Hindawi Publishing Corporation
Boundary Value Problems
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
Minghe Pei1 and Sung Kag Chang2
1
2
Department of Mathematics, Bei Hua University, JiLin 132013, China
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 conditions gxa, x a, . . . , xn−1 a 0; xi b μi , i 0, 1, . . . , n − 3; hxc, x c, . . . , xn−1 c 0, where a < b < c, f : a, c × Rn →
R −∞, ∞ is continuous, g, h : Rn → R are continuous, andμi ∈ 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.
Copyright q 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
instance, see 1–22 and references therein. However, most of all the boundary conditions in
the above-mentioned references are for two-point boundary conditions 2–11, 14, 17–22, and
three-point boundary conditions are rarely seen 1, 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
xn f t, x, x , . . . , xn−1 ,
n ≥ 3,
1.1
2
Boundary Value Problems
with nonlinear three point boundary conditions
g xa, x a, . . . , xn−1 a 0,
xi b μi , i 0, 1, . . . , n − 3,
h xc, x c, . . . , xn−1 c 0,
1.2
where a < b < c, f : a, c × Rn → R −∞, ∞ is a continuous function, g, h : Rn → R are
continuous functions, and μi ∈ 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. αt, βt ∈ Cn a, c are called lower and upper solutions of BVP 1.1, 1.2,
respectively, if
αn t ≥ f t, αt, α t, . . . , αn−1 t , t ∈ a, c,
g αa, α a, . . . , αn−1 a ≤ 0,
αi b ≤ μi , i 0, 1, . . . , n − 3,
h αc, α c, . . . , αn−1 c ≤ 0,
βn t ≤ f t, βt, β t, . . . , βn−1 t , t ∈ a, c,
g βa, β a, . . . , βn−1 a ≥ 0,
2.1
βi b ≥ μi , i 0, 1, . . . , n − 3,
h βc, β c, . . . , βn−1 c ≥ 0.
Definition 2.2. Let E be a subset of a, c × Rn . We say that ft, x0 , x1 , . . . , xn−1 satisfies the
Nagumo condition on E if there exists a continuous function φ : 0, ∞ → 0, ∞ such that
ft, x0 , x1 , . . . , xn−1 ≤ φ|xn−1 |,
∞
0
t, x0 , x1 , . . . , xn−1 ∈ E,
sds
∞.
φs
2.2
Lemma 2.3 see 10. Let f : a, c × Rn → R be a continuous function satisfying the Nagumo
condition on
E t, x0 , x1 , . . . , xn−1 ∈ a, c × Rn : γi t ≤ xi ≤ Γi t, i 0, 1, . . . , n − 2 ,
2.3
Boundary Value Problems
3
where γi t, Γi t : a, c → R are continuous functions such that
γi t ≤ Γi t,
i 0, 1, . . . , n − 2, t ∈ a, c.
2.4
Then there exists a constant r > 0 (depending only on γn−2 t, Γn−2 t and φt such that every
solution xt of 1.1 with
γi t ≤ xi t ≤ Γi t,
i 0, 1, . . . , n − 2, t ∈ a, c
2.5
satisfies xn−1 ∞ ≤ r.
Lemma 2.4. Let φ : 0, ∞ → 0, ∞ be a continuous function. Then boundary value problem
xn xn−2 φ xn−1 ,
t ∈ a, c,
xn−2 a xi b xn−2 c 0,
i 0, 1, . . . , n − 3
2.6
2.7
has only the trivial solution.
Proof. Suppose that x0 t is a nontrivial solution of BVP 2.6, 2.7. Then there exists t0 ∈
n−2
n−2
n−2
a, c such that x0 t0 > 0 or x0 t0 < 0. We may assume x0 t0 > 0. There exists
t1 ∈ a, c such that
n−2
max x0
t∈a,c
n−1
Then x0
n−2
t : x0
t1 > 0.
