- Research Article
- Open Access

# Recent Existence Results for Second-Order Singular Periodic Differential Equations

- Jifeng Chu
^{1, 2}Email author and - JuanJ Nieto
^{3}

**2009**:540863

https://doi.org/10.1155/2009/540863

© J. Chu and J. J. Nieto. 2009

**Received:**12 February 2009**Accepted:**29 April 2009**Published:**8 June 2009

## Abstract

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.

## Keywords

- Periodic Solution
- Existence Result
- Fixed Point Theorem
- Existence Condition
- Topological Method

## 1. Introduction

plays an important role in studying the Lyapunov stability of periodic solutions of Lagrangian equations [11–13].

is that the mean value of is negative, , here , which is a strong force condition in a terminology first introduced by Gordon [18]. Moreover, if , which corresponds to a weak force condition, they found examples of functions 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 in Sections 3 and 4 is that the Green function associated with the linear periodic equations is positive, and therefore the results cannot cover the critical case, for example, when is a constant, , , and is the first eigenvalue of the linear problem with Dirichlet conditions . 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 is nonnegative. All results in Sections 3–5 shed some lights on the differences between a strong singularity and a weak singularity.

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

with . Some open problems for (1.5) or (1.6) are posed.

In this paper, we will use the following notation. Given , we write if for a.e. and it is positive in a set of positive measure. For a given function essentially bounded, we denote the essential supremum and infimum of by and , respectively.

## 2. Preliminaries

In Sections 3 and 4, we assume that

(A)the Green function associated with (2.1)–(2.2), is positive for all .

In Section 5, we assume that

(B)the Green function associated with (2.1)–(2.2), is nonnegative for all

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

The explicit formula for is

where is the Gamma function; see [21, 38]

Lemma 2.1.

Obviously, and .

## 3. 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 is a relatively compact subset of a convex set in a normed space . Let be a compact map with . Then one of the following two conclusions holds:

(a) has at least one fixed point in

(b)thereexist and such that

Theorem 3.2.

Suppose that satisfies (A) and satisfies the following.

(H_{1})There exist constants
and
such that

(H_{2})There exist continuous, nonnegative functions
and
such that

is nonincreasing and is nondecreasing in .

(H_{3})There exists a positive number
such that
and

Then for each , (1.1) has at least one positive periodic solution with for all and .

Proof.

Since ( ) holds, we can choose such that and

We claim that any fixed point of (3.9) for any must satisfy . Otherwise, assume that is a fixed point of (3.9) for some such that . Note that

This is a contradiction to the choice of and the claim is proved.

From this claim, the Leray-Schauder alternative principle guarantees that

has a periodic solution with . Since for all and is actually a positive periodic solution of (3.17).

In the next lemma, we will show that there exists a constant such that

for large enough.

In order to pass the solutions of the truncation equations (3.17) to that of the original equation (3.4), we need the following fact:

The fact and (3.19) show that is a bounded and equicontinuous family on . Now the Arzela-Ascoli Theorem guarantees that has a subsequence, , converging uniformly on to a function . Moreover, satisfies the integral equation

where the uniform continuity of on is used. Therefore, is a positive periodic solution of (3.4).

Lemma 3.3.

There exist a constant and an integer such that any solution of (3.17) satisfies (3.18) for all .

Proof.

Take such that and let . For , let

We claim first that . Otherwise, suppose that for some . Then from (3.24), it is easy to verify

This is a contradiction. Thus .

Now we consider the minimum values . Let . Without loss of generality, we assume that , otherwise we have (3.18). In this case,

Thus for , we have As , for all and the function is strictly increasing on . We use to denote the inverse function of restricted to .

In order to prove (3.18) in this case, we first show that, for ,

Otherwise, suppose that for some . Then there would exist such that and

On the other hand, by the strong force condition ( ), we can choose large enough such that

for all . So (3.30) holds for

Finally, multiplying (3.17) by and integrating from to , we obtain

if Thus we know that for some constant .

From the proof of Theorem 3.2 and Lemma 3.3, we see that the strong force condition ( ) is only used when we prove (3.18). From the next theorem, we will show that, for the case , we can remove the strong force condition ( ), and replace it by one weak force condition.

Theorem 3.4.

Assume that ( ) and ( )–( ) are satisfied. Suppose further that

(H_{4})for each constant
, there exists a continuous function
such that
for all
.

Then for each with (1.1) has at least one positive periodic solution with for all and .

Proof.

Corollary 3.5.

Assume that satisfies ( ) and . Then

(i)if then for each (1.5) has at least one positive periodic solution for all ;

(ii)if , then for each (1.5) has at least one positive periodic solution for each here is some positive constant.

