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Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential

  • Chengfu Che
  • Xiaoping Xue
Open Access
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Abstract

In this article we obtain two sequence of infinitely many periodic solutions for discrete second order Hamiltonian systems with an oscillating potential. One sequence of solutions are local minimizers of the functional corresponding to the system, the other sequence are minimax type critical points of the functional.

Keywords

Periodic Solution Index Theory Critical Point Theory Multiplicity Result Finite Dimensional Hilbert Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Discrete problems arise in the study of combinatorial analysis, quantum physics, chemical reactions, population dynamics, and so forth. Besides, they are also natural consequences of the discretization of continuous problems. On the other hand, the critical point theory has been a powerful tool in dealing with the existence and multiplicity results. Thus, discrete problems have been studied by many scholars via critical point theory. For example, Pankov and Zakharchenko used a variational method known as Nehari manifolds and a discrete version of the Lions concentration-compactness principle in [1] to establish existence results of nontrivial standing wave solutions for discrete nonlinear Schrödinger equation. Lately in [2], Pankov and Rothos employed Nehari manifolds approach and the Mountain Pass argument to demonstrate the existence of solutions in the discrete nonlinear Schrödinger equation with saturable nonlinearity. While for discrete Hamiltonian systems, Yu and Guo established a variational structure and introduced variational technique to the study of periodic solutions in [3, 4, 5].

The aim of this article is to apply the critical point theory to deal with the problem of infinitely multiplicity of periodic solutions for the following discrete second order Hamiltonian systems:
Δ 2 u ( t - 1 ) + F ( t , u ( t ) ) = 0 , t Z [ 1 , T ] u ( 0 ) = u ( T ) , Open image in new window
(1)
where Δu(t) := u(t + 1) - u(t), Δ2u(t) = Δ(Δu(t)), and ∇F(t, x) denotes the gradient of F with respect to the second variable. Systems (1) can be considered as a discrete analog of the following Hamiltonian systems:
- ü ( t ) = F ( u ( t ) ) , u ( 0 ) - u ( T ) = u ˙ ( 0 ) - u ˙ ( T ) = 0 , Open image in new window

where ü(t) denotes the second derivative of u with respect to t. Throughout of this article F will be called potential function, (·, ·) and | · | denote the inner product and norm in R N respectively. A basic assumption we make on F is that F satisfies the following sublinear condition:

(F α ) F(t, x) ∈ C1 (R N , R) for any tZ[0, T] and F is T-periodic in the first variable. Moreover there exist f, g : Z[0, T] → R+ and 0 < α < 1 such that
F ( t , x ) f ( t ) x α + g ( t ) for any t Z [ 0 , T ] and x R N . Open image in new window
After the initial work of Yu and Guo, there appeared many results about Hamiltonian systems, such as [6, 7, 8, 9]. Among these articles, the results in [5, 6] have a close relation with the one in this article: they also considered (1) with sublinear or subquadratic potential. Especially, an existence result was obtained under the sublinear condition (F α ) and a partially coercive assumption:
x - 2 α t = 0 T F ( t , x ) as x for all t Z [ 0 , T ] . Open image in new window
(2)

But until now, no multiplicity results are obtained under subquadratic or sublinear condition.

On the other hand, the multiplicity problem was considered in [3, 7, 8, 9]. When one study the multiplicity problem, an effective method is index theory. The index measure the size of subset which is invariant under some group action, such as Z p action (for explicit definition, see [10]). If the functional is also invariant under this group action, The multiplicity of critical points can be obtained form the multiplicity of index. Guo and Yu [7] used the above mentioned Z p index theory to show that there are at least T - 1 distinct Z T -orbits for (1) when the potential function is autonomous and superquadratic, and at least 2(T - 1) distinct Z T -orbits when moreover the potential function is even. They also obtained a result about the lower bounds for the number of T-periodic solutions for the asymptotically linear potential case. Their approach was based on Z p index theory introduced in [10], so the autonomous condition is essential. However, infinitely many kinds of results can not be obtained form this method, since the index of the whole space if finite. For superquadratic system (1) where the potential may depend on time, Guo et al. obtained at least two nontrivial solutions in [3, 9]. Later Xue and Tang obtained the same result under a more general superquadratic condition in [8].

