Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem

Let a, b be two integers with b − a ≥ 5 and let T2 {a 2, a 3, . . . , b − 2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problem Δ4u t − 2 f t, u t , Δ2u t − 1 , t ∈ T2, u a 1 u b − 1 Δ2u a Δ2u b − 2 0 under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.


Introduction
The deformations of an elastic beam whose both ends are simply supported are described by a fourth-order two-point boundary value problem y g x y e x , 0 < x < 1, y 0 y 1 y 0 y 1 0 1.1 See studies by Aftabizadeh 1 and Gupta in 2 .The existence of solutions of nonlinear boundary value problems of fourth-order differential equations has been studied by many authors; see 1-12 and the references therein.For example, Aftabizadeh 1 proved an existence theorem for nonlinear boundary value problems y f x, y, y , 0 < x < 1, y 0 y 0 , y 0 y 1 , y 1 y 0 , y 1 y 1 , 1.2 under several conditions that f is a bounded function.Yang 3 obtained existence results of 1.2 under the following assumption.
A There are constants a, b, c ≥ 0 with a/π 4 b/π 2 < 1 such that f x, y, u ≤ a y b|u| c. for all k ∈ N and that there are positive constants a, b, and c such that for all x ∈ 0, 1 , u, v ∈ R, then 1.2 possesses at least one solution.
Of course, the natural question is whether or not the similar existence can be established for the corresponding discrete analog of 1.2 of the form where T 2 {a 2, a 3, . . ., b − 2}, r i ∈ R for i ∈ {1, 2, 3, 4}.The purpose of this paper is to show that the answer is yes.To this end, we state and prove a spectrum result of two-parameter linear eigenvalue problem This result is a slightly generalized version of Shi and Wang 13, Theorem 2.1 .In Section 3, we use Leray-Schauder principle to study the existence of solutions of 1.6 , 1.7 under some nonresonant conditions involving the spectrum of 1.8 , 1.9 .Section 4 is considered with some perturbations of resonant linear problems.We established some a priori bounds and used these together with bifurcation arguments to prove the existence and multiplicity of solutions.
Finally, we note that the existence of solutions of second-order discrete boundary value problems has also received much attention; see studies by Agarwal and Wong in 14 , Henderson in 15 , and the references therein.However, relatively little is known about the existence of solutions of fourth-order discrete boundary value problems.The likely reason may be that the structure of spectrum of the corresponding linear eigenvalue problem is not very clear.To our best knowledge, only He and Yu 16 as well as Zhang et al. 17 dealt with the discrete problem of the form As we will see in Section 2, 1.9 has more advantage than 1.11 in the study of the spectrum of two-parameter linear eigenvalue problems.

Spectrum of Two-Parameter Linear Eigenvalue Problem
Let a, b be two integers with b − a ≥ 5. Recall Let X be the Banach space

2.4
Let Y be the Banach space

2.12
For k ∈ Λ, let λ k be the kth-eigenvalue of the second-order linear eigenvalue problem

2.13
It is well known that λ k is simple, and the corresponding eigenfunction See the study by Kelly and Peterson in 18 .
The following result is considered with the spectrum of two-parameter eigenvalue problem: for some k ∈ Λ.
Define two second-order difference operators

2.19
Then, for y ∈ Y and t ∈ T 1 ,

2.20
We claim that if 2.15 , 2.16 possess a nontrivial solution y, then either This is

2.22
Thus, r 1 λ k for some k ∈ Λ, and

2.27
This implies that 2.25 , 2.26 have a unique solution

2.28
We show that r 1 λ k γ, y t ψ k t .

2.29
In fact, from 2.25 we have

2.30
which implies that γ r 1 − λ k , and, subsequently, y t ψ k t .Therefore, the claim is true.Now, 2.17 follows by substituting this solution into 2.15 , 2.16 .Reciprocally, if 2.17 holds, then, clearly, sin Remark 2.3.From the proof of Proposition 2.2, we see that if 2.15 subjects to 1.9 , then we can factor L| Y as follows:

2.31
However, this cannot be done if 2.15 subjects to 1.11 .So, 1.9 has more advantage than 1.11 in the study of the spectrum of two-parameter linear eigenvalue problems.
Next, for j ∈ N, let us set

2.32
In view of the Proposition 2.2, we call L j an eigenline of 2.15 , 2.16 .We note that an eigenvalue pair α, β can belong to at most two eigenlines.If α, β belongs to just one L j , then the corresponding eigenspace is spanned by sin Suppose that the pair α, β is not an eigenvalue pair of 2.15 , 2.16 , that is, for all k ∈ Λ, and that h ∈ Z:

2.34
From the Fredholm Alternative, it follows that the boundary value problem has a unique solution for each h ∈ Z.Moreover, this solution admits a Fourier series expansion of the form

2.36
Also, we have

2.39
Finally, as an immediate consequence of Proposition 2.2, we have the following.

Existence Results for Nonresonant Problems
Theorem 3.1.Assume that the pair α, β satisfies for all k ∈ Λ and that there are positive constants a * , b * , and c * such that for k ∈ Λ.It turns out that 3.4 is equivalent to the fact that the square α − a * , α a * × β − b * , β b * does not intersect any of the eigenlines L j of 2.15 , 2.16 .From this point of view, 3.1 , 3.2 can be thought of as a two-parameter nonresonance condition relative to the eigenlines L j .
Proof of Theorem 3.1.It is easy to check that the problem has a unique solution l t .Set y t : u t − l t , t ∈ T 0 .