(iii)if , then for each with (1.5) has at least one positive periodic solution for all ;

(iv)if , then for each with (1.5) has at least one positive periodic solution for each .

Proof.

Note that if and if . Thus we have (i)–(iv).

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

Theorem 4.1 (see [40]).

be a completely continuous operator such that

(a) for

(b)There exists such that and all

Then has a fixed point in

In applications below, we take with the supremum norm and define

Theorem 4.2.

Suppose that satisfies ( ) and satisfies ( )–( ). Furthermore, assume that

is nonincreasing and is nondecreasing in

Then (1.1) has one positive periodic solution with .

Proof.

As in the proof of Theorem 3.2, we only need to show that (3.4) has a positive periodic solution with and

Let be a cone in defined by (4.2). Define the open sets

For each , we have . Thus for all Since is continuous, then the operator is well defined and is continuous and completely continuous. Next we claim that:

(i) for and

(ii)there exists such that and all

We start with (i). In fact, if then and for all Thus we have

Next we consider (ii). Let then Next, suppose that there exists and such that Since then for all As a result, it follows from ( ) and ( ) that, for all

Hence this is a contradiction and we prove the claim.

Now Theorem 4.1 guarantees that has at least one fixed point with Note by (4.7).

Combined Theorem 4.2 with Theorems 3.2 or 3.4, we have the following two multiplicity results.

Theorem 4.3.

Suppose that satisfies ( ) and satisfies ( )–( ) and ( )–( ). Then (1.1) has two different positive periodic solutions and with .

Theorem 4.4.

Suppose that satisfies ( ) and satisfies ( )–( ). Then (1.1) has two different positive periodic solutions and with .

Corollary 4.5.

Assume that satisfies ( ) and . Then

(i)if , then for each (1.5) has at least two positive periodic solutions for each ;

(ii)if , then for each with (1.5) has at least two positive periodic solutions for each .

Proof.

Since , it is easy to see that the right-hand side goes to 0 as . Thus, for any given , it is always possible to find such that (4.9) is satisfied. Thus, (1.5) has an additional positive periodic solution .

## 5. 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].

Theorem 5.1.

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

(H_{7})there exists a positive constant
such that
and
here

Then (1.1) has at least one positive -periodic solution.

Proof.

A -periodic solution of (1.1) is just a fixed point of the map defined by (4.6). Note that is a completely continuous map.

Let be the positive constant satisfying ( ) and Then we have . Now we define the set

Obviously, is a closed convex set. Next we prove

In fact, for each , using that and condition ( ),

In conclusion, . 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 . The following corollary is a direct result of Theorem 5.1.

Corollary 5.2.

Assume that conditions ( ) and ( ), ( ) are satisfied. Furthermore, assume that

If then (1.1) has at least one positive -periodic solution.

Corollary 5.3.

Suppose that satisfies ( ) and , , then for each with one hasthe following:

(i)if then (1.5) has at least one positive periodic solution for each .

(ii)if then (1.5) has at least one positive -periodic solution for each where is some positive constant.

Proof.

for some .

So (1.5) has at least one positive -periodic solution for

Note that if and if . We have the desired results (i) and (ii).

Remark 5.4.

The validity of (ii) in Corollary 5.3 under strong force conditions remains still open to us. Such an open problem has been partially solved by Corollary 3.5. However, we do not solve it completely because we need the positivity of in Corollary 3.5, and therefore it is not applicable to the critical case. The validity for the critical case remains open to the authors.

The next results explore the case when .

Theorem 5.5.

Suppose that satisfies ( ) and satisfies condition ( ). Furthermore, assume that

If then (1.1) has at least one positive -periodic solution.

Proof.

We follow the same strategy and notation as in the proof of Theorem 5.1. Let be the positive constant satisfying ( ) and then since . Next we prove

For each , by the nonnegative sign of and , we have

In conclusion, and the proof is finished by Schauder's fixed point theorem.

Corollary 5.6.

Suppose that satisfies ( ) and , then for each with , one has the following:

(i)if then (1.5) has at least one positive -periodic solution for each

(ii)if , then (1.5) has at least one positive -periodic solution for each where is some positive constant.

Proof.

Note that if and if . We have the desired results (i) and (ii).

## Declarations

### Acknowledgments

The authors express their thanks to the referees for their valuable comments and suggestions. The research of J. Chu is supported by the National Natural Science Foundation of China (Grant no. 10801044) and Jiangsu Natural Science Foundation (Grant no. BK2008356). The research of J. J. Nieto is partially supported by Ministerio de Education y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.

## Authors’ Affiliations

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