Existence and Multiplicity of Solutions for Discrete Nonlinear Two-Point Boundary Value Problems

Abstract

By using Morse theory, the critical point theory, and the character of 𝐾^1/2, we consider the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem −Δ²𝑥(𝑘−1)=𝑓(𝑘,𝑥(𝑘)),𝑘∈ℤ(1,𝑇) subject to 𝑥(0)=0=Δ𝑥(𝑇), where 𝑇 is a positive integer, ℤ(1,𝑇)={1,2,…,𝑇},Δ is the forward difference operator defined by Δ𝑥(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), and 𝑓∶ℤ(1,𝑇)×ℝ→ℝ is continuous. In argument, Morse inequalities play an important role.

1. Introduction

In this paper, we discuss the existence and multiplicity results of solutions to the following discrete nonlinear two-point boundary value problem (BVP): −Δ²𝑥()((𝑘(),𝑥(0)(𝑇),𝑘−1=𝑓𝑘,𝑥)),𝑘∈ℤ1,𝑇=0=Δ𝑥 (1.1) where 𝑇 is a positive integer, ℤ(1,𝑇)={1,2,…,𝑇},Δ is the forward difference operator defined by Δ𝑥(𝑘)=𝑥(𝑘+1)−𝑥(𝑘), and 𝑓∶ℤ(1,𝑇)×ℝ→ℝ is continuous.

Recent years, there have been many papers studying the existence and multiplicity of solutions for differential equations. For example, by employing the strongly monotone operator principle and the critical point theory, F. Li et al. in [1] establish some conditions on 𝑓 which are able to guarantee a class of boundary value problem on differential equation has a unique solution, at least one nonzero solution and infinitely many solutions; by using the critical point theory, and Morse theory, Yang and Zhang in [2] obtained some existence results for differential equations with parameters; there are also many authors who studied the existence results of positive solutions to boundary value problem on differential equation by employing the cone expansion or compression fixed point theorem or the critical point theory, see [3–5]; Jiang and Zhou in [6] obtained variational framework of the BVP(1.1) first by virtue of Green's function and separation of linear operator and studied the existence of a unique solution or at least one nontrivial solution by employing the strongly monotone operator principle and the critical point theory respectively.

In this paper, the main difference from the ordinary literatures is that we apply Morse theory and critical point theory to deal with problems on discrete systems. Then we establish some conditions on 𝑓 which include sublinear, superlinear or asymptotical case to guarantee that BVP(1.1) has at least one solution, at least two nontrivial solutions, and infinitely many solutions.

2. Preliminary

In this section, we give some notations and lemmas.

Let 𝐵={𝑥∶ℤ(1,𝑇+1)→ℝ,𝑥(0)=0=Δ𝑥(𝑇)} is a 𝑇-dimensional Hilbert space with inner product (𝑥,𝑦)=∑^𝑇_𝑘=1𝑥(𝑘)𝑦(𝑘), and we denote the induced norm by ‖𝑥‖=(∑^𝑇_𝑘=1|𝑥(𝑘)|²)^1/2. 𝐵_𝜌={𝑥∈𝐵∶‖𝑥‖⩽𝜌},𝑆_𝜌=𝜕𝐵_𝜌. And 𝐽_𝑎={𝑥∈𝐻∶𝐽(𝑥)⩽𝑎} for any 𝐽∈𝐶(𝐵,ℝ¹) and 𝑎∈ℝ¹. Let 𝐸 be a real Banach space, and let 𝐶¹(𝐸,ℝ) denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on 𝐸.

Let 𝐺 [6] be the Green's function of the linear boundary value problem −Δ²𝑥()(),𝑥(0)(𝑇),𝑘−1=0,𝑘∈ℤ1,𝑇=0=Δ𝑥 (2.1) where 𝐺(𝑘,𝑡)=min{𝑘,𝑡}=𝑘,𝑘⩽𝑡,𝑡,𝑡⩽𝑘 (2.2) then the (BVP)(1.1) has a solution if and only if the following equation: 𝑥(𝑘)=^𝑇_𝑡=1𝐺(𝑘,𝑡)𝑓(𝑡,𝑥(𝑡)),𝑘∈ℤ(1,𝑇) (2.3) has a solution in 𝐵. Define operators 𝐾,𝐟∶𝐵→𝐵 by 𝐾𝑥(𝑘)=^𝑇_𝑡=1𝐺(𝑘,𝑡)𝑥(𝑡),𝐟𝑥(𝑘)=𝑓(𝑘,𝑥(𝑘)),𝑘∈ℤ(1,𝑇),∀𝑥∈𝐵. (2.4) It is easy to see that a solution of (2.3) is equivalent to a solution in 𝐵 of the following operator equation: 𝑥=𝐾𝐟𝑥. (2.5)

It is well know, that all eigenvalues of 𝐾 are 𝜆_𝑡=14sin²((2𝑡−1)𝜋/(4𝑇+2)),𝑡=1,2,…,𝑇, (2.6) which have the corresponding orthonormal eigenfunctions {𝑒_𝑡}^𝑇_𝑡=1()𝜋=sin2𝑡−12𝑇+1^𝑇_𝑡=1sin²(2𝑛−1)𝜋2𝑇+1^−1/2,𝑡∈ℤ(1,𝑇). (2.7)

