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

1. 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 n-dimensional discrete nonlinear boundary value problem:

Δ[p(k)ΔX(k-1)]+q(k)X(k)=f(k,X(k)),k∈ℤ(a+1,b),X(a)+αX(a+1)=A,X(b+1)+βX(b)=B,
where n∈ℕ, X(k)=(x1(k),x2(k),…,xn(k))T∈ℝn, Δ is the forward difference operator defined by ΔX(k)=X(k+1)-X(k), f:ℤ×ℝn→ℝn with f(k,U)=(f1(k,U),f2(k,U),…,fn(k,U))T, and is continuous for U. a,b∈ℤ with a<b, α and β are constants, A=(a1,a2,…,an)T and B=(b1,b2,…,bn)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 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:

ddt(p(t)dydt)+q(t)y=f(t,y),
which is a generalization of Emden-Fowler equation:

ddt(tρdydt)+tδyγ=0.
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 ℝn), BVP (1.1) are reduced to

-Δ2X(k-1)=f(k,X(k)),k∈ℤ(a+1,b),X(a)=-αX(a+1),X(b+1)=-βX(b),
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].

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

2. Preliminary and Variational Framework

Let E be a real Hilbert space, J∈C1(E,ℝ), which means that J is a continuously Frećhet-differentiable functional defined on E. J is said to satisfy Palais-Smale condition (P-S condition for short), if any sequence {xn}n=1∞⊂E for which {J(xn)}n=1∞ is bounded and J′(xn)→0 as n→∞ possesses a convergent subsequence in E.

Let Bρ be the open ball in E with radius ρ and centered at 0 and let ∂Bρ 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∈C1(E,ℝ) satisfies the P-S condition and the following conditions.

There exist constant ρ>0 and a>0 such that J(x)≥a,∀x∈∂Bρ, where Bρ={x∈E:∥x∥E<ρ}.

J(0)≤0 and there exists x0∉Bρ such that J(x0)≤0.

Then J possesses a critical value c≥a. Moreover, c=infh∈Γmaxs∈[0,1]J(h(s)), where
Γ={h∈C([0,1],E)∣h(0)=0,h(1)=x0}.Lemma 2.2.

Let ai≥0, γ>2, i∈ℤ(1,m), then
1m∑i=1mai≤∑i=1mai2≤∑i=1mai.

Without loss of generality, we assume that a=0, b=m where m∈ℕ, and there exists a function P(k,U)∈C1(ℤ(1,m)×ℝn,ℝ), such that for any (k,U)∈ℤ(1,m)×ℝn, ∇UP(k,U)=f(k,U), where ∇UP(k,U)=(∂P/∂u1,…,∂P/∂un)T is the gradient of P(k,U) in U, and P(k,θ)≥0 where θ is the zero vector in ℝn.

Let c(k)=q(k)-p(k)-p(k+1), then BVP (1.1) reduces to

p(k+1)X(k+1)+c(k)X(k)+p(k)X(k-1)=f(k,X(k)),k∈ℤ(1,m),X(0)+αX(1)=A,X(m+1)+βX(m)=B,
where X(k)=(x1(k),x2(k),…,xn(k))T∈ℝn, ∀k∈ℤ(0,m+1).

Define

H={X=(X(1),…,X(m))T∣X(k)=(x1(k),…,xn(k))∈ℝn,∀k∈ℤ(1,m)}.
Then, H can be equipped with the inner product

〈X,Y〉H=∑k=1m(X(k),Y(k))=∑k=1m∑i=1nxi(k)yi(k),
by which the norm ∥·∥H can be induced by

∥X∥H=∑k=1m∥X(k)∥n2,∀X∈H,
where X(k)=(x1(k),x2(k),…,xn(k))T, Y(k)=(y1(k),y2(k),…,yn(k))T∈ℝn, and ∥·∥n and (·,·) are the norm and the inner product in ℝn, respectively.

Define a linear mapping L:H→ℝmn by

L(X)=(x1(1),…,x1(m),x2(1),…,x2(m),…,xn(1),…,xn(m))T.
Then L is a linear and one to one mapping. Clearly, ∥LX∥mn=∥X∥H.

