Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 262139, 12 pages
doi:10.1155/2012/262139
Research Article
Existence of Solutions of Nonlinear Mixed
Two-Point Boundary Value Problems for
Third-Order Nonlinear Differential Equation
Yongxin Gao and Fengqin Wang
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Yongxin Gao, yongxingao@sina.com
Received 17 February 2012; Revised 18 June 2012; Accepted 2 July 2012
Academic Editor: Mehmet Sezer
Copyright q 2012 Y. Gao and F. Wang. 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.
The authors use the upper and lower solution method to study the existence of solutions
of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation y fx, y, y , y , y b hy a, pya, yb, y a, y b 0, gya,
yb, y a, y b, y a, y b 0. Some new existence results are obtained by developing the
upper and lower solution method. Some applications are also presented.
1. Introduction
It is well known that the upper and lower solution method is a powerful tool for proving
existence results for boundary value problems. The upper and lower solution method has
been used to deal with the multipoint boundary value problems for second-order ordinary
differential equations 1–4 and for higher-order ordinary differential equations 5–11. There
are fewer results on nonlinear mixed two-point boundary value problems for higher-order
equations in the literature of ordinary differential equations. For this reason, we consider the
third-order nonlinear ordinary differential equation:
y f x, y, y , y
1.1
together with the nonlinear mixed two-point boundary conditions
y b h y a ,
p ya, yb, y a, y b 0,
g ya, yb, y a, y b, y a, y b 0,
1.2
2
Journal of Applied Mathematics
where the functions f, p, and g are continuous and monotonic, h is a homeomorphic
mapping.
We will develop the upper and lower solution method for the boundary value problem
y f x, y, y , y ,
y b h y a ,
p ya, yb, y a, y b 0,
1.3
g ya, yb, y a, y b, y a, y b 0
and establish some new existence results. Furthermore, some applications are also presented.
2. Preliminaries
In this section, we will give some preliminary considerations and some lemmas which are
essential to our main results.
Definition 2.1. Suppose the functions αx and βx ∈ C3 a, b satisfy
α3 x ≥ f x, αx, α x, α x ,
β3 x ≤ f x, βx, β x, β x ,
α x ≤ β x,
αa < βa,
x ∈ a, b,
2.1
α a < β a.
Then αx and βx are, respectively, called the lower and upper solutions of the BVP 1.3.
Because of Definition 2.1, it is clear that αx ≤ βx, x ∈ a, b. Let D a, b ×
αx, βx × α x, β x.
Definition 2.2. Let CD × R, R denote the class of continuous functions from D × R into R,
and let fx, y, y , y ∈ CD × R, R and αx, βx ∈ C3 a, b be lower and upper solutions
of BVP 1.3. Suppose that there is a function Ws ∈ CR , 0, ∞ such that
f x, y, y , y ≤ W y ,
2.2
for every x, y, y , y ∈ D × R, where
∞
λ
s
ds > max β x − min α x,
x∈a,b
Ws
x∈a,b
2.3
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3
with
λb − a max α a − β b, α b − β a .
2.4
Then we say that f satisfies Nagumo’s condition on the set D relative to αx, βx.
We assume throughout this paper the following.
H1 There are lower and upper solutions αx and βx of BVP 1.3 as Definition 2.1.
H2 Function fx, y, y , y satisfies Nagumo’s condition on the set D relative to
αx, βx.
H3 Function fx, y, y , y ∈ Ca, b × R3 , R is nonincreasing in y.
H4 h : α a, β a → α b, β b is a homeomorphism with
h α a α b,
h β a β b.
2.5
H5 Function ps, t, u, v is continuous on R4 and nondecreasing in t, u, v and satisfies
p βa, βb, β a, β b ≤ 0,
p αa, αb, α a, α b ≥ 0.
2.6
H6 Function gx, y, z, p, q, r is continuous on R6 and nondecreasing in x, y, q and
nonincreasing in r, and it satisfies
g βa, βb, β a, β b, β a, β b ≤ 0,
g αa, αb, α a, α b, α a, α b ≥ 0.
2.7
It is not difficult to obtain the following lemma.
