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Volume 2009, Article ID 362983, 16 pagesdoi:10.1155/2009/362983 Research Article Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems 1 Department o

Volume 2009, Article ID 362983, 16 pages

doi:10.1155/2009/362983

Research Article

Existence and Uniqueness of

Solutions for Higher-Order Three-Point

Boundary Value Problems

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: x n 

f t, x, x, , x n−1 , n ≥ 3, with the three point boundary conditions gxa, xa, , x n−1 a  0; x i b  μ i, i  0, 1, , n − 3; hxc, xc, , x n−1 c  0, where a < b < c, f : a, c ×Rn →

R  −∞, ∞ is continuous, g, h : R n → R are 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

Copyrightq 2009 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

the above-mentioned references are for two-point boundary conditions2 11,14,17–22, and

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

x n  ft, x, x, , x n−1

with nonlinear three point boundary conditions

g

x a, xa, , x n−1 a 0,

x i b  μ i , i  0, 1, , n − 3,

h

x c, xc, , x n−1 c 0,

1.2

where a < b < c, f : a, c × R n → R  −∞, ∞ is a continuous function, g, h : R n → R are

continuous functions, and μ i ∈ R, i  0, 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 of a  b or b  c in the boundary conditions 1.2, 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 α t, βt ∈ C n a, c are called lower and upper solutions of BVP 1.1, 1.2, respectively, if

α n t ≥ ft, α t, αt, , α n−1 t, t ∈ a, c,

g

α a, αa, , α n−1 a≤ 0,

α i b ≤ μ i , i  0, 1, , n − 3,

h

α c, αc, , α n−1 c≤ 0,

β n t ≤ ft, β t, βt, , β n−1 t, t ∈ a, c,

g

β a, βa, , β n−1 a≥ 0,

β i b ≥ μ i , i  0, 1, , n − 3,

h

β c, βc, , β n−1 c≥ 0.

2.1

Definition 2.2 Let E be a subset of a, c × R n We say that ft, x0 , x1, , x n−1 satisfies the

f t, x0 , x1, , x n−1 ≤ φ|xn−1|, t, x0, x1, , x n−1 ∈ E,

∞

0

sds

2.2

Lemma 2.3 see 10 Let f : a, c × R n → R be a continuous function satisfying the Nagumo

condition on

Et, x0 , x1, , x n−1 ∈ a, c × Rn : γ i t ≤ x i≤ Γi t, i  0, 1, , n − 2 , 2.3

where γ i t, Γ i t : a, c → R are continuous functions such that

Then there exists a constant r > 0 (depending only on γ n−2t, Γ n−2t and φt such that every

solution x t of 1.1 with

satisfies x n−1∞≤ r.

Lemma 2.4 Let φ : 0, ∞ → 0, ∞ be a continuous function Then boundary value problem

x n−2 a  x i b  x n−2 c  0, i  0, 1, , n − 3 2.7

has only the trivial solution.

Proof Suppose that x0t is a nontrivial solution of BVP 2.6, 2.7 Then there exists t0 ∈

a, c such that x n−20 t0 > 0 or x n−20 t0 < 0 We may assume x n−20 t0 > 0 There exists

t1∈ a, c such that

max

Then x n−10 t1  0, x n0 t1 ≤ 0 From 2.6 we have

0≥ x0n t1  x n−20 t1φx n−1

0 t1

> 0, 2.9 which is a contradiction Hence BVP2.6, 2.7 has only the trivial solution

3 Main Results

We may now formulate and prove our main results on the existence and uniqueness of

solutions for nth-order three point boundary value problem1.1, 1.2

Theorem 3.1 Assume that

i there exist lower and upper solutions αt, βt of BVP 1.1, 1.2, respectively, such that

−1n −i α i t ≤ −1 n −i β i t, t ∈ a, b, i  0, 1, , n − 2,

ii ft, x0 , , x n−1 is continuous on a, c×Rn ,−1n −i f t, x0 , , x n−1 is nonincreasing in

x i i  0, 1, , n−3 on D b

a , and f t, x0 , , x n−1 is nonincreasing in x i i  0, 1, , n−

3 on Dc

b and satisfies the Nagumo condition on D c

a , where

ϕ i t  minα i t, β i t , ψ i t  maxα i t, β i t , i  0, , n − 2,

D b

at, x0 , , x n−1 ∈ a, b × Rn : ϕ i t ≤ x i ≤ ψ i t, i  0, , n − 2 ,

D c

b t, x0 , , x n−1 ∈ b, c × Rn : ϕ i t ≤ x i ≤ ψ i t, i  0, , n − 2 ,

D c

at, x0 , , x n−1 ∈ a, c × R n : ϕ i t ≤ x i ≤ ψ i t, i  0, , n − 2 ;

