Boundary value problems of a discrete generalized beam equation via variational methods

Xia Liu; Tao Zhou; Haiping Shi

doi:10.1515/math-2018-0121

Artikel Open Access

Boundary value problems of a discrete generalized beam equation via variational methods

Xia Liu , Tao Zhou und Haiping Shi

Veröffentlicht/Copyright: 26. Dezember 2018

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Abstract

The authors explore the boundary value problems of a discrete generalized beam equation. Using the critical point theory, some sufficient conditions for the existence of the solutions are obtained. Several consequences of the main results are also presented. Examples are given to illustrate the theorems.

Keywords: Boundary value problems; Discrete problems; Generalized beam equation; Saddle point theorem; Variational methods

MSC 2010: 39A10; 34B05; 58E05; 65L10

1 Introduction and statement of main results

Beam equations have historical importance, as they have been the focus of attention for prominent scientists such as Leonardo da Vinci (14th Century) and Daniel Bernoulli (18th Century) [9]. In this article, we study the existence of solutions to boundary value problem (BVP) of discrete generalized beam equation

Δ4x(t−2)−Δr(t−1)Δx(t−1)=f(t,x(t)),t∈[1,T]Z,(1.1)

with boundary value conditions

Δkx(−1)=Δkx(T−1),k=0,1,2,3,(1.2)

where 1 ≤ T ∈ ℕ, Δ is the forward difference operator defined by Δx(t) = x(t + 1) − x(t), Δ^kx(t) = Δ(Δ^k−1x(t)) (2 ≤ k ≤ 4), Δ⁰x(t) = x(t), define [1, T]_ℤ := [1, T] ∩ ℤ, r(t) ∈ ℝ^T+1 with r(0) = r(T), f(t, x) ∈ C([1, T]_ℤ × ℝ,ℝ).

(1.1) and (1.2) can be considered as a discrete analogue of

x(4)(s)−[r(s)x′(s)]′=f(s,x(s)),s∈(0,1),(1.3)

with boundary value conditions

x(k)(0)=x(k)(1),k=0,1,2,3.(1.4)

(1.3) is a generalization of the beam equation

x(4)(s)=f(s,x(s)),s∈R.

Practical applications of the beam equations [9] are evident in mechanical structures built under the premise of beam theory. In recent years, many researchers [6,10,12,21,22] have paid a lot of attention to equations similar to (1.3).

Difference equations [1,2,3,4,5,7,11,13,14,15,17,18,19,20,23,24,26,27] are widely found in mathematics itself and in its applications to combinatorial analysis, quantum physics, chemical reactions and so on. Many authors were interested in difference equations and obtained many significant conclusions.

Cabada and Dimitrov [4] studied the following nonlinear singular and non-singular fourth-order difference equation

x(t+4)+Mx(t)=λg(t)f(x(t))+c(t),t∈{0,1,⋯,T−1},

coupled with periodic boundary value conditions. They obtained some sufficient conditions on existence and nonexistence theorems.

In 2010, He and Su [11] considered boundary value problems of the fourth order nonlinear difference equation

Δ4x(t−1)+ηΔ2x(t−1)−ξx(t)=λf(t,x(t)),t∈Z[a+1,b+1],

with three parameters by using the critical point theory and monotone operator theory. They obtained some existence, multiplicity, and nonexistence of nontrivial solutions.

By using the Dancer’s global bifurcation theorem, Ma and Lu [15] investigated the boundary value problem to the following fourth order nonlinear difference equation

Δ4x(t−2)=λh(t)f(x(t)),t∈{2,3,⋯,T},

and gave the existence and multiplicity of positive solutions.

Fang and Zhao [7] in 2009 established a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equation

Δ4x(t−2)−r(t)x(t)+f(t,x(t+1),x(t),x(t−1))=0,t∈Z,

by using Mountain Pass Theorem, a weak convergence argument and a discrete version of Lieb’s lemma.

