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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

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

Xia Liu
• Corresponding author
• College of Continuing Education and Open College, Guangdong University of Foreign Studies, Guangzhou, 510420, China
• Science College, Hunan Agricultural University, Changsha, 410128, China
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• Other articles by this author:
/ Tao Zhou
/ Haiping Shi
• Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou, 510440, China
• School of Mathematics and Statistics, Central South University, Changsha, 410083, China
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• Other articles by this author:
Published Online: 2018-12-26 | DOI: https://doi.org/10.1515/math-2018-0121

## 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.

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), Δk x(t) = Δ(Δk−1 x(t)) (2 ≤ k ≤ 4), Δ0 x(t) = x(t), define [1, T] := [1, T] ∩ ℤ, r(t) ∈ ℝT+1 with r(0) = r(T), f(t, x) ∈ C([1, T] × 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 ℝN 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] × 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 PQ be an open subset of Q. A function I : PR is called Fréchet differentiable at xP if there exists a bounded linear operator Lx: QR such that

$limh→0I(x+h)−I(x)−Lx(h)RhQ=0.$

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

Let X be a real Banach space and IC1(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 $\begin{array}{}{\left\{{x}_{k}\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$X for which $\begin{array}{}{\left\{I\left({x}_{k}\right)\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ is bounded and I′(xk)arrow 0 as karrow ∞ possesses a convergent subsequence. Here, the sequence $\begin{array}{}{\left\{{x}_{k}\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ is called a Palais-Smale sequence.

Let X be a real Banach space. We denote by the symbol Br the open ball in X about 0 of radius r, Br its boundary, and 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 xX, 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 xX and q > 1.

As usual, we use ∥x∥ = ∥x2 for the Euclidean norm. Since ∥xq and ∥x∥ are equivalent, there are numbers τ1, τ2 such that τ2τ1 > 0, and

$τ1∥x∥≤∥x∥q≤τ2∥x∥,∀x∈X.$(2.2)

#### Remark 2.1

For any xX, 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, X is isomorphic toT. In the later sections of this article, when we write x = (x(1), x(2),⋯,x(T)) ∈ ℝT, we always imply that x can be extended to a vector in X so that (2.3) is satisfied.

For any xX, 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 IC1(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 xX 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]). Let X be a real Banach space, X = X1X2, where X1 ≠ {0} and is finite dimensional. Assume that IC1(X,ℝ) satisfies the Palais-Smale condition and

• (I1)

there exist two constants σ, ρ > 0 such that IBρX1σ;

• (I2)

there exists eBρX1 and a constant ω > σ such that Ie+X2ω.

Then I possesses a critical value cω, 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:

• (r)

for any t ∈ [0, T], r(t) ≥ 0;

• (G1)

there is a positive constant c1 such that

$|f(t,x)|≤c1,∀(t,x)∈[1,T]Z×R;$

• (G2)

G(t, x) → +∞ uniformly for t ∈ [1, T] asx∣ → +∞.

Then the BVP (1.1),(1.2) possesses at least one solution.

#### Remark 3.2

Condition (G1) means that there is a positve constant c2 such 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:

• (G3)

there are two constants 1 < μ < 2 and δ > 0 such that

$0

• (G4)

there are three constants c3 > 0, c4 > 0 and 1 < νμ such 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 constants c5 and c6 such that

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

In the case that f(t, x) is independent of x, 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)

where fC(ℝ, ℝ).

#### Corollary 3.5

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

• (H1)

there is a function H(x) ∈ C1(ℝ,ℝ) such that

$H′(x)=f(x);$

• (H2)

there is a positive constant κ1 such that

$|f(x)|≤κ1,∀x∈R;$

• (H3)

H(x) → +∞ asx∣ → +∞.

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:

• (H4)

there are two constants 1 < μ̃ < 2 and δ̃ > 0 such that

$0

• (H5)

there are three constants κ2 > 0, κ3 > 0 and ν̃ with 1 < ν̃μ̃ such that

$H(x)≥κ2|x|ν~−κ3,∀x∈R.$

Then the BVP (3.1),(1.2) possesses at least one solution.

Let Ω satisfy:

• (A1)

Ω is a symmetric and positive semidefinite matrix;

• (A2)

λ1 = 0 is a simple eigenvalue of Ω with multiplicity one and with the eigenvector e1 = [1, 1,⋯,1].

#### Remark 3.7

1. 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.

2. 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,⋯, λT.

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)Xd ∈ 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 {xk}k∈ℕX be such that {I(xk)}k∈ℕ is bounded and I′(xk) → 0 as k → ∞. Accordingly, for any k ∈ ℕ, there is a number c7 > 0 such that

$−c7≤Ixk≤c7.$

Let $\begin{array}{}{x}_{k}={x}_{k}^{\left(1\right)}+{x}_{k}^{\left(2\right)}\in {X}_{1}\oplus {X}_{2}.\end{array}$ 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 $\begin{array}{}{\left\{{x}_{k}^{\left(1\right)}\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ is bounded.

Then, we shall prove that $\begin{array}{}{\left\{{x}_{k}^{\left(2\right)}\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ 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 $\begin{array}{}\left\{\sum _{t=1}^{T}G\left(t,{x}_{k}^{\left(2\right)}\left(t\right)\right)\right\}\end{array}$ is bounded. Here ξ ∈ (0, 1).

It comes from condition (G2) that $\begin{array}{}{\left\{{x}_{k}^{\left(2\right)}\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ is bounded. If not, assume that $\begin{array}{}∥{x}_{k}^{\left(2\right)}∥\to +\mathrm{\infty }\end{array}$ as k → ∞. In that there are ck ∈ ℝ, k ∈ ℕ, such that $\begin{array}{}{x}_{k}^{\left(2\right)}\end{array}$ = (ck, ck,⋯, ck)X, then

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

Since $\begin{array}{}G\left(t,{x}_{k}^{\left(2\right)}\left(t\right)\right)=G\left(t,{c}_{k}\right),\end{array}$ then $\begin{array}{}G\left(t,{x}_{k}^{\left(2\right)}\left(t\right)\right)\to +\mathrm{\infty }\end{array}$ as k → ∞. This contradicts the fact $\begin{array}{}\left\{\sum _{t=1}^{T}G\left(t,{x}_{k}^{\left(2\right)}\left(t\right)\right)\right\}\end{array}$ 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 $\begin{array}{}\left({G}_{1}^{\prime }\right),\end{array}$ 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 {xk}k∈ℕX be such that {I(xk)}k∈ℕ is bounded and I′(xk) → 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 $\begin{array}{}\underset{k\to \mathrm{\infty }}{lim}{I}^{\prime }\left({x}_{k}\right)=0\end{array}$ 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] 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 $\begin{array}{}{\left\{∥{x}_{k}∥\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ is bounded. For the reason that X is a finite dimensional space, $\begin{array}{}{\left\{{x}_{k}\right\}}_{k=1}^{\mathrm{\infty }}\end{array}$ 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 Ω x(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

For t ∈ [1,3], 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 of Ω are 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.

## 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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Accepted: 2018-08-31

Published Online: 2018-12-26

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

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1412–1422, ISSN (Online) 2391-5455,

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