On the Solvability of Discrete Nonlinear Two-Point Boundary Value Problems

Abstract

We prove the existence and uniqueness of solutions for a family of discrete boundary value problems by using discrete's Wirtinger inequality. The boundary condition is a combination of Dirichlet and Neumann boundary conditions.

1. Introduction

In this paper, we study the following nonlinear discrete boundary value problem:−Δ(𝑎(𝑘−1,Δ𝑢𝑢((𝑘0)−=1)Δ))𝑢=(𝑇𝑓)=(𝑘)0,,𝑘∈ℤ[1,𝑇],(1.1) where 𝑇≥2 is a positive integer and Δ𝑢(𝑘)=𝑢(𝑘+1)−𝑢(𝑘) is the forward difference operator. Throughout this paper, we denote by ℤ[𝑎,𝑏] the discrete interval {𝑎,𝑎+1,…,𝑏}, where 𝑎 and 𝑏 are integers and 𝑎<𝑏.

We consider in (1.1) two different boundary conditions: a Dirichlet boundary condition (𝑢(0)=0) and a Neumann boundary condition (Δ𝑢(𝑇)=0). In the literature, the boundary condition considered in this paper is called a mixed boundary condition.

We also consider the function space 𝑊={𝑣∶ℤ[0,𝑇+1]⟶ℝ;suchthat𝑣(0)=Δ𝑣(𝑇)=0},(1.2) where 𝑊 is a T-dimensional Hilbert space [1, 2] with the inner product (𝑢,𝑣)=^𝑇_𝑘=1𝑢(𝑘)𝑣(𝑘),∀𝑢,𝑣∈𝑊.(1.3) The associated norm is defined by‖𝑢‖=^𝑇_𝑘=1||𝑢(𝑘)||²^1/2.(1.4) For the data 𝑓 and 𝑎, we assume the following.(𝐻₁)𝑓∶ℤ[1,𝑇]→ℝ. (𝐻₂)𝑎(𝑘,⋅)∶ℝ→ℝforall𝑘∈ℤ[0,𝑇] and their exists a mapping 𝐴∶ℤ[0,𝑇]×ℝ→ℝ which satisfies 𝑎(𝑘,𝜉)=(𝜕/𝜕𝜉)𝐴(𝑘,𝜉), for all 𝑘∈ℤ[0,𝑇] and 𝐴(𝑘,0)=0, for all 𝑘∈ℤ[0,𝑇].(𝐻₃)(𝑎(𝑘,𝜉)−𝑎(𝑘,𝜂))⋅(𝜉−𝜂)>0 for all 𝑘∈ℤ[0,𝑇] and 𝜉,𝜂∈ℝ such that 𝜉≠𝜂.(𝐻₄)|𝜉|^𝑝(𝑘)≤𝑎(𝑘,𝜉)𝜉≤𝑝(𝑘)𝐴(𝑘,𝜉) for all 𝑘∈ℤ[0,𝑇] and 𝜉∈ℝ.(𝐻₅)𝑝∶ℤ[0,𝑇]→(1,+∞).

The theory of difference equations occupies now a central position in applicable analysis. We just refer to the recent results of Agarwal et al. [1], Yu and Guo [3], Koné and Ouaro [4], Guiro et al. [5], Cai and Yu [6], Zhang and Liu [7], Mihăilescu et al. [8], Candito and D′Agui [9], Cabada et al. [10], Jiang and Zhou [11], and the references therein. In [7], the authors studied the following problem:Δ²𝑦(𝑘−1)𝑦+(0𝜆)𝑓=(𝑦𝑦((𝑘𝑇))+=10),=0𝑘,∈ℤ[1,𝑇],(1.5) where 𝜆>0 is a parameter, Δ²𝑦(𝑘)=Δ(Δ𝑦(𝑘)), 𝑓∶[0,+∞)→ℝ a continuous function satisfying the condition 𝑓(0)=−𝑎<0,where𝑎isapositiveconstant.(1.6)

The problem (1.5) is referred as the “semipositone” problem in the literature, which was introduced by Castro and Shivaji [2]. Semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources.

The studies regarding problems like (1.1) or (1.5) can be placed at the interface of certain mathematical fields such as nonlinear partial differential equations and numerical analysis. On the other hand, they are strongly motivated by their applicability in mathematical physics as mentioned above.