There are numerous papers dealing with similar problems to the one that we study (nonlinear difference equation with semidefinite linear parts, periodic boundary value problems) and many of these use similar techniques-matrix formulation in ℝ^N with various variational methods (see, e.g., [3,8,19,25]) and in many cases even the saddle point theorem (e.g., [17,23]). In this article, the boundary value problems of a discrete generalized beam equation are explored. Applying the critical point theory, we establish some criteria for the existence of the solutions for (1.1) and (1.2). The eigenvalues of some symmetric matrix associated with the problem are used in proving main results. The motivation for this article comes from the recent article [11] since it deals with boundary value problem to the fourth order difference equation by using the critical point theory.

Let

G(t,x)=∫0xf(t,s)ds,

for any (t, x) ∈ [1, T]_ℤ × ℝ.

The rest of this article is organized as follows. In Section 2, we state some preliminary lemmas and transfer the existence of the BVP of (1.1) and (1.2) into the existence of the critical points of some functionals. Our main results are given in Section 3. In Section 4, we prove our main results by making use of variational methods. Two examples are presented to illustrate our main results in Section 5.

2 Preliminary lemmas

Let Q and R be Banach spaces, and P ⊂ Q be an open subset of Q. A function I : P → R is called Fréchet differentiable at x ∈ P if there exists a bounded linear operator L_x: Q → R such that

limh→0I(x+h)−I(x)−Lx(h)RhQ=0.

We write I′(x) = L_x and call it the Fréchet derivative of I at x.

Let X be a real Banach space and I ∈ C¹(X,ℝ) be a continuously Fréchet differentiable functional defined on X. As usually, I is said to satisfy the Palais-Smale condition if any sequence xkk=1∞ ⊂ X for which Ixkk=1∞ is bounded and I′(x_k)arrow 0 as karrow ∞ possesses a convergent subsequence. Here, the sequence xkk=1∞ is called a Palais-Smale sequence.

Let X be a real Banach space. We denote by the symbol B_r the open ball in X about 0 of radius r, ∂B_r its boundary, and B̄_r its closure.

Denote a space X as

X:={x:[−1,T+2]Z→R|Δkx(−1)=Δkx(T−1),k=0,1,2,3}.

For any x ∈ X, we define

x,y:=∑t=1Tx(t)y(t),∀x,y∈X,

and

∥x∥:=∑t=1Tx2(t)12,∀x∈X.

Denote the norm ∥⋅∥_q on X by

∥x∥q=∑t=1T|x(t)|q1q,(2.1)

for all x ∈ X and q > 1.

As usual, we use ∥x∥ = ∥x∥₂ for the Euclidean norm. Since ∥x∥_q and ∥x∥ are equivalent, there are numbers τ₁, τ₂ such that τ₂ ≥ τ₁ > 0, and

τ1∥x∥≤∥x∥q≤τ2∥x∥,∀x∈X.(2.2)

Remark 2.1

For anyx ∈ X, it is obvious that

x(−1)=x(T−1),x(0)=x(T),x(1)=x(T+1),x(2)=x(T+2).(2.3)

In fact, Xis isomorphic to ℝ^T. In the later sections of this article, when we writex = (x(1), x(2),⋯,x(T)) ∈ ℝ^T, we always imply thatxcan be extended to a vector inXso that (2.3) is satisfied.

For any x ∈ X, we denote the functional I by

I(x):=−12∑t=1TΔ2x(t−2)2−12∑t=1Tr(t)(Δx(t))2+∑t=1TG(t,x(t)).(2.4)

Hence I ∈ C¹(X,ℝ). By computing, we have

∂I∂x(t)=−Δ4x(t−2)+Δr(t−1)Δx(t−1)+f(t,x(t)),t∈[1,T]Z.

Accordingly, I′(x) = 0 if and only if

Δ4x(t−2)−Δr(t−1)Δx(t−1)=f(t,x(t)),t∈[1,T]Z.

