Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 493639, 12 pages
doi:10.1155/2011/493639
Research Article
Multiple Positive Solutions for Singular
Periodic Boundary Value Problems of Impulsive
Differential Equations in Banach Spaces
Hongxia Fan1, 2 and Yongxiang Li1
1
Department of Mathematics, Northwest Normal University,
Lanzhou 730070, China
2
Department of Mathematics, Lanzhou Jiaotong University,
Lanzhou 730070, China
Correspondence should be addressed to Hongxia Fan, lzfanhongxia@163.com
Received 21 June 2011; Accepted 26 August 2011
Academic Editor: Elena Braverman
Copyright q 2011 H. Fan and Y. Li. 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.
By means of the fixed point theory of strict set contraction operators, we establish a new existence
theorem on multiple positive solutions to a singular boundary value problem for second-order
impulsive differential equations with periodic boundary conditions in a Banach space. Moreover,
an application is given to illustrate the main result.
1. Introduction
The theory of impulsive differential equations describes processes that experience a sudden
change of their state at certain moments. In recent years, a great deal of work has been done
in the study of the existence of solutions for impulsive boundary value problems, by which a
number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics,
and physics phenomena are described. For the general aspects of impulsive differential
equations, we refer the reader to the classical monograph 1. For some general and recent
works on the theory of impulsive differential equations, we refer the reader to 2–14.
Meanwhile, the theory of ordinary differential equations in abstract spaces has become a new
important branch see 15–18. So it is interesting and important to discuss the existence of
positive solutions for impulsive boundary value problem in a Banach space.
Let E, · be a real Banach space, J 0, 2π, 0 t0 < t1 < t2 < · · · < tm <
tm1 2π, J0 0, t1 , and Ji ti , ti1 , i 1, . . . , m. Note that P CJ, E {u : u is a
2
Abstract and Applied Analysis
map from J into E such that ut is continuous at t /
tk and left continuous at t tk and
utk exist, k 1, 2, . . . , m}, and it is also a Banach space with norm
uP C suput.
1.1
t∈J
Let the Banach space E be partially ordered by a cone P of E; that is, x ≤ y if and only
if y − x ∈ P , and P CJ, E is partially ordered by K {u ∈ P CJ, E : ut ∈ P, t ∈ J} : u ≤ v if
and only if v − u ∈ K; that is, ut ≤ vt for all t ∈ J.
In this paper, we consider the following singular periodic boundary value problem
with impulsive effects in Banach E
−u t M2 ut ft, ut,
t ∈ J, t / tk ,
Δu|ttk Ik utk ,
Δu ttk −I k utk ,
u0 u2π,
k 1, 2, . . . , m,
1.2
u 0 u 2π,
where M > 0 is constant, ft, u may be singular at t 0 and/or t 2π, f ∈ C0, 2π × P, P ,
Ik , I k ∈ CP, P , Δu|ttk utk − ut−k , Δu |ttk u tk − u t−k , k 1, 2, . . . , m, and ui tk resp., ui t−k denote the right limit resp., left limit of ui t at t tk , i 0, 1.
In the special case where E R 0, ∞, and Ik I k 0, k 1, 2, . . . , m, problem
1.2 is reduced to the usual second-order periodic boundary value problem. For example, in
19, the periodic boundary value problem:
−u t Mut ft, ut,
u0 u2π,
t ∈ 0, 2π,
u 0 u 2π,
1.3
was proved to have at least one positive solution, by Jiang 19 .
In 20, the authors studied the multiplicity of positive solutions for IBVP1.2 in E R ; the main tool is the theory of fixed point index.
In 21, the author considers the following periodic boundary value problem of
second-order integrodifferential equations of mixed type in Banach space:
−u ft, u, T u, Su,
Δu t ∈ 0, 2π,
Δu|ttk Ik utk ,
ttk
−I k utk ,
u0 u2π,
k 1, 2, . . . , m,
u 0 u 2π,
1.4
Abstract and Applied Analysis
3
where f ∈ CJ × E × E × E, E, Ik , I k ∈ CE, E, and the operators T , S are given by
T ut t
kt, susds,
0
Sut 2π
k1 t, susds,
1.5
0
with k ∈ CD, R, D {t, s ∈ R2 : 0 ≤ s ≤ t ≤ 2π}, k1 ∈ CJ × J, R. By applying
the monotone iterative technique and cone theory based on a comparison result, the author
obtained an existence theorem of minimal and maximal solutions for the IBVP1.4.