Remark 2.1 (see [6]). (i)𝐾∶𝐵→𝐵 is a linear continuous operator; furthermore, 𝐾 is symmetric in 𝐵.(ii)There are 𝑥=∑^𝑇_𝑘=1(𝑥,𝑒_𝑘)𝑒_𝑘,‖𝑥‖²=∑^𝑇_𝑘=1|𝑥,𝑒_𝑘|²,𝐾𝑥=∑^𝑇_𝑘=1𝜆_𝑘(𝑥,𝑒_𝑘)𝑒_𝑘,𝑥∈𝐵.(iii)The square root operator of 𝐾: 𝐾^1/2∶𝐵→𝐵 is bounded linear and symmetric, 𝐾^1/2𝑥=^𝑇_𝑘=1𝜆_𝑘^1/2𝑥,𝑒_𝑘𝑒_𝑘,𝑥∈𝐵. (2.8)(iv)(𝐾^1/2𝑥,𝑥)=∑^𝑇_𝑘=1𝜆_𝑘^1/2|(𝑥,𝑒_𝑘)|², 𝑥∈𝐵, this implies that 𝐾^1/2𝑥≠0 for all 𝑥∈𝐻 with 𝑥≠0. Therefore, 𝐾^1/2𝑥₁≠𝐾^1/2𝑥₂ for all 𝑥₁,𝑥₂∈𝐵 with 𝑥₁≠𝑥₂.

In next section, we will use the critical point theory and Morse theory to discuss the main results. Here, we state some necessary definitions and lemmas.

Definition 2.2. Let 𝐸 be a real Banach space; 𝐷 is an open subset of 𝐸. Suppose the functional 𝐽∶𝐷→ℝ is 𝐶¹(𝐸,ℝ) on 𝐷. If 𝑥₀∈𝐷 and the Fréchet derivative 𝑓^′(𝑥₀)=0, then we call that 𝑥₀ is a critical point of the functional 𝑓, and 𝑐=𝑓(𝑥₀) is a critical value of 𝑓.

Definition 2.3. Let 𝑓∈𝐶¹(𝐸,ℝ). If any sequence {𝑥_𝑚}⊂𝐸 for which {𝑓(𝑥_𝑚)} is bounded and 𝑓^′(𝑥_𝑚)→0 as 𝑚→∞ possesses a convergent subsequence, then we say that 𝑓 satisfies Palais-Smale condition (denoted by P.S. condition for short).

Lemma 2.4 (see [7, Definition 1.16, page 13]). Let 𝐸 be a real reflexive Banach space. Suppose that the functional 𝐽∶𝐸→ℝ¹ is 𝐶¹, and is bounded below and satisfies P.S. condition, then 𝐽 must have a minimum in 𝐸, that is, there exists a 𝑥^∗∈𝐸 such that 𝐽(𝑥^∗)=inf_𝑥∈𝐸𝐽(𝑥), therefore 𝑥^∗ is a critical point of the functional 𝐽.

Definition 2.5 (see [2]). Let 𝐽(0)=0, 𝐸=𝑉⊕𝑋 with dim𝑉<∞ be a real Banach space. Assume that there exists 𝜌>0 small, such that 𝐽(𝑥)‖𝑥‖𝐽(𝑥)‖𝑥‖⩽0,𝑥∈𝑉,⩽𝜌,>0,𝑥∈𝑋,0<⩽𝜌 (2.9) Then 𝐽 has a local linking at 0.

Definition 2.6. Let 𝐸 be a real Banach space, let 𝑢 be an isolated critical point of 𝐽 with 𝐽(𝑢)=𝑐 and let 𝑈 be a neighborhood of 𝑢, containing the unique critical point. We call 𝐶_𝑞(𝐽,𝑢)=𝐻_𝑞(𝐽_𝑐∩𝑈,𝐽_𝑐∩𝑈⧵{𝑢}) the 𝑞-th critical group of 𝐽 at 𝑢,𝑞=0,1,2,…, where 𝐻_𝑞(⋅,⋅) stands for the 𝑞-th singular relative homology group with integer coefficients. We say that 𝑢 is a homological nontrivial critical point of 𝐽 if at least one of its critical groups is nontrivial.

Lemma 2.7 (see [8]). Assume that 𝐽∈𝐶¹(𝐸,ℝ) satisfies P.S. condition and has a local linking at 0. Then 𝐶_𝑘(𝐽,𝜃)≇0, that is, 0 is a homological nontrivial critical point of 𝐽.

Lemma 2.8 (see [8]). Let 𝑀 be a Finsler manifold of 𝐶². Assume that 𝑓∈𝐶¹(𝑀,ℝ) satisfies the P.S. condition and 𝑎 is the only critical value of 𝑓 in [𝑎,𝑏). If connected component of 𝐾_𝑎 is only composed of isolated critical points, then 𝑓_𝑎 is a deformation retract of 𝑓_𝑏⧵𝐾_𝑏.

Lemma 2.9 (see [2]). (i) The operator equation𝑥=𝐾𝐟𝑥 (2.10) has a solution in 𝐵 if and only if the operator equation 𝑥=𝐾^1/2𝐟𝐾^1/2𝑥 (2.11) has a solution in 𝐵.
(ii) The uniqueness of solution for these two above equations is also equivalent.
(iii) If (2.11) has a nonzero solution in 𝐵, then (2.10) has a nonzero solution in 𝐵. If (2.11) has infinitely many solutions in 𝐵, then (2.10) has also infinitely many solutions in 𝐵.

Lemma 2.10 (see [9]). Suppose that the functional 𝐽(𝑥)=12‖𝑥‖²−^𝑇_𝑘=1𝐹𝑘,𝐾^1/2𝑥(𝑘),𝑥∈𝐵 (2.12) has a critical point 𝑥∈𝐵, where 𝐹(𝑘,𝑥)=∫^𝑥₀𝑓(𝑘,𝑠)𝑑𝑠, then (BVP)(1.1) has a solution in 𝐵.