As a result, a function x ∈ X is a critical point of the functional I on X if and only if x is a solution of the BVP (1.1) and (1.2). For convenience, we define two T × T matrices as follows.

When T = 1, define A = B = (0).

When T = 2, define

A=8−8−88,

and

B=r(0)+r(1)−r(0)−r(1)−r(0)−r(1)r(0)+r(1).

When T = 3, define

A=6−3−3−36−3−3−36.

When T = 4, define

A=6−42−4−46−422−46−4−42−46.

When T ≥ 5, define

A=6−4100⋯001−4−46−410⋯00011−46−41⋯000001−46−4⋯0000001−46⋯0000⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯00000⋯6−41000000⋯−46−4110000⋯1−46−4−41000⋯01−46.

When T ≥ 3, define

B=r(0)+r(1)−r(1)0⋯−r(0)−r(1)r(1)+r(2)−r(2)⋯00−r(2)r(2)+r(3)⋯0⋯⋯⋯⋯⋯000⋯−r(T−1)−r(0)00⋯r(T−1)+r(0).

Let Ω := A + B. Consequently, the functional I(x) can be rewritten as

I(x)=−12x∗Ωx+∑t=1TG(t,x(t)).(2.5)

Lemma 2.2

(Saddle Point Theorem [16]). LetXbe a real Banach space, X = X₁ ⊕ X₂, whereX₁ ≠ {0} and is finite dimensional. Assume thatI ∈ C¹(X,ℝ) satisfies the Palais-Smale condition and

there exist two constantsσ, ρ > 0 such thatI∣_{∂B_ρ∩X₁} ≤ σ;
there existse ∈ B_ρ ∩ X₁and a constantω > σsuch thatI_e+X₂ ≥ ω.
ThenIpossesses a critical valuec ≥ ω, where
c=infh∈Γmaxx∈Bρ∩X1I(h(x)),Γ={h∈C(B¯ρ∩X1,X)h|∂Bρ∩X1=id}
and id denotes the identity operator.

3 Main results

We shall give our main results in this section.

Theorem 3.1

Suppose that the following conditions are satisfied:

for anyt ∈ [0, T]_ℤ, r(t) ≥ 0;
there is a positive constantc₁such that
|f(t,x)|≤c1,∀(t,x)∈[1,T]Z×R;
G(t, x) → +∞ uniformly fort ∈ [1, T]_ℤas ∣x∣ → +∞.
Then the BVP (1.1),(1.2) possesses at least one solution.

Remark 3.2

Condition (G₁) means that there is a positve constantc₂such that

(G1′)|G(t,x)|≤c1|x|+c2,∀(t,x)∈[1,T]Z×R.

Theorem 3.3

Suppose that (r) and the following conditions are satisfied:

there are two constants 1 < μ < 2 andδ > 0 such that
0<xf(t,x)≤μG(t,x),∀t∈[1,T]Z,|x|≥δ;
there are three constantsc₃ > 0, c₄ > 0 and 1 < ν ≤ μsuch that
G(t,x)≥c3|x|ν−c4,∀(t,x)∈[1,T]Z×R.
Then the BVP (1.1),(1.2) possesses at least one solution.

Remark 3.4

Condition (G₃) means that there are two positve constantsc₅andc₆such that

(G3′)G(t,x)≤c5|x|μ+c6,∀(t,x)∈[1,T]Z×R.

In the case thatf(t, x) is independent ofx, we study the autonomous fourth order discrete system

Δ4x(t−2)−Δr(t−1)Δx(t−1)=f(x(t)),t∈[1,T]Z,(3.1)

wheref ∈ C(ℝ, ℝ).

