Abstract and Applied Analysis

Volume 2012 (2012), Article ID 929870, 13 pages

doi:10.1155/2012/929870

Research Article

On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

Qiang Li and Yongxiang Li

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 30 August 2012; Accepted 1 November 2012

Academic Editor: K. Sadarangani

Copyright © 2012 Qiang Li and Yongxiang Li. 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.

The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays , where is a positive ω-periodic function, is a continuous function which is ω-periodic in t, and are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

1. Introduction

In this paper, we deal with the existence of positive periodic solution of the second-order functional differential equation with multiple delays

where is a positive -periodic function, is a continuous function which is -periodic in , and ,…, are -periodic functions is a constant.

In recent years, the existence of periodic solutions for second-order functional differential equations has been researched by many authors see [1–8] and references therein. In some practice models, only positive periodic solutions are significant. In [4–8], the authors obtained the existence of positive periodic solutions for some second-order functional differential equations by using fixed-point theorems of cone mapping. Especially in [5], Wu considered the second-order functional differential equation

and obtained the existence results of positive periodic solutions by using the Krasnoselskii fixed-point theorem of cone mapping when the coefficient satisfies the condition that for every . And in [8], Li obtained the existence results of positive -periodic solutions for the second-order differential equation with constant delays

by employing the fixed-point index theory in cones. For the second-order differential equations without delay, the existence of positive periodic solutions has been discussed by more authors, see [9–14].

Motivated by the paper mentioned above, we research the existence of positive periodic solutions of (1.1). We aim to obtain the essential conditions on the existence of positive periodic solution of (1.1) by constructing a special cone and applying the fixed-point index theory in cones.

In this paper, we assume the following conditions:

(H1)  is -periodic function and there exists a constant such that

where is the -norm of in , is the conjugate exponent of defined by , and the function is defined by

in which is the Gamma function.(H2)  and is -periodic in .(H3)  are -periodic functions.

In Assumption (H1), if , since , then (1.4) implies that satisfies the condition

This condition includes the case discussed in [5].

The techniques used in this paper are completely different from those in [5]. Our results are more general than those in [5] in two aspects. Firstly, we relax the conditions of the coefficient appeared in an equation in [5] and expand the range of its values. Secondly, by constructing a special cone and applying the fixed-point index theory in cones, we obtain the essential conditions on the existence of positive periodic solutions of (1.1). The conditions concern the first eigenvalue of the associated linear periodic boundary problem, which improve and optimize the results in [5]. To our knowledge, there are very few works on the existence of positive periodic solutions for the above functional differential equations under the conditions concerning the first eigenvalue of the corresponding linear equation.

Our main results are presented and proved in Section 3. Some preliminaries to discuss (1.1) are presented in Section 2.

2. Preliminaries

In order to discuss (1.1), we consider the existence of -periodic solution of the corresponding linear differential equation

where is a -periodic function. It is obvious that finding an -periodic solution of (2.1) is equivalent to finding a solution of the linear periodic boundary value problem

In [14], Torres show the following existence resulted.

Lemma 2.1. Assume that (H1 ) holds, then for every , the linear periodic boundary problem (2.2) has a unique solution expressed by

where is the Green function of the linear periodic boundary problem (2.2), which satisfies the positivity: for every .

For the details, see [14, Theorem 2.1 and Corollary 2.3].

For , we use to denote the th-order continuous differentiable -periodic functions space. Let be the Banach space of all continuous -periodic functions equipped the norm .

Let

be the cone of all nonnegative functions in . Then is an ordered Banach space by the cone . has a nonempty interior

Let

Lemma 2.2. Assume that (H1) holds, then for every , (2.1) has a unique -periodic solution . Let , then is a completely continuous linear operator, and when , has the positivity estimate

Proof. Let . By Lemma 2.1, the linear periodic boundary problem (2.2) has a unique solution given by (2.3). We extend to a -periodic function, which is still denoted by , then is a unique -periodic solution of (2.1). By (2.3),

From this we see that maps every bounded set in to a bounded equicontinuous set of . Hence, by the Ascoli-Arzelà theorem, is completely continuous.

