Existence of Solutions to Boundary Value Problems for the Discrete Generalized Emden-Fowler Equation

Abstract

We consider the existence of solutions to boundary value problems for the discrete generalized Emden-Fowler equation. By means of the minimax methods in the critical point theory, some new results are obtained. Two examples are also given to illustrate the main results.

1. Introduction and Statements of Main Results

Let 𝐙 and 𝐑 be the sets of all integers and real numbers, respectively. For 𝑎,𝑏∈𝐙, define 𝐙[𝑎]={𝑎,𝑎+1,…},  𝐙[𝑎,𝑏]={𝑎,𝑎+1,…,𝑏} when 𝑎≤𝑏.

In this paper, we consider the following boundary value problem (BVP for short) consisting of the discrete generalized Emden-Fowler equation: Δ[𝑝(𝑡)Δ𝑢(𝑡−1)]+𝑞(𝑡)𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈𝐙[1,𝑘], (1.1) and the boundary value conditions: 𝑢(0)+𝛼𝑢(1)=𝐴,𝑢(𝑘+1)+𝛽𝑢(𝑘)=𝐵, (1.2) where 𝑘 is a positive integer, 𝛼, 𝛽, 𝐴, and 𝐵 are constants, and Δ𝑢(𝑡)=𝑢(𝑡+1)−𝑢(𝑡) is the forward difference operator. We assume that 𝑝(𝑡) is nonzero and realvalued for each 𝑡∈𝐙[1,𝑘+1]. We also assume that 𝑞(𝑡) is realvalued for each 𝑡∈𝐙[1,𝑘],  𝑓(𝑡,𝑥) is realvalued for each (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑, and 𝑓(𝑡,𝑥) is continuous in the second variable 𝑥.

Equation (1.1) has been extensively studied by many authors; for example, see [1–9] concerning its disconjugacy, disfocality, oscillation, asymptotic behaviour, existence of periodic solutions, and solutions to boundary value problem.

Recently, Yu and Guo in [10] employed the critical point theory to obtain the existence of solutions to the BVP (1.1)-(1.2). Motivated by this and the results in [11], the main purpose of this paper is to give some new sufficient conditions for the existence of solutions to the BVP (1.1)-(1.2) by applying the Saddle Point Theorem and the Least Action Principle.

Before giving the main results, we first set 𝐹(𝑡,𝑥)=^𝑥₀,𝑓(𝑡,𝑠)𝑑𝑠,𝑐(𝑡)=𝑞(𝑡)−𝑝(𝑡)−𝑝(𝑡+1),𝑀=𝑐(1)−𝛼𝑝(1)𝑝(2)0⋯00𝑝(2)𝑐(2)𝑝(3)⋯000𝑝(3)𝑐(3)⋯00⋯⋯⋯⋯⋯⋯000⋯𝑐(𝑘−1)𝑝(𝑘)000⋯𝑝(𝑘)𝑐(𝑘)−𝛽𝑝(𝑘+1)𝜂=(𝜂(1),𝜂(2),…,𝜂(𝑘))^𝜏=0⋮0.𝑝(1)𝐴𝑝(𝑘+1)𝐵 (1.3) The main results are as follows.

Theorem 1.1. Suppose that 𝑓, 𝑀 satisfy the following assumptions.(F₁) There are constants 𝐶₁>0,  𝐶₂>0,  1/2≤𝜃<1 such that for all (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑, |𝑓(𝑡,𝑥)|≤𝐶₁|𝑥|^𝜃+𝐶₂. (1.4)(F₂) One has(i)either ‖𝑢‖^−2𝜃∑^𝑘_𝑡=1𝐹(𝑡,𝑢(𝑡))→−∞  as ‖𝑢‖→∞, or(ii)‖𝑢‖^−2𝜃∑^𝑘_𝑡=1𝐹(𝑡,𝑢(𝑡))→+∞  as ‖𝑢‖→∞,where for all 𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝜏∈𝐑^𝑘, ‖𝑢‖=(∑^𝑘_𝑡=1|𝑢(𝑡)|²)^1/2. (P₁) The matrix 𝑀 is singular.Then the BVP (1.1)-(1.2) has at least one solution.

Remark 1.2. There  are functions 𝑝(𝑡), 𝑞(𝑡), and 𝑓(𝑡,𝑥) satisfying our Theorem 1.1 and not satisfying the results in [10]. For example, let 𝐴, 𝐵 be arbitrary constants, 𝑘=4,  𝛼=𝛽=−1,  𝜃=1/2 and 𝑥𝑝(𝑡)≡1,𝑞(𝑡)≡0,𝑓(𝑡,𝑥)=g(𝑡)₄√1+𝑥²+1, (1.5) where g(𝑡)<0 for every 𝑡∈𝐙[1,4]. It is easy to verify that (F₁), (F₂)(i), and (P₁) are satisfied. Then the BVP (1.1)-(1.2) has at least one solution. And it is easy to see that this solution is a nonzero solution since 𝑓(𝑡,0)≠0.

Theorem 1.3. Suppose that 𝑓, 𝑀 satisfy the following assumptions.(F₃) For any 𝑡∈𝐙[1,𝑘],  lim_|𝑥|→∞(𝑓(𝑡,𝑥)/𝑥)=0.(P₂) The matrix 𝑀 is nonsingular.Then the BVP (1.1)-(1.2) has at least one solution.

Corollary 1.4. Suppose that 𝑓, 𝑀 satisfy (F₁) and (P₂). Then the BVP (1.1)-(1.2) has at least one solution.

