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Multiple periodic solutions for resonant difference equations
Advances in Difference Equations volume 2014, Article number: 236 (2014)
Abstract
In this paper, we study the existence of multiple periodic solutions for nonlinear second-order difference equations with resonance at origin. The approach is based on critical point theory, minimax methods, homological linking and Morse theory.
1 Introduction
In this paper, we consider the existence of multiple periodic solutions for the following nonlinear difference equations:
where is a fixed integer, , , is a differential function satisfying
and is the th eigenvalue of the linear periodic boundary value problem
Since (1.1) implies that (P) possesses a trivial periodic solution , we are interested in finding nontrivial periodic solutions for (P). It follows from [1] that all the eigenvalues of () are , . Thus , for , where
For the convenience of later use, we denote by the distinct eigenvalues of (). Moreover, if N is odd then all eigenvalues of () are multiplicity two except , and let
be the corresponding orthonormal eigenvectors; if N is even then all eigenvalues are multiplicity two except and , and let
be the corresponding orthonormal eigenvectors.
Now we establish the variational framework associated with (P). Set
and
Then we can rewrite (P) and () as
respectively.
Let with inner product and norm . Then
For , define , then there exist positive numbers , such that
Define the functional by
where . Then with derivatives
where is the identity matrix of order N, . Hence the periodic solutions of (P) are exactly the critical points of J or −J in E.
We assume that the nonlinearity f satisfies the following conditions:
() for .
() There exists such that
() There exists such that
() There exist and such that
() There exist and such that
(f) For any a fixed number , there exists such that
Therefore we regard the problem (P) as resonance at origin under the assumption ().
Critical point theory has been widely used to study the existence of periodic solutions and solutions for nonlinear difference boundary value problems since the first result was established by using variational methods in 2003 (see [2]). Since then, by using critical point theory, minimax methods and Mores theory, the existence of solutions for non-resonant difference equations has been extensively investigated (see [3–9] and the references therein). As for resonant cases, Zhu and Yu [10] applied critical point theory to study the existence of positive solutions for a second order nonlinear discrete Dirichlet boundary value problem
when nonlinearity f is odd and resonant at infinity. Zheng and Xiao [11] employed critical groups and the mountain pass theorem to study the existence of nontrivial solutions for (1.7) when the nonlinearity and is resonant at infinity. Liu et al. [12] used Morse theory, critical point theory and minimax methods to study the existence of multiple solutions for (1.7) with resonance at both infinity and origin, one can refer to [13, 14].
However, we note that only a few papers concern the existence of periodic solutions for difference equations with resonance. In 2011, Zhang and Wang [15] used variational methods and Morse theory to study the multiplicity of periodic solutions for (P) with double resonance between two consecutive eigenvalues at infinity. The main aim of this paper is to study the multiplicity of nonzero periodic solutions for (P) with resonance at origin. The approach is based on critical theory, Morse theory and homological linking.
The rest of this paper is organized as follows. In Section 2, we collect some useful preliminary results about Morse theory. In Section 3, we give some auxiliary results. Our main results and proofs will be given in Section 4.
2 Preliminaries about Morse theory
In this section, we recall some facts about Morse theory and critical groups [16, 17]. Let E be a real Hilbert space. We say that J satisfies the (PS) condition if every sequence such that is bounded and as has a convergent subsequence.
Suppose that is a functional satisfying the (PS) condition. Let be an isolated critical point of J with , and let U be a neighborhood of . The group
is called the q th critical group of J at , where , denotes the singular relative homology group of the topological pair with coefficient field . Define
Assume that is a finite number. Take . The group
is called the q th critical group of J at infinity (see [18]). The Morse type numbers of the pair are defined by . Denote by the Betti numbers of the pair . By Morse theory, the relationship between and is described by
and
From , for each , it follows that if for some , then J must have a critical point with . If , then for all . Thus if for some , then J must have a new critical point.
For some , define
Then
Suppose that and . Then is a self-adjoint linear operator on E. The dimension of the largest negative space of is called the Morse index of J at , and the dimension of the kernel of is called the nullity of J at . We say that is nondegenerate if the nullity of J at is zero, i.e., has a bounded inverse. For an isolated critical point, the following important result is valid.
Suppose that is an isolated critical point of with finite Morse index and nullity .
-
(i)
for .
-
(ii)
If is nondegenerate, then .
-
(iii)
If , then for or .
Let 0 be an isolated critical point of with finite Morse index and nullity . Assume that J has a local linking at 0 with respect to a direct sum decomposition , , i.e., there exists such that
Then for either or .
Let E be a real Banach space with and suppose that is finite. Assume that satisfies the (PS) condition and
(H1) there exist and such that
where ,
(H2) there exist and with such that
where .
Then J has a critical point with and
3 Auxiliary results
We first show that the functional J satisfies the (PS) condition.
Lemma 3.1 Assume that f satisfies () or (), then J defined by (1.4) satisfies the (PS) condition.
Proof We only prove the case where () holds; the other case can be proved similarly. Let be such that
We only need to show that is bounded. Taking positive number , it follows from (3.1) that there exists such that
By (), there exist such that
Hence, by (1.2), (), (3.2) and (3.3), we have
Since and , we get that is bounded. □
Remark 3.1 By the proof of Lemma 3.1, we have that if f satisfies () or (), then −J satisfies the (PS) condition.
For , defining
then and
Now we construct a linking with respect to the decomposition or .
Lemma 3.2 Suppose that f satisfies () and (f). Then
-
(i)
for any fixed , there exist and such that
(3.4) -
(ii)
for any fixed , there exist and such that
(3.5)
Proof By () and (f), for any , there exists such that
-
(i)
For any , it follows from (3.6) that
Taking , we then have
Because , the function defined on achieves its maximum
at
Hence we get (3.4), where .
