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New potential condition on homoclinic orbits for a class of discrete Hamiltonian systems
Advances in Difference Equations volume 2014, Article number: 73 (2014)
In the present paper, we establish an existence criterion to guarantee that the second-order self-adjoint discrete Hamiltonian system has a nontrivial homoclinic solution, which does not need periodicity and coercivity conditions on .
MSC:39A11, 58E05, 70H05.
Consider the second-order self-adjoint discrete Hamiltonian system
where , , is the forward difference, and .
As usual, we say that a solution of system (1.1) is homoclinic (to 0) if as . In addition, if then is called a nontrivial homoclinic solution.
The existence and multiplicity of homoclinic solutions of system (1.1) or its special forms have been investigated by many authors. Papers [3–8] deal with the periodic case where p, L and W are periodic in n or independent of n. In contrast, if the periodicity is lost, because of lack of compactness of the Sobolev embedding, up to our knowledge, all existence results require a coercivity condition on L:
In this paper, we derive an existence result which does not need periodicity and coercivity conditions on . To state our results precisely, we make the following assumptions.
is real symmetric positive definite matrix for all .
is real symmetric nonnegative definite matrix for all , and there exist a positive integer and such that
where here and in the sequel, denotes the standard inner product in and is the induced norm.
(W1) is continuously differentiable in x for every , , for all .
(W2) uniformly for all .
(W3) uniformly for all .
(W4) , , and there exist , , and such that
Now, we are ready to state the main result of this paper.
Theorem 1.1 Assume that p, L and W satisfy (P), (L), (W1), (W2), (W3), and (W4). If there exist and such that
then system (1.1) possesses a nontrivial homoclinic solution.
In Theorem 1.1, we replace (L) and (W4) by the following assumptions:
(L′) is real symmetric nonnegative definite matrix for all , and it satisfies (1.2).
(W4′) , , and there exist and such that
Then we have the following corollary immediately.
Corollary 1.2 Assume that p, L and W satisfy (P), (L′), (W1), (W2), (W3) and (W4′). Then system (1.1) possesses a nontrivial homoclinic solution.
Remark 1.3 If satisfies the well-known global Ambrosetti-Rabinowitz superquadratic condition:
(AR) there exists such that
then there exists a constant such that
moreover for all , and
In addition, by virtue of (W2), there exists such that
These show that (W3) and (W4) hold with , and . Let and and choose and . In view of Theorem 1.1, if
then system (1.1) possesses a nontrivial homoclinic solution.
Example 1.4 Let , and
It is easy to see that for all , and
These show that (W3) and (W4) hold with , and . We choose and . Then
In view of Theorem 1.1, if , then system (1.1) possesses a nontrivial homoclinic solution.
Throughout this section, we always assume that p and L satisfy (P) and (L). Let
and for , let
Then E is a Hilbert space with the above inner product, and the corresponding norm is
As usual, for , set
and their norms are defined by
Lemma 2.1 Suppose that (L) is satisfied. Then
Proof Since , it follows that . Hence, there exists such that . There are two possible cases.
Case (i). . According to (L), one has
Case (ii). . Without loss of generality, we can assume that , then
Cases (i) and (ii) imply that (2.1) and (2.2) hold. □
Now we define a functional Φ on E by
For any , there exists an such that for . Hence, under assumptions (P), (L), (W1), and (W2), the functional Φ is of class . Moreover,
Let with and for .
Lemma 2.2 Suppose that (L), (W1) and (W2) are satisfied. Then
Proof From (2.4) and the definition of e, we get
Now the conclusion of Lemma 2.1 follows by (2.8). □
Applying the mountain-pass lemma without the (PS) condition, by standard arguments, we can prove the following lemma.
Lemma 2.3 Let , . Suppose that (P), (L), (W1), (W2) and (W3) are satisfied. Then there exist a constant and a sequence satisfying
Lemma 2.4 Suppose that (P), (L), (W1), (W2), (W3), and (W4) are satisfied. Then any sequence satisfying
is bounded in E.
