Existence of Multiple Solutions for a Class of n-Dimensional Discrete Boundary Value Problems

By using critical point theory, we obtain some new results on the existence of multiple solutions for a class of -dimensional discrete boundary value problems. Results obtained extend or improve existing ones.


Introduction
Let N, Z, R be the set of all natural numbers, integers, and real numbers, respectively.For any a, b ∈ Z, define Z a {a, a 1, a 2, . ..},Z a, b {a, a 1, . . ., b} when a < b.In this paper, we consider the existence of multiple solutions for the following ndimensional discrete nonlinear boundary value problem: where n ∈ N, X k x 1 k , x 2 k , . . ., x n k T ∈ R n , Δ is the forward difference operator defined by ΔX k that the uniqueness of solutions implies the existence of solutions for some conjugate boundary value problems.In recent years, by using various methods and techniques, such as nonlinear alternative of Leray-Schauder type, the cone theoretic fixed point theorem, and the method of upper and lower solution, a series of existence results for the solutions of the BVP 1.1 in some special cases, for example, n 1, α β 0, A B 0; n 1, α 0, β −1, A B 0 have been obtained in literatures.We refer to 3-7 .
The critical point theory has been an important tool for investigating the periodic solutions and boundary value problems of differential equations 8-10 .In recent years, it is applied to the study of periodic solutions 11-13 and boundary value problems 14-17 of difference equations.
For the case when n 1, the scalar BVP 1.1 was studied by Yu and Guo in 17 .By using critical point theorem, they obtained various conditions to guarantee the existence of one solution, but they did not obtain the existence conditions of multiple solutions.In this scalar case, the BVP can be viewed as the discrete analogue of the following self-adjoint differential equation: which is a generalization of Emden-Fowler equation: The Emden-Fowler equation originated from earlier theories of gaseous dynamics in astrophysics 18 , and later, found applications in the study of fluid mechanics, relative mechanics, nuclear physics, and in the study of chemical reaction system 19 .
For the case where p k ≡ −1, q k ≡ 0, A B θ the zero vector of R n , BVP 1.1 are reduced to which were studied by Jiang and Zhou in 15 .They obtained the existence results of multiple solutions by using critical point theory again.
In the aforementioned references, most of the difference equations involved are scalar.The purpose of this paper is further to demonstrate the powerfulness of critical point theory in the study of existence of discrete boundary value problems and obtain various conditions for the existence of at least two nontrivial solutions for the BVP 1.1 .
The remaining of this paper is organized as follows.First, in Section 2, we will establish the variational framework associated with 1.1 and transfer the problem of the existence of solutions of 1.1 into that of the existence of critical points of the corresponding functional.Some basic results will also be recalled.Then, in Section 3, we present various new conditions on the existence of at least two nontrivial solutions for the BVP 1.1 .Some examples are given to illustrate the conclusions.We mention that our results generalize the ones in 15 and improve the ones in 20 .

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To conclude the introduction, we refer to 21, 22 for the general background on difference equations.

Preliminary and Variational Framework
Let E be a real Hilbert space, J ∈ C 1 E, R , which means that J is a continuously Frećhetdifferentiable functional defined on E. J is said to satisfy Palais-Smale condition P-S condition for short , if any sequence {x n } ∞ n 1 ⊂ E for which {J x n } ∞ n 1 is bounded and J x n → 0 as n → ∞ possesses a convergent subsequence in E.
Let B ρ be the open ball in E with radius ρ and centered at 0 and let ∂B ρ denote its boundary.The following lemmas will be useful in the proofs of our main results.Lemma 2.1.(Mountain Pass Lemma [10]).Let E be a real Hilbert space, and assume that J ∈ C 1 E, R satisfies the P-S condition and the following conditions.
Then J possesses a critical value c ≥ a.Moreover, c inf h∈Γ max s∈ 0,1 J h s , where Without loss of generality, we assume that a 0, b m where m ∈ N, and there exists a function

International Journal of Mathematics and Mathematical Sciences
Define Then, H can be equipped with the inner product by which the norm • H can be induced by International Journal of Mathematics and Mathematical Sciences 5 with , j 1, 2, . . ., n.

2.11
Define a functional J on H as where where X 0 , X 1 , X m , X m 1 satisfying 2.4 .Thus, there is a one to one correspondence from H to Clearly, J X 0 if and only if X satisfies 2.3 and 2.4 .Therefore, the existence of solutions to BVP 2.3 and 2.4 is transferred to the existence of the critical point of the functional J on H.
By a solution {X k } m 1 k 0 of 2.3 and 2.4 , we mean that {X k } m k 1 satisfies 2.3 , and 2.4 holds.

Main Results
In this section, we will suppose that the matrix Q defined in 2.11 is positive definite, λ min , λ max are the minimal eigenvalue and maximal eigenvalue of Q, respectively.It is clear that λ min , λ max are also the minimal eigenvalue and maximal eigenvalue of M, respectively.The first result is as follows.
Proof.Since P k, θ ≥ 0, we know that W θ ≥ 0 where θ denotes the zero element of H.By 3.1 , we get W θ 0 and J θ 0. Let Then by 3.2 , we get where γ γ ω ≥ 0. Then for all X ∈ H, we have H N mn X H γ .

