Abstract and Applied Analysis

Volume 2013 (2013), Article ID 420835, 17 pages

http://dx.doi.org/10.1155/2013/420835

Review Article

Existence for Singular Periodic Problems: A Survey of Recent Results

Jifeng Chu,¹ Juntao Sun,² and Patricia J. Y. Wong³

¹Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China

²School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

³School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798

Received 13 December 2012; Accepted 11 January 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Jifeng Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

is that the mean value of is negative; that is, , here , which corresponds to a strong force condition, according to a terminology first introduced by Gordon [30]. Moreover, if , which corresponds to a weak force condition, they found examples of functions 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 by and , respectively, for a given function essentially bounded.

2. Second-Order Scalar Singular Equations

In this section, we recall some results for second-order singular differential equations

here are continuous, -periodic functions. The nonlinearity is continuous in and -periodic in and has a singularity at .

First we need to present some preliminary results on the linear equation

with periodic boundary conditions

We assume the following:

(A) the Green function , associated with (3)-(4), is positive for all , or(B) the Green function , associated with (3)-(4), is nonnegative for all .

When , condition (A) is equivalent to and condition (B) is equivalent to . In this case, we have

For a nonconstant function , there is an -criterion proved in [46], which is given in Lemma 1 for the sake of completeness. Let denote the best Sobolev constant in the following inequality:

The explicit formula for is

where is the gamma function, see [47, 48].

Lemma 1 (see [46, Corollary 2.3]). Assume that and for some . If

then the condition (A) holds. Moreover, condition (B) holds if

When the hypothesis (A) is satisfied, we denote

Obviously, and .

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 is an open subset of a convex set K in a normed linear space and . Let be a compact and continuous map. Then one of the following two conclusions holds.

(I) has at least one fixed point in .(II) There exists and such that .

Theorem 3 (see [37, Theorem 4.1]). Suppose that satisfies (A) and satisfies the following.

There exists a nonincreasing positive continuous function on and a constant such that for , where satisfies

There exist continuous, nonnegative functions and such that

  is nonincreasing and is nondecreasing in . There exists a positive number such that and

 here

Then for each , (2) has at least one positive periodic solution with for all and .

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 () 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 , 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 ()-() are satisfied. Suppose further the following condition.

() For each constant , there exists a continuous function such that for all .

Then for each with , (2) has at least one positive periodic solution with for all and .

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 be a cone in and is a subset of , we write and .

Theorem 5 (see [51]). Let be a Banach space and a cone in . Assume are open bounded subsets of X with . Let

be a completely continuous operator such that

(a) for ,(b)there exists such that and all .

Then has a fixed point in .

Theorem 6 (see [33, Theorem 3.2]). Suppose that satisfies (A) and satisfies ()-(). Furthermore, assume the following conditions.

() There exist continuous, nonnegative functions such that

  is nonincreasing and is non-decreasing in .() There exists with such that

Then (2) has one positive periodic solution with .

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

Theorem 7. Suppose that satisfies (A) and satisfies ()–() and ()-(). Then (2) has two different positive periodic solutions and with .

Theorem 8. Suppose that satisfies (A) and satisfies ()–(). Then (2) has two different positive periodic solutions and with .

To illustrate our results, we have selected the following singular equation:

here , , and is a given parameter. The corresponding results are also valid for the general case

with .

Corollary 9. Assume that satisfies (A) and . Then one has the following results.

(i)If , then for each , (18) has at least one positive periodic solution for all .(ii)If , then for each , (18) has at least one positive periodic solution for each ; here is some positive constant.(iii)If , then for each , (18) has at least two positive periodic solutions for each .(iv)If , then for each , with , (18) has at least one positive periodic solution for all .(v)If , then for each , with , (18) has at least one positive periodic solution for each .(vi)If , then for each , with , (18) has at least two positive periodic solutions for each .

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 () and () are satisfied. Furthermore, suppose that

() there exists a positive constant such that and

 here .

Then (2) has at least one positive -periodic solution.

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

Corollary 11 (see [31, Corollary 3.2]). Assume that conditions (B) and () and () are satisfied. Furthermore, assume that

() there exists a positive constant such that and

If , then (2) has at least one positive -periodic solution.