Remark 1.5. Since  for all (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑,  |𝑓(𝑡,𝑥)|≤𝐶₂ implies |𝑓(𝑡,𝑥)|≤𝐶₁|𝑥|^𝜃+𝐶₂, our Corollary 1.4 extends Theorem 3.2 in [10].

Theorem 1.6. Suppose that 𝑓, 𝑀 satisfy the following assumptions.(F₄) For any 𝑡∈𝐙[1,𝑘],  lim_|𝑥|→∞(𝐹(𝑡,𝑥)/𝑥²)=0.(F₅) One has(i)either 2𝐹(𝑡,𝑥)−(𝑓(𝑡,𝑥)+𝜂(𝑡))𝑥→−∞  as |𝑥|→∞ for all 𝑡∈𝐙[1,𝑘], or (ii)2𝐹(𝑡,𝑥)−(𝑓(𝑡,𝑥)+𝜂(𝑡))𝑥→+∞  as |𝑥|→∞ for all 𝑡∈𝐙[1,𝑘].(P₃) The matrix 𝑀 is singular and indefinite. Then the BVP (1.1)-(1.2) has at least one solution.

Remark 1.7. There are functions 𝑝(𝑡),  𝑞(𝑡), and 𝑓(𝑡,𝑥) satisfying our Theorem 1.6 and not satisfying the results in [10]. For example, let 𝐴, 𝐵 be arbitrary constants, 𝑘=4, 𝛼=1, 𝛽=0 and 𝑥𝑝(1)=𝑝(3)=−1,𝑝(2)=𝑝(4)=1,𝑝(5)=−2,𝑞(𝑡)≡0,𝑓(𝑡,𝑥)=g(𝑡)(√1+𝑥²ln(1+𝑥²)+√1+𝑥²2𝑥1+𝑥²+1), (1.6) where g(𝑡)<0 for every 𝑡∈𝐙[1,4]. It is easy to verify that (F₄), (F₅)(i) and (P₃) are satisfied. Then the BVP (1.1)-(1.2) has at least one solution. And it is easy to see that this solution is a nonzero solution since 𝑓(𝑡,0)≠0.

Corollary 1.8. Assume that (F₄) holds. If one of the following conditions is satisfied: (H₁) the matrix 𝑀 is negative semi-definite, and (F₅)(i) holds,  (H₂) the matrix 𝑀 is positive semi-definite, and (F₅)(ii) holds, then, the BVP (1.1)-(1.2) has at least one solution.

2. Variational Structure and Two Basic Lemmas

Let 𝐑^𝑘 be the real Euclidean space with dimension 𝑘. For any 𝑢,𝑣∈𝐑^𝑘, ‖𝑢‖ and (𝑢,𝑣) denote the usual norm and inner product in 𝐑^𝑘, respectively.

Define the functional 𝐽 on 𝐑^𝑘 as follows: 1𝐽(𝑢)=2(𝑀𝑢,𝑢)+(𝜂,𝑢)−^𝑘_𝑡=1𝐹(𝑡,𝑢(𝑡)),∀𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝜏∈𝐑^𝑘. (2.1) It is well known that 𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝜏∈𝐑^𝑘 is a critical point of 𝐽 if and only if {𝑢(𝑡)}^𝑘+1_𝑡=0=(𝑢(0),𝑢(1),𝑢(2),…,𝑢(𝑘),𝑢(𝑘+1))^𝜏 is a solution of the BVP (1.1)-(1.2), where 𝑢(0)=𝐴−𝛼𝑢(1) and 𝑢(𝑘+1)=𝐵−𝛽𝑢(𝑘). For details, see [10]. It follows from the continuity of 𝑓 that 𝐽 is continuously differentiable on 𝐑^𝑘. Moreover, one has (𝐽^′(𝑢),ℎ)=(𝑀𝑢,ℎ)+(𝜂,ℎ)−^𝑘_𝑡=1𝑓(𝑡,𝑢(𝑡))⋅ℎ(𝑡),∀𝑢,ℎ∈𝐑^𝑘. (2.2)

When the matrix 𝑀 is singular and indefinite, we suppose that 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑙 and 0>−𝜇₁≥−𝜇₂≥⋯≥−𝜇_𝑠 are the positive and negative eigenvalues of 𝑀, respectively, and 𝑙+𝑠=𝑘−1. We also suppose that 𝜍_𝑖, 1≤𝑖≤𝑙 and 𝜉_𝑗, 1≤𝑗≤𝑠 are the eigenvectors of 𝑀 corresponding to eigenvalues 𝜆_𝑖, 1≤𝑖≤𝑙 and −𝜇_𝑗, 1≤𝑗≤𝑠 satisfying (𝜍_𝑖1,𝜍_𝑖2)=0,𝑖₁≠𝑖₂,1,𝑖₁=𝑖₂,(𝜍_𝑖,𝜉_𝑗)=0,(𝜉_𝑗1,𝜉_𝑗2)=0,𝑗₁≠𝑗₂,1,𝑗₁=𝑗₂, (2.3) where 𝑖, 𝑖₁, 𝑖₂∈𝐙[1,𝑙],  𝑗, 𝑗₁, 𝑗₂∈𝐙[1,𝑠]. We denote 𝑅⁺=span{𝜍_𝑖∣𝑖∈𝐙[1,𝑙]},𝑅⁻=span{𝜉_𝑗𝑅∣𝑗∈𝐙[1,𝑠]},₀=(𝑅⁺⊕𝑅⁻)^⟂. (2.4)(2.5) Then 𝐑^𝑘 has the direct sum decomposition 𝐑^𝑘=𝑅⁺⊕𝑅⁰⊕𝑅⁻. So, for each 𝑢∈𝐑^𝑘, 𝑢 can be expressed by 𝑢=𝑢⁺+𝑢⁰+𝑢⁻, (2.6) where 𝑢^∗∈𝑅^∗, ∗ is +,0, or −, respectively. Furthermore, we have the following estimates: 𝜆₁‖𝑢⁺‖²≤(𝑀𝑢⁺,𝑢⁺)≤𝜆_𝑙‖𝑢⁺‖²,𝑢⁺∈𝑅⁺,−𝜇_𝑠‖𝑢⁻‖²≤(𝑀𝑢⁻,𝑢⁻)≤−𝜇₁‖𝑢⁻‖²,𝑢⁻∈𝑅⁻. (2.7)

