Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems

Abstract

We consider the existence and multiplicity of solutions to discrete conjugate boundary value problems. A generalized asymptotically linear condition on the nonlinearity is proposed, which includes the asymptotically linear as a special case. By classifying the linear systems, we define index functions and obtain some properties and the concrete computation formulae of index functions. Then, some new conditions on the existence and multiplicity of solutions are obtained by combining some nonlinear analysis methods, such as Leray-Schauder principle and Morse theory. Our results are new even for the case of asymptotically linear.

1. Introduction

Let ℕ,ℤ, and ℝ be the sets of all natural numbers, integers, and real numbers, respectively. For 𝑎,𝑏∈ℤ, define ℤ(𝑎,𝑏)={𝑎,𝑎+1,…,𝑏} when 𝑎≤𝑏. Δ is the forward difference operator defined by Δ𝑢(𝑛)=𝑢(𝑛+1)−𝑢(𝑛), and Δ²𝑢(𝑛)=Δ(Δ𝑢(𝑛)). Let 𝐴 be an 𝑚×𝑚 matrix. 𝐴^𝜏 or 𝑥^𝜏 denotes the transpose of matrix 𝐴 or vector 𝑥. The set of eigenvalues of matrix 𝐴 will be denoted by 𝜎(𝐴), and the determinant of matrix 𝐴 will be denoted by det𝐴.

Discrete boundary value problems (BVPs for short) arise in the study of solid state physics, combinatorial analysis, chemical reactions, population dynamics, and so forth. Besides, they are also natural consequences of the discretization of continuous BVPs. Thus, these problems have been studied by many scholars.

Discrete two-point BVPsΔ²𝑢(𝑛−1)+𝑓(𝑛,𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),𝑢(0)=𝐴,𝑢(𝑇+1)=𝐵 (1.1) often appear in electrical circuit analysis, mathematical physics, finite elasticity, and so forth as the mathematical models, where 𝑓∶ℤ(1,𝑇)×ℝ^𝑑→ℝ^𝑑 with 𝑑∈ℕ, 𝑇>0 is a given integer, and 𝐴,𝐵 are given constants.

We may think of (1.1) as being a discrete analogue of the continuous BVPs:𝑢^′′+𝑓(𝑡,𝑢)=0,𝑎≤𝑡≤𝑏,𝑢(𝑎)=𝐴,𝑢(𝑏)=𝐵, (1.2) which have been studied by many scholars because of its numerous applications in science and technology. In particular, Hale, Walter, Mawhin, and so forth have obtained some significant results on the existence, uniqueness, and multiplicity of solutions of (1.2). We refer the readers to [1–3] and references therein for further details.

Let 𝑦(𝑛)=𝐵𝑛−𝐴(𝑛−𝑇−1)𝑇+1,𝑧(𝑛)=𝑢(𝑛)−𝑦(𝑛). (1.3) Then (1.1) reduces to Δ²𝑧(𝑛−1)+𝑓(𝑛,𝑧(𝑛)+𝑦(𝑛))=0,𝑛∈ℤ(1,𝑇),𝑧(0)=0=𝑧(𝑇+1). (1.4) Hence, in the following, we can only consider the discrete conjugate BVPs, that is,Δ²𝑢(𝑛−1)+𝑓(𝑛,𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1). (1.5)

As being remarked in [4], the nature of the solution of a continuous problem is not identical with that of the solution of its discrete analogue. And since discrete analogs of continuous problems yield interesting dynamical systems in their own right, many scholars have investigated BVPs (1.5) independently. There are fundamental questions that arise for BVPs (1.5). Does a solution exist, is it unique, and how many solutions can be found if BVPs (1.5) have multiple solutions? How to find the lower bound or the upper bound of the number of solutions of BVPs (1.5)? Furthermore, how to obtain the precise number of solutions of BVPs (1.5)?

In recent years, the existence, uniqueness, and multiplicity of solutions of discrete BVPs have been studied by many authors. In fact, early in 1968, Lasota [5] studied the discretizations of (1.2) with 𝑓(𝑡,𝑢) replaced by 𝑓(𝑡,𝑢,𝑢^′) and proved that the discrete problem had one and only one solution with 𝑓 satisfying a Lipschitz condition. Note that under certain conditions the solution of a nonhomogeneous BVPs can be expressed in terms of Green's functions. For example, suppose that 𝑢(𝑛) is a solution of (1.1). Then 𝑢(𝑛)=^𝑇_𝑚=1𝐺(𝑛,𝑚)𝑓(𝑚,𝑢(𝑚))+𝑤(𝑛), (1.6) where 𝐺(𝑛,𝑚) is Green's function for Δ²𝑢(𝑛−1)=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1),𝑤(𝑛)=𝐴+𝐵−𝐴𝑇+1𝑛. (1.7) Let ℬ={𝑢∣𝑢 is a real-valued function defined on ℤ(0,𝑇+1),𝑢(0)=𝑢(𝑇+1)=0}, and define 𝒯∶ℬ→ℬ by 𝒯𝑢(𝑛)=^𝑇_𝑚=1𝐺(𝑛,𝑚)𝑓(𝑚,𝑢(𝑚))+𝑤(𝑛) (1.8) for 𝑛 in ℤ(0,𝑇+1). Then there is a one-to-one correspondence between the fixed points of 𝒯 and the solutions of BVPs (1.1). When the nonlinearity 𝑓 satisfies growth conditions known as Lipschitz conditions, a unique solution of BVPs (1.1) can be obtained by using Contraction Mapping Theorem see [6, 7] for more details.

Note that discrete BVPs model numerous physical phenomena in nature hence it is of fundamental importance to know the criteria that ensure the existence of at least one meaningful solution. And since discrete BVPs often have multiple solutions, it is useful to have a collection of results that yield existence of solutions without the implication that the solutions must be unique. To this end, many scholars have obtained some significant results on the existence and multiplicity of solutions of discrete BVPs by using various analytic techniques and various fixed-point theorems, for example, the upper and lower solution method [8–10], the conical shell fixed point theorems [11, 12], the Brouwer and Schauder fixed point theorems [9, 13, 14], and topological degree theory [15, 16]. As we know, critical-point theory (which includes the minimax method and Morse theory, etc.) has played an important role in dealing with the existence and multiplicity of solutions to continuous systems [2, 17]. It is natural for us to think that critical-point theory may be applied to study the existence and multiplicity of solutions to discrete systems. In fact, in recent papers [18–25], the authors have applied critical-point theory to study the existence and multiplicity of periodic solutions to discrete systems. We also refer to [26–31] for the discrete BVPs. In [26], Agarwal et al. employed the Mountain Pass Lemma to study (1.5) and obtained the existence of multiple solutions. Very recently, B. Zheng and Q. Zhang [32] studied discrete BVPs (1.5) with 𝑓(𝑛,𝑢(𝑛))=𝑉^′(𝑢(𝑛)) and obtained the existence of exactly three solutions by using both Morse theory and degree theory, and so forth. To the best of our knowledge, [32] is among a few works dealing with discrete BVPs by using Morse theory. Hence, further studies on application of Morse theory to discrete BVPs are still perspective.

Here, we consider the case 𝑓(𝑛,𝑢(𝑛))=∇𝑉(𝑛,𝑢(𝑛)) that is, we consider the following discrete conjugate BVPs:Δ²𝑢(𝑛−1)+∇𝑉(𝑛,𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1), (1.9) where 𝑉(𝑛,⋅)∈𝐶¹(ℝ^𝑑,ℝ) for every 𝑛∈ℤ(1,𝑇), ∇𝑉(𝑛,𝑧) denotes the gradient of 𝑉 with respect to 𝑧, and 𝑑≥2, 𝑇>0 are given integers.

Assume∇𝑉(𝑛,𝑧)=𝐴(𝑛,𝑧)𝑧+𝑜(|𝑧|) (1.10) as |𝑧|→∞, where 𝐴∶ℤ(1,𝑇)×ℝ^𝑑→GL_𝑠(ℝ^𝑑), and𝐴₁(𝑛)≤𝐴(𝑛,𝑧)≤𝐴₂(𝑛) (1.11) for every 𝑛∈ℤ(1,𝑇), and 𝑧∈ℝ^𝑑, 𝐴₁,𝐴₂∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), GL_𝑠(ℝ^𝑑) denotes the group of 𝑑×𝑑 real nonsingular symmetric matrices, and |𝑧| denotes the Euclidean norm of 𝑧 in ℝ^𝑑. Throughout this paper, for any 𝐴₁,𝐴₂∈GL_𝑠(ℝ^𝑑), we denote 𝐴₁≤𝐴₂ if 𝐴₂−𝐴₁ is semi-positive definite, and we denote 𝐴₁<𝐴₂ if 𝐴₂−𝐴₁ is positive definite. For any 𝐴₁,𝐴₂∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), we denote 𝐴₁≤𝐴₂ if 𝐴₁(𝑛)≤𝐴₂(𝑛) for every 𝑛∈ℤ(1,𝑇), and we denote 𝐴₁<𝐴₂ if 𝐴₁(𝑛)≤𝐴₂(𝑛) for every 𝑛∈ℤ(1,𝑇) and {𝑛∣𝐴₁(𝑛)<𝐴₂(𝑛)}≠∅.

If 𝐴(𝑛,𝑧)≡𝐴(𝑛) in (1.10), then (1.10) is usually called an asymptotically linear condition. So here we call (1.10) and (1.11) generalized asymptotically linear conditions. Our results are new even for the case of asymptotically linear case.

