Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems

Volume 2009, Article ID 362983,16pages doi:10.1155/2009/362983

Research Article

Existence and Uniqueness of

Solutions for Higher-Order Three-Point

Boundary Value Problems

Minghe Pei

_{and Sung Kag Chang}

1_{Department of Mathematics, Bei Hua University, JiLin 132013, China}

2_{Department of Mathematics, Yeungnam University, Kyongsan 712-749, South Korea}

Correspondence should be addressed to Sung Kag Chang,skchang@ynu.ac.kr

Received 5 February 2009; Accepted 14 July 2009

Recommended by Kanishka Perera

We are concerned with the higher-order nonlinear three-point boundary value problems:xn ft, x, x, . . . , xn−1_{, n≥3,with the three point boundary conditionsgxa, xa, . . . , x}n−1_a 0;xi_{b μ}

i, i0,1, . . . , n−3;hxc, xc, . . . , xn−1c 0,wherea < b < c, f:a, c×Rn → R −∞,∞is continuous,g, h : Rn _{→ Rare continuous, andμ}

i ∈ R, i 0,1, . . . , n−3 are

arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.

Copyrightq2009 M. Pei and S. K. Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Higher-order boundary value problems were discussed in many papers in recent years; for instance, see1–22and references therein. However, most of all the boundary conditions in the above-mentioned references are for two-point boundary conditions2–11,14,17–22, and three-point boundary conditions are rarely seen1,12,13,16,18. Furthermore works for nonlinear three point boundary conditions are quite rare in literatures.

The purpose of this article is to study the existence and uniqueness of solutions for higher order nonlinear three point boundary value problem

with nonlinear three point boundary conditions

gxa, xa, . . . , xn−1a0,

xib μi, i0,1, . . . , n−3,

hxc, xc, . . . , xn−1_c0,

1.2

wherea < b < c,f:a, c×Rn _{→ R −∞,∞is a continuous function,g, h:R}n _{→ Rare}

continuous functions, andμi∈R, i0,1, . . . , n−3 are arbitrary given constants. The tools we

mainly used are the method of upper and lower solutions and Leray-Schauder degree theory. Note that for the cases ofaborbcin the boundary conditions1.2, our theorems hold also true. However, for brevity we exclude such cases in this paper.

2. Preliminary

In this section, we present some definitions and lemmas that are needed to our main results.

Definition 2.1. αt, βt ∈ Cn_{a, care called lower and upper solutions of BVP1.1,1.2,}

respectively, if

αn_{t≥ft, αt, αt, . . . , α}n−1_{t, t∈a, c,}

gαa, αa, . . . , αn−1_a≤0, αi_b≤μ

i, i0,1, . . . , n−3,

hαc, αc, . . . , αn−1_c≤0,

βn_{t≤ft, βt, βt, . . . , β}n−1_{t, t∈a, c,}

gβa, βa, . . . , βn−1_a≥0, βi_b≥μ

i, i0,1, . . . , n−3,

hβc, βc, . . . , βn−1_c≥0.

2.1

Definition 2.2. Let Ebe a subset ofa, c×Rn_{. We say thatft, x}

0, x1, . . . , xn−1satisfies the

Nagumo condition onEif there exists a continuous functionφ:0,∞ → 0,∞such that

_{ft, x0}, x1, . . . , xn−1≤φ|xn−1|, t, x0, x1, . . . , xn−1∈E, _∞

0

sds φs ∞.

2.2

Lemma 2.3see10. Letf : a, c×Rn _{→ Rbe a continuous function satisfying the Nagumo}

condition on

whereγit,Γit:a, c → Rare continuous functions such that

γit≤Γit, i0,1, . . . , n−2, t∈a, c. 2.4

Then there exists a constant r > 0 (depending only on γn−2t,Γn−2tand φtsuch that every

solutionxtof 1.1with

γit≤xit≤Γit, i0,1, . . . , n−2, t∈a, c 2.5

satisfiesxn−1 ∞≤r.

Lemma 2.4. Letφ:0,∞ → 0,∞be a continuous function. Then boundary value problem

xn_xn−2_φxn−1_{, t∈a, c, 2.6}

xn−2a xib xn−2c 0, i0,1, . . . , n−3 2.7

has only the trivial solution.

