Journal of Applied Mathematics & Bioinformatics, vol.5, no.1, 2015, 45-51
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2015
Existence and multiplicity of solutions to boundary
value problems for nonlinear high-order
differential equations
Huijun Zheng1
Abstract
In this paper, we investigate the multiple positive solutions for the boundary value
problem of nonlinear differential equations. The arguments are based upon the
fixed point theorem of cone expansion and compression with norm type.
Mathematics Subject Classification: O175
Keywords: Higher order boundary value problem；Positive solution；cone fixed
point theorem；multiple solutions
1 Introduction
In recent years, many researchers study the existence and multiplicity of
1
Huijun Zheng School of Mathematics and Statistics , Northeast Petroleum
University, Daqing 163318, China. E-mail:zhenghuijun02@163.com
Article Info: Received : November 1, 2014. Revised : December 9, 2014.
Published online : January 15, 2015.
44
Existence and multiplicity of solutions to boundary value problems…
solutions to boundary value problems for nonlinear high-order ordinary
differential equations, especially for even number order equation. They have
obtained a lot of satisfactory results and most authors study the problems under
the conjugate boundary conditions or simpler boundary conditions [1]-[3].
However,it is not common that the study on the boundary problems have
high-order derivative right side boundary conditions. This paper study the
boundary value problems, for a class of differential equations with 2m order
right-hand side boundary conditions. In our paper, we give some sufficient
conditions for the existence of one or two positive solutions, for the boundary
value problems using the properties of the corresponding Green’s function and
cone expansion and compression fixed point theorem.
2 Preliminary Notes
In this paper, we concerned on the following nonlinear higher-order boundary
value problem
 − 1）m y (2 m ) (=
（
x) （
f x, y ( x)）
, a.e.x ∈ (0,1)
 (i )
 y (0)= 0, 0 ≤ i ≤ m − 1
 y ( j ) (1)= 0, m ≤ j ≤ 2m − 1

(1)
We assume that
（ H1 ） f ( x, y ) is continuous and nonnegative on [0,1] × [0, +∞) .
（ H 2 ） f ( x, y ) is not equal to zero any where for any compact subinterval
1
in [0,1] , and ∫ x m −1 f ( x, y )dx < +∞ .
0
Theorem 2.1
Let B be a Banach space and K ⊂ B a cone in B . Assume that
Huijun Zheng
45
Ω1 , Ω 2 are open subset of B with 0 ∈ Ω1 , Ω1 ⊂ Ω 2 .
Let Φ : K ∩ (Ω 2 \ Ω1 ) → K be a completely continuous operator such that either
(i) Φy ≤ y , y ∈ K ∩ ∂Ω1 , Φy ≥ y , y ∈ K ∩ ∂Ω 2 ; or
(ii) Φy ≥ y , y ∈ K ∩ ∂Ω1 , Φy ≤ y , y ∈ K ∩ ∂Ω 2 .
Then, Φ has a fixed point in K ∩ (Ω 2 \ Ω1 ) .
To obtain positive solutions for the problem (1). We state some properties of
Green’s function for (1).
As shown in [5], the problem (1) is equivalent to the integral equation
1
y ( x) = ∫ G ( x, s ) f ( s, y ( s ))ds
0
where
s
1

u m −1 (u + x − s ) m −1 du , 0 ≤ s ≤ x ≤ 1,
2 ∫0
 [(m − 1)！
]

G ( x, s ) = 
x
1

u m −1 (u + s − x) m −1 du , 0 ≤ x ≤ s ≤ 1.
2 ∫0
[(m − 1)！
]
(2)
Moreover, the following results have been recently offered by [5].
Lemma 2.1
For any x, s ∈ [0,1] , we have
α（x) g( s ) ≤ G ( x, s ) ≤ β ( x)g( s ) ， |
∂G ( x,s )
|≤ cs m −1 ，
∂x
Where
c=
Let
2m −1
xm
x m −1
1
；
； g (s) =
x
；
sm
α
(
x
)
=
β
(
)
=
2
2
m
[(m − 1)!]
2m − 1
[(m − 1)!]
.
y = sup | y ( x)| ， α = min α ( x) ， || β ||= max β ( x) . Define the cone K in
0≤ y ≤1
Banach
3
1
≤ x≤
4
4
0 ≤ x ≤1
46
Existence and multiplicity of solutions to boundary value problems…
space C[0,1] , given by
K=
{ y ∈ C[0,1] : y ( x) ≥ 0, min y ( x) ≥ α y / β } ，
1
3
≤ x≤
4
4
We define the operator Φ :K → K by
1
（Φy）
( x) =
∫ G( x, s) f (s, y(s))ds .
