Multiple Positive Solutions for Singular Periodic Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

and Applied Analysis 3 where f ∈ C J × E × E × E, E , Ik, Ik ∈ C E, E , and the operators T , S are given by Tu t ∫ t 0 k t, s u s ds, Su t ∫2π 0 k1 t, s u s ds, 1.5 with k ∈ C D,R , D { t, s ∈ R2 : 0 ≤ s ≤ t ≤ 2π}, k1 ∈ C J × 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 IBVP 1.4 . Motivated by the above facts, our aim is to study the multiplicity of positive solutions for IBVP 1.2 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 IBVP 1.2 . 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 ∈ PC J, E : ‖u‖PC ≤ r} r > 0 ; for D ⊂ PC J, E , we denote D t {u t : u ∈ D} ⊂ E t ∈ J .α denotes the Kuratowski measure of noncompactness. Let PC1 J, E {u | u be a map from J into E such that u t is continuously differentiable at t / tk and left continuous at t tk and u t k , u ′ tk , u ′ t k exist, k 1, 2, . . . , m}. Evidently, PC1 J, E is a Banach space with norm ‖u‖PC1 max {‖u‖PC, ∥ ∥u′ ∥ ∥ PC } . 2.1 Let J ′ J \ {t1, t2, . . . , tm}; a map u ∈ PC1 J, E ∩ C2 J ′, E is a solution of IBVP 1.2 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 A Ω ⊂ K. Then the fixed-point index i A,Ω, K 1. Lemma 2.2 see 21 . u ∈ PC1 J, E ∩ C2 J ′, E is a solution of IBVP 1.2 if and only if u ∈ PC J, E is a solution of the impulsive integral equation: u t ∫2π 0 G t, s f s, u s ds m ∑ k 1 [ G t, tk Ik u tk H t, tk Ik u tk ] , 2.2 4 Abstract and Applied Analysis where G t, s ( 2M ( e2πM − 1 ))−1 ⎧ ⎨ ⎩ e 2π−t s e t−s , 0 ≤ s ≤ t ≤ 2π, e 2π t−s e s−t , 0 ≤ t ≤ s ≤ 2π, H t, s ( 2 ( e2πM − 1 ))−1 ⎧ ⎨ ⎩ e 2π−t s − e t−s , 0 ≤ s ≤ t ≤ 2π, e s−t − e 2π t−s , 0 ≤ t < s ≤ 2π. 2.3 By simple calculations, we obtain that for t, s ∈ J × J , l0 : e M ( e2πM − 1) ≤ G t, s ≤ e2πM 1 2M ( e2πM − 1) : l1, 2.4 |H t, s | ≤ 1 2 , MG t, s H t, s > 0. 2.5 To establish the existence of multiple positive solutions in PC1 J, E ∩ C2 J ′, E of IBVP 1.2 , let us list the following assumptions: A1 ‖f t, x ‖ ≤ g t ‖h x ‖, t ∈ 0, 2π , x ∈ P , where g : 0, 2π → 0,∞ is continuous and h : P → P is bounded and continuous and satisfies ∫2π 0 g s ds < ∞. A2 h x in A1 satisfies cl1 ∫2π 0 g s ds l1 m ∑


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 In this paper, we consider the following singular periodic boundary value problem with impulsive effects in Banach E where M > 0 is constant, f t, u may be singular at t 0 and/or t 2π, f ∈ C 0, 2π × P, P , . ., m, and u i t k resp., u i t − k denote the right limit resp., left limit of u i t at t t k , i 0, 1.In the special case where E R 0, ∞ , and I k 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: was proved to have at least one positive solution, by Jiang 19 .
In 20 , the authors studied the multiplicity of positive solutions for IBVP 1.2 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: . 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 IBVP 1.4 .
Motivated by the above facts, our aim is to study the multiplicity of positive solutions for IBVP 1.2 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 IBVP 1.2 .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.

