Let N, Z and R denote sets of all natural numbers, integers and real numbers, respectively. We consider the existence of sign-changing solutions, positive solutions and negative solutions for the following second-order nonlinear difference equation with Dirichlet boundary value problem (BVP for short)
where is a given integer and , is continuous in the second variable, denotes the forward difference operator defined by , .
In recent years, many authors devoted to the study of (1.1) by employing various methods and obtained some interesting results. Here we mention a few. Employing critical point theory, Agarwal  established the existence results of multiple positive solutions. While the nonlinearity is discontinuous, Zhang  gained another new multiple solutions. Zhang and Sun  obtained two existence results of multiple solutions. By aid of algebra and Krasnoselskii fixed point theorem, Luo  investigated the existence of positive solutions.
Study on the sign-changing solutions is a very important research field both in differential equations and difference equations. As to the sign-changing solutions for differential equations, many scholars achieved excellent results  -  by making using of a variety of methods and techniques, such as Leray-Schauder degree theory, fixed point index theory, topological degree theory, invariant sets of descending flow, critical point theory and etc.. Among them, invariant sets of descending flow play an important role, which was first used by Sun  . However, to the authors’ knowledge, there are few literatures that considered sign-changing solutions for difference equations. Making use of invariant sets of descending flow,  studied periodic boundary value problem
In this paper, our purpose is to establish some sufficient conditions for the existence of solutions for (1.1). First, we will construct a functional I such that solutions of (1.1) correspond to critical points of I. Then, by using invariant sets of descending flow and Mountain pass lemma, we obtain sign-changing solutions, negative solutions and positive solutions for (1.1).
2. Preliminaries and Main Results
Given , let be a T-dimen- sional Hilbert space which is equipped with the inner product
then the norm can be induced by
Let H be the T-dimensional Hilbert space equipped with the usual inner product and the usual norm . It is not difficult to see that G is isomorphic to H, and are equivalent. Denote ， . Then for any , we find .
Define functional as
For any , can be rewritten as
Here is the transpose of the vector on and
In the following, we first consider the linear eigenvalue problem corresponding to (1.1)
By direct computation, we get eigenvalues of (2.3) as
Denote be the corresponding eigenvectors of , where
It is obvious that , and for all . Note that are also
eigenvalues of matrix .
Next, for , we consider BVP
where . It is not hard to know that (2.5) and the system of linear algebra equations are equivalent, then the unique solution of (2.5) can be expressed by
On the other side, we have
Lemma 2.1 The unique solution of (2.5) is
here can be written as
Proof. First consider the homogeneous equation of (2.5)
then the corresponding characteristic equation of (2.7) is
Since , which means we have
Two independent solutions of (2.7) can be expressed by and . Therefore, the general solution of (2.5) is .
The next step is to determine coefficients . Now using the method of variation of constant, it follows
Thus, the general solution of (2.5) is
Using initial conditions, we find and
Write , , then
Hence, we achieve the unique solution of (2.5)
here can be written as
Remark 2.1 From Lemma 2.1, we have
Define as follows
where is a completely continuous operator. Combining (2.6) with Lemma 2.1, we achieve that .
Remark 2.2 According to Lemma 2.1, it is not difficult to know that is a solution of (1.1) if and only if is a fixed point of .
Lemma 2.2 The functional I defined by (2.1) is Frechet differentiable on H and has the expression for .
Proof. For any , using the mean value theorem, it follows
Here . As f is continuous in x, we find
which leads to
thus we can immediately conclude that I is Frechet differentiable on H and
On the other side, for all and , there holds
Making use of the definition of inner product and Lemma 2.1, it follows
Then for all , that is to say, .
Remark 2.3 According to Lemma 2.2 and Remark 2.2, we find that critical points of I defined on H are precisely solutions of (1.1).
Now, we give some necessary lemmas and definitions.
Definition 2.1 (  ) Let , I is said to be satisfied Palais-Smale condition((PS)condition for short) if every sequence such that is bounded and has a convergent subsequence in H.
Definition 2.2 (  ) Assume . If any sequence for which is bounded and possesses a convergent subsequence in H, then we say that I satisfies the Cerami condition ((C) condition for short).
Lemma 2.3 (Mountain pass lemma  ) Let H be a real Hilbert space, assume that satisfies the (PS) condition and the following conditions:
(H1) There exist constants and such that for all .
