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

Abstract

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, (1.1) where 𝑛∈ℕ, 𝑋(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘))^𝑇∈ℝ^𝑛, Δ is the forward difference operator defined by Δ𝑋(𝑘)=𝑋(𝑘+1)−𝑋(𝑘), 𝑓∶ℤ×ℝ^𝑛→ℝ^𝑛 with 𝑓(𝑘,𝑈)=(𝑓₁(𝑘,𝑈),𝑓₂(𝑘,𝑈),…,𝑓_𝑛(𝑘,𝑈))^𝑇, and is continuous for 𝑈. 𝑎,𝑏∈ℤ with 𝑎<𝑏, 𝛼 and 𝛽 are constants, 𝐴=(𝑎₁,𝑎₂,…,𝑎_𝑛)^𝑇 and 𝐵=(𝑏₁,𝑏₂,…,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 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), (1.2) which is a generalization of Emden-Fowler equation:

𝑑𝑑𝑡(𝑡^𝜌𝑑𝑦𝑑𝑡)+𝑡^𝛿𝑦^𝛾=0. (1.3) 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

−Δ²𝑋(𝑘−1)=𝑓(𝑘,𝑋(𝑘)),𝑘∈ℤ(𝑎+1,𝑏),𝑋(𝑎)=−𝛼𝑋(𝑎+1),𝑋(𝑏+1)=−𝛽𝑋(𝑏), (1.4) 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 {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∈𝐶¹(𝐸,ℝ) satisfies the P-S condition and the following conditions. (1)There exist constant 𝜌>0 and 𝑎>0 such that 𝐽(𝑥)≥𝑎,∀𝑥∈𝜕𝐵_𝜌, where 𝐵_𝜌={𝑥∈𝐸∶‖𝑥‖_𝐸<𝜌}.(2)𝐽(0)≤0 and there exists 𝑥₀∉𝐵_𝜌 such that 𝐽(𝑥₀)≤0. Then 𝐽 possesses a critical value 𝑐≥𝑎. Moreover, 𝑐=inf_ℎ∈Γmax_𝑠∈[0,1]𝐽(ℎ(𝑠)), where Γ={ℎ∈𝐶([0,1],𝐸)∣ℎ(0)=0,ℎ(1)=𝑥₀}. (2.1)

Lemma 2.2. Let 𝑎_𝑖≥0, 𝛾>2, 𝑖∈ℤ(1,𝑚), then 1√𝑚^𝑚_𝑖=1𝑎_𝑖≤^𝑚_𝑖=1𝑎²_𝑖≤^𝑚_𝑖=1𝑎_𝑖. (2.2)

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)×ℝ^𝑛,ℝ), such that for any (𝑘,𝑈)∈ℤ(1,𝑚)×ℝ^𝑛, ∇_𝑈𝑃(𝑘,𝑈)=𝑓(𝑘,𝑈), where ∇_𝑈𝑃(𝑘,𝑈)=(𝜕𝑃/𝜕𝑢₁,…,𝜕𝑃/𝜕𝑢_𝑛)^𝑇 is the gradient of 𝑃(𝑘,𝑈) in 𝑈, and 𝑃(𝑘,𝜃)≥0 where 𝜃 is the zero vector in ℝ^𝑛.

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, (2.3)(2.4) where 𝑋(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘))^𝑇∈ℝ^𝑛, ∀𝑘∈ℤ(0,𝑚+1).

