Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 945467, 15 pages

doi:10.1155/2012/945467

Research Article

Positive Periodic Solutions for Second-Order Ordinary Differential Equations with Derivative Terms and Singularity in Nonlinearities

Yongxiang Li and Xiaoyu Jiang

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

Received 17 July 2012; Accepted 26 August 2012

Academic Editor: Gabriel N. Gatica

Copyright © 2012 Yongxiang Li and Xiaoyu Jiang. 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 -periodic solutions are obtained for the second-order ordinary differential equation where, is a continuous function, which is -periodic in and may be singular at . 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 of the second-order ordinary differential equation with first-order derivative term in the nonlinearity

where the nonlinearity is a continuous function, which is -periodic in t and may be singular at .

The existence problems of periodic solutions for nonlinear second-order ordinary differential equations have attracted many authors’ attention and concern, and most works are on the special equation

that does not contain explicitly first-order derivative term in nonlinearity. Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (1.2). These theorems and methods are mainly the upper and lower solutions method and monotone iterative technique [1–4], the continuation method of topological degree [5–7], variational method and critical point theory, [8–10] and so forth.

In recent years, the fixed point theorems of cone mapping, especially the fixed point theorem of Krasnoselskii’s cone expansion or compression type, have been extensively applied to two-point boundary value problems of second-order ordinary differential equations, and some results of existence and multiplicity of positive solutions have been obtained, see [11–15]. Lately, the authors of [16–18] have also applied the Krasnoselskii’s fixed point theorem to periodic problems of second-order nonlinear ordinary differential equations, and obtained existence results of positive periodic solutions. In these works, the new discovered positivity of Green function of the corresponding linear second-order periodic boundary value problems plays an important role. The positivity guarantees that the integral operators of the second-order periodic problems are cone-preserving in the cone

in the Banach space , where is a constant. Hence the fixed point theorems of cone mapping can be applied to the second-order periodic problems. For more precise results using the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of second-order ordinary differential equation, see [19–22]. However, all of these works are on the special second-order equation (1.2), and few people consider the existence of the positive periodic solutions for the general second-order equation (1.1) that explicitly contains the first order derivative term.

The purpose of this paper is to extend the results of [16–22] to the general second-order equation (1.1). We will use the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of (1.1). For the periodic problem of (1.1), since the corresponding integral operator has no definition on the cone in , the argument methods used in [16–22] are not applicable. We will use a completely different method to treat (1.1). Our main results will be given in Section 3. Some preliminaries to discuss (1.1) are presented in Section 2.

2. Preliminaries

Let denote the Banach space of all continuous -periodic function with norm . Let be the Banach space of all continuous differentiable -periodic function with the norm

Generally, denotes the th-order continuous differentiable -periodic function space for . Let be the cone of all nonnegative functions in .

Let be a constant. For , we consider the linear second-order differential equation

The -periodic solutions of (2.2) are closely related with the linear second-order boundary value problem

see [19]. It is easy to see that Problem (2.3) has a unique solution, which is explicitly given by

where . We have the following Lemma.

Lemma 2.1. Let . Then for every , the linear equation (2.2) has a unique -periodic solution , which is given by

Moreover, is a completely continuous linear operator.

Proof. Taking the derivative in (2.5) and using the boundary condition of , we obtain that

Therefore, satisfies (2.2). Let ; it follows from (2.5) that

Hence, is an -periodic solution of (2.2). From the maximum principle for second-order periodic boundary value problems [4], it is easy to see that is the unique -periodic solution of (2.2).

From (2.5) and (2.6), we easily see that is a linear bounded operator. By the compactness of the embedding , is a completely continuous operator.

Since for every , by (2.5), if and , then the -periodic solution of (2.2) for every , and we term it the positive -periodic solution. Let

Define the cone in by

We have the following Lemma.

Lemma 2.2. Let . Then for every , the positive -periodic solution of (2.2) . Namely, .

Proof. Let , . For every , from (2.5) it follows that

and therefore,

Using (2.5), we obtain that

For every , since

we have

Hence, .

Now we consider the nonlinear equation (1.1). Hereafter, we assume that the nonlinearity satisfies the following condition.

(F0) There exists such that

Let , then for ,  , and (1.1) is rewritten to

For , if , then and by the definition of , for every . Hence

is well defined, and we can define the integral operator by

By the definition of operator , the positive -periodic solution of (1.1) is equivalent to the nontrivial fixed point of . From Assumption (F0), Lemmas 2.1 and 2.2, we easily see the following Lemma.

Lemma 2.3. , and is completely continuous.

We will find the nonzero fixed point of by using the fixed point index theory in cones. Since the singularity of at implies that has no definition at , the fixed point index theory in the cone cannot be directly applied to . We need to make some Preliminaries.

We recall some concepts and conclusions on the fixed point index in [23, 24]. Let be a Banach space and 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 a definition. One important fact is that if , then has a fixed point in . The following two lemmas are needed in our argument.