3.2

iii gx0 , x1, , x n−1 is continuous on Rn , and−1n −i g x0 , x1, , x n−1 is nonincreasing

in x i i  0, 1, , n − 3 and nondecreasing in x n−1on n i0−2ϕ i a, ψ i a × R;

iv hx0 , x1, , x n−1 is continuous on Rn , and nonincreasing in x i i  0, 1, , n − 3 and

nondecreasing in x n−1on n i0−2ϕ i c, ψ i c × R.

Then BVP1.1, 1.2 has at least one solution xt ∈ C n a, c such that for each i  0, 1, , n − 2,

−1n −i α i t ≤ −1 n −i x i t ≤ −1 n −i β i t, t ∈ a, b,

Proof For each i  0, 1, , n − 2 define

w i t, x 

⎧

⎪

⎨

⎪

⎩

ψ i t, x > ψ i t,

x, ϕ i t ≤ x ≤ ψ i t,

ϕ i t, x < ϕ i t,

3.4

where ϕ i t  min{α i t, β i t}, ψ i t  max{α i t, β i t}.

x n t  λft, w0t, xt, , wn−2



t, x n−2 t, x n−1 t

x n−2 t − λw n−2



t, x n−2 tφx n−1 t

,

3.5

where φ is given by the Nagumo condition, with the boundary conditions

x n−2 a  λw n−2

a, x n−2 a− gw0a, xa, , wn−2



a, x n−2 a, x n−1 a,

x i b  λμ i , i  0, 1, , n − 3,

x n−2 c  λw n−2

c, x n−2 c− hw0c, xc, , wn−2



c, x n−2 c, x n−1 c.

3.6

Then we can choose a constant M n−2> 0 such that

−M n−2< α n−2 t ≤ β n−2 t < M n−2, t ∈ a, c, 3.7

f

t, α t, , α n−2 t, 0−M n−2 α n−2 tφ 0 < 0, t ∈ a, c,

f

t, β t, , β n−2 t, 0M n−2− β n−2 tφ 0 > 0, t ∈ a, c, 3.8



α n−2 a − gα a, , α n−2 a, 0 < M

n−2,



β n−2 a − gβ a, , β n−2 a, 0 < M

n−2,

3.9



α n−2 c − hα c, , α n−2 c, 0 < M

n−2,



β n−2 c − hβ c, , β n−2 c, 0 < M

n−2.

3.10

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

Step 1 Show that every solution x t of BVP 3.5, 3.6 satisfies



x n−2 t < M

Suppose that the estimate|x n−2 t| < M n−2is not true Then there exists t0 ∈ a, c such that x n−2 t0 ≥ M n−2or x n−2 t0 ≤ −M n−2 We may assume x n−2 t0 ≥ M n−2 There

exists t1 ∈ a, c such that

max

There are three cases to consider

Case 1 t1 ∈ a, c In this case, x n−1 t1  0 and x n t1 ≤ 0 For λ ∈ 0, 1, by 3.8, we get the following contradiction:

0≥ x n t1

 λft1, w0t1, x t1, , w n−2



t1, x n−2 t1, x n−1 t1

x n−2 t1 − λw n−2



t1, x n−2 t1φx n−1 t1

 λft1, w0t1, x t1, , w n−3



t1, x n−3 t1, β n−2 t1, 0

x n−2 t1 − λβ n−2 t1φ0

≥ λf

t1, β t1, , β n−2 t1, 0M n−2− β n−2 t1φ0> 0,

3.13

and for λ 0, we have the following contradiction:

Case 2 t1  a In this case,

max

and x n−1 a ≤ 0 For λ  0, by 3.6 we have the following contradiction:

For λ ∈ 0, 1, by 3.9 and condition iii we can get the following contradiction:

M n−2≤ x n−2 a,

 λw n−2



a, x n−2 a− gw0a, xa, , wn−2



a, x n−2 a, x n−1 a,

≤ λβ n−2 a − gβ a, , β n−2 a, 0< M n−2.

3.17

Case 3 t1  c In this case,

max

and x n−1 c ≥ 0 For λ  0, by 3.6 we have the following contradiction:

For λ ∈ 0, 1, by 3.10 and condition iv we can get the following contradiction:

M n−2≤ x n−2 c,

 λw n−2



c, x n−2 c− hw0c, xc, , wn−2



c, x n−2 c, x n−1 c

≤ λβ n−2 c − hβ c, , β n−2 c, 0< M n−2.