0
For any y ∈ K ,we have
1
≥ min ∫ α ( x)g (s) f ( s, y ( s ))ds
min Φy (x)
1
3 0
≤ x≤
4
4
1
3
≤ x≤
4
4
≥
1
α
α
max ∫ G( x,s) f ( s, y ( s ))ds =
Φy
0
≤
x
≤
0
1
β
β
This implies Φ ( K ) ⊂ K .
We can easily obtain it, Φ : K → K is a completely continuous mapping.
3 Main Results
Let A
=
1
3
4
1
4
g ( s )ds, B ∫
∫=
0
g ( s )ds , we denote Ω r = { y ∈ K , y < r} ( r > 0 ),
{ y ∈ K , y= r}.
then ∂Ω=
r
Theorem 3.1
Assume that ( H1 ) , ( H 2 ) hold，if there exist two positive numbers
m1 , m2 and m1 ≠ m2 ，let 0 < m1 , m2 < +∞ such that
（Ⅰ） ∀x ∈ [0,1] , y ∈ [0, m1 ] , f ( x, y ) ≤
m1
|| β ||
A−1 ；
Huijun Zheng
47
1 3
4 4
（Ⅱ） ∀x ∈ [ , ] , y ∈ [
m
α
m2 , m2 ] , f ( x, y ) ≥ 2 B −1 .
α
|| β ||
then the problem(1) has at least one positive solution, satisfying
min{m1 , m2 } ≤|| y ||≤ max{m1 , m2 } .
Proof. Let m1 < m2 ，then Ω m1 = { y ∈ K , y < m1} and Ω m2 = { y ∈ K , y < m2 }
1
1
0
0
∀y ∈ ∂Ω m1 , Φ y (x ) ≤ max ∫ β ( x)g(s) f ( s, y ( s ))ds ≤ m1 A−1 ∫ g(s)ds =
y
0 ≤ x ≤1
α m2
.
∀y ∈ ∂Ω m2 , then min y ( x) ≥ α y / β =
1
3
|| β ||
≤ x≤
4
4
It is easy to see
3
3
4
4
Φ y (x ) ≥ 1min3 ∫14 α ( x )g(s) f ( s, y ( s ))ds ≥ m2 B −1 ∫14 g(s)ds =
|| y || .
4
≤ x≤
4
In view of Theorem 2.1, we know that
Φ
has a fixed point
__
y ∈ Ωm2 \ Ωm1 .That is to say, y is a positive solution of (1)，and m1 ≤|| y ||≤ m2 .
Inference 3.1 Assume that ( H1 ) , ( H 2 ) hold. Our assumptions throughout are，
(Ⅲ) lim+ max
y →0 x∈[0,1]
f ( x, y ) B −1
f ( x, y ) A−1
and lim min
> 2 || β ||,
<
y →+∞ x∈[ 1 , 3 ]
y
|| β ||
α
y
4 4
then the problem(1) has at least one positive solution.
Proof.
Using (Ⅲ) we have, for any x ∈ [0,1] , there exists a sufficiently small
m1 > 0 , such that
f ( x, y ) ≤
Following from lim min
y →+∞ x∈[ 1 , 3 ]
4 4
A−1
m1 , y ∈ (0, m1 ]
|| β ||
(3)
f ( x, y ) B −1
> 2 || β || , that there exists a sufficiently large
y
α
48
Existence and multiplicity of solutions to boundary value problems…
1 3
m2 > 0 , for any x ∈ [ , ] such that,
4 4
f ( x, y ) ≥
B −1
α
2
|| β || y ≥
m2
α
B −1 ,
y ∈[
α
m2 , m2 ]
|| β ||
(4)
Thus, theorem 3.1 implies，then the problem(1) has at least one positive solution.
Inference 3.2 Assume that ( H1 ) , ( H 2 ) hold，Our assumptions throughout are，
（IV） lim+ min
1 3
y →0 x∈[ , ]
4 4
f ( x , y ) B −1
> 2 || β || ,
α
y
and
lim max
y →+∞ x∈[0,1]
f ( x, y ) A−1
,
<
y
|| β ||
then the problem (1) has at least one positive solution.