Lemma 2.11 (see [8]). Suppose 𝐽∈𝐶¹(𝐵,ℝ) satisfies P.S. and (A) conditions, then, one has 𝑀_𝑞−𝑀_𝑞−1+⋯+(−1)^𝑞𝑀⁰⩾𝛽_𝑞−𝛽_𝑞−1+⋯+(−1)^𝑞𝛽₀,𝑞=0,1,2,…, (2.13) where 𝛽_𝑞=𝛽_𝑞(𝑎,𝑏)=rank𝐻_𝑞(𝑓_𝑏,𝑓_𝑎),𝑞=0,1,2,…. Furthermore, if the left of the equation is convergent, then one has that ^∞_𝑞=0(−1)^𝑞𝑀_𝑞=^∞_𝑞=0(−1)^𝑞𝛽_𝑞. (2.14)
(A) Suppose that there are two regular values, then 𝐽 has at most finite critical points and rank of critical groups of every critical point is finite.

Lemma 2.12 (see [8, page 100, Theorem 3.2]). Let 𝐽∈𝐶²(𝐵,ℝ) be a functional and satisfy P.S. condition. Suppose that 𝐽^′=𝐼−𝐴, where 𝐴 is a compact map and 𝑝₀ is a isolated critical point of 𝐽, then we have ind𝐽^′,𝑝₀=^∞_𝑞=0(−1)^𝑞rank𝐶_𝑞𝐽,𝑝₀. (2.15)

3. Mail Results

Lemma 3.1. Suppose 𝑓 satisfies one of the following conditions:(a)(H₁)limsup_|𝑥|→∞𝐹(𝑘,𝑥)/𝑥²<𝑎for 𝑘∈ℤ(1,𝑇), where 𝑎∈[0,2sin²𝜋/(4𝑇+2)),(b)(H₂)lim_|𝑥|→∞𝐹(𝑘,𝑥)/𝑥²=𝑎 for 𝑘∈ℤ(1,𝑇), where 𝑎∈[0,2sin²(𝜋/(4𝑇+2)); and (H₃)lim_|𝑥|→∞(𝐹(𝑘,𝑥)−𝑎𝑥²)=−∞ for 𝑘∈ℤ(1,𝑇), where 𝑎∈[0,2sin²(𝜋/(4𝑇+2)),(c)(H₂) and (H₄)lim_|𝑥|→∞(𝑓(𝑘,𝑥)𝑥−2𝐹(𝑘,𝑥))=+∞, uniformly for 𝑘∈ℤ(1,𝑇), where 𝑎∈[0,2sin²(𝜋/(4𝑇+2)), then one has(i)𝐽 is coercive on 𝐵, that is 𝐽(𝑥)→+∞ as ‖𝑥‖→∞, (ii) 𝐽 satisfies P.S. condition.

Proof. (i)Let (a) hold. It follows from (H₁) that there is a constant 𝑐>0 such that 𝐹(𝑘,𝑥)⩽𝑎𝑥²+𝑐,𝑥∈ℝ,𝑘∈ℤ(1,𝑇). Therefore, 𝐽(𝑥)=12‖𝑥‖²−^𝑇_𝑘=1𝐹𝑘,𝐾^1/2𝑥(𝑘⩾1)2‖𝑥‖²−𝑎𝐾^1/2𝑥²=1+𝑐2‖𝑥‖²()⩾1−𝑎𝐾𝑥,𝑥−𝑐2‖𝑥‖²−𝑎4sin²))‖𝑥‖((𝜋/4𝑇+2²=1−𝑐2𝑎1−2sin²))𝑥‖((𝜋/4𝑇+2‖²−𝑐⟶+∞,‖𝑥‖⟶∞. (3.1)

Let (b) or (c) hold. By the condition (H₂), write 𝑓(𝑘,𝑥)=2𝑎𝑥+𝑔(𝑘,𝑥), 𝐹(𝑘,𝑥)=𝑎𝑥²+𝐺(𝑘,𝑥). If (H₃) holds, then lim_|𝑥|→∞𝐺(𝑘,𝑥)=−∞; if (H₄) holds, then lim_|𝑥|→∞𝐺(𝑘,𝑥)/𝑥²=0, lim_|𝑥|→∞(𝑔(𝑘,𝑥)𝑥−2𝐺(𝑘,𝑥))=+∞. It follows that for every 𝑀>0, there is 𝑅_𝑀>0 such that 𝑔(𝑘,𝑥)𝑥−2𝐺(𝑘,𝑥)⩾𝑀,|𝑥|⩾𝑅_𝑀. (3.2) Integrating the equality 𝑑𝐺()𝑑𝑥𝑘,𝑥𝑥²=𝑔(𝑘,𝑥)𝑥−2𝐺(𝑘,𝑥)𝑥³⩾𝑀𝑥³ (3.3) over the interval [𝑥,𝑦]⊂[𝑅_𝑀,+∞), we have 𝐺𝑘,𝑦𝑦²−𝐺(𝑘,𝑥)𝑥²𝑀⩾−21𝑦²−1𝑥². (3.4) Letting 𝑦→+∞, we see that 𝐺(𝑘,𝑥)⩽−𝑀/2, 𝑘∈ℤ(1,𝑇), 𝑥⩾𝑅_𝑀. In a similar way, we have 𝐺(𝑘,𝑥)⩽−𝑀/2, 𝑘∈ℤ(1,𝑇), 𝑥⩽−𝑅_𝑀. Hence, lim_|𝑥|→∞𝐺(𝑘,𝑥)=−∞.

