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Multiplicity of Periodic Solutions for a Higher Order Difference Equation
We study a higher order difference equation. By Lyapunov-Schmidt reduction methods and computations of critical groups, we prove that the equation has four -periodic solutions.
Considering the following higher order difference equation where , and are the sets of all positive integers and integers, respectively, , is the set of all real numbers, and there exists a positive integer such that, for any , , .
Throughout this paper, for , we define , when .
In recent years, much attention has been given to second order Hamiltonian systems and elliptic boundary value problems by a number of authors; see [1–3] and references therein. On one hand, there have been many approaches to study periodic solutions of differential equations or difference equations, such as critical point theory (which includes the minimax theory, the Kaplan-Yorke method, and Morse theory), fixed point theory, and coincidence theory; see, for example, [4–20].
Among these approaches, Morse theory is an important tool to deal with such problems. However, there are, at present, only a few papers dealing with higher order difference equation except [21–23]. On the other hand, under some assumptions, the functional may not satisfy the Palasis-Smale condition. Thus, we cannot apply the Morse theory to directly. To go around this difficulty, Tang and Wu  and Liu  obtain many interesting results of elliptic boundary value problems by combining Morse theory with Lyapunov-Schmidt reduction method or minimax principle. Inspired by this, we study the existence of periodic solutions of a higher order difference equation (1) by combining computations of critical groups with Lyapunov-Schmidt reduction method, and an existence theorem on multiple periodic solutions for such an equation is obtained.
For a given integer , let
We denote when is even, or when is odd. Because of , , then, has different values. Therefore, we can write these numbers in such a way: Assume , .
Combing Morse theory with Lyapunov-Schmidt reduction method, we have the following results.
Theorem 1. Suppose that , , and ; we assume that, ,, , where ;there exists a constant such that ;for any , Then (1) possesses at least four nontrivial -periodic solutions.
To apply Morse theory to study the existence of periodic solutions of (1), we will construct suitable variational structure.
Let be the set of sequences , where . For any and , is defined by Then is a vector space.
For any given positive integer , is defined as a subspace of by
can be equipped with inner product and norm as follows: where denotes the Euclidean Norm in , and denotes the usual scalar product in .
Define a linear map by
It is easy to see that the map defined in (10) is a linear homeomorphism with and is a finite dimensional Hilbert space, which can be identified with .
For (1), we consider the functional defined on by where , , .
Since is linearly homeomorphic to , by the continuity of , can be viewed as continuously differentiable functional defined on a finite dimensional Hilbert space. That is, . If we define , then where . Therefore, is a critical point of ; that is, if and only if
On the other hand, is -periodic in , and is -periodic in ; hence, is a critical point of if and only if for any , and is a -periodic solution of (1). Thus, we reduce the problem of finding -periodic solutions of (1) to that of seeking critical points of the functional in .
Apparently, . Consider for all . For convenience, we write as .
In view of , , , when , can be rewritten as
According to (18), for any positive integer with , we know that.
If , then (). That is, the matrix is positive definite.
3. Main Propositions
In order to prove our main results, we will give several propositions and notations as follows.
Definition 2 (see ). Let be a Banach space, let , and let be the th singular relative homology group of the topological pair with coefficients in an Abelian group G. is called the -dimension Betti number. Let be an isolated critical point of with , , and let be a neighborhood of in which has no critical points except . Then the group
is called the th critical group of at , here . Assume that satisfies PS condition; has no critical value less than ; then the th critical group at infinity of is defined as
If , then the Morse index of at is defined as the dimension of the maximal subspace of on which the quadratic form is negative definite. Define . We need the following condition.
Suppose that are two regular values of ; has at most finitely many critical points on and the rank of the critical group for every critical point is finite.
Definition 3 (see ). Assume that satisfies condition ; are all critical values of in and , . Choose . Define
Then is called the th Morse-type number of about the interval .
Here the critical groups of at an isolated critical point describe the local behavior of near , while the critical groups of at infinity describe the global property of . The Morse inequality gives the relation between them.
Proposition 4 (see ). Suppose that satisfies the PS condition and has only isolated critical points, and the critical values of are bounded below. Then we have where , ; is a formal series with nonnegative integer coefficients.
Now we recall the Lyapunov-Schmidt reduction method.
Proposition 5 (see ). Let be a separable Hilbert space with inner product and norm and let and be closed subspaces of such that . Let . If there is a real number such that, for all , , there holds then we have the following:(i) there exists a continuous function such that and is the unique member of such that(ii) the functional defined by and(iii) an element is a critical point of if and only if is a critical point of .
4. Proof of Theorem
Consider the following functional:
As we know, the PS condition is an important part of critical point theory. However, under our assumptions , the functional may not satisfy PS condition. Thus, we cannot apply the Morse theory directly. But the truncated functional does satisfy the PS condition. So we can obtain two critical points of via mountain pass lemma; then we can obtain other critical points by combing Morse theory with Lyapunov-Schmidt reduction method.
At first, we consider the truncated problem where where
Then the functional corresponding to (31) can be written as where . Apparently, .
We only consider the case of ; the case of is similar and omitted.
By , we know that
Then there exist real number (small enough) and such that
Lemma 8. Under the conditions of Theorem 1, the functional satisfies the PS condition.
Proof. Let be such a sequence; that is, there exists a positive constant such that , , and that as , .
That is, is a bounded sequence in the finite dimensional space . Consequently, it has a convergent subsequence. Thus, we obtain Lemma 8.
Let , , and .
Lemma 9. If is a local minimizer of , then must be a local minimizer of .
Proof. Let be a local minimizer of ; then for any sequence , , for big enough , we have .
Because , , and , so , .
For any , if , then .
If , by , , and , then for . Therefore, ; that is, .
The proof of Lemma 9 is complete.
It is easy to see that the zero function 0 is a local minimizer of , and as , where is a first eigenfunction corresponding to the first nonzero eigenvalue of . Thus, by the mountain pass lemma we obtain a critical point of . However, it is true that is a critical point of if is a critical point of ; then we deduce that is a critical point of with Similarly, we obtain another critical point of and
Next we will prove that has two more nonzero critical points. We decompose according to . We set
Since , for any and , we have where . Then, by Proposition 5, there exist a continuous map and a -functional such that
We need to show that has at least five critical points. Hence, we assume that has no critical value less than some .
Lemma 10. Suppose that satisfies , then the functional is anticoercive.
Proof. According to , there exists such that
Then, for any , we have
Assume that is a sequence in such that . Let , then . Because of , there exist some such that, up to subsequence , .
In particular, , . For , . Hence, by , By the above discussion, we have This concludes the proof.
Because is anticoercive, we choose and such that where . Since has no critical value in , .
Thus, we have the following commutative diagram with exact rows:(50)where all the homomorphisms except are induced by inclusions. The exactness of rows implies that , are isomorphisms. Hence is also an isomorphism, and we get
Because the anticoercive functional is defined on the -dimensional , it has a critical point , with
Let , , be the projection of , , in , respectively. Then they are all critical points of . By (11), (14), and Proposition 6, and 0 is a local minimizer of , we have If , , , are the only critical points of , then by Proposition 4 with , This is impossible. Thus has at least five critical points. So also has five critical points, four of which are nonzero. Therefore, (1) has at least four nontrivial solutions. This completes the proof of Theorem 1.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The research was supported by the Research Foundation of Education Bureau of Hunan Province, China (12C0632).
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