#### Abstract

The existence results of positive -periodic solutions are obtained for the third-order ordinary differential equation with delays where is -periodic function and is a continuous function which is -periodic in are positive constants. The discussion is based on the fixed-point index theory in cones.

#### 1. Introduction

In this paper, we discuss the existence of positive -periodic solutions for the third-order ordinary differential equation with delays where is a -periodic function, is a continuous function, and is -periodic in , and are positive constants.

In recent years, the existence of periodic solutions for first-order and second-order delay differential equations has been researched by many authors; see [1–5] for the first-order equations and see [6–12] for the second-order ones. In some practice models, only positive periodic solutions are significant. In [3, 8, 9, 11, 12], the authors obtained the existence of positive periodic solutions for some first-order and second-order delay differential equations by using Krasnoselskii’s fixed-point theorem of cone mapping. But, few people consider the existence of positive periodic solutions for third-order delay differential equations.

The third-order delay differential equations have their important physical contexts, for example, which can be formulated from the problem of the wave solution of the Korteweg-de Vries (KdV) equation with time delay. Recently, Zhao and Xu [13] pointed out that the KdV equation with time delay has more actual significance and they considered the solitary wave solution of the following KdV equation with time delay: where is a given constant and means the backward diffusion with time delay. They looked for a wave solution with and from (2) obtained the following third-order delay ordinary differential equation of the profile : Equation (3) is a special form of (1). If we look for a periodic wave solution of (2), we need to discuss the existence of the periodic solution of the delay ordinary differential equation (3). Hence, the existence problem of periodic solutions of the general third-order delay differential equation (1) is a significant topic.

For the third-order ordinary differential equations without delays, the existence of periodic solutions has been considered by several authors; see [14–23] and references therein. Some theorems and methods of nonlinear functional analysis have been applied to research on this problem, such as the methods of topological degree and Leray-Schauder fixed-point theorem [14, 19], the upper and lower solutions method and monotone iterative technique [15–17], the implicit function theorem [18], and Mawhin coincidence degree theory [20]. Especially, in recent years, the fixed-point theorem of Krasnoselskii’s cone expansion or compression type has been availably applied to some special third-order periodic boundary problems of ordinary differential equations, and some results of existence and multiplicity of positive periodic solutions have been obtained; see [21, 22]. In [21], Chu and Zhou considered the periodic boundary value problem for the third-order equation where is a constant and . Using the Krasnoselskii’s fixed-point theorem in cones, they obtained the existence results of positive solutions. Their results extended the one obtained by the Schauder fixed-point theorem in [19]. In [22], by the Krasnoselskii’s fixed-point theorem in cones, Feng established some existence and multiplicity results of positive periodic solutions for the third-order equation where and are positive constants and satisfy certain conditions. In [23], the present author extended and improved the results in [9, 10] to the general third-order equation that nonlinearity explicitly contains derivative terms and . However, all of these works are on the third-order equations without delays and the argument methods are not applicable to the delay equation (1).

Motivated by the facts mentioned above, we research the existence of positive periodic solutions of the third-order delay equation (1). We will use the fixed-point index theory in cones in a meticulous way to obtain the essential conditions on the existence of positive periodic solutions of (1). Our main results will be given in Section 3. Some preliminaries to discuss (1) are presented in Section 2.

#### 2. Preliminaries

Let denote the Banach space of all continuous -periodic function with norm . Generally, for , we use to denote the Banach space of all th-order continuous differentiable -periodic function with the norm . Let denote the cone of all nonnegative functions in .

Let be a constant. For , we consider the existence of -periodic solution of the linear third-order differential equation It is easy to verify that the linear third-order boundary value problem has a unique solution. We denote the solution by . By [16, Lemma 2.1], the -periodic solution of (7) can be expressed by . By [16, Lemma 2.1] or a direct calculation, we easily obtain the following lemma.