Define

𝐻={𝑋=(𝑋(1),…,𝑋(𝑚))^𝑇∣𝑋(𝑘)=(𝑥₁(𝑘),…,𝑥_𝑛(𝑘))∈ℝ^𝑛,∀𝑘∈ℤ(1,𝑚)}. (2.5) Then, 𝐻 can be equipped with the inner product

⟨𝑋,𝑌⟩_𝐻=^𝑚_𝑘=1(𝑋(𝑘),𝑌(𝑘))=^𝑚𝑛_𝑘=1𝑖=1𝑥_𝑖(𝑘)𝑦_𝑖(𝑘), (2.6) by which the norm ‖⋅‖_𝐻 can be induced by

‖𝑋‖_𝐻=^𝑚_𝑘=1‖𝑋(𝑘)‖²_𝑛,∀𝑋∈𝐻, (2.7) where 𝑋(𝑘)=(𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘))^𝑇, 𝑌(𝑘)=(𝑦₁(𝑘),𝑦₂(𝑘),…,𝑦_𝑛(𝑘))^𝑇∈ℝ^𝑛, 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)=(𝑥₁(1),…,𝑥₁(𝑚),𝑥₂(1),…,𝑥₂(𝑚),…,𝑥_𝑛(1),…,𝑥_𝑛(𝑚))^𝑇. (2.8) 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)), (2.9) where 𝑋∈𝐻, 𝐹(𝑋)=(𝑓(1,𝑋(1)),𝑓(2,𝑋(2)),…,𝑓(𝑚,𝑋(𝑚)))∈𝐻 and

𝑀=𝑄⋯0𝑄⋯⋮⋮⋱⋮0⋯𝑄_{𝑚𝑛×𝑚𝑛} (2.10) with

𝑄=𝑐(1)−𝛼𝑝(1)𝑝(2)0⋯00𝑝(2)𝑐(2)𝑝(3)⋯000𝑝(3)𝑐(3)⋯00⋯⋯⋯⋯⋯⋯000⋯𝑐(𝑚−1)𝑝(𝑚)000⋯𝑝(𝑚)𝑐(𝑚)−𝛽𝑝(𝑚+1)_𝑚×𝑚,𝜂𝑁=¹𝜂₂⋮𝜂_{𝑛𝑚𝑛×1},𝜂_𝑗=𝑝(1)𝑎_𝑗0⋮0𝑝(𝑚+1)𝑏_𝑗𝑚×1,𝑗=1,2,…,𝑛. (2.11)

Define afunctional 𝐽 on 𝐻 as

1𝐽(𝑋)=2(𝑀𝐿𝑋,𝐿𝑋)+(𝑁,𝐿𝑋)−𝑊(𝑋), (2.12) where 𝑊(𝑋)=∑^𝑚_𝑘=1𝑃(𝑘,𝑋(𝑘)). Then 𝐽∈𝐶¹(𝐻,ℝ).

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

𝑋=(𝑋(0),𝑋(1),…,𝑋(𝑚),𝑋(𝑚+1)), (2.13) where 𝑋(0),𝑋(1),𝑋(𝑚),𝑋(𝑚+1) satisfying (2.4). Thus, there is a one to one correspondence from 𝐻 to

𝐻={(𝑋(0),𝑋(1),…,𝑋(𝑚+1))∣𝑋(0)+𝛼𝑋(1)=𝐴,𝑋(𝑚+1)+𝛽𝑋(𝑚)=𝐵}. (2.14) 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. {𝑋(𝑘)}^𝑚+1_𝑘=0 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, 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.

Theorem 3.1. If there exist constants 𝛼₁, 𝛼₂∶0<𝛼₁<𝜆_min≤𝜆_max<𝛼₂, and 𝜌>0, 𝛾≥0, 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝜆_min−𝛼₁) satisfying 1𝑊(𝑋)≤2𝛼₁‖𝑋‖²_𝐻,∀𝑋∈𝐻,‖𝑋‖_𝐻<𝛿,1𝑊(𝑋)≥2𝛼₂‖𝑋‖²_𝐻−𝛾,∀𝑋∈𝐻,‖𝑋‖_𝐻>𝜌, (3.1)(3.2) then BVP (2.3) and (2.4) has at least two nontrivial solutions.