Motivated by the above facts, our aim is to study the multiplicity of positive solutions
for IBVP1.2 in a Banach space. By means of the fixed point index theory of strict set
contraction operators, we establish a new existence theorem on multiple positive solutions
for IBVP1.2. Moreover, an application is given to illustrate the main result.
The rest of this paper is organized as follows. In Section 2, we present some basic
lemmas and preliminary facts which will be needed in the sequel. Our main result and its
proof are arranged in Section 3. An example is given to show the application of the result in
Section 4.
2. Preliminaries
Let Tr {x ∈ E : x ≤ r}, Br {u ∈ P CJ, E : uP C ≤ r} r > 0; for D ⊂ P CJ, E,
we denote Dt {ut : u ∈ D} ⊂ Et ∈ J.α denotes the Kuratowski measure of noncompactness.
Let P C1 J, E {u | u be a map from J into E such that ut is continuously
differentiable at t /
tk and left continuous at t tk and utk , u t−k , u tk exist, k 1, 2, . . . , m}. Evidently, P C1 J, E is a Banach space with norm
uP C1 max uP C , u P C .
2.1
Let J J \ {t1 , t2 , . . . , tm }; a map u ∈ P C1 J, E ∩ C2 J , E is a solution of IBVP1.2 if
it satisfies 1.2.
Now, we first give the following lemmas in order to prove our main result.
Lemma 2.1 see 17. Let K be a cone in real Banach space E, and let Ω be a nonempty bounded
open convex subset of K. Suppose that A : Ω → K is a strict set contraction and AΩ ⊂ K. Then
the fixed-point index iA, Ω, K 1.
Lemma 2.2 see 21. u ∈ P C1 J, E ∩ C2 J , E is a solution of IBVP 1.2 if and only if u ∈
P CJ, E is a solution of the impulsive integral equation:
ut 2π
0
Gt, sfs, usds m Gt, tk I k utk Ht, tk Ik utk ,
k1
2.2
4
Abstract and Applied Analysis
where
⎧
−1 ⎨eM2π−ts eMt−s ,
Gt, s 2M e2πM − 1
⎩eM2πt−s eMs−t ,
⎧
−1 ⎨eM2π−ts − eMt−s ,
Ht, s 2 e2πM − 1
⎩eMs−t − eM2πt−s ,
0 ≤ s ≤ t ≤ 2π,
0 ≤ t ≤ s ≤ 2π,
2.3
0 ≤ s ≤ t ≤ 2π,
0 ≤ t < s ≤ 2π.
By simple calculations, we obtain that for t, s ∈ J × J,
l0 :
e2πM 1
eπM
2πM
≤ Gt, s ≤
: l1 ,
M e
−1
2M e2πM − 1
|Ht, s| ≤
1
,
2
MGt, s Ht, s > 0.
2.4
2.5
To establish the existence of multiple positive solutions in P C1 J, E ∩ C2 J , E of
IBVP1.2, let us list the following assumptions:
A1 ft, x ≤ gthx, t ∈ 0, 2π, x ∈ P , where g : 0, 2π → 0, ∞ is continuous
and
2π
h : P → P is bounded and continuous and satisfies 0 gsds < ∞.
A2 hx in A1 satisfies
2π
cl1
gsds l1
0
2π
dl1
0
m
hk k1
m
1
ck < 1,
2 k1
2.6
m
m
1
gsds l1 ek dk < 1,
2 k1
k1
where
hx
,
x → 0 x
c lim
ck lim
x → 0
hk lim
x → 0
Ik ,
x
I k x
,
d
hx
,
x → ∞ x
dk ek lim
lim
x → ∞
lim
x → ∞
Ik ,
x
I k x
2.7
.
Abstract and Applied Analysis
5
A3 For any r > 0 and a, b ⊂ 0, 2π, f is uniformly continuous on a, b × Tr .
A4 There exist L, Lk , Hk ≥ 0 such that αft, D ≤ LαD, αIk D ≤ Lk αD,
αI k D ≤ Hk αD k 1, . . . , m, and 4πLl1 l1 mk1 Hk 1/2 mk1 Lk < 1, for
t ∈ 0, 2π, and D ⊂ P is bounded.
