Sign-Changing Solutions for Discrete Dirichlet Boundary Value Problem ()
1. Introduction
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)
(1.1)
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 [1] established the existence results of multiple positive solutions. While the nonlinearity is discontinuous, Zhang [2] gained another new multiple solutions. Zhang and Sun [3] obtained two existence results of multiple solutions. By aid of algebra and Krasnoselskii fixed point theorem, Luo [4] 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 [5] - [14] 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 [10] . 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, [15] 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
(2.1)
For any
,
can be rewritten as
(2.2)
Here
is the transpose of the vector
on
and
In the following, we first consider the linear eigenvalue problem corresponding to (1.1)
(2.3)
By direct computation, we get eigenvalues of (2.3) as
(2.4)
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
(2.5)
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
(2.6)
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)
(2.7)
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
Then
Moreover
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
(2.8)
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 ( [16] ) 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 ( [17] ) 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 [16] ) 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
here
be the open ball in H with radius
and centered at 0,
denote boundary of
.
Lemma 2.4 ( [11] ) 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
,
and
then I has at least four critical points, one in
, one in
, one in
, and one in
.
Remark 2.4 By Theorem 5.1 [17] , we can replace (PS) condition by weaker (C) condition in Lemma 2.4.
Throughout this paper, we assume that
(J1)
(J2)
for
where
is a constant, or
,
and
satisfy
.
(J3) (i)
or
(ii)
.
where
.
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) [1] [2] [3] [4] 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 [11] [12] [13] or locally Lipschitz continuity in [7] [8] [14] , 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.
(i)
.
(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
(3.1)
furthermore,
.
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
(3.2)
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
. (3.3)
Let
,
,
for all
. Since
,
it follows
and
. (3.4)
By (J1) and (J2), there exist constants
,
and
such that
(3.5)
Choosing a positive constant D, since
is finite-dimensional, we have
(3.6)
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
and
.
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
(4.1)
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,
we obtain
.
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.
5. Applications
To illustrate Theorem 2.1 and Theorem 2.2, we will give two examples.
Example 5.1 Consider BVP
(5.1)
where
.
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
(5.2)
here
.
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).
6. Conclusion
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.
Acknowledgements
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.