The rest of this paper is organized as follows. In Section 2, firstly, we classify the linear systemsΔ²𝑢(𝑛−1)+𝐴(𝑛)𝑢(𝑛)=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1) (1.12) for every 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑). This classification gives a pair of integers (𝑖(𝐴),𝜈(𝐴))∈ℤ(0,𝑑𝑇)×ℤ(0,𝑑). We call 𝑖(𝐴) and 𝜈(𝐴) the index and nullity of 𝐴, respectively. Secondly, we give some properties of the index and nullity together with the concrete computation formulae. And finally, we introduce the definition of relative Morse index and give its precise description. By using both results in Section 2 and Leray-Schauder principle, we obtain some solvable conditions of (1.9) in Section 3. However, we cannot exclude the possibility that the solution we found is trivial. To this end, we make use of Morse theory to obtain the existence and multiplicity of nontrivial solutions to (1.9). Examples are also included to illustrate the results obtained.

2. Index Theory for Linear Systems

To establish the index theory for (1.12), we introduce the following finite dimensional sequence space: 𝐸={𝑢∣𝑢=(𝑢(0),𝑢(1),…,𝑢(𝑇),𝑢(𝑇+1))^𝜏,𝑢(0)=0=𝑢(𝑇+1)}, (2.1) where 𝑢(𝑛)=(𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑑(𝑛))^𝜏∈ℝ^𝑑 for every 𝑛∈ℤ(0,𝑇+1). Define the inner product on 𝐸 as follows: ⟨𝑢,𝑣⟩=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛)),∀𝑢,𝑣∈𝐸, (2.2) by which the norm ‖⋅‖ on 𝐸 can be induced by ‖𝑢‖=(^𝑇_𝑛=0|Δ𝑢(𝑛)|²)^1/2,∀𝑢∈𝐸, (2.3) where (⋅,⋅) is the usual inner product on ℝ^𝑑, and |⋅| is the usual norm on ℝ^𝑑.

Define a linear map Γ∶𝐸→ℝ^𝑑𝑇 byΓ𝑢=(𝑢₁(1),𝑢₁(2),…,𝑢₁(𝑇),𝑢₂(1),𝑢₂(2),…,𝑢₂(𝑇),…,𝑢_𝑑(1),𝑢_𝑑(2),…,𝑢_𝑑(𝑇))^𝜏. (2.4) It is easy to see that the map Γ defined in (2.4) is a linear homeomorphism, and (𝐸,⟨⋅,⋅⟩) is a Hilbert space, which can be identified with ℝ^𝑑𝑇.

Define 𝑞_𝐴(𝑢,𝑣)=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛))−^𝑇_𝑛=1(𝐴(𝑛)𝑢(𝑛),𝑣(𝑛)),∀𝑢,𝑣∈𝐸. (2.5) For any 𝑢,𝑣∈𝐸, if 𝑞_𝐴(𝑢,𝑣)=0, we say that 𝑢 and 𝑣 are 𝑞_𝐴 orthogonal. For any two subspaces 𝐸₁ and 𝐸₂ of 𝐸, if 𝑞_𝐴(𝑢,𝑣)=0 for any 𝑢∈𝐸₁ and 𝑣∈𝐸₂, we say that 𝐸₁ and 𝐸₂ are 𝑞_𝐴 orthogonal.

For any subspace 𝐸₁ of 𝐸, we say that 𝑞_𝐴 is positive definite (or negative definite) on 𝐸₁ if 𝑞_𝐴(𝑢,𝑢)>0 (or 𝑞_𝐴(𝑢,𝑢)<0) for all 𝑢∈𝐸₁⧵{0}. And if 𝑞_𝐴(𝑢,𝑢)=0 for all 𝑢∈𝐸₁, then 𝐸₁ is called a null subspace of 𝐸.

Proposition 2.1. For any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), the following results hold.(1)There are {𝜆_𝑖(𝐴)}^𝑚_𝑖=1⊂ℝ with 𝜆₁(𝐴)<𝜆₂(𝐴)<⋯<𝜆_𝑚(𝐴) such that Δ²𝑢(𝑛−1)+(𝐴(𝑛)+𝜆_𝑖(𝐴)𝐼_𝑑)𝑢(𝑛)=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1) (2.6) has a nontrivial solution. If 𝐸_𝑖(𝐴) denotes the subspace of solutions with respect to 𝜆_𝑖(𝐴), then dim𝐸_𝑖(𝐴)∶=𝑛_𝑖≤𝑑 and 𝐸=⨁^𝑚_𝑖=1𝐸_𝑖(𝐴).(2)The space 𝐸 has a 𝑞_𝐴 orthogonal decomposition 𝐸=𝐸⁺(𝐴)𝐸⁰(𝐴)𝐸⁻(𝐴) (2.7) such that 𝑞_𝐴 is positive definite, negative definite, and null on 𝐸⁺(𝐴), 𝐸⁻(𝐴), and 𝐸⁰(𝐴), respectively.

To prove Proposition 2.1, we need the following lemma.

Lemma 2.2. For any 𝑢={𝑢(𝑛)}^𝑇+1_𝑛=0∈𝐸, the following inequalities hold. 4sin²𝜋2(𝑇+1)^𝑇_𝑛=1|𝑢(𝑛)|²≤^𝑇_𝑛=0|Δ𝑢(𝑛)|²≤4cos²𝜋2(𝑇+1)^𝑇_𝑛=1|𝑢(𝑛)|². (2.8)

Proof. Note that ^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑢(𝑛))=^𝑇_𝑛=12(𝑢(𝑛),𝑢(𝑛))−^𝑇−1_𝑛=12(𝑢(𝑛),𝑢(𝑛+1))=(𝐴Γ𝑢,Γ𝑢), (2.9) where 𝐴=𝐵0⋯00𝐵⋯0⋯⋯⋯⋯00⋯𝐵_{𝑑𝑇×𝑑𝑇},𝐵=2−10⋯00−12−1⋯000−12⋯00⋯⋯⋯⋯⋯⋯000⋯2−1000⋯−12_𝑇×𝑇. (2.10) Assume that 𝜆 is an eigenvalue of 𝐵 and that 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑇)^𝜏 is an eigenvector associated to 𝜆. Define the sequence {𝑣(𝑛)}^𝑇+1_𝑛=0 as 𝑣(𝑖)=𝜉_𝑖,𝑖=1,2,…,𝑇,𝑣(0)=0=𝑣(𝑇+1). (2.11) Then {𝑣(𝑛)}^𝑇+1_𝑛=0 satisfies Δ²𝑣(𝑛−1)+𝜆𝑣(𝑛)=0,𝑛∈ℤ(1,𝑇),𝑣(0)=0=𝑣(𝑇+1). (2.12) Equation (2.12) has a nontrivial solution if and only if 𝜆=𝜆_𝑘=4sin²𝑘𝜋2(𝑇+1),𝑘=1,2,…,𝑇; (2.13) see [33]. So 𝜎(𝐴)=𝜎(𝐵)={𝜆₁,𝜆₂,…,𝜆_𝑇} with 𝜆₁<𝜆₂<⋯<𝜆_𝑇 and 𝜆_min=min{𝜆₁,𝜆₂,…,𝜆_𝑇}=4sin²𝜋2(𝑇+1)=𝜆₁,𝜆_max=max{𝜆₁,𝜆₂,…,𝜆_𝑇}=4cos²𝜋2(𝑇+1)=𝜆_𝑇. (2.14) Noticing that for any real symmetric 𝑑𝑇×𝑑𝑇 matrix 𝐴, we have 𝜆₁(Γ𝑢,Γ𝑢)≤(𝐴Γ𝑢,Γ𝑢)≤𝜆_𝑇(Γ𝑢,Γ𝑢),∀𝑢∈ℝ^𝑑𝑇. (2.15) Since (Γ𝑢,Γ𝑢)=∑^𝑇_𝑛=1|𝑢(𝑛)|², the inequalities (2.8) now follow from (2.9) and (2.15).

Remark 2.3. In the following, we rewrite (2.8) as 𝜆₁|𝑢|²≤‖𝑢‖²≤𝜆_𝑇|𝑢|² (2.16) for simplicity.