In [11], Jiang and Zhou studied the following problem: Δ²𝑢(𝑘−1)𝑢=(0𝑓)=(𝑘,Δ𝑢𝑢(𝑘(𝑇))),=𝑘0,∈ℤ[1,𝑇],(1.7) where 𝑇 is a fixed positive integer, 𝑓∶ℤ[1,𝑇]×ℝ→ℝ is a continuous function. Jiang and Zhou proved an existence of nontrivial solutions for (1.7) by using strongly monotone operator principle and critical point theory.

In this paper, we consider the same boundary conditions as in [11] but the main operator is more general than the one in [11] and involves variable exponent.

Problem (1.1) is a discrete variant of the variable exponent anisotropic problem −^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝑢𝑥=,𝜕𝜕0𝑥𝑢_𝑖on=Γ𝑓₁(𝑥)inΩ,𝜕𝑢𝜕𝑛=0onΓ²,(1.8) where Ω⊂ℝ^𝑁(𝑁≥3) is a bounded domain with smooth boundary, Γ₁∪Γ₂=𝜕Ω, 𝑓∈𝐿^∞(Ω), 𝑝_𝑖 continuous on Ω such that 1<𝑝_𝑖(𝑥)<𝑁 and ∑^𝑁𝑖=1(1/𝑝^−𝑖)>1 for all 𝑥∈Ω and all 𝑖∈ℤ[1,𝑁], where 𝑝^−𝑖∶=essinf_𝑥∈Ω𝑝_𝑖(𝑥).

The first equation in (1.8) was recently analyzed by Koné et al. [12] and Ouaro [13] and generalized to a Radon measure data by Koné et al. [14] for an homogeneous Dirichlet boundary condition (𝑢=0 on 𝜕Ω). The study of (1.8) will be done in a forthcoming work. Problems like (1.8) have been intensively studied in the last decades since they can model various phenomena arising from the study of elastic mechanics (see [15, 16]), electrorheological fluids (see [15, 17–19]), and image restoration (see [20]). In [20], Chen et al. studied a functional with variable exponent 1≤𝑝(𝑥)≤2 which provides a model for image denoising, enhancement, and restoration. Their paper created another interest for the study of problems with variable exponent.

Note that Mihăilescu et al. (see [21, 22]) were the first authors who studied anisotropic elliptic problems with variable exponent. In general, the interested reader can find more information about difference equations in [1–11, 23–25], more information about variable exponent in [12–22, 26].

Our goal in this paper is to use a minimization method in order to establish some existence results of solutions of (1.1). The idea of the proof is to transfer the problem of the existence of solutions for (1.1) into the problem of existence of a minimizer for some associated energy functional. This method was successfully used by Bonanno et al. [27] for the study of an eigenvalue nonhomogeneous Neumann problem, where, under an appropriate oscillating behavior of the nonlinear term, they proved the existence of a determined open interval of positive parameters for which the problem considered admits infinitely many weak solutions that strongly converges to zero, in an appropriate Orlicz-Sobolev space. Let us point out that, to our best knowledge, discrete problems like (1.1) involving anisotropic exponents have been discussed for the first time by Mihăilescu et al. (see [8]), in a second time by Koné and Ouaro [4], and in a third time by Guiro et al. [5]. In [8], the authors proved by using critical point theory, existence of a continuous spectrum of eigenvalues for the problem −Δ||Δ𝑢(𝑘−1)||^{𝑝(𝑘−1)−2}Δ𝑢𝑢((0𝑘)−=1𝑢)(𝑇=+𝜆1||𝑢)(=𝑘)0||,^{𝑞(𝑘)−2}𝑢(𝑘),𝑘∈ℤ[1,𝑇],(1.9) where 𝑇≥2 is a positive integer and the functions 𝑝∶ℤ[0,𝑇]→[2,∞) and 𝑞∶ℤ[1,𝑇]→[2,∞) are bounded while 𝜆 is a positive constant.

In [4], Koné and Ouaro proved, by using minimization method, existence and uniqueness of weak solutions for the following problem: −Δ(𝑎(𝑘−1,Δ𝑢𝑢((0𝑘)−=1𝑢)()𝑇)=+𝑓1)(=𝑘)0,,𝑘∈ℤ[1,𝑇],(1.10) where 𝑇≥2 is a positive integer.