Lemma 2.3. The boundary value problem
y f x, y, y , y ,
ya 0,
y a 0,
y b 0
2.8
has a Green function
⎧
x − s2 b − sa − x2
⎪
⎪
−
,
⎨
2
2b − a
Gx, s ⎪
−b − sa − x2
⎪
⎩
,
2b − a
a ≤ s ≤ x ≤ b,
a≤x≤s≤b
2.9
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with
b
|Gx, s|dt ≤
b − a3
,
12
|Gx x, s|dt ≤
b − a2
,
4
a
b
a
b
2.10
|Gxx x, s|dt ≤ b − a.
a
It is easy to prove the following lemma similarly to 12, page 25, Theorem 1.4.1.
Lemma 2.4. Assume that H1 , H2 hold. Then for any solution y of y fx, y, y , y with
αx ≤ yx ≤ βx, α x ≤ y x ≤ β x on a, b, there exists a constant N > 0 depending only
on α, β, W, such that
y x ≤ N,
x ∈ a, b
2.11
and one calls N is Nagumo’s constant.
Lemma 2.5. Assume that H1 –H3 hold. Then for any constant A ∈ αa, βa, B ∈
α a, β a, C ∈ α b, β b, the boundary value problem
y f x, y, y , y ,
ya A,
y a B,
2.12
y b C
at least has a solution y ∈ C3 a, b, with
αx ≤ yx ≤ βx,
α x ≤ y x ≤ β x
2.13
on a, b.
Proof. By Lemma 2.3, it is clear that BVP 2.12 is equivalent to integral equation
yx b
Gx, sf s, ys, y s, y s ds Wx,
a
where Wx is a polynomial satisfying y 0, ya A, y a B, y b C.
2.14
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5
Denote
m max max α x, max β x, N 1 ,
x∈a,b
x∈a,b
2.15
where N is Nagumo’s constant.
Assume
⎧ ⎪
f x, y, y , m ,
⎪
⎨
fm x, y, y , y f x, y, y , y ,
⎪
⎪
⎩f x, y, y , −m,
y > m,
−m ≤ y ≤ m,
y < −m,
⎧ ⎪
⎪
⎨fm x, βx, y ,y , y > βx,
G x, y, y , y fm x, y, y , y ,
αx ≤ y ≤ βx,
⎪
⎪
⎩f x, αx, y , y , y < αx,
m
⎧
y − β x
⎪
⎪
y
G
x,
y,
β
x,
⎪
2 , y > β x,
⎪
⎪
⎪
1 y
⎨ F x, y, y , y G x, y, y , y ,
α x ≤ y ≤ β x,
⎪
⎪
⎪
α x − y
⎪ ⎪
⎪
2 , y < α x.
⎩G x, y, α x, y −
1 y
2.16
Then Fx, y, y , y is bounded and continuous on a, b×R3 . Suppose |Fx, y, y , y | ≤
M; |W x| ≤ K i 0, 1, 2, x ∈ a, b.
Now, define an operator T on the set E C2 a, b, R by
i
T yx b
Gx, sF s, ys, y s, y s ds Wx.
2.17
a
If y ∈ E, the norm is defined by
y max yx y x y x .
x∈a,b
2.18
It is clear that
b − a3
T y x ≤
M K,
12
b − a2
M K,
T y x ≤
4
T y x ≤ b − aM K.
2.19
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Journal of Applied Mathematics
This shows that T maps the closed, bounded, and convex set
B
b − a3 M b − a2 M
b − aM 3K
y | y ∈ E, y ≤
12
4
2.20
into itself. Also, T is continuous and T y is bounded. All of these considerations imply that
T is completely continuous by Ascoli’s theorem. Schauder’ fixed point theorem then yields
the fixed point y of T on B. In other words, the following boundary value problem
y F x, y, y , y ,
ya A,
2.21
y a B,
y b C
has a solution y ∈ C3 a, b, and satisfying α x ≤ y x ≤ β x and αa ≤ ya ≤ βa, we
have αx ≤ yx ≤ βx, x ∈ a, b.
In the following we prove that
α x ≤ y x ≤ β x,
x ∈ a, b.