Proof of Proposition 2.1. (1) We claim that the norm ‖⋅‖_𝜆0 induced by the inner product (𝑢,𝑣)_𝜆0∶=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛))+^𝑇_𝑛=1((𝜆₀𝐼_𝑑−𝐴(𝑛))𝑢(𝑛),𝑣(𝑛)),∀𝑢,𝑣∈𝐸 (2.17) is equivalent to ‖⋅‖, where 𝜆₀ is a positive number satisfying 𝜆₀𝐼_𝑑>𝐴. In fact, it is easy to see that there exists 𝑐∈(0,+∞) such that 0≤^𝑇_𝑛=1((𝜆₀𝐼_𝑑−𝐴(𝑛))𝑢(𝑛),𝑢(𝑛))≤𝑐^𝑇_𝑛=1|𝑢(𝑛)|²≤𝑐𝜆₁^𝑇_𝑛=0|Δ𝑢(𝑛)|²=𝑐𝜆₁‖𝑢‖². (2.18) Hence ‖𝑢‖²≤‖𝑢‖²_𝜆0𝑐≤(1+𝜆₁)‖𝑢‖². (2.19) Define a bilinear function 𝑎(𝑢,𝑣)=^𝑇_𝑛=1(𝑢(𝑛),𝑣(𝑛)),∀𝑢,𝑣∈𝐸, (2.20) and then |𝑎(𝑢,𝑣)|≤(^𝑇_𝑛=1|𝑢(𝑛)|²)^1/2(^𝑇_𝑛=1|𝑣(𝑛)|²)^1/2≤1𝜆₁1‖𝑢‖‖𝑣‖≤𝜆₁‖𝑢‖_𝜆0‖𝑣‖_𝜆0. (2.21) Hence, there exists a unique continuous linear operator 𝐾_𝜆0∶𝐸→𝐸 satisfying ^𝑇_𝑛=1(𝑢(𝑛),𝑣(𝑛))=(𝑢,𝐾_𝜆0𝑣)_𝜆0. (2.22) It is easy to see that 𝐾_𝜆0 is self-adjoint, and hence all the eigenvalues of 𝐾_𝜆0 are real. Therefore, there exist 𝜇_𝑖, 𝑖=1,2,…,𝑚 and 𝑒_𝑖𝑗, 𝑗=1,2,…,𝑛_𝑖 such that (𝑒_𝑖𝑗,𝑒_𝑙𝑘)_𝜆0=𝐾1,𝑖=𝑙and𝑗=𝑘,0,𝑖≠𝑙or𝑗≠𝑘,^𝜆0𝑒_𝑖𝑗=𝜇_𝑖𝑒_𝑖𝑗, (2.23) where 𝑛_𝑖 is the multiplicity of 𝜇_𝑖 with ∑_𝑖𝑛_𝑖=𝑑𝑇. By (2.22) and (2.23) we have 𝜇_𝑖(𝑒_𝑖𝑗,𝑢)_𝜆0=^𝑇_𝑛=1(𝑒_𝑖𝑗(𝑛),𝑢(𝑛)),∀𝑢∈𝐸. (2.24) In particular, 𝜇_𝑖=∑^𝑇_𝑛=1|𝑒_𝑖𝑗(𝑛)|²>0. Without loss of generality we assume that 𝜇_𝑖 is strictly monotonously decreasing, that is, 𝜇₁>𝜇₂>⋯>𝜇_𝑚. Denote 𝜆_𝑖(𝐴)=1/𝜇_𝑖−𝜆₀ and 𝐸_𝑖(𝐴)=span{𝑒_𝑖𝑗}^𝑛𝑖_𝑗=1, then 𝐸=⨁^𝑚_𝑖=1𝐸_𝑖(𝐴). We claim that for every 𝜆_𝑖(𝐴), 𝑒_𝑖𝑗={𝑒_𝑖𝑗(𝑛)}^𝑇+1_𝑛=0∈𝐸 is a nontrivial solution of (2.6). In fact, by (2.24), for any 𝑢∈𝐸, we have 𝜇_𝑖^𝑇_𝑛=0(Δ𝑒_𝑖𝑗(𝑛),Δ𝑢(𝑛))+𝜇_𝑖^𝑇_𝑛=1((𝜆₀𝐼_𝑑−𝐴(𝑛))𝑒_𝑖𝑗(𝑛),𝑢(𝑛))=^𝑇_𝑛=1(𝑒_𝑖𝑗(𝑛),𝑢(𝑛)), (2.25) and since 𝜇_𝑖>0, the above equality means ^𝑇_𝑛=1(Δ²𝑒_𝑖𝑗1(𝑛−1)+(𝐴(𝑛)+𝜇_𝑖𝐼_𝑑−𝜆₀𝐼_𝑑)𝑒_𝑖𝑗(𝑛),𝑢(𝑛))=0,∀𝑢∈𝐸. (2.26) Therefore 𝑒_𝑖𝑗 satisfies (2.6). Now, we have proved the first result of Proposition 2.1 except dim𝐸_𝑖(𝐴)=𝑛_𝑖≤𝑑.
Set 𝑢(𝑛)=𝑦₁(𝑛),Δ𝑢(𝑛−1)=−𝑦₂(𝑛); then (2.6) is equivalent toΔ𝑦₁(𝑛)=−𝑦₂(𝑛+1),𝑛∈ℤ(0,𝑇),Δ𝑦₂(𝑛)=(𝐴(𝑛)+𝜆_𝑖(𝐴)𝐼_𝑑)𝑦₁𝑦(𝑛),𝑛∈ℤ(1,𝑇),₁(0)=0=𝑦₁(𝑇+1), (2.27) which is also equivalent to 𝑦𝑦(𝑛+1)=𝐵(𝑛)𝑦(𝑛),𝑛∈ℤ(1,𝑇),₁(0)=0=𝑦₁(𝑇+1), (2.28) where 𝑦𝑦(𝑛)=¹𝑦(𝑛)₂𝐼(𝑛),𝐵(𝑛)=^𝑑−𝜆_𝑖(𝐴)𝐼_𝑑−𝐴(𝑛)−𝐼_𝑑𝜆_𝑖(𝐴)𝐼_𝑑+𝐴(𝑛)𝐼_𝑑. (2.29) Since det(𝐵(𝑛))≡1, 𝐵(𝑛) is a nonsingular 2𝑑×2𝑑 matrix for every 𝑛. So, we can assume that Φ(𝑛) is the fundamental matrix of equation 𝑦(𝑛+1)=𝐵(𝑛)𝑦(𝑛) satisfying Φ(0)=𝐼_2𝑑. The general solution of 𝑦(𝑛+1)=𝐵(𝑛)𝑦(𝑛) can be given by 𝑦(𝑛)=Φ(𝑛)𝑐, where 𝑐=(^𝑐1_𝑐2)∈ℝ^2𝑑 and 𝑐_𝑖∈ℝ^𝑑,𝑖=1,2. Set ΦΦ(𝑛)=¹¹(𝑛)Φ₁₂Φ(𝑛)₂₁(𝑛)Φ₂₂(𝑛),whereΦ_𝑖𝑗(𝑛)is𝑑×𝑑matrix,𝑖,𝑗=1,2, (2.30) then 𝑦₁(𝑛)=Φ₁₁(𝑛)𝑐₁+Φ₁₂(𝑛)𝑐₂. By 𝑦₁(0)=0=𝑦₁(𝑇+1) and Φ(0)=𝐼_2𝑑, we have 𝑐₁=0 and 𝐸_𝑖(𝐴)≅{𝑐₂∈ℝ^𝑑∣Φ₁₂(𝑇+1)𝑐₂=0}⊆ℝ^𝑑. (2.31) Hence, dim𝐸_𝑖(𝐴)=𝑛_𝑖≤𝑑.
(2) For any 𝑢∈𝐸 with 𝑢=∑_𝑖,𝑗𝑐_𝑖𝑗𝑒_𝑖𝑗, by (2.23) and (2.24), we have𝑞_𝐴(𝑢,𝑢)=(𝑢,𝑢)_𝜆0−𝜆₀(𝑢,𝐾_𝜆0𝑢)_𝜆0=_𝑖,𝑗(|𝑐_𝑖𝑗|²−𝜆₀𝜇_𝑖|𝑐_𝑖𝑗|²)=_𝑖,𝑗𝜆_𝑖(𝐴)𝜇_𝑖|𝑐_𝑖𝑗|². (2.32) Hence, if we denote 𝐸⁺(𝐴)={𝑢=𝑐_𝑖𝑗𝑒_𝑖𝑗∣𝑐_𝑖𝑗=0,if𝜆_𝑖𝐸(𝐴)≤0},₀(𝐴)={𝑢=𝑐_𝑖𝑗𝑒_𝑖𝑗∣𝑐_𝑖𝑗=0,if𝜆_𝑖𝐸(𝐴)≠0},₋(𝐴)={𝑢=𝑐_𝑖𝑗𝑒_𝑖𝑗∣𝑐_𝑖𝑗=0,if𝜆_𝑖(𝐴)≥0}, (2.33) then the results hold.

Definition 2.4. For any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), define the index of 𝐴 as 𝑖(𝐴)∶=dim𝐸⁻(𝐴), and define the nullity of 𝐴 as 𝜈(𝐴)∶=dim𝐸⁰(𝐴).

In the following we shall discuss the properties of (𝑖(𝐴),𝑣(𝐴)).

Proposition 2.5. For any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), one has the following.(1)𝜈(𝐴) is the dimension of the solution subspace of (1.12), and 𝜈(𝐴)∈ℤ(0,𝑑).(2)𝑖(𝐴)=∑_{𝜆𝑖(𝐴)<0}𝑛_𝑖, where 𝜆_𝑖 and 𝑛_𝑖 are defined in the proof of Proposition 2.1.

Proof. (1) By Proposition 2.5, if 𝜆_𝑖(𝐴)≠0 for any 𝑖, then (1.12) has only a trivial solution, and hence 𝐸⁰(𝐴)={0}, 𝜈(𝐴)=0. If 𝜆_𝑖(𝐴)=0 for some 𝑖 with multiplicity 𝑛_𝑖, then by the proof of Proposition 2.1, 𝐸_𝑖(𝐴) is the solution subspace of (1.12) and 𝜈(𝐴)∶=dim𝐸_𝑖(𝐴)∈ℤ(1,𝑑).
(2) By the proof of Proposition 2.1, 𝐸⁻(𝐴)=⨁_{𝜆𝑖(𝐴)<0}𝐸_𝑖(𝐴), and 𝐸_𝑖(𝐴) and 𝐸_𝑗(𝐴) are 𝑞_𝐴 orthogonal if 𝑖≠𝑗. Hence the result holds.

Remark 2.6. By (1) of Proposition 2.5, 𝜈(𝐴)≥0, and 𝜈(𝐴)=0 if and only if 𝜆_𝑖(𝐴)≠0 for any 𝑖∈ℤ(1,𝑑𝑇) which holds if and only if (1.12) has only a trivial solution.

Proposition 2.7. For any 𝐴₁,𝐴₂∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), the following results hold.(1)If 𝐴₁≤𝐴₂, then 𝑖(𝐴₁)≤𝑖(𝐴₂).(2)If 𝐴₁≤𝐴₂, then 𝑖(𝐴₁)+𝜈(𝐴₁)≤𝑖(𝐴₂)+𝜈(𝐴₂).(3)If 𝐴₁<𝐴₂, then 𝑖(𝐴₁)+𝜈(𝐴₁)≤𝑖(𝐴₂).

To prove Proposition 2.7, we firstly prove the following lemma.