Set ‖𝑢‖_∞=max_{𝑡∈𝐙[1,𝑘]}|𝑢(𝑡)|,‖𝑢‖_𝑝=(^𝑘_𝑡=1|𝑢(𝑡)|^𝑝)^1/𝑝, (2.8) where 𝑢=(𝑢(1),𝑢(2),…,𝑢(𝑘))^𝜏∈𝐑^𝑘 and ∞>𝑝≥1. Then for any 𝑢∈𝐑^𝑘, 1√𝑘‖𝑢‖≤‖𝑢‖_∞1≤‖𝑢‖,√𝑘‖𝑢‖≤‖𝑢‖_𝑝≤𝑘^1/𝑝‖𝑢‖. (2.9)

We will make use of the least action principle and saddle point theorem to obtain the critical points of 𝐽. Let us first recall these theorems.

Lemma 2.1 (the least action principle, see [12]). Let 𝑋 be a real Banach space, and assume that 𝐽∈𝐶¹(𝑋,𝐑) is bounded from below in 𝑋 and satisfies the Palais-Smale condition ((PS) condition for short). Then 𝑐=inf_𝑢∈𝑋𝐽(𝑢) is a critical value of 𝐽.

Lemma 2.2 (saddle point theorem, see [13]). Let 𝑋 be a real Hilbert space, 𝑋=𝑋₁⊕𝑋₂, where 𝑋₁≠{0} and is finite dimensional. Suppose that 𝐽∈𝐶¹(𝑋,𝐑) satisfies the (PS) condition and(𝜑₁)there exist constants 𝜎,𝜌>0 such that 𝐽∣_{𝜕𝐵𝜌∩𝑋1}≤𝜎, where 𝐵_𝜌={𝑢∈𝑋∣‖𝑢‖<𝜌},  𝜕𝐵_𝜌 denotes the boundary of 𝐵_𝜌; (𝜑₂)there exist 𝑒∈𝐵_𝜌∩𝑋₁ and a constant 𝜔>𝜎 such that 𝐽∣_𝑒+𝑋2≥𝜔.Then 𝐽 possesses a critical value 𝑐≥𝜔 and 𝑐=inf_ℎ∈Φmax_{𝑢∈𝐵𝜌∩𝑋1}𝐽(ℎ(𝑢)), (2.10) where Φ={ℎ∈𝐶(𝐵_𝜌∩𝑋₁,𝑋)∣ℎ∣_{𝜕𝐵𝜌∩𝑋1}=id}.

Remark 2.3. As shown in [14], a deformation lemma can be proved with the weaker condition (𝐶) replacing the usual (PS) condition, and it turns out that the saddle point theorem holds under condition (𝐶).

3. Proofs of the Main Results

In order to prove Theorem 1.1, we need to prove the following lemma.

Lemma 3.1. Assume that conditions (F₁), (F₂), and (P₃) hold. Then the functional 𝐽 (see (2.1)) satisfies the (PS) condition; that is, for any sequence {𝑢_𝑚} such that 𝐽(𝑢_𝑚) is bounded and 𝐽^′(𝑢_𝑚)→0 as 𝑚→∞, there exists a subsequence of {𝑢_𝑚} which is convergent in 𝐑^𝑘.