Corollary 12 (see [31, Example 3.5]). Suppose that satisfies (B) and , , then for each , with , one has the following:

(i)if , then (18) has at least one positive periodic solution for each ,(ii)if , then (18) has at least one positive -periodic solution for each , where is some positive constant.

The next results explore the case when .

Theorem 13 (see [31, Theorem 3.6]). Suppose that satisfies (B) and satisfies condition (). Furthermore, assume that

() there exists such that

If , then (2) has at least one positive -periodic solution.

Corollary 14 (see [31, Example 3.8]). Suppose that satisfies (B) and , then for each , with , one has the following:

(i)if, then (18) has at least one positive -periodic solution for each ,(ii)if, then (18) has at least one positive -periodic solution for each , where is some positive constant.

3. Singular Damped Equations

In this section, we recall some results on second-order singular damped differential equations

where and the nonlinearity . In particular, the nonlinearity may have a repulsive singularity at , which means that

First we recall some results on the linear damped equation

associated to periodic boundary conditions (4). As in the last section, we say that (25)-(4) is nonresonant when its unique -periodic solution is the trivial one. When (25)-(4) is nonresonant, as a consequence of Fredholm's alternative, the nonhomogeneous equation

admits a unique -periodic solution which can be written as

where is the Green's function of problem (25)-(4). We also assume that the following standing hypothesis is satisfied.

(C) The Green's function , associated with (25)-(4), is positive for all .

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

Lemma 15 (see [52, Theorem 2.2]). Assume that and the following two inequalities are satisfied:

where . Then (C) holds.

For the special case and , one criterion has been developed by Cabada and Cid in [40].

Theorem 16 (see [40, Theorem 5.1]). Assume that and . Suppose further that there exists such that

where

Then () holds.

Theorem 17 (see [35, Theorem 3.2]). Suppose that (25) satisfies (C) and

Furthermore, assume that there exists a constant such that

there exists a continuous function such that for all , there exist continuous, nonnegative functions , , and such that

 where is nonincreasing, is non-decreasing in , and is non-decreasing in , the following inequality holds:

 where

then (23) has at least one positive -periodic solution with .

Corollary 18 (see [35, Corollary 3.3]). Let the nonlinearity in (23) be

where , , is a positive parameter.

(i)If , then (23) has at least one positive periodic solution for each .(ii)If , then (23) has at least one positive periodic solution for each , where is some positive constant.

Corollary 19 (see [35, Corollary 3.4]). Let the nonlinearity in (23) be

where with , is a positive parameter. Then there exists a positive constant such that (23) has at least one positive -periodic solution for each .

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:

under the impulse conditions

where are the instants where the impulses occur and , are continuous.

Our result is presented as follows.

Theorem 20 (see [19, Theorem 1.1]). Assume that and the following conditions hold.

() is -periodic and for all .() is -periodic and .() There exist two constants such that for any ,

where and .

() For any ,

Then problem (38)-(39) has at least one solution.

Remark 21. In fact, it is not difficult to find some functions satisfying () and (). For example,

Let

with the inner product

The corresponding norm is defined by

Then is a Banach space (in fact it is a Hilbert space).

If , then is absolutely continuous and . In this case, is not necessarily valid for every and the derivative may exist some discontinuities. It may lead to impulse effects.

Following the ideas of [53], take and multiply the two sides of the equality

by and integrate from to , so we have

Note that since , one has

Combining with (47), we get

As a result, we introduce the following concept of a weak solution for problem (38)-(39).

Definition 22. One says that a function is a weak solution of problem (38)-(39) if

holds for any .

Define the functional by

for every . Clearly, is well defined on , continuously Gáteaux differentiable functional whose Gáteaux derivative is the functional , given by

for any . Moreover, it is easy to verify that is weakly lower semicontinuous. Indeed, if , and , then converges uniformly to on and on , and combining the fact that , one has

By the standard discussion, the critical points of 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 be a Banach space and let . Assume that there exist and an open neighborhood of such that and

Let

If satisfies the (PS)-condition, that is, a sequence in satisfying is bounded and as has a convergent subsequence, then is a critical value of and .