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 1𝜔=max{|𝑊(𝑋)−2𝛼₂‖𝑋‖²_𝐻+𝛾|∶𝑋∈𝐻,‖𝑋‖_𝐻≤𝜌}. (3.3) Then by (3.2), we get 1𝑊(𝑋)≥2𝛼₂‖𝑋‖²_𝐻1−(𝛾+𝜔)=2𝛼₂‖𝑋‖²_𝐻−𝛾^′,∀𝑋∈𝐻, (3.4) where 𝛾^′=𝛾+𝜔≥0. Then forall𝑋∈𝐻, we have 1𝐽(𝑋)=2≤1(𝑀𝐿𝑋,𝐿𝑋)+(𝑁,𝐿𝑋)−𝑊(𝑋)2𝜆_max‖𝑋‖²_𝐻+‖𝑁‖_𝑚𝑛‖𝑋‖_𝐻1−(2𝛼₂‖𝑋‖²_𝐻−𝛾^′)=𝜆_max−𝛼₂2‖𝑋‖²_𝐻+‖𝑁‖_𝑚𝑛‖𝑋‖_𝐻+𝛾^′. (3.5)
Since 𝜆_max<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)=sup_𝑋∈𝐻𝐽(𝑋). So 𝑋₁ is a critical point of 𝐽(𝑋), and 𝑋₁ is a solution of BVP (2.3) and (2.4).
For any 𝑋∈𝐻 with 𝑑<‖𝑋‖_𝐻<𝛿, we have
𝜆𝐽(𝑋)≥^min−𝛼₁2‖𝑋‖²_𝐻−‖𝑁‖_𝑚𝑛‖𝑋‖_𝐻>0. (3.6) So 𝐽(𝑋₁)=sup_𝑋∈𝐻𝐽(𝑋)>0 and 𝑋₁ 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 −𝐾≤𝐽(𝑋^(𝑘)𝜆)≤^max−𝛼₂2‖𝑋^(𝑘)‖²_𝐻+‖𝑁‖_𝑚𝑛‖𝑋^(𝑘)‖_𝐻+𝛾^′, (3.7) that is, 𝜆_max−𝛼₂2‖𝑋_𝑘‖²_𝐻+‖𝑁‖_𝑚𝑛‖𝑋_𝑘‖_𝐻+𝛾^′+𝐾≥0. (3.8) Since 𝜆_max<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∈𝜕𝐵_𝑟={𝑋∈𝐻∶‖𝑋‖_𝐻=𝑟}, from (3.6), we get 𝜆𝐽(𝑋)≥^min−𝛼₁2𝑟²−‖𝑁‖_𝑚𝑛𝑟=𝑎>0. (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 𝑋₀∈𝐻 such that ‖𝑋₀‖_𝐻>𝑃>𝑟, then 𝑋₀∈𝐻⧵𝐵_𝑟, and 𝐽(𝑋₀)<0. So the condition (2) of the Mountain Pass Lemma is satisfied. Therefore, 𝐽 possesses a critical value 𝑐=inf_ℎ∈Γmax_𝑠∈[0,1]𝐽(ℎ(𝑠)), where
Γ={ℎ∈𝐶([0,1],𝐻)∶ℎ(0)=𝜃,ℎ(1)=𝑋₀}. (3.10) 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 max_𝑋∈𝐻𝐽(𝑋)=𝐽(𝑋₁)=𝐽(𝑋₂)=inf_ℎ∈Γmax_𝑠∈[0,1]𝐽(ℎ(𝑠)). (3.11) Pick ℎ(𝑠)=𝑠𝑋₀,𝑠∈[0,1], then ℎ∈Γ, and we have max_𝑋∈𝐻𝐽(𝑋)=𝐽(𝑋₁)=𝐽(𝑋₂)=inf_ℎ∈Γmax_𝑠∈[0,1]𝐽(ℎ(𝑠))≤max_𝑠∈[0,1]𝐽(𝑠𝑋₀). (3.12) 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^∗=inf_ℎ∈Γ^∗max_𝑠∈[0,1]𝐽(ℎ(𝑠)), where Γ^∗={ℎ∈𝐶([0,1],𝐻)∶ℎ(0)=𝜃,ℎ(1)=−𝑋₀}. (3.13) 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<𝜆_min≤𝜆_max<𝛼₂, and 𝜌>0, 𝛾≥0, 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝜆_min−𝛼₁) satisfying 1𝑃(𝑘,𝑈)≤2𝛼₁‖𝑈‖²_𝑛,‖𝑈‖_𝑛1<𝛿,𝑘∈ℤ(1,𝑚),𝑃(𝑘,𝑈)≥2𝛼₂‖𝑈‖²_𝑛−𝛾,‖𝑈‖_𝑛>𝜌,𝑘∈ℤ(1,𝑚). (3.14) 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)=^𝑚_𝑘=1𝑃(𝑘,𝑋(𝑘))≤^𝑚_𝑘=112𝛼₁‖𝑋(𝑘)‖²_𝑛=12𝛼₁‖𝑋‖²_𝐻, (3.15) and condition (3.1) of Theorem 3.1 is satisfied. Let 𝛾^′1=max{|𝑃(𝑘,𝑈)−2𝛼₂‖𝑈‖²_𝑛+𝛾|∶‖𝑈‖_𝑛≤𝜌,𝑘∈ℤ(1,𝑚)}. (3.16) Then 1𝑃(𝑘,𝑈)≥2𝛼₂‖𝑈‖²_𝑛−𝛾₁,𝑘∈ℤ(1,𝑚),𝑈∈ℝ^𝑛, (3.17) where 𝛾₁=𝛾+𝛾^′. So, for any 𝑋∈𝐻, 𝑊(𝑋)=^𝑚_𝑘=1𝑃(𝑘,𝑋(𝑘))≥^𝑚_𝑘=112[𝛼₂‖𝑋(𝑘)‖²_𝑛−𝛾₁1]=2𝛼₂‖𝑋‖²_𝐻−𝑚𝛾₁, (3.18) 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, (3.19) 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<𝜆_min, 𝛼₂>0, 𝜌>0, 𝛾>0, 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝜆_min−𝛼₁) and 𝜏>2 such that (3.1) holds and 1𝑊(𝑋)≥2𝛼₂‖𝑋‖^𝜏_𝐻−𝛾,∀𝑋∈𝐻,‖𝑋‖_𝐻>𝜌, (3.20) 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,((𝜆_max+1)/𝛼₂)^1/(𝜏−2)}, then 1𝑊(𝑋)≥2𝛼₂‖𝑋‖_𝐻^𝜏−2⋅‖𝑋‖²_𝐻1−𝛾≥2(𝜆_max+1)‖𝑋‖²_𝐻−𝛾 (3.21) 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<𝜆_min, 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝜆_min−𝛼₁) such that (3.1) holds and lim_{‖𝑋‖𝐻→∞}𝑊(𝑋)‖𝑋‖²_𝐻𝜆=𝛽>^max2. (3.22) Then BVP (2.3) and (2.4) has at least two nontrivial solutions.