Corollary 3.5

Suppose that (r) and the following conditions are satisfied:

there is a functionH(x) ∈ C¹(ℝ,ℝ) such that
H′(x)=f(x);
there is a positive constantκ₁such that
|f(x)|≤κ1,∀x∈R;
H(x) → +∞ as ∣x∣ → +∞.
Then the BVP (3.1),(1.2) possesses at least one solution.

Corollary 3.6

Suppose that (r), (H₁) and the following conditions are satisfied:

there are two constants 1 < μ̃ < 2 andδ̃ > 0 such that
0<xf(x)≤μ~H(x),∀|x|≥δ~;
there are three constantsκ₂ > 0, κ₃ > 0 andν̃with 1 < ν̃ ≤ μ̃such that
H(x)≥κ2|x|ν~−κ3,∀x∈R.
Then the BVP (3.1),(1.2) possesses at least one solution.

Let Ω satisfy:

Ω is a symmetric and positive semidefinite matrix;
λ₁ = 0 is a simple eigenvalue of Ω with multiplicity one and with the eigenvector e₁ = [1, 1,⋯,1]^⊤.

Remark 3.7

Our results could be extended to other problems with matrices satisfying (A1) − (A2). For example, it is obvious that we can also use the discrete beam equation with Neumann initial conditions.
On the other hand, we can directly use results from other papers for other nonlinearities and obtain, for example, multiplicity results for the beam equation with bistable nonlinearities [17] or nonlinearities satisfying Landesman-Lazer type conditions [23].

4 Proofs of the main results

In this section, we shall prove our main results by using variational methods.

Throughout this section, Ω is a symmetric and positive semidefinite matrix, 0 is a simple eigenvalue with multiplicity one and with the eigenvector (1, 1,⋯,1)^⊤. We denote the eigenvalues of Ω by λ₁, λ₂,⋯, λ_T.

Denote

λ_=minλi|λi≠0,i=1,2,⋯,T,(4.1)

and

λ¯=maxλi|λi≠0,i=1,2,⋯,T.(4.2)

Set X₂ = {(d, d,⋯, d)^⊤ ∈ X∣d ∈ ℝ}. Obviously, X₂ is an invariant subspace of X. We denote a subspace X₁ of X by

X=X1⊕X2.

Proof of Theorem 3.1

Let {x_k}_k∈ℕ ⊂ X be such that {I(x_k)}_k∈ℕ is bounded and I′(x_k) → 0 as k → ∞. Accordingly, for any k ∈ ℕ, there is a number c₇ > 0 such that

−c7≤Ixk≤c7.

Let xk=xk(1)+xk(2)∈X1⊕X2. On one hand, for k large enough, since

−∥x∥≤I′xk,x=−Ωxk,x+∑t=1Tft,xk(t)x(t),

combining with (G₁), we have

Ωxk,xk(1)≤∑t=1Tft,xk(t)xk(1)(t)+xk(1)≤c1∑t=1Txk(1)+xk(1)≤c1T+1xk(1).

On the other hand, we have

Ωxk,xk(1)=Ωxk(1),xk(1)≥λ_xk(1)2.

Consequently, we have

λ_xk(1)2≤c1T+1xk(1).(4.3)

(4.3) means that xk(1)k=1∞ is bounded.

Then, we shall prove that xk(2)k=1∞ is bounded.

As a matter of fact,

c7≥Ixk=−12xk⊤Ωxk+∑t=1TGt,xk(t)=−12xk(1)⊤Ωxk(1)+∑t=1TGt,xk(t)−Gt,xk(2)(t)+∑t=1TGt,xk(2)(t).

Thus,

∑t=1TGt,xk(2)(t)≤c7+12xk(1)⊤Ωxk(1)+∑t=1TGt,xk(t)−Gt,xk(2)(t)≤c7+λ¯2xk(1)2+∑t=1Tft,xk(1)(t)+ξxk(2)(t)⋅xk(1)(t)≤c7+λ¯2xk(1)2+c1Txk(1)(t),

which means that ∑t=1TGt,xk(2)(t) is bounded. Here ξ ∈ (0, 1).

