Journal of Applied Mathematics

Volume 2012 (2012), Article ID 615303, 13 pages

doi:10.1155/2012/615303

Research Article

Existence and Uniqueness of Homoclinic Solution for a Class of Nonlinear Second-Order Differential Equations

Lijuan Chen and Shiping Lu

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

Received 11 September 2012; Accepted 13 November 2012

Academic Editor: Julián López-Gómez

Copyright © 2012 Lijuan Chen and Shiping Lu. 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 authors study the existence and uniqueness of a set with -periodic solutions for a class of second-order differential equations by using Mawhin's continuation theorem and some analysis methods, and then a unique homoclinic orbit is obtained as a limit point of the above set of -periodic solutions.

1. Introduction

In this paper, we study the existence and uniqueness of homoclinic solutions for the following nonlinear second-order differential equations:

where ,  ,   and are all in .

As usual we say that a nonzero solution of (1.1) is homoclinic (to 0) if and as .

Equation (1.1) is important in the applied sciences such as nonlinear vibration of masses, see [1–3] and the references therein. But most of the authors in those papers are interested in the study of problems of periodic solutions. Recently, the existence of homoclinic solutions for some second-order ordinary differential equation (system) has been extensively studied by using critical point theory, see [4–13] and the references therein. For example, in [9], by using the Mountain Pass theorem, Lv et al. discussed the existence of homoclinic solutions for the following second-order Hamiltonian systems:

and in [13], the authors by means of variational method studied the problem of homoclinic solutions for the forced pendulum equation without the first derivative term. But, as far as we know, there were few papers studying the existence of homoclinic solution for the equation such as (1.1). This is due to the fact that (1.1) contains the first derivative term . This implies that the differential equation is not the Euler Lagrange equation associated with some functional . So the method of critical point theory (or variational method) in [4–13] cannot be applied directly. Although paper [13] discussed the existence of homoclinic solutions for the following equation containing the first derivative term:

the term containing the first derivative is only linear with respect to .

In order to investigate the homoclinic solutions to (1.1), firstly, we study the existence of -periodic solutions to the following equation for each :

where is a -periodic function such that

is a given constant, and is a constant independent of . Then a homoclinic solution to (1.1) is obtained as a limit point of the set , where is an arbitrary -periodic solution to (1.4) for each .

The significance of present paper is that we not only investigate the existence of homoclinic solution to (1.1), but also study the uniqueness of the homoclinic solution and, the existence of -periodic solutions to (1.4) is obtained by using Mawhin’s continuation theorem [14], not by using the methods of critical point theory, which is quite different from the approaches of [4–13, 15]. Furthermore, the method to obtain the homoclinic solution to (1.1) is also different from the corresponding ones of [15].

2. Main Lemmas

For each , let , , the norms of and are defined by and , respectively, then and are all Banach spaces. Furthermore for , , where .

Lemma 2.1 (see [12]). Let and , then for every , the following inequality holds:

where are constants.

Lemma 2.2 (see [12]). Let , then the following inequality holds:

where and are constants with and .

In order to use Mawhin’s continuation theorem for investigating the existence of -periodic solutions to (1.4), we give some definitions associated with Mawhin’s continuation theorem.

Definition 2.3 (see [14]). Let and be two Banach spaces with norms and , respectively. A linear operator

is said to be a Fredholm opeartor with index zero provided that

(1) is a closed subset of ; (2).

If is a Fredholm operator with index zero, then and . Let and be the continuous projectors. Clearly, , thus the restriction is invertible. Denote by the inverse of .

Definition 2.4 (see [14]). Let and be two Banach spaces with norms and , respectively, and the operator

is a Fredholm operator with index zero, is an open bounded set with . A continuous operator is said to be -compact in , provided that

(1) is a relative compact set of ; (2) is a bounded set of .

Lemma 2.5 (see [14]). Suppose that and are two Banach spaces, and is a Fredholm operator with index zero. Furthermore, is an open bounded subset and is -compact on . If all the following conditions hold:

(1), for all ;(2), for all ;(3),

where is an isomorphism. Then equation has a solution on .

Lemma 2.6. Assume that there are positive constants ,  ,  ,  , and with , such that the following conditions hold.

(A1), and .(A2)

(A3) with for all .

Then for every , (1.4) possesses a -periodic solution.

Remark 2.7. From (1.5), we see

which together with assumption (A1) yields that and are two constants independent of .

Similarly, we have that and are two constants independent of .

Proof. Set ,  ,  ,  ,   where , and

Clearly, , , which implies that is a closed subset of , and . So is a Fredholm operator with index zero. Let

be defined respectively by and let

Then has a unique continuous pseudo-inverse on defined by , where

For each open bounded set , from the above formula, it is easy to see that the mapper is -compact on .

Step  1. For each , let , that is,

We will show that is bounded in . Suppose that , then

Multiplying both sides of (2.12) by and integrating on the interval , we have from assumption (A2) that

By using hólder inequality, we get

which together with the conclusion of Remark 2.7 shows

Clearly, is a constant independent of and .

Multiplying both sides of (2.12) by and integrating on the interval , we have

It follows from (2.16) and assumption (A3) that

which implies

By using Lemma 2.2, we have

Clearly, is a constant independent of and .