Proof. First suppose that (F₁), (F₂)(i), and (P₃) hold. Recall that 𝐑^𝑘 is a finite dimensional Hilbert space. Consequently, in order to prove that 𝐽 satisfies the (PS) condition, we only need to prove that {𝑢_𝑚} is bounded. Let {𝑢_𝑚} be a sequence in 𝐑^𝑘 such that 𝐽(𝑢_𝑚) is bounded and 𝐽^′(𝑢_𝑚)→0 as 𝑚→∞. Then there exist 𝐶₃>0 and 𝑚₀∈𝐙[1] such that |𝐽(𝑢_𝑚)|≤𝐶₃,|(𝐽^′(𝑢_𝑚),ℎ)|≤‖ℎ‖ (3.1) for all 𝑚>𝑚₀,  ℎ∈𝐑^𝑘.
Since 𝑀 is singular and indefinite, we write 𝑢_𝑚=𝑢⁺_𝑚+𝑢⁰_𝑚+𝑢⁻_𝑚 with 𝑢^∗_𝑚∈𝑅^∗, where ∗=+,0,−, respectively. By (F₁), (2.9), and Hölder’𝑠 inequality (𝑝=1/𝜃,𝑞=1/(1−𝜃)), we have |^𝑘_𝑡=1𝑓(𝑡,𝑢_𝑚(𝑡))⋅(𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚≤(𝑡))|_𝑘𝑡=1(𝐶₁|𝑢⁰_𝑚(𝑡)+(𝑢⁺_𝑚(𝑡)+𝑢⁻_𝑚(𝑡))|^𝜃+𝐶₂)⋅|𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚(𝑡)|≤2𝐶₁‖𝑢⁰_𝑚‖^𝜃_∞^𝑘_𝑡=1|𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚(𝑡)|+2𝐶₁^𝑘_𝑡=1|𝑢⁺_𝑚(𝑡)+𝑢⁻_𝑚(𝑡)|^𝜃|𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚(𝑡)|+𝐶₂^𝑘_𝑡=1|𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚(𝑡)|≤2𝐶²₁𝑘𝜏₁‖𝑢⁰_𝑚‖_∞^2𝜃+𝜏₁2‖𝑢⁻_𝑚−𝑢⁺_𝑚‖²+2𝐶₁‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃₁‖𝑢⁻_𝑚−𝑢⁺_𝑚‖_𝑞+𝐶₂√𝑘‖𝑢⁻_𝑚−𝑢⁺_𝑚‖≤2𝐶²₁𝑘𝜏₁‖𝑢⁰_𝑚‖^2𝜃+𝜏₁2‖𝑢⁻_𝑚−𝑢⁺_𝑚‖²+2𝐶₁𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃‖𝑢⁻_𝑚−𝑢⁺_𝑚‖+𝐶₂√𝑘‖𝑢⁻_𝑚−𝑢⁺_𝑚‖, (3.2) where 𝜏₁=min{𝜆₁,𝜇₁}. On the other hand, by the fact that 𝑢⁺_𝑚 and 𝑢⁻_𝑚 are mutually orthogonal, one has ‖𝑢⁻_𝑚−𝑢⁺_𝑚‖=‖𝑢⁺_𝑚+𝑢⁻_𝑚‖. Hence we have |^𝑘_𝑡=1𝑓(𝑡,𝑢_𝑚(𝑡))⋅(𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚(𝑡))|≤2𝐶²₁𝑘𝜏₁‖𝑢⁰_𝑚‖^2𝜃+𝜏₁2‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²+2𝐶₁𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃+1+𝐶₂√𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖. (3.3) Similarly to (3.3), we have |^𝑘_𝑡=1[𝐹(𝑡,𝑢_𝑚(𝑡))−𝐹(𝑡,𝑢⁰_𝑚(𝑡))]|≤2𝐶²₁𝑘𝜇_𝑠‖𝑢⁰_𝑚‖^2𝜃+𝜇_𝑠2‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²+2𝐶₁𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃+1+𝐶₂√𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖. (3.4)
Take ℎ=𝑢⁻_𝑚−𝑢⁺_𝑚 in (2.2). Then −(𝑀𝑢_𝑚,𝑢⁻_𝑚−𝑢⁺_𝑚)=−(𝐽^′(𝑢_𝑚),𝑢⁻_𝑚−𝑢⁺_𝑚)+(𝜂,𝑢⁻_𝑚−𝑢⁺_𝑚)−^𝑘_𝑡=1𝑓(𝑡,𝑢_𝑚(𝑡))(𝑢⁻_𝑚(𝑡)−𝑢⁺_𝑚(𝑡)). (3.5) We know −(𝑀𝑢_𝑚,𝑢⁻_𝑚−𝑢⁺_𝑚)≥𝜆₁‖𝑢⁺_𝑚‖²+𝜇₁‖𝑢⁻_𝑚‖²≥𝜏₁‖𝑢⁺_𝑚+𝑢⁻_𝑚‖². (3.6) Thus, by (3.1) and (3.3), we have 𝜏₁‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²≤(1+‖𝜂‖)‖𝑢⁺_𝑚+𝑢⁻_𝑚‖+2𝐶²₁𝑘𝜏₁‖𝑢⁰_𝑚‖^2𝜃+𝜏₁2‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²+2𝐶₁𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃+1+𝐶₂√𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖, (3.7) that is, 𝜏₁2‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²−2𝐶₁𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃+1−(1+‖𝜂‖+𝐶₂√𝑘)‖𝑢⁺_𝑚+𝑢⁻_𝑚‖≤2𝐶²₁𝑘𝜏₁‖𝑢⁰_𝑚‖^2𝜃. (3.8) It follows from (3.8) and 1/2≤𝜃<1 that ‖𝑢⁺_𝑚+𝑢⁻_𝑚‖≤𝐶₄‖𝑢⁰_𝑚‖^𝜃+𝐶₅ (3.9) for all 𝑚>𝑚₀ and some positive constants 𝐶₄,𝐶₅.
By (3.1), (2.9), (3.4), and (3.9), we have 𝐶₃≥𝐽(𝑢_𝑚1)=2(𝑀𝑢_𝑚,𝑢_𝑚)+(𝜂,𝑢_𝑚)−^𝑘_𝑡=1𝐹(𝑡,𝑢_𝑚≥1(𝑡))2(𝜆₁‖𝑢⁺_𝑚‖²−𝜇_𝑠‖𝑢⁻_𝑚‖²)−‖𝜂‖(‖𝑢⁰_𝑚‖+‖𝑢⁺_𝑚+𝑢⁻_𝑚−‖)_𝑘𝑡=1(𝐹(𝑡,𝑢_𝑚(𝑡))−𝐹(𝑡,𝑢⁰_𝑚(𝑡)))−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰_𝑚𝜇(𝑡))≥−^𝑠2‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²−‖𝜂‖‖𝑢⁰_𝑚‖−2𝐶²₁𝑘𝜇_𝑠‖𝑢⁰_𝑚‖^2𝜃−𝜇_𝑠2‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²−2𝐶₁𝑘‖𝑢⁺_𝑚+𝑢⁻_𝑚‖^𝜃+1−(‖𝜂‖+𝐶₂√𝑘)‖𝑢⁺_𝑚+𝑢⁻_𝑚‖−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰_𝑚(𝑡))≥−𝜇_𝑠(𝐶²₄‖𝑢⁰_𝑚‖^2𝜃+2𝐶₄𝐶₅‖𝑢⁰_𝑚‖^𝜃+𝐶²₅)−‖𝜂‖‖𝑢⁰_𝑚‖−2𝐶²₁𝑘𝜇_𝑠‖𝑢⁰_𝑚‖^2𝜃−2𝐶₁𝑘(𝐶₄‖𝑢⁰_𝑚‖^𝜃+𝐶₅)^𝜃+1−(‖𝜂‖+𝐶₂√𝑘)(𝐶₄‖𝑢⁰_𝑚‖^𝜃+𝐶₅)−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰_𝑚(𝑡))=−(𝜇_𝑠𝐶²₄+2𝐶²₁𝑘𝜇_𝑠)‖𝑢⁰_𝑚‖^2𝜃−‖𝜂‖‖𝑢⁰_𝑚‖−2𝐶₁𝑘(𝐶₄‖𝑢⁰_𝑚‖^𝜃+𝐶₅)^𝜃+1−2𝜇_𝑠𝐶₄𝐶₅‖𝑢⁰_𝑚‖^𝜃−(‖𝜂‖+𝐶₂√𝑘)(𝐶₄‖𝑢⁰_𝑚‖^𝜃+𝐶₅)−𝜇_𝑠𝐶²₅−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰_𝑚(𝑡)). (3.10) Since 1/2≤𝜃<1, we deduce −𝐶₃≤‖𝑢⁰_𝑚‖^2𝜃(‖𝑢⁰_𝑚‖^𝑇−2𝜃_𝑡=1𝐹(𝑡,𝑢⁰_𝑚(𝑡))+𝐶₆)+𝐶₇ (3.11) for all 𝑚>𝑚₀ and some positive constants 𝐶₆,𝐶₇. The above inequality and (F₂)(i) imply that {𝑢⁰_𝑚(𝑡)} is bounded. Then it follows from (3.9) that {𝑢⁺_𝑚+𝑢⁻_𝑚} is bounded. Thus we conclude that {𝑢_𝑚} is bounded, and the (PS) condition is verified.
Now, suppose that (F₁), (F₂)(ii), and (P₃) hold. By a similar argument as above, we know also that 𝐽 satisfies the (PS) condition. The proof is complete.

