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 ∑