Let {𝑥_𝑛}⊂𝐵 be such that ‖𝑥_𝑛‖→∞ as 𝑛→∞ and 𝐽(𝑥_𝑛)⩽𝐶 for some constant 𝐶∈ℝ. Taking 𝑦_𝑛=𝑥_𝑛/‖𝑥_𝑛‖, then 𝑦_𝑛,𝑦_𝑛⇀𝑦₀, 𝑦₀∈𝐵 and ‖𝑦₀‖≤1. So, 𝐾^1/2𝑦_𝑛→𝐾^1/2𝑦₀ in 𝐵. 𝐶^∗𝑦_𝑛²≥𝐽𝑦_𝑛𝑦_𝑛²=12−1𝑥_𝑛^2𝑇_𝑘=1𝐹𝑘,𝐾^1/2𝑥_𝑛(𝑘)=12−1𝑥_𝑛^2𝑇_𝑘=1𝑎𝐾^1/2𝑥_𝑛(𝑘)²+𝐺𝑘,𝐾^1/2𝑥_𝑛(𝑘)=12−𝑎𝐾^1/2𝑥_𝑛𝑥_𝑛²−1𝑥_𝑛^2𝑇_𝑘=1𝐺𝑘,𝐾^1/2𝑥_𝑛(𝑘)=12−2sin²𝜋4𝑇+2(𝐾^1/2𝑦_𝑛)²−1𝑥_𝑛^2𝑇_𝑘=1𝐺𝑘,𝐾^1/2𝑥_𝑛(𝑘)≥12−2sin²𝜋4𝑇+2(𝐾^1/2𝑦_𝑛)²−𝐶𝑥_𝑛². (3.5) Then, 𝐾𝑦₀,𝑦₀=^𝑇_𝑘=1𝐾^1/2𝑦₀(𝑘)²=lim^𝑇_{𝑛⟶∞𝑘=1}𝐾^1/2𝑦_𝑛(𝑘)²=lim_𝑛⟶∞(𝐾^1/2𝑦_𝑛)²≥14sin²((𝜋/4𝑇)+2). (3.6) Hence |𝐾^1/2𝑦_𝑛(𝑘)|≠0 and |𝐾^1/2𝑥_𝑛(𝑘)|=|𝐾^1/2(‖𝑥_𝑛‖𝑦_𝑛(𝑘))|=‖𝑥_𝑛‖|𝐾^1/2𝑦_𝑛(𝑘)|→∞. And 𝐺(𝑘,𝐾^1/2𝑥_𝑛(𝑘))→−∞, as 𝑛→∞ for 𝑘∈ℤ(1,𝑇). Therefore, 𝐶^∗≥𝐽𝑥_𝑛=12𝑥_𝑛²−𝑎(𝐾^1/2𝑥_𝑛)²−^𝑇_𝑘=1𝐺𝑘,𝐾^1/2𝑥_𝑛(𝑘≥1)2𝑥_𝑛²−𝑎𝐾𝑥_𝑛,𝑥_𝑛−^𝑇_𝑘=1𝐺𝑘,𝐾^1/2𝑥_𝑛(𝑘)≥−^𝑇_𝑘=1𝐺𝑘,𝐾^1/2𝑥_𝑛(𝑘)⟶+∞,𝑛⟶∞. (3.7) This is impossible, so 𝐽 is coercive on 𝐵.(ii)Let {𝑥_𝑛}∈𝐵,{𝐽(𝑥_𝑛)} is bounded and 𝐽^′(𝑥_𝑛)→𝜃,𝑛→∞. By (i), {𝑥_𝑛} is bounded on 𝐵. Clearly, {𝑥_𝑛} possesses a convergent subsequence. Then, the P.S. condition is satisfied.

Lemma 3.2. Suppose that (H₅) there exist 𝛿,𝑎,𝑏∈(0,+∞) and an integer 𝑚∈ℤ(1,𝑇), which satisfy 2sin²(2𝑚−1)𝜋4𝑇+1⩽𝑏⩽𝑎⩽2sin²(2𝑚+1)𝜋4𝑇+1, (3.8) such that 𝑏𝑦²≤𝐹(𝑘,𝑦)≤𝑎𝑦²,|𝑦|≤𝛿,𝑘∈ℤ(1,𝑇).
Then the functional 𝐽 has a local linking with respect to 𝐵=𝑉₁⊕𝑉₂, where 𝑉₁=span{𝑒₁,𝑒₂,…,𝑒_𝑚},𝑉₂=𝑉^⟂₁.

