Abstract
In this paper, by using critical point theory, we obtain some new sufficientconditions on the existence and multiplicity of periodic and subharmonic solutions toa 2n th-order nonlinear difference equation containing both advance andretardation with ϕ-Laplacian. Some previous results have beengeneralized.
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1 Introduction
Let N, Z, and R denote the sets of all natural numbers,integers and real numbers, respectively. For , define , when . denotes the transpose of a vector.
Consider the following 2n th-order difference equation containing both advanceand retardation with ϕ-Laplacian of the type:
where , △ is forward difference operator defined by, , satisfied , , for each , and are T-periodic in k and T is agiven positive integer.
In this paper, given positive integer m, we will study the existence ofmT-periodic solutions for (1.1). As usual, such a mT-periodicsolution will be called a subharmonic solution.
We may think of (1.1) as a discrete analog of the following 2n th-orderfunctional differential equation:
Equations similar in structure to (1.2) have been studied by many authors. For example,for the case where , , Smets and Willem [1] have considered solitary waves with prescribed speed on infinite lattices ofparticles with nearest neighbor interaction for the following forward and backwarddifferential difference equation:
For the case where , , , Wang [2] has studied the existence of positive solutions of the equation
For the case where , , , Agarwal, Lu, and O’Regan [3] have studied the existence of positive solutions of the equation
For the case where , , Bonheure and Habets [4] have studied classical and non-classical solutions of a prescribed curvatureequation
In recent years, many authors have studied the existence of periodic solutions ofdifference equations. To mention a few, see [5–8] for second-order difference equations and [9, 10] for higher-order equations. Since 2003, critical point theory has beenemployed to establish sufficient conditions on the existence of periodic solutions ofdifference equations. By using the critical point theory, Guo and Yu [11–13] and Zhou et al.[14] established sufficient conditions on the existence of periodic solutions ofsecond-order nonlinear difference equations. In 2007, by using the Linking Theorem, Caiand Yu [15] obtained some criteria for the existence of periodic solutions of thefollowing equation:
for the case where f grows superlinearly at both 0 and ∞, where. In 2010, by using the Linking Theorem and the SaddlePoint Theorem, Zhou [16] extended f in (1.3) into sublinear or asymptotically linear andimproved the results of [15] when f is superlinear. In particular, a necessary and sufficientcondition for the existence of the unique periodic solution of (1.3) is also establishedin [16]. In 2013, by using the Linking Theorem, Deng [17] provided some sufficient conditions of the existence and multiplicity ofperiodic solutions and subharmonic solutions of the following equation:
where , is the p-Laplacian operator given by () and where f satisfies some growth conditionsnear both 0 and ∞. In 2012, Mawhin [18] considered T-periodic solutions of systems of difference equationsof the form
under various conditions upon and , where , , in which is continuously differentiable and strictly convex,satisfies and is a homeomorphism of onto the ball or of onto . By using direct variational method, he gave sufficientconditions for the existence of a minimizing sequence for the case of coercivepotential, or some averaged coercivity conditions of the Ahmad-Lazer-Paul type addingthe nonlinearity satisfies some growth conditions, or the convex potential. Using theSaddle Point Theorem, previously obtained results are extended to the case of anaveraged anticoercivity condition in [18]. However, the results on periodic solutions of higher-order nonlineardifference equations involving ϕ-Laplacian are very scarce in theliterature. Furthermore, since (1.1) contains both advance and retardation, there arevery few works dealing with this subject; see [10, 19]. The main purpose of this paper is to give some sufficient conditions for theexistence and multiplicity of periodic and subharmonic solutions of (1.1). Particularly,our results generalize the results in the literature [17, 20]; see Remark 3.4 and Remark 3.5 for details.
2 Preliminaries
Throughout this paper, we assume that,
(F1) there exists a functional with and satisfies
In this section, we first establish the variational setting associated with (1.1).
Let S be the set of all two-sided sequences, that is,
Then S is a vector space with for , . For any fixed positive integer m and T,we define the subspace of S as
Obviously, is isomorphic to and hence can be equipped with the inner product and norm as
and
On the other hand, we define the norm on as follows:
for all and . By Hölder’ inequality and Jensen’inequality, we have
Let
Therefore,
Clearly, . For all , define the functional J on as follows:
where
is the primitive function of .
