# Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems

Authors: Bo Zheng
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## 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.


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## Body

## 1. Introduction

Letℕ,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. Aτ or xτ 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 BVPs(1.1)Δ2u(n-1)+f(n,u(n))=0,n∈ℤ(1,T),u(0)=A,u(T+1)=B
often appear in electrical circuit analysis, mathematical physics, finite elasticity, and so forth as the mathematical models, where f:ℤ(1,T)×ℝd→ℝd 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:(1.2)u′′+f(t,u)=0,a≤t≤b,u(a)=A,u(b)=B,
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(1.3)y(n)=Bn-A(n-T-1)T+1,z(n)=u(n)-y(n).
Then (1.1) reduces to (1.4)Δ2z(n-1)+f(n,z(n)+y(n))=0,n∈ℤ(1,T),z(0)=0=z(T+1).
Hence, in the following, we can only consider the discrete conjugate BVPs, that is,(1.5)Δ2u(n-1)+f(n,u(n))=0,n∈ℤ(1,T),u(0)=0=u(T+1).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 (1.6)u(n)=∑m=1TG(n,m)f(m,u(m))+w(n),
where G(n,m) is Green's function for (1.7)Δ2u(n-1)=0,n∈ℤ(1,T),u(0)=0=u(T+1),w(n)=A+B-AT+1n.
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 (1.8)𝒯u(n)=∑m=1TG(n,m)f(m,u(m))+w(n)
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 casef(n,u(n))=∇V(n,u(n)) that is, we consider the following discrete conjugate BVPs:(1.9)Δ2u(n-1)+∇V(n,u(n))=0,n∈ℤ(1,T),u(0)=0=u(T+1),
where V(n,·)∈C1(ℝd,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(1.10)∇V(n,z)=A(n,z)z+o(|z|)
as |z|→∞, where A:ℤ(1,T)×ℝd→GLs(ℝd), and(1.11)A1(n)≤A(n,z)≤A2(n)
for every n∈ℤ(1,T), and z∈ℝd, A1,A2:ℤ(1,T)→GLs(ℝd), GLs(ℝd) denotes the group of d×d real nonsingular symmetric matrices, and |z| denotes the Euclidean norm of z in ℝd. Throughout this paper, for any A1,A2∈GLs(ℝd), 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)→GLs(ℝd), 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)}≠∅.IfA(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 Section2, firstly, we classify the linear systems(1.12)Δ2u(n-1)+A(n)u(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1)
for every A:ℤ(1,T)→GLs(ℝd). 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: (2.1)E={u∣u=(u(0),u(1),…,u(T),u(T+1))τ,u(0)=0=u(T+1)},
where u(n)=(u1(n),u2(n),…,ud(n))τ∈ℝd for every n∈ℤ(0,T+1). Define the inner product on E as follows: (2.2)〈u,v〉=∑n=0T(Δu(n),Δv(n)),∀u,v∈E,
by which the norm ∥·∥ on E can be induced by (2.3)∥u∥=(∑n=0T|Δu(n)|2)1/2,∀u∈E,
where (·,·) is the usual inner product on ℝd, and |·| is the usual norm on ℝd.Define a linear mapΓ:E→ℝdT by(2.4)Γu=(u1(1),u1(2),…,u1(T),u2(1),u2(2),…,u2(T),…,ud(1),ud(2),…,ud(T))τ.
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 ℝdT.Define(2.5)qA(u,v)=∑n=0T(Δu(n),Δv(n))-∑n=1T(A(n)u(n),v(n)),∀u,v∈E.
For any u,v∈E, if qA(u,v)=0, we say that u and v are qA orthogonal. For any two subspaces E1 and E2 of E, if qA(u,v)=0 for any u∈E1 and v∈E2, we say that E1 and E2 are qA orthogonal.For any subspaceE1 of E, we say that qA is positive definite (or negative definite) on E1 if qA(u,u)>0 (or qA(u,u)<0) for all u∈E1∖{0}. And if qA(u,u)=0 for all u∈E1, then E1 is called a null subspace of E.Proposition 2.1.
For anyA:ℤ(1,T)→GLs(ℝd), the following results hold.(1)
There are{λi(A)}i=1m⊂ℝ with λ1(A)<λ2(A)<⋯<λm(A) such that
(2.6)Δ2u(n-1)+(A(n)+λi(A)Id)u(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1)
has a nontrivial solution. If Ei(A) denotes the subspace of solutions with respect to λi(A), then dimEi(A):=ni≤d and E=⨁i=1mEi(A).(2)
The spaceE has a qA orthogonal decomposition
(2.7)E=E+(A)⨁E0(A)⨁E-(A)
such that qA is positive definite, negative definite, and null on E+(A), E-(A), and E0(A), respectively.To prove Proposition2.1, we need the following lemma.Lemma 2.2.
