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Nonlinear Discrete Periodic Boundary Value Problems at Resonance
Advances in Difference Equations volume 2009, Article number: 360871 (2010)
Abstract
Let be an integer with , and let . We study the existence of solutions of nonlinear discrete problems , where with is the th eigenvalue of the corresponding linear eigenvalue problem.
1. Introduction
Initialed by Lazer and Leach [1], much work has been devoted to the study of existence result for nonlinear periodic boundary value problem
where is an integer. Results from the paper have been extended to partial differential equations by several authors. The reader is referred, for detail, to Landesman and Lazer [2], Amann et al. [3], Brézis and Nirenberg [4], Fučík and Hess [5], and Iannacci and Nkashama [6] for some reference along this line. Concerning (1.1), results have been carried out by many authors also. Let us mention articles by Mawhin and Ward [7], Conti et al. [8], Omari and Zanolin [9], Ding and Zanolin [10], Capietto and Liu [11], Iannacci and Nkashama [12], Chu et al. [13], and the references therein.
However, relatively little is known about the discrete analog of (1.1) of the form
where , with , is continuous in . The likely reason is that the spectrum theory of the corresponding linear problem
was not established until [14]. In [14], Wang and Shi showed that the linear eigenvalue problem (1.3) has exactly real eigenvalues
Suppose that these above eigenvalues have different values , . Then (1.4) can be rewritten as
For each , we denote its eigenspace by . If , then we assume that in which is the eigenfunction of . If , then we assume that in which and are two linearly independent eigenfunctions of .
It is the purpose of this paper to prove the existence results for problem (1.2) when there occurs resonance at the eigenvalue and the nonlinear function may "touching" the eigenvalue . To have the wit, we have what follows.
Theorem 1.1.
Let with , is continuous in , and for some ,
where are two given functions. Suppose for some ,
Assume that for all , there exist a constant and a function such that
where is a given function satisfying
and for at least points in ,
where denotes the integer part of the real number .
Then (1.2) has at least one solution provided
where , , and
In [12], Iannacci and Nkashama proved the analogue of Theorem 1.1 for continuoustime nonlinear periodic boundary value problems (1.1). Our paper is motivated by Iannacci and Nkashama [12]. However, as we will see below, there are big differences between the continuous case and the discrete case. The main tool we use is the LeraySchauder continuation theorem (see Mawhin [15, Theorem ]).
Finally, we note that when in (1.2), the existence of odd solutions or even solutions was investigated by R. Ma and H. Ma [16] under some parity conditions on the nonlinearities. The existence of solutions of secondorder discrete problem at resonance was studied by Rodriguez in [17], in which the nonlinearity is required to be bounded. For other results on discrete boundary value problems, see Kelley and Peterson [18], Agarwal and O'Regan [19], Rachunkova and Tisdell [20], Yu and Guo [21], Atici and Cabada [22], Bai and Xu [23]. However, these papers do not address the problem under "asymptotic nonuniform resonance" conditions.
2. Preliminaries
Let
Let
Then is a Hilbert space under the inner product
and the corresponding norm is
Thus,
In the rest of the paper, we always assume that
Define a linear operator by
Lemma 2.1 (see [16]).
Let . Then
Similar to [12, Lemma ], we can prove the following.
Lemma 2.2 (see [12]).
Suppose that

(i)
there exist and real numbers , such that
(ii) there exist and a constant such that
Then for each real number , there is a decomposition
of satisfying
and there exists a function depending on and such that
3. Existence of Periodic Solutions
In this section, we need to give some lemmas first, which have vital importance to prove Theorem 1.1.
For convenience, we set
Thus, for any , we have the following Fourier expansion:
Let us write
where
Lemma 3.1.
Suppose that for , is an eigenvalue of (1.3) of multiplicity 2. Let be a given function satisfying
and for at least points in ,
Then there exists a constant such that for all , one has
Proof.
For ,
Taking into account the orthogonality of , , and in , we have
Set
Then,
where is a positive constant less than .