2.8
n
t1 0, x0 t1 ≤ 0. From 2.6 we have
n
n−2
0 ≥ x0 t1 x0
n−1
t1 φ x0 t1 > 0,
2.9
which is a contradiction. Hence BVP 2.6, 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 problem 1.1, 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 αi t ≤ −1n−i βi t,
αi t ≤ βi t,
t ∈ a, b, i 0, 1, . . . , n − 2,
t ∈ b, c, i 0, 1, . . . , n − 2;
3.1
4
Boundary Value Problems
ii ft, x0 , . . . , xn−1 is continuous on a, c×Rn , −1n−i ft, x0 , . . . , xn−1 is nonincreasing in
xi i 0, 1, . . . , n − 3 on Dab , and ft, x0 , . . . , xn−1 is nonincreasing in xi i 0, 1, . . . , n −
3 on Dbc and satisfies the Nagumo condition on Dac , where
ϕi t min αi t, βi t ,
ψ i t max αi t, βi t , i 0, . . . , n − 2,
Dab t, x0 , . . . , xn−1 ∈ a, b × Rn : ϕi t ≤ xi ≤ ψi t, i 0, . . . , n − 2 ,
Dbc t, x0 , . . . , xn−1 ∈ b, c × Rn : ϕi t ≤ xi ≤ ψi t, i 0, . . . , n − 2 ,
Dac t, x0 , . . . , xn−1 ∈ a, c × Rn : ϕi t ≤ xi ≤ ψi t, i 0, . . . , n − 2 ;
3.2
iii gx0 , x1 , . . . , xn−1 is continuous on Rn , and −1n−i gx0 , x1 , . . . , xn−1 is nonincreasing
in xi i 0, 1, . . . , n − 3 and nondecreasing in xn−1 on n−2
i0 ϕi a, ψi a × R;
iv hx0 , x1 , . . . , xn−1 is continuous on Rn , and nonincreasing in xi i 0, 1, . . . , n − 3 and
nondecreasing in xn−1 on n−2
i0 ϕi c, ψi c × R.
Then BVP 1.1, 1.2 has at least one solution xt ∈ Cn a, c such that for each i 0, 1, . . . , n − 2,
−1n−i αi t ≤ −1n−i xi t ≤ −1n−i βi t,
αi t ≤ xi t ≤ βi t,
t ∈ a, b,
t ∈ b, c.
3.3
Proof. For each i 0, 1, . . . , n − 2 define
⎧
⎪
ψi t,
⎪
⎪
⎨
wi t, x x,
⎪
⎪
⎪
⎩
ϕi t,
x > ψi t,
ϕi t ≤ x ≤ ψi t,
3.4
x < ϕi t,
where ϕi t min{αi t, βi t}, ψi t max{αi t, βi t}.
For λ ∈ 0, 1, we consider the auxiliary equation
xn t λf t, w0 t, xt, . . . , wn−2 t, xn−2 t , xn−1 t
xn−2 t − λwn−2 t, xn−2 t φ xn−1 t ,
3.5
where φ is given by the Nagumo condition, with the boundary conditions
xn−2 a λ wn−2 a, xn−2 a − g w0 a, xa, . . . , wn−2 a, xn−2 a , xn−1 a ,
xi b λμi , i 0, 1, . . . , n − 3,
xn−2 c λ wn−2 c, xn−2 c − h w0 c, xc, . . . , wn−2 c, xn−2 c , xn−1 c .
3.6
Boundary Value Problems
5
Then we can choose a constant Mn−2 > 0 such that
−Mn−2 < αn−2 t ≤ βn−2 t < Mn−2 , t ∈ a, c,
f t, αt, . . . , αn−2 t, 0 − Mn−2 αn−2 t φ0 < 0, t ∈ a, c,
3.7
f t, βt, . . . , βn−2 t, 0 Mn−2 − βn−2 t φ0 > 0,
3.8
t ∈ a, c,
n−2
a − g αa, . . . , αn−2 a, 0 < Mn−2 ,
α
n−2
a − g βa, . . . , βn−2 a, 0 < Mn−2 ,
β
n−2
c − h αc, . . . , αn−2 c, 0 < Mn−2 ,
α
n−2
c − h βc, . . . , βn−2 c, 0 < Mn−2 .