Proof of Theorem 1.1. Assume that (F₁), (F₂)(i), and (P₁) hold. The proof for the case when (F₁), (F₂)(ii), and (P₁) hold is similar and will be omitted here. Since 𝑝(𝑡) is nonzero for each 𝑡∈𝐙, the singular symmetric matrix 𝑀 has at least one nonzero eigenvalue and we will give the proof in three cases. (i)Suppose that the matrix 𝑀 is singular and indefinite. Then 𝐑^𝑘 has the direct sum decomposition: 𝐑^𝑘=𝑅⁺⊕𝑅⁰⊕𝑅⁻. In view of Lemma 3.1, we only to check that conditions (𝜑₁) and (𝜑₂) in the saddle point theorem hold. To this end, let 𝑋₁=𝑅⁺⊕𝑅⁰, 𝑋₂=𝑅⁻. For any 𝑢=𝑢⁺+𝑢⁰∈𝑋₁, by (F₁), (2.9), and the mean value theorem, we have 𝜆𝐽(𝑢)≥¹2‖𝑢⁺‖²−‖𝜂‖(‖𝑢⁰‖+‖𝑢⁺‖)−^𝑘_𝑡=1𝑓(𝑡,𝑢⁰(𝑡)+𝜃₁(𝑡)𝑢⁺(𝑡))⋅𝑢⁺(𝑡)−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰≥𝜆(𝑡))₁2‖𝑢⁺‖²−‖𝜂‖(‖𝑢⁰‖+‖𝑢⁺‖)−2𝐶₁√𝑘‖𝑢⁰‖^𝜃‖𝑢⁺‖−2𝐶₁𝑘‖𝑢⁺‖^𝜃+1−𝐶₂√𝑘‖𝑢⁺‖−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰≥𝜆(𝑡))₁2‖𝑢⁺‖²−‖𝜂‖(‖𝑢⁰‖+‖𝑢⁺‖)−4𝐶²₁𝑘𝜆₁‖𝑢⁰‖^2𝜃−𝜆₁4‖𝑢⁺‖²−2𝐶₁𝑘‖𝑢⁺‖^𝜃+1−𝐶₂√𝑘‖𝑢⁺‖−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰=𝜆(𝑡))₁4‖𝑢⁺‖²−2𝐶₁𝑘‖𝑢⁺‖^𝜃+1−(‖𝜂‖+𝐶₂√𝑘)‖𝑢⁺‖−‖𝑢⁰‖^2𝜃(‖𝑢⁰‖^𝑘−2𝜃_𝑡=1𝐹(𝑡,𝑢⁰(𝑡))+4𝐶²₁𝑘𝜆₁+‖𝜂‖‖𝑢⁰‖^−2𝜃+1), (3.12) where 0<𝜃₁(𝑡)<1, 𝑡=1,2,…,𝑘. Since 1/2≤𝜃<1 and (F₂)(i), we have 𝐽(𝑢)→+∞as‖𝑢‖→∞in𝑋₁. (3.13)
On the other hand, for any 𝑢=𝑢⁻∈𝑋₂, by (F₁), (2.9), and the mean value theorem, we have 1𝐽(𝑢)≤−2𝜇₁‖𝑢‖²+‖𝜂‖‖𝑢‖−^𝑘_𝑡=1𝑓(𝑡,𝜃₂(𝑡)𝑢(𝑡))⋅𝑢(𝑡)−^𝑘_𝑡=11𝐹(𝑡,0)≤−2𝜇₁‖𝑢‖²+‖𝜂‖‖𝑢‖+^𝑘_𝑡=1𝐶₁|𝑢(𝑡)|^𝜃+1+𝐶₂√1𝑘‖𝑢‖≤−2𝜇₁‖𝑢‖²+𝐶₁𝑘‖𝑢‖^𝜃+1+(‖𝜂‖+𝐶₂√𝑘)‖𝑢‖, (3.14) where 0<𝜃₂(𝑡)<1,  𝑡=1,2,…,𝑘. Since 1/2≤𝜃<1, we can obtain 𝐽(𝑢)→−∞as‖𝑢‖→∞in𝑋₂. (3.15)
Let 𝑒=0, then it follows from (3.13) and (3.15) that (𝜑₁) and (𝜑₂) are satisfied. By the saddle point theorem, 𝐽 has at least one critical point.
(ii) Suppose that 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑙 are the positive eigenvalues of 𝑀 and 𝑙=𝑘−1. Then 𝐑^𝑘 has the direct sum decomposition: 𝐑^𝑘=𝑅⁺⊕𝑅⁰, where 𝑅⁺ and 𝑅⁰ are defined as in (2.4) and (2.5), respectively. By a similar argument as in the proof of Lemma 3.1, we see that 𝐽 satisfies the (PS) condition. By (3.13), 𝐽 is bounded from below. Then, by the least action principle, 𝑐₁=inf_𝑢∈𝐑^𝑘𝐽(𝑢) is a critical value of 𝐽.
(iii) Suppose that 0>−𝜇₁≥−𝜇₂≥⋯≥−𝜇_𝑠 are the negative eigenvalues of 𝑀 and 𝑠=𝑘−1. Then 𝐑^𝑘 has the direct sum decomposition: 𝐑^𝑘=𝑅⁰⊕𝑅⁻. Following almost the same procedure as the proof of Lemma 3.1, we know also that 𝐽 satisfies the (PS) condition in this case.
For any 𝑢=𝑢⁻∈𝑅⁻, (3.15) holds. For any 𝑢=𝑢⁰∈𝑅⁰, since 𝑀𝑢=0, we have 𝐽(𝑢)≥−‖𝜂‖‖𝑢⁰‖−^𝑘_𝑡=1𝐹(𝑡,𝑢⁰(𝑡))=−‖𝑢⁰‖^2𝜃(‖𝑢⁰‖^𝑘−2𝜃_𝑡=1𝐹(𝑡,𝑢⁰(𝑡))+‖𝜂‖‖𝑢⁰‖^−2𝜃+1). (3.16) Due to (F₂)(i) and 1/2≤𝜃<1, 𝐽(𝑢)→+∞as‖𝑢‖→∞in𝑅⁰. (3.17) By the saddle point theorem, there exists at least one critical point of 𝐽.
Since 𝐽 has at least one critical point in all three cases, the BVP (1.1)-(1.2) has at least one solution.