1 3
Proof. Follows from (IV), there exists 0 < m1 < m2 ，for any x ∈ [ , ] , such that
4 4
f ( x, y ) B −1
≥ 2 || β || ， y ∈ [0, m1 ]
y
α
1 3
Consequently, we have for any x ∈ [ , ] ，
4 4
f ( x, y ) ≥
Following from
B −1
α
2
lim max
y →+∞ x∈[0,1]
|| β || y ≥
m1
α
B −1 ,
y ∈[
α
m1 , m1 ]
|| β ||
(5)
f ( x, y ) A−1
, there exists ε1 > 0 , such that
<
y
|| β ||
A−1
− ε1 > 0 , and
|| β ||
f ( x, y ) ≤ (
A−1
− ε1 ) y , y ≥ m2 .
|| β ||
Let c max{ f ( x, y ) : 0 ≤ y ≤ m2 } , then it is obvious to see
=
f ( x, y ) ≤ c + (
A−1
− ε1 ) y, y ∈ [0, +∞) .
|| β ||
Choose r > max{m2 , c(ε1 || β ||) −1} .Let Ω r = { y ∈ C[0,1] :|| y ||< r}. then for
Huijun Zheng
49
for any y ∈ ∂Ω r , we have
f ( x, y ) ≤ c + (
A−1
A−1
− ε1 ) || y || ≤
r.
|| β ||
|| β ||
(6)
Thus, we obtain there exist x ' ∈ [0,1] and r ≥ m 2 , for all x ∈ [0,1] and y ∈ [0, r ] ,
A−1
f ( x, y ) ≤
r . By Theorem 3.1 we know, then the problem(1) has at least one
|| β ||
positive solution.
Theorem 3.2
Assume that ( H1 ) , ( H 2 ) hold，for all m2 > 0 ，satisfy the (Ⅱ).
Let
f ( x, y ) A−1
f ( x, y ) A−1
, lim max
,
lim max
<
<
y →0+ x∈[0,1]
y
|| β || y →+∞ x∈[0,1] y
|| β ||
Then the problem (1) has two positive solutions.
Proof.
We can prove as the same as Corollary 1 and 2.Following from there
exist 0 < m1 < m2 < r1 ，for any x ∈ [0,1] ,
f ( x, y ) ≤
A−1
m1 , y ∈ (0, m1 ] .
|| β ||
and exist x ' ∈ [0,1] and r1 ≥ m 2 , such that
f ( x, y ) ≤
A−1
r1 ,
|| β ||
when x ∈ [0,1] and y ∈ [0, r1 ]
Then by (IV)，we know form Theorem 3.1，Then the problem (1) has two positive
solutions y1 , y2 , and satisfy 0 < m1 <|| y1 ||< m2 <|| y2 ||< r1 .
Theorem 3.3 Assume that ( H1 ) , ( H 2 ) hold，for all m1 > 0 ，satisfy the (IV). Let
50
Existence and multiplicity of solutions to boundary value problems…
lim min
y →0+ x∈[ 1 , 3 ]
4 4
f ( x, y ) || β || −1
> 2 B ,
α
y
and
lim min
y →+∞ x∈[ 1 , 3 ]
4 4
f ( x, y ) || β || −1
> 2 B .
α
y
Then the problem (1) has two positive solutions y1 , y2 , and satisfy
0 < m1" <|| y1 ||< m2 <|| y2 ||< m2" .
1 3
α ' '
Proof. There exist 0 < m1' < m1 < m2'' , for any x ∈ [ , ], y ∈ [
m1 , m1 ] , we
4 4
|| β ||
have
f ( x, y ) ≥
min
1 3
x∈[ , ]
4 4
B −1
α
2
|| β || y ≥
m1'
α
B −1 .
Hence we obtain for any
f ( x, y ) ≥
min
1 3
x∈[ , ]
4 4
B −1
α
2
|| β || y ≥
1 3
α '' ''
B −1 , x ∈ [ , ], y ∈ [
m2 , m2 ] .
α
4 4
|| β ||
m2"
Then by (IV), we know from Theorem 3.1. Then the problem (1) has two positive
solutions y1 , y2 , satisfy 0 < m1' <|| y1 ||< m1 <|| y2 ||< m2' .
4 Conclusion
We study a class of higher order nonlinear differential equation boundary valu
e problem and give some sufficient conditions for existence of positive solutions u
sing the cone fixed point theorems. It enriches the results of the related literatures
further.
Huijun Zheng
51
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