β
3.9
3.10
In the following, we will complete the proof in four steps.
Step 1. Show that every solution xt of BVP 3.5, 3.6 satisfies
n−2 t < Mn−2 ,
x
t ∈ a, c,
3.11
independently of λ ∈ 0, 1.
Suppose that the estimate |xn−2 t| < Mn−2 is not true. Then there exists t0 ∈ a, c
such that xn−2 t0 ≥ Mn−2 or xn−2 t0 ≤ −Mn−2 . We may assume xn−2 t0 ≥ Mn−2 . There
exists t1 ∈ a, c such that
max xn−2 t : xn−2 t1 ≥ Mn−2 > 0.
t∈a,c
3.12
There are three cases to consider.
Case 1 t1 ∈ a, c. In this case, xn−1 t1 0 and xn t1 ≤ 0. For λ ∈ 0, 1, by 3.8, we get
the following contradiction:
0 ≥ xn t1 λf t1 , w0 t1 , xt1 , . . . , wn−2 t1 , xn−2 t1 , xn−1 t1 xn−2 t1 − λwn−2 t1 , xn−2 t1 φ xn−1 t1 λf t1 , w0 t1 , xt1 , . . . , wn−3 t1 , xn−3 t1 , βn−2 t1 , 0
xn−2 t1 − λβn−2 t1 φ0
≥ λ f t1 , βt1 , . . . , βn−2 t1 , 0 Mn−2 − βn−2 t1 φ0 > 0,
3.13
6
Boundary Value Problems
and for λ 0, we have the following contradiction:
0 ≥ xn t1 xn−2 t1 φ0 ≥ Mn−2 φ0 > 0.
3.14
Case 2 t1 a. In this case,
max xn−2 t : xn−2 a≥ Mn−2 > 0,
t∈a,c
3.15
and xn−1 a ≤ 0. For λ 0, by 3.6 we have the following contradiction:
0 < Mn−2 ≤ xn−2 a 0.
3.16
For λ ∈ 0, 1, by 3.9 and condition iii we can get the following contradiction:
Mn−2 ≤ xn−2 a,
λ wn−2 a, xn−2 a − g w0 a, xa, . . . , wn−2 a, xn−2 a , xn−1 a ,
3.17
≤ λ βn−2 a − g βa, . . . , βn−2 a, 0 < Mn−2 .
Case 3 t1 c. In this case,
max xn−2 t : xn−2 c≥ Mn−2 > 0,
t∈a,c
3.18
and xn−1 c ≥ 0. For λ 0, by 3.6 we have the following contradiction:
0 < Mn−2 ≤ xn−2 c 0.
3.19
For λ ∈ 0, 1, by 3.10 and condition iv we can get the following contradiction:
Mn−2 ≤ xn−2 c,
λ wn−2 c, xn−2 c − h w0 c, xc, . . . , wn−2 c, xn−2 c , xn−1 c
3.20
≤ λ βn−2 c − h βc, . . . , βn−2 c, 0 < Mn−2 .
By 3.6, the estimates
i x t < Mi : c − aMi1 μi ,
are obtained by integration.
i 0, 1, . . . , n − 3, t ∈ a, c
3.21
Boundary Value Problems
7
Step 2. Show that there exists Mn−1 > 0 such that every solution xt of BVP 3.5, 3.6
satisfies
n−1 t < Mn−1 ,
x
t ∈ a, c,
3.22
independently of λ ∈ 0, 1.
