Recent Existence Results for Second-Order Singular Periodic Differential Equations

We present some recent existence results for second-order singular periodic differential equations. A nonlinear alternative principle of Leray-Schauder type, a well-known fixed point theorem in cones, and Schauder's fixed point theorem are used in the proof. The results shed some light on the differences between a strong singularity and a weak singularity.


Introduction
The main aim of this paper is to present some recent existence results for the positive Tperiodic solutions of second order differential equation x a t x f t, x e t , 1.1 where a t , e t are continuous and T -periodic functions. The nonlinearity f t, x is continuous in t, x and T -periodic in t. We are mainly interested in the case that f t, x has a repulsive singularity at x 0: It is well known that second order singular differential equations describe many problems in the applied sciences, such as the Brillouin focusing system 1 and nonlinear elasticity 2 . Therefore, during the last two decades, singular equations have attracted many researchers, and many important results have been proved in the literature; see, for Boundary Value Problems example, 3-10 . Recently, it has been found that a particular case of 1.1 , the Ermakov-Pinney equation plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations 11-13 . In the literature, two different approaches have been used to establish the existence results for singular equations. The first one is the variational approach 14-16 , and the second one is topological methods. Because we mainly focus on the applications of topological methods to singular equations in this paper, here we try to give a brief sketch of this problem. As far as the authors know, this method was started with the pioneering paper of Lazer and Solimini 17 . They proved that a necessary and sufficient condition for the existence of a positive periodic solution for equation x 1 x λ e t 1.4 is that the mean value of e is negative, e < 0, here λ ≥ 1, which is a strong force condition in a terminology first introduced by Gordon 18 . Moreover, if 0 < λ < 1, which corresponds to a weak force condition, they found examples of functions e with negative mean values and such that periodic solutions do not exist. Since then, the strong force condition became standard in the related works; see, for instance, 2, 8-10, 13, 19-21 , and the recent review 22 . With a strong singularity, the energy near the origin becomes infinity and this fact is helpful for obtaining the a priori bounds needed for a classical application of the degree theory. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity by topological methods is more recent but has also attracted many researchers 4, 6, 23-28 . In 27 , for the first time in this topic, Torres 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. Among topological methods, the method of upper and lower solutions 6, 29, 30 , degree theory 8, 20, 31 , some fixed point theorems in cones for completely continuous operators 25, 32-34 , and Schauder's fixed point theorem 27, 35, 36 are the most relevant tools.
In this paper, we select several recent existence results for singular equation 1.1 via different topological tools. The remaining part of the paper is organized as follows. In Section 2, some preliminary results are given. In Section 3, we present the first existence result for 1.1 via a nonlinear alternative principle of Leray-Schauder. In Section 4, the second existence result is established by using a well-known fixed point theorem in cones. The condition imposed on a t in Sections 3 and 4 is that the Green function G t, s associated with the linear periodic equations is positive, and therefore the results cannot cover the critical case, for example, when a is a constant, a t k 2 , 0 < k < λ 1 π/T, and λ 1 is the first eigenvalue of the linear problem with Dirichlet conditions x 0 x T 0. Different from Sections 3 and 4, the results obtained in Section 5, which are established by Schauder's fixed point theorem, can cover the critical case because we only need that the Green function G t, s is nonnegative. All results in Sections 3-5 shed some lights on the differences between a strong singularity and a weak singularity. Problems   3 To illustrate our results, in Sections 3-5, we have selected the following singular equation:

Boundary Value
x " a t x x −α μx β e t , 1.5 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 with b, c ∈ C 0, T . Some open problems for 1.5 or 1.6 are posed.
In this paper, we will use the following notation. Given ψ ∈ L 1 0, T , we write ψ 0 if ψ ≥ 0 for a.e. t ∈ 0, T , and it is positive in a set of positive measure. For a given function p ∈ L 1 0, T essentially bounded, we denote the essential supremum and infimum of p by p * and p * , respectively.

Preliminaries
Consider the linear equation x 0 x T , x 0 x T .