Lemma 1. *Let . Then, for every , the linear equation (7) has a unique -periodic solution which is given by
**
Moreover, is a completely continuous linear operator.*

Lemma 2. *Let . Then, the solution of the linear third-order boundary value (8) is positive on .*

*Proof. *Let . It is easy to prove that the linear second-order boundary value problem
has a unique solution which is given by
and the linear first-order boundary value problem
has a unique solution given by
By a direct calculation, we can verify that
is the unique solution of the linear third-order boundary value (8). When , and, by (11), on . Since on , from (14), we see that for every .

Let . Then, the solution of (8) for every . If and , by (9), the -periodic solution of (7) is positive. We will show that the -periodic solution has stronger positivity. Let where . Choose a cone in by We have the following lemma.

Lemma 3. *Let . Then, for every , the -periodic solution of (7) .*

*Proof. *Let and let . For every , from (9), it follows that
and, therefore,
Using (9) again, we obtain that
For every , since
we have
Hence, .

Now, we consider the periodic solution problem of the linear third-order differential equation with variable coefficient Let be a positive -periodic function and satisfy the assumption and set Then, , and the conclusion of Lemma 3 holds. For (22), we have the following lemma:

Lemma 4. *Let satisfy the assumption . Then, for every , the linear equation (22) has a unique -periodic solution . Moreover, is a completely continuous linear operator and .*

*Proof. *Let and be the positive constants defined by (24) and let be the -periodic solution operator of (7) given by (9). By Lemma 3, , and is a positive linear bounded operator. We rewrite (22) into the form of
Then, it is easy to see that the -periodic solution problem of (22) is equivalent to the operator equation in Banach space
where is the identity operator in and is the product operator defined by
which is a positive linear bounded operator. We prove that the norm of in satisfies .

For every and , by definition (9) of and the positivity of , we have
Therefore, . By the arbitrariness of , we have .

Thus, has a bounded inverse operator given by the series
Consequently, (26), equivalently (22), has a unique -periodic solution
where
By this and (29), we have
Hence, can be expressed in the form of
where
which is a linear bounded operator from into . By Lemma 1, is completely continuous. Thus, from (33), we see that is a completely continuous linear operator.

By the positivity of and , from the expression (34) of , we see that is a positive linear operator. Hence, for every , . By (33) and Lemma 3, . Thus, .

The proof of Lemma 4 is completed.

Let be a continuous function. For every , set Then, is continuous. Now, we define a mapping by where is the periodic solution operator of (22). By Lemma 4, we have the following lemma.

Lemma 5. *Let satisfy the assumption . Then, the operator defined by (36) is completely continuous.*

By the definition of operator and Lemma 4, the positive -periodic solution of (1) is equivalent to the nonzero fixed point of . We will find the nonzero fixed point of by using the fixed-point index theory in cones.

We recall some concepts and conclusions on the fixed-point index in [15, 16]. Let be a Banach space and be a closed convex cone in . Assume is a bounded open subset of with boundary and . Let be a completely continuous mapping. If for any , then the fixed-point index has definition. One important fact is that if , then has a fixed point in . The following two lemmas in [24, 25] are needed in our argument.

Lemma 6. *Let be a bounded open subset of with , and let be a completely continuous mapping. If for every and , then .*

Lemma 7. *Let be a bounded open subset of and let be a completely continuous mapping. If there exists an such that for every and , then .*

In next section, we will use Lemma 6 and Lemma 7 to discuss the existence of positive -periodic solutions of (1).

#### 3. Main Results

We consider the the existence of positive -periodic solutions of the third-order delay equation (1). Let satisfy the assumption and let be the positive constants defined by (24). Let , and be -periodic in . Let and be the constants defined by (15) and let . To be convenient, we introduce the notations Our main results are as follows.

Theorem 8. *Let satisfy the assumption , let be continuous, and let be -periodic in . If satisfies the condition
**
then (1) has at least one positive -periodic solution.*

Theorem 9. *Let satisfy the assumption , let be continuous, and let be -periodic in . If satisfies the condition
**
then (1) has at least one positive -periodic solution.*

In Theorem 8, the condition allows that is superlinear growth on , and . For the application, see Example 10. In Theorem 9, the condition allows sublinear growth on , and . See Example 11.

