In this paper, we consider the periodic boundary value problem with sign changing nonlinearity
subject to the boundary value conditions:
where
is a positive constant and
f
(
t
,
u
) is a continuous function. Using LeggettWilliams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term
f
may change sign.
AMS Mathematics Subject Classification : 34B18.
1. Introduction
In this paper, we are concerned with the multiplicity of positive solutions of the nonlinear threeorder periodic boundary value problem
throughout this paper, we assume that
(H
_{1}
)
is continuous and there exists a constant
L
> 0 such that
for all (
t
,
u
) ∈ [0, 2π]×[0, ∞];
(H
_{2}
)There exist continuous, nonnegative and nondecreasing functions
g
(
x
),
h
(
x
) on (0, ∞) such that
g
(
x
) ≤
f
(
t
,
x
) +
L
≤
h
(
x
).
Nonlinear periodic boundary value problem have been extensively studied by many authors. The existence of solutions is one of the most important aspects of periodic boundary value problem(see
[1]

[7]
,
[9]

[17]
and references therein). In recent years, many authors take more interested in the twoorder or fourorder periodic boundary value problem (
[4]
,
[11]
,
[13]

[17]
). However, for threeorder periodic boundary value problems , a few of authors have studied (
[9]
,
[10]
,
[12]
). For the periodic boundary value problem (1), different methods and techniques have been employed to discuss the existence of positive solutions. We recall the following three results. In
[12]
, Kong and Wang, by employing Schauder fixed point theorem together with priori estimates and perturbation technique, established the existence of at least one positive solution under suitable conditions of
f
. In
[9]
, by using Krasnoselskii fixed point theorem together with nonlinear alternative of LeraySchauder, the existence of positive periodic solutions have been discussed. Recently, In
[10]
, Yao obtained existence results for singular and multiple positive periodic solutions by applying GuoLakshmikantham fixed point index theory for cones.
Inspired and motivated by the work mentioned above, in this paper, we shall apply LeggettWilliams fixed point theorem to investigate the existence of at least three positive periodic solutions to (1). The interesting point is the nonlinear term
f
may change sign.
2. Background and definitions
The proof of our main result is based on the LeggettWilliams fixedpoint theorem, which deals with fixed points of a conepreserving operator defined on an ordered Banach space. For the convenience of readers, we present here the necessary definitions from cone theory in Banach spaces. These definitions can be found in the recent literature.
Definition 2.1.
Let
E
be a real Banach space over
R
. A nonempty closed set
P
⊂
E
is said to be a cone provided that

(i)αu+βv∈Pfor allu,v∈Pand allα≥ 0,β≥ 0, and

(ii)u,−u∈Pimpliesu= 0.
If
P
⊂
E
is a cone, we denote the order induced by
P
on
E
by ≤. For
u
,
v
∈
P
, we write
u
≤
v
if and only if
v
−
u
∈
P
.
Definition 2.2.
The map
ψ
is said to be a nonnegative continuous concave functional on a cone
P
of a real Banach space
E
provided that
ψ
:
P
→ [0, ∞) is continuous and
for all
x
,
y
∈
P
and 0 ≤
t
≤ 1.
Definition 2.3.
Let 0 <
a
<
b
be given and let
ψ
be a nonnegative continuous concave functional on the cone
P
. Define the sets
and
P
(
ψ
,
a
,
b
) by
Next we state the LeggettWilliams fixedpoint theorem. The proof can be found in Deimling’s text
[8]
.
Theorem 2.4
(LeggettWilliams FixedPoint Theorem).
Let E
= (
E
, ∥·∥)
be a Banach space
,
P
⊂
E
is a cone in E
.
Let
be a completely continuous operator and let ψ be a nonnegative continuous concave functional on P such that ψ
(
u
) ≤ ∥
u
∥,
Suppose that there exist
0 <
r
<
a
<
b
<
c such that

(S1) {u∈P(ψ,a,b) ψ(u) >a} ≠ ∅and ψ(Tu) >a for u∈P(ψ,a,b),

(S2)