3.6
Then 1.6 , 1.7 can be rewritten as it follows that 3.2 and 3.3 still hold except that c * is replaced by c * * .So, we may suppose that r 1 r 2 r 3 r 4 0 in 1.7 .
Let us define T : where A and B are the operators defined in 2.38 .The growth condition 3.3 together with the compactness A and B implies that T is a completely continuous operator.By Remark 2.1, the problem

3.12
We will study this fixed point problem by means of the well-known Leray-Schauder principle 18 .To do this, we show that there is a uniform bound independent of λ ∈ 0, 1 for the solutions of the equation u, v λT u, v .

3.13
Thus, let u, v be a solution of 3.13 .From the definition of T and 3.3 , we obtain the result that

3.15
Combining 3.14 and 3.15 and using 3.2 and 2.38 , we obtain the existence of a constant

3.16
By the Leray-Schauder principle 19 , we conclude the existence of at least one solution of 3.12 , and the theorem follows.

Existence and Multiplicity Results for Perturbations of Resonant Linear Problems
In this section, we consider the perturbations of resonant linear problems of the form where γ, δ ∈ 0, ∞ × 0, ∞ with γ δ > 0, μ k μ k γ, δ , and g and h satisfy the following.
H1 Sublinear growth condition g : T 2 × R → R is continuous, and there exist α ∈ 0, 1 , H2 There exists β > 0 such that We will establish some a priori bounds and use these together with Leray-Schauder continuation and bifurcation arguments to reduce results which say that there are multiple solutions of 4.1 μ , 4.1 for μ on one side of zero and guarantee the existence of at least one solution for μ 0 and μ on the other side of zero.To wit, we have the following.We have the following "dual" theorem if H2 is replaced by the assumption H2 that there exists β > 0 such that sg t, s < 0, for t ∈ T 2 , |s| > β. 4.5 Theorem 4.2.Let (H1), (H2'), and (H3) hold.Then there exist μ − < 0 < μ such that 4.1 μ , 4.1 have x s ψ k s , t ∈ T 0 , 4.8 where It is easy to show the following.It is clear that 4.11 where I represents the identity operator and X P , X I−P , Z I−E , and Z E are the images of P, I − P, I − E, and E, respectively.It is obvious that the restriction of L to X I−P is a bijection from X I−P onto Z E , the image of L. We define M : Z E → X I−P by M : L| X I−P −1 .

4.12
Since ker L span{ψ k }, we see that each x ∈ X can be uniquely decomposed into for some ρ ∈ R, and v ∈ X I−P .For z ∈ Z, we also have the decomposition Lemma 4.5.Equations 4.1 μ , 4.1 are equivalent to the system

4.15
Lemma 4.6.Let (H1) and (H2) hold.Then there exists R 0 such that any solution y of 4.1 μ , 4.1 satisfies as long as where J : X → Z is defined by

4.18
Discrete Dynamics in Nature and Society Proof.Obviously L μJ | X I−P : X I−P → Z E is invertible for |μ| ≤ δ.Moreover, by 4.17 ,

4.19
Let y ρψ k v be any solution of 4.1 μ , 4.1 .Then we have that, if ρ / 0, where If we assume that the conclusion of the lemma is false, we obtain a sequence {η n } with 0 ≤ η n ≤ δ and η n → 0, and a sequence of corresponding solutions {y n ρ n ψ k v n } of 4.1 η n , 4.1 such that y n X → ∞.From 4.24 , we conclude that it is necessary that |ρ n | → ∞.We may assume that

4.29
It is easy to see that Combining 4.30 and 4.26 , we conclude that there exists a positive constant Γ such that, for n ∈ N, where j : Z → X is the natural homomorphism, B R {u ∈ X u X < R}, and "deg" denotes Leray-Schauder degree when μ / 0 and coincidence degree when μ 0 (see the study by Gaines and Mawhin in [20]).Therefore 4.7 μ has a solution in B R for μ ∈ 0, δ .Proof.By Lemma 4.6 and the definition of L, the degree is well defined for μ ∈ 0, δ and is a constant with respect to μ.

4.39
Hence there exists R 0 > 0 such that y which completes the proof.
By a similar manner we may establish the following.

4.43
Then it is not difficult to check that τ 0 > 0. Hence if we take δ 1 so small that δ 1 R 1 < τ 0 , then for μ ∈ −δ This completes the proof.
Proof of Theorem 4.2.Using similar arguments, we may get the desired results.

1 . 3 Del
Pino and Manásevich 4 extended Yang's result and proved the following.Theorem A. Assume that the pair α, β satisfies

Lemma 4 . 3 .
P is a projection and Im P Ker L .Define an operator E : Z → Z by have the following.

Lemma 4 . 4 .
E is a projection and Im E Im L .

2 δΔ 2 2 s a 2 ψ
ρ n −→ ∞, ρ n ≥ C ∀n ∈ N 4.25since the other case can be treated by the same way.Thus 4.24 yields thatv n X :≤ C ρ n α ψ k s − 1 − γψ k s ψ k s b−k s f s, ρ n ψ k s v n s .