Proof. Let 𝑉₁=span{𝑒₁,𝑒₂,…,𝑒_𝑚},𝑉₂=𝑉^⟂₁. If 𝑥∈𝑉₁, then (𝐾𝑥,𝑥)≥𝜆_𝑚‖𝑥‖²; if 𝑥∈𝑉₂, then (𝐾𝑥,𝑥)≤𝜆_𝑚+1‖𝑥‖².
By Remark 2.1, we can assume that for the given 𝛿>0, there exists a 𝜌=𝛿/𝐿>0 such that 𝑥∈𝑉₁, ‖𝑥‖≤𝜌⇒|𝐾^1/2𝑥|≤𝐿‖𝑥‖≤𝛿, and thus, by (H₅), we have𝐽(𝑥)=12‖𝑥‖²−^𝑇_𝑘=1𝐹𝐾^1/2𝑥(𝑘≤1)2‖𝑥‖²−^𝑇_𝑘=1𝑏𝐾^1/2𝑥(𝑘)²=12‖𝑥‖²−𝑏𝐾^1/2𝑥,𝐾^1/2𝑥=12‖𝑥‖²()≤1−𝑏𝐾𝑥,𝑥2−𝑏4sin²))‖𝑥‖((2𝑚−1)(𝜋/4𝑇+1²≤0,𝑥∈𝑉₁,‖𝑥‖≤𝜌. (3.9) For 𝑥∈𝑉₂, consider the above 𝜌. By Remark 2.1, we still have ‖𝑥‖≤𝜌⇒|𝐾^1/2𝑥|≤𝐿‖𝑥‖≤𝛿; thus by (H₅), we have 𝐽(𝑥)=12‖𝑥‖²−^𝑇_𝑘=1𝐹𝐾^1/2𝑥(𝑘≥1)2‖𝑥‖²−^𝑇_𝑘=1𝐴(𝐾^1/2𝑥)²=12‖𝑥‖²−𝑎𝐾^1/2𝑥,𝐾^1/2𝑥=12‖𝑥‖²()≥1−𝑎𝐾𝑥,𝑥2−𝑎4sin²))‖𝑥‖((2𝑚+1)(𝜋/4𝑇+1²≥0,𝑥∈𝑉₂,‖𝑥‖≤𝜌. (3.10) This implies that 𝐽 has a local linking at 0 with respect to 𝐵=𝑉₁⊕𝑉₂.

Theorem 3.3. If condition (H₁) holds, then the BVP(1.1) has at least a solution.

Proof. We will verify that the functional 𝐽(𝑥) defined in Lemma 2.10 has a critical point 𝑥∈𝐵. By [6], we know that 𝐽∶𝐵→ℝ is 𝐶¹ functional. And by Lemma 3.1, it is easy to know that 𝐽 is bounded below and satisfies P.S. condition. It follows from Lemma 2.4 that 𝐽 has a critical point in 𝑥∈𝐵.

Theorem 3.4. If conditions (H₂) and (H₃) hold, then the BVP(1.1) has at least a solution.

Theorem 3.5. If conditions (H₂) and (H₄) hold, then the BVP(1.1) has at least a solution. The proof of Theorem 3.4 and Theorem 3.5 is similar to the proof of Theorem 3.3.

Theorem 3.6. If conditions (H₁) and (H₅) hold, then the BVP(1.1) has at least two nontrivial solutions.

Proof. By Lemma 3.1, we know that 𝐽 satisfies P.S. condition, and it follows from Lemmas 2.7 and 3.2 that 𝑥=𝜃 is a homological nontrivial critical point of 𝐽, then 𝐶_𝑘(𝐽,0)≇0. If inf_𝑥∈𝐵𝐽(𝑥)≥0, then 𝐽(𝑥)=inf_𝑥∈𝑉1𝐽(𝑥)=0, 𝑥∈𝑉₁, ‖𝑥‖<𝜌, which implies that all 𝑥∈𝑉₁ with ‖𝑥‖<𝜌 are solutions of the BVP(1.1). If inf_𝑥∈𝐵𝐽(𝑥)<0, that is, 0 is not a minimizer of 𝐽, by Lemma 3.1, it is easy to know that 𝐽 is bounded below and satisfies P.S. condition. Then by Lemma 2.4, 𝐽 has a minimizer 𝑥₀→𝐵, that is, a critical point and 𝑥₀≠0; Without loss of generality, we may suppose that the minimizer 𝑥₀ is unique. Let 𝑐=𝐽(𝑥₀), then Morse index of 𝑥₀ is 𝐶_𝑞𝐽,𝑥₀=𝐻_𝑞𝐽_𝑐∩𝑈_𝑥0,𝐽_𝑐⧵𝑥₀∩𝑈_𝑥0=𝐺,𝑞=0,0,𝑞≠0. (3.11)
We suppose, 𝐽 only have two critical points with 𝜃 and 𝑥₀, where 𝑐₁=𝑐, 𝑐₂=𝐽(𝜃), 𝑐₁<𝑐₂. Let 𝑏>max{𝑐₁,𝑐₂}, 𝜀>0 is small enough such that 𝑐₁+𝜀<𝑐₂−𝜀,𝑐₂+𝜀<𝑏.
If 𝜒(𝑋,𝑌)=∑^∞_𝑞=0(−1)^𝑞𝐻_𝑞(𝑋,𝑌) is finite, then 𝜒(𝑋,𝑌) satisfies additivity. So, 𝜒𝐽_𝑏,𝐽_𝑐1−𝜀=𝜒𝐽_𝑏,𝐽_𝑐2−𝜀+𝜒𝐽_𝑐2−𝜀,𝐽_𝑐1−𝜀. (3.12) It follows from the second deformation lemma, exactness, and excision that 𝐻_𝑞(𝐽_𝑏,𝐽_𝑐2−𝜀)≅𝐻_𝑞(𝐽_𝑐2+𝜀,𝐽_𝑐2−𝜀)≅𝐻_𝑞(𝐽_𝑐2+𝜀,𝐽_𝑐2⧵{𝜃})≅𝐻_𝑞(𝐽_𝑐2,𝐽_𝑐2⧵{𝜃})≅𝐻_𝑞(𝐽_𝑐2∩𝑈_𝜃,(𝐽_𝑐2⧵{𝜃})∩𝑈_𝜃). Thus, 𝐻_𝑞𝐽_𝑏,𝐽_𝑐2−𝜀=𝐻_𝑞𝐽_𝑐2∩𝑈_𝜃,𝐽_𝑐2⧵{𝜃}∩𝑈_𝜃=𝐶_𝑞()=𝐽,𝜃𝐺,𝑞=𝑚,0,𝑞≠𝑚. (3.13) Hence, 𝜒(𝐽_𝑏,𝐽_𝑐2−𝜀)=(−1)^𝑚.
By the same way,𝐻_𝑞𝐽_𝑐2+𝜀,𝐽_𝑐1−𝜀≅𝐻_𝑞𝐽_𝑐1+𝜀,𝐽_𝑐1−𝜀≅𝐻_𝑞𝐽_𝑐1,𝐽_𝑐1−𝜀≅𝐻_𝑞𝐽_𝑐1,𝐽_𝑐1⧵𝑥₀≅𝐶_𝑞𝐽,𝑥₀=𝐺,𝑞=0,0,𝑞≠0. (3.14) So, 𝜒(𝐽_𝑐2+𝜀,𝐽_𝑐1−𝜀)=1. But 𝜒𝐽_𝑏,𝐽_𝑐1−𝜀=^∞_𝑞=0(−1)^𝑞𝐻_𝑞𝐽_𝑏,𝐽_𝑐1−𝜀=^∞_𝑞=0(−1)^𝑞𝐻_𝑞𝐽_𝑏=1. (3.15) Then, (−1)^𝑚+1≠1 is a contradict with (3.12).
Thus, 𝐽 has at least three critical points, that is, the BVP (1.1) has at least two nontrivial solutions.