It comes from condition (G₂) that xk(2)k=1∞ is bounded. If not, assume that xk(2)→+∞ as k → ∞. In that there are c_k ∈ ℝ, k ∈ ℕ, such that xk(2) = (c_k, c_k,⋯, c_k)^⊤ ∈ X, then

xk(2)=∑t=1Txk(2)(t)212=∑t=1Tck212=Tck→+∞,k→∞.

Since Gt,xk(2)(t)=Gt,ck, then Gt,xk(2)(t)→+∞ as k → ∞. This contradicts the fact ∑t=1TGt,xk(2)(t) is bounded. Therefore, the functional I(x) satisfies the Palais-Smale condition. Therefore, it suffices to prove that I(x) satisfies the conditions (I₁) and (I₂) of Saddle Point Theorem.

First, we shall prove the condition (I₂). For any x⁽²⁾ ∈ X₂, x⁽²⁾ = (x⁽²⁾(1), x⁽²⁾(2),⋯, x⁽²⁾(T))^⊤, there is c ∈ ℝ such that

x(2)(i)=c,∀i∈[1,T]Z.

It comes from (G₂) that there is a constant c₈ > 0 such that G(t, c) > 0 for t ∈ ℤ and ∣c∣ > c₈.

Set

c9=minn∈[1,T]Z,|c|≤c8G(t,c),c10=min{0,c9}.

Thus,

G(t,c)≥c10,∀(t,c)∈[1,T]Z×R.

Then

Ix(2)=∑t=1TG(t,x(2)(t))=∑t=1TG(t,c)≥Tc10,∀x(2)∈X2.

Next, we shall prove the condition (I₁). For any x⁽¹⁾ ∈ X₁, by (G1′), we have

Ix(1)=−12x(1)⊤Ωx(1)+∑t=1TGt,x(1)(t)≤−λ_2x(1)2+c1∑t=1Tx(1)(t)+Tc2≤−λ_2x(1)2+c1Tx(1)+Tc2.

Take

ω=Tc10.

Then, there exists a constant ρ > 0 large enough such that

Ix(1)≤ω−1=σ<ω,∀x(1)∈X1,x(1)=ρ.

The conditions of (I₁) and (I₂) of Saddle Point Theorem are satisfied. In the light of Saddle Point Theorem, Theorem 3.1 holds. □

Proof of Theorem 3.3

Let {x_k}_k∈ℕ ⊂ X be such that {I(x_k)}_k∈ℕ is bounded and I′(x_k) → 0 as k → ∞. Accordingly, for any k ∈ ℕ, there is a number c₁₁ > 0 such that

−c11≤Ixk≤c11.

For k large enough, it comes from limk→∞I′xk=0 that

I′xk,xk≤xk.

Since

I′xk,xk=−xk⊤Ωxk+∑t=1Tft,xk(t)xk(t).

Accordingly, for k large enough, we have

c11+12xk≥Ixk−12I′xk,xk=∑t=1TGt,xk(t)−12ft,xk(t)xk(t).

Denote

Γ1=t∈[1,T]Z:xk(t)≥δ;Γ2=t∈[1,T]Z:xk(t)<δ.

Combining with (G₃), we have

c11+12xk≥∑t=1TGt,xk(t)−12∑t∈Γ1Tft,xk(t)xk(t)−12∑t∈Γ2Tft,xk(t)xk(t)≥∑t=1TGt,xk(t)−μ2∑t∈Γ1TGt,xk(t)−12∑t∈Γ2Tft,xk(t)xk(t)=1−μ2∑t=1TGt,xk(t)+12∑t∈Γ2TμGt,xk(t)−ft,xk(t)xk(t).

Set

L(t,x)=μGt,x−ft,xx.