3.20

By3.6, the estimates



x i t < M

i: c − aM i1μ i, i  0, 1, , n − 3, t ∈ a, c 3.21 are obtained by integration

Step 2 Show that there exists M n−1 > 0 such that every solution x t of BVP 3.5, 3.6 satisfies



x n−1 t < M

Let

E  {t, x0 , , x n−1 ∈ a, c × R n:|x i | ≤ M i , i  0, 1, , n − 2}, 3.23

and define the function F λ:a, c × R n → R as follows:

F λ t, x0 , , x n−1  λft, w0t, x0, , w n−2t, x n−2, x n−1

In the following, we show that F λ t, x0 , , x n−1 satisfies the Nagumo condition on E,

|F λ t, x0 , , x n−1| λf t, w0t, x0, , w n−2t, xn−2, xn−1

 x n−2− λw n−2t, xn−2φ|xn−1|

≤ 1  2M n−2φ|x n−1| : φ E |x n−1|.

3.25

Furthermore, we obtain

∞

0

s

φ E s ds

∞

0

s

ByStep 1andLemma 2.3, there exists M n−1> 0 such that |x n−1 t| < M n−1for t ∈ a, c Since

M n−2and φ E do not depend on λ, the estimate |x n−1 t| < M n−1ona, c is also independent

of λ.

Step 3 Show that for λ 1, BVP 3.5, 3.6 has at least one solution x1t.

Define the operators as follows:

L : C n a, c ⊂ C n−1a, c −→ Ca, c × R n , 3.28

Lxx n t, x n−2 a, xb, , x n−3 b, x n−2 c,

by

N λ xF λ



t, x t, , x n−1 t, A λ , λμ0, , λμ n−3, C λ



with

A λ: λw n−2

a, x n−2 a− gw0a, xa, , wn−2

a, x n−2 a, x n−1 a

C λ: λw n−2

c, x n−2 c− hw0c, xc, , wn−2

c, x n−2 c, x n−1 c. 3.31

defined by

Consider the setΩ  {x ∈ C n−1a, c : x i∞< M i , i  0, 1, , n − 1}.

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

by homotopy invariance, we get

theory we have

x n t  ft, w0t, xt, , wn−2



t, x n−2 t, x n−1 t

x n−2 t − w n−2



t, x n−2 tφx n−1 t

,

3.36

with the boundary conditions

x n−2 a  w n−2



a, x n−2 a− gw0a, xa, , wn−2



a, x n−2 a, x n−1 a,

x i b  μ i , i  0, 1, , n − 3,

x n−2 c w n−2



c, x n−2 c− hw0c, xc, , wn−2



c, x n−2 c, x n−1 c,

3.37

has at least one solution x1t in Ω.

Step 4 Show that x1t is a solution of BVP 1.1, 1.2

In fact, the solution x1t of BVP 3.36, 3.37 will be a solution of BVP 1.1, 1.2, if it satisfies

ϕ i t ≤ x1i t ≤ ψ i t, i  0, 1, , n − 2, t ∈ a, c. 3.38

By contradiction, suppose that there exists t0 ∈ a, c such that x1n−2 t0 > ψ n−2t0 There exists t1 ∈ a, c such that

max

t ∈a,c



x1n−2 t − ψ n−2t: xn−21 t1 − ψ n−2t1 > 0. 3.39

Now there are three cases to consider

Case 1 t1 ∈ a, c In this case, since ψ n−2t  β n−2 t on a, c, we have x1n−1 t1  β n−1 t1 and x n1 t1 ≤ β n t1 By conditions i and ii, we get the following contradiction:

0≥ x1n t1 − β n t1

≥ ft1, w0t1, x1t1, , wn−2



t1, x n−21 t1, x n−11 t1

x n−21 t1 − w n−2



t1, x1n−2 t1φx n−1

1 t1

− ft1, β t1, , β n−1 t1

≥ ft1, β t1, , β n−1 t1x n−21 t1 − β n−2 t1φx n−1

1 t1

− ft1, β t1, · · · , β n−1 t1

x n−21 t1 − β n−2 t1φx n−1

1 t1

> 0.