Lemma 2.8. Let 𝐸₁ be a subspace of 𝐸 satisfying 𝑞_𝐴(𝑢,𝑢)≤0,∀𝑢∈𝐸₁, (2.34) and then dim𝐸₁≤𝑖(𝐴)+𝜈(𝐴). (2.35) Moreover, if 𝑞_𝐴(𝑢,𝑢)<0,∀𝑢∈𝐸₁⧵{0}, (2.36) then dim𝐸₁≤𝑖(𝐴). (2.37)

Proof. Without loss of generality we can assume that dim𝐸₁=𝑘≥1 and 𝐸₁=span{𝑒₁,𝑒₂,…,𝑒_𝑘}. Let 𝑒_𝑖=𝑒⁺_𝑖+𝑒^∗_𝑖, where 𝑒^∗_𝑖∈𝐸⁻(𝐴)⊕𝐸⁰(𝐴), 𝑒⁺_𝑖∈𝐸⁺(𝐴). To prove 𝑖(𝐴)+𝜈(𝐴)≥𝑘, we only need to prove that 𝑒^∗₁,𝑒^∗₂,…,𝑒^∗_𝑘 is linear independent. If not, there exist not all zero constants 𝛼₁,𝛼₂,…,𝛼_𝑘 such that ∑^𝑘_𝑖=1𝛼_𝑖𝑒^∗_𝑖=0. So 𝑒∶=∑^𝑘_𝑖=1𝛼_𝑖𝑒_𝑖=∑^𝑘_𝑖=1𝛼_𝑖𝑒⁺_𝑖∈𝐸⁺(𝐴), and hence 𝑞_𝐴(𝑒,𝑒)>0. This is a contradiction to (2.34). This implies that 𝑒^∗₁,𝑒^∗₂,…,𝑒^∗_𝑘 is linear independent and 𝑖(𝐴)+𝜈(𝐴)∶=dim𝐸⁻(𝐴)+dim𝐸⁰(𝐴)=dim(𝐸⁻(𝐴)⊕𝐸⁰(𝐴))≥𝑘. The first part is proved.
To prove the second part, let 𝑒_𝑖=𝑒⁺_𝑖+𝑒⁻_𝑖+𝑒⁰_𝑖, where 𝑒^∗_𝑖∈𝐸^∗(𝐴), ∗=+,0,−. To prove 𝑖(𝐴)≥𝑘, we only need to prove that 𝑒⁻₁,𝑒⁻₂,…,𝑒⁻_𝑘 is linear independent. If not, there exist not all zero constants 𝑐₁,𝑐₂,…,𝑐_𝑘 such that ∑^𝑘_𝑖=1𝑐_𝑖𝑒⁻_𝑖=0. So, 𝑒∶=∑^𝑘_𝑖=1𝑐_𝑖𝑒_𝑖=∑^𝑘_𝑖=1(𝑐_𝑖𝑒⁺_𝑖+𝑐_𝑖𝑒⁰_𝑖)∈𝐸⁺(𝐴)⊕𝐸⁰(𝐴) and 𝑞_𝐴(𝑒,𝑒)≥0. This is a contradiction to (2.36).

Proof of Proposition 2.7. For any 𝑢∈𝐸, denote 𝑢=𝑢⁺+𝑢⁰+𝑢⁻, where 𝑢^∗∈𝐸^∗(𝐴₁), ∗=+,−,0.(1)From Lemma 2.8, we only need to show that 𝑞_𝐴2(𝑢,𝑢)<0,∀𝑢∈𝐸⁻(𝐴₁)⧵{0}. (2.38) In fact, for every 𝑢=𝑢⁻∈𝐸⁻(𝐴₁), if 𝑢⁻≠0, then 𝑞_𝐴2(𝑢,𝑢)≤𝑞_𝐴1(𝑢,𝑢)=𝑞_𝐴1(𝑢⁻,𝑢⁻)<0. (2.39)(2)From Lemma 2.8, we only need to show that 𝑞_𝐴2(𝑢,𝑢)≤0,∀𝑢∈𝐸⁻(𝐴₁)⊕𝐸⁰(𝐴₁). (2.40) In fact, for every 𝑢=𝑢⁻+𝑢⁰∈𝐸⁻(𝐴₁)⊕𝐸⁰(𝐴₁), one has 𝑞_𝐴2(𝑢,𝑢)≤𝑞_𝐴1(𝑢,𝑢)=𝑞_𝐴1(𝑢⁰,𝑢⁰)+𝑞_𝐴1(𝑢⁻,𝑢⁻)≤0. (2.41)(3)From Lemma 2.8, we only need to show that 𝑞_𝐴2(𝑢,𝑢)<0,∀𝑢∈𝐸⁻(𝐴₁)⊕𝐸⁰(𝐴₁)⧵{0}. (2.42) In fact, for every 𝑢=𝑢⁻+𝑢⁰ with 𝑢⁻∈𝐸⁻(𝐴₁), 𝑢⁰∈𝐸⁰(𝐴₁), if 𝑢⁻≠0, then 𝑞_𝐴2(𝑢,𝑢)≤𝑞_𝐴1(𝑢,𝑢)=𝑞_𝐴1(𝑢⁰,𝑢⁰)+𝑞_𝐴1(𝑢⁻,𝑢⁻)<0. (2.43) If 𝑢⁻=0, 𝑢⁰≠0, then {𝑢⁰(𝑛)}^𝑇+1_𝑛=0 is a nontrivial solution of Δ²𝑢(𝑛−1)+𝐴₁(𝑛)𝑢(𝑛)=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1). (2.44) From 𝐴₁<𝐴₂ we have 𝑞_𝐴2(𝑢⁰,𝑢⁰)<𝑞_𝐴1(𝑢⁰,𝑢⁰)=0. (2.45) Hence (2.42) holds.

Proposition 2.9. If 𝑈∈𝑂(𝑑), that is, 𝑈∈GL_𝑠(ℝ^𝑑) and 𝑈^𝜏𝑈=𝐼_𝑑, then for any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), 𝑖(𝑈^𝜏𝐴𝑈)=𝑖(𝐴),𝜈(𝑈^𝜏𝐴𝑈)=𝜈(𝐴). In particular, for any 𝐴∈GL_𝑠(ℝ^𝑑), we have 𝑖(𝐴)=^𝑑_𝑖=1♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘<𝑎_𝑖},𝜈(𝐴)=^𝑑_𝑖=1♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘=𝑎_𝑖}, (2.46) where 𝜆_𝑘 is given by (2.13), and {𝑎_𝑖}^𝑑_𝑖=1=𝜎(𝐴) is the set of eigenvalues of 𝐴.

Proof. Firstly, we claim that 𝜆_𝑖(𝑈^𝜏𝐴𝑈)=𝜆_𝑖(𝐴). (2.47) In fact, since 𝜆₀𝐼_𝑑>𝐴 if and only if 𝜆₀𝐼_𝑑>𝑈^𝜏𝐴𝑈, we can choose 𝜆₀(𝑈^𝜏𝐴𝑈)=𝜆₀(𝐴). By (2.22) and (2.23), it is easy to see that 𝜇_𝑖(𝑈^𝜏𝐴𝑈)=𝜇_𝑖(𝐴), and hence (2.47) holds. Therefore, by Definition 2.4, 𝑖(𝑈^𝜏𝐴𝑈)=𝑖(𝐴),𝜈(𝑈^𝜏𝐴𝑈)=𝜈(𝐴). (2.48) Since 𝐸^∗(diag{𝐴₁,𝐴₂})≅𝐸^∗(𝐴₁)⊕𝐸^∗(𝐴₂),∗=+,0,−, (2.49) we have 𝑖(diag{𝐴₁,𝐴₂})=𝑖(𝐴₁)+𝑖(𝐴₂),𝜈(diag{𝐴₁,𝐴₂})=𝜈(𝐴₁)+𝜈(𝐴₂). (2.50) Note that the scalar eigenvalue problem Δ²𝑦(𝑛−1)+𝜆𝑦(𝑛)=0,𝑛∈ℤ(1,𝑇),𝑦(0)=0=𝑦(𝑇+1) (2.51) has a nontrivial solution if and only if 𝜆=𝜆_𝑘=4sin²(𝑘𝜋/2(𝑇+1)), 𝑘∈ℤ(1,T). By Proposition 2.1 and Definition 2.4, we see that for any 𝛼∈ℝ, 𝑖(𝛼)=♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘<𝛼},𝜈(𝛼)=♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘=𝛼}. (2.52) Since {𝑎_𝑖}^𝑑_𝑖=1 is the set of eigenvalues of 𝐴, there exists an orthogonal matrix 𝑈 such that 𝑈^𝜏𝐴𝑈=diag{𝑎₁,𝑎₂,…,𝑎_𝑑}. (2.53) From (2.48), (2.50), and (2.52) we have 𝑖(𝐴)=^𝑑_𝑖=1♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘<𝑎_𝑖},𝜈(𝐴)=^𝑑_𝑖=1♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘=𝑎_𝑖}. (2.54) This completes the proof.

Proposition 2.10. For any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑) with 𝑖(𝐴)=0, one has ^𝑇_𝑛=0|Δ𝑢(𝑛)|²≥^𝑇_𝑛=1(𝐴(𝑛)𝑢(𝑛),𝑢(𝑛)),∀𝑢∈𝐸. (2.55) And the equality holds if and only if 𝑢∈𝐸⁰(𝐴).

Proof. For any 𝑢∈𝐸 with 𝑢=∑_𝑖,𝑗𝑐_𝑖𝑗𝑒_𝑖𝑗, we have ^𝑇_𝑛=0|Δ𝑢(𝑛)|²=^𝑇_𝑛=1(𝐴(𝑛)𝑢(𝑛),𝑢(𝑛))+_𝑖,𝑗𝜆_𝑖(𝐴)𝜇_𝑖|𝑐_𝑖𝑗|². (2.56) Because 𝑖(𝐴)=0, by definition, 𝜆_𝑖(𝐴)≥0 for any 𝑖. So the inequality holds. And the equality holds if and only if 𝑐_i𝑗=0 as 𝜆_𝑖(𝐴)≠0, that is, 𝑢∈𝐸⁰(𝐴).

By now, we have proved the monotonicity and have offered the computation formulae of the indices. These will play an important role in discussing nonlinear Hamiltonian systems in the next section. In the end of this section, we shall introduce the relative Morse index, which is a precise expression of the number 𝑖(𝐴₂)−𝑖(𝐴₁) as 𝐴₂>𝐴₁.