Proof. Suppose thatx0tis a nontrivial solution of BVP2.6,2.7. Then there existst0 ∈

a, csuch thatx₀n−2t0 > 0 orx₀n−2t0 < 0. We may assumex₀n−2t0 > 0. There exists

t1∈a, csuch that

max

t∈a,cx n−2

0 t:x

n−2

0 t1>0. 2.8

Thenx₀n−1t1 0,x₀nt1≤0. From2.6we have

0≥x₀nt1 x₀n−2t1φx₀n−1t1

>0, 2.9

which is a contradiction. Hence BVP2.6,2.7has only the trivial solution.

3. Main Results

We may now formulate and prove our main results on the existence and uniqueness of solutions fornth-order three point boundary value problem1.1,1.2.

Theorem 3.1. Assume that

ithere exist lower and upper solutionsαt, βtof BVP1.1,1.2, respectively, such that

−1n−iαi_t≤−1n−i_βi_{t, t∈a, b, i0,1, . . . , n−2,}

iift, x0, . . . , xn−1is continuous ona, c×Rn,−1n−ift, x0, . . . , xn−1is nonincreasing in

xii0,1, . . . , n−3onDba, andft, x0, . . . , xn−1is nonincreasing inxi i0,1, . . . , n−

3onDc

band satisfies the Nagumo condition onDca, where

ϕit min

αi_{t, β}i_{t , ψ}

it max

αi_{t, β}i_{t , i0, . . . , n−2,}

Db

a

t, x0, . . . , xn−1∈a, b×Rn :ϕit≤xi≤ψit, i0, . . . , n−2 ,

Dc

b

t, x0, . . . , xn−1∈b, c×Rn:ϕit≤xi≤ψit, i0, . . . , n−2 ,

Dc

a

t, x0, . . . , xn−1∈a, c×Rn:ϕit≤xi≤ψit, i0, . . . , n−2 ;

3.2

iiigx0, x1, . . . , xn−1is continuous onRn, and−1n−igx0, x1, . . . , xn−1is nonincreasing

inxi i0,1, . . . , n−3and nondecreasing inxn−1onni−02ϕia, ψia×R;

ivhx0, x1, . . . , xn−1is continuous onRn, and nonincreasing inxi i0,1, . . . , n−3and

nondecreasing inxn−1onin−02ϕic, ψic×R.

Then BVP1.1,1.2has at least one solutionxt∈Cn_{a, csuch that for eachi0,1, . . . , n−2,}

−1n−iαit≤−1n−ixit≤−1n−iβit, t∈a, b,

αit≤xit≤βit, t∈b, c.

3.3

Proof. For eachi0,1, . . . , n−2 define

wit, x ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ψit, x > ψit,

x, ϕit≤x≤ψit,

ϕit, x < ϕit,

3.4

whereϕit min{αit, βit},ψit max{αit, βit}.

Forλ∈0,1, we consider the auxiliary equation

xnt λft, w0t, xt, . . . , wn−2

t, xn−2t, xn−1t

xn−2t−λwn−2

t, xn−2tφxn−1t,

3.5

whereφis given by the Nagumo condition, with the boundary conditions

xn−2a λwn−2

a, xn−2a−gw0a, xa, . . . , wn−2

a, xn−2a, xn−1a,

xib λμi, i0,1, . . . , n−3,

xn−2c λwn−2

c, xn−2c−hw0c, xc, . . . , wn−2

Then we can choose a constantMn−2>0 such that

−Mn−2< αn−2t≤βn−2t< Mn−2, t∈a, c, 3.7

ft, αt, . . . , αn−2t,0−Mn−2αn−2t

φ0<0, t∈a, c,

ft, βt, . . . , βn−2t,0Mn−2−βn−2t

φ0>0, t∈a, c,

3.8

αn−2_{a−gαa, . . . , α}n−2_{a,0< M} n−2,

βn−2a−gβa, . . . , βn−2a,0< Mn−2,

3.9

αn−2c−hαc, . . . , αn−2c,0< Mn−2,

βn−2c−hβc, . . . , βn−2c,0< Mn−2.

3.10

In the following, we will complete the proof in four steps.

