Multiple Periodic Solutions for Difference Equations with Double Resonance at Infinity
© X. Zhang and D.Wang. 2011
Received: 7 February 2010
Accepted: 6 November 2010
Published: 19 January 2011
By using variational methods and Morse theory, we study the multiplicity of the periodic solutions for a class of difference equations with double resonance at infinity. To the best of our knowledge, investigations on double-resonant difference systems have not been seen in the literature.
Recently, many authors have studied the boundary value problems on nonlinear differential equations with double resonance(see [2–5]). It is well known that in different fields of research, such as computer science, mechanical engineering, control systems, artificial or biological neural networks, and economics, the mathematical modelling of important questions leads naturally to the consideration of nonlinear difference equations. For this reason, in recent years the solvability of nonlinear difference equations have been extensively investigated(see [1, 6–8] and the references cited therein). However, to the best of our knowledge, investigations on double resonant difference systems have not been seen in the literature.
In this paper, several theorems on the multiplicity of the periodic solutions to the double resonant system (1.1) are obtained via variational methods and Morse theory. The research here was mainly motivated by the works [2, 4].
Now, the main results of this paper are stated as follows.
In Section 3, we will prove the main results, before which some preliminary results on Morse theory will be collected in Section 2. Some fundamental facts relative to (1.1) revealed here will benefit the further investigations in this direction, which will be remarked in Section 4.
2. Preliminary Results on Critical Groups
The following results were due to Su .
3. Proofs of Main Results
In this section, we will establish the variational structure relative to problem (1.1) and prove the main results via Morse theory.
From the discussion in [1, Section 2], we see that , , for and if is even or if is odd.
Set and . The following Lemmas 3.5–3.7 benefit from .
This contradict to (3.15) and the proof is complete.
First we have the following claim:
The proof is completed.
Proof of Theorem 1.3.
Proof of Theorem 1.4.
Now, we consider the case where we have and with (3.56). Since , we know that either or . If , ( ) with reads contradiction . If , by similar argument, we can get (3.57). Thus and ( ) reads the contradiction . The proof is complete.
The proof of the following lemma is similar to that of () and is omitted.
Proof of Theorem 1.5.
4. Conclusion and Future Directions
It is known that there have been many investigations on the solvability of elliptic equations with double-resonance via variational methods, where the so called unique continuation property of the Laplace operator, proved by Robinson , plays an important role in proving the compactness of the corresponding functional (see [2–5] and the references cited therein). In this paper, the solvability of the periodic problem on difference equations with double resonance is first studied and the "unique continuation property" of the second-order difference operator is derived by proving Lemma 3.4.
In addition, under the double resonance assumption and , some fundamental facts relative to (1.1) are revealed in Lemmas 3.5–3.7, on which, further investigations, employing new restrictions different from (f3) and (f4), may be based.
On the observations as above, it is reasonable to believe that the research in this paper will benefit the future study in this direction.
The authors are grateful for the referee's careful reviewing and helpful comments. Also the authors would like to thank Professor Su Jiabao for his helpful suggestions. This work is supported by NSFC(10871005) and BJJW(KM200610028001).
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