*Proof of Theorem 8. *Choose working space . Let be the cone in defined by (16) and let be the completely continuous operator defined by (36). Then, the positive -periodic solution of (1) is equivalent to nontrivial fixed point of . Let and set
We show that the operator has a fixed point in when is small enough and is large enough.

By the assumption of and the definition of , there exist and , such that
Let . We now prove that satisfies the condition of Lemma 6 in ; namely, for every and . In fact, if there exist and such that , since , by Lemma 4 and the definition of and , satisfies the delay differential equation
Since , by the definitions of and , we have
Hence, from (41), it follows that
By this inequality and (42), we have
Integrating both sides of this inequality from to and using the periodicity of , we have
Hence, we obtain that
Since , from (47), it follows that , which is a contradiction. Hence, satisfies the condition of Lemma 6 in . By Lemma 6, we have

On the other hand, since , by the definition of , there exist and such that
Choose and . Clearly, . We show that satisfies the condition of Lemma 7 in ; namely, for every and . In fact, if there exist and such that , since , by Lemma 4 and the definition of and , satisfies the differential equation
Since , by the definition of and , we have
By the latter inequalities of (51), we have
These inequalities mean that
Hence, we obtain that
By (54) and the former inequality of (51), we have
From this, the latter inequalities of (51) and (49), it follows that
By this inequality and (50), we have
Integrating this inequality on and using the periodicity of , we have
Hence, we obtain that
Since , from (59), it follows that , which is a contradiction. This means that satisfies the condition of Lemma 7 in . By Lemma 7,

Now, by the additivity of fixed-point index, (48), and (60), we have
Hence, has a fixed point in , which is a positive -periodic solution of (1).

*Proof of Theorem 9. *Let be defined by (40). We prove that the operator defined by (36) has a fixed point in if is small enough and is large enough.

By the assumption of and the definition of , there exist and , such that
Let and let . We prove that satisfies the condition of Lemma 7 in ; namely, for every and . In fact, if there exist and such that , since , by Lemma 4 and the definition of and satisfies the differential equation
Since , by the definitions of and , satisfies (43). From (43) and (62), we see that
From this and (63), it follows that
Integrating this inequality on and using the periodicity of , we have
Consequently,
Since , from (67), it follows that , which is a contradiction. Hence, satisfies the condition of Lemma 7 in . By Lemma 7, we have

Since , by the definition of , there exist and such that
Choosing , we show that satisfies the condition of Lemma 6 in ; namely, for every and . In fact, if there exist and such that , since , by Lemma 4 and the definition of and satisfies the differential equation
Since , by the definition of satisfies (51). By (51) we can show that satisfies (54). By (51) and (54), we have,
Since satisfies (51) and (71), from (69), it follows that
By this and (70), we have
Integrating this inequality on and using the periodicity of , we obtain that
Hence, we have
Since , from (75), it follows that , which is a contradiction. This means that satisfies the condition of Lemma 6 in . By Lemma 6,

Now, from (68) and (76), it follows that
Hence, has a fixed point in , which is a positive -periodic solution of (1).

*Example 10. *Consider the superlinear third-order delay differential equation
where , , and satisfy the conditions
It is easy to verify that satisfies the assumption for and
satisfies the assumption with and . Hence, by Theorem 8, (78) has at least one positive -periodic solution.

*Example 11. *Consider the third-order delay differential equation
where , , and are positive -periodic functions. It is easy to verify that satisfies the assumption for . Let
Then, and . Hence, satisfies the assumption . By Theorem 9, (81) has at least one positive -periodic solution.

#### Conflict of Interests

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

#### Acknowledgments

The research is supported by NNSFs of China (11261053, 11061031) and the NSF of Gansu Province (1208RJZA129).