Abstract and Applied Analysis

Volume 2014 (2014), Article ID 527162, 10 pages

http://dx.doi.org/10.1155/2014/527162

Research Article

Multiplicity Results for Variable-Coefficient Singular Third-Order Differential Equation with a Parameter

Zhibo Cheng¹ and Yun Xin²

¹School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China

²College of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, China

Received 24 March 2014; Accepted 9 June 2014; Published 19 June 2014

Academic Editor: Narcisa C. Apreutesei

Copyright © 2014 Zhibo Cheng and Yun Xin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate a class of variable coefficients singular third-order differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. By applications of Green’s function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.

1. Introduction

Generally speaking, differential equations with singularities have been considered from the very beginning of the discipline. The main reason is that singular forces are ubiquitous in applications, the most obvious examples being gravitational and electromagnetic forces. In 1965, Ding [1] discussed the Brillouin electron beam focusing system:

and obtained the existence of positive periodic solution for the model if .

Ding’s work has attracted the attention of many specialists in differential equations. More recently, the method of lower and upper solutions, Poincaré-Birkhoff twist theorem, Mawhin’s topological degree theorem, Schauder’s fixed point theorem, and Krasnoselskii fixed point theorem in a cone have been employed to investigate the existence of positive periodic solutions of singular second order differential equations (see, e.g., [2–13]). For example, in 2007, Torres [10] studied singular forced semilinear differential equation:

By Schauder’s fixed point theorem, the author has shown that the additional assumption of a weak singularity enabled the obtention of new criteria for the existence of periodic solutions. Afterwards, Wang [13] investigated the existence and multiplicity of positive periodic solutions of the singular systems (2) by Krasnoselskii fixed point theorem. The conditions he presented to guarantee the existence of positive periodic solutions are beautiful.

At the beginning, most of work concentrated on second-order singular differential equations, as in the references we mentioned above. Recently, there have been published some results on third-order singular differential equation (see [14–19]). For example, in [14], Chu and Zhou considered the following third-order singular differential equation:

with periodic boundary conditions . Here, is a positive constant and nonlinearity may be singular at . They discussed (3) by transforming it into a first-order equation and a second-order equation. Restricted by Green’s function of the second-order differential equation, they obtained existence theorem of periodic solutions for (3) in a small range of , and, to be concrete, . Afterward, Li [16] investigated the third-order ordinary differential equation:

where is -periodic in , and may be singular at . By applying a fixed point theorem in cones, the author obtained existence results of positive -periodic solutions for (4). Recently, Ren et al. [19] studied the third-order singular nonlinear differential equation:

where takes positive values. Using Green’s function for third-order differential equation and some fixed-point theorems, that is, Leray-Schauder alternative principle and Schauder’s fixed point theorem, they established three new existence results of positive periodic solutions for (5).

In the above papers, the authors investigated singular third-order equations with constant coefficients. However, the study on the singular third-order equation with variable coefficients is relatively infrequent. Motivated by Torres et al. [10, 13, 14, 16, 19], in this paper, we consider the singular third-order differential equation with variable coefficients:

where is a positive parameter, and may take positive value or negative value. are -periodic functions; , are -periodic continuous scalar functions in . The nonlinear term of (6) can be with a singularity at origin; that is,

It is said that (6) is of repulsive type (resp., attractive type) if (resp., ) as .

As far as we know, studies on third-order differential equation with variable coefficients are rather infrequent, especially those focused on the research of singular third-order differential equations with variable coefficients. The main difficulty lies in the calculation of Green’s function of the third-order differential equation with variable coefficients, being more complicated than in the constant-coefficient case. Therefore, in Section 2, we first study Green’s function of the above mentioned third-order differential equation. In Section 3, we define a cone and discuss several properties of the equivalent operator on the cone. In order to simplify the proof in Section 3, we establish a series of lemmas and corollaries to estimate the operator. All the corollaries are the corresponding results for taking negative values. In Section 4, by employing Green’s function and Krasnoselskii fixed point theorem, we state and prove the existence of positive periodic solutions for singular third-order differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The result is applicable to the case of a strong singularity as well as the case of a weak singularity. Our results improve and extend the results in [10, 14, 19].

2. Green’s Function of Third-Order Differential Equation

Let with the maximum norm . Obviously, is a Banach space. For a given function , we denote the essential supremum and infimum by and , if they exist.

Firstly, we consider

where is an -periodic function. Obviously, the calculation of Green’s function of (8) is very complicated, so, by analysis of the third-order differential equation (8), we consider only the following two cases.

Case 1. There exist differentiable -periodic functions and and a positive real constant such that , , and . Then, (8) is transformed into

Then, solution of (9) is written as

where

Solution of (10) is written as

Next, we will consider , which can be found in [20].

Suppose that

where

Then there are continuous -periodic functions and such that and

Suppose (14) holds and ; then the equation

has an -periodic solution. Moreover, the periodic solution can be expressed by

where

with

Let and . If

then we have

Moreover,

Therefore, we know that the solution of (8) is written as

Therefore, by writing

we can get

Lemma 1. Assume that (14) and (21) hold. Then for all .