Step 1. Show that every solutionxtof BVP3.5,3.6satisfies

xn−2t< Mn−2, t∈a, c, 3.11

independently ofλ∈0,1.

Suppose that the estimate|xn−2_{t| < M}

n−2 is not true. Then there existst0 ∈ a, c

such thatxn−2_t

0 ≥Mn−2 orxn−2t0≤ −Mn−2. We may assumexn−2t0 ≥Mn−2 . There

existst1∈a, csuch that

max

t∈a,cx

n−2_t:xn−2_t

1≥Mn−2>0. 3.12

There are three cases to consider.

Case 1t1 ∈a, c. In this case,xn−1t1 0 andxnt1≤0. Forλ ∈0,1, by3.8, we get

the following contradiction:

0≥xnt1

λft1, w0t1, xt1, . . . , wn−2

t1, xn−2t1

, xn−1t1

xn−2t1−λwn−2

t1, xn−2t1

φxn−1t1

λft1, w0t1, xt1, . . . , wn−3

t1, xn−3t1

, βn−2t1,0

xn−2t1−λβn−2t1

φ0

≥λft1, βt1, . . . , βn−2t1,0

Mn−2−βn−2t1

φ0>0,

and forλ0, we have the following contradiction:

0≥xnt1 xn−2t1φ0≥Mn−2φ0>0. 3.14

Case 2t1a. In this case,

max

t∈a,cx

n−2_t:xn−2_a≥M

n−2>0, 3.15

andxn−1_{a≤0. Forλ0, by3.6we have the following contradiction:}

0< Mn−2≤xn−2a 0. 3.16

Forλ∈0,1, by3.9and conditioniiiwe can get the following contradiction:

Mn−2≤xn−2a,

λwn−2

a, xn−2a−gw0a, xa, . . . , wn−2

a, xn−2a, xn−1a,

≤λβn−2a−gβa, . . . , βn−2a,0< Mn−2.

3.17

Case 3t1c. In this case,

max

t∈a,cx

n−2_t:xn−2_c≥M

n−2>0, _3.18

andxn−1_{c≥0. Forλ0, by3.6we have the following contradiction:}

0< Mn−2≤xn−2c 0. 3.19

Forλ∈0,1, by3.10and conditionivwe can get the following contradiction:

Mn−2≤xn−2c,

λwn−2

c, xn−2c−hw0c, xc, . . . , wn−2

c, xn−2c, xn−1c

≤λβn−2c−hβc, . . . , βn−2c,0< Mn−2.

3.20

By3.6, the estimates

xit< Mi: c−aMi1μi, i0,1, . . . , n−3, t∈a, c 3.21

Step 2. Show that there exists Mn−1 > 0 such that every solution xt of BVP 3.5, 3.6

satisfies

xn−1t< Mn−1, t∈a, c, 3.22

independently ofλ∈0,1. Let

E{t, x0, . . . , xn−1∈a, c×Rn:|xi| ≤Mi, i0,1, . . . , n−2}, 3.23

and define the functionFλ:a, c×Rn → Ras follows:

Fλt, x0, . . . , xn−1 λft, w0t, x0, . . . , wn−2t, xn−2, xn−1

xn−2−λwn−2t, xn−2φ|xn−1|.

3.24

In the following, we show that Fλt, x0, . . . , xn−1 satisfies the Nagumo condition on E,

independently ofλ∈0,1. In fact, sincefsatisfies the Nagumo condition onDc

a, we have

|Fλt, x0, . . . , xn−1|λft, w0t, x0, . . . , wn−2t, xn−2, xn−1

xn−2−λwn−2t, xn−2φ|xn−1| ≤12Mn−2φ|xn−1|:φE|xn−1|.

3.25

Furthermore, we obtain

_∞

0

s φEsds

_∞

0

s

12Mn−2φsds ∞. 3.26

Thus,Fλsatisfies the Nagumo condition onE, independently ofλ∈0,1. Let

γit −Mi, Γit Mi, i0,1, . . . , n−2, t∈a, c. 3.27

ByStep 1andLemma 2.3, there existsMn−1>0 such that|xn−1t|< Mn−1fort∈a, c. Since

Mn−2andφEdo not depend onλ, the estimate|xn−1t|< Mn−1ona, cis also independent

ofλ.