Proof To prove the boundedness of , arguing by contradiction, suppose that . Let . Then . By virtue of (2.5), (2.6), and (2.10), we have
If , then it follows from (L), (W4) and (2.11) that
Combining (2.12) with (2.13) and using (2.5) and (2.10), we have
This contradiction shows that .
Going if necessary to a subsequence, we may assume the existence of such that
Let , then
Now we define . Then and . Passing to a subsequence, we have in , then for all . Clearly, (2.15) implies that .
It is obvious that implies . Hence, it follows from (2.5), (2.10), and (W3) that
which is a contradiction. Thus is bounded in E. □
3 Proof of theorem
Proof of Theorem 1.1 Applying Lemmas 2.3 and 2.4, we deduce that there exists a bounded sequence satisfying (2.9). By Lemma 2.2 and (1.3), one has
Going if necessary to a subsequence, we can assume that in E and . Next, we prove that .
Arguing by contradiction, suppose that , i.e. in E, and so for every . Hence,
According to (W4) and (3.2), one gets
By virtue of (2.5), (2.6), and (2.9), we have
Using (W4), (2.1), (3.1), (3.2), and (3.4), we obtain
which, together with (2.6), (2.9), and (3.3), yields
resulting in the fact that . Consequently, it follows from (W1), (2.5), and (2.9) that
This contradiction shows . By standard arguments, we easily prove that is a nontrivial solution of (1.1). □
Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications. 2nd edition. Dekker, New York; 2000.
Ahlbran CD, Peterson AC: Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations. Kluwer Academic, Dordrecht; 1996.
Deng XQ, Cheng G: Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign. Acta Appl. Math. 2008, 103: 301–314. 10.1007/s10440-008-9237-z
Deng XQ, Cheng G, Shi HP: Subharmonic solutions and homoclinic orbits of second order discrete Hamiltonian systems with potential changing sign. Comput. Math. Appl. 2009, 58: 1198–1206. 10.1016/j.camwa.2009.06.045
Ma M, Guo ZM: Homoclinic orbits and subharmonics for nonlinear second order difference equations. Nonlinear Anal. 2007, 67: 1737–1745. 10.1016/j.na.2006.08.014
Tang XH, Lin XY, Xiao L: Homoclinic solutions for a class of second order discrete Hamiltonian systems. J. Differ. Equ. Appl. 2010, 16: 1257–1273. 10.1080/10236190902791635
Zhang, Q: Homoclinic orbits for a class of discrete periodic Hamiltonian systems. Proc. Am. Math. Soc. (in press)
Zhou Z, Yu JS, Chen Y: Homoclinic solutions in periodic difference equations with saturable nonlinearity. Sci. China Math. 2011, 54: 83–93. 10.1007/s11425-010-4101-9
Chen WX, Yang MB, Ding YH: Homoclinic orbits of first order discrete Hamiltonian systems with super linear terms. Sci. China Math. 2011, 54: 2583–2596. 10.1007/s11425-011-4276-8
Lin XY, Tang XH: Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. J. Math. Anal. Appl. 2011, 373: 59–72. 10.1016/j.jmaa.2010.06.008
Ma M, Guo ZM: Homoclinic orbits for second order self-adjoint difference equations. J. Math. Anal. Appl. 2006, 323(1):513–521. 10.1016/j.jmaa.2005.10.049
Tang XH, Chen J: Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems. Adv. Differ. Equ. 2013., 2013: Article ID 242 10.1186/1687-1847-2013-242
Tang XH, Lin XY: Homoclinic solutions for a class of second order discrete Hamiltonian systems. Acta Math. Sin. 2012, 28: 609–622. 10.1007/s10114-012-9233-0
Tang XH, Lin XY: Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential. J. Differ. Equ. Appl. 2013, 19: 796–813. 10.1080/10236198.2012.691168
This work is partially supported by Scientific Research Fund of Hunan Provincial Education Department (13A093, 07A066) and supported by Hunan Provincial Natural Science Foundation of China (No. 14JJ2133).
The author declares that they have no competing interests.