Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 808175, 12 pages
doi:10.1155/2011/808175
Research Article
The Existence of Positive Solutions for
Singular Impulse Periodic Boundary Value Problem
Zhaocai Hao and Tanggui Chen
Department of Mathematics, Qufu Normal University, Qufu,
Shandong 273165, China
Correspondence should be addressed to Zhaocai Hao, zchjal@163.com
Received 15 May 2011; Accepted 1 July 2011
Academic Editor: Jian-Ping Sun
Copyright q 2011 Z. Hao and T. Chen. 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.
We obtain new result of the existence of positive solutions of a class of singular impulse periodic
boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require
the monotonicity of functions in paper Zhang and Wang, 2003. An example is also given to
illustrate our result.
1. Introduction
Because of wide interests in physics and engineering, periodic boundary value problems have
been investigated by many authors see 1–19. In most real problems, only the positive
solution is significant.
In this paper, we consider the following periodic boundary value problem PBVP in
short with impulse effects:
Δu|ttk
−u t Mut ft, ut, t ∈ J ,
Ik utk , −Δu ttk Jk utk , k 1, 2, . . . l,
u0 u2π,
1.1
u 0 u 2π.
Here, J 0, 2π, 0 < t1 < t2 < · · · < tl < 2π, J J \{t1 , t2 , . . . , tl }, M > 0, f ∈ CJ ×R , R , Ik ∈
CR , R, Jk ∈√CR , R , R 0, ∞, R 0, ∞ with −1/mJk u < Ik u < 1/mJk u,
u ∈ R , m M. Δu|ttk utk − ut−k , Δu |ttk u tk − u t−k , where ui tk and ui t−k ,
i 0, 1, respectively, denote the right and left limit of ui t at t tk .
2
International Journal of Differential Equations
In 7, Liu applied Krasnoselskii’s and Leggett-Williams fixed-point theorem to establish the existence of at least one, two, or three positive solutions to the first-order periodic
boundary value problems
x t atxt ft, xt,
a.e. t ∈ 0, T \ t1 , . . . , tp ,
Δx|ttk Ik xtk ,
k 1, . . . , p,
1.2
x0 xT .
Jiang 5 has applied Krasnoselskii’s fixed point theorem to establish the existence of positive
solutions of problem
x t Mxt ft, xt,
x0 x2π,
t ∈ 0, 2π,
x 0 x 2π.
1.3
The work 5 proved that periodic boundary value problem PBVP in short 1.3 without
singularity have at least one positive solutions provided ft, x is superlinear or sublinear
at x 0 and x ∞. In 14, Tian et al. researched PBVP 1.1 without singularity.
They obtained the existence of multiple positive solutions of PBVP 1.1 by replacing
the suplinear condition or sublinear condition of 4 with the following limit inequality
condition:
A1 l
2πf0 J0 i σ > 2πM,
l
2πf∞ J∞ i σ > 2πM,
i1
1.4
i1
A2 2πf 0
l
i1
J i σ < 2πM,
0
2πf
∞
l
∞
J i σ < 2πM.
1.5
i1
Nieto 10 introduced the concept of a weak solution for a damped linear equation with
Dirichlet boundary conditions and impulses. These results will allow us in the future to deal
with the corresponding nonlinear problems and look for solutions as critical points of weakly
lower semicontinuous functionals.
We note that the function f involved in above papers 5, 7, 10, 14 does not have
singularity. Xiao et al. 16 investigate the multiple positive solutions of singular boundary
value problem for second-order impulsive singular differential equations on the halfline,
where the function ft, u is singular only at t 0 and/or t 1. Reference 19 studied
PBVP 1.3, where the function f has singularity at x 0. The authors present the existence
of multiple positive solutions via the Krasnoselskii’s fixed point theorem under the following
conditions.
International Journal of Differential Equations
3
A1 There exist nonnegative valued ξx, ηx ∈ C0, ∞ and P t, Qt ∈ L1 0, 2π
such that
0 ≤ ft, x ≤ P tξx Qtηx, a.e. t, x ∈ 0, 2π × 0, ∞,
⎧
⎫
⎨
⎬
x
> Bj ,
sup
2π
2π
x∈0,∞⎩
P tdtξx/ηx 0 Qtdt η δj t ⎭
0
1.6
where ηx is nonincreasing and ξx/ηx is nondecreasing on 0, ∞,
A2 min
lim inf
2π
0
fx, wdx : δj t ≤ w ≤ t
t
t → 0
>
1
,
Aj
1.7
>
1
.