Definition 2.11. For any 𝐴₁,𝐴₂∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑) with 𝐴₁<𝐴₂, define 𝐼(𝐴₁,𝐴₂)=_𝜆∈[0,1)𝜈(𝐴₁+𝜆(𝐴₂−𝐴₁)). (2.57)

If 𝐴₁=𝛼₁𝐼_𝑑, 𝐴₂=𝛼₂𝐼_𝑑, where 𝛼₁<𝛼₂ are two real numbers, then by Proposition 2.9, we have 𝐼(𝛼₁𝐼_𝑑,𝛼₂𝐼_𝑑)=_𝜆∈[0,1)𝜈(𝛼₁𝐼_𝑑+𝜆(𝛼₂−𝛼₁)𝐼_𝑑)=𝑑♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘∈[𝛼₁,𝛼₂)},𝑖(𝛼₁𝐼_𝑑)=𝑑♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘<𝛼₁},𝑖(𝛼₂𝐼_𝑑)=𝑑♯{𝑘∈ℤ(1,𝑇)∣𝜆_𝑘<𝛼₂}. (2.58)

So 𝐼(𝛼₁𝐼_𝑑,𝛼₂𝐼_𝑑)=𝑖(𝛼₂𝐼_𝑑)−𝑖(𝛼₁𝐼_𝑑). (2.59) This gives us a steer toward the following result.

Proposition 2.12. For any 𝐴₁,𝐴₂∶ℤ(1,𝑇)→𝐺𝐿_𝑠(ℝ^𝑑) with 𝐴₁<𝐴₂, one has 𝐼(𝐴₁,𝐴₂)=𝑖(𝐴₂)−𝑖(𝐴₁). (2.60)

Proof. Denote 𝑖(𝜆)∶=𝑖(𝐴₁+𝜆(𝐴₂−𝐴₁)) for 𝜆∈[0,1], 𝜈(𝜆)∶=𝜈(𝐴₁+𝜆(𝐴₂−𝐴₁)); then to prove (2.60), we only need to prove that 𝑖(𝜆+0)=𝑖(𝜆)+𝜈(𝜆),∀𝜆∈[0,1),𝑖(𝜆−0)=𝑖(𝜆),∀𝜆∈(0,1] (2.61)(2.62) hold. In fact, if (2.61) and (2.62) hold, then the function 𝜆→𝑖(𝜆) is integer-valued, left continuous, and nondecreasing. So, for any 𝜆₁∈[0,1], 𝑖(𝜆₁)−𝑖(0) must equal to the sum of the jumps 𝑖(𝜆) incurred in [0,𝜆₁). By (2.61) and (2.62), this is precisely the sum of 𝜈(𝜆), 0≤𝜆<𝜆₁, that is, 𝑖(𝜆₁)−𝑖(0)=_{𝜆∈[0,𝜆1)}𝜈(𝜆). (2.63) Hence, if we choose 𝜆₁=1, then (2.60) holds.
From (3) of Proposition 2.7, to prove (2.61), we only need to prove 𝑖(𝜆+0)≤𝑖(𝜆)+𝜈(𝜆) which is also equivalent to 𝑑𝑇−𝑖(𝜆)−𝜈(𝜆)≤𝑑𝑇−𝑖(𝜆+0). For any 𝑠∈[0,1], set 𝑚⁺(𝑠)=𝑑𝑇−𝑖(𝑠)−𝜈(𝑠), we only need to prove𝑚⁺(𝜆)≤𝑚⁺(𝜆+0)+𝜈(𝜆+0). (2.64) Similar to the proof of Lemma 2.8, it is easy to know that for 𝜀>0 sufficiently small, if 𝑞_{𝐵𝜆+𝜀}(𝑢,𝑢)≥0,∀𝑢∈𝐸⁺(𝜆), (2.65) then (2.64) holds, where 𝐸⁺(𝜆)=𝐸⁺(𝐴₁+𝜆(𝐴₂−𝐴₁)), 𝐵_𝜆+𝜀(𝑛)=𝐴₁(𝑛)+(𝜆+𝜀)(𝐴₂(𝑛)−𝐴₁(𝑛)). While as 𝜀>0 is sufficiently small and 𝑢∈𝐸⁺(𝜆), we have 𝑞_{𝐵𝜆+𝜀}(𝑢,𝑢)=𝑞_𝐵𝜆(𝑢,𝑢)−𝜀^𝑇_𝑛=1((𝐴₂(𝑛)−𝐴₁(𝑛))𝑢(𝑛),𝑢(𝑛))=𝑞_𝐵𝜆(𝑢⁺,𝑢⁺)−𝜀^𝑇_𝑛=1((𝐴₂(𝑛)−𝐴₁(𝑛))𝑢(𝑛),𝑢(𝑛))≥0, (2.66) where 𝐵_𝜆(𝑛)=𝐴₁(𝑛)+𝜆(𝐴₂(𝑛)−𝐴₁(𝑛)). Hence 𝑖(𝜆+0)≤𝑖(𝜆)+𝜈(𝜆).
On the other hand, from (1) of Proposition 2.7, to prove (2.62), we only need to prove 𝑖(𝜆)≤𝑖(𝜆−0). By Lemma 2.8, to prove 𝑖(𝜆)≤𝑖(𝜆−0), we only need to prove𝑞_{𝐵𝜆−𝜀}(𝑢,𝑢)<0,∀𝑢∈𝐸⁻(𝜆)⧵{0}, (2.67) where 𝜀>0 is sufficiently small, 𝐵_𝜆−𝜀(𝑛)=𝐴₁(𝑛)+(𝜆−𝜀)(𝐴₂(𝑛)−𝐴₁(𝑛)). And as 𝜀>0 is sufficiently small, we have 𝑞_{𝐵𝜆−𝜀}(𝑢,𝑢)=𝑞_𝐵𝜆(𝑢,𝑢)+𝜀^𝑇_𝑛=1((𝐴₂(𝑛)−𝐴₁(𝑛))𝑢(𝑛),𝑢(𝑛))=𝑞_𝐵𝜆(𝑢⁻,𝑢⁻)+𝜀^𝑇_𝑛=1((𝐴₂(𝑛)−𝐴₁(𝑛))𝑢(𝑛),𝑢(𝑛))<0, (2.68) where 𝐵_𝜆(𝑛)=𝐴₁(𝑛)+𝜆(𝐴₂(𝑛)−𝐴₁(𝑛)). This completes the proof.

Proposition 2.13. For any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑), there exists 𝜀₀>0 such that for any 𝜀∈(0,𝜀₀], one has 𝜈(𝐴+𝜀𝐼_𝑑)=0,𝜈(𝐴−𝜀𝐼_𝑑)=0,𝑖(𝐴−𝜀𝐼_𝑑)=𝑖(A),𝑖(𝐴+𝜀𝐼_𝑑)=𝑖(𝐴)+𝜈(𝐴). (2.69)(2.70)(2.71)(2.72)

Proof. From Proposition 2.12 we have 𝑖(𝐴+𝐼_𝑑)=𝑖(𝐴)+𝐼(𝐴,𝐴+𝐼_𝑑). From Definition 2.11 and Proposition 2.12, we know that 𝐼(𝐴,𝐴+𝐼_𝑑)=∑_𝜆∈[0,1)𝜈(𝐴+𝜆𝐼_𝑑) is finite, and then there must exist some 𝜀₀>0 such that for any 𝜀∈(0,𝜀₀], 𝜈(𝐴+𝜀𝐼_𝑑)=0 and 𝑖(𝐴+𝜀𝐼_𝑑)=𝑖(𝐴)+_𝜆∈[0,1)𝜈(𝐴+𝜆𝜀𝐼_𝑑)=𝑖(𝐴)+𝜈(𝐴). (2.73) This proves (2.69) and (2.72).
To prove (2.70) and (2.71), note that 𝐼(𝐴−𝐼_𝑑,𝐴)=𝑖(𝐴)−𝑖(𝐴−𝐼_𝑑) and𝐼(𝐴−𝐼_𝑑,𝐴)=_𝜆∈[0,1)𝜈(𝐴−(1−𝜆)𝐼_𝑑)=_𝜆∈(0,1]𝜈(𝐴−𝜆𝐼_𝑑). (2.74) Since 𝐼(𝐴−𝐼_𝑑,𝐴) is finite, there exists 𝜀₀>0 such that for any 𝜀∈(0,𝜀₀], 𝜈(𝐴−𝜀𝐼_𝑑)=0. Hence 𝑖(𝐴−𝜀𝐼_𝑑)=𝑖(𝐴)−𝐼(𝐴−𝜀𝐼_𝑑,𝐴)=𝑖(𝐴)−_𝜆∈(0,1]𝜈(𝐴−𝜆𝜀𝐼_𝑑)=𝑖(𝐴). (2.75) This proves (2.70) and (2.71).

3. Main Results

In this section, firstly, we shall obtain the existence of solutions to (1.9) by using both the index theory in Section 2 and Leray-Schauder principle. Then, we obtain the multiplicity of solutions to (1.9) by using Morse theory.

Theorem 3.1. Assume that (1)there exist 𝐴∶ℤ(1,𝑇)×ℝ^𝑑→GL_𝑠(ℝ^𝑑) and ℎ∶ℤ(1,𝑇)×ℝ^𝑑→ℝ^𝑑 which are both continuous with respect to the second variable, where ℎ(𝑛,𝑧)=𝑜(|𝑧|) as |𝑧|→∞ for every 𝑛∈ℤ(1,𝑇) and ∇𝑉(𝑛,𝑧)=𝐴(𝑛,𝑧)𝑧+ℎ(𝑛,𝑧); (3.1)(2)there exist 𝐴₁,𝐴₂∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑) satisfying 𝐴₁≤𝐴₂, 𝑖(𝐴₁)=𝑖(𝐴₂), 𝜈(𝐴₂)=0 such that 𝐴₁(𝑛)≤𝐴(𝑛,𝑧)≤𝐴₂(𝑛),∀𝑧∈ℝ^𝑑 (3.2) for every 𝑛∈ℤ(1,𝑇). Then (1.9) has at least one solution.