Clearly, and for any , by using for , we can compute the partial derivative as
Thus, u is a critical point of J on if and only if
Due to the periodicity of and in the first variable k, we reduce the existenceof periodic solutions of (1.1) to the existence of critical points of Jon .
Let M be the matrix defined by
By matrix theory, we see that the eigenvalues of M are
Thus, , , , …, . Therefore,
For convenience, we identify with . Let
Then
Let be the direct orthogonal complement of to , i.e., .
For and , we have
For and , we have
Let H be a Hilbert space and denote the set of functionals that are Fréchetdifferentiable and their Fréchet derivatives are continuous on H. Let. A sequence is called a Palais-Smale sequence (P. S. sequence forshort) for J if is bounded and as . We say J satisfies the Palais-Smale condition(P. S. condition for short) if any P. S. sequence for J possesses a convergentsubsequence.
Let be the open ball in H with radius r andcenter 0, and let denote its boundary. Lemma 2.1 is taken from [21].
Lemma 2.1 (Linking Theorem)
Let H be a real Hilbert space and, whereis a finite-dimensional subspaceof H. Assume thatsatisfies the P. S. condition and thefollowing conditions.
(J1) There exist constantsandsuch that;
(J2) There exist anand a constantsuch thatwhere.
Then J possesses a critical value. Moreover, c can be characterized as
whereandis the identity operator on ∂Q.
3 Main results
Let
Here we give some conditions.
() There exist constants , and such that
() There exist constants , , and such that
(F2) There exist constants , and such that
(F3) There exist constants , , and such that
() and .
() and .
() .
() .
Remark 3.1 By () it is easy to see that there exists a constant such that
Remark 3.2 By (F3) it is easy to see that there exists a constant such that
Remark 3.3 The p-Laplacian operator given by (), the curvature-type operator given by () and the identity operator given by satisfy () and ().
Our main results are as follows.
Theorem 3.1 Assume that (), (), (F1), (F2), (F3)are satisfied. If one of the following four cases is satisfied:
-
(1)
Assume that () and () are satisfied.
-
(2)
Assume that () and () are satisfied.
-
(3)
Assume that () and () are satisfied.
-
(4)
Assume that () and () are satisfied.
Then for any given positive integer m, (1.1) has at leastthree mT-periodic solutions.
Remark 3.4 If (), and , Theorem 3.1 reduces to Theorem 3.1 in [20].
Remark 3.5 If (), Theorem 3.1 reduces to Theorem 1.1 in [17].
Corollary 3.1 Assume that (F1) and the followingconditions are satisfied.
() There exists constantsuch that.
() There exist constantsandsuch that
() There exists constantsuch that
() There exist constantsandsuch that
Then for any given positive integer m, (1.1) has at leastthree mT-periodic solutions.
4 Proof of the main results
Lemma 4.1 Assume that (), (F1), (F3), and() are satisfied. Then thefunctional J is bounded from above in.
Proof By (2.1), (2.3), (3.1), and (3.2), for any , we have
where and .
The proof of Lemma 4.1 is complete. □
Remark 4.1 The case is trivial. For the case , M has a different form, namely,
However, in this special case, the argument need not be changed and we omit it.
Lemma 4.2 Assume that (), (F1), (F3), and() are satisfied. Then thefunctional J satisfies the P. S. conditionin.
Proof Let be a P. S. sequence, then there exists a positive constant such that
By (4.1), it is easy to see that
Therefore,
Since , it is not difficult to see that is a bounded sequence in . As a consequence, possesses a convergence subsequence in. Thus the P. S. condition is verified. □
Lemma 4.3 Assume that (), (F1), (F3), and() are satisfied. Then thefunctional J is bounded from above in.
Proof Similar to the proof of Lemma 4.1, we have
where . Since , we have
The proof of Lemma 4.3 is complete. □
Lemma 4.4 Assume that (), (F1), (F3), and() are satisfied. Then thefunctional J satisfies the P. S. conditionin.