The function 𝑎(𝑘−1,Δ𝑢(𝑘−1)) which appears in the left-hand side of problem (1.1) is more general than the one which appears in (1.9). Indeed, as examples of functions which satisfy the assumptions (𝐻₂)–(𝐻₅), we can give the following. (i)𝐴(𝑘,𝜉)=(1/𝑝(𝑘))|𝜉|^𝑝(𝑘), where 𝑎(𝑘,𝜉)=|𝜉|^{𝑝(𝑘)−2}𝜉, for all 𝑘∈ℤ[0,𝑇] and 𝜉∈ℝ.(ii)𝐴(𝑘,𝜉)=1/𝑝(𝑘)[(1+|𝜉|²)^𝑝(𝑘)/2−1], where 𝑎(𝑘,𝜉)=(1+|𝜉|²)^{(𝑝(𝑘)−2)/2}𝜉, for all 𝑘∈ℤ[0,𝑇] and 𝜉∈ℝ.

In [5], Guiro et al. studied the following two-point boundary value problems −Δ(𝑎(𝑘−1,Δ𝑢(𝑘−1Δ))𝑢)(+0)||𝑢=(𝑘Δ)||𝑢(𝑇^𝑝(𝑘)𝑢)(𝑘=)0=.𝑓(𝑘),𝑘∈ℤ[1,𝑇],(1.11)

The function 𝑎(𝑘−1,Δ𝑢(𝑘−1)) has the same properties as in [4], but the boundary conditions are different. For this reason, Guiro et al. defined a new norm in the Hilbert space considered in order to get, by using minimization methods, existence of a unique weak solution (which is also a classical solution since the Hilbert space associated is of finite dimension). Indeed, they used the following norm:‖𝑢‖=^𝑇+1_𝑘=1||Δ𝑢(𝑘−1)||²+^𝑇_𝑘=1||𝑢(𝑘)||²^1/2,(1.12) which is associated to the Hilbert space 𝑊={𝑣∶ℤ[0,𝑇+1]⟶ℝ;suchthatΔ𝑣(0)=Δ𝑣(𝑇)=0}.(1.13) In order to get the coercivity of the energy functional, the authors of [5] assumed the following on the exponent: 𝑝∶ℤ[0,𝑇]⟶(2,+∞).(1.14) The assumption above allowed them to exploit the convexity property of the map 𝑥→𝑥^𝑝−/2.

Problem (1.11) is a discrete variant of the following problem: −^𝑁_𝑖=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝜕𝑥𝑢,𝜕𝜕𝑥𝑢_𝑖=𝑓(𝑥,𝑢)inΩ,𝜕𝑛=0onΩ,(1.15) which was studied by Boureanu and Radulescu in [26] with an additional condition that 𝑢≥0. Note that, in [26], the Neumann condition is more general than the one in problem (1.11). In this paper, we use the discrete Wirtinger inequality (see [23]) which allows us to assume that the exponent 𝑝∶ℤ[0,𝑇]→(1,+∞). The discrete Wirtinger inequality is a discrete variant of the well-known Poincaré-Wirtinger inequality (see [28]). Another difference of the present paper compared to [5] is on the boundary condition.

The remaining part of this paper is organized as follows. Section 2 is devoted to mathematical preliminaries. The main existence and uniqueness result is stated and proved in Section 3. In Section 4, we discuss some extensions, and, finally, in Section 5, we apply our theoretical results to an example.

2. Preliminaries

From now, we will use the following notations: 𝑝⁻=min_{[]𝑘∈ℤ0,𝑇}𝑝(𝑘),𝑝⁺=max_{[]𝑘∈ℤ0,𝑇}𝑝(𝑘).(2.1) Moreover, it is useful to introduce other norms on 𝑊, namely, |𝑢|_𝑚=^𝑇_𝑘=1||𝑢(𝑘)||^𝑚^1/𝑚∀𝑢∈𝑊,𝑚≥2.(2.2) We have the following inequalities (see [6, 8]) which are used in the proof of Lemma 2.1: 𝑇^{(2−𝑚)/(2𝑚)}|𝑢|₂≤|𝑢|_𝑚≤𝑇^1/𝑚|𝑢|₂∀𝑢∈𝑊,𝑚≥2.(2.3) In the sequel, we will use the following auxiliary result.

Lemma 2.1 (see [5]). There exist two positive constants 𝐶₁, 𝐶₂ such that ^𝑇+1_𝑘=1||Δ𝑢(𝑘−1)||^{𝑝(𝑘−1)}≥𝐶₁^𝑇+1_𝑘=1|Δ𝑢(𝑘−1)|²^𝑝−/2−𝐶₂,(2.4) for all 𝑢∈𝑊𝑤𝑖𝑡ℎ‖𝑢‖>1.

We have the following result.