2.22
In fact, if it is invalid, there is no harm in setting the right inequality to be not true the case
that the left inequality is not true can be proved in the same way. By the assumption, if
y x > β x, for some x ∈ a, b, then there is a x0 ∈ a, b such that
y x0 − β x0 max y x − β x > 0,
x∈a,b
y x0 β x0 ,
y x0 ≤ β x0 .
2.23
2.24
Now, let
⎧
⎪
⎪
⎨βx0 , yx0 > βx0 ,
ωx0 yx0 , αx0 ≤ yx0 ≤ βx0 ,
⎪
⎪
⎩αx , yx < αx .
0
0
0
2.25
They imply that αx0 ≤ ωx0 ≤ βx0 and
G x0 , yx0 , β x0 , β x0 fm x0 , ωx0 , β x0 , β x0 .
2.26
Journal of Applied Mathematics
7
We have
y x0 − β x0 ≥ F x0 , yx0 , y x0 , y x0 − f x0 , βx0 , β x0 , β x0 F x0 , yx0 , y x0 , β x0 − f x0 , βx0 , β x0 , β x0 y x0 − β x0 G x0 , yx0 , β x0 , β x0 − f x0 , βx0 , β x0 , β x0 2
1 y x0 > G x0 , yx0 , β x0 , β x0 − f x0 , βx0 , β x0 , β x0 fm x0 , ωx0 , β x0 , β x0 − f x0 , βx0 , β x0 , β x0 f x0 , ωx0 , β x0 , β x0 − f x0 , βx0 , β x0 , β x0 2.27
≥0
which contradicts 2.24; hence, 2.22 is true.
Further, by the definition of F, y is a solution of the boundary value problem
y fm x, y, y , y ,
ya A,
y a B,
2.28
y b C.
Because there is a ξ ∈ a, b, such that
y b − y a
y ξ ≤ λ < m,
b−a
2.29
so a, b has a maximal subinterval c, d with interior point ξ, for any x ∈ c, d, |y x| ≤ m.
Hence, for x ∈ c, d, yx is the solution of BVP 2.12. And by Lemma 2.4, we have |y x| ≤
N < m; this contracts that c, d is the maximal subinterval, so we know c, d a, b.
Consequently, y is a solution of BVP 2.12.
3. Main Results
Theorem 3.1. Assume H1 –H4 , H6 hold, then BVP
y f x, y, y , y ,
ya A,
y b h y a ,
g ya, yb, y a, y b, y a, y b 0
3.1
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Journal of Applied Mathematics
has a solution y ∈ C3 a, b, satisfying
αx ≤ yx ≤ βx,
α x ≤ y x ≤ β x
3.2
on a, b.
Proof. By Lemma 2.5, we know that the boundary value problem
y f x, y, y , y ,
ya A,
y a B,
3.3
y b hB
has a solution y ∈ C3 a, b, with
αx ≤ yx ≤ βx,
α x ≤ y x ≤ β x
3.4
on a, b. For any A ∈ αa, βa, B ∈ α a, β a.
For fixed A, if B α a, then y a ≥ α a, y b ≤ α b. By H6 , we know
g ya, yb, y a, y b, y a, y b ≥ g αa, αb, α a, α b, α a, α b ≥ 0.
3.5
On the other hand, if B β a, then y a ≤ β a, y b ≥ β b. By H6 , we have
g ya, yb, y a, y b, y a, y b ≤ g βa, βb, β a, β b, β a, β b ≤ 0.
3.6
Define the following sets:
Π y y | y f x, y, y , y ,
ya A, y a B, y b hB; A ∈ αa, βa , B ∈ α a, β a ,
M1 B | B ∈ α a, β a , yx ∈ Π y ,
g ya, yb, y a, y b, y a, y b > 0 ,
M2 B | B ∈ α a, β a , yx ∈ Π y ,
g ya, yb, y a, y b, y a, y b < 0 .
3.7
Journal of Applied Mathematics
9
Obviously, Πy is nonempty. If the theorem is not true, we know that M1 and M1 are all
nonempty, and M1 M2 α a, β a. we claim that M1 is closed. To see this, let Bn ∈ M1 ,
with Bn → B0 n → ∞. Consider the following boundary value problem:
y f x, y, y , y ,
ya A,
3.8
y a Bn ,
y b hBn .