For anyu={u(n)}n=0T+1∈E, the following inequalities hold. 							(2.8)4sin2π2(T+1)∑n=1T|u(n)|2≤∑n=0T|Δu(n)|2≤4cos2π2(T+1)∑n=1T|u(n)|2.Proof.
Note that(2.9)∑n=0T(Δu(n),Δu(n))=∑n=1T2(u(n),u(n))-∑n=1T-12(u(n),u(n+1))=(AΓu,Γu),
where
(2.10)A=(B0⋯00B⋯0⋯⋯⋯⋯00⋯B)dT×dT,B=(2-10⋯00-12-1⋯000-12⋯00⋯⋯⋯⋯⋯⋯000⋯2-1000⋯-12)T×T.
Assume that λ is an eigenvalue of B and that ξ=(ξ1,ξ2,…,ξT)τ is an eigenvector associated to λ. Define the sequence {v(n)}n=0T+1 as
(2.11)v(i)=ξi,i=1,2,…,T,v(0)=0=v(T+1).
Then {v(n)}n=0T+1 satisfies
(2.12)Δ2v(n-1)+λv(n)=0,n∈ℤ(1,T),v(0)=0=v(T+1).
Equation (2.12) has a nontrivial solution if and only if
(2.13)λ=λk=4sin2kπ2(T+1),k=1,2,…,T;
see [33]. So σ(A)=σ(B)={λ1,λ2,…,λT} with λ1<λ2<⋯<λT and
(2.14)λmin=min{λ1,λ2,…,λT}=4sin2π2(T+1)=λ1,λmax=max{λ1,λ2,…,λT}=4cos2π2(T+1)=λT.
Noticing that for any real symmetric dT×dT matrix A, we have
(2.15)λ1(Γu,Γu)≤(AΓu,Γu)≤λT(Γu,Γu),∀u∈ℝdT.
Since (Γu,Γu)=∑n=1T|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
(2.16)λ1|u|2≤∥u∥2≤λT|u|2
for simplicity.Proof of Proposition2.1.
(1) We claim that the norm∥·∥λ0 induced by the inner product
(2.17)(u,v)λ0:=∑n=0T(Δu(n),Δv(n))+∑n=1T((λ0Id-A(n))u(n),v(n)),∀u,v∈E
is equivalent to ∥·∥, where λ0 is a positive number satisfying λ0Id>A. In fact, it is easy to see that there exists c∈(0,+∞) such that
(2.18)0≤∑n=1T((λ0Id-A(n))u(n),u(n))≤c∑n=1T|u(n)|2≤cλ1∑n=0T|Δu(n)|2=cλ1∥u∥2.
Hence
(2.19)∥u∥2≤∥u∥λ02≤(1+cλ1)∥u∥2.
Define a bilinear function
(2.20)a(u,v)=∑n=1T(u(n),v(n)),∀u,v∈E,
and then
(2.21)|a(u,v)|≤(∑n=1T|u(n)|2)1/2(∑n=1T|v(n)|2)1/2≤1λ1∥u∥∥v∥≤1λ1∥u∥λ0∥v∥λ0.
Hence, there exists a unique continuous linear operator Kλ0:E→E satisfying
(2.22)∑n=1T(u(n),v(n))=(u,Kλ0v)λ0.
It is easy to see that Kλ0 is self-adjoint, and hence all the eigenvalues of Kλ0 are real. Therefore, there exist μi, i=1,2,…,m and eij, j=1,2,…,ni such that
(2.23)(eij,elk)λ0={1,i=landj=k,0,i≠lorj≠k,Kλ0eij=μieij,
where ni is the multiplicity of μi with ∑ini=dT. By (2.22) and (2.23) we have
(2.24)μi(eij,u)λ0=∑n=1T(eij(n),u(n)),∀u∈E.
In particular, μi=∑n=1T|eij(n)|2>0. Without loss of generality we assume that μi is strictly monotonously decreasing, that is, μ1>μ2>⋯>μm. Denote λi(A)=1/μi-λ0 and Ei(A)=span{eij}j=1ni, then E=⨁i=1mEi(A). We claim that for every λi(A), eij={eij(n)}n=0T+1∈E is a nontrivial solution of (2.6). In fact, by (2.24), for any u∈E, we have
(2.25)μi∑n=0T(Δeij(n),Δu(n))+μi∑n=1T((λ0Id-A(n))eij(n),u(n))=∑n=1T(eij(n),u(n)),
and since μi>0, the above equality means
(2.26)∑n=1T(Δ2eij(n-1)+(A(n)+1μiId-λ0Id)eij(n),u(n))=0,∀u∈E.
Therefore eij satisfies (2.6). Now, we have proved the first result of Proposition 2.1 except dimEi(A)=ni≤d.