Lemma 2.2 (discrete Wirtinger's inequality, see Theorem 12.6.2, page 860 in [23]). For any function 𝑢(𝑘), 𝑘∈ℤ[0,𝑇] satisfying 𝑢(0)=0, the following inequality holds: 4sin²2(2𝑇𝜋+1)^𝑇_𝑘=1||𝑢(𝑘)||²≤^𝑇−1_𝑘=0||Δ𝑢(𝑘)||².(2.5)

3. Existence and Uniqueness of Weak Solution

In this section, we study the existence and uniqueness of weak solution of (1.1).

Definition 3.1. A weak solution of (1.1) is a function 𝑢∈𝑊 such that ^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)=^𝑇_𝑘=1𝑓(𝑘)𝑣(𝑘)forany𝑣∈𝑊.(3.1)
Note that, since 𝑊 is a finite dimensional space, the weak solutions coincide with the classical solutions of the problem (1.1).
We have the following result.

Theorem 3.2. Assume that (𝐻₁)–(𝐻₅) hold. Then, there exists a unique weak solution of (1.1).
The energy functional corresponding to problem (1.1) is defined by 𝐽∶𝑊→ℝ such that 𝐽(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1𝑓(𝑘)𝑢(𝑘).(3.2) We first present some basic properties of 𝐽.

Proposition 3.3. The functional 𝐽 is well defined on 𝑊 and is of class 𝐶¹(𝑊,ℝ) with the derivative given by ⟨𝐽′(𝑢),𝑣⟩=^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)−^𝑇_𝑘=1𝑓(𝑘)𝑣(𝑘),(3.3) for all 𝑢,𝑣∈𝑊.

The proof of Proposition 3.3 can be found in [5].

We now define the functional 𝐼 by 𝐼(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1)).(3.4)

We need the following lemma for the proof of Theorem 3.2.

Lemma 3.4. The functional 𝐼 is weakly lower semicontinuous.

Proof. 𝐴 is convex with respect to the second variable according to (𝐻₂). Thus, it is enough to show that 𝐼 is lower semicontinuous. For this, we fix 𝑢∈𝑊 and 𝜖>0. Since 𝐼 is convex, we deduce that, for any 𝑣∈𝑊, 𝐼(𝑣)≥≥𝐼𝐼((𝑢𝑢))++𝐼^_𝑇+1(𝑢),𝑣−𝑢_𝑘=1≥𝐼(𝑢)−_𝑇+1𝑎(𝑘−1,Δ𝑢(𝑘−1))(Δ𝑣(𝑘−1)−Δ𝑢(𝑘−1))_𝑘=1≥𝐼(𝑢)−_𝑇+1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||||Δ𝑣(𝑘−1)−Δ𝑢(𝑘−1)||_𝑘=1≥𝐼(𝑢)−_𝑇+1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||||𝑣(𝑘)−𝑢(𝑘)+𝑢(𝑘−1)−𝑣(𝑘−1)||_𝑘=1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||||𝑣(𝑘)−𝑢(𝑘)||+||𝑣(𝑘−1)−𝑢(𝑘−1)||.(3.5) We define 𝐻 and 𝐵 by 𝐻=^𝑇+1_𝑘=1𝐵=_𝑇+1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||||𝑣(𝑘)−𝑢(𝑘)||,_𝑘=1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||||𝑣(𝑘−1)−𝑢(𝑘−1)||.(3.6) By using Schwartz inequality, we get 𝐻≤^𝑇+1_𝑘=1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||²^1/2^𝑇+1_𝑘=1||𝑣(𝑘)−𝑢(𝑘)||²^1/2≤^𝑇+1_𝑘=1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||²^1/2‖𝑣−𝑢‖.(3.7) The same calculus gives 𝐵≤^𝑇+1_𝑘=1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||²^1/2‖𝑣−𝑢‖.(3.8) Finally, we have 𝐼(𝑣)≥𝐼(𝑢)−1+2^𝑇+1_𝑘=1||𝑎(𝑘−1,Δ𝑢(𝑘−1))||²^1/2‖𝑣−𝑢‖≥𝐼(𝑢)−𝜖(3.9) for all 𝑣∈𝑊 with ‖𝑣−𝑢‖<𝛿=𝜖/𝐾(𝑇,𝑢), where 𝐾(𝑇,𝑢)=(1+2∑^𝑇+1𝑘=1|𝑎(𝑘−1,Δ𝑢(𝑘−1))|²)^1/2.
We conclude that 𝐼 is weakly lower semicontinuous.

We also have the following result.

Proposition 3.5. The functional 𝐽 is bounded from below, coercive, and weakly lower semicontinuous.

