Abstract and Applied AnalysisVolume 2012 (2012), Article ID 185692, 12 pagesdoi:10.1155/2012/185692

Research Article

Positive Periodic Solutions for First-Order Neutral Functional Differential Equations with Periodic Delays

Zeqing Liu,¹ Xin Li,¹ Shin Min Kang,² and Young Chel Kwun³

¹Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China²Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea³Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea

Received 15 February 2012; Accepted 20 March 2012

Academic Editor: István Györi

Copyright © 2012 Zeqing Liu et al. 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.

In this paper, two classes of first-order neutral functional differential equations with periodic delays are considered. Some results on the existence of positive periodic solutions for the equations are obtained by using the Krasnoselskii fixed point theorem. Four examples are included to illustrate our results.

1. Introduction and Preliminaries

In recent years, there have been a few papers written on the existence of periodic solutions, nontrivial periodic solutions, maximal and minimal periodic solutions and positive periodic solutions for several classes of functional differential equations with periodic delays, which arise from a number of mathematical ecological models, economical and control models, physiological and population models, and other models, see, for example, [1–5] and the references therein.

In 2004, Wan et al. [5] studied the first-order functional differential equation with periodic delays where are -periodic, and is -periodic with respect to the first variable. By using a fixed point theorem in cones, they proved the existence of a periodic solution and a positive periodic solution of (1.1), respectively, under certain conditions. In 2005, Kang and Zhang [2] used the partial ordering and topological degree theory to establish the existence of a nontrivial periodic solution of (1.1). In 2010, Kang et al. [1] gave the existence of maximal and minimal periodic solutions of (1.1) by utilizing the method of lower and upper solutions. By means of the continuation theorem of coincidence degree principle, Serra [4] discussed the existence of periodic solutions for the following neutral functional differential equation where and are constants. In 2008, Luo et al. [3] employed the Krasnoselskii fixed point theorem to prove the existence of positive periodic solutions for two kinds of neutral functional differential equations with periodic delays where and are constants, , , , , and are -periodic and is -periodic with respect to the first variable, , and .

Motivated by the papers [1–5] and the references therein, we consider two new kinds of first-order neutral functional differential equations with periodic delays: where is a constant, , , , , , , , , and are -periodic functions and is -periodic with respect to the first variable, , and . It is evident that (1.4) and (1.5) include, respectively, (1.1)–(1.3) as special cases. To the best of our knowledge, the existence of periodic solutions for (1.4) and (1.5) have not been investigated till now. The aim of this paper is, by applying the Krasnoselskii fixed point theorem and some new techniques, to establish a set of sufficient conditions which guarantee the existence of positive periodic solutions of (1.4) and (1.5). Four examples are given to show the efficiency and applications of our results.

Throughout this paper, we assume that , , , denotes the set of all positive integers, , It is well known that is a Banach space with the norm Let It is easy to see that is a bounded closed and convex subset of the Banach space .

Lemma 1.1 (the Krasnoselskii fixed point theorem). Let be a nonempty bounded closed convex subset of a Banach space and mappings from into such that for every pair . If is a contraction mapping and is completely continuous, then the equation has at least one solution in .

2. Main Results

Now we use the Krasnoselskii fixed point theorem to show the existence of positive solutions for (1.4) and (1.5).

Theorem 2.1. Assume that there exist constants , , , , , and satisfying Then (1.5) has at least one positive -periodic solution in .

Proof. It is obvious that (1.4) has a solution if and only if the integral equation has a solution . Define two mappings and by for each . It follows from (2.5) that for any and which mean that Using (2.1)–(2.3) and (2.5), we infer that for all and which imply that Now we show that is a completely continuous mapping in . First, we claim that is continuous in . Let and with . Note that . It follows from the uniform continuity of in that for given , there exists satisfying Since , it follows that there exists satisfying In view of (2.1), (2.2), (2.5), (2.12), and (2.13), we get that which yields that , that is, is continuous in .Second, we claim that is relatively compact. It is sufficient to show that is uniformly bounded and equicontinuous in . Notice that (2.1)–(2.3) and (2.5) ensure that which give that is uniformly bounded and equicontinuou sin , which together with (2.7), (2.11), and Lemma 1.1 yields that there is with . It follows from (2.4) and (2.5) that is a positive -periodic solution of (1.4). This completes the proof.

Theorem 2.2. Assume that there exist constants , , , , , , and satisfying (2.2), (2.3): and either or Then (1.4) has at least one positive -periodic solution with for each .

Proof. As in the proof of Theorem 2.1, we conclude similarly that (1.4) has an -periodic solution . Now we assert that for all . Otherwise, there exists satisfying . In view of (2.4), (2.5), and (2.16), we have which implies that Assume that (2.17) holds. By means of (2.2), (2.3), (2.17), and the continuity of , , , , , , , and , we get that which contradicts (2.20).Assume that (2.18) holds. In light of (2.2), (2.3), (2.18), and the continuity of , , , , , ,, and , we infer that which contradicts (2.20). This completes the proof.

Theorem 2.3. Assume that there exist constants , , , , , and satisfying (2.1), (2.2), and Then (1.5) has at least one positive -periodic solution in .

Proof. It is obvious that (1.5) has a solution if and only if the integral equation has a solution . Define two mappings and by for each . The rest of the proof is similar to that of Theorem 2.1, and is omitted. This completes the proof.

Theorem 2.4. Assume that there exist constants , , , , , , and satisfying (2.2), (2.16), (2.23), and either (2.18) or Then (1.5) has at least one positive -periodic solution with for each . The proof of Theorem 2.4 is similar to that of Theorems 2.2 and 2.3 and is omitted.

Remark 2.5. Even if , and for all , the conditions of Theorems 2.2 and 2.4 in this paper are different from these conditions of Theorems 2.1–2.4 in [3], respectively.

3. Examples

Now we construct four examples which illustrate the results obtained in Section 2. Note that none of the known results can be applied to the examples.

Example 3.1. Consider the first-order neutral functional differential equation with periodic delays Let , , , , , , , and It is easy to see that (2.1) and (2.2) hold. Notice that that is, (2.3) is satisfied. Thus Theorem 2.1 yields that (3.1) has a positive -periodic solution in .

Example 3.2. Consider the first-order neutral functional differential equation with periodic delays Let , , , , , , , , and It is clear that (2.2), (2.16), and (2.18) hold. It follows that that is, (2.3) holds. Obviously (2.17) follows from the above inequalities. Hence, Theorem 2.2 ensures that (3.4) has a positive -periodic solution with for all .

Example 3.3. Consider the first-order neutral functional differential equation with periodic delays Let , , , , , , , and Clearly, (2.1) and (2.2) hold. Note that that is, (2.23) is fulfilled. Thus Theorem 2.3 yields that (3.7) has a positive -periodic solution in .

Example 3.4. Consider the first-order neutral functional differential equation with periodic delays Let , , , , , , , , and Obviously, (2.2), (2.16), and (2.18) hold. A simple calculation yields that that is, (2.23) holds. Clearly (2.26) follows from the above inequalities. Thus Theorem 2.4 ensures that (3.10) has a positive -periodic solution with for all .

Acknowledgment

This study was supported by research funds from the Dong-A University.

References

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