Corollary 3.6. If there exists constants 𝛼₁∶0<𝛼₁<𝜆_min, 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝜆_min−𝛼₁) such that (3.14) holds and lim_{‖𝑈‖𝑛→∞}𝑃(𝑘,𝑈)‖𝑈‖²_𝑛>𝜆_max2,∀𝑘∈ℤ(1,𝑚). (3.23) 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) (3.24) with 𝑋(0)+𝑋(1)=𝐴,𝑋(6)+2𝑋(5)=𝐵, (3.25) where 9𝑓(𝑘,𝑈)=400𝑘𝑈(4‖𝑈‖²₂−2sin‖𝑈‖²₂) (3.26) 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, 𝑄𝑄,9𝑀=,𝑄=8200021020002102000210200026𝑃(𝑘,𝑈)=400𝑘[‖𝑈‖⁴₂−(1−cos‖𝑈‖²₂)]. (3.27) 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. (3.28) Thus 𝑄 is positive definite and 4.9<𝜆_min<5, 13.1<𝜆_max<13.2. Take 𝛼₁=4.5, we find that 48=5−4.5<𝑑=2‖𝑁‖₁₀𝜆_min−𝛼₁<44.9−4.5=10. (3.29) Pick 𝛼₂=14, 𝛿=10>𝑑, 𝜌=100, then for any 𝑈∈ℝ²∶‖𝑈‖₂<𝛿=10, 𝑘∈ℤ(1,5), we have 9𝑃(𝑘,𝑈)≤400‖𝑈‖⁴₂<94‖𝑈‖²₂=𝛼₁2‖𝑈‖²₂. (3.30) For any 𝑈∈ℝ², ‖𝑈‖₂>𝜌=100, 𝑘∈ℤ(1,5), 9𝑃(𝑘,𝑈)=400𝑘[‖𝑈‖⁴₂−(1−cos‖𝑈‖²₂9)]≥2000(‖𝑈‖⁴₂𝛼−2)>²2‖𝑈‖²₂−9.1000 (3.31)