To prove Theorem 3.1, we need the following Leray-Schauder principle; see [34] for detailed proof.

Lemma 3.2. Assume that (𝑋,‖⋅‖_𝑋) is a Banach space and that Φ∶𝑋→𝑋 is completely continuous. If the set {‖𝑥‖_𝑋∣𝑥∈𝑋,𝑥=𝜆Φ𝑥,0<𝜆<1} is bounded, then Φ must have a fixed point in a closed ball 𝐵_𝑅 in 𝑋, where 𝐵_𝑅={𝑥∣𝑥∈𝑋,‖𝑥‖_𝑋≤𝑅},𝑅=sup{‖𝑥‖_𝑋∣𝑥=𝜆Φ𝑥,0<𝜆<1}. (3.3)

Proof of Theorem 3.1. Assume that (3.2) holds. Since 𝜈(𝐴₂)=0, from (1) of Proposition 2.5, we know that the system Δ²𝑢(𝑛−1)+𝐴₂(𝑛)𝑢(𝑛)=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1) (3.4) has only a trivial solution. Define Γ₁∶𝐸→𝐸 as (Γ₁𝑢)(𝑛)=Δ²𝑢(𝑛−1)+𝐴₂(𝑛)𝑢(𝑛); (3.5) then Γ₁ is an invertible operator. Define Γ₂∶𝐸→𝐸 as (Γ₂𝑢)(𝑛)=𝐴₂(𝑛)𝑢(𝑛)−∇𝑉(𝑛,𝑢(𝑛)); (3.6) then finding the solutions to (1.9) is equivalent to finding solutions to Γ₁𝑢−Γ₂𝑢=0 (3.7) in 𝐸, which is also equivalent to finding the fixed points of Γ₁⁻¹Γ₂ in 𝐸 since Γ₁ is invertible. By Lemma 3.2, we only need to prove that the possible solutions to Δ²𝑢(𝑛−1)+(1−𝜆)𝐴₂(𝑛)𝑢(𝑛)+𝜆∇𝑉(𝑛,𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1) (3.8) are priori bounded with respect to the norm ‖⋅‖ in 𝐸, where 𝜆∈(0,1). If not, there exist {𝑢_𝑘}⊂𝐸,{𝜆_𝑘}⊂(0,1) with ‖𝑢_𝑘‖→∞ such that Δ²𝑢_𝑘(𝑛−1)+(1−𝜆_𝑘)𝐴₂(𝑛)𝑢_𝑘(𝑛)+𝜆_𝑘∇𝑉(𝑛,𝑢_𝑘𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),_𝑘(0)=0=𝑢_𝑘(𝑇+1). (3.9) Denote 𝑣_𝑘(𝑛)=𝑢_𝑘(𝑛)/‖𝑢_𝑘‖, 𝐵_𝑘(𝑛)=(1−𝜆_𝑘)𝐴₂(𝑛)+𝜆_𝑘𝐴(𝑛,𝑢_𝑘(𝑛)), 𝑒_𝑘(𝑛)=𝜆_𝑘(∇𝑉(𝑛,𝑢_𝑘(𝑛))−𝐴(𝑛,𝑢_𝑘(𝑛))𝑢_𝑘(𝑛))/‖𝑢_𝑘‖; then (3.9) is equivalent to Δ²𝑣_𝑘(𝑛−1)+𝐵_𝑘(𝑛)𝑣_𝑘(𝑛)+𝑒_𝑘𝑣(𝑛)=0,𝑛∈ℤ(1,𝑇),_𝑘(0)=0=𝑣_𝑘(𝑇+1). (3.10) From (3.1), 𝑒_𝑘→0 as ‖𝑢_𝑘‖→∞. We may assume that 𝑣_𝑘→𝑣₀,𝜆_𝑘→𝜆₀, and 𝑏^𝑘_𝑖𝑗→𝑏_𝑖𝑗, where 𝐵_𝑘(𝑛)=(𝑏^(𝑘)_𝑖𝑗)_𝑑×𝑑(𝑛). Denote 𝐵₀(𝑛)=(𝑏_𝑖𝑗(𝑛))_𝑑×𝑑; let 𝑘→∞ in (3.10); we have Δ²𝑣₀(𝑛−1)+𝐵₀(𝑛)𝑣₀𝑣(𝑛)=0,𝑛∈ℤ(1,𝑇),₀(0)=0=𝑣₀(𝑇+1). (3.11) On the other hand, (3.2) implies that 𝐴₁≤𝐵_𝑘≤𝐴₂, and hence 𝐴₁≤𝐵₀≤𝐴₂. By 𝑖(𝐴₂)=𝑖(𝐴₁), 𝜈(𝐴₂)=0, and Proposition 2.7, we have 𝜈(𝐴₁)=𝜈(𝐴₂)=𝜈(𝐵₀)=0. This contradicts the fact that (3.11) has a nontrivial solution.

Example 3.3. Let 𝑉(𝑛,𝑧)=^𝑑_𝑖=1𝐹_𝑖(𝑧_𝑖1)+2^𝑑_𝑖=1𝑎_𝑖𝑖(𝑛)𝑧²_𝑖+_{1≤𝑖<𝑗≤𝑑}𝑎_𝑖𝑗(𝑛)𝑧_𝑖𝑧_𝑗+(^𝑑_𝑖=1𝑧²_𝑖+1)^3/4sin𝑛, (3.12) where 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑑)^𝜏, 𝐹_𝑖(𝑡)=∫^𝑡₀𝑠𝑓_𝑖(𝑠)𝑑𝑠, 𝑓_𝑖∶ℝ→[0,𝛼] is continuous and 𝛼>0, 𝑎_𝑖𝑗(𝑛)=𝑎_𝑗𝑖(𝑛), 𝑖=1,2,…,𝑑, 𝑗=1,2,…,𝑑. Set 𝐴(𝑛)=(𝑎_𝑖𝑗(𝑛))_𝑑×𝑑=𝑈^𝜏diag{𝛼₁(𝑛),𝛼₂(𝑛),…,𝛼_𝑑(𝑛)}𝑈, (3.13) where 𝑈∈𝑂(𝑑). Since ∇𝑉(𝑛,𝑧)=(𝑧₁𝑓₁(𝑧₁),𝑧₂𝑓₂(𝑧₂),…,𝑧_𝑑𝑓_𝑑(𝑧_𝑑))^𝜏3+𝐴(𝑛)𝑧+2(^𝑑_𝑖=1𝑧²_𝑖+1)^−1/4𝑧sin𝑛=diag{𝑓₁(𝑧₁),𝑓₂(𝑧₂),…,𝑓_𝑑(𝑧_𝑑3)}𝑧+𝐴(𝑛)𝑧+2(^𝑑_𝑖=1𝑧²_𝑖+1)^−1/4𝑧sin𝑛, (3.14)𝑉(𝑛,𝑧) satisfies (3.1) with 𝐴(𝑛,𝑧)=𝐴(𝑛)+diag{𝑓₁(𝑧₁),𝑓₂(𝑧₂),…,𝑓_𝑑(𝑧_𝑑𝐴)},₁(𝑛)=𝐴(𝑛),𝐴₂(𝑛)=𝐴(𝑛)+𝛼𝐼_𝑑,3ℎ(𝑛,𝑧)=2(^𝑑_𝑖=1𝑧²_𝑖+1)^−1/4𝑧sin𝑛. (3.15) If 𝜈(𝛼_𝑖)=0 for every 𝑖∈ℤ(1,𝑑), then 𝜈(𝐴₁)=0. By Proposition 2.13, if 𝛼>0 is small enough, then 𝜈(𝐴₂)=0 and 𝑖(𝐴₁)=𝑖(𝐴₂). Hence, by Theorem 3.1, (1.9) has at least one solution. In particular, if we choose 𝑓_𝑖(𝑡)=𝛼(sin𝑡)^2𝑖 and 𝛼_𝑖(𝑛)≡𝛼_𝑖∈ℝ, [𝛼_𝑖,𝛼_𝑖+𝛼]∩{𝜆_𝑘}^𝑇_𝑘=1=∅ for every 𝑖=1,2,…,𝑑, then 𝑖(𝐴₁)=𝑖(𝐴₂), 𝜈(𝐴₁)=𝜈(𝐴₂)=0. And hence (1.9) has at least one solution.

Theorem 3.4. In assumption (3.1) if 𝐴(𝑛,𝑧)=𝐴(𝑛) satisfying 𝜈(𝐴)≠0 and (𝑧,ℎ(𝑛,𝑧))≥𝑐₁|𝑧|^𝛼−𝑏₁,|ℎ(𝑛,𝑧)|≤𝑐₂|𝑧|^𝛼−1+𝑏₂, (3.16) where 𝑐₁,𝑐₂,𝑏₁,𝑏₂ are all positive constants and 1≤𝛼<2, then (1.9) has at least one solution.

