Abstract and Applied Analysis

Volume 2013 (2013), Article ID 218346, 6 pages

http://dx.doi.org/10.1155/2013/218346

Research Article

Unbounded Solutions of Asymmetric Oscillator

Tieguo Ji and Zhenhui Zhang

College of Science, Hebei University of Engineering, Handan 056038, China

Received 4 July 2013; Revised 26 September 2013; Accepted 1 October 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Tieguo Ji and Zhenhui Zhang. 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.

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation , where are positive constants, and is continuous, bounded, and -periodic in for some .

1. Introduction

In 1960s, Lazer and Leach [1] considered the existence of -periodic solution of the following differential equation:

where ,   is continuous and -periodic, and is bounded and continuous. If exist, they showed that (1) has at least one -periodic solution provided:

Assume are positive constants satisfying

Fabry and Mawhin [2] considered the following asymmetric oscillator:

where are continuous and bounded, is continuous and -periodic, , and . Moreover, the limits exist; has a sublinear primitive; that is,

They introduced the function

where is the solution of the following initial value problem:

and gave approximations of solutions of (4) with large initial values. Moreover, they proved that (4) has unbounded solutions if the function has only simple zeros.

Later, Kunze et al. [3] considered the following resonant oscillator:

where , is bounded, is -periodic, and . They showed that if

where is bounded, , then every solution of (8) is unbounded.

Wang [4] considered the unboundedness of solutions of the equation:

where and is continuous and -periodic. By using the well-known Birkhoff Ergodic theorem [5], he showed that if the function is bounded and the limits exist, then is a sufficient condition for the existence of unbounded solutions of (10).

Liu [6] discussed the boundedness of all solutions of the following nonlinear equation:

where

with ,  . Let ; he defined a -periodic function as

where exist, , , and is the solution of

with the initial value . Under some smoothness conditions, he showed that all solutions of (11) are bounded if the function has no zero for all .

Recently, Li and Zhang [7] studied the unboundedness of solutions of the following asymmetric oscillator:

where are positive constants satisfying nonresonance condition:

and are bounded and is -periodic in . By using the similar method used in [4], they obtained some sufficient conditions for the existence of unbounded solutions of (15). For more recent results on the boundedness or unboundedness of solutions of differential equations of second order, we refer to [6, 8–18] and the references therein.

Assume are positive constants. Let be the solution of the following initial value problem:

then it is well-known that is -periodic with

In this paper, we consider the following asymmetric oscillator:

where ,  , is continuous, bounded, and -periodic in , and the limits exist. We assume that satisfies the following resonance condition:

A solution of (19) is called to have large initial value if . We will give some sufficient conditions for the unboundedness of solutions of (19). Especially, if , we will obtain higher order approximation and corresponding sufficient conditions for the unboundedness of solutions of (19) with large initial values. The results of this paper are new which improve some relative results on the literature in some sense. Throughout this paper, we assume with .

The main results of this paper are the following.

Theorem 1. Introduce a -periodic function

where

Then all solutions of (19) with large initial values are unbounded provided that the function has only finite number of zeros in and all its zeros are simple.

Corollary 2. Assume is -periodic and define a -periodic function as

If has only finite number of zeros in and all its zeros are simple, then all solutions of (19) with large initial values are unbounded.

Theorem 3. Consider (19) with is -periodic. Let be given in Corollary 2 and assume . Define -periodic functions and as

Then all solutions of (19) with large initial values are unbounded provided one of the following conditions holds:

(i) and has only finite number of zeros in and all of them are simple;(ii) and the function has no zero in .

2. Generalized Polar Coordinate Transformation

Let be the solution of the following initial value problem:

and define a function as

then it is easy to verify that ; is -periodic and satisfies the identity:

Moreover, one can verify that is the solution of (17).

For ,   , define the generalized polar coordinate transformation by

Under the mapping and by using (27), it is not difficult to verify that (19) can be transformed into the following planar system:

where are given by (28).

Lemma 4. Let be the solution of (29) satisfying initial condition , then for ; the Poincaré mapping satisfies

where is given by (21).