Let
E {t, x0 , . . . , xn−1 ∈ a, c × Rn : |xi | ≤ Mi , i 0, 1, . . . , n − 2},
3.23
and define the function Fλ : a, c × Rn → R as follows:
Fλ t, x0 , . . . , xn−1 λft, w0 t, x0 , . . . , wn−2 t, xn−2 , xn−1 xn−2 − λwn−2 t, xn−2 φ|xn−1 |.
3.24
In the following, we show that Fλ t, x0 , . . . , xn−1 satisfies the Nagumo condition on E,
independently of λ ∈ 0, 1. In fact, since f satisfies the Nagumo condition on Dac , we have
|Fλ t, x0 , . . . , xn−1 | λft, w0 t, x0 , . . . , wn−2 t, xn−2 , xn−1 xn−2 − λwn−2 t, xn−2 φ|xn−1 |
3.25
≤ 1 2Mn−2 φ|xn−1 | : φE |xn−1 |.
Furthermore, we obtain
∞
0
s
ds φE s
∞
0
s
ds ∞.
1 2Mn−2 φs
3.26
Thus, Fλ satisfies the Nagumo condition on E, independently of λ ∈ 0, 1. Let
γi t −Mi ,
Γi t Mi ,
i 0, 1, . . . , n − 2, t ∈ a, c.
3.27
By Step 1 and Lemma 2.3, there exists Mn−1 > 0 such that |xn−1 t| < Mn−1 for t ∈ a, c. Since
Mn−2 and φE do not depend on λ, the estimate |xn−1 t| < Mn−1 on a, c is also independent
of λ.
Step 3. Show that for λ 1, BVP 3.5, 3.6 has at least one solution x1 t.
Define the operators as follows:
L : Cn a, c ⊂ Cn−1 a, c −→ Ca, c × Rn ,
3.28
8
Boundary Value Problems
by
Lx xn t, xn−2 a, xb, . . . , xn−3 b, xn−2 c ,
Nλ : Cn−1 a, c −→ Ca, c × Rn ,
3.29
by
Nλ x Fλ t, xt, . . . , xn−1 t , Aλ , λμ0 , . . . , λμn−3 , Cλ ,
3.30
with
Aλ : λ wn−2 a, xn−2 a − g w0 a, xa, . . . , wn−2 a, xn−2 a , xn−1 a
Cλ : λ wn−2 c, xn−2 c − h w0 c, xc, . . . , wn−2 c, xn−2 c , xn−1 c .
3.31
Since L−1 is compact, we have the following compact operator:
Tλ : Cn−1 a, c −→ Cn−1 a, c,
3.32
Tλ x L−1 Nλ x.
3.33
defined by
Consider the set Ω {x ∈ Cn−1 a, c : xi ∞ < Mi , i 0, 1, . . . , n − 1}.
By Steps 1 and 2, the degree degI − Tλ , Ω, 0 is well defined for every λ ∈ 0, 1, and
by homotopy invariance, we get
degI − T0 , Ω, 0 degI − T1 , Ω, 0.
3.34
Since the equation x T0 x has only the trivial solution from Lemma 2.4, by the degree
theory we have
degI − T1 , Ω, 0 degI − T0 , Ω, 0 ±1.
3.35
Hence, the equation x T1 x has at least one solution. That is, the boundary value problem
xn t f t, w0 t, xt, . . . , wn−2 t, xn−2 t , xn−1 t
xn−2 t − wn−2 t, xn−2 t φ xn−1 t ,
3.36
Boundary Value Problems
9
with the boundary conditions
xn−2 a wn−2 a, xn−2 a − g w0 a, xa, . . . , wn−2 a, xn−2 a , xn−1 a ,
xi b μi , i 0, 1, . . . , n − 3,
xn−2 c wn−2 c, xn−2 c − h w0 c, xc, . . . , wn−2 c, xn−2 c , xn−1 c ,
3.37
has at least one solution x1 t in Ω.
Step 4. Show that x1 t is a solution of BVP 1.1, 1.2.
