Nonlinear Discrete Periodic Boundary Value Problems at Resonance
© R. Ma and H. Ma. 2009
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.
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 , Amann et al. , Brézis and Nirenberg , Fučík and Hess , and Iannacci and Nkashama  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 , Conti et al. , Omari and Zanolin , Ding and Zanolin , Capietto and Liu , Iannacci and Nkashama , Chu et al. , 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
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.
where denotes the integer part of the real number .
Then (1.2) has at least one solution provided
In , 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 . 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  under some parity conditions on the nonlinearities. The existence of solutions of second-order discrete problem at resonance was studied by Rodriguez in , in which the nonlinearity is required to be bounded. For other results on discrete boundary value problems, see Kelley and Peterson , Agarwal and O'Regan , Rachunkova and Tisdell , Yu and Guo , Atici and Cabada , Bai and Xu . However, these papers do not address the problem under "asymptotic nonuniform resonance" conditions.
Then is a Hilbert space under the inner product
In the rest of the paper, we always assume that
Lemma 2.1 (see ).
Similar to [12, Lemma ], we can prove the following.
Lemma 2.2 (see ).
there exist and real numbers , such that
(ii) there exist and a constant 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.
Thus, for any , we have the following Fourier expansion:
Let us write
Then there exists a constant such that for all , one has
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,
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.
Proof of Theorem 1.1.
The proof is motivated by Iannacci and Nkashama .
Let be associated to the function by Lemma 3.1. Then, by assumption (1.8), there exist and , 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 , 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
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 ,
and accordingly, is bounded in .
Setting , we have
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
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
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
However, this contradicts (1.11).
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 [ ]).
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