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On the Existence and Stability of Periodic Solutions for a Nonlinear Neutral Functional Differential Equation
This paper deals with the existence and stability of periodic solutions for the following nonlinear neutral functional differential equation By using Schauder-fixed-point theorem and Krasnoselskii-fixed-point theorem, some sufficient conditions are obtained for the existence of asymptotic periodic solutions. Main results in this paper extend the related results due to Ding (2010) and Lopes (1976).
In recent years, the existence and stability of periodic solutions for differential equation has been extensively studied. Many researchers used the Lyapunov functional method, Hopf bifurcation techniques, and Mawhin continuation theorems to obtain the existence and stability of periodic solutions for neutral functional differential equation (NFDE); see papers [1–14] and their references therein. However, researches on the existence and stability of periodic solutions for NFDE by using fixed-point theorem are relatively rare [15, 16]. The reason lies in the fact that it is difficult to construct an appropriate completely continuous operator and an appropriate bounded closed convex set.
In this paper, we will investigate the existence and stability of periodic solutions for the following nonlinear NFDE where , and . Such a kind of NFDE has been used for the study of distributed networks containing a transmission line [17, 18]. For example, suppose a system consists of a long electrical cable of length , and one end of which isconnected to a power source with resistance , while the other end is connected to an oscillating circuit formed of a condenser and a nonlinear element, the volt-ampere characteristic of which is . Let be the parameters of the transmission line, respectively, the characteristic impedance of the line, the propagation velocity and assume the losses can not be disregarded. The process of the final end volt in such a system can be described by the following NFDES: or where , , , , , , , , and is a given nonlinear function. If , then , . Obviously, we see that (2) (or (3)) is a special case of (1). The aim of this paper is to establish some criteria to guarantee the existence and stability of periodic solution for (1) by using Schauder's fixed-point theorem and Krasnoselskii's fixed-point theorem. The interesting is that main results obtained in this paper extend the related existing results. Furthermore, our results can also be applied to solve the problem of the existence and stability of periodic solutions for (2) and (3).
2. Main Results and Proofs
In this section, let () denote the set of all continuously differentiable functions (all continuous functions) , where is a Banach space with the supremum norm , with the norm in a period interval, and is a positive constant. The next lemma will be used in the sequel.
Lemma 1. If , , then the scalar equation has a unique -periodic solution:
Proof. It is easy to prove. We can find it in many ODE textbooks (e.g., see Example 2 on page 35 of ).
Theorem 2. Suppose that and . If there exists a constant such that then (1) has a -periodic solution.
Proof. According to the condition (5), we can find a sufficiently small such that
Let , , , and ; then (1) can be rewritten as
where with . It suffices to prove that (7) has a -periodic solution. Let
Then is a bounded closed convex set of the Banach space . For any given , consider the nonhomogeneous equation:
According to Lemma 1, (9) has a unique -periodic solution:
Define an operator by
In order to prove that (7) has a periodic solution, we shall make sure that satisfies the conditions of Schauder's fixed-point theorem (see Lemma 2.2.4 on page 40 of ).
Note that for any , we have and Therefore, . Meanwhile, we get
By (6), we have Thus, .
For any , , . According to Arzela-Ascoli Theorem (see Theorm 4.9.6 on page 84 of ), the subset of is a precompact set; therefore, is a compact operator.
Suppose that , , then and as . Also, we have
When as , for uniformly. And since is continuous, , . Consequently, is continuous.
Thus, by Schauder-fixed-point theorem (see Lemma 2.2.4 on page 40 of ), there is a such that . Therefore, (7) has at least one -periodic solution. Since and , (1) has at least one -periodic solution. The proof is completed.
Next, we explore the stability of this -periodic solution for (1). We assume that theconditions of Theorem 2 are satisfied. Therefore, (1) has at least one -periodic solution . Let then (1) is transformed as where and . Clearly, (16) has trivial solution . Now we use Krasnoselskiis-fixed-point theorem (see  or [15, Lemma 2.2]) to prove that trivial solution to (16) is asymptotically stable.