Setu(n)=y1(n),Δu(n-1)=-y2(n); then (2.6) is equivalent to(2.27)Δy1(n)=-y2(n+1),n∈ℤ(0,T),Δy2(n)=(A(n)+λi(A)Id)y1(n),n∈ℤ(1,T),y1(0)=0=y1(T+1),
which is also equivalent to
(2.28)y(n+1)=B(n)y(n),n∈ℤ(1,T),y1(0)=0=y1(T+1),
where
(2.29)y(n)=(y1(n)y2(n)),B(n)=(Id-λi(A)Id-A(n)-Idλi(A)Id+A(n)Id).
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)=I2d. The general solution of y(n+1)=B(n)y(n) can be given by y(n)=Φ(n)c, where c=(c1c2)∈ℝ2d and ci∈ℝd,i=1,2. Set
(2.30)Φ(n)=(Φ11(n)Φ12(n)Φ21(n)Φ22(n)),whereΦij(n)isd×dmatrix,i,j=1,2,
then y1(n)=Φ11(n)c1+Φ12(n)c2. By y1(0)=0=y1(T+1) and Φ(0)=I2d, we have c1=0 and
(2.31)Ei(A)≅{c2∈ℝd∣Φ12(T+1)c2=0}⊆ℝd.
Hence, dimEi(A)=ni≤d.
(2) For anyu∈E with u=∑i,jcijeij, by (2.23) and (2.24), we have(2.32)qA(u,u)=(u,u)λ0-λ0(u,Kλ0u)λ0=∑i,j(|cij|2-λ0μi|cij|2)=∑i,jλi(A)μi|cij|2.
Hence, if we denote
(2.33)E+(A)={u=∑cijeij∣cij=0,ifλi(A)≤0},E0(A)={u=∑cijeij∣cij=0,ifλi(A)≠0},E-(A)={u=∑cijeij∣cij=0,ifλi(A)≥0},
then the results hold.Definition 2.4.
For anyA:ℤ(1,T)→GLs(ℝd), 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 anyA:ℤ(1,T)→GLs(ℝd), 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)<0ni, where λi and ni are defined in the proof of Proposition 2.1.Proof.
(1) By Proposition2.5, if λi(A)≠0 for any i, then (1.12) has only a trivial solution, and hence E0(A)={0}, ν(A)=0. If λi(A)=0 for some i with multiplicity ni, then by the proof of Proposition 2.1, Ei(A) is the solution subspace of (1.12) and ν(A):=dimEi(A)∈ℤ(1,d).
(2) By the proof of Proposition2.1, E-(A)=⨁λi(A)<0Ei(A), and Ei(A) and Ej(A) are qA orthogonal if i≠j. Hence the result holds.Remark 2.6.
By (1) of Proposition2.5, ν(A)≥0, and ν(A)=0 if and only if λi(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 anyA1,A2:ℤ(1,T)→GLs(ℝd), the following results hold.(1)
IfA1≤A2, then i(A1)≤i(A2).(2)
IfA1≤A2, then i(A1)+ν(A1)≤i(A2)+ν(A2).(3)
IfA1<A2, then i(A1)+ν(A1)≤i(A2).To prove Proposition2.7, we firstly prove the following lemma.Lemma 2.8.
LetE1 be a subspace of E satisfying
(2.34)qA(u,u)≤0,∀u∈E1,
and then
(2.35)dimE1≤i(A)+ν(A).
Moreover, if
(2.36)qA(u,u)<0,∀u∈E1∖{0},
then
(2.37)dimE1≤i(A).Proof.
Without loss of generality we can assume thatdimE1=k≥1 and E1=span{e1,e2,…,ek}. Let ei=ei++ei*, where ei*∈E-(A)⊕E0(A), ei+∈E+(A). To prove i(A)+ν(A)≥k, we only need to prove that e1*,e2*,…,ek* is linear independent. If not, there exist not all zero constants α1,α2,…,αk such that ∑i=1kαiei*=0. So e:=∑i=1kαiei=∑i=1kαiei+∈E+(A), and hence qA(e,e)>0. This is a contradiction to (2.34). This implies that e1*,e2*,…,ek* 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, letei=ei++ei-+ei0, where ei*∈E*(A), *=+,0,-. To prove i(A)≥k, we only need to prove that e1-,e2-,…,ek- is linear independent. If not, there exist not all zero constants c1,c2,…,ck such that ∑i=1kciei-=0. So, e:=∑i=1kciei=∑i=1k(ciei++ciei0)∈E+(A)⊕E0(A) and qA(e,e)≥0. This is a contradiction to (2.36).Proof of Proposition2.7.
For anyu∈E, denote u=u++u0+u-, where u*∈E*(A1), *=+,-,0.(1)
From Lemma2.8, we only need to show that
(2.38)qA2(u,u)<0,∀u∈E-(A1)∖{0}.
