Research Article | Open Access
Qiongfen Zhang, X. H. Tang, "Existence of Homoclinic Orbits for a Class of Asymptotically -Linear Difference Systems with -Laplacian", Abstract and Applied Analysis, vol. 2011, Article ID 351562, 17 pages, 2011. https://doi.org/10.1155/2011/351562
Existence of Homoclinic Orbits for a Class of Asymptotically -Linear Difference Systems with -Laplacian
Abstract
By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptotically -linear difference system with -Laplacian , where , , , are not periodic in , and W is asymptotically -linear at infinity.
1. Introduction
Consider the following -Laplacian difference system: where is the forward difference operator defined by , , , , , , : are not periodic in , is asymptotically -linear at infinity, and and are continuously differentiable in . As usual, we say that a solution of (1.1) is homoclinic (to 0) if as . In addition, if , then is called a nontrivial homoclinic solution.
When , (1.1) can be regarded as a discrete analogue of the following second-order Hamiltonian system:
The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by PoincarΓ© [1]. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. For the existence of homoclinic solutions of problem (1.2), one can refer to the papers [2β5].
Difference equations usually describe evolution of certain phenomena over the course of time. For example, if a certain population has discrete generations, the size of the th generation is a function of the th generation . In fact, difference equations provide a natural description of many discrete models in real world. Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control, it is of practical importance to investigate the solutions of difference equations. For more details about difference equations, we refer the readers to the books [6β8].
In some recent papers [9β20], the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Along this direction, several authors [21β28] used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated by the above papers, we consider the existence of homoclinic orbits for problem (1.1) by using the variant version of Mountain Pass Theorem. Our result is new, which seems not to have been considered in the literature. Here is our main result.
Theorem 1.1. Suppose that and satisfy the following conditions. (K1)There are two positive constants and such that (K2)There is a positive constant such that (W1), as uniformly for .(W2)There exists a constant such that (W3)There exists a function such that (W4), and for any fixed , Then problem (1.1) has at least one nontrivial homoclinic solution.
Remark 1.2. The function in this paper is asymptotically -linear at infinity. The behavior of the gradient of at infinity is like that of a function , where is a real function but not a matrix function. To the best of our knowledge, similar results of this kind of -Laplacian difference systems with asymptotically -linear at infinity cannot be found in the literature. From this point, our result is new.
2. Preliminaries
Let and for , let Then is a uniform convex Banach space with this norm. As usual, for , let and their norms are given by respectively.
For any , let
To prove our results, we need the following generalization of Lebesgue's dominated convergence theorem.
Lemma 2.1 (see [29]). Let and be two sequences of measurable functions on a measurable set , and let If then
Lemma 2.2. For ,
Proof. Since , it follows that . Hence, there exists such that Hence, we have
Lemma 2.3. Suppose that (K1), (K2), and (W2) hold. If in , then and in , where satisfies .
Proof. From (K1) and (K2), we have Hence, from (2.12), we have Moreover, since in and for almost every , hence, It follows from Lemmaββ2.1, (2.13), and the previous equations that This shows that in . By a similar proof, we can prove that in . The proof is complete.
Lemma 2.4. Under the conditions of Theorem 1.1, one has for , which yields that Moreover, is continuously FrΓ©chet-differential defined on ; that is, and any critical point of on is classical solution of (1.1) with .
Proof. Firstly, we show that . Let , by (2.9) and (K1), we have By (W2), we get Hence, from (2.9) and (2.19), we have It follows from (2.5), (2.18), and (2.20) that . Next we prove that . Rewrite as follows: where It is easy to check that and Next, we prove that , and For any and for any function , by (K2), we have Then by the previous equations and Lebesgue's dominated convergence theorem, we have Similarly, we can prove that (2.25) holds by using (W2) instead of (K2). Finally, we prove that . Let in ; then by Lemma 2.3, we have This shows that . Similarly, we can prove that . Furthermore, by a standard argument, it is easy to show that the critical points of in are classical solutions of (1.1) with . The proof is complete.