Set as the Banach space of bounded continuous function with the supremum norm . Also, Given the initial function , denote the norm of by , which should not cause confusion with the same symbol for the norm in .
Theorem 3. Let be as in Theorem 2. Assume that all conditions of Theorem 2 are satisfied. Suppose that satisfies the Lipschitz condition and If there exists such that then the solution to (16) through satisfies .
Proof. By (18), we have
Given the initial function , by [20, Theorem 12.2.3], there exists a unique solution for (16). Let then is a bounded convex closed set of . We write (16) as then we have
For all , define the operators and by
For any , , as , and , . Therefore, we have Thus, . Again by (17) and (19), we have Thus, .
Since , and here, the derivative of at zero means the left-hand side derivative when and the right-hand side derivative when , one can see is bounded for all . Therefore, is a precompact set of . Thus, is compact.
Suppose that , , as ; then uniformly for as . Since and is continuous, we have as . Thus, is continuous. Due to the fact that and , we know that is a contractive operator.
According to Krasnoselskii's fixed-point theorem (see  or [15, Lemma 2.2]), there is a such that . Therefore, is a solution for (16). Because the solution through for the equation is unique, the solution as .
When satisfies the Lipschitz condition, then there is a constant such that Since satisfies then that is Therefore, if , then there clearly exists a for any such that for all if . Thus, we have the following.
Theorem 4. If the Lipschitz constant for corresponding to satisfies then the zero solution for (16) is stable.
When is constant and the equation has only one solution , then is an equilibrium of (1) and (1) can be transformed to the following equation: where and . Now, we consider the stability of the zero solution for (34).
Theorem 5. Suppose that is and ; then the zero solution of (34) is exponentially asymptotically stable.
Proof. For all in , let Then is stable, and and are linear and continuous. Consider the equation . Let Then Thus, according to the last conclusion of Theorem 12.7.1 in [20, Page 297], the zero solution of is uniformly asymptotically stable. On the other hand, one can see that Thus, . According to [20, Theorem 12.9.1], the zero solution of (34) is exponentially asymptotically stable.
In this section, we present two examples to illustrate the applicability of our main results.
Example 6 (Lopes et al. [8, 9, 13, 15, 23]). Consider the NFDE (2) which arises from a transmission line model, where , and is a given nonlinear function. Now, let . It is not difficult to see that (2) is a special case of (1). Therefore, by Theorems 2–5, we have the following.
Theorem 7. Suppose that and . If there exists a constant such that then (2) has a -periodic solution.
Theorem 9. Let be as in Theorem 7. Assume that all conditions of Theorem 7 are satisfied. If satisfies the Lipschitz condition, and there exists such that then the solution through to (2) satisfying as , where is a -periodic solution of (2).
Remark 10. The sufficient conditions for the existence of periodic solutions in  are very complicated. For example, they need extra condition , and where .
Theorem 12. Suppose that is constant, the equation has only one solution , , and ; then the equilibrium of (2) is exponentially asymptotically stable.
Example 13 (Lopes ). Consider the NFDE (3) which arises from a transmission line model, where and is a given nonlinear function. Let , and ; then (3) can be rewritten as Now, let . It is not difficult to see that (43) is a special case of (1). Therefore, by Theorems 2–5, we have the following.
Theorem 14. Suppose that and . If there exists a constant such that then (3) has a -periodic solution.
Theorem 15. Let be as in Theorem 14. Assume that all conditions of Theorem 14 are satisfied. If satisfies the Lipschitz condition, , and there exists such that then the solution through of (3) as , where is a -periodic solution of (3).
Theorem 17. Suppose that is constant, the equation has only one solution , , and , then the equilibrium of (3) is exponentially asymptotically stable.
This work was partially supported by the National Natural Science Foundation of China nos. 11201411 and 11031002.
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