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We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discrete

Difference equations represent the discrete counterpart of ordinary differential equations and are usually studied in connection with numerical analysis. As it is well known, the critical point theory is used to deal with the existence of solutions of difference equations. For example, in 2003, Guo and Yu [

In this paper, we consider the following second order nonlinear difference equations with

Assume further that

Moreover, we may regard (

The study of homoclinic solutions for (

Throughout this paper, we always suppose that the following conditions hold:

function

In many studies of

In this paper, instead of (

The rest of the paper is organized as follows. In Section

We will establish the corresponding variational framework associated with (

Consider the real sequence spaces

Define the space

Now we consider the variational functional

The following lemma plays an important role in this paper; it was established in [

If

The main result is as follows.

Suppose conditions

Equation (

If

To prove the multiplicity results, we need the following lemma.

Let

We define the Nehari manifold:

To prove the main results, we need some lemmas.

Suppose conditions

Let

Note that, from

Let

Suppose conditions

Let

Firstly, we prove that

Suppose

According to (

Finally, we show that there exists a convergent subsequence of

By

(1) Now we need five steps to finish this proof.

By (

Note that, by

Therefore, for all

Then we define a mapping

Let

Let

Let

(2) If

This completes Theorem

Finally, we exhibit examples to demonstrate the applicability of Theorem

Consider the second order difference equation:

Consider the

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous referee for his/her valuable suggestions. This work is supported by Program for the National Natural Science Foundation of China (no. 11371313) and Biomathematics Laboratory of Yuncheng University (SWSX201302, SWSX201305).