Until now, as the authors knows there are no results of infinitely many kinds appeared for system (1). In this article, we are going to give some sufficient conditions to ensure (1) has infinitely many periodic solutions. Roughly speaking, instead of coercive assumption (2), we suppose F has a suitable oscillating behavior at infinity:
lim sup r inf x R N , x = r t = 0 T F ( t , x ) = + , Open image in new window
(3)
lim inf R sup x R N , x = R x - 2 α t = 0 T F ( t , x ) = - . Open image in new window
(4)

Then we obtain two sequence of infinitely many periodic solutions by minimax methods. One sequence of solutions is local minimizer of the functional φ corresponding to system (1), and the other are minimax type critical points of φ. The explicit form of φ will be given in Section 2. The idea in this article was inspired by Habets et al. [11] and Zhang and Tang [12], where Dirichlet type and periodic type boundary value problem for continuous systems were studied.

The following are main results:

Theorem 1.1 Assume that F satisfies (F α ), (3) and (4). Then

(a) there exists a sequence {u n } of solutions of (1) such that {u n } is a critical point of φ and limn→∞φ(u n ) = +∞;

(b) there exists a sequence { u n * } Open image in new windowof solutions of (1) such that { u n * } Open image in new windowis a local minimum of φ and lim n φ ( u n * ) = - Open image in new window.

In the rest of this article, we first give some preliminaries in Section 2, then give the proof of Theorem 1.1 in Section 3.

2 Preliminaries

In this section, we first introduce some notations. Let R, Z, N be the sets of real numbers, integers and natural numbers, respectively. For a, bZ and cR, Z [a, b] denotes the discrete interval {a, a + 1,..., b} when a < b and [c] denote the largest integer less than c. In order to apply critical point theory, we then introduced the variational structure corresponding to system (1). For any given positive integer T, the linear space H T is defined by
H T = { u : Z R N | u ( t ) = u ( t + T ) for all t Z } . Open image in new window
H T can be equipped with inner product
u , v = t = 0 T ( u ( t ) , v ( t ) ) , Open image in new window
and the corresponding norm reads as
u = t = 0 T u ( t ) 2 1 2 . Open image in new window
It is easy to see that H T is a finite dimensional Hilbert space and is linear homeomorphic to R NT . Define another norm ∥ · ∥ by
u = max t Z [ 0 , T ] u ( t ) . Open image in new window
Since H T is finite dimensional, this norm is equivalent with ∥ · ∥:
1 T u u u . Open image in new window
Consider the functional defined on H T
φ ( u ) = 1 2 t = 0 T Δ u ( t ) 2 - t = 0 T F ( t , u ( t ) ) , u H T . Open image in new window

One can easily check that uH T is a critical point of φ if and only if u is a solution of (1).

The following lemma give a new decomposition of H T according to spectra of operator Δ2 with periodic boundary condition.

Lemma 2.1 [[8], Lemma 1] As a subspace of H T , N k is is defined by
N k : = { u H T | - Δ 2 u ( t - 1 ) = λ k u ( t ) } , Open image in new window

where λ k = 2 - 2 cos k ω , ω = 2 π 2 , k Z [ 0 , [ T 2 ] ] Open image in new window. Then we have:

(a) N k N j for any kj and j , k Z [ 0 , [ T 2 ] ] Open image in new window.

(b) H T = k = 0 [ T 2 ] Open image in new window.

Set V = N0 and W = k = 1 [ T 2 ] N k Open image in new window. Then it is easy to see H T = VW and
t = 0 T Δ u ( t ) 2 λ 1 u for any u W . Open image in new window

The element u of V is just the eigenvector corresponding to λ0 = 0 which satisfy u(t) = u(0) for tZ [0, T].

Now we introduce a minimax theorem which include many well know results. This theorem not only asserts the existence of a Palais Smale sequence, but also gives the location information of the Palais Smale sequence. This will play an important role in our proof of Theorem 1.1.

Proposition 2.1 [[13], Corollary 4.3] Let K be a compact metric space, K0K a closed set, X a Banach space, χC (K0, X) and let us define the complete metric space M by
M = { g C ( K , X ) | g ( s ) = χ ( s ) , s K 0 } Open image in new window
with the usual distance d. Let φC 1 (X, R) and let us define
c = inf g M max s K φ ( g ( s ) ) , c 1 = max s χ ( K 0 ) φ ( s ) . Open image in new window
If c > c1, then for each sequence {f k } ⊂ M such that max K φ (f k ) → c, there exists a sequence {v k } ⊂ X such that
φ ( v k ) c , d i s t ( v k , f k ( K ) ) 0 , φ ( v k ) 0 Open image in new window

as k → ∞.