Proof . By Lemma 3.4, 𝐽 is weakly lower semicontinuous. We will only prove the coerciveness of the energy functional since the boundedness from below of 𝐽 is a consequence of coerciveness. The other proofs can be found in [5]. By (𝐻₄), we deduce that 𝐽(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1≥𝑓(𝑘)𝑢(𝑘)_𝑇+1_𝑘=11𝑝(𝑘)||Δ𝑢(𝑘−1)||^{𝑝(𝑘−1)}−^𝑇_𝑘=1≥1||𝑓(𝑘)𝑢(𝑘)||𝑝^+𝑇+1_𝑘=1||Δ𝑢(𝑘−1)||^{𝑝(𝑘−1)}−^𝑇_𝑘=1||𝑓(𝑘)||²^1/2^𝑇_𝑘=1||𝑢(𝑘)||²^1/2.(3.10) To prove the coercivity of 𝐽, we may assume that ‖𝑢‖>1, and we get from the above inequality and Lemma 2.1, the following: 𝐽(𝑢)≥𝐶¹𝑝⁺^𝑇+1_𝑘=1|Δ𝑢(𝑘−1)|²^𝑝−/2−𝐶₂−^𝑇_𝑘=1||𝑓(𝑘)||²^1/2^𝑇_𝑘=1||𝑢(𝑘)||²^1/2.(3.11) Using Wirtinger's discrete inequality, we obtain 𝐽(𝑢)≥𝐶¹𝑝⁺4sin²2(2𝑇𝜋+1)^𝑇_𝑘=1||𝑢(𝑘)||²^𝑝−/2−𝐶₂−^𝑇_𝑘=1||𝑓(𝑘)||²^1/2^𝑇_𝑘=1||𝑢(𝑘)||²^1/2≥𝐶¹2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)^𝑇_𝑘=1||𝑢(𝑘)||²^𝑝−/2−𝐶₂−^𝑇_𝑘=1||𝑓(𝑘)||²^1/2^𝑇_𝑘=1||𝑢(𝑘)||²^1/2≥𝐶¹2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)‖𝑢‖^𝑝−−𝐾₁‖𝑢‖−𝐶₂,(3.12) where 𝐾₁ is positive constant. Hence, since 𝑝⁻>1, 𝐽 is coercive.

We can now give the proof of Theorem 3.2.

Proof of Theorem 3.2. By Proposition 3.5, 𝐽 has a minimizer which is a weak solution of (1.1).
In order to end the proof of Theorem 3.2, we will prove the uniqueness of the weak solution.
Let 𝑢₁ and 𝑢₂ be two weak solutions of problem (1.1), then we have ^𝑇+1_𝑘=1𝑎𝑘−1,Δ𝑢₁(𝑘−1)Δ𝑢₁−𝑢₂(𝑘−1)=^𝑇_𝑘=1𝑓(𝑘)𝑢₁−𝑢₂(𝑘),_𝑇+1_𝑘=1𝑎𝑘−1,Δ𝑢₂(𝑘−1)Δ𝑢₁−𝑢₂(𝑘−1)=^𝑇_𝑘=1𝑓(𝑘)𝑢₁−𝑢₂(𝑘).(3.13) Adding the two equalities of (3.13), we obtain ^𝑇+1_𝑘=1𝑎𝑘−1,Δ𝑢₁(𝑘−1)−𝑎𝑘−1,Δ𝑢₂(𝑘−1)Δ𝑢₁−𝑢₂(𝑘−1)=0.(3.14) Using (𝐻₃), we deduce from (3.14) that Δ𝑢₁(𝑘−1)=Δ𝑢₂(𝑘−1)∀𝑘=1,…,𝑇+1.(3.15) Therefore, by using discrete's Wirtinger inequality, we get 4sin²2(2𝑇𝜋+1)^𝑇_𝑘=1||𝑢₁−𝑢₂(𝑘)||²≤^𝑇+1_𝑘=1||Δ𝑢₁−𝑢₂(𝑘−1)||²=0,(3.16) which implies that (∑^𝑇𝑘=1|(𝑢₁−𝑢₂)(𝑘)|²)^1/2=0. It follows that 𝑢₁=𝑢₂.