2.2
In Sections 3 and 4, we assume that A the Green function G t, s , associated with 2.1 -2.2 , is positive for all t, s ∈ 0, T × 0, T .
In Section 5, we assume that B the Green function G t, s , associated with 2.1 -2.2 , is nonnegative for all t, s ∈ 0, T × 0, T .
When a t k 2 , condition A is equivalent to 0 < k 2 < λ 1 π/T 2 and condition B is equivalent to 0 < k 2 ≤ λ 1 . In this case, we have

4 Boundary Value Problems
For a nonconstant function a t , there is an L p -criterion proved in 37 , which is given in the following lemma for the sake of completeness. Let K q denote the best Sobolev constant in the following inequality: The explicit formula for K q is where Γ is the Gamma function; see 21, 38

Existence Result (I)
In this section, we state and prove the first existence result for 1.1 . The proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in 39 . This part can be regarded as the scalar version of the results in 4 .

Lemma 3.1.
Assume Ω is a relatively compact subset of a convex set K in a normed space X. Let T : Ω → K be a compact map with 0 ∈ Ω. Then one of the following two conclusions holds: a T has at least one fixed point in Ω; b thereexist x ∈ ∂Ω and 0 < λ < 1 such that x λT x.
Suppose that a t satisfies (A) and f t, x satisfies the following.
H 3 There exists a positive number r such that σr γ * > 0, and Then for each e ∈ C R/T Z, R , 1.1 has at least one positive periodic solution x with x t > γ t for all t and 0 < x − γ < r.
Proof. The existence is proved using the Leray-Schauder alternative principle, together with a truncation technique. The idea is that we show that has a positive periodic solution x satisfying x t γ t > 0 for t and 0 < x < r. If this is true, it is easy to see that u t x t γ t will be a positive periodic solution of 1.1 with Since H 3 holds, we can choose n 0 ∈ {1, 2, · · · } such that 1/n 0 < σr γ * and Let N 0 {n 0 , n 0 1, · · · }. Consider the family of equations x a t x λf n t, x t γ t a t n , 3 Boundary Value Problems where λ ∈ 0, 1 , n ∈ N 0 , and

3.8
Problem 3.7 is equivalent to the following fixed point problem: where T n is defined by We claim that any fixed point x of 3.9 for any λ ∈ 0, 1 must satisfy x / r. Otherwise, assume that x is a fixed point of 3.9 for some λ ∈ 0, 1 such that x r. Note that

3.13
Boundary Value Problems 7 Thus we have from condition H 2 , for all t ∈ 0, T , 3.14 Therefore, This is a contradiction to the choice of n 0 , and the claim is proved. From this claim, the Leray-Schauder alternative principle guarantees that has a fixed point, denoted by x n , in B r {x ∈ X : x < r}, that is, equation x a t x f n t, x t γ t a t n 3.17 has a periodic solution x n with x n < r. Since x n t ≥ 1/n > 0 for all t ∈ 0, T and x n is actually a positive periodic solution of 3.17 .
In the next lemma, we will show that there exists a constant δ > 0 such that for n large enough.
In order to pass the solutions x n of the truncation equations 3.17 to that of the original equation 3.4 , we need the following fact:

3.21
The fact x n < r and 3.19 show that {x n } n∈N 0 is a bounded and equicontinuous family on 0, T . Now the Arzela-Ascoli Theorem guarantees that {x n } n∈N 0 has a subsequence, {x n k } k∈N , converging uniformly on 0, T to a function x ∈ X. Moreover, x n k satisfies the integral equation G t, s f s, x n k s γ s ds 1 n k .

3.22
Letting k → ∞, we arrive at  Proof. The lower bound in 3.18 is established using the strong force condition H 1 of f t, x . By condition H 1 , there exists c 0 ∈ 0, 1 small enough such that Boundary Value Problems 9 Take n 1 ∈ N 0 such that 1/n 1 ≤ c 0 and let N 1 {n 1 , n 1 1, · · · }. For n ∈ N 1 , let α n min 0≤t≤T x n t γ t , β n max 0≤t≤T x n t γ t .