In fact, the solution x1 t of BVP 3.36, 3.37 will be a solution of BVP 1.1, 1.2, if it
satisfies
i
ϕi t ≤ x1 t ≤ ψi t,
3.38
i 0, 1, . . . , n − 2, t ∈ a, c.
n−2
By contradiction, suppose that there exists t0 ∈ a, c such that x1
exists t1 ∈ a, c such that
t0 > ψn−2 t0 . There
n−2
n−2
max x1 t − ψn−2 t : x1 t1 − ψn−2 t1 > 0.
3.39
t∈a,c
Now there are three cases to consider.
n−1
Case 1 t1 ∈ a, c. In this case, since ψn−2 t βn−2 t on a, c, we have x1 t1 βn−1 t1 n
and x1 t1 ≤ βn t1 . By conditions i and ii, we get the following contradiction:
n
0 ≥ x1 t1 − βn t1 n−2
n−1
≥ f t1 , w0 t1 , x1 t1 , . . . , wn−2 t1 , x1 t1 , x1 t1 n−1
n−2
n−2
x1 t1 − wn−2 t1 , x1 t1 φ x1 t1 − f t1 , βt1 , . . . , βn−1 t1 n−1
n−2
≥ f t1 , βt1 , . . . , βn−1 t1 x1 t1 − βn−2 t1 φ x1 t1 3.40
− f t1 , βt1 , · · · , βn−1 t1 n−1
n−2
x1 t1 − βn−2 t1 φ x1 t1 > 0.
Case 2 t1 a. In this case, we have
n−2
n−2
max x1 t − ψn−2 t : x1 a − βn−2 a > 0,
t∈a,c
3.41
10
Boundary Value Problems
n−1
and x1 a ≤ βn−1 a. By 3.37 and conditions i and iii we can get the following
contradiction:
n−2
βn−2 a < x1
a,
n−2
n−2
n−1
wn−2 a, x1 a − g w0 a, x1 a, . . . , wn−2 a, x1 a , x1 a
3.42
≤ βn−2 a − g βa, . . . , βn−2 a, βn−1 a ≤ βn−2 a.
Case 3 t1 c. In this case, we have
n−2
n−2
max x1 t − ψn−2 t : x1 c − βn−2 c > 0,
t∈a,c
3.43
n−1
and x1 c ≥ βn−1 c. By 3.37 and conditions i and iv we can get the following
contradiction:
n−2
βn−2 c < x1
c
n−2
n−2
n−1
wn−2 c, x1 c − h w0 c, x1 c, . . . , wn−2 c, x1 c , x1 c
3.44
≤ βn−2 c − h βc, . . . , βn−2 c, βn−1 c ≤ βn−2 c.
n−2
Similarly, we can show that ϕn−2 t ≤ x1
n−2
αn−2 t ϕn−2 t ≤ x1
t on a, c. Hence
t ≤ ψn−2 t βn−2 t,
t ∈ a, c.
3.45
Also, by boundary condition 3.37 and condition i, we have
i
αi b x1 b βi b,
i
α b ≤
i
x1 b
i
≤ β b,
n−1
,
2
n−2
.
i n − 2 − 2j, j 1, 2, . . . ,
2
i n − 1 − 2j, j 1, 2, . . . ,
3.46
Therefore by integration we have for each i 0, 1, . . . , n − 2,
i
−1n−i αi t ≤ −1n−i x1 t ≤ −1n−i βi t,
i
αi t ≤ x1 t ≤ βi t,
t ∈ b, c,
t ∈ a, b,
3.47
Boundary Value Problems
11
that is,
i
ϕi t ≤ x1 t ≤ ψi t,
i 0, 1, . . . , n − 2, t ∈ a, c.
3.48
Hence x1 t 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
functions f, g and h, and a kind of estimating condition in Theorem 3.1.