Step 3. Show that forλ1, BVP3.5,3.6has at least one solutionx1t.

Define the operators as follows:

by

Lxxnt, xn−2a, xb, . . . , xn−3b, xn−2c,

Nλ:Cn−1a, c−→Ca, c×Rn,

3.29

by

Nλx

Fλ

t, xt, . . . , xn−1_{t, A}

λ, λμ0, . . . , λμn−3, Cλ

, 3.30

with

Aλ:λ

wn−2

a, xn−2_a−gw

0a, xa, . . . , wn−2

a, xn−2_{a, x}n−1_a

Cλ:λ

wn−2

c, xn−2c−hw0c, xc, . . . , wn−2

c, xn−2c, xn−1c. 3.31

SinceL−1_{is compact, we have the following compact operator:}

Tλ:Cn−1a, c−→Cn−1a, c, 3.32

defined by

Tλx L−1Nλx. 3.33

Consider the setΩ {x∈Cn−1_{a, c:x}i

∞< Mi, i0,1, . . . , n−1}.

By Steps1and2, the degree degI−Tλ,Ω,0is well defined for everyλ∈0,1,and

by homotopy invariance, we get

degI−T0,Ω,0 degI−T1,Ω,0. 3.34

Since the equationx T0x has only the trivial solution fromLemma 2.4, by the degree

theory we have

degI−T1,Ω,0 degI−T0,Ω,0 ±1. 3.35

Hence, the equationxT1xhas at least one solution. That is, the boundary value problem

xnt ft, w0t, xt, . . . , wn−2

t, xn−2t, xn−1t

xn−2_t−w n−2

t, xn−2_tφxn−1_t,

with the boundary conditions

xn−2a wn−2

a, xn−2a−gw0a, xa, . . . , wn−2

a, xn−2a, xn−1a,

xib μi, i0,1, . . . , n−3,

xn−2cwn−2

c, xn−2c−hw0c, xc, . . . , wn−2

c, xn−2c, xn−1c,

3.37

has at least one solutionx1tinΩ.

Step 4. Show thatx1tis a solution of BVP1.1,1.2.

In fact, the solutionx1tof BVP3.36,3.37will be a solution of BVP1.1,1.2, if it

satisfies

ϕit≤x₁it≤ψit, i0,1, . . . , n−2, t∈a, c. 3.38

By contradiction, suppose that there existst0 ∈ a, csuch thatx₁n−2t0 > ψn−2t0. There

existst1∈a, csuch that

max

t∈a,c

x₁n−2t−ψn−2t

:x₁n−2t1−ψn−2t1>0. 3.39

Now there are three cases to consider.

Case 1t1∈a, c. In this case, sinceψn−2t βn−2tona, c, we havex₁n−1t1 βn−1t1

andx₁nt1≤βnt1. By conditionsiandii, we get the following contradiction:

0≥x₁nt1−βnt1

≥ft1, w0t1, x1t1, . . . , wn−2

t1, x1n−2t1

, x₁n−1t1

x₁n−2t1−wn−2

t1, x₁n−2t1

φx₁n−1t1

−ft1, βt1, . . . , βn−1t1

≥ft1, βt1, . . . , βn−1t1

x₁n−2t1−βn−2t1

φx₁n−1t1

−ft1, βt1,· · ·, βn−1t1

x₁n−2t1−βn−2t1

φx₁n−1t1

>0.

3.40

Case 2t1a. In this case, we have

max

t∈a,c

x₁n−2t−ψn−2t

:x₁n−2a−βn−2_a>0,

and x₁n−1a ≤ βn−1_{a. By 3.37 and conditionsi and iii we can get the following} contradiction:

βn−2_{a< x}n−2

1 a,

wn−2

a, x₁n−2a−gw0a, x1a, . . . , wn−2

a, x₁n−2a, x₁n−1a

≤βn−2a−gβa, . . . , βn−2a, βn−1a≤βn−2a.

3.42

Case 3t1c. In this case, we have

max

t∈a,c

x₁n−2t−ψn−2t

:x₁n−2c−βn−2c>0, 3.43

and x₁n−1c ≥ βn−1_{c. By 3.37 and conditions i and iv we can get the following} contradiction:

βn−2c< x₁n−2c

wn−2

c, x₁n−2c−hw0c, x1c, . . . , wn−2

c, x₁n−2c, x₁n−1c

≤βn−2c−hβc, . . . , βn−2c, βn−1c≤βn−2c.