Next we consider -periodic solution for another impulsive singular problem:

under impulsive conditions

where , is -periodic, with ; are the instants where the impulses occur, and are continuous and .

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

Theorem 24 (see [29]). Assume that is -periodic. Then problem (56) has a positive -periodic weak solution if and only if .

From Theorem 24, if , then problem (52) does not have a positive -periodic weak solution. However, if the impulses happen, for this singular problem may exist a positive -periodic weak solution. Inspired by the above facts, our aim is to reveal a new existence result on positive -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 for some , but it is trivial when for all . For example, if problem (56)-(57) does not possess positive periodic solution when for all , then a positive periodic solution of problem (56)-(57) with for some 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:

() is -periodic and ;() there exist two constants such that for any ,

where .

Then problem (56)-(57) has at least a positive -periodic solution.

5. Singular Differential Systems

In this section, we will consider the system of Hill's equations

Here, and are -periodic in the variable , , and the nonlinearities can be singular at where .

Throughout, let . We are interested in establishing the existence of continuous -periodic solutions of the system (59), that is, and for all . Moreover, we are concerned with constant-sign solutions , by which we mean for all and , where is fixed. Note that positive solution, the usual consideration in the literature, is a special case of constant-sign solution when for .

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

Here, is the partial derivative of with respect to the second variable, and is a norm in . The particular case (60) occurs in the problem [36]

where the potential

and presents a singularity of the repulsive type, that is, uniformly in . 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

where 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 as in (60). In [45] it is also shown that the system

has a solution when , and . We will generalize the system (64) in Examples 46–48 to allow 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

Here, and is a norm in . Moreover, , , and are -periodic in , , and can be singular at .

To seek a -periodic solution of the system (65), we first obtain a solution of the following system of boundary value problems:

Then, set

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

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

To begin, let be Green's function of the boundary value problem

Throughout, we will assume that the functions are such that

(C1) the Hill's equation is nonresonant (i.e., the unique periodic solution is the trivial solution), and for all .

Note that Torres [46] has a result on that ensures that condition (C1) is satisfied. In fact, if , then (C1) holds if ; if is not a constant, then (C1) is valid if the norm of is bounded above by some specific constant.

Let be fixed. Define

We also let

We now present our main result which tackles (65) when the norm in is the norm or the norm.

Theorem 27. Assume that the following conditions hold for each ,

(); () let ; for any numbers with , the function is an -Carathéodory function, that is,

(i) the map is continuous for almost all ,(ii) the map is measurable for all ,(iii) for any , there exists such that implies for almost all ;

() for and ;() there exist and such that

()the norm is the norm where is fixed, and

 where

Then, (65) has a -periodic constant-sign solution such that

where

Theorem 27 is proved using Theorem 26; in fact we will seek a constant-sign solution of (66) in and then extend it to a -periodic constant-sign solution of (65) as in (67). Here, let be the closed convex set given by

where is chosen as in (75), and define the operator as

where

Clearly, a fixed point of is a solution of (66). We can show that ; that is, for each . Further, we can prove that is continuous and compact; that is, is bounded and is equicontinuous for any and . By Theorem 26, the system (66) has a constant-sign solution . Now, a -periodic constant-sign solution of (65) can be obtained as in (67).

Remark 28. The constants that appear in (C5) determine the upper bounds of the solution . Noting (75), we see that a smaller (bigger) gives a smaller (bigger) , and hence a smaller (bigger) set where the solution lies.

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

Theorem 29. Assume that (C1)–(C5) hold for each . The norm is the norm where is fixed. Then (65) has a -periodic constant-sign solution such that

where, for we have ,

Remark 30. A similar remark as Remark 28 also holds for Theorem 29. Moreover, we note that the upper bounds that fulfill (80)–(82) are independent of , thus the information of being a particular norm is not used. On the other hand, in Theorem 27, the upper bounds that satisfy (75) depend on . 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 for some and some .

Theorem 31. Suppose that

(C7) there exists such that .