With this mapping, BVP (2.3) and (2.4) can be represented by the matrix equation:

ML(X)+N=L(F(X)),
where X∈H, F(X)=(f(1,X(1)),f(2,X(2)),…,f(m,X(m)))∈H and

J(X)=12(MLX,LX)+(N,LX)-W(X),
where W(X)=∑k=1mP(k,X(k)). Then J∈C1(H,ℝ).

For any X=(X(1),X(2),…,X(m))∈H, denote

X̅=(X(0),X(1),…,X(m),X(m+1)),
where X(0),X(1),X(m),X(m+1) satisfying (2.4). Thus, there is a one to one correspondence from H to

H̅={(X(0),X(1),…,X(m+1))∣X(0)+αX(1)=A,X(m+1)+βX(m)=B}.
Clearly, J'(X)=0 if and only if X̅ 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)}k=0m+1 of (2.3) and (2.4), we mean that {X(k)}k=1m satisfies (2.3), and (2.4) holds. {X(k)}k=0m+1 is nontrivial if ∥(X(1),…,X(m))∥H≠0.

3. Main Results

In this section, we will suppose that the matrix Q defined in (2.11) is positive definite, λmin, λmax are the minimal eigenvalue and maximal eigenvalue of Q, respectively. It is clear that λmin, λmax are also the minimal eigenvalue and maximal eigenvalue of M, respectively. The first result is as follows.

Theorem 3.1.

If there exist constants α1, α2:0<α1<λmin≤λmax<α2, and ρ>0, γ≥0, δ>d=2∥N∥mn/(λmin-α1) satisfying
W(X)≤12α1∥X∥H2,∀X∈H,∥X∥H<δ,W(X)≥12α2∥X∥H2-γ,∀X∈H,∥X∥H>ρ,
then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Proof.

Since P(k,θ)≥0, we know that W(θ̅)≥0 where θ̅ denotes the zero element of H. By (3.1), we get W(θ̅)=0 and J(θ̅)=0. Let
ω=max{|W(X)-12α2∥X∥H2+γ|:X∈H,∥X∥H≤ρ}.
Then by (3.2), we get
W(X)≥12α2∥X∥H2-(γ+ω)=12α2∥X∥H2-γ′,∀X∈H,
where γ′=γ+ω≥0. Then for allX∈H, we have
J(X)=12(MLX,LX)+(N,LX)-W(X)≤12λmax∥X∥H2+∥N∥mn∥X∥H-(12α2∥X∥H2-γ′)=λmax-α22∥X∥H2+∥N∥mn∥X∥H+γ′.

Since λmax<α2, 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 X1∈H, such that J(X1)=supX∈HJ(X). So X1 is a critical point of J(X), and X̅1 is a solution of BVP (2.3) and (2.4).

For any X∈H with d<∥X∥H<δ, we have

J(X)≥λmin-α12∥X∥H2-∥N∥mn∥X∥H>0.
So J(X1)=supX∈HJ(X)>0 and X̅1 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 limk→∞J′(X(k))=0, then there exists K>0 such that J(X(k))>-K, and it follows from (3.5) that
-K≤J(X(k))≤λmax-α22∥X(k)∥H2+∥N∥mn∥X(k)∥H+γ′,
that is,
λmax-α22∥Xk∥H2+∥N∥mn∥Xk∥H+γ′+K≥0.
Since λmax<α2, 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∈∂Br={X∈H:∥X∥H=r}, from (3.6), we get
J(X)≥λmin-α12r2-∥N∥mnr=a>0.
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 X0∈H such that ∥X0∥H>P>r, then X0∈H∖Br, and J(X0)<0. So the condition (2) of the Mountain Pass Lemma is satisfied. Therefore, J possesses a critical value c=infh∈Γmaxs∈[0,1]J(h(s)), where

Γ={h∈C([0,1],H):h(0)=θ̅,h(1)=X0}.
A critical point corresponding to c is nontrivial as c≥a>0. Let X2 be a critical point corresponding to the critical value c of J. If X2≠X1, then we are done. Otherwise, X2=X1, which gives
maxX∈HJ(X)=J(X1)=J(X2)=infh∈Γmaxs∈[0,1]J(h(s)).
Pick h(s)=sX0,s∈[0,1], then h∈Γ, and we have
maxX∈HJ(X)=J(X1)=J(X2)=infh∈Γmaxs∈[0,1]J(h(s))≤maxs∈[0,1]J(sX0).
Thus, there exists s0∈(0,1) such that J(X2)=J(s0X0), and X3=s0X0 is also a critical point of J(X) in H.