3.5
Since λ max < α 2 , we see that J X → −∞ as X H → ∞.Thus J X is bounded from above on H, and J X can achieve its maximum on H.In other words, there exists X 1 ∈ H, such that J X 1 sup X∈H J X .So X 1 is a critical point of J X , and X 1 is a solution of BVP 2.3 and 2.4 .
For any X ∈ H with d < X H < δ, we have So J X 1 sup X∈H J X > 0 and X 1 is a nontrivial solution to BVP 2.3 and 2.4 .To obtain another nontrivial solution of BVP 2.3 and 2.4 , we will use Mountain Pass Lemma.We first show that J X satisfies P-S condition.
In fact, for any sequence {X k } ∞ k 1 in H, {J X k } ∞ k 1 is bounded and lim k → ∞ J X k 0, then there exists K > 0 such that J X k > −K, and it follows from 3.5 that that is, Since λ max < α 2 , this implies that {X k } ∞ k 1 is bounded and possesses a convergent subsequence.So J X satisfies P-S condition on H.
Choosing r ∈ d, δ , then for X ∈ ∂B r {X ∈ H : X H r}, from 3.6 , we get 3.9 This shows that J satisfies condition 1 of the Mountain Pass Lemma.On the other hand, from 3.5 , J X → −∞ as X H → ∞, so there exists P > r such that J X < 0 for X H > P. Pick X 0 ∈ H such that X 0 H > P > r, then X 0 ∈ H \ B r , and J X 0 < 0. So the condition 2 of the Mountain Pass Lemma is satisfied.Therefore, J possesses a critical value c inf h∈Γ max s∈ 0,1 J h s , where A critical point corresponding to c is nontrivial as c ≥ a > 0. Let X 2 be a critical point corresponding to the critical value c of J.If X 2 / X 1 , then we are done.Otherwise, X 2 X 1 , which gives max

3.11
Pick h s sX 0 , s ∈ 0, 1 , then h ∈ Γ, and we have max Thus, there exists s 0 ∈ 0, 1 such that J X 2 J s 0 X 0 , and X 3 s 0 X 0 is also a critical point of J X in H.
If X 3 / X 1 , then Theorem 3.1 holds.Otherwise, X 3 X 1 s 0 X 0 .In this situation, we replace X 0 with −X 0 in the above arguments; then J possesses a critical value c * ≥ a and c * inf h∈Γ * max s∈ 0,1 J h s , where Assume that X 4 is a critical point corresponding to c * .If X 4 / X 1 , then the proof is complete.Otherwise, X 4 X 1 .Similarly, we can find a critical point X 5 of J such that X 5 −s 1 X 0 holds for some s 1 ∈ 0, 1 .Clearly, X 5 / X 3 .The proof of Theorem 3.1 is now complete.
From Theorem 3.1, we have the following corollaries.

3.14
Then BVP 2.3 and 2.4 has at least two nontrivial solutions.
Proof.For any and condition 3.1 of Theorem 3.1 is satisfied.Let

3.16
Then where and condition 3.2 of Theorem 3.1 is satisfied.The conclusion follows from Theorem 3.1.

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Remark 3.3.For the special case where p k ≡ −1, q k ≡ 0, A B θ, the BVP 2.3 and 2.4 was studied in 15 .Here, the corresponding matrix Q becomes which is positive definite when α > −1, β ≥ −1 and N 0, . . ., 0 T .So, d 0. In this case, Corollary 3.2 reduces to Theorem 3.1 of 15 .Therefore, our results extend the ones in 15 .Corollary 3.2 also improves the conclusion of Theorem 1 in 20 .

Corollary 3.4. If there exist constants
then BVP 2.3 and 2.4 has at least two nontrivial solutions.
Proof.It suffices to prove that 3.20 implies 3.2 .In fact, pick ρ max{ρ, for X H > ρ , so condition 3.2 of Theorem 3.1 is satisfied, and the proof are complete.

3.27
So, N 10 2. With the Matlab software, we can get the approximate eigenvalue of matrix Q:

3.33
Then BVP 2.3 and 2.4 has at least two nontrivial solutions.
Corollary 3.9.Assume that P k, θ According to Corollary 3.9, we know that the given BVP in this example has at least two nontrivial solutions.
Remark 3.14.When Q is negative definite, we can get similar conclusions.We do not repeat here.
and is continuous for U. a, b ∈ Z with a < b, α and β are constants, A a 1 , a 2 , . . ., a n T and B b 1 , b 2 , . . ., b n T are n-dimensional vectors, p k and q k are real value functions defined on Z a 1, b 1 and Z a 1, b , respectively, and p k / 0. Existence of solutions of discrete boundary value problems has been the subject of many investigations.Motivated by Hartman's landmark paper 1 , Henderson 2 showed International Journal of Mathematics and Mathematical Sciences LX mn X H .With this mapping, BVP 2.3 and 2.4 can be represented by the matrix equation: and • n and •, • are the norm and the inner product in R n , respectively.Define a linear mapping L : H → R mn by L X x 1 1 , ..., x 1 m , x 2 1 , ..., x 2 m , ..., x n 1 , ..., x n m T .2.8Then L is a linear and one to one mapping.Clearly,