A5 For any x ∈ P , I k x ≥ MIk x;
◦
A6 P is a solid cone, and there exist u0 ∈ P , J0 a , b ⊂ J such that t ∈ J 0 , x ≥ u0
b
imply ft, x ≥ htu0 , h ∈ CJ0 , 0, ∞, and l : l0 a hsds > 1.
Define an operator A as follows:
Aut 2π
0
Gt, sfs, usds m Gt, tk I k utk Ht, tk Ik utk ,
t ∈ J.
k1
2.8
Lemma 2.3. Assuming (A1) and (A4) hold, then, for any r > 0, A : P CJ, P ∩ Br → P CJ, P is
bounded and continuous.
Proof. According to A1 and A4, we obtain that A is a bounded operator. In the following,
we will show that A is continuous.
Let {un }, {u} ⊂ P CJ, P ∩Br , and un − uP C → 0. Next we show that Aun − AuP C →
0. By A1, {Aun t} is equicontinuous on each Ji i 0, . . . , m. By the Lebesgue
dominated convergence theorem and 2.4, we have
Aun t − Aut
m
2π
≤
Gt, s fs, un s − fs, us ds Gt, tk I k un tk − I k utk 0
k1
m
|Ht, tk |Ik un tk − Ik utk k1
≤ l1
2π
m fs, un s − fs, usds l1 I k un tk − I k utk 0
m
1
Ik un tk − Ik utk −→ 0
2 k1
k1
n −→ ∞.
2.9
In view of the Ascoli-Arzela theorem, {Aun } is a relatively compact set in P CJ, E. In the
following we will verify that Aun − AuP C → 0 n → ∞.
If this is not true, then there are ε0 > 0 and {uni } ⊂ {un } such that Auni − AuP C ≥
ε0 i 1, 2, . . .. Since {Aun } is a relatively compact set, there exists a subsequence of
{Auni } which converges to v ∈ P CJ, P , without loss of generality, and we assume that
limi → ∞ Auni v, that is, limi → ∞ Auni − vP C 0, so v Au, which imply a contradiction.
Therefore A is continuous.
6
Abstract and Applied Analysis
Lemma 2.4. Assuming (A1), (A3), and (A4) hold, then, for any R > 0, A : P CJ, P ∩ BR →
P CJ, P is a strict set contraction operator.
Proof. For any R > 0, S ⊂ P CJ, P ∩ BR , by A1, AS is bounded and equicontinuous on each
Ji , i 0, . . . , m, and by 17,
2.10
αP C AS sup αASt,
t∈J
where ASt {Aut : u ∈ S, t ∈ J}.
Let
2π
D
Gt, sfs, usds : u ∈ S ,
0
Dδ 2π−δ
2.11
Gt, sfs, usds : u ∈ S ,
0 < δ < min{π, t1 , 2π − tm }.
δ
By A1 and 2.4, for any u ∈ S,
2π
2π−δ
Gt, sfs, usds −
Gt, sfs, usds
δ
0
≤ l1 maxhu
x∈TR
δ
gsds l1 maxhx
2π
u∈TR
0
2.12
gsds.
2π−δ
In view of 2.12 and A1, we have dH Dδ , D → 0 δ → 0 , where dH Dδ , D denotes the
Hausdorff distance of D and Dδ .
Therefore,
lim αDδ αD.
2.13
δ → 0
Next we will estimate αDδ . Since
2π−δ
Gt, sfs, usds ∈ 2π − 2δco
Gt, sfs, us : s ∈ δ, 2π − δ ,
2.14
δ
thus
2π−δ
αDδ α
Gt, sfs, usds : u ∈ S
δ
≤ 2π − δα co Gt, sfs, us : s ∈ δ, 2π − δ, u ∈ S
≤ 2πα Gt, sfs, us : s ∈ δ, 2π − δ, u ∈ S
≤ 2πl1 α fIδ × SIδ ,
where Iδ δ, 2π − δ, SIδ {ut : t ∈ Iδ , u ∈ S}.
2.15
Abstract and Applied Analysis
7
By A3 and A4, it is not difficult to prove that
α fIδ × SIδ max α ft, SIδ ≤ LαSIδ ≤ LαSJ.
t∈Iδ
2.16
By 17, we have
LαSJ ≤ 2LαP C S.
2.17
Let δ → 0 , and making use of the fact that limδ → 0 αDδ αD, we obtain
αD ≤ 2πl1 2LαP C S 4πLl1 αP C S.