Theorem 3.7. If condition (H₂), (H₃), and (H₅) hold, then the BVP(1.1) has at least two nontrivial solutions.

Theorem 3.8. If condition (H₂), (H₄), and (H₅) hold, then the BVP(1.1) has at least two nontrivial solutions.

The proof of Theorems 3.5, and 3.6 is similar to the proof of Theorem 3.6.

Theorem 3.9. Suppose that (H₆) there exists 𝑚∈ℤ(1,𝑇), such that 4sin²(2𝑚−1)𝜋4𝑇+1<𝑓^′_𝑥(𝑘,0)<4sin²(2𝑚+1)𝜋()4𝑇+1,𝑘∈ℤ1,𝑇, (3.16) (H₇) there exist 𝜇∈(0,1/2) and 𝑅>0 such that 0<𝐹(𝑘,𝑥)=∫^𝑥₀𝑓(𝑘,𝑠)𝑑𝑠⩽𝜇𝑥𝑓(𝑘,𝑥) for all 𝑘∈ℤ(1,𝑇) and 𝑥∈ℝ with |𝑥|⩾𝑅. Then the BVP(1.1) has at least one nontrivial solution.

Proof. Obviously, 𝑑𝐽(𝜃)=𝜃 and 𝑑²𝐽(𝜃)=𝐼−𝜆𝐾^1/2𝐟^′_𝑥(𝜃)𝐾^1/2∶=𝐼−Ψ, where Ψ=𝐾^1/2𝐟^′_𝑥(𝜃)𝐾^1/2. It follows from (H₆) that there exists a sufficiently small 𝜀₀>0 such that 𝑎=4sin²(2𝑚−1)𝜋4𝑇+1+𝜀₀⩽𝑓^′_𝑥(𝑘,0)⩽4sin²(2𝑚+1)𝜋4𝑇+1−𝜀₀=𝑏,𝑘∈ℤ(1,𝑇). (3.17) Since (Ψ𝑦,𝑦)=(𝑓^′_𝑥(⋅,0)𝐾^1/2𝑦,𝐾^1/2𝑦) for all 𝑦∈𝐵, 𝑎(𝐾𝑦,𝑦)⩽(Ψ𝑦,𝑦)⩽𝑏(𝐾𝑦,𝑦) for 𝑦∈𝐵. So, for every 𝑚∈ℤ(1,𝑇), 𝑎inf_𝐸𝑚−1sup_{𝑦∈𝐸⟂𝑚−1,𝑦=1}𝐾𝑦,𝑦⩽inf_𝐸𝑚−1sup_{𝑦∈𝐸⟂𝑚−1,𝑦=1}Ψ𝑦,𝑦⩽𝑏inf_𝐸𝑚−1sup_{𝑦∈𝐸⟂𝑚−1,𝑦=1}𝐾𝑦,𝑦, (3.18) where 𝐸_𝑚−1 is (𝑚−1)-dimension subspace of 𝐵. Therefore, 4𝑎sin²((2𝑚−1)𝜋/(4𝑇+1)𝜇_𝑘⩽4𝑏sin²((2𝑚−1)(𝜋/4𝑇)+1), where 𝑚-th eigenvalues of Ψ are 𝜇_𝑚. If there exists 𝑚₀ such that 𝜇_𝑚0=1, then 4𝑎sin²((2𝑚₀−1)(𝜋/4𝑇)+1)⩽1⩽4𝑏sin²((2𝑚₀−1)(𝜋/4𝑇)+1). Thus, 1/4sin²((2𝑚−1)𝜋/(4𝑇+1))+𝜀₀⩽1/4𝑎sin²((2𝑚₀−1)𝜋/(4𝑇+1))⩽1/4sin²((2𝑚+1)𝜋/(4𝑇+1))−𝜀₀. It is not true. So, 1 is not eigenvalues of Ψ. Thus 𝜃 is a nondegenerate critical point of 𝐽 and Morse index number is 𝑛. Then, 𝐶_𝑞(𝐽,𝜃)=𝛿_𝑞𝑛𝐺.
