Sign-Changing Solutions for Discrete Dirichlet Boundary Value Problem

Using invariant sets of descending flow and variational methods, we establish some sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions for second-order nonlinear difference equations with Dirichlet boundary value problem. Some results in the literature are improved.


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) where 2 T ≥ is a given integer and [ ] { } continuous in the second variable, ∆ denotes the forward difference operator defined by ( ) ( ) ( ) ( ) 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).
In the following, we first consider the linear eigenvalue problem corresponding to (1.1) By direct computation, we get eigenvalues of (2.3) as ( ) ( ) It is not hard to know that (2.5) and the system of linear algebra equations ( ) A mI x h + = 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 then the corresponding characteristic equation of (2.7) is ( ) Two independent solutions of (2.7) can be expressed by ( ) ( ) x k p = .Therefore, the general solution of (2.5) is , a k a k .Now using the me- thod of variation of constant, it follows Using initial conditions, we find ( ) ( ) x k P P P P P Hence, we achieve the unique solution of (2.5) . , Proof.For any , x y H ∈ , using the mean value theorem, it follows On the other side, for all , , gent 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: (H 1 ) There exist constants 0 ρ > and 0 α > such that ( ) Then I has a critical value c a ≥ , moreover, c can be characterized as Assume there is a path then I has at least four critical points, one in ( ) Throughout this paper, we assume that where At last, we state our main results as following.
+∞ 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 (J 3 ) holds, then the conclusion of (i) is true.

Existence of Sign-Changing Solutions of (1.1)
In this section, we shall make use of Lemma  (i) r = +∞ .(ii) r < +∞ is not an eigenvalue of (2.3), here r is defined by (J 2 ).
Then the functional I defined by (2.1) satisfies (PS) condition.
Proof .(i) Assume r = +∞ γ > , 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 { } n x is bounded and the proof is finished.Lemma 3.2 I satisfies (C) condition under (J 3 ).
Proof .First assume (J 3 ) (i) be satisfied.There exists and there has a positive constant ∑ which contradicts to (3.2).Then I satisfies (C) condition.When (J 3 ) (ii) holds, we can prove I satisfies (C) condition in a similar way.
Then Lemma 3.2 is verified.Proof.(i) According to (J 1 ) and (J 2 ).For all 0 u ≠ and [ ] ( ) By (J 1 ) and (J 2 ), there exist constants 0 Choosing a positive constant D, since x H ∈ is finite-dimensional, we have It is obviously that  , we obtain 3) and remark 2.1, we have ( ) 0 x k < .Consequently, x is a negative solution of (1.1).(ii) can be discussed similarly, we only need to change y + to y − to prove (ii).For simplicity, we omit its proof.Lemma 3.4 Suppose 1 2 , z z be eigenvectors corresponding to eigenvalues , Proof.(1) If r = +∞ .From (3.1), we can see that ( ) (2) Assume ( ) , 0 z z = .Thus 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 ( ) ( ) 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.

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.  ( ) It is easy to find that critical points of the function ( ) correspond to positive solutions (negative solutions) of (1.1).Consequently, I + satisfies (PS) condition.Similarly, it is not difficult to know that I − satisfies (PS) condition.Lemma 4.1 is proved.

Proof of Theorem 2.2 From
[ ] ( ) Then we can choose a positive constant 2 C such that In view of Lemma 2.3 and 4.1, we yield that there exists H µ ∈ such that ( ) 0 ( ) Consequently, 0 µ − = .Thus , it vanishes identically.By ( ) . Therefore, µ is a posi- tive solution of (1.1).
In a similar way as above, if we consider the case of I − , a negative solution can be obtained.Then the proof of Theorem 2.2 is finished.

Applications
To illustrate Theorem 2.
By direct calculation, we get ( ) . According to (2.4), we obtain ( )  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).

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.
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, general solution of (2.5) is )[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.

1
Suppose one of the following conditions holds.

→
as n → ∞ .So we claim { } n x + is bounded.We assume, by contradiction, that there has a subsequence of { } as n → ∞ .Denoting 1 0 z > , the eigenvector associated with 1 λ , we obtain 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 { } n x has a convergent subsequence.

For
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.arepositive 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 3According to Lemma 2.2 and Remark 2.2, we find that critical points of I defined on H are precisely solutions of (1.1).
, I is said to be satisfied Palais-Smale condition((PS)condition for short) if every sequence { } n I x is bounded and ( ) ( ) open ball in H with radius ρ and centered at 0, B ρ ∂ denote boundary of B ρ .Lemma 2.4 ([11]) Let H be a Hilbert space, there are two open convex subsets and one in 1 . Let { } n x is bounded.If r = +∞ , choosing a con- stant 0 1 and Theorem 2.2, we will give two examples.
Y. H. Long, B. L. Zeng DOI: 10.4236/jamp.2017.5111822242 Journal of Applied Mathematics and Physics it follows from Theorem 2.2 that (5.2) has at least a positive solution and a negative solution.