# Nonlinear Discrete Periodic Boundary Value Problems at Resonance

- Ruyun Ma
^{1}Email author and - Huili Ma
^{2}

**2009**:360871

https://doi.org/10.1155/2009/360871

© R. Ma and H. Ma. 2009

**Received: **25 June 2009

**Accepted: **6 December 2009

**Published: **27 January 2010

## Abstract

## Keywords

## 1. Introduction

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

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.

where denotes the integer part of the real number .

Then (1.2) has at least one solution provided

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

## 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

where

Lemma 3.1.

Then there exists a constant such that for all , one has

Proof.

where is a positive constant less than .

Let

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

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.

Lemma 3.2.

Proof.

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

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

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 .

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

Set

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).

Example 3.3.

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.

## Declarations

## Authors’ Affiliations

## References

- Lazer AC, Leach DE:
**Bounded perturbations of forced harmonic oscillators at resonance.***Annali di Matematica Pura ed Applicata*1969,**82:**49–68. 10.1007/BF02410787MATHMathSciNetView ArticleGoogle Scholar - Landesman EM, Lazer AC:
**Nonlinear perturbations of linear elliptic boundary value problems at resonance.***Journal of Applied Mathematics and Mechanics*1970,**19:**609–623.MATHMathSciNetGoogle Scholar - 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/BF01320867MATHMathSciNetView ArticleGoogle Scholar - 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.MATHGoogle Scholar - 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)90082-8MATHMathSciNetView ArticleGoogle Scholar - 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)90064-2MATHMathSciNetView ArticleGoogle Scholar - 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/BF01371406MATHMathSciNetView ArticleGoogle Scholar - 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/BF01766859MATHMathSciNetView ArticleGoogle Scholar - 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/BF01763171MATHMathSciNetView ArticleGoogle Scholar - 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-DMATHMathSciNetView ArticleGoogle Scholar - 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.2003.09.001MATHMathSciNetView ArticleGoogle Scholar - 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)90121-5MATHMathSciNetView ArticleGoogle Scholar - 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.2007.05.007MATHMathSciNetView ArticleGoogle Scholar - 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.2004.12.010MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar - 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.2008.03.047MATHMathSciNetView ArticleGoogle Scholar - Rodriguez J:
**Nonlinear discrete Sturm-Liouville problems.***Journal of Mathematical Analysis and Applications*2005,**308**(1):380–391. 10.1016/j.jmaa.2005.01.032MATHMathSciNetView ArticleGoogle Scholar - Kelley WG, Peterson AC:
*Difference Equations*. Academic Press, Boston, Mass, USA; 1991:xii+455.MATHGoogle Scholar - Agarwal RP, O'Regan D:
**Boundary value problems for discrete equations.***Applied Mathematics Letters*1997,**10**(4):83–89. 10.1016/S0893-9659(97)00064-5MATHMathSciNetView ArticleGoogle Scholar - 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.2006.07.010MATHMathSciNetView ArticleGoogle Scholar - 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.2006.08.011MATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetView ArticleGoogle Scholar - 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.2006.02.091MATHMathSciNetView ArticleGoogle Scholar

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