Proof of Theorem 1.3. Since the matrix 𝑀 is nonsingular, we will give the proof in three cases.(i) Suppose that 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑙 and 0>−𝜇₁≥−𝜇₂≥⋯≥−𝜇_𝑠 are the positive and negative eigenvalues of 𝑀, respectively, and 𝑙+𝑠=𝑘. Then 𝐑^𝑘 has the direct sum decomposition: 𝐑^𝑘=𝑅⁺⊕𝑅⁻. Denote 𝜏₁=min{𝜆₁,𝜇₁}. It follows from (F₃) that there exists a positive constant 𝑎₁ such that 1|𝑓(𝑡,𝑥)|≤4𝜏₁|𝑥|+𝑎₁ (3.18) for any (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑.
We now prove that the functional 𝐽 satisfies the (PS) condition. Let {𝑢_𝑚} be a sequence in 𝐑^𝑘 such that 𝐽(𝑢_𝑚) is bounded and 𝐽^′(𝑢_𝑚)→0 as 𝑚→∞. Write 𝑢_𝑚=𝑢⁺_𝑚+𝑢⁻_𝑚, where 𝑢⁺_𝑚∈𝑅⁺,  𝑢⁻_𝑚∈𝑅⁻. Similarly to (3.8), we have, by (3.18), 3𝜏₁4‖𝑢⁺_𝑚+𝑢⁻_𝑚‖²≤(1+‖𝜂‖+𝑎₁√𝑘)‖𝑢⁺_𝑚+𝑢⁻_𝑚‖. (3.19) Thus {𝑢_𝑚} is bounded, and the (PS) condition is verified.
For any 𝑢=𝑢⁺∈𝑅⁺, by (3.18) and the mean value theorem, we have 1𝐽(𝑢)≥2𝜆₁‖𝑢‖²−‖𝜂‖‖𝑢‖−^𝑘_𝑡=1𝑓(𝑡,𝜃₃(𝑡)𝑢(𝑡))⋅𝑢(𝑡)−^𝑘_𝑡=1𝜆𝐹(𝑡,0)≥(¹2−𝜏₁4)‖𝑢‖²−‖𝜂‖‖𝑢‖−𝑎₁√𝑘‖𝑢‖, (3.20) where 0<𝜃₃(𝑡)<1,  𝑡=1,2,…,𝑘. For any 𝑢=𝑢⁻∈𝑅⁻, Similarly to (3.20), we have 𝜇𝐽(𝑢)≤−(¹2−𝜏₁4)‖𝑢‖²+‖𝜂‖‖𝑢‖+𝑎₁√𝑘‖𝑢‖. (3.21)
Let 𝑒=0, then it follows from (3.20) and (3.21) that (𝜑₁) and (𝜑₂) are satisfied. By the saddle point theorem, 𝐽 has at least one critical point.
(ii) Suppose that 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑙 are the positive eigenvalues of 𝑀 and 𝑙=𝑘. Then 𝐸 has the direct sum decomposition: 𝐑^𝑘=𝑅⁺. It follows from (3.20) that 𝐽 satisfies the (PS) condition and is bounded from below. Then, by the least action principle, 𝑐₂=inf_𝑢∈𝐑^𝑘𝐽(𝑢) is a critical value of 𝐽.
(iii) Suppose that 0>−𝜇₁≥−𝜇₂≥⋯≥−𝜇_𝑠 are the negative eigenvalues of 𝑀 and 𝑠=𝑘. Then 𝐑^𝑘 has the direct sum decomposition: 𝐑^𝑘=𝑅⁻. It follows from (3.21) that 𝐽 satisfies the (PS) condition and is bounded from above. Then, by the least action principle, 𝑐₃=−inf_𝑢∈𝐑^𝑘𝐽(𝑢) is a critical value of −𝐽.
Since 𝐽 has a critical point in all three cases, the BVP (1.1)-(1.2) has at least one solution.