3 Proof of Theorem 1.1

In this section we give the proof of Theorem 1.1. Before that, we need to establish some basic lemmas.

Lemma 3.1 Suppose (F α ) holds, then φ is coercive in the subspace W, that is φ(u) asu∥ → ∞ in W.

Proof: For any uW, it follows from (F α ) that
φ ( u ) = 1 2 t = 0 T Δ u ( t ) 2 - t = 0 T F ( t , u ( t ) ) 1 2 λ 1 u 2 - t = 0 T ( f ( t ) u ( t ) α + 1 + g ( t ) u ( t ) ) 1 2 λ 1 u 2 - u α + 1 t = 0 T f ( t ) - u t = 0 T g ( t ) 1 2 λ 1 u 2 - u α + 1 t = 0 T f ( t ) - u t = 0 T g ( t ) . Open image in new window

Since α < 1, we have φ(u) → ∞ as ∥u∥ → ∞ in W.

Lemma 3.2 Suppose (3) holds. Then there exists a positive sequence {a n } such that
lim n a n = + a n d lim n sup u V , u = a n φ ( u ) = - . Open image in new window
Proof. For uV, we have u(0) = u(1) = ⋯ = u(T), ∥u2 = T|u(0)|2 and
φ ( u ) = - t = 0 T F ( t , u ( t ) ) = - t = 0 T F ( t , u ( 0 ) ) . Open image in new window
By (3) there exists a sequence {d n } such that
lim n inf x R N , x = d n t = 0 T F ( t , x ) = + . Open image in new window
So if we choose a n = T d n Open image in new window, then we have
sup u V , u = a n φ ( u ) = sup u V , u = a n - t = 0 T F ( t , u ( 0 ) ) = - inf u ( 0 ) R N , u ( 0 ) = d n t = 0 T F ( t , u ( 0 ) ) - Open image in new window

as n → ∞.

Lemma 3.3 Suppose (F α ) and (4) hold. Then there exists a positive sequence {b m } such that
lim m b m = + a n d lim m inf u H b m φ ( u ) = + , Open image in new window

where H b m = { u V | u = b m } W Open image in new window.

Proof. For any u H b m Open image in new window, let u = ū + ũ Open image in new window with ū V Open image in new window and ũ W Open image in new window. It follows form (F α ) that
t = 0 T F ( t , u ( t ) ) - t = 0 T F ( t , ū ( t ) ) = t = 0 T 0 1 ( F ( t , ū ( 0 ) + s ũ ( t ) ) , ũ ( t ) ) t = 0 T ( f ( t ) ū ( 0 ) + ũ ( t ) α + g ( t ) ) ũ ( t ) 2 t = 0 T f ( t ) ( ū ( 0 ) α + ũ ( t ) α ũ ( t ) ) + t = 0 T g ( t ) ũ ( t ) 2 ( ū ( 0 ) α + ũ α ) ) ũ t = 0 T f ( t ) + ũ t = 0 T g ( t ) 2 λ 1 8 ũ 2 + 8 λ 1 ū 2 α + ũ 1 + α t = 0 T f ( t ) + ũ t = 0 T g ( t ) λ 1 4 ũ 2 + C ũ 1 + α + C ũ + C ū 2 α . Open image in new window
Substitute the above inequality into φ(u), we have
φ ( u ) = 1 2 t = 0 T Δ u ( t ) 2 - t = 0 T F ( t , u ( t ) ) = 1 2 t = 0 T Δ ũ ( t ) 2 - t = 0 T F ( t , u ( t ) ) - t = 0 T F ( t , ū ( t ) ) + t = 0 T F ( t , ū ( t ) λ 1 4 ũ 2 - C ũ 1 + α - C ũ + ū 2 α t = 0 T F ( t , ū ( t ) ) ū 2 α + C . Open image in new window
The sum of the first three terms is bounded form below. On the other hand, it follows form (F-) there exists sequence e m → ∞ such that
lim m sup x R N , x = e m x - 2 α t = 0 T F ( t , x ) = - . Open image in new window
Hence, if we choose b m = T e m Open image in new window, we have
lim m inf u H b m φ ( u ) = + . Open image in new window

After the above preparations we give the proof of our main result.

