Open Access

Nonlinear Discrete Periodic Boundary Value Problems at Resonance

Advances in Difference Equations20102009:360871

DOI: 10.1155/2009/360871

Received: 25 June 2009

Accepted: 6 December 2009

Published: 27 January 2010


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 continuous-time 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 Leray-Schauder 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 second-order 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


Then is a Hilbert space under the inner product

and the corresponding norm is

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
  1. (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




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



For ,
Taking into account the orthogonality of , , and in , we have

where is a positive constant less than .



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


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


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 Leray-Schauder 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

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

Rewrite , and let, taking a subsequence and relabeling if necessary,



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
Obviously, , , and . If we take that

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.



This work was supported by the NSFC (no. 10671158), the NSF of Gansu Province (no. 3ZS051-A25-016), NWNU-KJCXGC-03-17, NWNU-KJCXGC-03-18, the Spring-Sun program (no. Z2004-1-62033), SRFDP (no. 20060736001), and the SRF for ROCS, SEM (2006 [ ]).

Authors’ Affiliations

Department of Mathematics, Northwest Normal University
College of Economics and Management, Northwest Normal University


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© R. Ma and H. Ma. 2009

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