By Lemma 2.5, it is known that, for every n ∈ N, BVP 3.8 has a solution yn x ∈ C3 a, b,
satisfying
αx ≤ yn x ≤ βx,
α x ≤ yn x ≤ β x,
x ∈ a, b
3.9
and, by Lemma 2.4, we know |yn x| ≤ N.
Clearly, sequences {yn x}, {yn x}, {yn x} are uniformly bounded and equicontinuous on a, b. Consequently, there exists a subsequence of yn x which converges uniformly
on a, b, to a solution y0 x of the BVP:
y f x, y, y , y , y ,
ya A,
y a B0 ,
3.10
y b hB0 with
g y0 a, y0 b, y0 a, y0 b, y0 a, y0 b ≥ 0.
3.11
By assumption, equality cannot occur, so that
g y0 a, y0 b, y0 a, y0 b, y0 a, y0 b > 0,
3.12
and thus B0 ∈ M1 . Consequently, M1 is closed. Likewise, we may show M2 is closed. This is
a contradiction and proves the theorem.
Similar to the proof of Theorem 3.1, we can obtain the following theorem.
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Journal of Applied Mathematics
Theorem 3.2. Assume H1 –H6 hold, then BVP
y f x, y, y , y ,
y b h y a ,
p ya, yb, y a, y b 0,
3.13
g ya, yb, y a, y b, y a, y b 0
has a solution y ∈ C3 a, b, satisfying
αx ≤ yx ≤ βx,
α x ≤ y x ≤ β x
3.14
on a, b.
4. Applications
We all know it is difficult to find a solution of some nonlinear ordinary differential equation.
But according to Theorem 3.2, we can know whether a boundary value problem, especially
a nonlinear boundary value problem, has a solution and we also can know the existence
regions of the solution and its derivative.
Example 4.1. Consider the following linear boundary value problem
y x −yx 2y x,
x ∈ 0, 1,
y 1 − 2ey 0 0,
−3e − 2y0 y1 y 0 y 1 0,
4.1
y0 y1 − 4y 0 − 5y 1 y 0 − y 1 0.
It is easy to know that αx −xex − 1, βx xex 1 are lower and upper solutions
of the linear boundary value problem, respectively, where
hs 2es,
ps, t, u, v −3e − 2s t u v,
g x, y, z, p, q, r x y − 4z − 5p q − r,
4.2
Journal of Applied Mathematics
11
and all assumptions of Theorem 3.2 hold. So the linear boundary value problem has a solution
yx satisfying
−xex − 1 ≤ yx ≤ xex 1,
−ex − xex ≤ y x ≤ ex xex .
4.3
Obviously, the trivial solution of the linear boundary value problem is one.
Example 4.2. Consider nonlinear boundary value problem
2
y x −yx y x − y x,
y
π √
2πy
π π π ,
,
4 2
0,
4
π 1 π 1 π π y
y
y
0,
y2
4
4
2
8
4
2
π 1 π π 2 π 2
π 1 π y
y
y
− y
0.
y
y
4
8
2
4
2
4
8
2
2
− sin
x∈
4.4
It is easy to verify that αx − sinx, βx 0 are lower and upper solutions of the
nonlinear boundary value problem, respectively, where
hs sin
√
2πs ,
1
1
ps, t, u, v s2 t u v,
4
8
1
1
g x, y, z, p, q, r x y z2 p2 q − r,
8
8
4.5
and all assumptions of Theorem 3.2 hold, so the BVP has a solution yx satisfying
− sinx ≤ yx ≤ 0,
− cosx ≤ y x ≤ 0.
4.6
5. Conclusion
In this paper, we study a nonlinear mixed two-point boundary value problem for a thirdorder nonlinear ordinary differential equation. Some new existence results are obtained by
developing the upper and lower solution method. Furthermore, some applications are also
presented.
Acknowledgments
The work is supported by the Fundamental Research Funds for the Central Universities no.
ZXH2012 K004 and Civil Aviation University of China Research Funds no. 2012KYM05.
The authors would like to thank the referees for their valuable comments.
12
Journal of Applied Mathematics
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