In fact, for every u=u-∈E-(A1), if u-≠0, then
(2.39)qA2(u,u)≤qA1(u,u)=qA1(u-,u-)<0.(2)
From Lemma2.8, we only need to show that
(2.40)qA2(u,u)≤0,∀u∈E-(A1)⊕E0(A1).
In fact, for every u=u-+u0∈E-(A1)⊕E0(A1), one has
(2.41)qA2(u,u)≤qA1(u,u)=qA1(u0,u0)+qA1(u-,u-)≤0.(3)
From Lemma2.8, we only need to show that
(2.42)qA2(u,u)<0,∀u∈E-(A1)⊕E0(A1)∖{0}.
In fact, for every u=u-+u0 with u-∈E-(A1), u0∈E0(A1), if u-≠0, then
(2.43)qA2(u,u)≤qA1(u,u)=qA1(u0,u0)+qA1(u-,u-)<0.
If u-=0, u0≠0, then {u0(n)}n=0T+1 is a nontrivial solution of
(2.44)Δ2u(n-1)+A1(n)u(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1).
From A1<A2 we have
(2.45)qA2(u0,u0)<qA1(u0,u0)=0.
Hence (2.42) holds.Proposition 2.9.
IfU∈O(d), that is, U∈GLs(ℝd) and UτU=Id, then for any A:ℤ(1,T)→GLs(ℝd), i(UτAU)=i(A),ν(UτAU)=ν(A). In particular, for any A∈GLs(ℝd), we have
(2.46)i(A)=∑i=1d♯{k∈ℤ(1,T)∣λk<ai},ν(A)=∑i=1d♯{k∈ℤ(1,T)∣λk=ai},
where λk is given by (2.13), and {ai}i=1d=σ(A) is the set of eigenvalues of A.Proof.
Firstly, we claim that(2.47)λi(UτAU)=λi(A).
In fact, since λ0Id>A if and only if λ0Id>UτAU, we can choose λ0(UτAU)=λ0(A). By (2.22) and (2.23), it is easy to see that μi(UτAU)=μi(A), and hence (2.47) holds. Therefore, by Definition 2.4,
(2.48)i(UτAU)=i(A),ν(UτAU)=ν(A).
Since
(2.49)E*(diag{A1,A2})≅E*(A1)⊕E*(A2),*=+,0,-,
we have
(2.50)i(diag{A1,A2})=i(A1)+i(A2),ν(diag{A1,A2})=ν(A1)+ν(A2).
Note that the scalar eigenvalue problem
(2.51)Δ2y(n-1)+λy(n)=0,n∈ℤ(1,T),y(0)=0=y(T+1)
has a nontrivial solution if and only if λ=λk=4sin2(kπ/2(T+1)), k∈ℤ(1,T). By Proposition 2.1 and Definition 2.4, we see that for any α∈ℝ,
(2.52)i(α)=♯{k∈ℤ(1,T)∣λk<α},ν(α)=♯{k∈ℤ(1,T)∣λk=α}.
Since {ai}i=1d is the set of eigenvalues of A, there exists an orthogonal matrix U such that
(2.53)UτAU=diag{a1,a2,…,ad}.
From (2.48), (2.50), and (2.52) we have
(2.54)i(A)=∑i=1d♯{k∈ℤ(1,T)∣λk<ai},ν(A)=∑i=1d♯{k∈ℤ(1,T)∣λk=ai}.
This completes the proof.Proposition 2.10.
For anyA:ℤ(1,T)→GLs(ℝd) with i(A)=0, one has
(2.55)∑n=0T|Δu(n)|2≥∑n=1T(A(n)u(n),u(n)),∀u∈E.
And the equality holds if and only if u∈E0(A).Proof.
For anyu∈E with u=∑i,jcijeij, we have
(2.56)∑n=0T|Δu(n)|2=∑n=1T(A(n)u(n),u(n))+∑i,jλi(A)μi|cij|2.
Because i(A)=0, by definition, λi(A)≥0 for any i. So the inequality holds. And the equality holds if and only if cij=0 as λi(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 numberi(A2)-i(A1) as A2>A1.Definition 2.11.
For anyA1,A2:ℤ(1,T)→GLs(ℝd) with A1<A2, define
(2.57)I(A1,A2)=∑λ∈[0,1)ν(A1+λ(A2-A1)).IfA1=α1Id, A2=α2Id, where α1<α2 are two real numbers, then by Proposition 2.9, we have (2.58)I(α1Id,α2Id)=∑λ∈[0,1)ν(α1Id+λ(α2-α1)Id)=d♯{k∈ℤ(1,T)∣λk∈[α1,α2)},i(α1Id)=d♯{k∈ℤ(1,T)∣λk<α1},i(α2Id)=d♯{k∈ℤ(1,T)∣λk<α2}.So(2.59)I(α1Id,α2Id)=i(α2Id)-i(α1Id).