Let
We claim that with the equality holding only if , where are constants.
In fact, we have from Lemma 2.1 that
Obviously, implies that , and accordingly for some .
Next we prove that implies . Suppose to the contrary that .
We note that has at most zeros in . Otherwise, must have two consecutive zeros in , and subsequently, in by (1.3). This is a contradiction.
Using (3.6) and the fact that has at most zeros in , it follows that
which contradicts . Hence, .
We claim that there is a constant such that
Assume that the claim is not true. Then we can find a sequence and , such that, by passing to a subsequence if necessary,
From (3.17), it follows that
By (3.12), (3.16), and (3.17), we obtain, for ,
and hence
that is,
By the first part of the proof, , so that, by (3.19), , a contradiction with the second equality in (3.16).
Set and observing that the proof is complete.
Lemma 3.2.
Let be as in Lemma 3.1 and let be associated with by that lemma. Let . Let be a function satisfying
Then for all , one has
Proof.
Using the computations in the proof of Lemma 3.1 and (3.22), we obtain
So that, using (3.7), (3.8), the relation , and Lemma 2.1, it follows that
Proof of Theorem 1.1.
The proof is motivated by Iannacci and Nkashama [12].
Let be associated to the function by Lemma 3.1. Then, by assumption (1.8), there exist and , such that
for all and all with . Hence, (1.2) is equivalent to
where and satisfy (2.12) and (2.14) with . Moreover, by (2.13)
Let , so that
It follows from (3.28) and (3.29) that
Define by
So we have
Define
Then there exists such that
Therefore, (1.2) is equivalent to
To prove that (1.2) has at least one solution in , it suffices, according to the LeraySchauder continuation method [15], to show that all of the possible solutions of the family of equations
(in which , with , fixed) are bounded by a constant which is independent of and .
Notice that, by (3.32), we have
It is clear that for , (3.36) has only the trivial solution. Now if is a solution of (3.36) for some , using Lemma 3.2 and Cauchy's inequality, we obtain
where
So we conclude that
for some constant , depending only on and (but not on or ). Taking , we get
We claim that there exists , independent of and , such that for all possible solutions of (3.36)
Suppose on the contrary that the claim is false. Then there exists with and for all ,
From (3.41), it can be shown that
and accordingly, is bounded in .
Setting , we have
Define an operator by
Then is completely continuous since is finite dimensional. Now, (3.45) is equivalent to
By (3.26), it follows that is bounded. Using (3.47), we may assume that (taking a subsequence and relabeling if necessary) in , and , .
On the other hand, using (3.41), we deduce immediately that
Therefore,
Rewrite , and let, taking a subsequence and relabeling if necessary,
Set
Since in , or .
We claim that
We may assume that , and only deal with the case . The other case can be treated by similar method.
It follows from (3.50) that
which implies that for all sufficiently large,
On the other hand, we have from (3.44), (3.55), and the fact that there exists such that for and ,
This together with (3.55) implies that for ,
Therefore, (3.52) holds.
Now let us come back to (3.43). Multiplying both sides of (3.43) by and summing from to , we get that
Combining this with (3.52) and (3.53), it follows that
However, this contradicts (1.11).
Example 3.3.
By [16], the eigenvalues and eigenfunctions of
can be listed as follows:
Let us consider the nonlinear discrete periodic boundary value problem
where
Obviously, , , and . If we take that
then
Now, it is easy to verify that satisfies all conditions of Theorem 1.1. Consequently, for any periodic function , (3.62) has at least one solution.
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Acknowledgments
This work was supported by the NSFC (no. 10671158), the NSF of Gansu Province (no. 3ZS051A25016), NWNUKJCXGC0317, NWNUKJCXGC0318, the SpringSun program (no. Z2004162033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006 []).
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Keywords
 Differential Equation
 Real Number
 Partial Differential Equation
 Ordinary Differential Equation
 Functional Analysis