- Research Article
- Open Access
Nonlinear Discrete Periodic Boundary Value Problems at Resonance
Advances in Difference Equations volume 2009, Article number: 360871 (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.
Initialed by Lazer and Leach , 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 , 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
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.
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 , 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
and the corresponding norm is
In the rest of the paper, we always assume that
Define a linear operator by
Lemma 2.1 (see ).
Let . Then
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
Then for each real number , there is a decomposition
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
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
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 ,
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.
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 .
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
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 , 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).
By , 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.
Lazer AC, Leach DE: Bounded perturbations of forced harmonic oscillators at resonance. Annali di Matematica Pura ed Applicata 1969, 82: 49–68. 10.1007/BF02410787
Landesman EM, Lazer AC: Nonlinear perturbations of linear elliptic boundary value problems at resonance. Journal of Applied Mathematics and Mechanics 1970, 19: 609–623.
Amann H, Ambrosetti A, Mancini G: Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Mathematische Zeitschrift 1978,158(2):179–194. 10.1007/BF01320867
Brézis H, Nirenberg L: Characterizations of the ranges of some nonlinear operators and applications to boundary value problems. Annali della Scuola Normale Superiore di Pisa 1978,5(2):225–326.
Fučík S, Hess P: Nonlinear perturbations of linear operators having nullspace with strong unique continuation property. Nonlinear Analysis 1979,3(2):271–277. 10.1016/0362-546X(79)90082-8
Iannacci R, Nkashama MN: Nonlinear boundary value problems at resonance. Nonlinear Analysis: Theory, Methods & Applications 1987,11(4):455–473. 10.1016/0362-546X(87)90064-2
Mawhin J, Ward JR Jr.: Periodic solutions of some forced Liénard differential equations at resonance. Archiv der Mathematik 1983,41(4):337–351. 10.1007/BF01371406
Conti G, Iannacci R, Nkashama MN: Periodic solutions of Liénard systems at resonance. Annali di Matematica Pura ed Applicata 1985, 139: 313–327. 10.1007/BF01766859
Omari P, Zanolin F: Existence results for forced nonlinear periodic BVPs at resonance. Annali di Matematica Pura ed Applicata 1985, 141: 127–157. 10.1007/BF01763171
Ding TR, Zanolin F: Time-maps for the solvability of periodically perturbed nonlinear Duffing equations. Nonlinear Analysis: Theory, Methods & Applications 1991,17(7):635–653. 10.1016/0362-546X(91)90111-D
Capietto A, Liu B: Quasi-periodic solutions of a forced asymmetric oscillator at resonance. Nonlinear Analysis: Theory, Methods & Applications 2004,56(1):105–117. 10.1016/j.na.2003.09.001
Iannacci R, Nkashama MN: Unbounded perturbations of forced second order ordinary differential equations at resonance. Journal of Differential Equations 1987,69(3):289–309. 10.1016/0022-0396(87)90121-5
Chu J, Torres PJ, Zhang M: Periodic solutions of second order non-autonomous singular dynamical systems. Journal of Differential Equations 2007,239(1):196–212. 10.1016/j.jde.2007.05.007
Wang Y, Shi Y: Eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. Journal of Mathematical Analysis and Applications 2005,309(1):56–69. 10.1016/j.jmaa.2004.12.010
Mawhin J: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics. Volume 40. American Mathematical Society, Providence, RI, USA; 1979:v+122.
Ma R, Ma H: Unbounded perturbations of nonlinear discrete periodic problem at resonance. Nonlinear Analysis: Theory, Methods & Applications 2009,70(7):2602–2613. 10.1016/j.na.2008.03.047
Rodriguez J: Nonlinear discrete Sturm-Liouville problems. Journal of Mathematical Analysis and Applications 2005,308(1):380–391. 10.1016/j.jmaa.2005.01.032
Kelley WG, Peterson AC: Difference Equations. Academic Press, Boston, Mass, USA; 1991:xii+455.
Agarwal RP, O'Regan D: Boundary value problems for discrete equations. Applied Mathematics Letters 1997,10(4):83–89. 10.1016/S0893-9659(97)00064-5
Rachunkova I, Tisdell CC: Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions. Nonlinear Analysis: Theory, Methods & Applications 2007,67(4):1236–1245. 10.1016/j.na.2006.07.010
Yu J, Guo Z: On boundary value problems for a discrete generalized Emden-Fowler equation. Journal of Differential Equations 2006,231(1):18–31. 10.1016/j.jde.2006.08.011
Atici FM, Cabada A: Existence and uniqueness results for discrete second-order periodic boundary value problems. Computers & Mathematics with Applications 2003,45(6–9):1417–1427.
Bai D, Xu Y: Nontrivial solutions of boundary value problems of second-order difference equations. Journal of Mathematical Analysis and Applications 2007,326(1):297–302. 10.1016/j.jmaa.2006.02.091
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 ).