By the continuity of L(t, x) with respect to the first and second variables, we have that there is a constant c₁₂ > 0 such that

L(t,x)≥−c12,

for all t ∈ [1, T]_ℤ and ∣x∣ ≥ δ. Hence,

c11+12xk≥1−μ2∑t=1TGt,xk(t)−12Tc12,x≥δ.

It follows from (G₄) and (2.2) that

c11+12xk≥1−μ2c3∑t=1Txk(t)ν−1−μ2c4T−12Tc12≥1−μ2c3τ1νxkν−1−μ2c4T−12Tc12.

Let

c13=−1−μ2c4T−12Tc12.

Then,

c11+12xk≥1−μ2c3τ1νxkν+c13.

Thus,

1−μ2c3τ1νxkν−12xk≤c11−c13,

which means that xkk=1∞ is bounded. For the reason that X is a finite dimensional space, xkk=1∞ possesses a convergent subsequence. Accordingly, the Palais-Smale condition is proved.

To exploit the Saddle Point Theorem, we shall prove that the functional I satisfies the conditions (I₁) and (I₂).

First, we shall prove that the functional I satisfies the condition (I₂). For any x⁽²⁾ ∈ X₂, in that Ωx⁽²⁾ = 0, we have

Ix(2)=∑t=1TGt,x(2)(t).

On account of (G₄),

Ix(2)≥c3∑t=1Tx(2)(t)ν−c4T≥−c4T.

Then, we shall prove the condition (I₁). For any x⁽¹⁾ ∈ X₁, combining with (G₄), we have

Ix(1)=−12x(1)⊤Ωx(1)+∑t=1TGt,x(1)(t)≤−λ_2x(1)2+c3∑t=1Tx(1)(t)ν+Tc4≤−λ_2x(1)2+c3τ2νTx(1)ν+Tc4.

Take

ω=−c4T.

Owing to 1 < ν < 2, there exists a constant ρ > 0 large enough such that

Ix(1)≤ω−1=σ<ω,∀x(1)∈X1,x(1)=ρ.

Hence, the condition (I₁) is satisfied.

As a result of Saddle Point Theorem, the BVP (1.1),(1.2) possesses at least one solution. The proof is complete. □

Remark 4.1

In the light of Theorems 3.1 and 3.3, the results of Corollaries 3.5 and 3.6 are obviously true.

5 Example

In this section, an example is given to illustrate our main result.

Example 5.1

Fort ∈ [1,3]_ℤ, suppose that

Δ4x(t−2)−2Δ(t−2)2Δx(t−1)=νx(t)|x(t)|ν−2+μx(t)|x(t)|μ−2,(5.1)

satisfies the boundary value conditions

x(−1)=x(2),Δx(−1)=Δx(2),Δ2x(−1)=Δ2x(2),Δ3x(−1)=Δ3x(2).(5.2)

We have

r(t)=2t2,t∈[1,3]Z,

with

r(0)=18,

and

f(t,x)=νx|x|ν−2+μx|x|μ−2,G(t,x)=|x|ν+|x|μ.

Besides,

Ω=26−5−21−516−11−21−1132,

and the eigenvalues ofΩare λ₁ = 0, λ₂ = 23 and λ₃ = 51. It is obvious that all the conditions of Theorem 3.3 are satisfied and then the BVP (5.1),(5.2) possesses at least one nontrivial solution.

Funding This project is supported by the National Natural Science Foundation of China (No. 11501194).

Acknowledgement

The authors are extremely grateful to the referees and the editors for their careful reading and making some valuable comments and suggestions on the manuscript. This work was carried out while visiting Central South University. The author Haiping Shi wishes to thank Professor Xianhua Tang for his invitation.

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Received: 2018-07-09

Accepted: 2018-08-31

Published Online: 2018-12-26

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Artikel in diesem Heft

https://doi.org/10.1515/math-2018-0121

Schlagwörter für diesen Artikel

Boundary value problems; Discrete problems; Generalized beam equation; Saddle point theorem; Variational methods

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