This gives us a steer toward the following result.Proposition 2.12.
For anyA1,A2:ℤ(1,T)→GLs(ℝd) with A1<A2, one has
(2.60)I(A1,A2)=i(A2)-i(A1).Proof.
Denotei(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
(2.61)i(λ+0)=i(λ)+ν(λ),∀λ∈[0,1),(2.62)i(λ-0)=i(λ),∀λ∈(0,1]
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,
(2.63)i(λ1)-i(0)=∑λ∈[0,λ1)ν(λ).
Hence, if we choose λ1=1, then (2.60) holds.
From (3) of Proposition2.7, 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 prove(2.64)m+(λ)≤m+(λ+0)+ν(λ+0).
Similar to the proof of Lemma 2.8, it is easy to know that for ε>0 sufficiently small, if
(2.65)qBλ+ε(u,u)≥0,∀u∈E+(λ),
then (2.64) holds, where E+(λ)=E+(A1+λ(A2-A1)), Bλ+ε(n)=A1(n)+(λ+ε)(A2(n)-A1(n)). While as ε>0 is sufficiently small and u∈E+(λ), we have
(2.66)qBλ+ε(u,u)=qBλ(u,u)-ε∑n=1T((A2(n)-A1(n))u(n),u(n))=qBλ(u+,u+)-ε∑n=1T((A2(n)-A1(n))u(n),u(n))≥0,
where Bλ(n)=A1(n)+λ(A2(n)-A1(n)). Hence i(λ+0)≤i(λ)+ν(λ).
On the other hand, from (1) of Proposition2.7, 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 prove(2.67)qBλ-ε(u,u)<0,∀u∈E-(λ)∖{0},
where ε>0 is sufficiently small, Bλ-ε(n)=A1(n)+(λ-ε)(A2(n)-A1(n)). And as ε>0 is sufficiently small, we have
(2.68)qBλ-ε(u,u)=qBλ(u,u)+ε∑n=1T((A2(n)-A1(n))u(n),u(n))=qBλ(u-,u-)+ε∑n=1T((A2(n)-A1(n))u(n),u(n))<0,
where Bλ(n)=A1(n)+λ(A2(n)-A1(n)). This completes the proof.Proposition 2.13.
For anyA:ℤ(1,T)→GLs(ℝd), there exists ε0>0 such that for any ε∈(0,ε0], one has
(2.69)ν(A+εId)=0,(2.70)ν(A-εId)=0,(2.71)i(A-εId)=i(A),(2.72)i(A+εId)=i(A)+ν(A).Proof.
From Proposition2.12 we have i(A+Id)=i(A)+I(A,A+Id). From Definition 2.11 and Proposition 2.12, we know that I(A,A+Id)=∑λ∈[0,1)ν(A+λId) is finite, and then there must exist some ε0>0 such that for any ε∈(0,ε0], ν(A+εId)=0 and
(2.73)i(A+εId)=i(A)+∑λ∈[0,1)ν(A+λεId)=i(A)+ν(A).
This proves (2.69) and (2.72).
To prove (2.70) and (2.71), note that I(A-Id,A)=i(A)-i(A-Id) and(2.74)I(A-Id,A)=∑λ∈[0,1)ν(A-(1-λ)Id)=∑λ∈(0,1]ν(A-λId).
Since I(A-Id,A) is finite, there exists ε0>0 such that for any ε∈(0,ε0], ν(A-εId)=0. Hence
(2.75)i(A-εId)=i(A)-I(A-εId,A)=i(A)-∑λ∈(0,1]ν(A-λεId)=i(A).
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 existA:ℤ(1,T)×ℝd→GLs(ℝd) and h:ℤ(1,T)×ℝd→ℝd 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
(3.1)∇V(n,z)=A(n,z)z+h(n,z);(2)
there existA1,A2:ℤ(1,T)→GLs(ℝd) satisfying A1≤A2, i(A1)=i(A2), ν(A2)=0 such that
(3.2)A1(n)≤A(n,z)≤A2(n),∀z∈ℝd
for every n∈ℤ(1,T). Then (1.9) has at least one solution.To prove Theorem3.1, 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 BR in X, where
(3.3)BR={x∣x∈X,∥x∥X≤R},R=sup{∥x∥X∣x=λΦx,0<λ<1}.Proof of Theorem3.1.