Proof of Corollary 1.4. This is immediate from Theorem 1.3.

The following lemma is useful for proving Theorem 1.6 and Corollary 1.8.

Lemma 3.2. Under the condition (F₅), the functional 𝐽 satisfies condition (𝐶); that is, for any sequence {𝑢_𝑚} such that 𝐽(𝑢_𝑚) is bounded and ‖𝐽^′(𝑢_𝑚)‖(1+‖𝑢_𝑚‖)→0 as 𝑚→∞, there exists a subsequence of {𝑢_𝑚} which is convergent in 𝐑^𝑘.

Proof. First suppose that (F₅)(i) holds. Let {𝑢_𝑚} be a sequence in 𝐑^𝑘 such that 𝐽(𝑢_𝑚) is bounded and ‖𝐽^′(𝑢_𝑚)‖(1+‖𝑢_𝑚‖)→0 as 𝑚→∞. Then there exists a constant 𝐿₁>0 such that |𝐽(𝑢_𝑚)|≤𝐿₁,‖𝐽^′(𝑢_𝑚)‖(1+‖𝑢_𝑚‖)≤𝐿₁ (3.22) for all 𝑚∈𝐙[1]. Hence, we have −3𝐿₁≤−‖𝐽^′(𝑢_𝑚)‖⋅(1+‖𝑢_𝑚‖)−2𝐽(𝑢_𝑚)≤(𝐽^′(𝑢_𝑚),𝑢_𝑚)−2𝐽(𝑢_𝑚)=^𝑘_𝑡=1[2𝐹(𝑡,𝑢_𝑚(𝑡))−(𝑓(𝑡,𝑢_𝑚(𝑡))+𝜂(𝑡))𝑢_𝑚(𝑡)]. (3.23) Then, {𝑢_𝑚} is bounded. In fact, if {𝑢_𝑚} is unbounded, there exist a subsequence of {𝑢_𝑚} (still denoted by {𝑢_𝑚}) and 𝑡₀∈𝐙[1,𝑘] such that |𝑢_𝑚(𝑡₀)|→∞as𝑚→∞. By (F₅)(i), we have 2𝐹(𝑡₀,𝑢_𝑚(𝑡₀))−(𝑓(𝑡₀,𝑢_𝑚(𝑡₀))+𝜂(𝑡₀))𝑢_𝑚(𝑡₀)→−∞as𝑚→∞. (3.24) The continuity of 2𝐹(𝑡,𝑥)−(𝑓(𝑡,𝑥)+𝜂(𝑡))𝑥 with respect to 𝑥 and (F₅)(i) implies that there exists a constant 𝐿₂>0 such that for any (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑, 2𝐹(𝑡,𝑥)−(𝑓(𝑡,𝑥)+𝜂(𝑡))𝑥≤𝐿₂. Then, we get ^𝑘_𝑡=1[2𝐹(𝑡,𝑢_𝑚(𝑡))−(𝑓(𝑡,𝑢_𝑚(𝑡))+𝜂(𝑡))𝑢_𝑚(𝑡)]≤2𝐹(𝑡₀,𝑢_𝑚(𝑡₀))−(𝑓(𝑡₀,𝑢_𝑚(𝑡₀))+𝜂(𝑡₀))𝑢_𝑚(𝑡₀)+(𝑘−1)𝐿₂. (3.25) Thus, ^𝑘_𝑡=1[2𝐹(𝑡,𝑢_𝑚(𝑡))−(𝑓(𝑡,𝑢_𝑚(𝑡))+𝜂(𝑡))𝑢_𝑚(𝑡)]→−∞as𝑚→∞, (3.26) which contradicts (3.23). Therefore, {𝑢_𝑚} is bounded in 𝐑^𝑘 and 𝐽 satisfies condition (𝐶).
Now, suppose that (F₅)(ii) holds. By a similar argument as above, we know also that 𝐽 satisfies condition (𝐶). The proof is complete.