Proof. From (23), we know . Since , from (25) we see that for all .

Case 2. There exist an -periodic differentiable function and a positive real constant such that , , and . Then, (8) is transformed into

Then, solution of (27) is written as

Solution of (28) is written as

By the following lemma, which can be found in [21], we will consider the sign of .

Lemma 2 (see [21]). Let us define

where is the Gamma function. Assume that and for some . If

where if and if , then for all .

Similarly to (26), we know that the solution of (8) can be written as

Here, . And we get the following Lemma.

Lemma 3. Assume for some . and (32) hold. Then, for all .

Proof. From Lemma 10, we know . Since , from (25) we see that for all .

3. Preliminary Lemmas

Firstly, we establish the existence of positive periodic solutions for third-order differential equation (6) by using fixed point theorem, which can be found in [22].

Lemma 4 (see [22]). Let be a Banach space and a cone in . Assume that , are bounded open subsets of with , and let

be a completely continuous operator such that either

(i) and ; or(ii) and .

Then has a fixed point in .

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:

 () is a scalar continuous function defined for , and for .  () .  () for .

Under Lemmas 1 and 3, we always denote

Obviously, and .

Case 1. Let . The following are the main existence results in this section.

Define the cone in by

We take with . Also, for , let

Define the operator as follows:

where is nonnegative and is nonnegative. If takes negative values, we will choose so that is nonnegative. This is possible because or .

Now, if is a fixed point of in , then is a positive solution of (6). Also note that each component of any nonnegative periodic solution is strictly positive for all because of the positiveness of the Green functions and assumptions and . We now look at several properties of the operator.

Lemma 5. Assume that (14), (21), , and hold and . Then, and is completely continuous.

Proof. If , then , and then is defined. Now we have that

Thus, . It is easy to verify that is completely continuous.

If takes negative values, we need to choose appropriate domains so that become nonnegative. The proof of and in Lemma 6 is the same as in Lemma 5.

Lemma 6. Assume that (14), (21), , and hold.

(a)If  , there is a such that if , then is defined on , and is completely continuous.(b)If  , there is a such that if , then is defined on , , and is completely continuous.

Proof. We split into the two terms and . The first term is always nonnegative and used to carry out the estimates of the operator in the lemma and corollaries in this section. We will make the second term nonnegative by choosing appropriate domains of . The choice of the even split of here is not necessarily optimal in terms of obtaining maximal -intervals for the existence of periodic solutions of the equation.

Noting that is positive on , , implies that there exists a constant such that

for . Now for and , note that

and, therefore, we have, for ,

Thus, it is clear that in (38) is well defined and positive, and now it is easy to see that and is completely continuous.

On the other hand, if  , there is an such that

for . Now let . Then for , we have that , and, therefore,

Now, in (38) is well defined and positive. It is clear that and is completely continuous.

Now let

Lemma 7. Assume that (14), (21), , and hold and . Let and if there exist such that

for , then the following inequality holds:

Proof. From the definition of , it follows that

If takes negative values, we need to adjust and in Lemma 6 to guarantee that is nonnegative.

Corollary 8. Assume that (14), (21), , and hold.

(a)If  , then Lemma 7 holds assuming that , where is defined by Lemma 6.(b)If  , then Lemma 7 holds assuming that , where is defined by Lemma 6.

Proof. We split into the two terms and . By choosing and in Lemma 6, become nonnegative. The estimate in Corollary 8 can be carried out by the first terms as in Lemma 7.

Let be the function given by

It is easy to see that is a nondecreasing function on . The following lemma is essentially the same as Lemma  2.8 in [23].

Lemma 9 (see [23]). Assume holds. If   exists (which can be infinity), then exists and .

Lemma 10. Assume that (14), (21), , and hold and . Let and if there exists an such that

then

where the constant .

Proof. From the definition of , we have, for ,

If takes negative values, we need to restrict the domain of to guarantee that is nonnegative.

Corollary 11. Assume that (14), (21), , and hold. If  , Lemma 10 holds assuming that , where is defined by Lemma 6.

Proof. If we choose defined in Lemma 6, then is well defined and is nonnegative, and Corollary 11 can be shown in the same way as Lemma 10.

The conclusions of Lemmas 7 and 10 are based on the inequality assumptions between and . If these assumptions are not necessarily true, we will have the following results.

Lemma 12. Assume that (14), (21), , and hold and . Let . Then

for all , where .

Proof. If , then , for . Therefore for . By the definition of , we have

Now we consider the cases that may take negative values. We need to restrict the domain of to guarantee that is nonnegative. is used to carry out the estimates as Lemma 12.

Corollary 13. Assume that (14), (21), , and hold.

(a)If  , then Lemma 12 holds assuming that , where is defined by Lemma 6.(b)If  , then Lemma 12 holds assuming that , where is defined by Lemma 6.

Proof. By selecting and defined in Lemma 6, is well defined and is nonnegative, and then Corollary 13 can be shown as Lemma 12.

Lemma 14. Assume that (14), (21), , and hold and ; let . Then

for all , where .