3.44

Similarly, we can show thatϕn−2t≤x1n−2tona, c. Hence

αn−2t ϕn−2t≤x₁n−2t≤ψn−2t βn−2t, t∈a, c. 3.45

Also, by boundary condition3.37and conditioni, we have

αib x₁ib βib, in−1−2j, j1,2, . . . ,

n−1 2

,

αib≤x₁ib≤βib, in−2−2j, j1,2, . . . ,

n−2 2

.

3.46

Therefore by integration we have for eachi0,1, . . . , n−2,

−1n−iαit≤−1n−ix₁it≤−1n−iβit, t∈a, b,

αi_t≤xi

1 t≤βit, t∈b, c,

that is,

ϕit≤x₁it≤ψit, i0,1, . . . , n−2, t∈a, c. 3.48

Hencex1tis a solution of BVP1.1,1.2and satisfies3.3.

Now we give a uniqueness theorem by assuming additionally the differentiability for functionsf,gandh, and a kind of estimating condition inTheorem 3.1.

Theorem 3.2. Assume that

ithere exist lower and upper solutionsαt, βtof BVP1.1,1.2, respectively, such that

−1n−iαi_t≤−1n−i_βi_{t, t∈a, b, i0,1, . . . , n−2,}

αi_t≤βi_{t, t∈b, c, i0,1, . . . , n−2;} 3.49

iift, x0, . . . , xn−1 and its first-order partial derivatives in xi i 0,1, . . . , n −1 are

continuous ona, c×Rn_,−1n−i_∂f/∂x

i ≤ 0i 0,1, . . . , n−3onDab,∂f/∂xi ≤

0 i0,1, . . . , n−3onDc

band satisfy the Nagumo condition onDca;

iiigx0, x1, . . . , xn−1 is continuous on Rn and continuously partially differentiable on n−2

i0ϕia, ψia×R, and

−1n−i∂g ∂xi ≤

0, i0,1, . . . , n−3,

∂g

∂xn−1 ≤0, on n−2

i0

ϕia, ψia

×R;

3.50

ivhx0, x1, . . . , xn−1 is continuous on Rn and continuously partially differentiable on n−2

i0ϕic, ψic×R, and

∂h ∂xi ≤

0, i0,1, . . . , n−3,

∂h ∂xn−1 ≥

0, on

n−2

i0

ϕic, ψic

×R;

vthere exists a functionγt∈Cn_{a, csuch thatγ}n−2_{t>0ona, c,and}

γnt<

n−1

i0

∂f ∂xi ·

γit, onDc_a

n−1

i0

∂g ∂xi ·

γia>0, on

n−2

i0

ϕia, ψia

×R,

n−1

i0

∂h ∂xi ·

γic>0, on

n−2

i0

ϕic, ψic

×R,

γib 0, ifn−i: odd, i0,1, . . . , n−3,

γib≥0, ifn−i: even, i0,1, . . . , n−3.

3.52

Then BVP1.1,1.2has a unique solutionxtsatisfying3.3.

Proof. The existence of a solution for BVP 1.1, 1.2 satisfying 3.3 follows from

Theorem 3.1.

Now, we prove the uniqueness of solution for BVP1.1,1.2. To do this, we letx1t

andx2tare any two solutions of BVP1.1,1.2satisfying3.3. Letzt x2t−x1t. It

is easy to show thatztis a solution of the following boundary value problem

znt

n−1

i0

ditzit, 3.53

n−1

i0

aizia 0,

n−1

i0

cizic 0, 3.54

zib 0, i0,1, . . . , n−3, 3.55

where for eachi0,1, . . . , n−1,

dit ₁

0

∂ ∂xi

ft, x1t θzt, x1t θzt, . . . , x n−1

1 t θz

n−1_tdθ,

ai ₁

0

∂ ∂xi

gx1a θza, x1a θza, . . . , x n−1

1 a θzn−1a dθ, ci ₁ 0 ∂ ∂xi

hx1c θzc, x1c θzc, . . . , x n−1

1 c θzn−1c

dθ.