Proof Let be a P. S. sequence, then there exists a positive constant such that
By (4.2), it is easy to see that
Therefore,
Since , we know that is a bounded sequence in . As a consequence, possesses a convergence subsequence in. Thus the P. S. condition is verified. □
Proof of Theorem 3.1 Assumptions (F1) and (F2) imply that and for . Adding , then is a trivial mT-periodic solution of (1.1).
By Lemma 4.1 or Lemma 4.3, J is bounded from above on . We define . Equation (4.1) implies . This means that −J is coercive. By thecontinuity of J, there exists such that . Clearly, is a critical point of J.
Case 1. Assume that () and () are satisfied. We claim that .
Let
By (), (F2), and (), for any , , we have
where .
Take . Then and
Therefore, . From (4.4), we have also proved that J satisfiesthe condition (J1) of the Linking Theorem.
For all , we have
Thus, the critical point of J corresponding to the critical value is a nontrivial mT-periodic solution of (1.1). Inthe following, we will verify the condition (J2).
Take , for any and , let . Then
where .
Let , . We have and , and , are bounded from above, and for . Thus there exists a constant such that where .
Case 2. Assume that () and () are satisfied. We claim that .
Let . By (), (), and (F2), for any , , we have
where .
Take . Then and
Therefore, . From (4.7), we have also proved that J satisfiesthe condition (J1) of the Linking Theorem.
For all , we have
Thus, the critical point of J corresponding to the critical value is a nontrivial mT-periodic solution of (1.1). Inthe following, we will verify the condition (J2).
Take , for any and , let . Then
where .
Since , and for , thus there exists a constant such that where .
Case 3. Assume that () and () are satisfied. Similar to Case 1, by (4.6), we see that. Similar to Case 2, by (4.5), we see that there exists aconstant such that where . We have also proved that J satisfies thecondition (J1) and (J2) of the Linking Theorem.
Case 4. Assume that () and () are satisfied. Similar to Case 1, by (4.3), we see that. Similar to Case 2, by (4.8), we see that there exists aconstant such that where . We have also proved that J satisfies thecondition (J1) and (J2) of the Linking Theorem.
By one of the above four cases and the Linking Theorem, J possesses criticalvalue . Moreover, α can be characterized as
where and is the identity operator on ∂Q. Let be a critical point associated to the critical valueα of J, i.e., . If , then The proof is complete. Otherwise,. Then , i.e., . Choosing , we have . Take . Similarly, there exists a positive number, , where . Again, by the Linking Theorem, J possesses acritical value . Moreover, can be characterized as
where and is the identity operator on . If , then the proof is finished. If , then . Due to the fact that and , J attains its maximum at some points in theinterior of sets Q and . However, and . Therefore, there must be a point , and .
The proof of Theorem 3.1 is complete. □
Proof of Corollary 3.1 By (), there exists constant such that , for . Hence () implies (). By (), there exist constants , and such that , for . So () implies ().
By (), there exist constants and such that
So () implies (F2).
By (), there exist constants , and such that
So () implies (F3). Since , () implies ().
If , then () implies (). If , then by (), there exist constants and such that
we have . So, if , then () implies ().
So, by Theorem 3.1, Corollary 3.1 holds. □
5 Example
As an application of Theorem 3.1, we give an example to illustrate our result.
Example 5.1 For a given positive integer T, consider the following2n th-order difference equation:
where
Let
It is easy to verify that all the assumptions of Theorem 3.1 are satisfied. So, for anygiven positive integer m, (5.1) has at least three mT-periodicsolutions.
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Acknowledgements
This work is supported by the Program for Changjiang Scholars and Innovative ResearchTeam in University (no. IRT1226), the National Natural Science Foundation of China(no. 11171078), the Specialized Fund for the Doctoral Program of Higher Education ofChina (no. 20114410110002), and the Project for High Level Talents of GuangdongHigher Education Institutes.
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Lin, G., Zhou, Z. Periodic and subharmonic solutions for a 2nth-order difference equationcontaining both advance and retardation with ϕ-Laplacian. Adv Differ Equ 2014, 74 (2014). https://doi.org/10.1186/1687-1847-2014-74
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DOI: https://doi.org/10.1186/1687-1847-2014-74