Proof of Theorem 1.1 Denote the ball in V with radius a n by B a n Open image in new window. Then we define a family of maps
Γ n = { γ C ( B a n , H ) γ B a n = I d | B a n } Open image in new window
and corresponding minimax values
c n = inf γ Γ n max u B a n φ ( γ ( u ) ) Open image in new window
for each n. By Lemma 3.1, the functional φ is coercive on W, then there exists a constant M such that infuWφ(u) ≥ M. On the other hand, it is well know that B a n Open image in new window and W are linked (see [[14], Theorem 4.6]), i.e., for any γ Γ n , γ ( B a n ) W Open image in new window. It follows that max u B a n φ ( γ ( u ) ) inf u W φ ( u ) Open image in new window for any γ ∈ Γ n . Hence we have c n ≥ infuWφ(u) ≥ M. In view of Lemma 3.2,
c n > max u B a n φ ( u ) Open image in new window

holds for large values of n, where B a n Open image in new window denote the boundary of B a n Open image in new window in V : {uV|∥u∥ = a n }.

For such n, there exists a sequence {γ k } ⊂ Γ n such that
max u B a n φ ( γ k ( u ) ) c n as k . Open image in new window
Applying Proposition 2.1 with X = H , K = B a n , K 0 = B a n , χ = I d Open image in new window, we know there exists a sequence {v k } ⊂ H such that
φ ( v k ) c n , dist ( v k , γ k ( B a n ) ) 0 , φ ( v k ) 0 Open image in new window
(5)

as k → ∞. If we can show {v k } is bounded, then from the fact that H is finite dimensional we know there is a subsequence, which is still be denoted by {v k } such that v k converge to some point u n . By the continuity of φ and φ', we know φ(u n ) = c n and φ'(u n ) = 0. That is, u n is a critical point of φ.

Now, let us show the sequence {v k } is bounded. For large enough k, by (5), we have
c n max u B a n φ ( γ k ( u ) ) c n + 1 , Open image in new window
and we can find w k γ k ( B a n ) Open image in new window such that ∥v k - w k ∥ ≤ 1. By Lemma 3.3, we can find a large enough m such that b m > a n and inf u H b m φ ( u ) > c n + 1 Open image in new window. This implies that γ k ( B a n ) Open image in new window can not intersect the hyperplane H b m Open image in new window for each k. Let w k = w ̄ k + w ̃ k Open image in new window with w ̄ k V Open image in new window and w ̃ k W Open image in new window. Then w ̄ k < b m Open image in new window for each k. Besides, by (F α ), it is obvious that
c n + 1 φ ( w k ) = 1 2 t = 0 T Δ w k ( t ) 2 - t = 0 T F ( t , w k ( t ) ) 1 2 λ 1 w ̃ k 2 - t = 0 T ( f ( t ) w k ( t ) α + 1 + g ( t ) w k ( t ) ) 1 2 λ 1 w ̃ k 2 - 4 t = 0 T f ( t ) ( w ̄ k ( 0 ) α + 1 + w ̃ k ( t ) α + 1 ) - t = 0 T g ( t ) ( w ̄ k ( 0 ) + w ̃ k ( t ) ) 1 2 λ 1 w ̃ k 2 - 4 w ̃ k α + 1 t = 0 T f ( t ) - w ̃ k t = 0 T g ( t ) - 4 b m α + 1 t = 0 T f ( t ) - b m t = 0 T g ( t ) . Open image in new window
(6)
This implies that w ̃ k Open image in new window is bounded too. From w k C ( w ̄ k + w ̃ k ) Open image in new window we know w k is bounded. Hence {v k } is bounded. From previous discussion we know that the accumulation point u n of {v k } is a critical point and c n is critical value of φ. In order to prove part (a), we still have to show
lim n c n = + . Open image in new window
(7)
Note that if we choose large enough n such that a n > b m , then γ ( B a n ) Open image in new window intersect the hyperplane H b m Open image in new window for any γ ∈ Γ n . It follows that
max u B a n φ ( γ ( u ) ) inf u H b m φ ( u ) . Open image in new window

This inequality and Lemma 3.3 implies (7).