(H2) There exists such that .
Then I has a critical value , moreover, c can be characterized as
be the open ball in H with radius and centered at 0, denote boundary of .
Lemma 2.4 (  ) Let H be a Hilbert space, there are two open convex subsets and on H with , and . If satisfies the (PS) condition and for all . Assume there is a path such that
then I has at least four critical points, one in , one in , one in , and one in .
Remark 2.4 By Theorem 5.1  , we can replace (PS) condition by weaker (C) condition in Lemma 2.4.
Throughout this paper, we assume that
(J2) for where is a constant, or , and satisfy
At last, we state our main results as following.
Theorem 2.1 Suppose (J1) and (J2) and . Then one has the following.
(i) If is not an eigenvalue of (2.3), then (1.1) has at least three nontrivial solutions, one sign-changing, one positive and one negative.
(ii) If r is an eigenvalue of (2.3) and (J3) holds, then the conclusion of (i) is true.
Theorem 2.2 If and for all
. Then (1.1) has at least two nontrivial solutions, one negative and one positive.
From Theorem 2.2, we can get
Corollary 2.3 Suppose for any , we have:
(i) If and for any , then
(1.1) has at least a negative solution.
(ii) If and for any , then
(1.1) has at least a positive solution.
Our results improve previous work in the following way:
(1)     considered Dirichlet boundary value problem, but it is unknown whether the solutions are sign-changing. While in this paper, the nonlinear term f can change sign.
(2) The nonlinearity f satisfies classical Ambrosett-Rabinowitz superlinear condition in    or locally Lipschitz continuity in    , which are not used in our results.
3. Existence of Sign-Changing Solutions of (1.1)
In this section, we shall make use of Lemma 2.4 to complete the proof of Theorem 2.2. Let convex cones and . The distance respecting to in H is written by . For arbitrary , we denote
then are open convex subsets on H with . In addition, contains only sign-changing functions.
Lemma 3.1 Suppose one of the following conditions holds.
(ii) is not an eigenvalue of (2.3), here r is defined by (J2).
Then the functional I defined by (2.1) satisfies (PS) condition.
Proof . (i) Assume . Let be a (PS) sequence, i.e., is bounded and as . Since H is a finite dimensional Hilbert space, we only need to show that is bounded. If , choosing a constant , we have for all . Then
Since is bounded, we conclude that is a bounded sequence and (PS) condition is satisfied.
(ii) suppose is not an eigenvalue of (2.3). We are now ready to prove that is bounded. Arguing by contradiction, we suppose there is a subsequence of with and for each , either
is bounded or . Put . Clearly, . Then
there have a subsequence of and satisfying that as . Write
Since for all and , we get
Because of as , we have . In view of Lem-
ma 2.2, we find that r is an eigenvalue of matrix A, which contradicts to the assumption. So is bounded and the proof is finished.
Lemma 3.2 I satisfies (C) condition under (J3).
Proof . First assume (J3) (i) be satisfied. There exists , if be a sequence such that and , there holds
Then we claim is bounded. Actually, if is unbounded, there possesses a subsequence of and some satisfying . According to (J3) (i), we get
and there has a positive constant such that for any and . Therefore,
which contradicts to (3.2). Then I satisfies (C) condition.
When (J3) (ii) holds, we can prove I satisfies (C) condition in a similar way. Then Lemma 3.2 is verified.
Lemma 3.3 If (J1) and (J2) hold, there exist and such that for , we have
(i) if is a nontrivial critical point of I and , then x is a negative solution of (1.1);
(ii) if is a nontrivial critical point of I and , then x is a positive solution of (1.1).
Proof. (i) According to (J1) and (J2). For all and , there exists such that
Let , , for all . Since
it follows and
By (J1) and (J2), there exist constants , and such that
Choosing a positive constant D, since is finite-dimensional, we have
It is obviously that . Moreover, and imply
Making use of (3.4)-(3.6), we get
here . Hence
When , there holds
Since , we obtain
If is a nontrivial critical point of I, it is clear that . It follows from (3.7) that . Combining (3.3) and remark 2.1, we have . Consequently, x is a negative solution of (1.1).
(ii) can be discussed similarly, we only need to change to to prove (ii). For simplicity, we omit its proof.
Lemma 3.4 Suppose be eigenvectors corresponding to eigenvalues of (2.3) and . If , then as .