Asumme that 𝐽 is no other critical points, then Morse index number of (𝐻,𝐽_𝑎) is𝑀_𝑛=1,𝑀_𝑞=0,𝑞≠𝑛,^∞_𝑞=0(−1)^𝑞𝑀_𝑞=(−1)^𝑛. (3.19)
It follows from (H₇) that there exists 𝐶₁,𝐶₂>0 such that 𝐹(𝑘,𝑥)⩾𝐶₁|𝑥|^1/𝜇−𝐶₂, (𝑘,𝑥)∈ℤ(1,𝑇)×ℝ¹. So,𝐽()=1𝜏𝑥2𝜏²‖𝑥‖²−^𝑇_𝑡=1𝐹𝑘,𝜏𝐾^1/2𝑥(𝑘⩽1)2𝜏²‖𝑥‖²−𝐶₁𝜏^1/𝜇𝐾^1/2𝑥^1/𝜇_1/𝜇+𝐶₂,𝑥∈𝐵. (3.20) This implies lim_𝜏→+∞𝐽(𝜏𝑥)=−∞ for all 𝑥∈𝑆^∞. So, there exists 𝛾(𝑥)>0 such that 𝐽(𝛾(𝑥)𝑥)=𝑎 for all 𝑥∈𝑆^∞. Then, we will prove that there exists 𝐴>0 such that 𝑎<−𝐴, and if 𝐽(𝜏𝑥)⩽𝑎, then 𝑑/𝑑𝜏[𝐽(𝜏𝑥)]<0.
Indeed, let 𝐴=𝐶₃, then for all 𝑥∈𝑆^∞,𝑑𝐽()=𝐽𝑑𝜏𝜏𝑥^′(𝜏𝑥),𝑥=𝜏−^𝑇_𝑡=1𝐾^1/2𝑥(𝑘)𝑓𝑘,𝜏𝐾^1/2𝑥(𝑘)1⩽𝜏−𝜏𝜇^𝑇_𝑡=1𝐹𝑘,𝜏𝐾^1/2𝑥(𝑘+𝐶)³⩽1𝜏𝜇1𝜏𝜇2𝜏²−^𝑇_𝑡=1𝐹𝑘,𝜏𝐾^1/2𝑥(𝑘)+𝐶₃=1𝜏𝜇𝐽()𝜏𝜇𝜏𝑥+𝐶₃<0. (3.21) So, 𝛾(𝑥) satisfies 𝐽(𝛾(𝑥)𝑥)=𝑎 and for all 𝑥∈𝑆^∞ is unique. It follows from implicit function theorem that we have 𝛾∈𝐶(𝑆^∞,ℝ¹₊), where ℝ¹₊=[0,+∞).
At last, 𝐽(𝜃)=0 and there exists 𝜀₀>0 such that 𝐵_𝜀0∩𝐽_𝑎=∅ and 𝛾(𝑥)⩾𝜀₀ for all 𝑥∈𝑆^∞. We define a deformation retract 𝜂∶[0,1]×(𝐵⧵𝐵_𝜀0)→𝐵⧵𝐵_𝜀0 by𝜂()=()𝑥𝑠,𝑥1−𝑠𝑥+𝑠𝛾‖𝑥‖𝑥‖𝑥‖,𝑥∈𝐵⧵𝐵_𝜀0⧵𝐽_𝑎,𝑥,𝑥∈𝐽_𝑎. (3.22) This proves that 𝐵⧵𝐵_𝜀0≃𝐽_𝑎. The,n 𝑆^∞≃𝐽_𝑎. Therefore, 𝐻_𝑞(𝐻,𝐽_𝑎)≃𝐻_𝑞(𝐻,𝑆^∞)≃0. Since 𝑆^∞ is contractible, we have Betti number 𝛽_𝑞=0,𝑞⩾0. Thus, ^∞_𝑞=0(−1)^𝑞𝛽_𝑞=0. (3.23) Then by (3.19) and (3.23), we have ∑^∞_𝑞=0(−1)^𝑞𝑀_𝑞≠∑^∞_𝑞=0(−1)^𝑞𝛽_𝑞. Thus, according to the Morse inequality, 𝐽 has at least one nontrivial critical point.

Theorem 3.10. Suppose that condition (H₇) is satisfied and that (H₈)𝑓(𝑘,𝑥) is odd in 𝑥, that is, 𝑓(𝑘,𝑥)=−𝑓(𝑘,−𝑥) for all (𝑘,𝑥)∈[0,1]×ℝ¹,. Then the BVP(1.1) has infinitely many solutions.