Let . For every , from (2.8) it follows that

and therefore,

Using (2.8) and this inequality, we have that

Hence, by the periodicity of , (2.7) holds for every .

From (2.7) we easily see that ; namely, is a strongly positive linear operator. By the well-known Krein-Rutman theorem, the spectral radius is a simple eigenvalue of , and has a corresponding positive eigenfunction ; that is,

Since can be replaced by , where is a constant, we can choose such that

Set , then . By Lemma 2.2 and the definition of , satisfies the differential equation

Thus, is the minimum positive real eigenvalue of the linear equation (2.1) under the -periodic condition. Summarizing these facts, we have the following lemma.

Lemma 2.3. Assume that (H1) holds, then there exist such that (2.13) and (2.14) hold.

Let satisfy the assumption (H2). For every , set

Then is continuous. Define a mapping by

By the definition of operator , the -periodic solution of (1.1) is equivalent to the fixed point of . Choose a subcone of by

By the strong positivity (2.7) of and the definition of , we easily obtain the following.

Lemma 2.4. Assume that (H1) holds, then and is completely continuous.

Hence, the positive -periodic solution of (1.1) is equivalent to the nontrivial 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 a closed convex cone in . Assume that 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 [16] are needed in our argument.

Lemma 2.5. Let be a bounded open subset of with , and a completely continuous mapping. If for every and , then .

Lemma 2.6. Let be a bounded open subset of and a completely continuous mapping. If there exists an such that for every and , then .

3. Main Results

We consider the existence of positive -periodic solutions of (1.1). Assume that satisfy (H2). To be convenient, we introduce the notations

where and . Our main results are as follows.

Theorem 3.1. Suppose that (H1)–(H3) hold. If satisfies the condition

(H4) ,

then (1.1) has at least one positive -periodic solution.

Theorem 3.2. Suppose that (H1)–(H3) hold. If satisfies the condition

(H5) ,

then (1.1) has at least one positive -periodic solution.

In Theorem 3.1, the condition (H4) allows to be superlinear growth on , . For example,

satisfies (H4) with and , where are positive -periodic functions.

In Theorem 3.2, the condition (H5) allows to be sublinear growth on , . For example,

satisfies (H5) with and , where are positive -periodic solution.

Applying Theorems 3.1 and 3.2 to (1.2), we have the following.

Corollary 3.3. Suppose that (H1)–(H3) hold. If the parameter satisfies one of the following conditions

(1), (2),

then (1.2) has at least one positive -periodic solution.

This result improves and extends [5, Theorem 1.3].

Proof of Theorem 3.1. Let be the cone defined by (2.17) and the operator defined by (2.16). Then the positive -periodic solution of (1.1) is equivalent to the nontrivial fixed point of . Let and set

We show that the operator has a fixed point in when is small enough and large enough.

Since , by the definition of , there exist and , such that

where and . Choosing , we prove that satisfies the condition of Lemma 2.5 in ; namely, for every and . In fact, if there exist and such that and since , by definition of and Lemma 2.2, satisfies the differential equation

where is defined by (2.15). Since , by the definitions of and , we have

This implies that

From this and (3.5), it follows that

By this inequality and (3.6), we have

Let be the function given in Lemma 2.4. Multiplying the inequality (3.10) by and integrating on , we have

For the left side of the above inequality using integration by parts, then using the periodicity of and and (2.14), we have

Consequently, we obtain that

Since , by the definition of and (2.13),

From this and (3.13), we conclude that , which is a contradiction. Hence, satisfies the condition of Lemma 2.5 in . By Lemma 2.5, we have

On the other hand, since , by the definition of , there exist and such that

where and . Choose and . Clearly, . We show that satisfies the condition of Lemma 2.6 in ; namely, for every and . In fact, if there exist and such that , since , by the definition of and Lemma 2.2, satisfies the differential equation

Since , by the definitions of and , we have

This means that

Combining this with (3.16), we have that

From this inequality and (3.17), it follows that

Multiplying this inequality by and integrating on , we have

For the left side of the above inequality using integration by parts and (2.14), we have

From this and (3.22), it follows that

Since , by the definition of and (2.13), we have

Hence, from (3.24) it follows that , which is a contradiction. Therefore, satisfies the condition of Lemma 2.6 in . By Lemma 2.6, we have

Now by the additivity of the fixed-point index (3.15), and (3.26), we have

Hence has a fixed point in , which is a positive -periodic solution of (1.1).