By conditionsii,iii, andiv, we have thatdit∈Ca, c, i0,1, . . . , n−3,and

−1n−idit≤0, i0,1, . . . , n−3, t∈a, b,

dit≤0, i0,1, . . . , n−3, t∈b, c,

−1n−iai≤0, i0,1, . . . , n−3, an−1≤0,

ci≤0, i0,1, . . . , n−3, cn−1≥0.

3.57

Now suppose that there exists t0 ∈ a, c such that zn−2t0/0. Without loss of

generality assumezn−2_t

0>0, and let

Ω M:Mzn−2t< γn−2t, t∈a, c. 3.58

It is easy to see that 0 ∈ Ωby condition v, hence Ω/∅. Let M0 supΩ. We have that

0 < M0 < ∞,M0zn−2t ≤ γn−2tona, c, and there exists a pointt1 ∈a, csuch that

M0zn−2t1 γn−2t1. Furthermoret1/a, c. In fact, ift1 a, thenM0zn−1a≤ γn−1a.

By conditionvand3.55we can easily show that

−1n−iM0zit−γit

≤0, i0,1, . . . , n−3, t∈a, b. 3.59

In particular

−1n−iM0zia−γia

≤0, i0,1, . . . , n−3. 3.60

Hence

n−1

i0

M0aizia≥ n−1

i0

aiγia>0, 3.61

which contradicts to 3.54. Thus t1/a. Similarly we can show that t1/c. Consequently

M0zn−1t1 γn−1t1.

Now, there are two cases to consider, that is

t1∈a, b or t1∈b, c. 3.62

Ift1∈a, b, then by3.59we have

−1n−iM0zit1−γit1

Thus, by3.53and conditionvwe have

M0znt1 n−1

i0

M0dit1zit1≥ n−1

i0

dit1γit1> γnt1. 3.64

Consequently, by Taylor’s theorem there existst2∈t1, csuch that

M0zn−2t> γn−2t, ∀t∈t1, t2, 3.65

which is a contradiction.

A similar contradiction can be obtained ift1 ∈b, c. Hencezn−2t ≡0 ona, c. By

3.55, we obtainzt≡0 ona, c. This completes the proof of the theorem.

Next we give two examples to demonstrate the application ofTheorem 3.2.

Example 3.3. Consider the following third-order three point BVP:

x−tx2t2_1x1

3x

3_−t4_{sintx, t∈−1,1, 3.66}

13x−1 x−13−x−1 13 0,

x0 0,

−1−x1 2x1 x13x1 13 0.

3.67

Let

ft, x0, x1, x2 −tx0

2t2_1x 1

1 3x

3

1−t4sintx2,

gx0, x1, x2 13x1x31−x213,

hx0, x1, x2 −1−x02x1x31 x213.

3.68

Choose αt −t, βt t and γt t. It is easy to check that αt −t, and βt t are lower and upper solutions of BVP3.66,3.67respectively, and all the assumptions in

Theorem 3.2are satisfied. Therefore byTheorem 3.2BVP3.66,3.67has a unique solution xxtsatisfying

t≤xt≤ −t, t∈−1,0, −t≤xt≤t, t∈0,1,

Example 3.4. Consider the following fourth-order three point BVP:

x4−t2xxx3, t∈−1,1, 3.70

−x−1 x−1313x−1 0, x0 0, x0 0,

−x1−4x1 x129x1 0.

3.71

Let

ft, x0, x1, x2, x3 −t2x0x2x32,

gx0, x1, x2, x3 −x0x3113x2,

hx0, x1, x2, x3 −x0−4x1x2₁9x2.

3.72

Chooseαt −t2_{, βt t}2 _{andγt t}2_{. It is easy to check thatαt −t}2_{, andβt t}2

are lower and upper solutions of BVP3.70,3.71, respectively, and all the assumptions in

Theorem 3.2are satisfied. Therefore byTheorem 3.2BVP3.70,3.71has a unique solution xxtsatisfying

−2≤xt≤2, t∈−1,1,

2t≤xt≤ −2t, t∈−1,0, −2t≤xt≤2t, t∈0,1,

−t2≤xt≤t2, t∈−1,1.

3.73

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