Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems

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 andmultiplicity 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.


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 Δ 2 u 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 det A.
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 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: 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 Then 1.1 reduces to

1.4
Hence, in the following, we can only consider the discrete conjugate BVPs, that is,

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 where G n, m is Green's function for , and define 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, criticalpoint 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: and for every n ∈ Z 1, T , and

and we denote
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 systems 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.

Index Theory for Linear Systems
To establish the index theory for 1.12 , we introduce the following finite dimensional sequence space: 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 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 E 1 and E 2 of E, if q A u, v 0 for any u ∈ E 1 and v ∈ E 2 , we say that E 1 and E 2 are q A orthogonal.
For any subspace E 1 of E, we say that q A is positive definite or negative definite on

has a nontrivial solution. If E i A denotes the subspace of solutions with respect to
The space E has a q A orthogonal decomposition such that q A is positive definite, negative definite, and null on E A , E − A , and E 0 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. 2.8 Proof.Note that where

2.10
Assume that λ is an eigenvalue of B and that ξ ξ 1 , ξ 2 , . . ., ξ T τ is an eigenvector associated to λ. Define the sequence {v n } T 1 n 0 as

2.12
Equation 2.12 has a nontrivial solution if and only if

2.14
Noticing that for any real symmetric dT × dT matrix A, we have 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 for simplicity.
Proof of Proposition 2.1.1 We claim that the norm • λ 0 induced by the inner product u, v λ 0 : In fact, it is easy to see that there exists c ∈ 0, ∞ such that

2.19
Define a bilinear function and then Hence, there exists a unique continuous linear operator It is easy to see that K λ 0 is self-adjoint, and hence all the eigenvalues of K λ 0 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 λ 0 ⎧ ⎨ ⎩ 1, i l and j k, where n i is the multiplicity of μ i with i n i dT .By 2.22 and 2.23 we have In particular, Without loss of generality we assume that μ i is strictly monotonously decreasing, that is, 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 and since μ i > 0, the above equality means Therefore e ij satisfies 2.6 .Now, we have proved the first result of Proposition 2.1 except dim which is also equivalent to where Hence, dim E i A n i ≤ d. 2 For any u ∈ E with u i,j c ij e ij , by 2.23 and 2.24 , we have

2.32
Hence, if we denote then the results hold.

Definition 2.4. For any
and define the nullity of A as ν A : dim E 0 A .
In the following we shall discuss the properties of i A , v A .

one has the following.
1 ν A is the dimension of the solution subspace of 1.12 , and ν A ∈ Z 0, d .
, 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 E 0 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 : dim 2 By the proof of Proposition 2.1, E − A λ i 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 A 1 , A 2 : Z 1, T → GL s R d , the following results hold.
To prove Proposition 2.7, we firstly prove the following lemma.
Lemma 2.8.Let E 1 be a subspace of E satisfying and then

2.35
Moreover, if To prove the second part, let e i e i e − i e 0 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 c 1 , c 2 , . . ., c k such that k i 1 c i e − i 0. So, e : , e ≥ 0. This is a contradiction to 2.36 .

Proof of Proposition 2.7. For any
1 From Lemma 2.8, we only need to show that

2.38
In fact, for every u u 2 From Lemma 2.8, we only need to show that

2.40
In fact, for every 3 From Lemma 2.8, we only need to show that

2.42
In fact, for every
where λ k is given by 2.13 , and Proof.Firstly, we claim that

2.47
In fact, since λ 0 I d > A if and only if λ 0 I d > U τ AU, we can choose λ 0 U τ AU λ 0 A .By 2.22 and 2.23 , it is easy to see that μ i U τ AU μ i A , and hence 2.47 holds.Therefore, by Definition 2.4,

2.50
Note that the scalar eigenvalue problem

2.54
This completes the proof.

Proposition 2.10. For any
A n u n , u n , ∀u ∈ E.

2.55
And the equality holds if and only if u ∈ E 0 A .

Discrete Dynamics in Nature and Society 13
Proof.For any u ∈ E with u i,j c ij e ij , we have

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 ∈ E 0 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 A 2 − i A 1 as A 2 > A 1 .

2.57
If , where α 1 < α 2 are two real numbers, then by Proposition 2.9, we have

2.58
So

2.59
This gives us a steer toward the following result.
Proposition 2.12.For any then to prove 2.60 , we only need to prove that 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 λ 1 ∈ 0, 1 , i λ 1 − i 0 must equal to the sum of the jumps i λ incurred in 0, λ 1 .By 2.61 and 2.62 , this is precisely the sum of ν λ , 0 ≤ λ < λ 1 , that is, Hence, if we choose λ 1 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 prove m 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 where B λ n A 1 n λ A 2 n − A 1 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 prove where And as ε > 0 is sufficiently small, we have where

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 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 :

