Research Article | Open Access
Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation
By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.
Neutral functional differential equations manifest themselves in many fields including biology, mechanics, and economics [1–4]. For example, in population dynamics, since a growing population consumes more (or less) food than a matured one, depending on individual species, this leads to neutral functional equations . These equations also arise in classical “cobweb” models in economics where current demand depends on price but supply depends on the previous periodic solutions . The study on neutral functional differential equations is more intricate than ordinary delay differential equations. In recent years, there has been a good amount of work on periodic solutions for neutral differential equations (see [5–12] and the references cited therein). For example, in , Wu and Wang discussed the second-order neutral delay differential equation By a fixed point theorem, they obtain some existence results of positive periodic solutions for (1.1). Recently, in , Cheung et al. considered second-order neutral functional differential equation By choosing available operators and applying Krasnoselskii's fixed point theorem, they obtained sufficient conditions for the existence of periodic solutions to (1.2).
In general, most of the existing results are concentrated on first-order and second-order neutral functional differential equations, while studies on third-order neutral functional differential equations are rather infrequent, especially on the positive periodic solutions for third-order neutral functional differential equations. In the study of high-order (in particular third-order) differential equations, the naive idea to translate the equation into a first-order differential system by defining , , , works well for showing existence of periodic solutions, however, it does not obviously lead to existence proofs for positive periodic solutions, since the condition of positivity for the higher order equation is different from the natural positivity condition for the corresponding system. Another approach, which will be used in this paper, is to transform the third-order equation into a corresponding integral equation and to establish the existence of positive periodic solutions based on a fixed point theorem in cones. Following this path one needs an explicit representation of Green's function which is rather intricate to compute.
In this paper, we consider the following third-order neutral functional differential equation: Here is a positive parameter; , and for ; , , , , , , and are -periodic functions.
Notice that here neutral operator is a natural generalization of the familiar operator . But possesses a more complicated nonlinearity than . For example, the neutral operators is homogeneous in the following senses , whereas the neutral operator in general is inhomogeneous. As a consequence many of the new results for differential equations with the neutral operator will not be a direct extension of known theorems for neutral differential equations.
The paper is organized as follows. In Section 2, we first analyze qualitative properties of the generalized neutral operator which will be helpful for further studies of differential equations with this neutral operator; in Section 3, we consider two types of third-order constant coefficient linear differential equations and present their Green's functions and properties for those equation; in Section 4, by an application of the fixed point index theorem we obtain sufficient conditions for the existence, multiplicity and nonexistence of positive periodic solutions to third-order neutral differential equation. We will give an example to illustrate our results; in Section 5, the Lyapunov stability of periodic solutions for the equation will then be established. And an example is also given in this section.
2. Analysis of the Generalized Neutral Operator
Let with norm , and let , . Then is a Banach space. A cone in is defined by , , where is a fixed positive number with . Moreover, define operators by
Lemma 2.1. If , then the operator has a continuous inverse on , satisfying (1)(2)(3)
Proof. We have the following cases. Case 1 (). Let and , . Therefore Since , we get from that has a continuous inverse with Here . Then and consequently Moreover, Case 2 (). Let By definition of the linear operator , we have Here is defined as in Case 1. Summing over yields Since , we obtain that the operator has a bounded inverse , and for all we get On the other hand, from , we have that is, Let be arbitrary. We are looking for such that that is Therefore and hence proving that exists and satisfies Statements (1) and (2) are proved. From the above proof, (3) can easily be deduced.
Lemma 2.2. If and , we have for that
Proof. Since and , by Lemma 2.1, one has for that
Lemma 2.3. If and , then for , one has
Proof. Since , , and , by Lemma 2.1, we have for that
3. Green's Functions
Theorem 3.1. For and , the equation has a unique solution which is of the form where where denotes denotes .
Proof. It is easy to check that the associated homogeneous equation of (3.1) has the solution . The only periodic solution of the associated homogeneous equation of (3.1) is the trivial solution, that is, . This follows by assuming that is periodic; we immediately get that and by assuming that and choosing such that , , we get
which for contradicts periodicity of , proving that .
Applying the method of variation of parameters, we get and then Noting that , we obtain where denotes . Therefore where is defined as in (3.3).
