Positive Solutions for a Class of Discrete Mixed Boundary Value Problems with the (p, q)-Laplacian Operator

In recent years, with the development of mechanical engineering, control system, computer science, and economics, the existence of solutions of difference equations has attracted wide attention (see [1–6]). For example, applying Ricceri variational principle to obtain the existence of multiple solutions [7–9], taking the invariant sets of descending flow to prove the existence of sign-changing solutions [10], making the linking theorem to get the existence and multiplicity of periodic solutions [11], and using critical point theory to obtain the existence of homoclinic solutions [12–15] and heteroclinic solutions [16]. As we know, the fixed-point method and upper and lower solution techniques are important tools to solve the existence of solutions for boundary value problems (see [17, 18]). But recently, it is more common to use critical point theory to study Dirichlet boundary value problems (see [19–23]). More result on difference equations by using critical point theories can be referred to [24–27]. In [28], D’Aguı̀ et al. established the existence of at least two positive solutions for the following discrete Dirichlet boundary value problem:


Introduction
In recent years, with the development of mechanical engineering, control system, computer science, and economics, the existence of solutions of difference equations has attracted wide attention (see [1][2][3][4][5][6]). For example, applying Ricceri variational principle to obtain the existence of multiple solutions [7][8][9], taking the invariant sets of descending flow to prove the existence of sign-changing solutions [10], making the linking theorem to get the existence and multiplicity of periodic solutions [11], and using critical point theory to obtain the existence of homoclinic solutions [12][13][14][15] and heteroclinic solutions [16].
As we know, the fixed-point method and upper and lower solution techniques are important tools to solve the existence of solutions for boundary value problems (see [17,18]). But recently, it is more common to use critical point theory to study Dirichlet boundary value problems (see [19][20][21][22][23]). More result on difference equations by using critical point theories can be referred to [24][25][26][27].
In this paper, under suitable assumptions on the nonlinearity f, we use the theory of two nonzero critical points (see [30]) to ensure that there are at least two nonzero solutions for problem (D f λ ). e two nonzero critical points theorem is an appropriate combination of local minimum theorem (see [31]) and classical Ambrosetti-Rabinowitz theorem (see [32]). An important hypothesis of mountain pass theorem is Palais-Smale condition. It satisfies the application of infinite dimensional space by requiring the condition that the nonlinear term is stronger than psuperlinearity at infinity. In order to obtain the existence of two nonzero solutions, we can assume the classical Ambrosetti-Rabinowitz condition and nonlinear algebraic condition (see (40) in eorem 2) hold, that is, more widespread than the p-sublinearity at zero. Moreover, when we require that f(k, 0) ≥ 0 for all k ∈ Z(1, N), we can use strong maximum principle to obtain the existence of positive solutions, which has been proved in Lemma 2.
Let s * � min s(k): k ∈ Z(1, N) { }, a special case of our main result is stated as follows.
Theorem 1. Let f: R ⟶ R be a continuous function such that then, for each λ ∈ (0, (1 + s admits at least two positive solutions. e structure of the article is as follows. In Section 2, some basic definitions and properties are given. In Section 3, we give the main results. Under suitable hypothesis, Lemma 1 is used to obtain that the problem (D

Preliminaries
In this section, we recall some definitions, notations, and properties. Consider the N-dimensional Banach space: and define the norm and ‖u‖ ∞ � max |u(k)| : k ∈ Z(1, N) { } is another norm in S. Proposition 1. e following inequality holds: Proof. Let u ∈ S, then there exist k * ∈ Z(1, N) such that Since 2 Discrete Dynamics in Nature and Society If ||u|| ∞ > 1, then that is, If ‖u‖ ∞ ≤ 1, then that is, In summary, we have Put (16) and consider the function J λ : S ⟶ R for all λ > 0 by where It is clear that Φ 1 , Φ 2 , Ψ ∈ C 1 (S, R) and their Gâteaux derivatives at the point u ∈ S are given by Discrete Dynamics in Nature and Society for all u, v ∈ S. So, we have Hence, a critical point u of J λ is a solution of problem (D f λ ). Now, we recall a definition and a two nonzero critical points theorem for the reader's convenience. □ Definition 1. Let X be a real Banach space; we say that a Gâteaux differentiable function J λ : X ⟶ R satisfies the (PS)-condition, if any sequence u n n∈N ⊆ X such that and for each the functional J λ � Φ − λΨ satisfies the (PS)-condition and it is unbounded from below.
In order to obtain the positive solution of problem (D f λ ), we establish the following strong maximum principle.
If u(j) > 0, then it is easy know that u > 0 in Z (1, N).
where ξ + � max ξ, 0 u(k)). Standard arguments show that J + λ ∈ C 1 (S, R) and the critical points of J + λ are precisely the solutions of the following problem:  Discrete Dynamics in Nature and Society Next, we suppose that f(k, 0) ≥ 0 and f(k, x) � f(k, 0) for all x ≤ 0 and for all k ∈ Z(1, N). Put we have the following result. Proof. Let λ > 2 p N + s − 2 p− 1 /qL ∞ . We consider a sequence u n n∈N ⊆ S such that J λ (u n ) ⟶ c ∈ R and J λ ′ (u n ) ⟶ 0, as n ⟶ + ∞. Let u + n � max u n , 0 and u − n � max −u n , 0 for all n ∈ N. We first prove that u − n is bounded. On one hand, we have for all k ∈ Z (1, N). So, On the other hand, we assume that for each k ∈ Z(1, N), then for all n ∈ N, which leads to ‖u − n ‖ p− 1 ⟶ 0 as n ⟶ +∞. So, we have ‖u − n ‖ ⟶ 0 as n ⟶ +∞. It means that there exists an M > 0 such that u − n ≤ M. From (10) we know that ‖u − n ‖ ∞ ≤ N 1− 1/p M � c, for all k ∈ Z(1, N). Next, we suppose that the sequence u n is unbounded, that is, u + n is unbounded. As L ∞ > 0, we know that there exists an l ∈ R such that L ∞ > l > 2 p N + s − 2 p− 1 /λq. From the definition of L ∞ , there is δ k > 0 such that F(k, t) > l|t| p for all t > δ k . Furthermore, since F(k, t) is a continuous function, there exists a constant N). We can obtain that Hence, for all u n such that ‖u n ‖ ∞ ≥ 1, we conclude that Discrete Dynamics in Nature and Society (39) we can get lim n⟶+∞ J λ (u n ) � −∞ and this is absurd. Hence, J λ satisfies (PS)-condition.
Let u n be such that u − n is bounded and u + n is unbounded. From the proof above we can see that J λ is unbounded from below.

Main Results
e main results of this paper are as follows. function satisfying f(k, 0) ≥ 0 for all k ∈ Z(1, N). If there are two constants c and d with d < c such that (40) en, for each λ ∈ Λ 1 with the problem (D f λ ) admits at least two positive solutions.
Proof. Put Φ, Ψ as in (18). It is clear that inf X (Φ) � Φ(0) � Ψ(0) � 0. According to Lemma 3, we know that a nonzero critical point in S of the functional J + λ is precisely a positive solution of problem (D f λ ). Next, we just need to prove condition (21) of Lemma 1.