3.3
Proof of Theorem 3.1.Assume that 3.2 holds.Since ν A 2 0, from 1 of Proposition 2.5, we know that the system has only a trivial solution.Define Γ 1 : E → E as then finding the solutions to 1.9 is equivalent to finding solutions to in E, which is also equivalent to finding the fixed points of Γ −1 1 Γ 2 in E since Γ 1 is invertible.By Lemma 3.2, we only need to prove that the possible solutions to are priori bounded with respect to the norm • in E, where λ ∈ 0, 1 .If not, there exist

3.10
From 3.1 , e k → 0 as u k → ∞.We may assume that

3.11
On the other hand, 3.2 implies that A 1 ≤ B k ≤ A 2 , and hence 0, and Proposition 2.7, we have ν A 1 ν A 2 ν B 0 0. This contradicts the fact that 3.11 has a nontrivial solution.
z sin n,

3.14
Discrete Dynamics in Nature and Society V n, z satisfies 3.1 with z sin n.

3.15
If ν α i 0 for every i ∈ Z 1, d , then ν A 1 0. By Proposition 2.13, if α > 0 is small enough, then ν A 2 0 and i A 1 i A 2 .Hence, by Theorem 3.1, 1.9 has at least one solution.In particular, if we choose f i t α sin t 2i and α i n And hence 1.9 has at least one solution.
where c 1 , c 2 , b 1 , b 2 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 A 1 A εI d , by Lemma 3.2, we only need to prove that the possible solutions to are priori bounded with respect to the norm which implies that ν A ελ 0 I d / 0. We claim that λ 0 0. If not, λ 0 ∈ 0, 1 , then

3.20
From 3.18 , we have

3.22
Dividing u k α−1 at both sides, we have

3.23
This is a contradiction since v 0 / 0 and c 1 > 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 respect to the second variable, and for every n ∈ Z 1, T with ν A 1 0.
2 ∇V n, 0 ≡ 0, A 0 n V n, 0 , and i A 1 / ∈ Z i A 0 , i A 0 ν A 0 .Then 1.9 has at least one nontrivial solution.
3 Moreover, if we further assume that ν A 0 0, |i A 1 − i A 0 | ≥ 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 ∈ C 1 E, R .As in 2 , denote 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 u 0 be an isolated critical point of f with f u 0 c ∈ R, and let U be a neighborhood of u 0 ; the group is called the qth critical group of f at u 0 , 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 {u 1 , u 2 , . . ., u l }, denote M q l i 1 dim C q f, u i and β q dim H 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 ∈ C 2 E, R satisfies the (PS) condition.Then one has the following Morse inequalities: for q 0, 1, 2, . . . .One also has the following Morse equality: where Q t is a polynomial with nonnegative integer coefficients. Then So, as u > R 0 , we have that is, no critical point is outside the ball B R 0 .

3.45
Then the function f u satisfies 2 and 3 .In fact, 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,

3.47
Hence, let R 1 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 A 1 F.

3.48
Proof.Define a < min where a, b are finite.Noticing that all critical points of f lie in B R 0 , if u is a critical point of f, then

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 b 1 , f a 1 .However, ν A 1 0 implies that f 1 has only critical point 0 with Morse index i A 1 .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 A 1 f, u 0. Since i A 1 / ∈ Z i A 0 , i A 0 ν A 0 , then from Lemma 3.8, we have

3.51
And hence u / 0 is a critical point of f; that is, 1.9 has at least one nontrivial solution.

3.52
Now we assume that |i A 1 − i A 0 | ≥ d and that u is the only nonzero critical point of f.Then from Morse equality, we have dim C q f, u t q t i A 1 1 t Q t .ii If dim C i A 0 1 f, u ≥ 1, then similar to the above proof we have |i A 0 − i A 1 | ≤ d − 1, also a contradiction.
Therefore, f has at least two nonzero critical points and hence 1.9 has at least two nontrivial solutions.

3 . 53
We necessarily have dimC i A 1 f, u ≥ 1, and either dim C i A 0 −1 f, u ≥ 1, or dim C i A 0 1 f, u ≥ 1. 3.54 i If dim C i A 0 −1 f, u ≥ 1, then by assumption we have i A 0 − 1 / i A 1 .Since the nullity of u is less or equal to d, from Lemma 3.8, we have|i A 0 − 1 − i A 1 | ≤ d − 2 3.55 which implies that |i A 0 − i A 1 | ≤ d − 1.This is impossible since |i A 0 − i A 1 | ≥ d.
z denotes the gradient of V with respect to z, and d ≥ 2, T > 0 are given integers.
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 A 1 29Since 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 If not, there exist not all zero constants α 1 , α 2 , . . ., α k such that k i 1 α i e * , and hence q A e, e > 0. This is a contradiction to 2.34 .This implies that e * i ∈ E − A ⊕ E 0 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.i 0. So e : k i 1 α i e i k i 1 α i e i ∈ E A * 1 , e * 2 , . . ., e * k is linear independent and i A ν For any A : Z 1, T → GL s R d , there exists ε 0 > 0 such that for any ε ∈ 0, ε 0 , This completes the proof.Proposition 2.13.To prove 2.70 and 2.71 , note that I A − I d , A i A − i A − I d and 5, then there exists c 1 ≥ 0 such that