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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 707686, 8 pages
http://dx.doi.org/10.1155/2013/707686
Research Article

Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

School of Science, Jimei University, Xiamen 361021, China

Received 1 April 2013; Accepted 5 June 2013

Academic Editor: Zidong Wang

Copyright © 2013 Honghua Bin and Zhenkun Huang. 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.

Abstract

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

1. Introduction

In this paper we are interested in the lower bound of the number of periodic solutions for second-order autonomous difference system where , , , and is a fixed positive integer.

Discrete systems have been investigated by many authors using various methods, and many interesting results have obtained; see [17] and references therein. The critical point theory [8, 9] is a useful tool to investigate differential equations, which is developed to study difference equations. Using minimax methods in critical point theory, Guo and Yu [10, 11] investigated the existence of periodic and subharmonic solutions of system (1), where nonlinearity is either sublinear or superlinear. In this paper, we assume that(P1) there exist a function and a symmetric matrix such that , , and where denotes the usual norm in . Moreover there exist functions , such that , , where denotes the gradient of function.

System (1) can be regarded as discrete analogous of the following differential system: A great deal of research has been devoted to (3). For example, by using minimax theory, Rabinowitz [12] has given some interesting results, and Mawhin and Willem [9] obtained some results using the critical point theory. Moreover, there is a vast literature on the problems concerning periodic solutions, BVP, asymptotically behavior of solutions, and so forth.

Morse theory [8, 9, 1316] has been used to solve the asymptotically linear problem. Chang [17], Amann and Zehnder [18] obtained the existence of three distinct solutions via Morse theory, where (3) was nonresonant at infinity. Moreover, the resonant case has been considered in [1923]. The estimate of number of periodic solutions of (3) was established in [24]. Motivated by [24], we will use Morse theory to consider the lower bound of number of periodic solutions for system (1).

Throughout this paper we employ some standard notations. Denote by the real number and the integer sets, respectively. is the real space with dimension . if and . or denotes the transpose of matrix or vector .

If and are bounded on , and system (1) is -resonant at , then functional does not satisfy the compactness condition of the Palais-Smale type. Therefore our discussion will be divided into three cases. Moreover, we assume that(P2) has a finite number of nondegenerated critical points;(P3)all -periodic solutions of system (1) are not -resonant;(P4)for , , where and .

Now we state the main results as follows.

Theorem 1. Assume that (P1)–(P4) hold, and system (1) is not -resonant at . Then where is the number of the nonconstant -periodic solutions of system (1), is the global twist number (see (32)), and will be defined in Section 3.

Theorem 2. Assume that (P1)–(P4) hold, system (1) is -resonant at , and is bounded in , . Then (4) is valid.

Theorem 3. Assume that (P1)–(P4) hold, system (1) is -resonant at , and are bounded in . Then

Remark 4. Benci and Fortunato [24] studied asymptotically linear equation (3). Theorem 1 extends and generalizes the analogous results in [24], and Theorems 2-3 are new results.

The organization of this paper is organized as follows. In Section 2 we study the compactness condition for functional . Some facts about Morse theory and necessary preliminaries are given in Section 3. In Section 4 the main results are proved.

2. (PS) Condition

We say that a -functional on Hilbert space satisfies the Palais-Smale (PS) condition, if every sequence in such that is bounded and as , contains a convergent subsequence.

Here we first introduce space .

Let , where . For any . Then is a linear space. Let equip with inner product and norm as follows: where and are the usual inner product and norm in , respectively. Obviously, is a Hilbert space with dimension and homeomorphism to .

By the variational method, the -periodic solutions of (1) are same as the critical points of the -functional By assumption (P1), the functional can be rewritten as and we write .

Consider eigenvalue problem that is, , . By the periodicity, the difference system has complexity solution for , where , . Moreover, .

Let denote the real eigenvector corresponding to the eigenvalues , , and , where stands for the greatest-integer function. In terms of eigenvalue for some , we can split space as follows: where Moreover, there exists such that

Let us recall the definition of resonance (see [24]).

A -periodic solution of (1) is called -resonance, if there exists , where denotes the Hessian matrix of and is the spectrum of matrix. We say that (1) is -resonant at , if there exists .

Lemma 5. Assume that (P1) and (P4) hold, and system (1) is not -resonant at . Then functional (see (8)) satisfies the (PS) condition.

Proof. Let be the (PS) sequence for functional ; that is, is bounded, and as . Therefore, for any , we have
By , we write with . To show that satisfies (PS) condition, it is enough to prove that is bounded in . That is, we need only to prove that and are bounded in . By contradiction, without loss of generality, there exists such that Therefore, for all , by assumption (P1), there exist and such that for large . Thus there is , . Taking in (13), by previous argument, it follows a contradiction. Therefore and are bounded in . This completes the proof.