Proof. From Proposition 2.13, there exists 𝜀>0 such that 𝑖(𝐴+𝜀𝐼_𝑑)=𝑖(𝐴)+𝜈(𝐴) and 𝜈(𝐴+𝜀𝐼_𝑑)=0 for any 𝐴∶ℤ(1,𝑇)→GL_𝑠(ℝ^𝑑). Denote 𝐴₁=𝐴+𝜀𝐼_𝑑, by Lemma 3.2, we only need to prove that the possible solutions to Δ²𝑢(𝑛−1)+𝜆𝐴₁(𝑛)𝑢(𝑛)+(1−𝜆)𝐴(𝑛)𝑢(𝑛)+(1−𝜆)ℎ(𝑛,𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),𝑢(0)=0=𝑢(𝑇+1) (3.17) are priori bounded with respect to the norm ‖⋅‖ in 𝐸. If not, there exist {𝑢_𝑘}⊂𝐸, 𝜆_𝑘∈(0,1) with ‖𝑢_𝑘‖→∞ such that Δ²𝑢_𝑘(𝑛−1)+𝜆_𝑘𝐴₁(𝑛)𝑢_𝑘(𝑛)+(1−𝜆_𝑘)𝐴(𝑛)𝑢_𝑘(𝑛)+(1−𝜆_𝑘)ℎ(𝑛,𝑢_𝑘𝑢(𝑛))=0,𝑛∈ℤ(1,𝑇),_𝑘(0)=0=𝑢_𝑘(𝑇+1). (3.18) Denote 𝑣_𝑘(𝑛)=𝑢_𝑘(𝑛)/‖𝑢_𝑘‖, we may assume that 𝑣_𝑘→𝑣₀ and 𝜆_𝑘→𝜆₀. Hence, 𝑣=𝑣₀ is a nontrivial solution to Δ²𝑣₀(𝑛−1)+𝜆₀𝐴₁(𝑛)𝑣₀(𝑛)+(1−𝜆₀)𝐴(𝑛)𝑣₀𝑣(𝑛)=0,𝑛∈ℤ(1,𝑇),₀(0)=0=𝑣₀(𝑇+1) (3.19) which implies that 𝜈(𝐴+𝜀𝜆₀𝐼_𝑑)≠0. We claim that 𝜆₀=0. If not, 𝜆₀∈(0,1], then 𝐴<𝐴+𝜆₀𝜀𝐼_𝑑≤𝐴+𝜀𝐼_𝑑. From Proposition 2.7, we have 𝑖(𝐴+𝜀𝐼_𝑑)=𝑖(𝐴)+𝜈(𝐴)≤𝑖(𝐴+𝜆₀𝜀𝐼_𝑑) and 𝑖(𝐴+𝜀𝐼_𝑑)=𝑖(𝐴+𝜆₀𝜀𝐼_𝑑). However, from Proposition 2.7, we also have 𝑖(𝐴+𝜆₀𝜀𝐼_𝑑)+𝜈(𝐴+𝜆₀𝜀𝐼_𝑑)≤𝑖(𝐴+𝜀𝐼_𝑑)+𝜈(𝐴+𝜀𝐼_𝑑) which implies that 𝜈(𝐴+𝜆₀𝜀𝐼_𝑑)=0, a contradiction! Hence 𝜆₀=0 and Δ²𝑣₀(𝑛−1)+𝐴(𝑛)𝑣₀𝑣(𝑛)=0,𝑛∈ℤ(1,𝑇),₀(0)=0=𝑣₀(𝑇+1). (3.20) From (3.18), we have ^𝑇_𝑛=1(Δ²𝑢_𝑘(𝑛−1)+𝐴(𝑛)𝑢_𝑘(𝑛),𝑣₀+(n))_𝑇𝑛=1𝜀𝜆_𝑘(𝑢_𝑘(𝑛),𝑣₀(𝑛))+^𝑇_𝑛=1(1−𝜆_𝑘)(ℎ(𝑛,𝑢_𝑘(𝑛)),𝑣₀(𝑛))=0. (3.21) Therefore, from (3.16), (3.20), and (3.21), for 𝑘 large enough, 0≥^𝑇_𝑛=1(ℎ(𝑛,𝑢_𝑘(𝑛)),𝑣₀=(𝑛))_𝑇𝑛=1(ℎ(𝑛,𝑢_𝑘𝑢(𝑛)),^𝑘(𝑛)‖𝑢_𝑘‖)+^𝑇_𝑛=1(ℎ(𝑛,𝑢_𝑘(𝑛)),𝑣₀(𝑛)−𝑣_𝑘(𝑛))≥‖𝑢_𝑘‖^𝑇−1_𝑛=1(𝑐₁|𝑢_𝑘(𝑛)|^𝛼−𝑏₁)−‖𝑣₀−𝑣_𝑘‖_∞^𝑇_𝑛=1(𝑐₂|𝑢_𝑘(𝑛)|^𝛼−1+𝑏₂). (3.22) Dividing ‖𝑢_𝑘‖^𝛼−1 at both sides, we have 0≥^𝑇_𝑛=1(𝑐₁|𝑣_𝑘(𝑛)|^𝛼−𝑏₁‖𝑢_𝑘‖^−𝛼)−‖𝑣₀−𝑣_𝑘‖_∞^𝑇_𝑛=1(𝑐₂|𝑣_𝑘(𝑛)|^𝛼−1+𝑏₂‖𝑢_𝑘‖^1−𝛼)→𝑐₁^𝑇_𝑛=1|𝑣₀(𝑛)|^𝛼. (3.23) This is a contradiction since ‖𝑣₀‖≠0 and 𝑐₁>0. The proof is complete.

If ∇𝑉(𝑛,0)≡0, then 𝑢≡0 is a solution to (1.9). As usual we call this solution the trivial solution. It is much regretted that we do not know if the solution we found is not the trivial one in Theorems 3.1 and 3.4. In the following, we will obtain the existence of nontrivial solutions to (1.9) by using Morse theory.

Theorem 3.5. Assume the following (1)𝑉∶ℤ(1,𝑇)×ℝ^𝑑→ℝ is 𝐶² with respect to the second variable, and ∇𝑉(𝑛,𝑧)=𝐴₁(𝑛)𝑧+𝑜(|𝑧|),as|𝑧|→∞ (3.24) for every 𝑛∈ℤ(1,𝑇) with 𝜈(𝐴₁)=0.(2)∇𝑉(𝑛,0)≡0,𝐴₀(𝑛)=𝑉^′′(𝑛,0), and 𝑖(𝐴₁)∉ℤ(𝑖(𝐴₀),𝑖(𝐴₀)+𝜈(𝐴₀)). Then (1.9) has at least one nontrivial solution. (3)Moreover, if we further assume that 𝜈(𝐴₀)=0, |𝑖(𝐴₁)−𝑖(𝐴₀)|≥𝑑, then (1.9) has at least two nontrivial solutions.

To prove Theorem 3.5, we need some results on Morse theory. Let 𝐸 be a real Hilbert space and 𝑓∈𝐶¹(𝐸,ℝ). As in [2], denote 𝑓^𝑐={𝑢∈𝐸∣𝑓(𝑢)≤𝑐},𝒦_𝑐={𝑢∈𝐸∣𝑓^′(𝑢)=0,𝑓(𝑢)=𝑐} (3.25) for 𝑐∈ℝ. The following is the definition of the Palais-Smale condition (the (PS) condition for short).

Definition 3.6. The functional 𝑓 satisfies the (PS) condition if any sequence {𝑢_𝑚}⊂𝐸 such that {𝑓(𝑢_𝑚)} is bounded and 𝑓^′(𝑢_𝑚)→0 as 𝑚→∞ has a convergent subsequence.

Let 𝑢₀ be an isolated critical point of 𝑓 with 𝑓(𝑢₀)=𝑐∈ℝ, and let 𝑈 be a neighborhood of 𝑢₀; the group 𝐶_𝑞(𝑓,𝑢₀)∶=𝐻_𝑞(𝑓^𝑐∩𝑈,𝑓^𝑐∩𝑈⧵{𝑢₀}),𝑞=1,2,… (3.26) is called the 𝑞th critical group of 𝑓 at 𝑢₀, where 𝐻_𝑞(𝐴,𝐵) denotes 𝑞th singular relative homology group of the pair (𝐴,𝐵) over a field 𝐹, which is defined to be quotient 𝐻_𝑞(𝐴,𝐵)=𝑍_𝑞(𝐴,𝐵)/𝐵_𝑞(𝐴,𝐵), where 𝑍_𝑞(𝐴,𝐵) is the 𝑞th singular relative closed chain group and 𝐵_𝑞(𝐴,𝐵) is the 𝑞th singular relative boundary chain group.

For any two regular values 𝑎<𝑏, if 𝒦∩𝑓⁻¹[𝑎,𝑏]={𝑢₁,𝑢₂,…,𝑢_𝑙}, denote 𝑀_𝑞=∑^𝑙_𝑖=1dim𝐶_𝑞(𝑓,𝑢_𝑖) and 𝛽_𝑞=dim𝐻_𝑞(𝑓^𝑏,𝑓^𝑎). The following results play an important role in proving Theorem 3.5; see [2] for the detailed proof.

Lemma 3.7. Assume that 𝑓∈𝐶²(𝐸,ℝ) satisfies the (PS) condition. Then one has the following Morse inequalities: 𝑀_𝑞−𝑀_𝑞−1+⋯+(−1)^𝑞𝑀₀≥𝛽_𝑞−𝛽_𝑞−1+⋯+(−1)^𝑞𝛽₀ (3.27) for 𝑞=0,1,2,…. One also has the following Morse equality: ^∞_𝑞=0𝑀_𝑞𝑡^𝑞=^∞_𝑞=0𝛽_𝑞𝑡^𝑞+(1+𝑡)𝑄(𝑡), (3.28) where 𝑄(𝑡) is a polynomial with nonnegative integer coefficients.

Lemma 3.8. Assume that 𝑓∈𝐶²(𝐸,ℝ) and that 𝑢₀ is an isolated critical point of 𝑓 with finite Morse index 𝜇(𝑢₀) and nullity 𝜈(𝑢₀). Thenone has the following. (1)For any 𝑞∉ℤ(𝜇(𝑢₀),𝜇(𝑢₀)+𝜈(𝑢₀)), 𝐶_𝑞(𝑓,𝑢₀)≅0.(2)If 𝑢₀ is a nondegenerate critical point, then 𝐶_𝑞(𝑓,𝑢₀)≅𝛿_{𝑞,𝜇(𝑢0)}𝐹,𝑞=0,1,2,…. (3.29)(3)If 𝑓 has a minimal value at 𝑢₀, then 𝐶_𝑞(𝑓,𝑥₀)≅𝛿_𝑞,0𝐹,𝑞=0,1,2,…. (3.30)(4)If there exist integers 𝑞₁≠𝑞₂ such that 𝐶_𝑞1(𝑓,𝑢₀)≇0 and 𝐶_𝑞2(𝑓,𝑢₀)≇0, then |𝑞₁−𝑞₂|≤𝜈(𝑢₀)−2.