Next we prove part (b) of Theorem 1.1. For fixed mN, define the subset P m of H by
P m = { u = ū + ũ H | ū V , ū b m , ũ W } . Open image in new window
It follows form (6) that φ is bounded form below on P m . Let us set
μ m = inf u P m φ ( u ) Open image in new window
and choose a minimizing sequence {u k } in P m , that is,
φ ( u k ) μ m as m . Open image in new window
From (6) we know that {u k } is bounded in H. Then there exists a subsequence, which is still be denoted by {u k } such that u k u m * Open image in new window as k → ∞. Form the fact that P m is a closed subset of H and that φ is continuous we know u m * P m Open image in new window and
μ m = lim k φ ( u k ) = φ ( u m * ) . Open image in new window
If we can show u m * Open image in new window is in the interior of P m , then u m * Open image in new window is a critical point of φ. Let u m * = ū m * + ũ m * Open image in new window with ū m * V Open image in new window and ũ m * W Open image in new window. If a n < b m , then B a n P m Open image in new window. This implies that
φ ( u m * ) = inf u P m φ ( u ) sup u B a n φ ( u ) . Open image in new window

It follows from Lemma 3.2 that φ ( u m * ) - Open image in new window as m → ∞. By Lemma 3.3 we have ū m * b m Open image in new window for large values of m, which means that u m * Open image in new window is in the interior of P m , and u m * Open image in new window is a critical point of φ with φ ( u m * ) - Open image in new window as m → ∞. The proof of Theorem 1.1 is finished.

Example 1 Now we give an example of potential function which satisfies condition (F α ), (3) and (4). For simplicity, we drop the dependence in t:
F ( x ) = x 1 + α sin ( log ( 1 + x ) ) . Open image in new window
Note that F is continuously differentiable and its gradient reads as
F ( x ) = x α - 1 sin ( log ( 1 + x ) ) x + x α cos ( log ( 1 + x ) ) 1 + x x . Open image in new window

Then it is easy to see F satisfies condition (F α ), (3) and (4). A routine application of Theorem 1.1 shows that system (1) with potential function F has infinitely many periodic solutions.

Notes

Acknowledgements

The authors sincerely thanks the editors and the referee for their many valuable comments which helped improving the article. This research was partly supported by the NNSF of China (Grant No. 10971043), by the Heilongjiang Province Foundation for Distinguished Young Scholars (Grant No. JC200810), and by the Program of Excellent Team at Harbin Institute of Technology.

References

  1. 1.
    Pankov A, Zakharchenko N: On some discrete variational problems. Acta Appl Math 2000, 65: 295–303.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Pankov A, Rothos V: Solitons in discrete nonlinear Schrodinger equation with saturable non-linearity. Proc R Soc Lond Ser A 2008, 464: 3219–3236.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Guo Z, Yu J: Existence of periodic and subharmonic solutions for second-order superlinear difference equations. Sci China Ser A 2003, 46(4):506–515.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Guo Z, Yu J: Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems. Nonlinear Anal 2003, 55(7–8):969–983.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Guo Z, Yu J: The existence of periodic and subharmonic solutions of subquadratic second-order difference equations. J Lond Math Soc 2003, 68(2):419–430.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Xue YF, Tang CL: Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system. Nonlinear Anal 2007, 6(7):2072–2080.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guo Z, Yu J: Multiplicity results for periodic solutions to second-order difference equations. J Dyn Diff Equ 2006, 18(4):943–960.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Xue YF, Tang CL: Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems. Appl Math Comput 2008, 196: 494–500.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhou Z, Guo Z, Yu J: Periodic solutions of higher-dimensional discrete systems. Proc R Soc Edinburgh 2004, 134A: 1013–1022.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Liu J: A geometrical index for the group Zp. Acta Math Sinica, New Ser 1989, 5(3):193–196.MATHCrossRefGoogle Scholar
  11. 11.
    Habets P, Manásevich R, Zanolin F: A nonlinear boundary value problem with potential oscillating aroud the first eigenvalue. J Diff Equ 1995, 117: 428–445.MATHCrossRefGoogle Scholar
  12. 12.
    Zhang P, Tang CL: Infinitely many periodic solutions for nonautonomous sublinear second order Hamiltonian systems. Abstr Appl Anal 2010, 2010: 1–10.Google Scholar
  13. 13.
    Mawhin J, Willem M: Critical Point Theory and Hamiltonian System. Springer-Verlag, New York; 1989.CrossRefGoogle Scholar
  14. 14.
    Rabinowitz P: Minimax Methods in Critical Point Theory with Application to differential Equations. AMS, Providence 1986.Google Scholar

Copyright information

© Che and Xue; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople's Taiwan

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