On the other hand, multiplying both sides of (2.12) by and integrating on the interval , we have

It follows from assumption (A2) that

which together with (2.19) and results in

that is,

Therefore

where is a constant independent of and . By using Lemma 2.2 again, we get

Obviously, is a constant independent of and . Therefore, if , then by (2.19) we see that

Clearly, is a constant independent of and ; that is, is uniformly bounded for all and .

Step  2. From assumptions (A2) and (A3), we see that there must be a constant such that and . Set , where . We will show that , for all .

In fact, by assumption (A2), we see that , and if , then or . So

where . This implies that , for all .

Step  3. Set , , we will show .

Let , for all , when , we have and

So for all ,  , and then

Therefore, by Lemma 2.5, (1.4) has a -periodic solution .

Remark 2.8. Suppose that all the conditions in Lemma 2.6 hold. We see that for each , (1.4) has a -periodic solution . This implies that

Furthermore, as same as the proof of step  1 in Lemma 2.6 with replacing by , we have

where and are two positive constants independent of .

Lemma 2.9 (see [12]). Let be the -periodic solution for (1.4) and satisfies (2.30) and (2.31) for all . Then there exists a function such that for each interval , there is a subsequence of with uniformly on .

3. Main Results

Theorem 3.1. Suppose that assumptions (A1), (A2), and (A3) in Lemma 2.6 hold. Then (1.1) has a unique homoclinic solution.

Proof. Since assumptions (A1), (A2), onsisting of Kuratowski operations we used following principles and (A3) in Lemma 2.6 hold, by using Lemma 2.6, we see that (1.4) has a -periodic solution satisfying (2.30) and (2.31) for each . It follows from Lemma 2.9 that there exists a such that for each interval , there is a subsequence of satisfying uniformly on . Below, we will show that is just a unique homoclinic solution to (1.1).

Step  1. We show that is a solution of (1.1).

In view of being a -periodic solution to (1.4), we have

Take such that , there exists such that for all

Integrating (3.2) from to , we have

Since Lemma 2.9 shows that uniformly on and uniformly on as , let in (3.3), we get

In view of and are arbitrary, (3.4) shows that is a solution of (1.1).

Step  2. We prove that , as .

Obviously, for every , there exists such that for all , we have

It follows that

and then

which yields

By using Lemma 2.1, as ,

So we have , as .

Step  3. We will show that

From the Remark 2.8 and Lemma 2.9, we have

which together with (1.1) implies that

where and . If , as , then there exist a and a sequence such that

From this, we have for

It follows that

which contradicts (3.6), and so (3.10) holds.

Step  4. Finally, we will prove that (1.1) possesses a unique homoclinic solution. In order to do it, let , where and are two arbitrary homoclinic solutions of (1.1). Then

We will show that

If (3.17) does not hold, then there must be a such that

or

If , then from (3.16), we see that there is a constant such that and for . Let such that , then

that is,

So and , and then from (1.1), we see

By using the condition (A3), we have that

which contradicts to (3.20). This contradiction implies that (3.18) does not hold. Similarly, we can prove that (3.19) does not hold, either. So .

As an application, we consider the following example:

where are constants and, . Corresponding to (1.1), we can chose and such that assumptions (A2) and (A3) hold. Furthermore, by the direct calculation, we can easily obtain that

This implies that assumption (A1) also holds. So by applying Theorem 3.1, we know that (3.25) possesses a unique homoclinic solution.

Acknowledgments

The authors are very grateful to the referee for her/his careful reading of the original paper and for her/his valuable suggestions for improving this paper. This work was sponsored by the key NNSF of China (no. 11271197) and Science Foundation of NUIST (no. 20090202; 2012r101).

References

E. N. Dancer, “On the ranges of certain damped nonlinear differential equations,” Annali di Matematica Pura ed Applicata, vol. 119, pp. 281–295, 1979.

P. Girg and F. Roca, “On the range of certain pendulum-type equations,” Journal of Mathematical Analysis and Applications, vol. 249, no. 2, pp. 445–462, 2000.

P. Amster and M. C. Mariani, “Some results on the forced pendulum equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 7, pp. 1874–1880, 2008.

M. Izydorek and J. Janczewska, “Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1119–1127, 2007.

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.

X. Lv, S. Lu, and P. Yan, “Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 7-8, pp. 3484–3490, 2010.

S. Lu, “Homoclinic solutions for a class of second-order -Laplacian differential systems with delay,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 780–788, 2011.

X. H. Tang and L. Xiao, “Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 586–594, 2009.

X. Lv, S. Lu, and P. Yan, “Existence of homoclinic solutions for a class of second-order Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 72, no. 1, pp. 390–398, 2010.

Y. Xu and M. Huang, “Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity,” Journal of Differential Equations, vol. 244, no. 3, pp. 582–598, 2008.

X. H. Tang and L. Xiao, “Homoclinic solutions for a class of second-order Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 3-4, pp. 1140–1152, 2009.

X. H. Tang and L. Xiao, “Homoclinic solutions for ordinary -Laplacian systems with a coercive potential,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 3-4, pp. 1124–1132, 2009.

D. C. Offin and H. F. Yu, “Homoclinic orbit in the forced pendulum system,” Fields Institute of Communication, vol. 8, pp. 113–126, 1996.

R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.

C. Vladimirescu, “An existence result for homoclinic solutions to a nonlinear second-order ODE through differential inequalities,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 68, no. 10, pp. 3217–3223, 2008.