Proof. First, it follows from (H₇) that 𝐽 satisfies P.S. condition in 𝐵 [6].
It follows from (H₈) that 𝐽 is even. Asumme that 𝐽 has only finitely many critical points {𝑥₁,−𝑥₁,𝑥₂,−𝑥₂,…,𝑥_𝑚,−𝑥_𝑚}. We choose two constant numbers 𝑎<0 and 𝑏>0, such that they satisfy 𝑎<𝑚𝑖𝑛{𝐽(𝑥₁),𝐽(𝑥₂),…,𝐽(𝑥_𝑚),−𝐶₄} and 𝑏>𝑚𝑖𝑛{𝐽(𝑥₁),𝐽(𝑥₂),…,𝐽(𝑥_𝑚)}.
On the one hand, it follows from Theorem 3.9 that 𝐵⧵𝐵_𝜀0≃𝐽_𝑎. Then, 𝑆^∞≃𝐽_𝑎. Since 𝑆^∞ is contractible, we can deduce that Betti number 𝛽_𝑞=0,𝑞⩾0.
Therefore, ^∞_𝑞=0(−1)^𝑞𝛽_𝑞=0. (3.24) On the other hand, we choose enough small 𝑟>0 such that 𝐵(𝜃,𝑟), 𝐵(𝑦_𝑖,𝑟), 𝐵(−𝑦_𝑖,𝑟), 𝑖=1,2,…,𝑚 is mutually disjoint. So, 𝑀_𝑞=𝑀_𝑞(𝑎,𝑏)=^𝑚_𝑖=1rank𝐶_𝑞𝐽,𝑦_𝑖+rank𝐶_𝑞𝐽,−𝑦_𝑖+rank𝐶_𝑞(𝐽,𝜃),𝑞⩾0. (3.25) It follows from Borsuk theorem that ^∞_𝑞=0(−1)^𝑞𝑀_𝑞=^𝑚_𝑖=1^∞_𝑞=0(−1)^𝑞rȧnk𝐶_𝑞𝐽,𝑦_𝑖+rank𝐶_𝑞𝐽,−𝑦_𝑖+^∞_𝑞=0(−1)^𝑞rank𝐶_𝑞()=𝐽,𝜃_𝑚𝑖=1deg𝐼−𝐴,𝐵𝑦_𝑖,𝑟,𝜃+deg𝐼−𝐴,𝐵−𝑦_𝑖,𝑟,𝜃+deg(𝐼−𝐴,𝐵(𝜃,𝑟),𝜃)=deg𝐼−𝐴,^𝑚_𝑖=1𝐵𝑦_𝑖,𝑟∪𝐵−𝑦_𝑖,𝑟∪𝐵(𝜃,𝑟),𝜃, (3.26) that is ^∞_𝑞=0(−1)^𝑞𝑀_𝑞=anoddnumber. (3.27) By (3.24) and (3.27), we obtain that ^∞_𝑞=0(−1)^𝑞𝑀_𝑞≠^∞_𝑞=0(−1)^𝑞𝛽_𝑞. (3.28) It is a contradiction. Thus, 𝐽 has infinitely many critical points in 𝐵.

Example 3.11. Consider the (BVP) −Δ²𝑥(𝑘−1)=cos𝑥(),𝑥(0)(𝑇),1+sin𝑥,𝑘∈ℤ1,𝑇=0=Δ𝑥 (3.29) Here, 𝑓(𝑘,𝑥)=cos𝑥/(1+sin𝑥),𝐹(𝑘,𝑥)=ln(1+sin𝑥). By simple calculation, we have lim_|𝑥|→∞(𝐹(𝑘,𝑥)/𝑥²)<2(sin²)(𝜋/(4𝑇+2)). Thus, the condition (H₁) holds. Then by Theroem 3.3., the (BVP) has at least one solution.

Example 3.12. Consider the (BVP) of Example 3.11, here 𝑓(𝑘,𝑥)=4𝑥sin²2𝜋();4𝑇+1,−1<𝑥<1,𝑘∈ℤ1,𝑇𝑥sin²𝜋();4𝑇+𝑘+2+132,𝑥⩾1,𝑘∈ℤ1,𝑇𝑥sin²𝜋(),𝐹()=4𝑇+𝑘+2−132,𝑥⩽−1,𝑘∈ℤ1,𝑇𝑘,𝑥2𝑥²sin²2𝜋();4𝑇+1,−1<𝑥<1,𝑘∈ℤ1,𝑇2𝑥²sin²𝜋();4𝑇+𝑘+2+132𝑥−66,𝑥⩾1,𝑘∈ℤ1,𝑇2𝑥²sin²𝜋().4𝑇+𝑘+2−132𝑥−66,𝑥⩽−1,𝑘∈ℤ1,𝑇 (3.30)

By simple calculation, we have lim_|𝑥|⟶∞𝐹()sup𝑘,𝑥𝑥²=2sin²𝜋4𝑇+𝑘+2<2sin²𝜋()4𝑇+2,𝑘∈ℤ1,𝑇, (3.31) and there exists 𝑚=1 and 𝛿=1>0 such that 2sin²2𝜋−1𝑥4𝑇+1²⩽𝐹(𝑘,𝑥)⩽2sin²2𝜋+1𝑥4𝑇+1²,|𝑥|⩽1. (3.32) Let 𝑎=2sin²((2𝜋+1)/(4𝑇+1)), 𝑏=2sin²((2𝜋−1)/(4𝑇+1)) then 𝑎,𝑏 satisfy 2sin²𝜋4𝑇+1⩽𝑏⩽𝑎⩽2sin²3𝜋4𝑇+1. (3.33) Thus, the conditions (H₁) and (H₅) hold. Hence, by Theorem 3.6, the (BVP) has at least two nontrivial solutions.

Example 3.13. Consider the (BVP) of Example 3.11, where 𝑓()=()𝑘,𝑥ln1+𝑥1+𝑥+10𝑥⁵(𝑘,𝑥)∈ℤ(1,𝑇)×ℝ¹. (3.34)

It is obvious that 𝑓(𝑘,𝑥)∈𝐶¹(ℤ(1,𝑇)×ℝ¹,ℝ¹) is odd with respect to 𝑥. On the other hand,𝐹(𝑘,𝑥)=^𝑥₀𝑓()1𝑘,𝑠𝑑𝑠=2ln²()+51+𝑥3𝑥⁶,𝑘∈ℤ(1,𝑇), (3.35) and we have lim_|𝑥|⟶∞𝐹(𝑘,𝑥)()=1𝑥𝑓𝑘,𝑥6. (3.36) It is easy to prove that condition (H₇) holds. Hence, by Theorem 3.10, the (BVP) has infinitely many solutions.

Acknowledgment

The authors greatly thank Professor Fuyi Li for his help and many valuable discussions.