Preliminaries
Let PC 1 J, E {u | u be a map from J into E such that u t is continuously differentiable at t / t k and left continuous at t t k and u t k , u t − k , u t k exist, k 1, 2, . . ., m}.Evidently, PC 1 J, E is a Banach space with norm 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
where

2.3
By simple calculations, we obtain that for t, s ∈ J × J, To establish the existence of multiple positive solutions in PC 1 J, E ∩ C 2 J , E of IBVP 1.2 , let us list the following assumptions: A1 f t, x ≤ g t h x , t ∈ 0, 2π , x ∈ P , where g : 0, 2π → 0, ∞ is continuous and h : P → P is bounded and continuous and satisfies where A6 P is a solid cone, and there exist u 0 ∈
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 {u n }, {u} ⊂ PC J, P ∩B r , and u n − u PC → 0. Next we show that Au n − Au PC → 0. By A1 , { Au n t } is equicontinuous on each J i i 0, . . ., m .By the Lebesgue dominated convergence theorem and 2.4 , we have

2.9
In view of the Ascoli-Arzela theorem, {Au n } is a relatively compact set in PC J, E .In the following we will verify that Au n − Au PC → 0 n → ∞ .If this is not true, then there are ε 0 > 0 and {u ni } ⊂ {u n } such that Au ni − Au PC ≥ ε 0 i 1, 2, . . . .Since {Au n } is a relatively compact set, there exists a subsequence of {Au ni } which converges to v ∈ PC J, P , without loss of generality, and we assume that lim i → ∞ Au ni v, that is, lim i → ∞ Au ni − v PC 0, so v Au, which imply a contradiction.Therefore A is continuous.
Proof.For any R > 0, S ⊂ PC J, P ∩ B R , by A1 , AS is bounded and equicontinuous on each J i , i 0, . . ., m, and by 17 , where

2.11
By A1 and 2.4 , for any u ∈ S,

2.12
In view of 2.12 and A1 , we have

2.13
Next we will estimate α D δ .Since

2.17
Let δ → 0 , and making use of the fact that lim δ → 0 α D δ α D , we obtain

2.18
It is clear that

2.21
By A4 and Lemma 2.3, A is a strict set contraction operator from PC J, P into PC J, P .

Main Result
Theorem 3.1.Assuming that (A1)-(A6) hold, then the IBVP 1.2 has at least two positive solutions u 1 and u 2 satisfying where l was specified in (A6).
Proof.First we verify that there exists δ > 0 such that v ≥ δ for v ≥ u 0 .If this is not true, then there exists {v n } ⊂ Ewhich satisfies v n ≥ u 0 and v n < 1/n n 1, 2, . . ., so we have and e k > e k , and For x ≥ r 2 and x ∈ P ,

3.6
Therefore, for any x ∈ P , we have where 0 g s ds ml 1 m/2 , U 1 {u ∈ PC J, P : u PC < r 1 }, U 2 {u ∈ PC J, P : u PC < r 3 }, U 3 {u ∈ PC J, P : u PC < r 3 , u t ≥ lu 0 for t ∈ J 0 and l > 1}.It is clear that U 1 , U 2 , U 3 are nonempty, bounded, and convex open sets in PC J, P , and U 1 PC J, P ∩ B r 1 , U 2 PC J, P ∩ B r 3 , and U 3 {u ∈ U 2 : u t ≥ lu 0 , t ∈ J 0 }.
From 3.2 , we obtain

3.11
Similarly, A : U 1 → PC J, P is a strict set contraction operator, and for u ∈ U 1 , by 3.3 and 3.5 , we obtain

so
A U 1 ⊂ U 1 .

3.17
Thus, A has two fixed points u 1 and u 2 in U 3 and U 2 \ U 1 ∪ U 3 , respectively, which means u 1 t and u 2 t are positive solution of the IBVP 1.2 , where u 1 t ≥ lu 0 , for t ∈ J 0 and l > 1.

Example
To illustrate how our main result can be used in practice, we present an example.

and l : l 0 b a h s ds > 1 .
Define an operator A as follows: d H D δ , D → 0 δ → 0 , where d H D δ , D denotes the Hausdorff distance of D and D δ .

Example 4 . 1 .
Consider the following problem: Abstract and Applied Analysis map from J into E such that u t is continuous at t / t k and left continuous at t t k and u t k exist, k 1, 2, . . ., m}, and it is also a Banach space with norm 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 PC J, E is partially ordered by t 1 , and J i t i , t i 1 , i 1, . . ., m.Note that PC J, E {u : u is a 2