Aj
1.8
A3 min
lim inf
2π
0
fx, wdx : δj t ≤ w ≤ t
t → ∞
t
Here, δj , Aj , Bj are some constants.
In this paper, the nonlinear term ft, u is singular at u 0, and positive solution of
PBVP 1.1 is obtained by a nonlinear alternative principle of Leray-Schauder type in cone.
We do not require the monotonicity of functions η, ξ/η used in 19. An example is also given
to illustrate our result.
This paper is organized as follows. In Section 1, we give a brief overview of recent
results on impulsive and periodic boundary value problems. In Section 2, we present some
preliminaries such as definitions and lemmas. In Section 3, the existence of one positive
solution for PBVP 1.1 will be established by using a nonlinear alternative principle of LeraySchauder type in cone. An example is given in Section 4.
2. Preliminaries
Consider the space P CJ, R {u : u is a map from J into R such that ut is continuous at
t/
tk , left continuous at t tk , and utk exists, for k 1, 2, . . . l.}. It is easy to say that P CJ, R
is a Banach space with the norm upc supt∈J |ut|. Let P C1 J, R {u ∈ P CJ, R : u t
tk , and u tk , u t−k exist and u t is left continuous at
exists at t /
tk and is continuous at t /
t tk , for k 1, 2, . . . l.} with the norm upc1 max{upc , u pc }. Then, P C1 J, R is also
a Banach space.
4
International Journal of Differential Equations
Lemma 2.1 see 15. u ∈ P C1 J, R ∩ C2 J , R is a solution of PBVP 1.1 if and only if u ∈
P CJ is a fixed point of the following operator T :
T ut 2π
Gt, sfs, usds 0
l
k1
l
∂Gt, s Gt, tk Jk utk ∂s k1
Ik utk ,
2.1
stk
where Gt, s is the Green’s function to the following periodic boundary value problem:
−u Mu 0,
u0 u2π,
Gt, s :
u 0 u 2π,
2.2
⎧⎧
⎪⎪emt−s em2π−ts ,
1 ⎨⎨
0 ≤ s ≤ t ≤ 2π,
Γ⎪
⎩⎪
⎩ems−t em2π−st ,
0 ≤ t ≤ s ≤ 2π,
here, Γ 2me2mπ − 1. It is clear that
2emπ
e2mπ 1
Gπ ≤ Gt, s ≤ G0 .
Γ
Γ
2.3
K u ∈ P CJ, R : ut ≥ σupc , t ∈ J ,
2.4
Define
where
σ
1
e2mπ
.
2.5
The following nonlinear alternative principle of Leray-Schauder type in cone is very
important for us.
Lemma 2.2 see 4. Assume that Ω is a relatively open subset of a convex set K in a Banach space
P CJ, R. Let T : Ω → K be a compact map with 0 ∈ Ω. Then, either
i T has a fixed point in Ω, or,
ii there is a u ∈ ∂Ω and a λ < 1 such that u λT u.
3. Main Results
In this section, we establish the existence of positive solutions of PBVP 1.1.
International Journal of Differential Equations
5
Theorem 3.1. Assume that the following three hypothesis hold:
H1 there exists nonnegative functions ξu, ηu, γu ∈ C0, ∞ and pt, qt ∈
L1 0, 2π such that
ft, u ≤ ptξu qtηu,
max Jk u ≤ γu,
t, u ∈ 0, 2π × 0, ∞,
3.1
t, u ∈ 0, 2π × 0, ∞,
1≤k≤l
3.2
H2 there exists a positive number r > 0 such that
A
2
2π
max ξx
x∈σr,r
psds max ηx
2π
x∈σr,r
0
qsds Alγr < r,
3.3
0
H3 for the constant r in (H2 ), there exists a function Φr > 0 such that
ft, u > Φr t,
2π
t, u ∈ 0, 2π × 0, r,
Φr sds > 0.
3.4
0
Then PBVP 1.1 has at least one positive periodic solution with 0 < u < r, where
√
e2mπ 1
e2π M 1
A
.
√
√
me2mπ − 1
M e2π M − 1
3.5
Proof. The existence of positive solutions is proved by using the Leray-Schauder alternative
principle given in Lemma 2.2. We divide the rather long proof into six steps.
Step 1. From 3.3, we may choose n0 ∈ {1, 2, . . .} such that
A
2
2π
max ξx
x∈σr,r
psds max ηx
x∈σr,r
0
2π
qsds Alγr 0
1
< r.
n0
3.6
Let N0 {n0 , n0 1, . . .}. For n ∈ N0 . We consider the family of equations
−u t Mut λfn t, ut Δu|ttk Ik utk ,
M
,
n
t ∈ J ,
− Δu ttk Jk utk ,
u0 u2π,
u 0 u 2π,
k 1, 2, . . . l,
3.7
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International Journal of Differential Equations
where λ ∈ 0, 1 and
1
,
fn t, u f t, max u,
n
t, u ∈ J × 0, ∞.