3.40

Case 2 t1  a In this case, we have

max

t ∈a,c



x n−21 t − ψ n−2t: xn−21 a − β n−2 a > 0, 3.41

and x n−11 a ≤ β n−1 a By 3.37 and conditions i and iii we can get the following contradiction:

β n−2 a < x n−21 a,

 w n−2



a, x n−21 a− gw0a, x1a, , wn−2



a, x n−21 a, x n−11 a

≤ β n−2 a − gβ a, , β n−2 a, β n−1 a≤ β n−2 a.

3.42

Case 3 t1  c In this case, we have

max

t ∈a,c



x n−21 t − ψ n−2t: xn−21 c − β n−2 c > 0, 3.43

and x n−11 c ≥ β n−1 c By 3.37 and conditions i and iv we can get the following contradiction:

β n−2 c < x n−21 c

 w n−2



c, x n−21 c− hw0c, x1c, , wn−2



c, x n−21 c, x n−11 c

≤ β n−2 c − hβ c, , β n−2 c, β n−1 c≤ β n−2 c.

3.44

Similarly, we can show that ϕ n−2t ≤ xn−21 t on a, c Hence

α n−2 t  ϕ n−2t ≤ x n−21 t ≤ ψ n−2t  β n−2 t, t ∈ a, c. 3.45

α i b  x i1 b  β i b, i  n − 1 − 2j, j  1, 2, ,



n− 1 2



,

α i b ≤ x i1 b ≤ β i b, i  n − 2 − 2j, j  1, 2, ,



n− 2 2



.

3.46

Therefore by integration we have for each i  0, 1, , n − 2,

−1n −i α i t ≤ −1 n −i x i1 t ≤ −1 n −i β i t, t ∈ a, b,

that is,

ϕ i t ≤ x1i t ≤ ψ i t, i  0, 1, , n − 2, t ∈ a, c. 3.48

Hence x1t is a solution of BVP 1.1, 1.2 and satisfies 3.3

Now we give a uniqueness theorem by assuming additionally the differentiability for

Theorem 3.2 Assume that

i there exist lower and upper solutions αt, βt of BVP 1.1, 1.2, respectively, such that

−1n −i α i t ≤ −1 n −i β i t, t ∈ a, b, i  0, 1, , n − 2,

ii ft, x0 , , x n−1 and its first-order partial derivatives in xi i  0, 1, , n − 1 are

continuous on a, c × R n ,−1n −i ∂f/∂x i  ≤ 0 i  0, 1, , n − 3 on D b

a , ∂f/∂x i ≤

0 i  0, 1, , n − 3 on D c

b and satisfy the Nagumo condition on D c

a;

iii gx0 , x1, , x n−1 is continuous on Rn and continuously partially differentiable on

n−2

i0ϕ i a, ψ i a × R, and

−1n −i ∂g

∂x i ≤ 0, i  0, 1, , n − 3,

∂g

∂x n−1 ≤ 0, onn−2

i0



ϕ i a, ψ i a× R;

3.50

iv hx0 , x1, , x n−1 is continuous on Rn and continuously partially differentiable on

n−2

i0ϕ i c, ψ i c × R, and

∂h

∂x i ≤ 0, i  0, 1, , n − 3,

∂h

∂x n−1 ≥ 0, onn−2

i0



ϕ i c, ψ i c× R;

3.51

v there exists a function γt ∈ C n a, c such that γ n−2 t > 0 on a, c, and

γ n t < n−1

i0

∂f

∂x i · γ i t, on D c

a

n−1



i0

∂g

∂x i · γ i a > 0, onn−2

i0



ϕ i a, ψ i a× R,

n−1



i0

∂h

∂x i · γ i c > 0, onn−2

i0



ϕ i c, ψ i c× R,

γ i b  0, if n − i : odd, i  0, 1, , n − 3,

γ i b ≥ 0, if n − i : even, i  0, 1, , n − 3.

3.52

Then BVP1.1, 1.2 has a unique solution xt satisfying 3.3.

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 let x1t and x2t are any two solutions of BVP 1.1, 1.2 satisfying 3.3 Let zt  x2t − x1t It

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

z n t n−1

i0

n−1



i0

a i z i a  0, n−1

i0

where for each i  0, 1, , n − 1,

d i t 

1

0

∂

∂x i

f

t, x1t  θzt, x 

1t  θzt, , x n−11 t  θz n−1 tdθ,

a i

1

0

∂

∂x i

g

x1a  θza, x 

1a  θza, , x n−11 a  θz n−1 adθ,

c i

1

0

∂

∂x i

h

x1c  θzc, x 

1c  θzc, , x n−11 c  θz n−1 cdθ.

3.56