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<𝜆_min≤𝜆_max<𝛼₁, 𝜌>0, 𝛾≥0, and 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝛼₁−𝜆_max) satisfying 1𝑊(𝑋)≥2𝛼₁‖𝑋‖²_𝐻,∀𝑋∈𝐻,‖𝑋‖_𝐻<𝛿,1𝑊(𝑋)≤2𝛼₂‖𝑋‖²_𝐻+𝛾,∀𝑋∈𝐻,‖𝑋‖_𝐻>𝜌. (3.32)(3.33) 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<𝜆_min≤𝜆_max<𝛼₁, 𝜌>0, 𝛾≥0, and 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝛼₁−𝜆_max) satisfying 1𝑃(𝑘,𝑈)≥2𝛼₁‖𝑈‖²_𝑛,‖𝑈‖_𝑛<𝛿,𝑘∈ℤ(1,𝑚),1𝑃(𝑘,𝑈)≤2𝛼₂‖𝑈‖²_𝑛+𝛾,‖𝑈‖_𝑛>𝜌,𝑘∈ℤ(1,𝑚), (3.34)(3.35) 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>𝜆_max, 𝛼₂>0, 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝛼₁−𝜆_max) and 1<𝜏<2 such that (3.32) holds and 1𝑊(𝑋)≤2𝛼₂‖𝑋‖^𝜏_𝐻+𝛾,∀𝑋∈𝐻,‖𝑋‖_𝐻>𝜌, (3.36) 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>𝜆_max, and 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝛼₁−𝜆_max) such that (3.32) holds and lim_{‖𝑋‖𝐻→∞}𝑊(𝑋)‖𝑋‖_𝐻𝜆=𝛽<^min2, (3.37) 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>𝜆_max, and 𝛿>𝑑=2‖𝑁‖_𝑚𝑛/(𝛼₁−𝜆_max) such that (3.34) holds and lim_{‖𝑈‖𝑛→∞}𝑃(𝑘,𝑈)‖𝑈‖²_𝑛<𝜆_min2,𝑘∈ℤ(1,𝑚), (3.38) 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. (3.39) Clearly, 𝑃(𝑘,𝑈)=800‖𝑈‖²₂𝑘(2+‖𝑈‖₂)+(1−cos‖𝑈‖²₂), (3.40) and 𝑃(𝑘,𝜃)=0. Pick 𝛼₁=13.4, 𝛼₂=4.8, then we have 0<𝛼₂<𝜆_min≤𝜆_max<𝛼₁, and 403=413.4−13.1<𝑑=2‖𝑁‖_𝑚𝑛𝛼₁−𝜆_max<413.4−13.2=20. (3.41) 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‖²₂=5(2+20)80‖𝑈‖²₂>𝛼112¹‖𝑈‖²₂. (3.42) 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‖²₂𝛼202+2<²2‖𝑈‖²₂+2. (3.43) 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.