Nonlinear Discrete Periodic Boundary Value Problems at Resonance

Abstract

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.

1. Introduction

Initialed by Lazer and Leach [1], much work has been devoted to the study of existence result for nonlinear periodic boundary value problem

(1.1)

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

(1.2)

where , with , is continuous in . The likely reason is that the spectrum theory of the corresponding linear problem

(1.3)

was not established until [14]. In [14], Wang and Shi showed that the linear eigenvalue problem (1.3) has exactly real eigenvalues

(1.4)

Suppose that these above eigenvalues have different values , . Then (1.4) can be rewritten as

(1.5)

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.

Let with , is continuous in , and for some ,

(1.6)

where are two given functions. Suppose for some ,

(1.7)

Assume that for all , there exist a constant and a function such that

(1.8)

where is a given function satisfying

(1.9)

and for at least points in ,

(1.10)

where denotes the integer part of the real number .

Then (1.2) has at least one solution provided

(1.11)

where , , and

(1.12)

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

Let

(2.1)

Let

(2.2)

Then is a Hilbert space under the inner product

(2.3)

and the corresponding norm is

(2.4)

Thus,

(2.5)

In the rest of the paper, we always assume that

(2.6)

Define a linear operator by

(2.7)

Lemma 2.1 (see [16]).

Let . Then

(2.8)

Similar to [12, Lemma ], we can prove the following.

Lemma 2.2 (see [12]).

Suppose that

• (i) there exist and real numbers , such that

(2.9)

 (ii) there exist and a constant such that

(2.10)

Then for each real number , there is a decomposition

(2.11)

of satisfying

(2.12)

(2.13)

and there exists a function depending on and such that

(2.14)

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

(3.1)

Thus, for any , we have the following Fourier expansion:

(3.2)

Let us write

(3.3)

where

(3.4)

Lemma 3.1.

Suppose that for , is an eigenvalue of (1.3) of multiplicity 2. Let be a given function satisfying

(3.5)

and for at least points in ,

(3.6)

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

(3.7)

Proof.

For ,

(3.8)

Taking into account the orthogonality of , , and in , we have

(3.9)

Set

(3.10)

Then,

(3.11)

where is a positive constant less than .

Let

(3.12)

We claim that with the equality holding only if , where are constants.

In fact, we have from Lemma 2.1 that

(3.13)

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

(3.14)

which contradicts . Hence, .

We claim that there is a constant such that

(3.15)

Assume that the claim is not true. Then we can find a sequence and , such that, by passing to a subsequence if necessary,

(3.16)

(3.17)

From (3.17), it follows that

(3.18)

By (3.12), (3.16), and (3.17), we obtain, for ,

(3.19)

and hence

(3.20)

that is,

(3.21)

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.

Let be as in Lemma 3.1 and let be associated with by that lemma. Let . Let be a function satisfying

(3.22)

Then for all , one has

(3.23)

Proof.

Using the computations in the proof of Lemma 3.1 and (3.22), we obtain

(3.24)

So that, using (3.7), (3.8), the relation , and Lemma 2.1, it follows that

(3.25)

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

(3.26)

for all and all with . Hence, (1.2) is equivalent to

(3.27)

where and satisfy (2.12) and (2.14) with . Moreover, by (2.13)

(3.28)

Let , so that

(3.29)

It follows from (3.28) and (3.29) that

(3.30)

Define by

(3.31)

So we have

(3.32)

Define

(3.33)

Then there exists such that

(3.34)

Therefore, (1.2) is equivalent to

(3.35)

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

(3.36)

(in which , with , fixed) are bounded by a constant which is independent of and .

Notice that, by (3.32), we have

(3.37)

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

(3.38)

where

(3.39)

So we conclude that

(3.40)

for some constant , depending only on and (but not on or ). Taking , we get

(3.41)

We claim that there exists , independent of and , such that for all possible solutions of (3.36)

(3.42)

Suppose on the contrary that the claim is false. Then there exists with and for all ,

(3.43)

From (3.41), it can be shown that

(3.44)

and accordingly, is bounded in .

Setting , we have

(3.45)

Define an operator by

(3.46)

Then is completely continuous since is finite dimensional. Now, (3.45) is equivalent to

(3.47)

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

(3.48)

Therefore,

(3.49)

Rewrite , and let, taking a subsequence and relabeling if necessary,

(3.50)

Set

(3.51)

Since in , or .

We claim that

(3.52)

(3.53)

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

(3.54)

which implies that for all sufficiently large,

(3.55)

On the other hand, we have from (3.44), (3.55), and the fact that there exists such that for and ,

(3.56)

This together with (3.55) implies that for ,

(3.57)

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

(3.58)

Combining this with (3.52) and (3.53), it follows that

(3.59)

However, this contradicts (1.11).

Example 3.3.

By [16], the eigenvalues and eigenfunctions of

(3.60)

can be listed as follows:

(3.61)

Let us consider the nonlinear discrete periodic boundary value problem

(3.62)

where

(3.63)

Obviously, , , and . If we take that

(3.64)

then

(3.65)

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.