Now, Define 1𝑓(𝑢)=2^𝑇_𝑛=0|Δ𝑢(𝑛)|²−^𝑇_𝑛=1𝑉(𝑛,𝑢(𝑛)),𝑢∈𝐸. (3.31) Then the functional 𝑓 is 𝐶² with ⟨𝑓^′(𝑢),𝑣⟩=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛))−^𝑇_𝑛=1(∇𝑉(𝑛,𝑢(𝑛)),𝑣(𝑛))=−^𝑇_𝑛=1(Δ²𝑢(𝑛−1)+∇𝑉(𝑛,𝑢(𝑛)),𝑣(𝑛)),∀𝑢,𝑣∈𝐸. (3.32) So solutions to (1.9) are precisely the critical points of 𝑓.

Lemma 3.9. Under assumptions of Theorem 3.5, there exist 𝑅₀>0,𝑅₁>0 and 𝑓 satisfying the following conditions. (1)𝑓^′(𝑢)=0 implies ‖𝑢‖≤𝑅₀.(2)𝑓 and 𝑓 have the same critical set.(3)If ‖𝑢‖≥𝑅₁, then 𝑓(𝑢)=(1/2)⟨𝐿𝑢,𝑢⟩.

Proof. Define 𝐿 and 𝑔 on 𝐸 as ⟨𝐿𝑢,𝑣⟩=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛))−^𝑇_𝑛=1(𝐴₁1(𝑛)𝑢(𝑛),𝑣(𝑛)),𝑔(𝑢)=𝑓(𝑢)−2⟨𝐿𝑢,𝑢⟩. (3.33) Then ⟨𝑓^′(𝑢),𝑣⟩=⟨𝐿𝑢,𝑣⟩+⟨𝑔^′(𝑢),𝑣⟩. (3.34) Assumption 𝜈(𝐴₁)=0 implies that 𝐿 is invertible. Taking 𝜀₁=‖𝐿⁻¹‖⁻¹/2, there exists 𝑅₀>0 such that if ‖𝑢‖>𝑅₀, then ‖𝑔^′(𝑢)‖<𝜀₁‖𝑢‖. (3.35) So, as ‖𝑢‖>𝑅₀, we have ‖𝑓^′(𝑢)‖≥‖𝐿⁻¹‖⁻¹‖𝑢‖−‖𝑔^′(𝑢)‖>𝜀₁‖𝑢‖, (3.36) that is, no critical point is outside the ball 𝐵_𝑅0.
To prove (2) and (3), let 𝜌∈𝐶^∞(ℝ,[0,1]) satisfy𝜌(𝑡)=1,𝑡≤0,0,𝑡≥1 (3.37) with 0≤𝜌(𝑡)≤1 and max|𝜌^′(𝑡)|≤3/2. Let 𝜀=‖𝐿⁻¹‖⁻¹/5, then there exists 𝑐₁≥0 such that ‖𝑔^′(𝑢)‖<𝜀‖𝑢‖+𝑐₁. (3.38) Hence, |𝑔(𝑢)|≤¹₀|⟨𝑔^′≤(𝑠𝑢),𝑢⟩|𝑑𝑠+|𝑔(0)|₁₀(𝜀𝑠‖𝑢‖²+𝑐₁=𝜀‖𝑢‖)𝑑𝑠+|𝑔(0)|2‖𝑢‖²+𝑐₁‖𝑢‖+|𝑔(0)|. (3.39) Therefore, there exists 𝑐₂>0 such that |𝑔(𝑢)|≤𝜀‖𝑢‖²+𝑐₂,∀𝑢∈𝐸. (3.40) Define 𝑅>max{1,𝑅₀,𝑐₁+3/2𝜖}, (3.41) where 𝑅₀ is defined above and 𝜆=max{𝑐₂,1}⋅𝑅. (3.42) If 𝑅≤‖𝑢‖≤𝜆+𝑅, then 𝑐₂𝜆<1,‖𝑢‖𝜆≤2. (3.43) Let 𝜌_𝜆(𝑡)=𝜌(𝑡−𝑅𝜆), (3.44) and define 1𝑓(𝑢)=2⟨𝐿𝑢,𝑢⟩+𝜌_𝜆(‖𝑢‖)𝑔(𝑢). (3.45) Then the function 𝑓(𝑢) satisfies (2) and (3). In fact, 𝑓(𝑢)=1𝑓(𝑢),‖𝑢‖≤𝑅,2⟨𝐿𝑢,𝑢⟩,‖𝑢‖≥𝜆+𝑅. (3.46) The only thing we have to check is that 𝑓^′(𝑢)≠0 as 𝑅≤‖𝑢‖≤𝜆+𝑅. However, ‖𝑓^′1(𝑢)‖=‖𝐿𝑢+𝜆𝜌^′(‖𝑢‖−𝑅𝜆𝑢)𝑔(𝑢)‖𝑢‖+𝜌(‖𝑢‖−𝑅𝜆)𝑔^′(𝑢)‖≥‖𝐿⁻¹‖⁻¹3‖𝑢‖−2𝜆(𝜀‖𝑢‖²+𝑐₂)−(𝜀‖𝑢‖+𝑐₁)=5𝜀‖𝑢‖−𝜀‖𝑢‖−3𝜀2𝜆‖𝑢‖²−(𝑐₁+3𝑐₂)2𝜆≥𝜀‖𝑢‖−(𝑐₁+32)>0. (3.47) Hence, let 𝑅₁=𝜆+𝑅; the proof is completed.

From Lemma 3.9, 𝑓^′(𝑢)=0 if and only if 𝑓^′(𝑢)=0. Thus, in order to find solutions to (1.9), it suffices to find the critical points of 𝑓. Moreover, 𝑓 satisfies the (PS) condition by Lemma 3.9.

Lemma 3.10. Under the assumptions of Theorem 3.5, there exist 𝑎,𝑏 with 𝑎<𝑏 such that the critical points of 𝑓 belong to 𝑓⁻¹((𝑎,𝑏))∶={𝑢∣𝑎<𝑓(𝑢)<𝑏} and 𝐻_𝑞(𝑓^𝑏,𝑓^𝑎)≅𝛿_{𝑞,𝑖(𝐴1)}𝐹. (3.48)

Proof. Define 𝑎<min_𝐵12𝑅𝑓−1,𝑏>max_𝐵12𝑅𝑓+1,𝑓₁1(𝑢)=2⟨𝐿𝑢,𝑢⟩, (3.49) where 𝑎,𝑏 are finite. Noticing that all critical points of 𝑓 lie in 𝐵_𝑅0, if 𝑢 is a critical point of 𝑓, then 𝑎<min_𝐵𝑅0𝑓≤𝑓(𝑢)≤max_𝐵𝑅0𝑓<𝑏. (3.50) This implies that {𝑢∣𝑎<𝑓(𝑢)<𝑏} contains all critical points of 𝑓. By the properties of the raltive singular homology group, we have 𝐻_𝑞(𝑓^𝑏,𝑓^𝑎)≅𝐻_𝑞(𝑓^𝑏₁,𝑓^𝑎₁). However, 𝜈(𝐴₁)=0 implies that 𝑓₁ has only critical point 0 with Morse index 𝑖(𝐴₁). From Lemma 3.8 the conclusion holds.

Proof of Theorem 3.5. (1) By Lemma 3.10 and the Morse inequalities, 𝑓 must have a critical point 𝑢 with 𝐶_𝑖(𝐴1)(𝑓,𝑢)≇0. Since 𝑖(𝐴₁)∉ℤ(𝑖(𝐴₀),𝑖(𝐴₀)+𝜈(𝐴₀)), then from Lemma 3.8, we have 𝐶_𝑖(𝐴1)(𝑓,0)≅0. (3.51) And hence 𝑢≠0 is a critical point of 𝑓; that is, (1.9) has at least one nontrivial solution.
(2) Since 𝜈(𝐴₀)=0, we have𝐶_𝑞(𝑓,0)≅𝛿_{𝑞,𝑖(𝐴0)}𝐹,𝑞=0,1,2,…. (3.52) Now we assume that |𝑖(𝐴₁)−𝑖(𝐴₀)|≥𝑑 and that 𝑢 is the only nonzero critical point of 𝑓. Then from Morse equality, we have 𝑡^𝑖(𝐴0)+^∞_𝑞=0dim𝐶_𝑞(𝑓,𝑢)𝑡^𝑞=𝑡^𝑖(𝐴1)+(1+𝑡)𝑄(𝑡). (3.53) We necessarily have dim𝐶_𝑖(𝐴1)(𝑓,𝑢)≥1, and eitherdim𝐶_{𝑖(𝐴0)−1}(𝑓,𝑢)≥1,ordim𝐶_{𝑖(𝐴0)+1}(𝑓,𝑢)≥1. (3.54)(i)If dim𝐶_{𝑖(𝐴0)−1}(𝑓,𝑢)≥1, then by assumption we have 𝑖(𝐴₀)−1≠𝑖(𝐴₁). Since the nullity of 𝑢 is less or equal to 𝑑, from Lemma 3.8, we have |𝑖(𝐴₀)−1−𝑖(𝐴₁)|≤𝑑−2 (3.55) which implies that |𝑖(𝐴₀)−𝑖(𝐴₁)|≤𝑑−1. This is impossible since |𝑖(𝐴₀)−𝑖(𝐴₁)|≥𝑑.(ii)If dim𝐶_{𝑖(𝐴0)+1}(𝑓,𝑢)≥1, then similar to the above proof we have |𝑖(𝐴₀)−𝑖(𝐴₁)|≤𝑑−1, also a contradiction.
Therefore, 𝑓 has at least two nonzero critical points and hence (1.9) has at least two nontrivial solutions.

Acknowledgments

The author would like to express her thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project (2009).