Proof of Theorem 3.2. Let be defined by (3.4). We prove that the operator defined by (2.16) has a fixed point in if is small enough and is large enough.

By and the definition of , there exist and , such that

where . Let and . We prove that satisfies the condition of Lemma 2.6 in ; namely, for every and . In fact, if there exist and such that and since , by the definition of and Lemma 2.2, satisfies the differential equation

Since , by the definitions of and , satisfies (3.7), and hence (3.8) holds. From (3.8) and (3.28), it follows that

By this and (3.29), we obtain that

Multiplying this inequality by and integrating on , we have

For the left side of this inequality using integration by parts and (2.14), we have

From this and (3.32), it follows that

Since , from the definition of and (2.13) it follows that (3.14) holds. By (3.14) and (3.34), we see that , which is a contradiction. Hence, satisfies the condition of Lemma 2.6 in . By Lemma 2.6, we have

Since , by the definition of , there exist and such that

where . Choosing , we show that satisfies the condition of Lemma 2.5 in ; namely, for every and . In fact, if there exist and such that , since , by the definition of and Lemma 2.2, satisfies the differential equation

Since , by the definitions of and , satisfies (3.18), and hence (3.19) holds. By (3.19) and (3.36), we have

From this inequality and (3.37), it follows that

Multiplying this inequality by and integrating on , we have

For the left side of this inequality using integration by parts and (2.14), we have

Consequently, we obtain that

Since , by the definition of and (2.13) we see that (3.25) holds. From (3.25) and (3.42), we see that , which is a contradiction. Hence, satisfies the condition of Lemma 2.5 in . By Lemma 2.5 we have

Now, from (3.35) and (3.43) it follows that

Hence, has a fixed point in , which is a positive -periodic solution of (1.1).

Acknowledgment

This research supported by NNSF of China (11261053 and 11061031).

References

B. Liu, “Periodic solutions of a nonlinear second-order differential equation with deviating argument,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 313–321, 2005.

J.-W. Li and S. S. Cheng, “Periodic solutions of a second order forced sublinear differential equation with delay,” Applied Mathematics Letters, vol. 18, no. 12, pp. 1373–1380, 2005.

Y. Wang, H. Lian, and W. Ge, “Periodic solutions for a second order nonlinear functional differential equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 110–115, 2007.

J. Wu and Z. Wang, “Two periodic solutions of second-order neutral functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 677–689, 2007.

Y. Wu, “Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 1, pp. 433–443, 2009.

C. Guo and Z. Guo, “Existence of multiple periodic solutions for a class of second-order delay differential equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3285–3297, 2009.

B. Yang, R. Ma, and C. Gao, “Positive periodic solutions of delayed differential equations,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4538–4545, 2011.

Y. Li, “Positive periodic solutions of second-order differential equations with delays,” Abstract and Applied Analysis, vol. 2012, Article ID 829783, 13 pages, 2012.

Y. X. Li, “Positive periodic solutions of nonlinear second order ordinary differential equations,” Acta Mathematica Sinica, vol. 45, no. 3, pp. 481–488, 2002.

Y. Li, “Positive periodic solutions of first and second order ordinary differential equations,” Chinese Annals of Mathematics B, vol. 25, no. 3, pp. 413–420, 2004.

Y. Li and H. Fan, “Existence of positive periodic solutions for higher-order ordinary differential equations,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1715–1722, 2011.

F. Li and Z. Liang, “Existence of positive periodic solutions to nonlinear second order differential equations,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1256–1264, 2005.

B. Liu, L. Liu, and Y. Wu, “Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3337–3345, 2010.

P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643–662, 2003.

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, NY, USA, 1985.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, New York, NY, USA, 1988.