Lemma 2.4 (see [24]). Let be a bounded open subset of with and a completely continuous mapping. If for every and , then .

Lemma 2.5 (see [24]). Let be a bounded open subset of and a completely continuous mapping. If there exists an such that for every and , then .

We use Lemmas 2.4 and 2.5 to show the following fixed-point theorem in cones which is applicable to the operator defined by (2.18).

Theorem 2.6. Let be a Banach space and a closed convex cone. Assume and are bounded open subsets of with , . Let be a completely continuous mapping. If satisfies the following conditions:

(1)  for , ; (2) there exists such that for , , or the following conditions:(3) there exists such that for , ; (4)  for , ,

then has a fixed-point in .

Proof. By Dugundji’s extension theorem, the operator can be extended into a completely continuous operator from to , says .

If satisfies conditions (1) and (2) of Theorem 2.6, then also satisfies them. By Lemmas 2.4 and 2.5, respectively, we have

By the additivity of the fixed point index, we have

Hence has a fixed-point in . Since is an extension of , it follows that has a fixed-point in .

If satisfies conditions (3) and (4) of Theorem 2.6, with a similar count, we obtain that

This means that has a fixed-point in . Hence, has a fixed-point in .

Theorem 2.6 is an improvement of the fixed point theorem of Krasnoselskii’s cone expansion or compression. We will use it to discuss the existence of positive -periodic solutions of (1.1) in the next section.

3. Main Results

We consider the the existence of positive -periodic solutions of (1.1). Let satisfy Assumption (F0) and be -periodic in . Let be the constant defined by (2.8) and . To be convenient, we introduce the notations

Our main results are as follows.

Theorem 3.1. Let be continuous and be -periodic in t. If satisfies Assumption (F0) and the condition

(F1),

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

Theorem 3.2. Let be continuous and be -periodic in t. If satisfies Assumption (F0) and the conditions

(F2),

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

Noting that is an eigenvalue of the associated linear eigenvalue problems of (1.1) with periodic boundary condition, if one inequality concerning comparison with in (F1) or (F2) of Theorem 3.1 or Theorem 3.2 is not true, the existence of periodic solution to (1.1) cannot be guaranteed. Hence, the is the optimal value in condition (F1) and (F2).

In Theorem 3.1, the condition (F1) allows to have superlinear growth on and . For example,

satisfies (F0) with and (F1) with and .

In Theorem 3.2, the condition (F2) allows that has singularity at . For example,

satisfies (F0) with , and (F2) with and . The existence of periodic solutions for singular ordinary differential equations has been studied by several authors, see [20, 25, 26]. But the equations considered by these authors do not contain derivative term .

Proof of Theorem 3.1. Choose the working space . Let be the closed convex cone in defined by (2.9) and the operator defined by (2.18). 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 by Theorem 2.6 when is small enough and large enough.

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

Let . We now prove that satisfies the Condition (1) of Theorem 2.6, namely, for every and . In fact, if there exist and such that , then by definition of and Lemma 2.1, satisfies the differential equation

Since , by the definitions of and , we have

Hence from (3.5) it follows that

By this, (3.6), and the definition of we have

Integrating both sides of this inequality from to and using the periodicity of , we obtain that

Since , it follows that , which is a contradiction. Hence the Condition (1) of Theorem 2.6 holds.

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

Define a function by

Then is continuous. By (3.5) and Assumption (F0),

This implies that

Hence for every , , and , we have

Combining this with (3.11), it follows that

Choose . Clearly,. We show that satisfies the Condition (2) of Theorem 2.6 if is large enough, namely, for every and . In fact, if there exist and such that , since , by definition of and Lemma 2.1, satisfies the differential equation

From (3.17) and (3.16), it follows that

Integrating this inequality on and using the periodicity of , we get that

Since , by the definition of , we have

By the first inequality of (3.20), we have

From this and (3.19), it follows that

By this and the second inequality of (3.20), we have

Therefore, choose , then satisfies the Condition (2) of Theorem 2.6.

Now by the first part of Theorem 2.6, 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 use Theorem 2.6 to prove that the operator has a fixed point in if is small enough and large enough.

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

Let and . We prove that satisfies the Condition (3) of Theorem 2.6, namely, for every and . In fact, if there exist and such that , since , by definition of and Lemma 2.1, satisfies the differential equation

Since , by the definitions of and , satisfies (3.7). From (3.7), and (3.24) it follows that

By this, (3.25), and the definition of , we have

Integrating this inequality on and using the periodicity of , we obtain that

Since , from this inequality it follows that , which is a contradiction. Hence satisfies the Condition (3) of Theorem 2.6.