3.25
We claim first that β n > c 0 for alln ∈ N 1 . Otherwise, suppose that β n ≤ c 0 for some n ∈ N 1 . Then from 3.24 , it is easy to verify f n t, x n t γ t > r a 1 . 3.26 Integrating 3.17 from 0 to T , we deduce that This is a contradiction. Thus β n > c 0 for n ∈ N 1 . Now we consider the minimum values α n . Let n ≥ n 1 . Without loss of generality, we assume that α n < c 0 , otherwise we have 3.18 . In this case, α n min 0≤t≤T x n t γ t x n t n γ t n < c 0 3.28 for some t n ∈ 0, T . As β n > c 0 , there exists c n ∈ 0, 1 without loss of generality, we assume t n < c n such that x n c n γ c n c 0 and x n t γ t ≤ c 0 for t n ≤ t ≤ c n . By 3.24 , it can be checked that f n t, x n t γ t > a t x n t γ t e t , ∀t ∈ t n , c n .

3.29
Thus for t ∈ t n , c n , we have x " n t γ " t > 0. As x n t n γ t n 0, x n t γ t > 0 for all t ∈ t n , c n and the function y n : x n γ is strictly increasing on t n , c n . We use ξ n to denote the inverse function of y n restricted to t n , c n .
In order to prove 3.18 in this case, we first show that, for n ∈ N 1 , x n t γ t ≥ 1 n .

3.30
Otherwise, suppose that α n < 1/n for some n ∈ N 1 . Then there would exist b n ∈ t n , c n such that x n b n γ b n 1/n and Multiplying 3.17 by x n t γ t and integrating from b n to c n , we obtain a t x n t − a t n x n t γ t dt.

3.32
By the facts x n < r and x n ≤ H, one can easily obtain that the right side of the above equality is bounded. As a consequence, there exists L > 0 such that 1/n f ξ n y , y dy ≤ L.

3.33
On the other hand, by the strong force condition H 1 , we can choose n 2 ∈ N 1 large enough such that x n t a t x n t − a t n x n t γ t dt.

3.35
We notice that the estimate 3.30 is used in the second equality above . In the same way, one may readily prove that the right-hand side of the above equality is bounded. On the other hand, if n ∈ N 2 , by H 1 , if α n → 0 . Thus we know that α n ≥ δ for some constant δ > 0.
From the proof of Theorem 3.2 and Lemma 3.3, we see that the strong force condition H 1 is only used when we prove 3.18 . From the next theorem, 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 3.4. Assume that (A) and (H 2 )-( H 3 ) are satisfied. Suppose further that
Then for each e t with γ * ≥ 0, 1.1 has at least one positive periodic solution x with x t > γ t for all t and 0 < x − γ < r.
Proof. We only need to show that 3.18 is also satisfied under condition H 4 and γ * ≥ 0. The rest parts of the proof are in the same line of Theorem 3.2. Since H 4 holds, there exists a continuous function φ r γ * 0 such that f t, x ≥ φ r γ * t for all t, x ∈ 0, T × 0, r γ * . Let x r γ * be the unique periodic solution to the problems 2.1 -2.2 with h φ r γ * . That is G t, s φ r γ * s ds.

3.37
Then we have ii if α ≥ 1, β ≥ 1, then for each e ∈ C R/T Z, R , 1.5 has at least one positive periodic solution for each 0 < μ < μ 1 , here μ 1 is some positive constant.
Proof. We apply Theorems 3.2 and 3.4. Take then H 2 is satisfied, and the existence condition H 3 becomes for some r > 0. Note that condition H 1 is satisfied when α ≥ 1, while H 4 is satisfied when α > 0. So 1.5 has at least one positive periodic solution for

Existence Result (II)
In this section, we establish the second existence result for 1.1 using a well-known fixed point theorem in cones. We are mainly interested in the superlinear case. This part is essentially extracted from 24 . First we recall this fixed point theorem in cones, which can be found in 40 . 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.
be a completely continuous operator such that b There exists υ ∈ K \ {0} such that x / Tx λυ for all x ∈ ∂ K Ω 2 and all λ > 0.
Then T has a fixed point in Boundary Value Problems