2.18
It is clear that
α
α
m
Gt, tk I k utk : u ∈ S
k1
m
≤
Ht, tk Ik utk : u ∈ S
k1
≤
Hk αP C S,
2.19
m
1
Lk αP C S.
2 k1
2.20
l1
m
k1
Hence, according to 2.18–2.20, we have
m
1
Lk αP C S.
4πLl1 l1 Hk 2 k1
k1
αP C AS ≤
m
2.21
By A4 and Lemma 2.3, A is a strict set contraction operator from P CJ, P into
P CJ, P .
3. Main Result
Theorem 3.1. Assuming that (A1)–(A6) hold, then the IBVP 1.2 has at least two positive solutions
u1 and u2 satisfying
u1 t ≥ lu0 t,
for t ∈ J0 a , b , l > 1,
3.1
where l was specified in (A6).
Proof. First we verify that there exists δ > 0 such that v ≥ δ for v ≥ u0 . If this is not true,
then there exists {vn } ⊂ Ewhich satisfies vn ≥ u0 and vn < 1/n n 1, 2, . . ., so we have
◦
u0 ≤ θ, which is a contradiction with u0 ∈ P .
By A2, there exist c > c, ck > ck , d > d, dk > dk , hk > hk , and ek > ek , and
0 < r1 < δ,
r2 > max{δ, lu0 }
3.2
8
Abstract and Applied Analysis
satisfy
c l1
2π
gsds l1
0
m
m
1
c < 1,
2 k1 k
hk k1
2π
b : d l1
gsds l1
0
m
ek k1
m
1
d < 1.
2 k1 k
3.3
3.4
For x ∈ Tr1 ∩ P ,
hx ≤ c x,
Ik x ≤ ck x,
I k x ≤ hk x.
3.5
Ik x ≤ dk x,
I k x ≤ ek x.
3.6
For x ≥ r2 and x ∈ P ,
hx ≤ d x,
Therefore, for any x ∈ P , we have
hx ≤ d x M ,
Ik x ≤ dk x M ,
I k x ≤ ek x M ,
3.7
where
M max{M0 , M1 , . . . , Mm , K1 , . . . , Km },
M0 sup{hx : x ∈ Tr2 ∩ P },
Mk sup{Ik x : x ∈ Tr2 ∩ P }, Kk sup I k x : x ∈ Tr2 ∩ P
k 1, 2, . . . , m.
3.8
2π
Let r3 r2 1 − b−1 G, G M l1 0 gsds ml1 m/2, U1 {u ∈ P CJ, P :
uP C < r1 }, U2 {u ∈ P CJ, P : uP C < r3 }, U3 {u ∈ P CJ, P : uP C < r3 , ut ≥ lu0 for
t ∈ J0 and l > 1}. It is clear that U1 , U2 , U3 are nonempty, bounded, and convex open sets in
P CJ, P , and U1 P CJ, P ∩ Br1 , U2 P CJ, P ∩ Br3 , and U3 {u ∈ U2 : ut ≥ lu0 , t ∈ J0 }.
From 3.2, we obtain
U1 ⊂ U2 ,
U3 ⊂ U2 ,
U1 ∩ U3 ∅.
3.9
According to Lemma 2.4, A : U2 → P CJ, P is a strict set contraction operator, and for
u ∈ U2 , by 2.4 and 3.7, we obtain
m 2π
Gt, tk I k utk Ht, tk Ik utk Gt, sfs, usds Aut 0
k1
≤ l1
2π
0
gsdshu m 1
l1 I k Ik 2
k1
Abstract and Applied Analysis
9
2π
m
m
1
gsds d u M l1
ek u M dk u M
2
0
k1
k1
2π
2π
m
m
1
m
gsds l1 ek d u M l1
gsds ml1 d l1
2 k1 k
2
0
0
k1
≤ l1
bu G
≤ br3 G < r3 .
3.10
Hence
A U2 ⊂ U2 .
3.11
Similarly, A : U1 → P CJ, P is a strict set contraction operator, and for u ∈ U1 , by
3.3 and 3.5, we obtain
Aut
m 2π
Gt, tk I k utk Ht, tk Ik utk Gt, sfs, usds 0
k1
≤ c l1
2π
gsdsu 0
c l1
m l1 hk u
k1
2π
0
gsds l1
m
hk k1
1
2
m
1 ck u
2
3.12
ck u
k1
< u ≤ r1 ,
so
A U1 ⊂ U1 .
