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.

The main aim of this paper is to present some recent existence results for the positive -periodic solutions of second order differential equation

where are continuous and -periodic functions. The nonlinearity is continuous in and -periodic in . We are mainly interested in the case that has a repulsive singularity at :

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 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

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.

To illustrate our results, in Sections 3–5, we have selected the following singular equation:

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.

Consider the linear equation

with periodic boundary conditions

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

For a nonconstant function , there is an -criterion proved in [37], which is given in the following lemma 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 [21, 38]

Lemma 2.1.

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 .

Throughout this paper, we define the function by

which corresponds to the unique -periodic solution of

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.

(H₁)There exist constants and such that

(H₂)There exist continuous, nonnegative functions and such that

is nonincreasing and is nondecreasing in .

(H₃)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.

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 satisfying for and If this is true, it is easy to see that will be a positive periodic solution of (1.1) with since

Since () holds, we can choose such that and

Let . Consider the family of equations

where and

Problem (3.7) is equivalent to the following fixed point problem:

where is defined by

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

By the choice of , . Hence, for all , we have

Therefore,

Thus we have from condition (), for all ,

Therefore,

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 fixed point, denoted by , in , that is, equation

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:

for some constant and for all . To this end, by the periodic boundary conditions, for some . Integrating (3.17) from 0 to , we obtain

Therefore

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

Letting , we arrive at

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.

The lower bound in (3.18) is established using the strong force condition () of . By condition (), there exists small enough such that

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

Integrating (3.17) from 0 to , we deduce that

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,

for some . As , there exists (without loss of generality, we assume ) such that and for By (3.24), it can be checked that

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

Multiplying (3.17) by and integrating from to , we obtain

By the facts and one can easily obtain that the right side of the above equality is bounded. As a consequence, there exists such that

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

(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 by (),

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

(H₄)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.

We only need to show that (3.18) is also satisfied under condition () and The rest parts of the proof are in the same line of Theorem 3.2. Since () holds, there exists a continuous function such that for all . Let be the unique periodic solution to the problems (2.1)–(2.2) with . That is

Then we have

here

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.

We apply Theorems 3.2 and 3.4. Take

then () is satisfied, and the existence condition () becomes

for some . Note that condition () is satisfied when , while () is satisfied when . So (1.5) has at least one positive periodic solution for

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

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]).

Let be a Banach space and a cone in . Assume are open bounded subsets of with Let

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

(H₅)there exist continuous nonnegative functions such that

is nonincreasing and is nondecreasing in

(H₆)there exists with such that

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

and the operator by

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.

Take Then () is satisfied and the existence condition () becomes

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 .

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

(H₇)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 (),

On the other hand, by conditions () and (), we have

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

(H₈)there exists a positive constant such that and

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.

We apply Corollary 3.5 and follow the same notation as in the proof of Corollary 3.5. Then () and () are satisfied, and the existence condition () becomes

for some with . Note that

Therefore, (5.5) becomes

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

(H₉)there exists such 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

On the other hand, by () 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.

We apply Theorem 5.5 and follow the same notation as in the proof of Corollary 3.5. Then () is satisfied, and the existence condition () becomes

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).

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.

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