Here and in the sequel, the letter will be indiscriminately used to denote various positive constants whose exact values are irrelevant, and is arbitrarily small. Moreover we also denote by the various positive constants in this paper.

Lemma 6. Assume that (P1) and (P4) hold. System (1) is -resonant at , is bounded in , and . Then satisfies the (PS) condition.

Proof. Let be the (PS) sequence for functional ; that is, is bounded, and as .
Since system (1) is -resonant at , . Similarly, let with , , and. By the same method as proof of Lemma 5, it also follows that and are bounded in . Next we prove that is bounded in .
by , , and are bounded in , and it follows that is bounded. On the other hand, , so is bounded. It is easy to see from assumption that is bounded. The proof is completed.

If we assume that are bounded and system (1) is -resonant at , then functional does not satisfy the (PS) condition. In order to overcome the difficult arising from the lack of compactness condition, we use a suitable penalization technique (one can refer to [20, 24]) and add a perturbation term to the functional . Define where is a positive real number and the penalized functional is given by where . Obviously, if is a critical point of with , then is also the critical point of .

Lemma 7. Assume that (P1) and (P4) hold, are bounded in , and system (1) is -resonant at . Then satisfies the (PS) condition. Moreover, for any critical point of , there exists such that , where , and .

Proof. Let be the (PS) sequence for functional ; that is, is bounded in , and for any , Similarly to the proof of Lemma 5, we need only to prove that is bounded in .
Taking in (19), it follows that . By the definition of , it follows that is bounded. Therefore the penalized functional satisfies the (PS) condition.
Let be the critical point of , then So there is a such that , and the proof is completed.

3. Preliminaries

Let be a real Hilbert space, and let be a -functional on . We denote by the set of critical points of , the level set of , and . In the following we suppose that is a -functional on which satisfies the (PS) condition.

Definition 8 (see [9, 14]). Let be a critical point of . The Morse index of by is defined as the supremum of the dimensions of the vector subspace of on which is negative definite. The nullity of by is defined as the dimension of . A critical point will be said to be nondegenerate if is invertible.
Denote by , the Morse index and nullity of for functional . By (10), .
A set is called critical set if for some . A critical set is called discrete nondegenerate critical manifold, if is connected and does not depend on .

Definition 9. The Poincare polynomial of the pair is defined by where denotes the th singular relative homology of the pair with coefficients in field . Define the topological Morse index of critical set as .
For simplicity, we write and instead of and , respectively. It is well known that if is a nondegenerate critical point and is finite, then . If is a nondegenerate critical manifold and is finite, then , where is a polynomial with nonnegative integer coefficients (see [13, 15]).

Next we investigate and use functional (see (8)) or (see (18)) instead of , instead of .

Lemma 10 (see [19, 24]). Assume that (P1) and (P4) hold, and system (1) is not -resonant at . Then there exists , such that

Lemma 11. Assume that (P1) and (P4) hold, system (1) is -resonant at , , and is bounded in . Then there exists , and (21) is valid.

Proof. Write with . Then there exist such that as , as . Let , . By previous argument, it follows that has no critical points in .
On the other hand, for all , Therefore there exists , such that . For , we have hence as , which implies that is bounded from the following in . Let , then , and is a strong deformation retraction of . By Lemma 6, satisfies (PS) condition, and we have So we obtain (21).

Lemma 12. Under the assumption of Theorem 3, there exists such that .

Proof. Let with , , and . Then there exist such that all critical points of are in , where and are the same as in proof of Lemma 11, and . In fact,
Similarly, for , and as , which implies that is bounded from the following in . Let . If , by satisfies (PS) condition, and methods of strong deformation retract, we have . The proof is completed.

Assume that on Hilbert space there is an action of discrete group , and denote by the fixed points set for the action; that is, . The functional is called invariant, if , , and . In the following, denotes a cyclic group of order. In terms of Proposition 8.2 and Proposition  8.5 in [13], we have following lemma.

Lemma 13. Assume that is a -functional on an Hilbert space and satisfies (PS) condition. Let   ( possible ) be two regular values of . Assume that consists only of critical sets, and then the following Morse relation holds: where is a polynomial with nonnegative integer coefficients. If all the critical points of in are nondegenerate and have finite Morse index, then (26) can be written as Now if is invariant, and consists only of nondegenerate critical points having finite Morse index, then (26) becomes where is a formal series with nonnegative integer coefficients. Moreover if consists only of nondegenerate critical manifolds having finite Morse index, then

Remark 14. By (29), our main goal in this paper is to estimate which gives a lower bound of the number of the nonconstant critical points of in .