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

Choosing , we show that satisfies the Condition (4) of Theorem 2.6, namely, for every and . In fact, if there exist and such that , then by the definition of and Lemma 2.1, satisfies the differential equation

Since , by the definition of , satisfies (3.20). By the second inequality of (3.20), we have

Consequently,

By (3.32) and the first inequality of (3.20), we have

From this, the second inequality of (3.20) and (3.29), it follows that

By this and (3.30), we have

Integrating this inequality on and using the periodicity of , we obtain that

Since , from this inequality it follows that , which is a contradiction. This means that satisfies the Condition (4) of Theorem 2.6.

By the second part of Theorem 2.6, has a fixed point in , which is a positive -periodic solution of (1.1).

Example 3.3. Consider the second-order differential equation

where . If and for , then satisfies the conditions (F0) and (F1). By Theorem 3.1, (3.37) has at least one positive -periodic solution.

Example 3.4. Consider the singular differential equation:

where . If and for , then satisfies the conditions (F0) and (F2). By Theorem 3.2, the (3.38) has a positive -periodic solution.

4. Remarks

Our discussion on the existence of the positive -periodic solutions to (1.1) is applicable to the following ordinary differential equation:

where the nonlinearity is continuous and is -periodic in . For (4.1), we need the following assumption.

There exists such that

Similarly to Lemma 2.1, we have the following conclusion.

Lemma 4.1. Let be a constant. Then for every , the linear second order differential equation

has a unique -periodic solution , which is given by

where is the unique solution of the linear second-order boundary value problem

which is explicitly given by

with .

Since

we renew to define and by

Now, using the similar arguments to Theorems 3.1 and 3.2, we can obtain the following results.

Theorem 4.2. Let be continuous and be -periodic in t. If satisfies Assumption (F0)* and the condition

(F1),

then (4.1) has at least one positive -periodic solution.

Theorem 4.3. Let be continuous and be -periodic in t. If satisfies Assumption (F0)* and the conditions

(F2),

then (4.1) has at least one positive -periodic solution.

Theorems 4.2 and 4.3 improve and extend some results in References [18, 19, 22].

Acknowledgment

This paper is supported by NNSFs of China (11261053 and 11061031).

References

S. Leela, “Monotone method for second order periodic boundary value problems,” Nonlinear Analysis, vol. 7, no. 4, pp. 349–355, 1983.

J. J. Nieto, “Nonlinear second-order periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 130, no. 1, pp. 22–29, 1988.

A. Cabada and J. J. Nieto, “A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 181–189, 1990.

A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302–320, 1994.

J. P. Gossez and P. Omari, “Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance,” Journal of Differential Equations, vol. 94, no. 1, pp. 67–82, 1991.

P. Omari, G. Villari, and F. Zanolin, “Periodic solutions of the Liénard equation with one-sided growth restrictions,” Journal of Differential Equations, vol. 67, no. 2, pp. 278–293, 1987.

W. G. Ge, “On the existence of harmonic solutions of Liénard systems,” Nonlinear Analysis, vol. 16, no. 2, pp. 183–190, 1991.

J. Mawhin and M. Willem, “Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations,” Journal of Differential Equations, vol. 52, no. 2, pp. 264–287, 1984.

V. C. Zelati, “Periodic solutions of dynamical systems with bounded potential,” Journal of Differential Equations, vol. 67, no. 3, pp. 400–413, 1987.

L. Lassoued, “Periodic solutions of a second order superquadratic system with a change of sign in the potential,” Journal of Differential Equations, vol. 93, no. 1, pp. 1–18, 1991.

H. Wang, “On the existence of positive solutions for semilinear elliptic equations in the annulus,” Journal of Differential Equations, vol. 109, no. 1, pp. 1–7, 1994.

L. H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations,” Proceedings of the American Mathematical Society, vol. 120, no. 3, pp. 743–748, 1994.

Z. Liu and F. Li, “Multiple positive solutions of nonlinear two-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 203, no. 3, pp. 610–625, 1996.

J. Henderson and H. Wang, “Positive solutions for nonlinear eigenvalue problems,” Journal of Mathematical Analysis and Applications, vol. 208, no. 1, pp. 252–259, 1997.

Y. Li, “Positive solutions of second-order boundary value problems with sign-changing nonlinear terms,” Journal of Mathematical Analysis and Applications, vol. 282, no. 1, pp. 232–240, 2003.

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

F. M. Atici and G. Sh. Guseinov, “On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 341–356, 2001.

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.

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. X. Li, “Oscillatory periodic solutions of nonlinear second order ordinary differential equations,” Acta Mathematica Sinica, vol. 21, no. 3, pp. 491–496, 2005.

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.

J. R. Graef, L. Kong, and H. Wang, “Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,” Journal of Differential Equations, vol. 245, no. 5, pp. 1185–1197, 2008.

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

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

I. Rachůnková, M. Tvrdý, and I. Vrkoč, “Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems,” Journal of Differential Equations, vol. 176, no. 2, pp. 445–469, 2001.

L. Kong, S. Wang, and J. Wang, “Positive solution of a singular nonlinear third-order periodic boundary value problem,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 247–253, 2001.