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 αi t ≤ −1n−i βi t,
αi t ≤ βi t,
t ∈ a, b, i 0, 1, . . . , n − 2,
t ∈ b, c, i 0, 1, . . . , n − 2;
3.49
ii ft, x0 , . . . , xn−1 and its first-order partial derivatives in xi i 0, 1, . . . , n − 1 are
continuous on a, c × Rn , −1n−i ∂f/∂xi ≤ 0 i 0, 1, . . . , n − 3 on Dab , ∂f/∂xi ≤
0 i 0, 1, . . . , n − 3 on Dbc and satisfy the Nagumo condition on Dac ;
iii gx0 , x1 , . . . , xn−1 is continuous on Rn and continuously partially differentiable on
n−2
i0 ϕi a, ψi a × R, and
−1n−i
∂g
≤ 0,
∂xi
∂g
≤ 0,
∂xn−1
i 0, 1, . . . , n − 3,
n−2
on
ϕi a, ψi a × R;
3.50
i0
iv hx0 , x1 , . . . , xn−1 is continuous on Rn and continuously partially differentiable on
n−2
i0 ϕi c, ψi c × R, and
∂h
≤ 0,
∂xi
∂h
≥ 0,
∂xn−1
i 0, 1, . . . , n − 3,
n−2
on
ϕi c, ψi c × R;
i0
3.51
12
Boundary Value Problems
v there exists a function γt ∈ Cn a, c such that γ n−2 t > 0 on a, c, and
γ n t <
n−1
∂g
i0
∂xi
n−1
∂h
i0
∂xi
n−1
∂f
∂xi
i0
· γ i a > 0,
· γ i t,
on
on Dac
n−2
ϕi a, ψi a × R,
i0
i
· γ c > 0,
n−2
on
ϕi c, ψi c × R,
3.52
i0
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.
Then BVP 1.1, 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 BVP 1.1, 1.2. To do this, we let x1 t
and x2 t are any two solutions of BVP 1.1, 1.2 satisfying 3.3. Let zt x2 t − x1 t. It
is easy to show that zt is a solution of the following boundary value problem
zn t n−1
di tzi t,
3.53
i0
n−1
ai zi a 0,
n−1
i0
i0
zi b 0,
ci zi c 0,
i 0, 1, . . . , n − 3,
3.54
3.55
where for each i 0, 1, . . . , n − 1,
di t ai ci 1
∂ n−1
f t, x1 t θzt, x1 t θz t, . . . , x1 t θzn−1 t dθ,
0 ∂xi
1
∂ n−1
g x1 a θza, x1 a θz a, . . . , x1 a θzn−1 a dθ,
0 ∂xi
1
∂ n−1
h x1 c θzc, x1 c θz c, . . . , x1 c θzn−1 c dθ.
0 ∂xi
3.56
Boundary Value Problems
13
By conditions ii, iii, and iv, we have that di t ∈ Ca, c, i 0, 1, . . . , n − 3, and
−1n−i di t ≤ 0,
di t ≤ 0,
i 0, 1, . . . , n − 3, t ∈ b, c,
−1n−i ai ≤ 0,
ci ≤ 0,
i 0, 1, . . . , n − 3, t ∈ a, b,
i 0, 1, . . . , n − 3, an−1 ≤ 0,
3.57
i 0, 1, . . . , n − 3, cn−1 ≥ 0.
Now suppose that there exists t0 ∈ a, c such that zn−2 t0 /
0. Without loss of
generality assume zn−2 t0 > 0, and let
Ω M : Mzn−2 t < γ n−2 t, t ∈ a, c .
3.58
It is easy to see that 0 ∈ Ω by condition v, hence Ω /
∅. Let M0 sup Ω. We have that
0 < M0 < ∞, M0 zn−2 t ≤ γ n−2 t on a, c, and there exists a point t1 ∈ a, c such that
a, c. In fact, if t1 a, then M0 zn−1 a ≤ γ n−1 a.