Lemma 2.5 (see [30]). Let be a real Banach space with its dual space and suppose that satisfies for some , , and with . Let be characterized by where is the set of continuous paths joining 0 to ; then there exists such that
3. Proof of Theorem 1.1
Proof of Theorem 1.1. We divide the proof of Theorem 1.1 into three steps.
Step 1. From (W1), there exists such that
where . From (3.1), we have
Let and ; then from (2.9), we obtain
which together with (2.9), (3.2), and (K1) implies that
Step 2. From (K1), we have
By (W2) and (W3), we get
Let ; it follows from (W2), (W3), (2.19), and (3.6) that
Define with
By an easy calculation, we have
In what follows, we prove that for some with , as . Otherwise, there exist a sequence with as and a positive constant such that for all . From (3.5), we obtain
It follows from (3.7) that
Hence, from Lebesgue's dominated theorem and (3.11), we have
It follows from (3.8), (3.9), (3.10), and (3.12) that
which is a contradiction. Hence, there exists with such that .Step 3. From Step 1, Step 2, and Lemma 2.5, we know that there is a sequence such that
where is the dual space of . In the following, we will prove that is bounded in . Otherwise, assume that as . Let ; we have . It follows from (2.5), (2.16), (3.14), and (K2) that
Set for . Then from (3.15), we have
From (K1), (K2), and (3.14), we get
which implies that
Let . From (W1), there exists such that
Since , it follows from (2.9) and (3.19) that
For , let
Thus, from (W4), we have as , which together with (3.16) implies that
Hence, we can take sufficiently large such that
The previous inequality and (W2) imply that
Next, for the previous , let
From (W4), we have and
From (3.15) and (3.26), we get
which implies that
Therefore, there exists such that
It follows from (3.20), (3.24), and (3.29) that
which implies that
but this contradicts to (3.18). Hence, is bounded in .
Going to a subsequence if necessary, we may assume that there exists such that as . In order to prove our theorem, it is sufficient to show that . For any with , let for and let for . Then from (2.16), we have
Since as and in , it follows from (3.14) that
It follows from (3.32) and (3.33) that as .
For any , and assume that for some with , . Since
then, we have
as . Noting that
Hence, we have
which implies that ; that is, is a critical point of . From (K1) and (W1), we know that . In fact, if , we have from (2.5), (K1), and (W1) that . On the other hand, from Step 1, Step 2, and Lemma 2.5, we know that . This is a contradiction. The proof of Theorem 1.1 is complete.
4. An Example
Example 4.1. In problem (1.1), let , and where with . One can easily check that satisfies conditions (K1) and (K2) with , , and . An easy computation shows that Then it is easy to check that satisfies (W1)β(W4). Hence, and satisfy all the conditions of Theorem 1.1 and then problem (1.1) has at least one nontrivial homoclinic solution.
Acknowledgment
This work is partially supported by the NNSF (no. 10771215) of China.
References
- H. PoincarΓ©, Les MΓ©thodes Nouvelles de la MΓ©canique CΓ©leste, Gauthier-Villars, Pairs, France, 1899.
- M. Izydorek and J. Janczewska, βHomoclinic solutions for a class of the second order Hamiltonian systems,β Journal of Differential Equations, vol. 219, no. 2, pp. 375β389, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- X. H. Tang and L. Xiao, βHomoclinic solutions for a class of second-order Hamiltonian systems,β Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1140β1152, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- L. L. Wan and C. L. Tang, βHomoclinic orbits for a class of the second order Hamiltonian systems,β Acta Mathematica Scientia Series B, vol. 30, no. 1, pp. 312β318, 2010. View at: Publisher Site | Google Scholar
- Q. F. Zhang and X. H. Tang, βExistence of homoclinic solutions for a class of second-order non-autonomous Hamiltonian systems,β Mathematica Slovaca. In press. View at: Google Scholar
- R. P. Agarwal, Difference Equations and Inequalities, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at: Zentralblatt MATH
- C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic Publishers Group, Dordrecht, 1996, Difference equations, continued fractions, and Riccati equation. View at: Zentralblatt MATH
- S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1999.