3.8
For every λ and n ∈ N0 , define an operator as follows:
2π
l
Gt, sfn s, usds Gt, tk Jk utk Tλ,n ut λ
0
l
∂Gt, s ∂s k1
k1
Ik utk ,
3.9
u ∈ K.
stk
Then, we may verify that
Tλ,n : K −→ K is completely continuous.
3.10
To find a positive solution of 3.7 is equivalent to solve the following fixed point problem in
P CJ, R:
u Tλ,n u 1
.
n
3.11
Let
Ω {x ∈ K : x < r},
3.12
then Ω is a relatively open subset of the convex set K.
Step 2. We claim that any fixed point u of 3.11 for any λ ∈ 0, 1 must satisfies u / r.
Otherwise, we assume that u is a solution of 3.11 for some λ ∈ 0, 1 such that u r.
Note that fn t, u ≥ 0. ut ≥ 1/n for all t ∈ J and r ≥ ut ≥ 1/n σu − 1/n. By the choice
of n0 , 1/n ≤ 1/n0 < r. Hence, for all t ∈ J, we get
1
1
1 1
1
1
r ≥ ut ≥ σ u − ≥ σ u − ≥ σ r −
> σr.
n
n
n
n
n
n
3.13
From 3.2, we have
Jk utk ≤ max Jk utk ≤ γutk ≤ γr.
3.14
1≤k≤l
Consequently, for any fixed point u of 3.11, by 3.8, 3.13, and 3.14, we have
ut λ
2π
0
≤
2π
0
l
l
∂Gt, s 1
Gt, sfn s, usds k 1 Gt, tk Jk utk Ik utk n
∂s
stk
k1
k1
Gt, sfs, usds l
k1
l
∂Gt, s Gt, tk Jk utk ∂s k1
Ik utk stk
1
n
International Journal of Differential Equations
≤
7
2π
Gt, sfs, usds
0
1 mt−tk e
em2π−ttk Jk utk Γ tk ≤t
emtk −t em2π−tk t Jk utk −emt−tk em2π−ttk mIk utk tk ≤t
tk >t
tk >t
1
emtk −t − em2π−tk t mIk utk n
2π
Gt, sfs, usds
0
1 mt−tk e
Jk utk − mIk utk Γ tk ≤t
em2π−ttk Jk utk mIk utk emtk −t Jk utk mIk utk tk ≤t
tk >t
1
em2π−tk t Jk utk − mIk utk .
n
tk >t
3.15
It follows from −1/mJk u < Ik u < 1/mJk u that
Jk utk − mIk utk > 0,
Jk utk mIk utk > 0.
3.16
So, we get from 3.1, 3.2, and 3.3 that
ut ≤
2π
0
l
2 e2mπ 1 1
Gt, sfs, usds Jk utk Γ
n
k1
2π
!
1
Gt, s psξus qsηus ds Alγr n
0
0
2π
2π
1
A
≤
psds max ξx qsds max ξx Alγr .
2 0
x∈σr,r
x∈σr,r
n
0
0
≤
3.17
Therefore,
A
r u ≤
2
2π
0
psds max ξx x∈σr,r
2π
qsds max ξx Alγr 0
This is a contraction, and so the claim is proved.
x∈σr,r
1
< r.
n0
3.18
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International Journal of Differential Equations
Step 3. From the above claim and the Leray-Schauder alternative principle, we know that
operator 3.9 with λ 1 has a fixed point denoted by un in Ω. So, 3.7 with λ 1 has a
positive solution un with
un t ≥
un < r,
1
,
n
t ∈ J.
3.19
Step 4. We show that {un } have a uniform positive lower bound; that is, there exists a constant
δ > 0, independent of n ∈ N0 , such that
min{un t} ≥ δ.
3.20
t
In fact, from 3.4, 3.8, 3.16, and 3.19, we get
un t 2π
Gt, sfn s, un sds 0
k1
Ik un tk stk
Gt, sfs, un sds 0
k1
≥
1
n
l
Gt, tk Jk un tk k1
l
∂Gt, s ∂s 2π
Gt, tk Jk un tk k1
l
∂Gt, s ∂s 2π
l
Ik un tk stk
1
n
Gt, sΦr sds
0
1 mt−tk e
Jk un tk − mIk un tk Γ tk ≤t
em2π−ttk Jk un tk mIk un tk tk ≤t
emtk −t Jk un tk mIk un tk tk >t
1
m2π−tk t
e
Jk un tk − mIk un tk n
tk >t
≥
2π
Gt, sΦr sds
0
≥
2emπ
Γ
2π
Φr sds : δ > 0.