(S3)ψ(Tu) >a for u∈P(ψ,a,c)with∥Tu∥ >b.
Then T has at least three fixed points
such that
∥
u
_{1}
∥ <
r
,
a
<
ψ
(
u
_{2}
), ∥
u
_{3}
∥ >
r
,
ψ
(
u
_{3}
) <
a
.
3. Some preliminary results
Lemma 3.1
(
[12]
).
If ρ
∈ (0, +∞),
then the linear problem
has a unique positive solution
For every function
u
∈
C
[0, 2π], we define the operator
where
By a direct calculation, we can easily obtain
Now, we consider the problem
If
u
is a positive solution of problem (3), i.e.
u
(
t
) > 0 for
t
∈ [0, 2π], it is easy to verify that
y
(
t
) = (
Ju
)(
t
) is a positive solution of problem (1). Therefore, we will concentrate the problem (3) for which we have the following result.
Lemma 3.2
(
[9
,
12]
).
Let w
(
t
)
be a unique solution of
(2),
then problem
(3)
is equivalent to integral equation
where
Lemma 3.3
(
[9
,
12]
).
Let
then we have the estimates
From
[9]
, we know that
is a positive solution of (3) when
y
(
t
) is the solution of
if the nonlinear term
f
satisfies the condition (H
_{2}
). So, we can transform the problem into discussing the existence of positive solutions for (5).
4. Existence of triple positive solutions
Let
E
=
C
[0, 2π] be endowed with the maximum norm,
Define the cone
P
⊂
E
by
where
σ
=
m/M
.
Finally, let the nonnegative continuous concave functional
ψ
:
P
→ [0, ∞) be defined by
We notice that, for each
u
∈
P
,
ψ
(
u
) ≤ ∥
u
∥.
Theorem 4.1.
Assume that
(H
_{1}
), (H
_{2}
)
hold. There exist constants
such that
Then the boundary value problem
(1)
has at least three positive solutions u
_{1}
,
u
_{2}
and u
_{3}
satisfying u_{i}
=
J
(
y_{i}
−
ρω
), ∥
y
_{1}
∥ <
r
,
a
<
ψ
(
y
_{2}
)
and
∥
y
_{3}
∥ >
r with ψ
(
y
_{3}
) <
a where y_{i}
(
t
)
is the solution of
(5)(i=1, 2, 3).
Proof.
Define the operator
T
:
P
→
P
by
where
G
(
t
,
s
) is the Green function given by (4). The boundary value problem (5) has a solution
u
=
u
(
t
) if and only if
u
solves the operator equation
u
=
Tu
. Thus we set out to verify that the operator
T
satisfies Theorem 2.1.
Firstly, we show that
. In fact, if
u
∈
P
, then (
Ju
)(
t
) −
ω
≥
ρω/ρ
−
ω
= 0. Thus from Lemma 3.3 and (H
_{1}
), it follows that (
Tu
)(
t
) ≥ 0, 0 ≤
t
≤ 2π. Let
u
∈
P
, then from Lemma 3.3, we have
On the other hand, by Lemma 3.3 and
(H
_{3}
)
, for all
u
∈
P
,
So
thus we have
TP
⊂
P
.
If
then (
Ju
)(
t
) −
ω
≤
c/ρ
−
ω
. Therefore, by (H
_{2}
),
(H
_{4}
)
and Lemma 3.3, for 0 ≤
t
≤ 2π,
Then
is well defined. It is easy to see that
T
is continuous and completely continuous since
f
: [0, 2π] × [0, ∞] →
R
is a continuous function.
In the same way, we can obtain that
by
(H
_{6}
)
. So, condition
(S
_{2}
)
of Theorem 2.1 is satisfied.
Next we prove
(S
_{1}
)
of Theorem 2.1 holds. Choose
It is easy to see that
u
_{0}
(
t
) ∈
P
(
ψ
,
a
,
b
) and
so {
u
∈
P
(
ψ
,
a
,
b
) 
ψ
(
u
) >
a
} ≠ ∅.
In fact, if
u
∈
P
(
ψ
,
a
,
b
), then
Thus
As a result, it follows from (H
_{2}
),
(H
_{5}
)
and Lemma 3.3 that, for 0 ≤
t
≤ 2π,
Consequently condition
(S
_{1}
)
of Theorem 2.1 is satisfied.
We finally show that
(S
_{3}
)
of Theorem 2.1 also holds.
Suppose that
u
∈
P
(
ψ
,
a
,
c
) with ∥
Tu
∥ >
b
. Then by Lemma 3.3, we have
Thus condition
(S
_{3}
)
of Theorem 2.1 is also satisfied. Therefore an application of Theorem 2.1 leads to the conclusion that the boundary value problem (5) has at least three positive solutions
y
_{1}
,
y
_{2}
and y
_{3}
satisfying ∥
y
_{1}
∥ <
r
,
a
<
ψ
(
y
_{2}
) and ∥
y
_{3}
∥ >
r
with
ψ
(
y
_{3}
) <
a
. Thus the boundary value problem (1) has at least three positive solutions
u
_{1}
,
u
_{2}
and u
_{3}
satisfying
u_{i}
=
J
(
y_{i}
−
ρω
).
BIO