Proof. If , then . Therefore, for . Thus we have that

Again, if takes negative values, we need to restrict and to guarantee is nonnegative.

Corollary 15. Assume that (14), (21), , and hold.

(a)If  , then Lemma 14 holds assuming that , where is defined by Lemma 6.(b)If  , then Lemma 14 holds assuming that , where is defined by Lemma 6.

Proof. By selecting and defined in Lemma 6, is well defined and is nonnegative, and then Corollary 15 can be shown as Lemma 14.

4. Main Results

In this section, we present out main results for the existence and multiplicity of positive periodic solutions of singular third-order equation of repulsive type (6). We state our theorems as follows.

Theorem 16. Let (14), (21), , and hold and . Assume that .

(a)If  , then, for all , (6) has a positive periodic solution.(b)If  , then, for all sufficiently small , (6) has two positive periodic solutions.(c)There exists a such that (6) has a positive periodic solution for .

Proof. (a) Since , is defined on and is nonnegative. Noting , it follows from Lemma 9 that . Therefore, we can choose so that , where the constant satisfies

and is the positive constant defined in Lemma 10. We have by Lemma 10 that

On the other hand, by the condition , there is a positive number such that

for and , where is chosen so that

It is easy to see that, for ,

Lemma 7 implies that

By Lemma 4, has a fixed point . The fixed point is the desired positive periodic solution of (6).

(b) Again, since , is defined on and is nonnegative. Fix two numbers ; there exists a such that

where and are defined in Lemma 14, which implies that, for ,

On the other hand, in view of the assumptions and , there are positive numbers such that

for and or where is chosen so that

Thus if , then

Let . If , then

which implies that

Thus, Lemma 7 implies that

It follows from Lemma 4 that has two fixed points and such that and , which are the desired distinct positive periodic solutions of (6) for satisfying

(c) First we note that is defined on and is nonnegative since . Fix a number . Lemma 14 implies that there exists a such that we have, for ,

On the other hand, in view of the assumption , there is a positive number such that

for and , where is chosen so that

Thus, if , then

Thus, Lemma 7 implies that

Lemma 4 implies that has a fixed point . The fixed point is the desired positive periodic solution of (6).

When takes negative values, we give the following theorem.

Theorem 17. Let (14), (21), , and hold. Assume that .

(a)If   and  , then there exists such that (6) has a positive periodic solution for .(b)If  , then, for all sufficiently small , (6) has two positive periodic solutions.(c)There exists a such that (6) has a positive periodic solution for .

Proof. (a) Since , by Lemma 6, there is a such that if ; then is nonnegative and is defined. Now, for a fixed number , Corollary 13 implies that there exists a such that, for ,

On the other hand, since , it follows from Lemma 2 that . Therefore, we can choose

so that , where the constant satisfies

We have, by Corollary 11, that

By Lemma 4, has a fixed point . The fixed point is the desired positive periodic solution of (6).

(b) First, since , by Lemma 6, there is such that if , is defined on and is nonnegative. Furthermore, . Now for a fixed number , Corollary 15 implies that there exists a such that we have, for ,

In view of the assumption , there is a positive number such that

for and , where is chosen so that

Thus, if , then

Thus, Corollary 8 implies that

It follows from Lemma 4 that has a fixed point which is a positive periodic solution of (6) for satisfying

On the other hand, since , by Lemma 6, there is such that if is defined on and is nonnegative. Furthermore, . For a fixed number , and Corollary 15 implies that there exists a such that we have, for ,

Since , there is a positive number such that

for and , where is chosen so that

Let . If , then

which implies that

Again, Corollary 8 implies that

It follows from Lemma 4 that has a fixed point , which is a positive periodic solution of (6) for satisfying

Noting that

we can conclude that and are the desired distinct positive solutions of (6) for .

(c) Since , by Lemma 6, there is a such that if , then is defined and is nonnegative. Now for a fixed number , Corollary 15 implies that there exists a such that we have, for ,

On the other hand, in view of the assumption , there is a positive number such that

for and , where is chosen so that

Thus, if , then

Thus, Corollary 8 implies that

Lemma 4 implies that has a fixed point . The fixed point is the desired positive periodic solution of (6).

Case 2. In this case, replacing assumptions (14) and (21) by assumption (32), we can get similar existence results which we omit here.

We illustrate our results with some examples.

Example 18. Consider the following singular equation:

where is a constant and .

Comparing (100) to (6), we see that . Take ; then Case 1 holds. By calculation, we get , and we have ; then (14) and (21) hold. Moreover, , ; then and hold. So, by Theorem 16(a), we can get that (100) has positive periodic solution.

Example 19. Consider the following singular equation:

where is a constant and , is 1-periodic function and , is continuous, and ; here, .

Comparing (101) to (6), we see that , , . Take ; then Case 1 holds. By calculation, we get , , . From , we have , and ; then (14) and (21) hold. Moreover, , , ; then and hold. So, by Theorem 16(b), we can get that (100) has two positive periodic solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

Research is supported by NSFC Project (nos. 11326124 and 11271339).

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