝑑𝑠. (126)Then, (104) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝑟≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,1≤𝑖≤𝑛, (127) where 0<𝑟≤𝑅 are given by 𝑛𝑟=^{−𝛽/(1+𝛽)𝑝}𝑊^{−𝛽/(1−𝛽2)}𝑉𝛽^{21/(1−𝛽2)}𝑊,1≤𝑝<∞_{−𝛽/(1−𝛽2)}𝑉𝛽^{21/(1−𝛽2)}𝑛,𝑝=∞,𝑅=^{−𝛽/(1+𝛽)𝑝}𝑊^{1/(1−𝛽2)}𝑉𝛽^{2−𝛽/(1−𝛽2)}𝑊,1≤𝑝<∞_{1/(1−𝛽2)}𝑉𝛽^{2−𝛽/(1−𝛽2)},𝑝=∞. (128)

Theorem 42 is obtained by considering the closed convex set Ω=𝑢∈(𝐶[0,𝑇])^𝑛∣𝑟≤𝜃_𝑖𝑢_𝑖(𝑡)[],},≤𝑅,𝑡∈0,𝑇1≤𝑖≤𝑛 (129) where 𝑅≥𝑟>0 are determined later as those given in (128).

Remark 43. As seen from (128), the functions 𝑤_𝑖 and 𝑣_𝑖 that appear in (C15) determine the lower and upper bounds of the solution 𝑢^𝑇_𝑖,1≤𝑖≤𝑛.

We have so far established the results when (i) 𝜂_𝑖^min>0, (ii) 𝜂_𝑖^min=0, and (iii) 𝜂_𝑖^max<0 for all 1≤𝑖≤𝑛. However, it could be that we only have 𝜂_𝑖(𝑡)≥0 for some 𝑖 and 𝜂_𝑗(𝑡)<0 for some 𝑗, which results in 𝜂_𝑖^min≥0 and 𝜂_𝑗^max<0 for some 1≤𝑖,𝑗≤𝑛. We present two results for such a case as follows. Note that Theorem 44 is obtained by applying Theorems 38–42, while Theorem 45 is obtained by applying Theorems 36, 40, and 42.

Theorem 44. Let (C1) and (C10) hold for each 1≤𝑖≤𝑛. Assume the following:(C18) conditions (C11) and (C12) hold for some 𝑖∈𝐼⊆{1,2,…,𝑛};(C19) conditions (C13) and (C14) hold for some 𝑖∈𝐽⊆{1,2,…,𝑛};(C20) conditions (C15), (C16), and (C17) hold for some 𝑖∈𝐾⊆{1,2,…,𝑛};
where 𝐼∪𝐽∪𝐾={1,2,…,𝑛}. Then, (104) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝑟≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,1≤𝑖≤𝑛, (130) where 0<𝑟≤𝑅 satisfy (a)(118) and (119) for 𝑖∈𝐼;(b)𝑟=1/𝑅,𝑅≥1, (123) for 𝑖∈𝐽;(c)(128) for 𝑖∈𝐾.

Theorem 45. Let (C1) and (C10) hold for each 1≤𝑖≤𝑛. Assume that (C18)–(C20) hold with 𝐼∪𝐽∪𝐾={1,2,…,𝑛}. Then, (104) has a 𝑇-periodic constant-sign solution 𝑢^𝑇∈(𝐶(ℝ))^𝑛 such that 𝑟_𝑖≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅_𝑖,𝑡∈ℝ,𝑖∈𝐼, (131) where 0<𝑟_𝑖≤𝑅_𝑖 satisfy (112) and (113) for 𝑖∈𝐼, and 𝑟≤𝜃_𝑖𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,𝑖∈𝐽∪𝐾, (132) where 0<𝑟≤𝑅 satisfy conclusions (b) and (c) of Theorem 44.

We will now apply the results obtained to the following system of Hill's equations, a particular form of it (see (64)) that has been discussed in [45], 𝑢₁^′′(𝑡)+𝑎₁(𝑡)𝑢₁(𝑡)=𝑢²₁+𝑢²₂^−𝛽1+𝜈₁𝑢²₁+𝑢²₂^𝛾1,𝑢₂^′′(𝑡)+𝑎₂(𝑡)𝑢₂(𝑡)=𝑢²₁+𝑢²₂^−𝛽2+𝜈₂𝑢²₁+𝑢²₂^𝛾2. (133) Clearly, the system (133) corresponds to (104) where 𝑛=2 and 𝐹_𝑖(𝑡,𝑢)=𝑢²₁+𝑢²₂^−𝛽𝑖+𝜈_𝑖𝑢²₁+𝑢²₂^𝛾𝑖,𝑖=1,2. (134) We will assume that 𝑎₁,𝑎₂∈𝐿¹[0,𝑇] satisfy (C1). Note that condition (C10) is clearly satisfied. Further, let 𝜃₁=𝜃₂=1, that is, we are interested in positive periodic solutions of (133).