Let u ∈ U3 , by 3.11, we have AuP C < r3 .
By 2.5, A5, and A6, for t ∈ J0 ,
Aut 2π
Gt, sfs, usds 0
≥
2π
k1
Gt, sfs, usds 0
≥
m
MGt, tk Ht, tk Ik utk k1
2π
Gt, sfs, usds
0
m Gt, tk I k utk Ht, tk Ik utk 3.13
10
Abstract and Applied Analysis
≥ l0
≥ l0
b
a
fs, usds
b
a
hsds u0
lu0 .
3.14
So Au ∈ U3 , and
A U3 ⊂ U3 .
3.15
According to 3.11–3.15 and Lemma 2.1, we have
i A, Uj , P CJ, P 1 j 1, 2, 3 .
3.16
Hence
i A, U2 \ U1 ∪ U3 , P CJ, P iA, U2 , P CJ, P − iA, U1 , P CJ, P − iA, U3 , P CJ, P 3.17
−1.
Thus, A has two fixed points u1 and u2 in U3 and U2 \ U1 ∪ U3 , respectively, which
means u1 t and u2 t are positive solution of the IBVP 1.2, where u1 t ≥ lu0 , for t ∈ J0 and
l > 1.
4. Example
To illustrate how our main result can be used in practice, we present an example.
Example 4.1. Consider the following problem:
x
1
999
n
2
t ln 1 xn1
,
−xn t 4xn √ 3 2 π
3
t
1
1
,
Δxn |t1/3 xn
8
3
1
1
Δxn t1/3 − xn
,
4
3
xn 0 xn 2π,
where xmn xn n 1, 2, . . . , m.
xn 0 xn 2π,
t ∈ J,
4.1
Abstract and Applied Analysis
11
Conclusion
IBVP 4.1 has at least two positive solutions {x1n t} and {x2n t} such that x1n t > 1 for
t ∈ π, 2π, n 1, 2, . . . , m.
Proof. Let J 0, 2π, E Rm {x x1 , x2 , . . . , xm : xn ∈ R, n 1, 2, . . . , m};
then, E is a Banach space with norm x max1≤n≤m |xn |. Let P {x1 , x2 , . . . , xm :
to IBVP 1.2, ft, x xn ≥ 0, n 1, 2, . . . , m}; then,
√ P is a solid cone in E. Compared
2
f1 , f2 , . . . , fm , fn t, x 1/ t32 999/πt ln1 xn1 xn /3 is singular at t 0.
Ix I1 x, I2 x, . . . , Im x, and In x 1/8xn 1/3. Ix I 1 x, . . . , I m x, and
I n x 1/4xn 1/3, n 1, 2, . . . , m.
Next we will√
verify that the conditions in Theorem 3.1 are satisfied.
2
xn /3.
Let gt 1/ t, hx h1 x, h2 x, . . . , hm x, and hn x 6000 ln1 xn1
It is clear that ft, x ≤ gthx, for t ∈ 0, 2π and x ∈ E, so A1 is satisfied.
By simple calculations, we have M 2, c d 1/3, c1 d1 1/8, h1 e1 1/4,
2π
2π
gsds
5.01326, l0 0.00093, and l1 0.25. Hence, l1 0 gsdsc l1 h1 1/2c1 < 1; that
0
is, A2 is satisfied.
Since E is a finite-dimensional space, it is obvious that A3 and A4 are satisfied.
It is clear that I n x 1/4xn 1/3 and MIn x 2 × 1/8xn 1/3 1/4xn 1/3,
so I n x MIn x; that is, A5 is satisfied.
◦
Let u0 1, 1, . . . , 1 ∈ P and J0 π, 2π ⊂ 0, 2π; for t ∈ J0 and x ≥ u0 , we have
1
fn t, x √
t
x 3000 ln 2
999
n
2
t ln 1 xn1 >
3 2
.
√
π
3
t
4.2
√
Let ht 3000 ln 2/ t; then, for t ∈ J0 and x ≥ u0 , we obtain that ft, x ≥ htu0 and
2π
l0 π hsds > 1. Therefore A6 is satisfied.
By Theorem 3.1, IBVP 4.1 has at least two positive solutions {x1n t} and {x2n t}
and satisfies x1n t > 1, n 1, 2, . . . , m.
Acknowledgments
This work is supported by the NNSF of China no. 10871160 and project of NWNU-KJCXGC3-47.
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