- R. P. Agarwal and J. Popenda, βPeriodic solutions of first order linear difference equations,β Mathematical and Computer Modelling, vol. 22, no. 1, pp. 11β19, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- R. P. Agarwal, K. Perera, and D. O'Regan, βMultiple positive solutions of singular discrete p-Laplacian problems via variational methods,β Advances in Difference Equations, no. 2, pp. 93β99, 2005. View at: Google Scholar
- Z. M. Guo and J. S. Yu, βExistence of periodic and subharmonic solutions for second-order superlinear difference equations,β Science in China Series A, vol. 46, no. 4, pp. 506β515, 2003. View at: Publisher Site | Google Scholar
- Z. M. Guo and J. S. Yu, βPeriodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,β Nonlinear Analysis: Theory, Methods & Applications, vol. 55, no. 7-8, pp. 969β983, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- Z. M. Guo and J. S. Yu, βThe existence of periodic and subharmonic solutions of subquadratic second order difference equations,β Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419β430, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- H. H. Liang and P. X. Weng, βExistence and multiple solutions for a second-order difference boundary value problem via critical point theory,β Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 511β520, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- J. Rodriguez and D. L. Etheridge, βPeriodic solutions of nonlinear second-order difference equations,β Advances in Difference Equations, no. 2, pp. 173β192, 2005. View at: Google Scholar | Zentralblatt MATH
- Y.-F. Xue and C.-L. Tang, βExistence of a periodic solution for subquadratic second-order discrete Hamiltonian system,β Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2072β2080, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- J. S. Yu, Z. M. Guo, and X. Zou, βPeriodic solutions of second order self-adjoint difference equations,β Journal of the London Mathematical Society, vol. 71, no. 2, pp. 146β160, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- J. S. Yu, Y. H. Long, and Z. M. Guo, βSubharmonic solutions with prescribed minimal period of a discrete forced pendulum equation,β Journal of Dynamics and Differential Equations, vol. 16, no. 2, pp. 575β586, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- J. S. Yu, X. Q. Deng, and Z. M. Guo, βPeriodic solutions of a discrete Hamiltonian system with a change of sign in the potential,β Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1140β1151, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- Z. M. Zhou, J. S. Yu, and Z. M. Guo, βPeriodic solutions of higher-dimensional discrete systems,β Proceedings of the Royal Society of Edinburgh, vol. 134, no. 5, pp. 1013β1022, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- P. Chen and X. H. Tang, βExistence of infinitely many homoclinic orbits for fourth-order difference systems containing both advance and retardation,β Applied Mathematics and Computation, vol. 217, no. 9, pp. 4408β4415, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- P. Chen and X. H. Tang, βExistence of homoclinic solutions for a class of nonlinear difference equations,β Advances in Difference Equations, Article ID 470375, 19 pages, 2010. View at: Google Scholar | Zentralblatt MATH
- P. Chen and X. Tang, βExistence of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations,β Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 485β505, 2011. View at: Publisher Site | Google Scholar
- H. Fang and D. P. Zhao, βExistence of nontrivial homoclinic orbits for fourth-order difference equations,β Applied Mathematics and Computation, vol. 214, no. 1, pp. 163β170, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- X. Lin and X. H. Tang, βExistence of infinitely many homoclinic orbits in discrete Hamiltonian systems,β Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 59β72, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- M. Ma and Z. M. Guo, βHomoclinic orbits and subharmonics for nonlinear second order difference equations,β Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1737β1745, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- M. Ma and Z. M. Guo, βHomoclinic orbits for second order self-adjoint difference equations,β Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 513β521, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- J. S. Yu, H. P. Shi, and Z. M. Guo, βHomoclinic orbits for nonlinear difference equations containing both advance and retardation,β Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 799β806, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
- H. L. Royden, Real Analysis, The Macmillan, New York, NY, USA, 2nd edition, 1968.
- G. Cerami, βAn existence criterion for the critical points on unbounded manifolds,β Istituto Lombardo Accademia di Scienze e Lettere, vol. 112, no. 2, pp. 332β336, 1978. View at: Google Scholar | Zentralblatt MATH
Copyright
Copyright Β© 2011 Qiongfen Zhang and X. H. Tang. 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.