-
(ii)
For any , by (3.6), we obtain
Taking , we then obtain
As , the function defined on achieves its minimum
at
Therefore, we obtain (3.5), where . □
Now we define
Lemma 3.3 (i) Suppose that f satisfies (). Then, for any fixed , there exist and such that when ,
where .
-
(ii)
Suppose that f satisfies (). Then, for any fixed , there exist and such that when ,
where .
Proof We only prove (i); the proof of (ii) is similar. For any , it follows from () and (3.2) that
As , we get
Therefore, there exists such that
For ,
Taking , when ,
By (3.8) and (3.9), we get (3.7). □
4 Main results and proofs
In this section, we give our main results and proofs. First, we compute the critical groups of J at both infinity and the origin.
Lemma 4.1 Assume that f satisfies (). Then
-
(i)
if f satisfies ().
-
(ii)
if f satisfies ().
Proof We only prove (i); (ii) can be proved similarly. Fix . Defining , we then have . For any and , it follows from (3.2) and (1.3) that
As , this implies that
On the other hand, by (), we have
where . Hence, for any fixed number a with , we obtain that
By , (4.1) and (4.2), we get that for any , there exists a unique such that
By (4.3) and the implicit function theorem, we obtain that . Define as
Then . Now define as
Clearly, η is continuous, and for any with , it follows from (4.3) that
Therefore
So is a strong deformation retract of . Notice that . Hence we have
□
Lemma 4.2 Assume that f satisfies ().
-
(i)
If f satisfies (), then for ,
-
(ii)
If f satisfies (), then for ,
Proof By () and (1.6), we have .
It follows from () that is a degenerate critical point of J with Morse index and nullity . This implies that is a degenerate critical point of −J with Morse index and nullity .
-
(i)
By (), we can verify that J has a local linking structure at 0 with respect to (see [12]). That implies −J has a local linking structure at 0 with respect to . Notice that and for . Using Proposition 2.2, we get
-
(ii)
Similarly, by (), we can verify that J has a local linking structure at 0 with respect to (see [12]). That implies −J has a local linking structure at 0 with respect to . Notice that for . By Proposition 2.2, we have
□
Next, we give our main results.
Theorem 4.1 Let f satisfy () and (f).
-
(i)
If f satisfies (), then for any fixed , there exists such that when , (P) has at least one nonzero periodic solution satisfying
(4.4) -
(ii)
If f satisfies (), then for any fixed , there exists such that when , (P) has at least one nonzero periodic solution satisfying
(4.5)
Theorem 4.2 Let f satisfy () and (f).
-
(i)
If f satisfies () and (), then, for any fixed , there exists such that when , (P) has at least three nonzero periodic solutions.
-
(ii)
If f satisfies () and (), then, for any fixed , there exists such that when , (P) has at least three nonzero periodic solutions.
Finally, we prove our main results.
Proof of Theorem 4.1 We only prove (i); the proof of (ii) is similar. It follows from () and Lemma 3.1 that J satisfies the (PS) condition. By Lemmas 3.2 and 3.3, J satisfies (H1) and (H2). This implies that and homologically link with respect to the direct sum decomposition (see Example 3 of Chapter II in [16]). Notice that . Applying Proposition 2.3, we get that J has a critical point such that and (4.4). Moreover, it follows from () that . Hence . □
Proof of Theorem 4.2 We only prove (i); (ii) can be proved similarly. It follows from Lemma 4.1(i) that
Using Lemma 4.2(i), we have
By Theorem 4.1, we know that there exists such that when , (P) has at least one nonzero periodic solution satisfying (4.4).
By Proposition 2.1(i), we have
Combining with (4.4), we get that . Note that .
-
(1)
If , then by Proposition 2.1(ii) and (4.4),
-
(2)
If , then or . By Proposition 2.1(iii),
-
(3)
If , then or or for . It follows from Proposition 2.1(iii) and (4.4) that
Thus, we conclude that
Assume that . Then the 2m th Morse inequality (2.1) is expressed as . This is impossible. Thus J must have another nonzero critical point . By Morse theory, we have either
or
Suppose that (4.9) holds. Then it follows from (4.9) and Proposition 2.1(i), (iii) that
Noticing that , we have that
Assume that . We will divide the consideration into four cases.
-
(1)
and . By (4.6)-(4.8), (4.11) and (4.12), we get
(4.13)
These are impossible.
-
(2)
and . By the 2m th and th Morse inequalities, we get that
(4.14)
It then follows from (4.6)-(4.8), (4.11), (4.12) and (4.14) that
These contradict (4.8).
-
(3)
and . We have (4.14) and (4.13). These are impossible.
-
(4)
and . Then we have (4.15), which contradicts (4.8).
Hence, J must have a third nonzero critical point . The proof of the case where (4.10) holds is similar. This completes the proof. □
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Acknowledgements
The authors thank the reviewers for their valuable comments and suggestions. In addition, this work is partly supported by the National Natural Science Foundation of China (No. 51278325) and by the Natural Science Foundations of Shanxi (Nos. 2011011002-4, 2012011004-3).
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Zhang, J., Wang, S., Liu, J. et al. Multiple periodic solutions for resonant difference equations. Adv Differ Equ 2014, 236 (2014). https://doi.org/10.1186/1687-1847-2014-236
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DOI: https://doi.org/10.1186/1687-1847-2014-236
Keywords
- difference equations
- periodic solutions
- homological linking
- critical group
- Morse theory