By direct calculation, we get the solution satisfies the periodic boundary value condition of the problem (3.1).
Theorem 3.2. For and , the equation has a unique -periodic solution where where denotes denotes .
Proof. It is similar to the proof of Theorem 3.1 and can therefore be omitted.
Theorem 3.3. , and if holds, then for all and .
Proof. One has the following:
A direct computation shows that . It is easy to see that for and for and .
For convenience, we denote If and , then obviously , , and .
For , Since , we have(i)For , then , , we get ,(ii)For , we have , , and
For , (i)For , we have , , and then .(ii)For , we have , , and
If , we get and , proving that for all and .
Next we compute a lower and an upper bound for for . We have The proof is complete.
Similarly, the following dual theorem can be proved.
Theorem 3.4. , and if holds, then for all and .
4. Positive Periodic Solutions for (1.3)
Define the Banach space as in Section 2. Denote It is easy to see that .
In the following we discuss (1.3) in two cases, namely, the case where , and (note that implies , implies ); and the case where and , (note that implies , implies ). Obviously, we have which makes Lemma 2.1 applicable for both cases, and also Lemmas 2.2 or 2.3, respectively.
Let denote the cone in , where is just as defined above. We also use and .
Now we discuss the two cases separately.
4.1. Case I: and
Now we consider and define operators , by Clearly , are completely continuous, for and . By Theorem 3.2, the solution of (4.6) can be written in the following form: In view of and , we have and hence Define an operator by Obviously, for any , if hold, is the unique positive -periodic solution of (4.6).
Lemma 4.1. is completely continuous, and
Proof. By the Neumann expansion of , we have Since and are completely continuous, so is . Moreover, by (4.13) and recalling that , we get
Define an operator by
Lemma 4.2. One has that .
Proof. From the definition of , it is easy to verify that . For , we have from Lemma 4.1 that On the other hand, Therefore that is, .
From the continuity of , it is easy to verify that is completely continuous in . Comparing (4.5) to (4.6), it is obvious that the existence of periodic solutions for (4.5) is equivalent to the existence of fixed points for the operator in . Recalling Lemma 4.2, the existence of positive periodic solutions for (4.5) is equivalent to the existence of fixed points of in . Furthermore, if has a fixed point in , it means that is a positive -periodic solutions of (1.3).
Lemma 4.3. If there exists such that then
Lemma 4.4. If there exists such that then
Lemma 4.5. If , then
Lemma 4.6. If , then
We quote the fixed point theorem on which our results will be based.
Lemma 4.7 (see ). Let be a Banach space and a cone in . For , define . Assume that is completely continuous such that for . (i)If for , then .(ii)If for , then .
Now we give our main results on positive periodic solutions for (1.3).
(a) If or 2, then (1.3) has positive -periodic solutions for ,
(b) if or 2, then (1.3) has positive -periodic solutions for ,
(c) if or , then (1.3) has no positive -periodic solutions for sufficiently small or sufficiently large , respectively.
Proof. (a) Choose . Taking , then for all , we have from Lemma 4.5 that Case 1. If , we can choose , so that for , where the constant satisfies Letting , we have for . By Lemma 2.2, we have for . In view of Lemma 4.4 and (4.30), we have for that It follows from Lemma 4.7 and (4.29) that thus and has a fixed point in , which means is a positive -positive solution of (1.3) for .Case 2. If , there exists a constant such that for , where the constant satisfies Letting , we have for . By Lemma 2.2, we have for . Thus by Lemma 4.4 and (4.33), we have for that Recalling Lemma 4.7 and (4.29) and that then and has a fixed point in , which means is a positive -positive solution of (1.3) for .Case 3. If , from the above arguments, there exist such that has a fixed point in and a fixed point in . Consequently, and are two positive -periodic solutions of (1.3) for .(b)Let . Taking , then by Lemma 4.6 we know if , then Case 1. If , we can choose so that for , where the constant satisfies Letting , we have for . By Lemma 2.2, we have for . Thus by Lemma 4.3 and (4.37), It follows from Lemma 4.7 and (4.36) that which implies and has a fixed point in . Therefore is a positive -periodic solution of (1.3) for .Case 2. If , there exists a constant such that for , where the constant satisfies Let , we have for . By Lemma 2.2, we have