Let and let . Assume that the following conditions hold for each (C1), (C3), (C4), and

(C8)there exist such that

 where for and for .

Further, let the following hold for each

(C9) the norm is the norm where is fixed, and

 where

Then, (65) has a -periodic solution such that

where

To derive Theorem 31, we let the closed convex set be

where and are chosen as in (87)–(89). Next, we define the operator as in (78) and show that Theorem 26 is applicable.

Remark 32. From the conclusion of Theorem 29, we see that the solution is “partially” of constant sign, in the sense that for , but may not be so for . Further, the constants that appear in (C8) determine the upper bounds of the solution . From (87) and (88), we see that a smaller (bigger) gives a smaller (bigger) , and hence a smaller (bigger) set where the solution lies.

Using similar arguments as in the derivation of Theorems 31 and 29 (in getting for and ), we obtain the following result.

Theorem 33. Suppose that (C7) hold. Let and let . Assume the following conditions hold for each (C1), (C3), (C4), and (C8). Then, (65) has a -periodic solution such that

where

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

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

Example 35. Consider (65) when

Fix , that is, we are seeking positive solutions. The corresponding Green's function has the explicit expression [36]

Condition (C1) is satisfied. By direct computation, we get and for . Thus, (C2) is fulfilled with

Moreover, we have

and so it is clear that (C4) and (C5) are satisfied with

Finally, we compute

Since for and , we check that (C6) holds for all .

All the conditions of Theorem 27 are satisfied, thus we conclude that the problem (65) with (94) has a positive -periodic solution such that

where (from (75))

We can also apply Theorem 29 to conclude that the problem (65) with (94) has a positive -periodic solution satisfying (100) and (from (82))

As mentioned in Remark 30, in general we cannot compare and . In fact, a direct calculation gives

5.2. Existence Results for (59)

In this section we will consider the general system of Hill's equations

Here, and are -periodic in the variable , , and the nonlinearities can be singular at where .

Once again, to obtain a -periodic solution of the system (104), we first seek a solution of the following system of boundary value problems:

The periodic solution is then given by

With being the Green's function of the boundary value problem (68), throughout we will assume that (C1) is satisfied. Moreover, for fixed and -periodic functions , we define

and also

For and , we denote the interval

A similar definition is valid for .

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

Theorem 36. Assume the following conditions hold for each (C1);

(C10) for any numbers with , the function is a -Carathéodory function, that is,

(i)the map is continuous for almost all ,(ii)the map is measurable for all ,(iii)for any , there exists such that implies for almost all ;

(C11)there exist , and -periodic functions with and for such that

 (here is the norm where is fixed);(C12).

Then, (104) has a -periodic constant-sign solution such that

where, for one has

(here is the norm of , likewise is the norm of ).

In proving Theorem 36, we actually seek a constant-sign solution of (105) in and then extend it to a -periodic constant-sign solution of (104) as in (106). Let be the closed convex set given by

where are chosen as in (112) and (113), and define the operator as

where

Clearly, a fixed point of 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 and that appear in (C11) determine the lower and upper bounds of the solution .

Theorem 38. Assume that the following conditions hold for each (C1), (C10), (C11), and (C12). Then, (104) has a -periodic constant-sign solution such that

where , and for all ,

Theorem 38 is obtained by similar arguments used in the derivation of Theorem 36, with a new defined as

where are chosen as in (118) and (119).

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

Our next result tackles the case when .

Theorem 40. Assume that the following conditions hold for each (C1), (C10),

(C13) there exist , and -periodic functions with , and for such that

 (here is the norm where is fixed);(C14).

Then, (104) has a -periodic constant-sign solution such that

where , and for all ,

The closed convex set used to get Theorem 40 is given by

where and satisfies (123).

Remark 41. As seen from (123), the functions , and that appear in (C13) determine the lower and upper bounds of the solution .

Finally, the next result tackles the case when .

Theorem 42. Assume that the following conditions hold for each (C1), (C10),

(C15) there exist and -periodic functions with , and for such that

 (here is the norm where is fixed);(C16); (C17) where

Then, (104) has a -periodic constant-sign solution such that

where are given by

Theorem 42 is obtained by considering the closed convex set

where are determined later as those given in (128).