M0 zn−2 t1 γ n−2 t1 . Furthermore t1 /
By condition v and 3.55 we can easily show that
−1n−i M0 zi t − γ i t ≤ 0,
i 0, 1, . . . , n − 3, t ∈ a, b.
3.59
In particular
−1n−i M0 zi a − γ i a ≤ 0,
i 0, 1, . . . , n − 3.
3.60
Hence
n−1
M0 ai zi a ≥
n−1
i0
ai γ i a > 0,
3.61
i0
a. Similarly we can show that t1 /
c. Consequently
which contradicts to 3.54. Thus t1 /
M0 zn−1 t1 γ n−1 t1 .
Now, there are two cases to consider, that is
t1 ∈ a, b
or
t1 ∈ b, c.
3.62
If t1 ∈ a, b, then by 3.59 we have
−1n−i M0 zi t1 − γ i t1 ≤ 0,
i 0, 1, . . . , n − 3.
3.63
14
Boundary Value Problems
Thus, by 3.53 and condition v we have
M0 zn t1 n−1
n−1
M0 di t1 zi t1 ≥
di t1 γ i t1 > γ n t1 .
i0
3.64
i0
Consequently, by Taylor’s theorem there exists t2 ∈ t1 , c such that
M0 zn−2 t > γ n−2 t,
3.65
∀t ∈ t1 , t2 ,
which is a contradiction.
A similar contradiction can be obtained if t1 ∈ b, c. Hence zn−2 t ≡ 0 on a, c. By
3.55, we obtain zt ≡ 0 on a, c. This completes the proof of the theorem.
Next we give two examples to demonstrate the application of Theorem 3.2.
Example 3.3. Consider the following third-order three point BVP:
1
x −tx 2t2 1 x x 3 − t4 sint x , t ∈ −1, 1,
3
3 3
1 3x −1 x −1 − x −1 1 0,
x0 0,
3.66
3.67
3 3
−1 − x1 2x 1 x 1 x 1 1 0.
Let
1
ft, x0 , x1 , x2 −tx0 2t2 1 x1 x13 − t4 sint x2 ,
3
gx0 , x1 , x2 1 3x1 x13 − x2 13 ,
3.68
hx0 , x1 , x2 −1 − x0 2x1 x13 x2 13 .
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 BVP 3.66, 3.67 respectively, and all the assumptions in
Theorem 3.2 are satisfied. Therefore by Theorem 3.2 BVP 3.66, 3.67 has a unique solution
x xt satisfying
t ≤ xt ≤ −t,
t ∈ −1, 0,
−1 ≤ x t ≤ 1,
−t ≤ xt ≤ t,
t ∈ −1, 1.
t ∈ 0, 1,
3.69
Boundary Value Problems
15
Example 3.4. Consider the following fourth-order three point BVP:
3
x4 −t2 x x x ,
3.70
t ∈ −1, 1,
3
−x−1 x −1 13x −1 0,
x0 0,
x 0 0,
3.71
2
−x1 − 4x 1 x 1 9x 1 0.
Let
ft, x0 , x1 , x2 , x3 −t2 x0 x2 x23 ,
3.72
gx0 , x1 , x2 , x3 −x0 x13 13x2 ,
hx0 , x1 , x2 , x3 −x0 − 4x1 x12 9x2 .
Choose αt −t2 , βt t2 and γt t2 . It is easy to check that αt −t2 , and βt t2
are lower and upper solutions of BVP 3.70, 3.71, respectively, and all the assumptions in
Theorem 3.2 are satisfied. Therefore by Theorem 3.2 BVP 3.70, 3.71 has a unique solution
x xt satisfying
−2 ≤ xt ≤ 2,
2t ≤ x t ≤ −2t,
t ∈ −1, 0,
−t2 ≤ x t ≤ t2 ,
t ∈ −1, 1,
−2t ≤ x t ≤ 2t,
t ∈ 0, 1,
3.73
t ∈ −1, 1.
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