If X3≠X1, then Theorem 3.1 holds. Otherwise, X3=X1=s0X0. In this situation, we replace X0 with -X0 in the above arguments; then J possesses a critical value c*≥a and c*=infh∈Γ*maxs∈[0,1]J(h(s)), where
Γ*={h∈C([0,1],H):h(0)=θ̅,h(1)=-X0}.
Assume that X4 is a critical point corresponding to c*. If X4≠X1, then the proof is complete. Otherwise, X4=X1. Similarly, we can find a critical point X5 of J such that X5=-s1X0 holds for some s1∈(0,1). Clearly, X5≠X3. The proof of Theorem 3.1 is now complete.

From Theorem 3.1, we have the following corollaries.

Corollary 3.2.

If there exist constants α1, α2:0<α1<λmin≤λmax<α2, and ρ>0, γ≥0, δ>d=2∥N∥mn/(λmin-α1) satisfying
P(k,U)≤12α1∥U∥n2,∥U∥n<δ,k∈ℤ(1,m),P(k,U)≥12α2∥U∥n2-γ,∥U∥n>ρ,k∈ℤ(1,m).
Then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Proof.

For any X∈H, ∥X∥H<δ, then ∥X(k)∥n≤∥X∥H<δ, k∈ℤ(1,m). So
W(X)=∑k=1mP(k,X(k))≤∑k=1m12α1∥X(k)∥n2=12α1∥X∥H2,
and condition (3.1) of Theorem 3.1 is satisfied. Let
γ'=max{|P(k,U)-12α2∥U∥n2+γ|:∥U∥n≤ρ,k∈ℤ(1,m)}.
Then
P(k,U)≥12α2∥U∥n2-γ1,k∈ℤ(1,m),U∈ℝn,
where γ1=γ+γ'. So, for any X∈H,
W(X)=∑k=1mP(k,X(k))≥∑k=1m12[α2∥X(k)∥n2-γ1]=12α2∥X∥H2-mγ1,
and condition (3.2) of Theorem 3.1 is satisfied. The conclusion follows from Theorem 3.1.

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
(2+α-10…00-12-1…000-12…00………………000…2-1000…-12+β)m×m,
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 α1<λmin, α2>0, ρ>0, γ>0, δ>d=2∥N∥mn/(λmin-α1) and τ>2 such that (3.1) holds and
W(X)≥12α2∥X∥Hτ-γ,∀X∈H,∥X∥H>ρ,
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,((λmax+1)/α2)1/(τ-2)}, then
W(X)≥12α2∥X∥Hτ-2·∥X∥H2-γ≥12(λmax+1)∥X∥H2-γ
for ∥X∥H>ρ', so condition (3.2) of Theorem 3.1 is satisfied, and the proof are complete.

The following corollaries is obvious.

Corollary 3.5.

If there exist α1:0<α1<λmin, δ>d=2∥N∥mn/(λmin-α1) such that (3.1) holds and
lim∥X∥H→∞W(X)∥X∥H2=β>λmax2.
Then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Corollary 3.6.

If there exists constants α1:0<α1<λmin, δ>d=2∥N∥mn/(λmin-α1) such that (3.14) holds and
lim∥U∥n→∞P(k,U)∥U∥n2>λmax2,∀k∈ℤ(1,m).
Then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Now we will give an example to illustrate our results.

Example 3.7.

Consider the discrete boundary problem:2X(k+1)+10X(k)+2X(k-1)=f(k,X(k)),k∈ℤ(1,5)
with
X(0)+X(1)=A,X(6)+2X(5)=B,
where
f(k,U)=9400kU(4∥U∥22-2sin∥U∥22)
for k∈ℤ(1,5) and A=(1,0)T, B=(0,0)T.