Remark 43. As seen from (128), the functions and that appear in (C15) determine the lower and upper bounds of the solution .

We have so far established the results when (i) , (ii) , and (iii) for all . However, it could be that we only have for some and for some , which results in and for some . 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 . Assume the following:

(C18) conditions (C11) and (C12) hold for some ;(C19) conditions (C13) and (C14) hold for some ;(C20) conditions (C15), (C16), and (C17) hold for some ;

where . Then, (104) has a -periodic constant-sign solution such that

where satisfy

(a)(118) and (119) for ;(b), (123) for ;(c)(128) for .

Theorem 45. Let (C1) and (C10) hold for each . Assume that (C18)–(C20) hold with . Then, (104) has a -periodic constant-sign solution such that

where satisfy (112) and (113) for , and

where 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],

Clearly, the system (133) corresponds to (104) where and

We will assume that satisfy (C1). Note that condition (C10) is clearly satisfied. Further, let , that is, we are interested in positive periodic solutions of (133).

Example 46. Consider the system (133) with

Clearly, (C11) is satisfied with and . Thus, (C12) also holds since

Theorem 38 (or Theorem 36) is applicable and we conclude that the system (133) with (135) has a -periodic positive solution such that

where are such that

To illustrate numerically, suppose

Green's function is given in (95) and

Hence, (138) yields . Let , then (139) reduces to

which is satisfied by . Let , then from (137) we conclude that the system (133) with (140) has a -periodic positive solution such that

Example 47. Consider the system (133) with

Here, (C13) is satisfied with and . Subsequently, (C14) also holds since

Employing Theorem 40, we conclude that the system (133) with (144) has a -periodic positive solution such that

where , and from (123), we have for ,

Combining the inequalities, we see that should satisfy

Example 48. Consider the system (133) with

where

Obviously, (C15) is satisfied with and . Then, (C16) also holds since

Moreover, condition (C17) is simply (150). Hence, we conclude from Theorem 42 that the system (133) with (149) and (150) has a -periodic positive solution such that

where are given by

Remark 49. In [45], it is shown that (64) has a solution when and . As seen from Examples 46–48, we have generalized the system (64) to allow 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.

References

R. P. Agarwal, D. Franco, and D. O'Regan, “Singular boundary value problems for first and second order impulsive differential equations,” Aequationes Mathematicae, vol. 69, no. 1-2, pp. 83–96, 2005.

R. P. Agarwal, D. O'Regan, and B. Yan, “Multiple positive solutions of singular Dirichlet second-order boundary-value problems with derivative dependence,” Journal of Dynamical and Control Systems, vol. 15, no. 1, pp. 1–26, 2009.

A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Birkhäuser Boston Inc., Boston, Mass, USA, 1993.

A. Bahri and P. H. Rabinowitz, “A minimax method for a class of Hamiltonian systems with singular potentials,” Journal of Functional Analysis, vol. 82, no. 2, pp. 412–428, 1989.

I. V. Barteneva, A. Cabada, and A. O. Ignatyev, “Maximum and anti-maximum principles for the general operator of second order with variable coefficients,” Applied Mathematics and Computation, vol. 134, no. 1, pp. 173–184, 2003.

A. Boucherif and N. Daoudi-Merzagui, “Periodic solutions of singular nonautonomous second order differential equations,” NoDEA. Nonlinear Differential Equations and Applications, vol. 15, no. 1-2, pp. 147–158, 2008.

J. Chu and J. J. Nieto, “Recent existence results for second-order singular periodic differential equations,” Boundary Value Problems, vol. 2009, Article ID 540863, 20 pages, 2009.

P. Habets and L. Sanchez, “Periodic solutions of some Liénard equations with singularities,” Proceedings of the American Mathematical Society, vol. 109, no. 4, pp. 1035–1044, 1990.

X. Li and Z. Zhang, “Periodic solutions for damped differential equations with a weak repulsive singularity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 6, pp. 2395–2399, 2009.

I. Rachunkova and M. Tvrdý, “Existence results for impulsive second-order periodic problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 59, no. 1-2, pp. 133–146, 2004.