Assume that (3.2) holds. Since ν(A2)=0, from (1) of Proposition 2.5, we know that the system
(3.4)Δ2u(n-1)+A2(n)u(n)=0,n∈ℤ(1,T),u(0)=0=u(T+1)
has only a trivial solution. Define Γ1:E→E as
(3.5)(Γ1u)(n)=Δ2u(n-1)+A2(n)u(n);
then Γ1 is an invertible operator. Define Γ2:E→E as
(3.6)(Γ2u)(n)=A2(n)u(n)-∇V(n,u(n));
then finding the solutions to (1.9) is equivalent to finding solutions to
(3.7)Γ1u-Γ2u=0
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
(3.8)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)
are priori bounded with respect to the norm ∥·∥ in E, where λ∈(0,1). If not, there exist {uk}⊂E,{λk}⊂(0,1) with ∥uk∥→∞ such that
(3.9)Δ2uk(n-1)+(1-λk)A2(n)uk(n)+λk∇V(n,uk(n))=0,n∈ℤ(1,T),uk(0)=0=uk(T+1).
Denote vk(n)=uk(n)/∥uk∥, Bk(n)=(1-λk)A2(n)+λkA(n,uk(n)), ek(n)=λk(∇V(n,uk(n))-A(n,uk(n))uk(n))/∥uk∥; then (3.9) is equivalent to
(3.10)Δ2vk(n-1)+Bk(n)vk(n)+ek(n)=0,n∈ℤ(1,T),vk(0)=0=vk(T+1).
From (3.1), ek→0 as ∥uk∥→∞. We may assume that vk→v0,λk→λ0, and bijk→bij, where Bk(n)=(bij(k))d×d(n). Denote B0(n)=(bij(n))d×d; let k→∞ in (3.10); we have
(3.11)Δ2v0(n-1)+B0(n)v0(n)=0,n∈ℤ(1,T),v0(0)=0=v0(T+1).
On the other hand, (3.2) implies that A1≤Bk≤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(3.12)V(n,z)=∑i=1dFi(zi)+12∑i=1daii(n)zi2+∑1≤i<j≤daij(n)zizj+(∑i=1dzi2+1)3/4sinn,
where z=(z1,z2,…,zd)τ, Fi(t)=∫0tsfi(s)ds, fi:ℝ→[0,α] is continuous and α>0, aij(n)=aji(n), i=1,2,…,d, j=1,2,…,d. Set
(3.13)A(n)=(aij(n))d×d=Uτdiag{α1(n),α2(n),…,αd(n)}U,
where U∈O(d). Since
(3.14)∇V(n,z)=(z1f1(z1),z2f2(z2),…,zdfd(zd))τ+A(n)z+32(∑i=1dzi2+1)-1/4zsinn=diag{f1(z1),f2(z2),…,fd(zd)}z+A(n)z+32(∑i=1dzi2+1)-1/4zsinn,V(n,z) satisfies (3.1) with
(3.15)A(n,z)=A(n)+diag{f1(z1),f2(z2),…,fd(zd)},A1(n)=A(n),A2(n)=A(n)+αId,h(n,z)=32(∑i=1dzi2+1)-1/4zsinn.
If ν(αi)=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 fi(t)=α(sint)2i and αi(n)≡αi∈ℝ, [αi,αi+α]∩{λk}k=1T=∅ 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
(3.16)(z,h(n,z))≥c1|z|α-b1,|h(n,z)|≤c2|z|α-1+b2,
where c1,c2,b1,b2 are all positive constants and 1≤α<2, then (1.9) has at least one solution.Proof.
From Proposition2.13, there exists ε>0 such that i(A+εId)=i(A)+ν(A) and ν(A+εId)=0 for any A:ℤ(1,T)→GLs(ℝd). Denote A1=A+εId, by Lemma 3.2, we only need to prove that the possible solutions to
(3.17)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)
are priori bounded with respect to the norm ∥·∥ in E. If not, there exist {uk}⊂E, λk∈(0,1) with ∥uk∥→∞ such that
(3.18)Δ2uk(n-1)+λkA1(n)uk(n)+(1-λk)A(n)uk(n)+(1-λk)h(n,uk(n))=0,n∈ℤ(1,T),uk(0)=0=uk(T+1).
Denote vk(n)=uk(n)/∥uk∥, we may assume that vk→v0 and λk→λ0. Hence, v=v0 is a nontrivial solution to
(3.19)Δ2v0(n-1)+λ0A1(n)v0(n)+(1-λ0)A(n)v0(n)=0,n∈ℤ(1,T),v0(0)=0=v0(T+1)
which implies that ν(A+ελ0Id)≠0. We claim that λ0=0. If not, λ0∈(0,1], then A<A+λ0εId≤A+εId. From Proposition 2.7, we have i(A+εId)=i(A)+ν(A)≤i(A+λ0εId) and i(A+εId)=i(A+λ0εId). However, from Proposition 2.7, we also have i(A+λ0εId)+ν(A+λ0εId)≤i(A+εId)+ν(A+εId) which implies that ν(A+λ0εId)=0, a contradiction! Hence λ0=0 and
(3.20)Δ2v0(n-1)+A(n)v0(n)=0,n∈ℤ(1,T),v0(0)=0=v0(T+1).
