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Existence for Singular Periodic Problems: A Survey of Recent Results
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.
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 , and gravitational forces . 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 . 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 . 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 , 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 .
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 , 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  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 . 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 . Let be a cone in and is a subset of , we write and .
Theorem 5 (see ). 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 .
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
Then (2) has at least one positive -periodic solution.
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 .
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 .
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:
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 , 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).
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  proved a famous result as follows.
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 ,
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  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 , 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  are not applicable to (59) with as in (60). In  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 .
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 ). 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  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
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.
Theorem 33. Suppose that (C7) hold. Let and let . Assume the following conditions hold for each