In this example, m=5,n=2, p(k)≡2, c(k)≡10, α=1, β=2, η1=(2,0,0,0,0)T, η2=(0,0,0,0,0)T, N=(2,0,0,0,0,0,0,0,0,0)T,
M=(QQ),Q=(8200021020002102000210200026),P(k,U)=9400k[∥U∥24-(1-cos∥U∥22)].
So, ∥N∥10=2. With the Matlab software, we can get the approximate eigenvalue of matrix Q:
λ1=4.9957,λ2=6.3645,λ3=8.4127,λ4=11.0445,λ5=13.1826.
Thus Q is positive definite and 4.9<λmin<5, 13.1<λmax<13.2. Take α1=4.5, we find that
8=45-4.5<d=2∥N∥10λmin-α1<44.9-4.5=10.
Pick α2=14, δ=10>d, ρ=100, then for any U∈ℝ2:∥U∥2<δ=10, k∈ℤ(1,5), we have
P(k,U)≤9400∥U∥24<94∥U∥22=α12∥U∥22.
For any U∈ℝ2, ∥U∥2>ρ=100, k∈ℤ(1,5),
P(k,U)=9400k[∥U∥24-(1-cos∥U∥22)]≥92000(∥U∥24-2)>α22∥U∥22-91000.

In view of Corollary 3.2, we see that the boundary value problem (3.24) and (3.25) has at least two nontrivial solutions.

By a similar method, we can obtain the following results.

Theorem 3.8.

Assume that P(k,θ)=0 for any k∈ℤ(1,m). If there exist constants α1, α2:0<α2<λmin≤λmax<α1, ρ>0, γ≥0, and δ>d=2∥N∥mn/(α1-λmax) satisfying
W(X)≥12α1∥X∥H2,∀X∈H,∥X∥H<δ,W(X)≤12α2∥X∥H2+γ,∀X∈H,∥X∥H>ρ.
Then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Corollary 3.9.

Assume that P(k,θ)=0 for any k∈ℤ(1,m). If there exist constants α1, α2:0<α2<λmin≤λmax<α1, ρ>0, γ≥0, and δ>d=2∥N∥mn/(α1-λmax) satisfying
P(k,U)≥12α1∥U∥n2,∥U∥n<δ,k∈ℤ(1,m),P(k,U)≤12α2∥U∥n2+γ,∥U∥n>ρ,k∈ℤ(1,m),
then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Corollary 3.10.

Assume that P(k,θ)=0 for any k∈ℤ(1,m). If there exist constants α1>λmax, α2>0, δ>d=2∥N∥mn/(α1-λmax) and 1<τ<2 such that (3.32) holds and
W(X)≤12α2∥X∥Hτ+γ,∀X∈H,∥X∥H>ρ,
then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Corollary 3.11.

Assume that P(k,θ)=0 for any k∈ℤ(1,m). If there exist constants α1>λmax, and δ>d=2∥N∥mn/(α1-λmax) such that (3.32) holds and
lim∥X∥H→∞W(X)∥X∥H=β<λmin2,
then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Corollary 3.12.

Assume that P(k,θ)=0 for any k∈ℤ(1,m). If there exist α1>λmax, and δ>d=2∥N∥mn/(α1-λmax) such that (3.34) holds and
lim∥U∥n→∞P(k,U)∥U∥n2<λmin2,k∈ℤ(1,m),
then BVP (2.3) and (2.4) has at least two nontrivial solutions.

At last, we will give another example.

Example 3.13.

Consider the boundary value problem (3.24) and (3.25) where f(k,u) is replaced by f(k,U)=(800(4+∥U∥2)k(2+∥U∥2)2+2sin∥U∥22)U.
Clearly,
P(k,U)=800∥U∥22k(2+∥U∥2)+(1-cos∥U∥22),
and P(k,θ)=0. Pick α1=13.4, α2=4.8, then we have 0<α2<λmin≤λmax<α1, and
403=413.4-13.1<d=2∥N∥mnα1-λmax<413.4-13.2=20.
Pick δ=20>d, ρ=200, then for any U∈ℝ2:∥U∥2<δ=20, we can see that
P(k,U)=800∥U∥22k(2+∥U∥2)+(1-cos∥U∥22)≥800∥U∥225(2+20)=80∥U∥2211>α12∥U∥22.
For any U∈ℝ2:∥U∥2>ρ=200, we can get
P(k,U)=800∥U∥22k(2+∥U∥2)+(1-cos∥U∥22)≤800∥U∥222+200+2=80∥U∥22202+2<α22∥U∥22+2.
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.

Acknowledgments

This work is supported by the Specialized Fund for the Doctoral Program of Higher Eduction. (no. 20071078001), by the Natural Science Foundation of GuangXi (no. 0991279), by the Foundation of Education Department of GuangXi Province (no. 200807MS121), and by the project of Scientific Research Innovation Academic Group for the Education System of Guangzhou City.

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