From (3.18), we have
(3.21)∑n=1T(Δ2uk(n-1)+A(n)uk(n),v0(n))+∑n=1Tελk(uk(n),v0(n))+∑n=1T(1-λk)(h(n,uk(n)),v0(n))=0.
Therefore, from (3.16), (3.20), and (3.21), for k large enough,
(3.22)0≥∑n=1T(h(n,uk(n)),v0(n))=∑n=1T(h(n,uk(n)),uk(n)∥uk∥)+∑n=1T(h(n,uk(n)),v0(n)-vk(n))≥∥uk∥-1∑n=1T(c1|uk(n)|α-b1)-∥v0-vk∥∞∑n=1T(c2|uk(n)|α-1+b2).
Dividing ∥uk∥α-1 at both sides, we have
(3.23)0≥∑n=1T(c1|vk(n)|α-b1∥uk∥-α)-∥v0-vk∥∞∑n=1T(c2|vk(n)|α-1+b2∥uk∥1-α)→c1∑n=1T|v0(n)|α.
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)×ℝd→ℝ is C2 with respect to the second variable, and
(3.24)∇V(n,z)=A1(n)z+o(|z|),as|z|→∞
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 thatν(A0)=0, |i(A1)-i(A0)|≥d, then (1.9) has at least two nontrivial solutions.To prove Theorem3.5, we need some results on Morse theory. Let E be a real Hilbert space and f∈C1(E,ℝ). As in [2], denote (3.25)fc={u∈E∣f(u)≤c},𝒦c={u∈E∣f′(u)=0,f(u)=c}
for c∈ℝ. The following is the definition of the Palais-Smale condition (the (PS) condition for short).Definition 3.6.
The functionalf satisfies the (PS) condition if any sequence {um}⊂E such that {f(um)} is bounded and f'(um)→0 as m→∞ has a convergent subsequence.Letu0 be an isolated critical point of f with f(u0)=c∈ℝ, and let U be a neighborhood of u0; the group (3.26)Cq(f,u0):=Hq(fc∩U,fc∩U∖{u0}),q=1,2,…
is called the qth critical group of f at u0, where Hq(A,B) denotes qth singular relative homology group of the pair (A,B) over a field F, which is defined to be quotient Hq(A,B)=Zq(A,B)/Bq(A,B), where Zq(A,B) is the qth singular relative closed chain group and Bq(A,B) is the qth singular relative boundary chain group.For any two regular valuesa<b, if 𝒦∩f-1[a,b]={u1,u2,…,ul}, denote Mq=∑i=1ldimCq(f,ui) and βq=dimHq(fb,fa). The following results play an important role in proving Theorem 3.5; see [2] for the detailed proof.Lemma 3.7.
Assume thatf∈C2(E,ℝ) satisfies the (PS) condition. Then one has the following Morse inequalities:
(3.27)Mq-Mq-1+⋯+(-1)qM0≥βq-βq-1+⋯+(-1)qβ0
for q=0,1,2,…. One also has the following Morse equality:
(3.28)∑q=0∞Mqtq=∑q=0∞βqtq+(1+t)Q(t),
where Q(t) is a polynomial with nonnegative integer coefficients.Lemma 3.8.
Assume thatf∈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 anyq∉ℤ(μ(u0),μ(u0)+ν(u0)), Cq(f,u0)≅0.(2)
Ifu0 is a nondegenerate critical point, then
(3.29)Cq(f,u0)≅δq,μ(u0)F,q=0,1,2,….(3)
Iff has a minimal value at u0, then
(3.30)Cq(f,x0)≅δq,0F,q=0,1,2,….(4)
If there exist integersq1≠q2 such that Cq1(f,u0)≇0 and Cq2(f,u0)≇0, then |q1-q2|≤ν(u0)-2.Now, Define(3.31)f(u)=12∑n=0T|Δu(n)|2-∑n=1TV(n,u(n)),u∈E.
Then the functional f is C2 with (3.32)〈f′(u),v〉=∑n=0T(Δu(n),Δv(n))-∑n=1T(∇V(n,u(n)),v(n))=-∑n=1T(Δ2u(n-1)+∇V(n,u(n)),v(n)),∀u,v∈E.
So solutions to (1.9) are precisely the critical points of f.Lemma 3.9.
Under assumptions of Theorem3.5, there exist R0>0,R1>0 and f̃ satisfying the following conditions. (1)
f'(u)=0 implies ∥u∥≤R0.(2)
f and f̃ have the same critical set.(3)
If∥u∥≥R1, then f̃(u)=(1/2)〈Lu,u〉.Proof.