P. Yan and M. Zhang, “Higher order non-resonance for differential equations with singularities,” Mathematical Methods in the Applied Sciences, vol. 26, no. 12, pp. 1067–1074, 2003.

V. Bevc, J. L. Palmer, and C. Susskind, “On the design of the transition region of axisymmetric, magnetically focused beam valves,” Journal of the British Institution of Radio Engineers, vol. 18, no. 12, pp. 696–708, 1958.

J. Ren, Z. Cheng, and S. Siegmund, “Positive periodic solution for Brillouin electron beam focusing system,” Discrete and Continuous Dynamical Systems B, vol. 16, no. 1, pp. 385–392, 2011.

P. J. Torres, “Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system,” Mathematical Methods in the Applied Sciences, vol. 23, no. 13, pp. 1139–1143, 2000.

M. A. del Pino and R. F. Manásevich, “Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity,” Journal of Differential Equations, vol. 103, no. 2, pp. 260–277, 1993.

J. Chu and M. Zhang, “Rotation numbers and Lyapunov stability of elliptic periodic solutions,” Discrete and Continuous Dynamical Systems A, vol. 21, no. 4, pp. 1071–1094, 2008.

J. Chu and M. Li, “Twist periodic solutions of second order singular differential equations,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 830–838, 2009.

M. Zhang, “Periodic solutions of equations of Emarkov-Pinney type,” Advanced Nonlinear Studies, vol. 6, no. 1, pp. 57–67, 2006.

J. Sun and D. O'Regan, “Impulsive periodic solutions for singular problems via variational methods,” Bulletin of the Australian Mathematical Society, vol. 86, no. 2, pp. 193–204, 2012.

J. Sun, J. Chu, and H. Chen, “Periodic solution generated by impulses for singular differential equations,” In press

D. Bonheure and C. De Coster, “Forced singular oscillators and the method of lower and upper solutions,” Topological Methods in Nonlinear Analysis, vol. 22, no. 2, pp. 297–317, 2003.

J. L. Bravo and P. J. Torres, “Periodic solutions of a singular equation with indefinite weight,” Advanced Nonlinear Studies, vol. 10, no. 4, pp. 927–938, 2010.

J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,” Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.

A. Fonda and R. Toader, “Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach,” Journal of Differential Equations, vol. 244, no. 12, pp. 3235–3264, 2008.

A. Fonda and A. J. Ureña, “Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force,” Discrete and Continuous Dynamical Systems A, vol. 29, no. 1, pp. 169–192, 2011.

P. J. Torres, “Non-collision periodic solutions of forced dynamical systems with weak singularities,” Discrete and Continuous Dynamical Systems. Series A, vol. 11, no. 2-3, pp. 693–698, 2004.

M. Zhang, “Periodic solutions of damped differential systems with repulsive singular forces,” Proceedings of the American Mathematical Society, vol. 127, no. 2, pp. 401–407, 1999.

M. Zhang, “A relationship between the periodic and the Dirichlet BVPs of singular differential equations,” Proceedings of the Royal Society of Edinburgh A, vol. 128, no. 5, pp. 1099–1114, 1998.

A. C. Lazer and S. Solimini, “On periodic solutions of nonlinear differential equations with singularities,” Proceedings of the American Mathematical Society, vol. 99, no. 1, pp. 109–114, 1987.

W. B. Gordon, “Conservative dynamical systems involving strong forces,” Transactions of the American Mathematical Society, vol. 204, pp. 113–135, 1975.

J. Chu and P. J. Torres, “Applications of Schauder's fixed point theorem to singular differential equations,” Bulletin of the London Mathematical Society, vol. 39, no. 4, pp. 653–660, 2007.

J. Chu, P. J. Torres, and M. Zhang, “Periodic solutions of second order non-autonomous singular dynamical systems,” Journal of Differential Equations, vol. 239, no. 1, pp. 196–212, 2007.

J. Chu and M. Li, “Positive periodic solutions of Hill's equations with singular nonlinear perturbations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 1, pp. 276–286, 2008.

