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
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
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.
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.
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 ).
3. Existence of Periodic Solutions
In this section, we need to give some lemmas first, which have vital importance to prove Theorem 1.1.
Let us write
In fact, we have from Lemma 2.1 that
Proof of Theorem 1.1.
The proof is motivated by Iannacci and Nkashama .
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
Notice that, by (3.32), we have
On the other hand, using (3.41), we deduce immediately that
We claim that
It follows from (3.50) that
Therefore, (3.52) holds.
However, this contradicts (1.11).
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