Proof. (1) If . From (3.1), we can see that as for any .
(2) Assume . For , we have . In general, we can suppose . Thus and there exists
satisfying . From for any
and , there exists such that
Then for , it follows
Since and , we find as . This completes the proof.
Now we are in the position to prove Theorem 2.1 by using Lemma 2.4.
Proof of Theorem 2.1 From (3.5), we get
which combine with (3.6) gives that
It follows from (3.4) that for any . Then there has such that . Moreover, in
view of Lemma 3.4, we can choose such that for all and . To apply Lemma 2.4, we define a path as
By direct computation, we get
Combining Lemmas 3.1, 3.3 and 2.4, we find there has a critical point in
corresponding to a sign-changing solution of (1.1). Moreover, we also have a critical point in corresponding to a positive
solution (a negative solution) of (1.1). The proof of (i) is completed.
Notice Lemma 3.2 and Remark 2.4, the proof of (ii) is analogous to (i) and we omit it.
4. Existence of Positive Solutions of (1.1)
In this section, we are now ready to prove existence of positive solutions of (1.1) using Lemma 2.3. Denote and . Assume for all . To prove Theorem 1.2, we consider functionals
It is easy to find that critical points of the function correspond to positive solutions (negative solutions) of (1.1).
Lemma 4.1 If for all , then and
satisfy (PS) condition.
Proof. Suppose be a sequence with is bounded and as . Denote for and . In view of (2.6) and , there holds
thus as . So we claim is bounded. We assume, by contradiction, that there has a subsequence of with as . For each , either is bounded or . Put . then . Moreover, there has a subsequence of and satisfying as .
Denoting , the eigenvector associated with , we obtain
Dividing by , it follows immediately that
Since and , then passing to the limit in
(4.1), we get a contradiction. Hence, our claim is true. Since H is finite dimensional, the above argument means that has a convergent subsequence. Consequently, satisfies (PS) condition.
Similarly, it is not difficult to know that satisfies (PS) condition. Lemma 4.1 is proved.
Proof of Theorem 2.2 From , there exist and such that
Now if we denote , then for , there holds
Because of , there exists a constant such that
Then we can choose a positive constant such that
for all . If is sufficiently large,
In view of Lemma 2.3 and 4.1, we yield that there exists such that
and . Hence
Consequently, . Thus .
If for some , we find
then . If somewhere in , it vanishes identically. By , we obtain for . Therefore, is a positive solution of (1.1).
In a similar way as above, if we consider the case of , a negative solution can be obtained. Then the proof of Theorem 2.2 is finished.
To illustrate Theorem 2.1 and Theorem 2.2, we will give two examples.
Example 5.1 Consider BVP
By direct calculation, we get
and for all . According to (2.4), we obtain
In addition, and
. From above argument, we find all conditions of
Theorem 2.1 are satisfied, thus (5.1) has at least a sign-changing solution, a positive solution and a negative solution.
For a certain case, fix , here , , then we can choose ， . After not very complicated calculation, we find
are positive solution, sign-changing solution, sign-changing solution and negative solution of (5.1), respectively.
Remark 5.1 From above example, we can get at least three nontrivial solutions of (1.1), one sign-changing, one positive and one negative if the nonlinearity f satisfy all the conditions of Theorem 2.1.
Example 5.2 Consider BVP
From (2.4), it is easy to see that . Moreover, , and for all . Therefore,
it follows from Theorem 2.2 that (5.2) has at least a positive solution and a negative solution.
In the case of , because of , we can choose . After direct computation, we get that and are
positive solution and negative solution of (5.2), respectively.
Remark 5.2 From example 5.2, it is not difficult to know that if the nonlinearity f satisfy all the conditions of Theorem 2.2, we can obtain at least a positive solution and a negative solution of (1.1).
In this manuscript, some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for a class of second-order nonlinear difference equations were established with Dirichlet boundary value problem by using invariant sets of descending flow and variational methods. Our results improve some existed ones in some literatures, because we not only establish some sufficient conditions on the existence of sign-changing solutions, but also we allow the nonlinearity f to dissatisfy Ambrosett-Rabinowitz type condition or locally Lipschitz continuity and to change sign.
This work was supported by Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institute. The authors would like to thank the reviewer for the valuable comments and suggestions, thanks.
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