0
3.21
International Journal of Differential Equations
9
Step 5. We prove that
un < H,
n ≥ n0
3.22
for some constant H > 0. Equations 3.19 and 3.20 tell us that δ ≤ un t ≤ r, so we may let
M1 M2 maxGt t, s,
max ft, u,
t, s∈J
t∈J, u∈δ,r
M3 max
u∈δ,r
l
Jk u.
3.23
k1
Then,
un supun t
t∈J
l
2π
sup
Gt t, sfs, un sds Gt t, tk Jk un tk t∈J 0
k1
l
∂
∂t
k1
"
#
∂Gt, s I
u
t
k
n
k
∂s
stk
2π
sup
Gt t, sfs, un sds
0
t∈J
m mt−tk e
Jk un tk − mIk un tk Γ tk ≤t
− em2π−ttk Jk un tk mIk un tk tk ≤t
− emtk −t Jk un tk mIk un tk tk >t
m2π−tk t
e
Jk un tk − mIk un tk t >t
k
l
2m e2mπ 1 ≤ sup
Gt t, s fs, un sds Jk un tk Γ
0
t∈J
k1
2m e2mπ 1
≤ 2πM1 M2 M3 : H.
Γ
2π
3.24
Step 6. Now, we pass the solution un of the truncation equation 3.7 with λ 1 to that of the
original equation 1.1. The fact that un < r and 3.22 show that {un }n∈N0 is a bounded and
equi-continuous family on 0, 2π. Then, the Arzela-Ascoli Theorem guarantees that {un }n∈N0
has a subsequence {unj }j∈N , converging uniformly on 0, 2π. From the fact un < r and
10
International Journal of Differential Equations
3.20, u satisfies δ ≤ ut ≤ r for all t ∈ J. Moreover, unj also satisfies the following integral
equation:
unj t 2π
0
Gt, sf s, unj s ds
l
l
1
∂Gt, s Gt, tk Jk unj tk Ik unj tk n .
∂s
j
stk
k1
k1
3.25
Let j → ∞, and we get
ut 2π
Gt, sfs, usds
0
l
k1
l
∂Gt, s Gt, tk Jk utk ∂s k1
3.26
Ik utk ,
stk
where the uniform continuity of ft, u on J × δ, r is used. Therefore, u is a positive solution
of PBVP 1.1. This ends the proof.
4. An Example
Consider the following impulsive PBVP:
|sin u|
−u t Mut t 1 3/2
t1 |cos u|,
u
Δu|ttk 2
min{c1 , c2 , . . . , cl }
utk ,
√
2 M
u0 u2π,
−Δu ttk ck utk ,
t ∈ J ,
k 1, 2, . . . , l,
4.1
u 0 u 2π,
where ck > 0 are constants. Then, PBVP 4.1 has at least one positive solution u with 0 <
u < 1.
To see this, we will apply Theorem 3.1.
Let
|sin u|
t1 |cos u|,
ft, u t2 1 3/2
u
4.2
then ft, u has a repulsive singularity at u 0
lim ft, u ∞,
u → 0
uniformaly in t.
4.3
International Journal of Differential Equations
11
Denote
pt t2 ,
qt t,
ξu 1 |sin u|
,
u3/2
ηu 1 |cos u|,
4.4
γu max{c1 , c2 , . . . , cl }u,
r 1,
Φr t t t2 .
Then, it is easy to say that 3.1, 3.2, and 3.3 hold. From 3.5, we know
lim A lim √
M → ∞
M → ∞
e2π
M
√
M
1
√
e2π M
−1
0.
4.5
So, we may choose M large enough to guarantee that 3.3 holds. Then, the result follows
from Theorem 3.1.
Remark 4.1. Functions ξ, η in example 4.1 do not have the monotonicity required as in 19.
So, the results of 19 cannot be applied to PBVP 4.1.
Acknowledgments
The authors are grateful to the anonymous referees for their helpful suggestions and
comments. Zhaocai Hao acknowledges support from NSFC 10771117, Ph.D. Programs
Foundation of Ministry of Education of China 20093705120002, NSF of Shandong Province
of China Y2008A24, China Postdoctoral Science Foundation 20090451290, ShanDong
Province Postdoctoral Foundation 200801001, and Foundation of Qufu Normal University
BSQD07026.
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