Proof. For , , by the boundedness of and , we obtain from (29)

which implies

Substituting (32) into (29) and integrating it over , we obtain

Substituting (32) and (33) into (29) and integrating it over , we obtain

where

, and are given by (21). Moreover, it is easy to verify from (21) and (35) that for all ,

Lemma 5. Let and be -periodic. Assume , where is given in Corollary 2 and is the Poincaré mapping of the solution of (29). Then for , there exists the following higher order approximation:

where and are given in Theorem 3.

Proof. Since in (29), we integrate the first equation of (29) from to . For , we obtain

Substituting (38) into the second equation of (29) and integrating it over , we get

where

Substituting (38) and (40) into (29) and integrating over we get

where

By (41) we get the following approximation

Substituting (43) into (29) and integrating it over , we get

where

Substituting (40) and (42) into (45) and then integrating it over , we obtain

where

Let

then (46) reduces to (37).

Lemma 6. If , then

Proof. It follows from the second equation of (48) that

which, together with ,   ,   implies (50).

3. Unboundedness Motions of Planar Mappings

In this section, we adopt the notations used in [9]. Given , let the set be the exterior of the open ball centered at the origin with radius ; that is,

then .

Define ; then the points and the group distance in can be described, respectively, by

Let be a one to one and continuous mapping. We denote its lift by in the form

where is -periodic, is a constant, and are continuous, -periodic and satisfy .

Given a point , let be the unique solution of the initial value problem for the differential equation:

This solution is defined in a maximal interval

where are certain numbers in the set satisfying

The solution is said to be defined in the future if and is said to be defined in the past if .

Let be the ordered sequence of zeros of in such that

Lemma 7. If has a simple zero ; that is, , then there exists orbits of (54) which are to be defined in the future and satisfy

or they are defined in the past and satisfy

Moreover, if and all zeros of are simple, then every orbit of (54) with large initial value is either to be defined in the future satisfying

or is defined in the past satisfying

The proof of the above lemma is similar to that of Proposition  3.1 in [9].

Lemma 8. Assume that for ,  ,   has the following expression:

where ,   , and are continuous, -periodic in and satisfy ,   uniformly in . Moreover If ,   for  all  , then all orbits of (63) with large initial values are defined in the future and satisfy . If ,   for  all  , then all orbits of (63) with large initial values are defined in the past and satisfy .

Proof. Let ,  for  all  , then and it follows from (63) that for ,

By induction, we get from (65)

and replacing by by we get from (64)

Obviously, (66) implies that the orbit is defined in the future. Next we claim that the orbit is unbounded in the future. In fact, (67) implies that is monotone increasing, hence the limit exists. If , taking the limit on both sides of (67) we obtain

which is a contradiction. Hence the orbit of (63) is defined in the future and satisfies . Similarly, if ,   for  all  , we can prove that the orbit of (63) is defined in the past and satisfies

4. Proofs of Theorems and an Example

Proof of Theorem 1. Since , by Lemma 4, for , the mapping has the approximate expression of (30) with , where is given by Theorem 1 and . Now Lemma 7 implies that all solutions of (30) with go to infinity either in the future or in the past, which implies that the solutions of (19) with large initial values are unbounded.

Proof of Theorem 3. By Lemma 5, for ,   , the mapping has the form (37) with and and are given by Theorem 3. If , then it follows from Lemma 6 that

Now, Lemma 7 implies that all orbits of (37) with go to infinity either in the future or in the past, which means that all solutions of (19) with large initial values are unbounded.

If , then by assumption of Theorem 3, and the function has no sign-changing. Lemma 8 implies that all orbits of (37) with go to infinity either in the future or in the past, which implies that all solutions of (19) with large initial values are unbounded.

Example 1. For (19), we assume ,   ,   ,  , and for some . We write in Fourier series form as

where

It follows from Corollary 2 that

and it is not difficult to show that . Then has only finite number of zeros in and all of them are simple. Corollary 2 implies that all solutions of (19) with large initial values are unbounded.

Acknowledgments

This paper is supported by Chinese NSF (11371117) and Hebei Province NSF (A2012402036).

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