4. Some Extensions

4.1. Extension 1

In this section, we show that the existence result obtained for (1.1) can be extended to more general discrete boundary value problem of the form −Δ(𝑎(𝑘−1,Δ𝑢(𝑘−1))𝑢)(+0)||𝑢=(𝑘Δ)𝑢||(𝑇)^{𝑞(𝑘)−2}𝑢=(𝑘0,)=𝑓(𝑘),𝑘∈ℤ[1,𝑇](4.1) where 𝑇≥2 is a positive integer, and we assume that (𝐻₆)𝑞∶ℤ[1,𝑇]→(1,+∞).By a weak solution of problem (4.1), we understand a function 𝑢∈𝑊 such that, for any 𝑣∈𝑊, ^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)+^𝑇_𝑘=1||𝑢(𝑘)||^{𝑞(𝑘)−2}𝑢(𝑘)𝑣(𝑘)=^𝑇_𝑘=1𝑓(𝑘)𝑣(𝑘).(4.2)

We have the following result.

Theorem 4.1. Under assumptions (𝐻₁)–(𝐻₆), there exists a unique weak solution of problem (4.1).

Proof . For 𝑢∈𝑊, 𝐽(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))+^𝑇_𝑘=11𝑝(𝑘)||𝑢(𝑘)||^𝑞(𝑘)−^𝑇_𝑘=1𝑓(𝑘)𝑢(𝑘)(4.3) is such that 𝐽∈𝐶¹(𝑊;ℝ) and is weakly lower semicontinuous, and we have 𝐽^(𝑢),𝑣=+^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)_𝑇_𝑘=1||𝑢(𝑘)||^{𝑞(𝑘)−2}𝑢(𝑘)𝑣(𝑘)−^𝑇_𝑘=1𝑓(𝑘)𝑣(𝑘),(4.4) for all 𝑢,𝑣∈𝑊.
This implies that the weak solutions of problem (4.1) coincide with the critical points of 𝐽. We then have to prove that 𝐽 is bounded below and coercive in order to complete the proof.
As ^𝑇_𝑘=11𝑞(𝑘)||𝑢(𝑘)||^𝑞(𝑘)≥0,(4.5) then 𝐽(𝑢)≥^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1𝑓(𝑘)𝑣(𝑘).(4.6) Using Proposition 3.5, we deduce that 𝐽 is bounded below and coercive.
Let 𝑢₁ and 𝑢₂ be two weak solutions of problem (4.1), then we have ^𝑇+1_𝑘=1𝑎𝑘−1,Δ𝑢₁(𝑘−1)Δ𝑢₁−𝑢₂(𝑘−1)+^𝑇_𝑘=1=||𝑢₁(𝑘)||^{𝑞(𝑘)−2}𝑢₁(𝑘)𝑢₁(𝑘)−𝑢₂(𝑘)_𝑇_𝑘=1𝑓(𝑘)𝑢₁−𝑢₂(𝑘),_𝑇+1_𝑘=1𝑎𝑘−1,Δ𝑢₂(𝑘−1)Δ𝑢₁−𝑢₂(𝑘−1)+^𝑇_𝑘=1=||𝑢₂(𝑘)||^{𝑞(𝑘)−2}𝑢₂(𝑘)𝑢₁(𝑘)−𝑢₂(𝑘)_𝑇_𝑘=1𝑓(𝑘)𝑢₁−𝑢₂(𝑘).(4.7) Adding these two equalities, we obtain ^𝑇+1_𝑘=1𝑎𝑘−1,Δ𝑢₁+(𝑘−1)−𝑎𝑘−1,Δ𝑢₂(𝑘−1)Δ𝑢₁−𝑢₂(𝑘−1)_𝑇_𝑘=1||𝑢¹(𝑘)||^{𝑞(𝑘)−2}𝑢₁(𝑘)−||𝑢₂(𝑘)||^{𝑞(𝑘)−2}𝑢₂(𝑘)𝑢₁(𝑘)−𝑢₂(𝑘)=0.(4.8) We deduce that ^𝑇_𝑘=1||𝑢¹(𝑘)||^{𝑞(𝑘)−2}𝑢₁(𝑘)−||𝑢₂(𝑘)||^{𝑞(𝑘)−2}𝑢₂(𝑘)𝑢₁(𝑘)−𝑢₂(𝑘)=0,(4.9) which implies that 𝑢₁(𝑘)−𝑢₂(𝑘)=0∀𝑘=1,…,𝑇,(4.10) and we get 𝑢₁=𝑢₂.

4.2. Extension 2

In this section, we show that the existence result obtained for (1.1) can be extended to more general discrete boundary value problem of the form −Δ(𝑎(𝑘−1,Δ𝑢(𝑘−1)))+𝑢𝜆(||0𝑢)(𝑘=)||Δ𝑢^𝛽+−2(𝑇𝑢()𝑘=)=0,𝑓(𝑘,𝑢(𝑘)),𝑘∈ℤ[1,𝑇],(4.11) where 𝑇≥2 is a positive integer, 𝜆∈ℝ⁺, and 𝑓∶ℤ[1,𝑇]×ℝ→ℝ is a continuous function with respect to the second variable for all (𝑘,𝑧)∈ℤ[1,𝑇]×ℝ.