Example 46. Consider the system (133) with 𝜈_𝑖>0,𝛽_𝑖>0,𝛾_𝑖∈[0,1),𝑖=1,2. (135)

Clearly, (C11) is satisfied with 𝑝=2,𝑞_𝑖=𝜈_𝑖 and 𝑤_𝑖=1,𝑖=1,2. Thus, (C12) also holds since 𝜂_𝑖^min=min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝜃_𝑖𝑞_𝑖(𝑠)𝑑𝑠=𝜈_𝑖min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠>0. (136) Theorem 38 (or Theorem 36) is applicable and we conclude that the system (133) with (135) has a 𝑇-periodic positive solution 𝑢^𝑇∈(𝐶(ℝ))² such that 𝑟≤𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,𝑖=1,2, (137) where 0<𝑟≤𝑅 are such that 𝑟≤min_𝑖=1,2𝜂_𝑖^min2^{𝛾𝑖/21/(1−𝛾𝑖)},𝑅≥max_𝑖=1,2𝑅^𝛾𝑖𝜂_𝑖^max2^𝛾𝑖/2+𝑟^−𝛽𝑖2^{−𝛽𝑖/2}max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠. (138)(139)

To illustrate numerically, suppose 𝑎₁(𝑡)=𝑎₂(𝑡)=14,𝑇=2𝜋,𝜈₁=14,𝜈₂=1,𝛾₁=𝛾₂=12,𝛽₁=𝛽₂=1. (140) Green's function is given in (95) and 𝜂_𝑖^min=𝜈_𝑖min_{𝑡∈[0,𝑇]}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠=4𝜈_𝑖. (141) Hence, (138) yields 𝑟≤√2. Let 𝑟=√2, then (139) reduces to 𝑅≥max_𝑖=1,2𝑅^1/24𝜈_𝑖2^1/4+𝑟⁻¹2^−1/24𝜈_𝑖=𝑅^1/22^9/4+2, (142) which is satisfied by 𝑅≥26.48. Let 𝑅=26.48, then from (137) we conclude that the system (133) with (140) has a 2𝜋-periodic positive solution 𝑢∈(𝐶(ℝ))² such that √2≤𝑢_𝑖(𝑡)≤26.48,𝑡∈ℝ,𝑖=1,2. (143)

Example 47. Consider the system (133) with 𝜈_𝑖=0,𝛽_𝑖∈(0,1),𝛾_𝑖∈[0,1),𝑖=1,2. (144)

Here, (C13) is satisfied with 𝑝=2,𝑞_𝑖=𝜈_𝑖=0 and 𝑤_𝑖=𝑣_𝑖=1,𝑖=1,2. Subsequently, (C14) also holds since 𝜂_𝑖^min=min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝜃_𝑖𝑞_𝑖(𝑠)𝑑𝑠=𝜈_𝑖min_{𝑡∈[0,𝑇]}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠=0. (145) Employing Theorem 40, we conclude that the system (133) with (144) has a 𝑇-periodic positive solution 𝑢^𝑇∈(𝐶(ℝ))² such that 1𝑅≤𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,𝑖=1,2, (146) where 𝑅≥1, and from (123), we have for 𝑖=1,2, 𝑅≥2^{𝛽𝑖/2(1−𝛽𝑖)}min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠^{−1/(1−𝛽𝑖)},𝑅≥2^{−𝛽𝑖/2}max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠^{1/(1−𝛽𝑖)}. (147) Combining the inequalities, we see that 𝑅 should satisfy 𝑅≥max1,max_𝑖=1,22^{𝛽𝑖/2(1−𝛽𝑖)}×min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠^{−1/(1−𝛽𝑖)},max_𝑖=1,22^{−𝛽𝑖/2}max_{𝑡∈[0,𝑇]}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠^{1/(1−𝛽𝑖)}. (148)

Example 48. Consider the system (133) with 𝜈_𝑖<0,𝛽_𝑖=𝛽∈(0,1),𝛾_𝑖𝜈=0,𝑖=1,2,_𝑖min_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠≥2^{−𝛽/2(1+𝛽)}𝑊^{−𝛽/(1−𝛽2)}𝑉𝛽^{21/(1−𝛽2)}×11−𝛽²,𝑖=1,2, (149)(150) where 𝑊=max_𝑘=1,2max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑑𝑠,𝑉=min_𝑘=1,2min_{𝑡∈[0,𝑇]}^𝑇₀𝑔_𝑘(𝑡,𝑠)𝑑𝑠. (151)

Obviously, (C15) is satisfied with 𝑝=2,𝑞_𝑖=𝜈_𝑖<0 and 𝑤_𝑖=𝑣_𝑖=1,𝑖=1,2. Then, (C16) also holds since 𝜂_𝑖^max=max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝜃_𝑖𝑞_𝑖(𝑠)𝑑𝑠=𝜈_𝑖max_{[]𝑡∈0,𝑇}^𝑇₀𝑔_𝑖(𝑡,𝑠)𝑑𝑠<0. (152) Moreover, condition (C17) is simply (150). Hence, we conclude from Theorem 42 that the system (133) with (149) and (150) has a 𝑇-periodic positive solution 𝑢^𝑇∈(𝐶(ℝ))² such that 𝑟≤𝑢^𝑇_𝑖(𝑡)≤𝑅,𝑡∈ℝ,𝑖=1,2, (153) where 0<𝑟≤𝑅 are given by 𝑟=2^{−𝛽/2(1+𝛽)}𝑊^{−𝛽/(1−𝛽2)}𝑉𝛽^{21/(1−𝛽2)},𝑅=2^{−𝛽/2(1+𝛽)}𝑊^{1/(1−𝛽2)}𝑉𝛽^{2−𝛽/(1−𝛽2)}. (154)

Remark 49. In [45], it is shown that (64) has a solution when 𝛽>0,𝛾∈[0,1) and 𝜈>0. As seen from Examples 46–48, we have generalized the system (64) to allow 𝜈 to be zero or negative.

Acknowledgments

J. Chu was supported by the National Natural Science Foundation of China (Grant nos. 11171090 and 11271078), the Program for New Century Excellent Talents in University (Grant no. NCET-10-0325), and China Postdoctoral Science Foundation funded project (Grant no. 2012T50431). J. Sun was supported by the National Natural Science Foundation of China (Grant nos. 11201270 and 11271372), Shandong Natural Science Foundation (Grant no. ZR2012AQ010), and Young Teacher Support Program of Shandong University of Technology.