J. Chu and Z. Zhang, “Periodic solutions of second order superlinear singular dynamical systems,” Acta Applicandae Mathematicae, vol. 111, no. 2, pp. 179–187, 2010.

J. Chu, N. Fan, and P. J. Torres, “Periodic solutions for second order singular damped differential equations,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 665–675, 2012.

D. Franco and P. J. Torres, “Periodic solutions of singular systems without the strong force condition,” Proceedings of the American Mathematical Society, vol. 136, no. 4, pp. 1229–1236, 2008.

D. Jiang, J. Chu, and M. Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 282–302, 2005.

I. Rachunkova, M. Tvrdý, and I. Vrkoč, “Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems,” Journal of Differential Equations, vol. 176, no. 2, pp. 445–469, 2001.

P. J. Torres, “Weak singularities may help periodic solutions to exist,” Journal of Differential Equations, vol. 232, no. 1, pp. 277–284, 2007.

A. Cabada and J. Cid, “On the sign of the Green's function associated to Hill's equation with an indefinite potential,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 303–308, 2008.

Z. Cheng and J. Ren, “Periodic and subharmonic solutions for Duffing equation with a singularity,” Discrete and Continuous Dynamical Systems. Series A, vol. 32, no. 5, pp. 1557–1574, 2012.

J. Chu and Z. Zhang, “Periodic solutions of singular differential equations with sign-changing potential,” Bulletin of the Australian Mathematical Society, vol. 82, no. 3, pp. 437–445, 2010.

D. Franco and J. R. L. Webb, “Collisionless orbits of singular and non singular dynamical systems,” Discrete and Continuous Dynamical Systems A, vol. 15, no. 3, pp. 747–757, 2006.

R. Hakl and P. J. Torres, “On periodic solutions of second-order differential equations with attractive-repulsive singularities,” Journal of Differential Equations, vol. 248, no. 1, pp. 111–126, 2010.

X. Lin, D. Jiang, D. O'Regan, and R. P. Agarwal, “Twin positive periodic solutions of second order singular differential systems,” Topological Methods in Nonlinear Analysis, vol. 25, no. 2, pp. 263–273, 2005.

P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643–662, 2003.

M. Zhang and W. Li, “A Lyapunov-type stability criterion using norms,” Proceedings of the American Mathematical Society, vol. 130, no. 11, pp. 3325–3333, 2002.

M. Zhang, “Optimal conditions for maximum and antimaximum principles of the periodic solution problem,” Boundary Value Problems, vol. 2010, Article ID 410986, 26 pages, 2010.

A. Granas, R. B. Guenther, and J. W. Lee, “Some general existence principles in the Carathéodory theory of nonlinear differential systems,” Journal de Mathématiques Pures et Appliquées, vol. 70, no. 2, pp. 153–196, 1991.

A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, NY, USA, 2003.

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen, TheNetherlands, 1964.

R. Hakl and P. J. Torres, “Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7599–7611, 2011.

J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.

J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, “The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 12, pp. 4575–4586, 2010.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74, Springer-Verlag, New York, NY, USA, 1989.

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, “Constant-sign solutions of a system of integral equations: the semipositone and singular case,” Asymptotic Analysis, vol. 43, no. 1-2, pp. 47–74, 2005.

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, “Constant-sign solutions of a system of integral equations with integrable singularities,” Journal of Integral Equations and Applications, vol. 19, no. 2, pp. 117–142, 2007.

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, “Constant sign solutions for systems of Fredholm and Volterra integral equations: the singular case,” Acta Applicandae Mathematicae, vol. 103, no. 3, pp. 253–276, 2008.

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, “Constant-sign solutions for systems of singular integral equations of Hammerstein type,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 999–1025, 2009.

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, “Constant-sign solutions for singular systems of Fredholm integral equations,” Mathematical Methods in the Applied Sciences, vol. 33, no. 15, pp. 1783–1793, 2010.

R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, “Periodic constant-sign solutions for systems of Hill's equations,” Asymptotic Analysis, vol. 67, no. 3-4, pp. 191–216, 2010.

D. O'Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer, Dordrecht, The Netherlands, 1998.