Proof of Theorem 1.6. Assume that (F₄), (F₅)(i), and (P₃) hold. The proof for the case when (F₄), (F₅)(ii), and (P₃) hold is similar and will be omitted here. Due to (P₃), 𝐑^𝑘 has the direct sum decomposition: 𝐑^𝑘=𝑅⁺⊕𝑅⁰⊕𝑅⁻, where 𝑅⁺,𝑅⁻ and 𝑅⁰ are defined as in (2.4) and (2.5), respectively. We claim that for every 𝑡∈𝐙[1,𝑘], 𝐹(𝑡,𝑥)−𝜂(𝑡)𝑥→−∞as|𝑥|→∞. (3.27) Indeed, according to (F₅)(i), we can obtain that for any given 𝜀>0, there exists a positive constant 𝐺₁ such that for 𝑡∈𝐙[1,𝑘],|𝑥|>𝐺₁, 12𝐹(𝑡,𝑥)−(𝑓(𝑡,𝑥)+𝜂(𝑡))𝑥<−𝜀. (3.28) Obviously, 1(𝑓(𝑡,𝑠𝑥)+𝜂(𝑡))𝑠𝑥−2𝐹(𝑡,𝑠𝑥)>𝜀, (3.29) where 𝑠>1,  |𝑥|>𝐺₁. We have 𝑑(𝑑𝑠𝐹(𝑡,𝑠𝑥)−𝜂(𝑡)𝑠𝑥𝑠²)=(𝑓(𝑡,𝑠𝑥)+𝜂(𝑡))𝑠𝑥−2𝐹(𝑡,𝑠𝑥)𝑠³≥𝑑1𝑑𝑠(−2𝜀𝑠²). (3.30) By integrating both sides of the above inequality from 1 to 𝑠, we get 𝐹(𝑡,𝑠𝑥)−𝜂(𝑡)𝑠𝑥𝑠²1−𝐹(𝑡,𝑥)+𝜂(𝑡)𝑥≥−2𝜀𝑠²+12𝜀. (3.31) Let 𝑠→+∞ in the above inequality, and it follows from (F₄) that 1𝐹(𝑡,𝑥)−𝜂(𝑡)𝑥≤−2𝜀 (3.32) for 𝑡∈𝐙[1,𝑘],|𝑥|>𝐺₁. From the arbitrariness of 𝜀, we can conclude that (3.27) holds, proving our claim.
Now we prove 𝐽(𝑢)→+∞as‖𝑢‖→∞in𝑅⁺⊕𝑅⁰. (3.33) If (3.33) does not hold, there exist a constant 𝐿₃>0 and a sequence {𝑢_𝑚} in 𝑅⁺⊕𝑅⁰ such that ‖𝑢_𝑚‖→∞ as 𝑚→∞ and 𝐽(𝑢_𝑚)≤𝐿₃ (3.34) for all 𝑚. Since ‖𝑢_𝑚‖→∞ as 𝑚→∞, there exist a subsequence of {𝑢_𝑚} (still denoted by {𝑢_𝑚}) and 𝑡₀∈𝐙[1,𝑘] such that |𝑢_𝑚(𝑡₀)|→∞as𝑚→∞. By (3.27), we have 𝐹(𝑡₀,𝑢_𝑚(𝑡₀))−𝜂(𝑡₀)𝑢_𝑚(𝑡₀)→−∞as𝑚→∞. (3.35) The continuity of 𝐹(𝑡,𝑥)−𝜂(𝑡)𝑥 with respect to 𝑥 and (3.27) implies that there exists a constant 𝐿₄>0 such that for any (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑, 𝐹(𝑡,𝑥)−𝜂(𝑡)𝑥≤𝐿₄. Then, we get 𝐽(𝑢_𝑚𝜆)≥¹2‖𝑢⁺_𝑚‖²−^𝑘_𝑡=1[𝐹(𝑡,𝑢_𝑚(𝑡))−𝜂(𝑡)𝑢_𝑚(𝑡)]≥−𝐹(𝑡₀,𝑢_𝑚(𝑡₀))+𝜂(𝑡₀)𝑢_𝑚(𝑡₀)−(𝑘−1)𝐿₄. (3.36) Thus, 𝐽(𝑢_𝑚)→+∞as𝑚→∞, (3.37) which contradicts (3.34). Hence, (3.33) follows.
On the other hand, by (F₄), there exists a positive constant 𝑎₂ such that for any (𝑡,𝑥)∈𝐙[1,𝑘]×𝐑, 1|𝐹(𝑡,𝑥)|≤4𝜇₁|𝑥|²+𝑎₂. (3.38) Then we have 𝜇𝐽(𝑢)≤−¹2‖𝑢⁻‖²+‖𝜂‖‖𝑢⁻𝜇‖+¹4‖𝑢⁻‖²+𝑎₂𝑘 (3.39) for all 𝑢=𝑢⁻∈𝑅⁻. Thus, we can conclude that 𝐽(𝑢)→−∞as‖𝑢‖→∞in𝑅⁻. (3.40)
It follows from (3.33) and (3.40) that 𝐽 satisfies conditions (𝜑₁) and (𝜑₂). Hence Theorem 1.6 follows from Lemma 3.2, the saddle point theorem, and Remark 2.3.

The proof of Corollary 1.8 is similar to that of Theorem 1.6 and is omitted.

Acknowledgments

The authors thank the referees for their valuable comment and careful corrections. This project is supported by Science and Technology Plan Foundation of Guangzhou (No. 2006J1-C0341).