Lemma 15. Let be a critical point of functional . Denote by the positive eigenvalues (repeated according to their multiplicity) of , where . Under the assumption (P2), we have , where denotes the greatest-integer function and is the number of eigenvalues such that .

Proof. By , we consider the equation , , where is the critical point of . It is easy to see that are eigenvalues of on , where . Therefore the number of negative eigenvalues is just what we are looking for; the proof is completed.

Definition 16. For any critical point of , there are positive eigenvalues (repeated according to their multiplicity) of , which will be denoted by . The number is called twist number of . Moreover the twist number of is defined by , where is the number of the positive eigenvalues (repeated according to their multiplicity) of .

Let be a constant critical point of functional ; that is, . By Lemma 15 and Definition 16, it is easy to deduce the following relation between the Morse index and the twist number as follows:

In view of the number or of the positive eigenvalues (repeated according to their multiplicity) of or , the constant critical point is called -positive (resp., -negative) if is even (resp., odd). On the contrary, the virtual critical point is called -positive (resp., -negative) if is odd (resp., even), see [24].

We denote by and the number of -positive and -negative critical points of . If is invertible, then . Thus, if we consider as a virtual critical point, we have that the number of -positive critical points equals the number of -negative critical points. However, if is singular, the result is not hold in general. If we introduce virtual critical points having twist number zero, where they are considered as -positive if and as -negative if , then the number of -positive critical points is also equal to the number of -negative critical points.

Let , which has been used in (4) and (5). We denote by the -positive critical points and by the -negative critical points such that Then the global twist number of the system (1) is defined by

4. Proof of Main Results

Proof of Theorem 1. The argument is analogous to one used by Benci and Fortunato in [24]. Set . Under the assumption (P2), let be the nondegenerate constant critical points of .
By Lemmas 5 and 10, functional satisfies (PS) condition, and there exists sufficiently small such that , where . Since is and invariant functional on , then by assumption (P3), we have ; that is,
Let , () denote the Morse indices of the -positive and -negative critical points (including ) of , and without loss of generalities, assume that is -negative. So for some , where is referred to (31). Then (33) becomes Set , , and , where are nonnegative integer and .
By Remark 14, the lower bound of the number of nonconstant -periodic solutions for system (1) is to estimate . Since , then Let . By (35), we turn our attention to estimate .
If is even (resp., odd), by Lemma 15, is also even (resp., odd). Therefore by the definition of -positive and -negative critical points of , () are even numbers, are odd numbers for , , and is a even number.
Set , , and By (34), we have , and where , . Meanwhile, if , , . If , , . Clearly if .
A straight analysis shows that . By (30) and the definition of global twist number that refer to (32), we have . It completes the proof of Theorem 1.

Proof of Theorem 2. Under the assumptions of Theorem 2, by Lemmas 6 and 11, functional satisfies (PS) condition, and there exists such that , for .
Similarly, we have , where () are nondegenerate critical points of . The remainder is the same as that of Theorem 1.

The following lemma is needed to prove Theorem 3.

Lemma 17. If all assumptions in Theorem 3 hold, then there exists (independent of ) such that where with , and denote the Morse index and nullity of critical point for functional , respectively.

Proof. Let be a critical point of . By Lemma 7, we have . Therefore if and only if . Since by assumption (P1), there exists such that as . Therefore there exists such that as . It follows that relation (38) is valid, and the proof is completed.

Proof of Theorem 3. Let , and denote by the constant critical points of . We assume, without loss of generalities, as , as . Clearly .
Set , , and , where is the decomposition of with and is large enough.
By Lemma 17, contains only critical points of which have . , since as . , since when . Moreover by assumption (P3), contains only nondegenerate critical manifolds.
Since satisfies (PS) condition, by Lemma 13, relation (28) reads that is, where . For , we set , where . And analogous notation can be introduced for . Then, considering the terms of degree in (42), we have where . Clearly that is, is the absolute value of the sum of the negative coefficients of . Next we estimate the number .
Let and be the -positive and -negative critical points of with nonzero twist numbers, whose order satisfies (31), and , , . Without loss of generalities, assume , and introduce virtual -negative critical points () having twist number and Morse index ; that is, For , set , where , . Then (43) can be written as . Setting , then if is odd, and if is even. So A straight analysis shows that if is even, and if is odd. Therefore By (30) and (45), we have In view of (45), (47), and (48), we have The proof is completed.

Remark 18. Although is invertible under the assumptions of Theorem 3, we do not make use of (42) directly, because we consider only the terms of degree in proof of Theorem 3.

Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grants (11101187), NCETFJ (JA11144), the Excellent Youth Foundation of Fujian Province (2012J06001), and the Foundation of Education of Fujian Province (JA09152).

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