For every 𝑘∈ℤ[1,𝑇] and every 𝑡∈ℝ, we put 𝐹(𝑘,𝑡)=∫^𝑡0𝑓(𝑘,𝜏)𝑑𝜏.

By a weak solution of problem (4.11), we understand a function 𝑢∈𝑊 such that ^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)+𝜆^𝑇_𝑘=1=||𝑢(𝑘)||^𝛽+−2𝑢(𝑘)𝑣(𝑘)_𝑇_𝑘=1𝑓(𝑘,𝑢(𝑘))𝑣(𝑘),forany𝑣∈𝑊.(4.12) We assume the following.(𝐻₇)𝑓(𝑘,⋅)∶ℝ→ℝ is continuous for all 𝑘∈ℤ[1,𝑇].(𝐻₈) There exists a positive constant 𝐶₃ such that |𝑓(𝑘,𝑡)|≤𝐶₃(1+|𝑡|^{𝛽(𝑘)−1}), for all 𝑘∈ℤ[1,𝑇] and 𝑡∈ℝ.(𝐻₉)1<𝛽⁻<𝑝⁻.

Remark 4.2. The hypothesis (𝐻₈) implies that there exists one constant 𝐶′>0 such that |𝐹(𝑘,𝑡)|≤𝐶′(1+|𝑡|^𝛽(𝑘)).
We have the following result.

Theorem 4.3. Under assumptions (𝐻₂)–(𝐻₅) and (𝐻₇)–(𝐻₉), there exists 𝜆^∗>0 such that, for 𝜆∈[𝜆^∗,+∞), the problem (4.11) has at least one weak solution.

Proof. Let 𝑔(𝑢)=∑^𝑇𝑘=1𝐹(𝑘,𝑢(𝑘)), then 𝑔^∶𝑊→𝑊 is completely continuous, and, thus, 𝑔 is weakly lower semicontinuous.
Therefore, for 𝑢∈𝑊, 𝐽(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))+𝜆𝛽^+𝑇_𝑘=1||𝑢(𝑘)||^𝛽+−^𝑇_𝑘=1𝐹(𝑘,𝑢(𝑘))(4.13) is such that 𝐽∈𝐶¹(𝑊;ℝ) and is weakly lower semicontinuous, and we have 𝐽^(𝑢),𝑣=−^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)+𝜆^𝑇_𝑘=1||𝑢(𝑘)||^𝛽+−2𝑢(𝑘)𝑣(𝑘)_𝑇_𝑘=1𝑓(𝑘,𝑢(𝑘))𝑣(𝑘),(4.14) for all 𝑢,𝑣∈𝑊.
This implies that the weak solutions of problem (4.11) coincide with the critical points of 𝐽. We then have to prove that 𝐽 is bounded below and coercive in order to complete the proof.
Then, for 𝑢∈𝑊 such that ‖𝑢‖>1, 𝐽(𝑢)≥𝐶¹2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)‖𝑢‖^𝑝−+𝜆𝛽^+𝑇_𝑘=1||𝑢(𝑘)||^𝛽+−𝐶₂−^𝑇_𝑘=1≥𝐶𝐹(𝑘,𝑢(𝑘))₁2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)‖𝑢‖^𝑝−+𝜆𝛽^+𝑇_𝑘=1||𝑢(𝑘)||^𝛽+−𝐶₂−𝐶′^𝑇≥𝐶₁_𝑘=11+||𝑢(𝑘)||^𝛽(𝑘)2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)‖𝑢‖^𝑝−+𝜆𝛽^+𝑇_𝑘=1||𝑢(𝑘)||^𝛽+−𝐶₂−𝐶′𝑇−𝐶′^𝑇_𝑘=1||𝑢(𝑘)||^𝛽(𝑘)≥𝐶₁2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)‖𝑢‖^𝑝−+𝜆𝛽⁺−𝐶^^𝑇_𝑘=1||𝑢(𝑘)||^𝛽+−𝐶₂−𝐶′𝑇−𝐶′‖𝑢‖^𝛽−≥𝐶¹2^𝑝−𝑝⁺sin^𝑝−2(2𝑇𝜋+1)‖𝑢‖^𝑝−−𝐶₂−𝐶′𝑇−𝐶′‖𝑢‖^𝛽−,(4.15) where we put 𝜆^∗=𝐶′𝛽⁺ with 𝐶′ a positive constant.
Furthermore, by the fact that 1<𝛽⁻<𝑝⁻, it turns out that 𝐽(𝑢)≥𝐶𝑝⁺‖𝑢‖^𝑝−−𝐶₂−𝐶^𝑇−𝐶^‖𝑢‖^𝛽−⟶+∞as‖𝑢‖⟶+∞.(4.16) Therefore, 𝐽 is coercive.