Huixuan Tan received M.SC. from Dalian University of Thechnology. She is currently a instructor in the Department of Mathematics at Shijiazhuang Mechanical Engineering College. Her research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
email: tanhuixuan 2004@163.com
Hanying Feng received M.SC. from Anhui University and Ph.D. from Beijing Institute of Thechnology.He is currently a professor in the Department of Mathematics at Shijiazhuang Mechanical Engineering College. His research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
email: fhanying@yahoo.com.cn
Xingfang Feng received M.SC. from Hebei Normal University. She is currently a instructor in Department of Mathematics at Shijiazhuang Mechanical Engineering College. Her research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
email: fxfg651@163.com
Yatao Du received M.SC. and Ph.D. from Hebei Normal University. She is currently a instructor in Department of Mathematics at Shijiazhuang Mechanical Engineering College. Her research interests focus on the theory and applications of differential equations.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China.
email: dyt77@sina.com
Nieto J.J.
(2002)
Differential inequalities for functional perturbations of firstorder ordinary differential equation
Appl. Math. Lett.
15
173 
179
DOI : 10.1016/S08939659(01)001148
Amster P.
,
De Npoli P.
,
Mariani M.C.
(2005)
Periodic solutions of a resonant thirdorder equation
Nonlinear Anal.
60
399 
410
DOI : 10.1016/j.na.2003.03.001
Cabada A.
(1994)
The method of lower and upper solutions for second third, fourth and higher order boundary value problem
J. Math. Anal. Appl.
185
302 
320
DOI : 10.1006/jmaa.1994.1250
Jiang D.
(1998)
On the existence of positive solutions to second order periodic BVPs
Acta Math. Sinica(New Ser).
18
31 
35
Jiang D.
,
Chu J.
,
O’Reganb D.
,
Agarwal R.P.
(2003)
Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces
J. Math. Anal. Appl.
286
563 
576
DOI : 10.1016/S0022247X(03)004931
Nieto J.J.
,
RodriguezLopez R.
(2003)
Remarks on periodic boundary value problems for functional differential equations
J. Comput. Appl. Math.
158
339 
353
DOI : 10.1016/S03770427(03)004527
Deimling K.
1985
Nonlinear Functional Analysis
SpringerVerlag
New York
Chu J.F.
,
Zhou Z.H.
(2006)
Positive solutions for singular nonlinear thirdorder periodic boundary problems
Nonlinear Anal.
64
1528 
1542
DOI : 10.1016/j.na.2005.07.005
Yao Q.L.
(2010)
Positive solutions of nonlinear threeorder periodic boundary value problem
Acta Math. Sci. Ser. A Chin. Ed.
30
1495 
1502
Cao J.Y.
,
Wang Q.Y.
(2010)
Existence of positive solutions to a class of twoorder differential equation with twopoint boundary value problems
J. Huaqiao Univ. Nat. Sci. Ed.
31
113 
117
Kong L.B.
,
Wang S.T.
,
Wang J.Y.
(2001)
Positive solution of a singular nonlinear thirdorder periodic boundary value problem
J. Comput. Appl. Math.
132
247 
253
DOI : 10.1016/S03770427(00)003253
Cabada A.
,
Nieto J.J.
(1990)
A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems
J. Math. Anal.
151
181 
189
DOI : 10.1016/0022247X(90)90249F
Wang M.X.
,
Cabada A.
,
Nieto J.J.
(1993)
Monotone methed for nonlinear second order periodic boundary value problems with Caratheodory functions
Ann. Polon. Math.
58
221 
235
Kong L.B.
,
Jiang D.Q.
(1998)
Multiple positive solutions of a nonlinear fourth order periodic boundary value problem
Ann. Polon. Math.
69
265 
270
Wang J.Y.
,
Jiang D.Q.
(1998)
A singular nonliear secondorder periodic boundary value problem
J. Tohoku Math.
50
203 
210
DOI : 10.2748/tmj/1178224974