DefineL and g on E as
(3.33)〈Lu,v〉=∑n=0T(Δu(n),Δv(n))-∑n=1T(A1(n)u(n),v(n)),g(u)=f(u)-12〈Lu,u〉.
Then
(3.34)〈f′(u),v〉=〈Lu,v〉+〈g′(u),v〉.
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
(3.35)∥g′(u)∥<ε1∥u∥.
So, as ∥u∥>R0, we have
(3.36)∥f'(u)∥≥∥L-1∥-1∥u∥-∥g'(u)∥>ε1∥u∥,
that is, no critical point is outside the ball BR0.
To prove (2) and (3), letρ∈C∞(ℝ,[0,1]) satisfy(3.37)ρ(t)={1,t≤0,0,t≥1
with 0≤ρ(t)≤1 and max|ρ'(t)|≤3/2. Let ε=∥L-1∥-1/5, then there exists c1≥0 such that
(3.38)∥g'(u)∥<ε∥u∥+c1.
Hence,
(3.39)|g(u)|≤∫01|〈g′(su),u〉|ds+|g(0)|≤∫01(εs∥u∥2+c1∥u∥)ds+|g(0)|=ε2∥u∥2+c1∥u∥+|g(0)|.
Therefore, there exists c2>0 such that
(3.40)|g(u)|≤ε∥u∥2+c2,∀u∈E.
Define
(3.41)R>max{1,R0,c1+3/2ϵ},
where R0 is defined above and
(3.42)λ=max{c2,1}·R.
If R≤∥u∥≤λ+R, then
(3.43)c2λ<1,∥u∥λ≤2.
Let
(3.44)ρλ(t)=ρ(t-Rλ),
and define
(3.45)f̃(u)=12〈Lu,u〉+ρλ(∥u∥)g(u).
Then the function f̃(u) satisfies (2) and (3). In fact,
(3.46)f̃(u)={f(u),∥u∥≤R,12〈Lu,u〉,∥u∥≥λ+R.
The only thing we have to check is that f̃'(u)≠0 as R≤∥u∥≤λ+R. However,
(3.47)∥f̃'(u)∥=∥Lu+1λρ′(∥u∥-Rλ)g(u)u∥u∥+ρ(∥u∥-Rλ)g′(u)∥≥∥L-1∥-1∥u∥-32λ(ε∥u∥2+c2)-(ε∥u∥+c1)=5ε∥u∥-ε∥u∥-3ε2λ∥u∥2-(c1+3c22λ)≥ε∥u∥-(c1+32)>0.
Hence, let R1=λ+R; the proof is completed.From Lemma3.9, f'(u)=0 if and only if f̃'(u)=0. Thus, in order to find solutions to (1.9), it suffices to find the critical points of f̃. Moreover, f̃ satisfies the (PS) condition by Lemma 3.9.Lemma 3.10.
Under the assumptions of Theorem3.5, there exist a,b with a<b such that the critical points of f̃ belong to f̃-1((a,b)):={u∣a<f̃(u)<b} and
(3.48)Hq(f̃b,f̃a)≅δq,i(A1)F.Proof.
Define(3.49)a<minB2R1f̃-1,b>maxB2R1f̃+1,f1(u)=12〈Lu,u〉,
where a,b are finite. Noticing that all critical points of f̃ lie in BR0, if u is a critical point of f̃, then
(3.50)a<minBR0f̃≤f̃(u)≤maxBR0f̃<b.
This implies that {u∣a<f̃(u)<b} contains all critical points of f̃. By the properties of the raltive singular homology group, we have Hq(f̃b,f̃a)≅Hq(f1b,f1a). 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 Theorem3.5.
(1) By Lemma3.10 and the Morse inequalities, f must have a critical point u with Ci(A1)(f,u)≇0. Since i(A1)∉ℤ(i(A0),i(A0)+ν(A0)), then from Lemma 3.8, we have
(3.51)Ci(A1)(f,0)≅0.
And hence u≠0 is a critical point of f; that is, (1.9) has at least one nontrivial solution.
(2) Sinceν(A0)=0, we have(3.52)Cq(f,0)≅δq,i(A0)F,q=0,1,2,….
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
(3.53)ti(A0)+∑q=0∞dimCq(f,u)tq=ti(A1)+(1+t)Q(t).
We necessarily have dimCi(A1)(f,u)≥1, and
(3.54)eitherdimCi(A0)-1(f,u)≥1,ordimCi(A0)+1(f,u)≥1.(i)
IfdimCi(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
(3.55)|i(A0)-1-i(A1)|≤d-2
which implies that |i(A0)-i(A1)|≤d-1. This is impossible since |i(A0)-i(A1)|≥d.(ii)
IfdimCi(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.



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*Source: 364079-2010-06-24.xml*