4.3. Extension 3

We consider the problem −Δ(𝑎(𝑘−1,Δ𝑢(𝑘𝑢(−0)1=)))Δ=𝑢𝑓(𝑇()𝑘=,𝑢0(𝑘,)),𝑘∈ℕ[1,𝑇](4.17) where 𝑇≥2.

We suppose the following.(𝐻₁₀)There exist two positive constants 𝐶₅ and 𝐶₆ such that 𝑓⁺(𝑘,𝑡)≤𝐶₅+𝐶₆|𝑡|^𝛽−1, for all (𝑘,𝑡)∈ℤ[1,𝑇]×ℝ, where 1<𝛽<𝑝⁻.

Definition 4.4. A weak solution of problem (4.17) is a function 𝑢∈𝑊 such that ^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)=^𝑇_𝑘=1𝑓(𝑘,𝑢(𝑘))𝑣(𝑘),∀𝑣∈𝑊.(4.18)
We have the following result.

Theorem 4.5. Under the hypothesis (𝐻₂)–(𝐻₅) and (𝐻₁₀), problem (4.3) admits at least one weak solution.

Proof. We consider𝐽(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1𝐹(𝑘,𝑢(𝑘)),∀𝑢∈𝐻.(4.19)𝐽 is such that 𝐽∈𝐶¹(𝑊,ℝ) and 𝐽^(𝑢),𝑣=^𝑇+1_𝑘=1𝑎(𝑘−1,Δ𝑢(𝑘−1))Δ𝑣(𝑘−1)−^𝑇_𝑘=1𝑓(𝑘,𝑢(𝑘))𝑣(𝑘),(4.20) for all 𝑢,𝑣∈𝑊.
As 𝑓=𝑓⁺−𝑓⁻, then 𝐹⁻(𝑘,𝑡)=∫^𝑇0𝑓⁺(𝑘,𝜏)𝑑𝜏.
By (𝐻₁₀), there exists 𝐶>0 such that ||𝐹⁺(𝑘,𝑡)||≤𝐶1+|𝑡|^𝛽.(4.21) For all 𝑢∈𝑊 such that ‖𝑢‖>1, we have 𝐽(𝑢)=^𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1=𝐹(𝑘,𝑢(𝑘))_𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1𝐹⁺(𝑘,𝑢(𝑘))+^𝑇_𝑘=1≥𝐹⁻(𝑘,𝑢(𝑘))_𝑇+1_𝑘=1𝐴(𝑘−1,Δ𝑢(𝑘−1))−^𝑇_𝑘=1𝐹⁺(𝑘,𝑢(𝑘)).(4.22) Therefore, similar to the proof of Theorem 4.3, Theorem 4.5 follows immediately.

5. Example

We consider the following problem: −Δ||Δ𝑢(0)||²Δ𝑢(0)+𝜆||𝑢(1)||²𝑢(1)=15(𝑢(1))²−Δ||Δ𝑢(1)||₃,Δ𝑢(1)+𝜆||𝑢(2)||²𝑢(2)=15(𝑢(2))³𝑢(0)=0,𝑢(2)=𝑢(3).,(5.1) Then, 𝑇=2, 𝑝(0)=4, 𝑝(1)=5, 𝛽(1)=3, 𝛽(2)=4, 𝑝⁻=4, 𝑝⁺=5, 𝛽⁻=3, 𝛽⁺=4,𝑓(1,𝑡)=(1/5)𝑡², and 𝑓(2,𝑡)=(1/5)𝑡³. Thus, 𝐹(1,𝑡)=115𝑡³,𝐹(2,𝑡)=120𝑡⁴.(5.2) After computation, we can take 𝐶′=1/15 and we deduce that 𝜆^∗=4/15.

Therefore, by Theorem 4.3, for any 𝜆≥4/15, Problem (5.1) admits at least one weak solution.

Acknowledgment

The authors want to express their deepest thanks to the editor and anonymous referees for comments and suggestions on the paper.