13
In applications below, we take X C 0, T with the supremum norm · and define that a t satisfies (A) and f t, x satisfies (H 2 )-(H 3 ). Furthermore, assume that Then 1.1 has one positive periodic solution x with r < x − γ ≤ R.
Proof. As in the proof of Theorem 3.2, we only need to show that 3.4 has a positive periodic solution u ∈ X with u t γ t > 0 and r < u ≤ R.
Let K be a cone in X defined by 4.2 . Define the open sets and the operator T : For each x ∈ Ω 2 K \ Ω 1 K , we have r ≤ ||x|| ≤ R. Thus 0 < σr γ * ≤ x t γ t ≤ R γ * for all t ∈ 0, T . Since f : 0, T × σr γ * , R γ * → 0, ∞ is continuous, then the operator K → K is well defined and is continuous and completely continuous. Next we claim that: i Tx ≤ x for x ∈ ∂ K Ω 1 , and ii there exists υ ∈ K \ {0} such that x / Tx λυ for all x ∈ ∂ K Ω 2 and all λ > 0.

Boundary Value Problems
We start with i . In fact, if x ∈ ∂ K Ω 1 , then x r and σr γ * ≤ x t γ t ≤ r γ * for all t ∈ 0, T . Thus we have

4.8
Hence min 0≤t≤T x t > σR, this is a contradiction and we prove the claim. Now Theorem 4.1 guarantees that T has at least one fixed point Combined Theorem 4.2 with Theorems 3.2 or 3.4, we have the following two multiplicity results.  ii if α > 0, then for each e ∈ C R/T Z, R with γ * ≥ 0, 1.5 has at least two positive periodic solutions for each 0 < μ < μ 1 .
Proof. Take g 1 x x −α , h 1 x μx β . Then H 5 is satisfied and the existence condition H 6 becomes Since β > 1, it is easy to see that the right-hand side goes to 0 as R → ∞. Thus, for any given 0 < μ < μ 1 , it is always possible to find such R r that 4.9 is satisfied. Thus, 1.5 has an additional positive periodic solution x.

Existence Result (III)
In this section, we prove the third existence result for 1.1 by Schauder's fixed point theorem. We can cover the critical case because we assume that the condition B is satisfied. This part comes essentially from 35 , and the results for the vector version can be found in 4 .
Obviously, Ω is a closed convex set. Next we prove T Ω ⊂ Ω.
In fact, for each x ∈ Ω, using that G t, s ≥ 0 and condition H 4 ,

5.3
In conclusion, T Ω ⊂ Ω. By a direct application of Schauder's fixed point theorem, the proof is finished.
As an application of Theorem 5.1, we consider the case γ * 0. The following corollary is a direct result of Theorem 5.1.
If γ * 0, then 1.1 has at least one positive T -periodic solution.
Proof. We apply Corollary 3.5 and follow the same notation as in the proof of Corollary 3.5. Then H 2 and H 4 are satisfied, and the existence condition H 8 becomes Therefore, 5.5 becomes for some R > 0.
Boundary Value Problems 17 So 1.5 has at least one positive T -periodic solution for 5.8 Note that μ 2 ∞ if α β < 1 − α 2 and μ 2 < ∞ if α β ≥ 1 − α 2 . We have the desired results i and ii . The next results explore the case when γ * > 0.
Proof. We follow the same strategy and notation as in the proof of Theorem 5.1. Let R be the positive constant satisfying H 9 and r γ * , then R > r > 0 since R > γ * . Next we prove T Ω ⊂ Ω.
For each x ∈ Ω, by the nonnegative sign of G t, s and f t, x , we have

5.11
In conclusion, T Ω ⊂ Ω, and the proof is finished by Schauder's fixed point theorem.

Boundary Value Problems
Corollary 5.6. Suppose that a t satisfies (B) and α, β ≥ 0, then for each e ∈ C R/T Z, R with γ * > 0, one has the following: i if α β < 1, then 1.5 has at least one positive T -periodic solution for each μ ≥ 0; ii if α β ≥ 1, then 1.5 has at least one positive T -periodic solution for each 0 ≤ μ < μ 3 , where μ 3 is some positive constant.