Existence for Singular Periodic Problems: A Survey of Recent Results

and Applied Analysis 3 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 (H 1 ) corresponds to the classical strong force condition, whichwas first introduced by Gordon in [30]. In fact, condition (H 1 ) 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 (H 1 ) and replace it by one weak force condition. Theorem 4 (see [33, Theorem 3.1]). Assume that (A) and (H 2 )-(H 3 ) are satisfied. Suppose further the following condition. (H 4 ) 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 solutionxwithx(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 ∂ K D = (∂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

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][22][23][24][25][26][27][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,  < 0, here  ≥ 1, which corresponds to a strong force condition, according to a terminology first introduced by Gordon [30].Moreover, if 0 <  < 1, 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][32][33][34][35][36][37][38][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.
When () =  2 , condition (A) is equivalent to 0 <  2 <  1 = (/) 2 and condition (B) is equivalent to 0 <  2 ≤  1 .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 K() denote the best Sobolev constant in the following inequality: The explicit formula for K() is where Γ is the gamma function, see [47,48].
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  and  ∈ Ω.Let  : Ω →  be a compact and continuous map.Then one of the following two conclusions holds.
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 (H 1 ) corresponds to the classical strong force condition, which was first introduced by Gordon in [30].In fact, condition (H 1 ) is only used when we try to obtain a prior lower bound.In Theorem 4, we will show that, for the case  * ≥ 0, we can remove the strong force condition (H 1 ) and replace it by one weak force condition.
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    = () ∩ .
To illustrate our results, we have selected the following singular equation: here ,  ∈ C[0, ], ,  > 0, and  ∈ R is a given parameter.
The corresponding results are also valid for the general case with ,  ∈ C[0, ].
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.
The next results explore the case when  * > 0.

Singular Damped Equations
In this section, we recall some results on second-order singular damped differential equations where ℎ,  ∈ C(R/Z, R) and the nonlinearity  ∈ C((R/Z) × (0, ∞) × R, R).In particular, the nonlinearity may have a repulsive singularity at  = 0, which means that lim  → 0 +  (, , ) = +∞, uniformly in (, ) ∈ R 2 .(24) 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  2 (, ) is the Green's function of problem ( 25)-( 4).We also assume that the following standing hypothesis is satisfied.
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.

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   ,  = 1, 2, . . ., −1 are the instants where the impulses occur and Our result is presented as follows.
Remark 21.In fact, it is not difficult to find some functions   satisfying ( 3 ) and ( 4 ).For example, Let with the inner product The corresponding norm is defined by Then  1  is a Banach space (in fact it is a Hilbert space).If  ∈  1   , then  is absolutely continuous and   ∈  2 ([0, ], R).In this case, Δ  () =   ( + ) −   ( − ) = 0 is not necessarily valid for every  ∈ (0, ) and the derivative   may exist some discontinuities.It may lead to impulse effects.
Following the ideas of [53], take  ∈  1  and multiply the two sides of the equality by  and integrate from 0 to , so we have Note that since   (0) −   () = 0, 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  ∈  1  is a weak solution of problem ( 38)- (39) if holds for any  ∈  1  .
The following version of the mountain pass theorem will be used in our argument.
Next we consider -periodic solution for another impulsive singular problem: under impulsive conditions where In 1987, Lazer and Solimini [29] proved a famous result as follows.
From Theorem 24, if ∫  0 () ≥ 0, 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   ̸ ≡ 0 for some 1 <  <  − 1, but it is trivial when   ≡ 0 for all 1 <  <  − 1.For example, if problem ( 56)-( 57) does not possess positive periodic solution when   ≡ 0 for all 1 <  <  − 1, then a positive periodic solution  of problem ( 56)-( 57) with   ̸ ≡ 0 for some 1 <  <  − 1 is called a positive periodic solution generated by impulses.
Our result is presented as follows.
Throughout, let  = ( 1 ,  2 , . . .,   ).We are interested in establishing the existence of continuous -periodic solutions  of the system (59), that is,  ∈ ((R))  and () = ( + ) for all  ∈ R.Moreover, we are concerned with constant-sign solutions , by which we mean     () ≥ 0 for all  ∈ R and 1 ≤  ≤ , where   ∈ {1, −1} is fixed.Note that positive solution, the usual consideration in the literature, is a special case of constant-sign solution when   = 1 for 1 ≤  ≤ .
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,  2 ℎ  is the partial derivative of ℎ  with respect to the second variable, and | ⋅ | is a norm in R  .The particular case (60) occurs in the problem [36] ü () + ∇   (,  ()) =  () , where the potential and ℎ presents a singularity of the repulsive type, that is, lim || → 0 ℎ(, ) = ∞ 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  > 0,  ∈ [0, 1), and  > 0. 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][57][58][59][60].This section is based on the work in [61].(60).In this section we will consider the system of Hill's equations  Our main tool is Schauder's fixed point theorem, which is stated below for completeness.

Existence Results for
Theorem 26 (see [62]).Let Ω be a convex subset of a Banach space  and  : Ω → Ω a continuous and compact map.Then  has a fixed point.
Remark 28.The constants   that appear in (C5) determine the upper bounds   of the solution    , 1 ≤  ≤ .Noting (75), we see that a smaller (bigger)   gives a smaller (bigger)   , 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   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     () ≤   ,  ∈ [0, ].

Theorem 29. Assume that (C1)-(C5) hold for each 1 ≤ 𝑖 ≤ 𝑛.
The norm | ⋅ | is the   norm where 1 ≤  ≤ ∞ is fixed.Then (65) has a -periodic constant-sign solution   ∈ ((R))  such that where, for 1 ≤  ≤  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.
Remark 32.From the conclusion of Theorem 29, we see that the solution   is "partially" of constant sign, in the sense that      () ≥ 0 for  ∈ , but may not be so for  ∈   .Further, the constants   that appear in (C8) determine the upper bounds   of the solution    , 1 ≤  ≤ .From (87) and (88), we see that a smaller (bigger)   gives a smaller (bigger)   , 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    ∈ Ω * for  ∈  and  ∈ Ω * ), we obtain the following result.
We will now present an example that illustrates Theorems 27 and 29.
All the conditions of Theorem 27 are satisfied, thus we conclude that the problem (65) with (94) has a positive 2periodic solution  = ( 1 ,  2 ) such that where (from (75)) We can also apply Theorem 29 to conclude that the problem (65) with (94) has a positive 2-periodic solution  = ( 1 ,  2 ) satisfying (100) and (from (82)) As mentioned in Remark 30, in general we cannot compare    and   .In fact, a direct calculation gives  = 1 : For  ≥   ≥ 0 and 1 ≤  ≤ , 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).
In proving Theorem 36, we actually seek a constant-sign solution of (105) in ([0, ])  and then extend it to a periodic constant-sign solution of (104) as in (106).Let Ω be the closed convex set given by where   ≥   > 0 are chosen as in (112) and ( 113 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    , 1 ≤  ≤ .
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], We will assume that  1 ,  2 ∈  1 [0, ] satisfy (C1).Note that condition (C10) is clearly satisfied.Further, let  1 =  2 = 1, that is, we are interested in positive periodic solutions of (133).(154) Remark 49.In [45], it is shown that (64) has a solution when  > 0,  ∈ [0, 1) and  > 0. As seen from Examples 46-48, we have generalized the system (64) to allow  to be zero or negative.