Triple positive solutions of nth order impulsive integro-differential equations ∗

In this paper, we prove the existence of at least three positive solutions of boundary value problems for nth order nonlinear impulsive integrodifferential equations of mixed type on infinite interval with infinite number of impulsive times. Our results are obtained by applying a new fixed point theorem introduced by Avery and Peterson.


Introduction
The branch of modern applied analysis known as "impulsive" differential equations furnishes a natural framework to mathematically describe some "jumping processes".Consequently, the area of impulsive differential equations has been developing at a rapid rate(see [2][3][4][5]).Most of the works in this area discussed the first-and second-order problems (see e. g. [2,3,[6][7][8][9][10][11][12]), though the theory of nth order nonlinear impulsive integro-differential equations of mixed type has received attention and some significant results have been obtained in very recent years (see [4,5,13,14]).For instance, Guo [5] has established the existence of solutions for a class of nth order problems on infinite interval with infinite number of impulsive times in Banach spaces by means of the Schauder fixed point theorem.By using the fixed point index theory of completely continuous operators, in [4] Guo has investigated the existence of twin positive solutions of a boundary value problem (BVP) for nth-order nonlinear impulsive integro-differential equation of mixed type as follows: where K ∈ C[D, R + ] with D = {(t, s) ∈ J × J : t ≥ s}, H ∈ C[J × J, R + ], R + denotes the set of all nonnegative numbers.∆u (i) | t=t k denotes the jump of u (i) (t) at t = t k , i. e.
Here, (E, | • |) is a real Banach spaces, the nonempty convex closed set P ⊂ E is a cone, that is, au ∈ P for all u ∈ P and all a ≥ 0, and u, −u ∈ P implies u = 0.
But to our best knowledge, there are no results on triple positive solutions for such impulsive equations.The purpose for us to present this paper is to obtain sufficient conditions for the existence of at least three positive solutions for (1).This is also an application of a new fixed point theorem introduced by Avery and Peterson [1] which has been used to verify the existence of three positive solutions for ordinary differential equations in [15] and for p-Laplacian dynamic equations on time scales in [16].
For the Banach space E, by a cone P ⊂ E we introduce a partial ordering in E, that is, and, in (1) and in what follows, An operator is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.
For a given cone P in a real Banach space E, the map χ : P → [0, ∞) is called a nonnegative continuous concave function on P provided that χ is continuous and For x, y ∈ P and 0 ≤ t ≤ 1. Dual to this, we call the map ϕ : P → [0, ∞) a nonnegative continuous convex function on P provided that ϕ is continuous and For x, y ∈ P and 0 ≤ t ≤ 1.
Let θ and γ be nonnegative continuous convex functions on P , α a nonnegative continuous concave function on P and ψ a nonnegative continuous function on P .Let a, b, c and d be positive real numbers.We define the following convex sets.
The following Lemma 1 is due to Avery and Peterson [1] which play an important role in this paper.
Lemma 1.Let P be a cone in E and θ, γ, α, ψ be defined as above, moreover, ψ satisfy ψ(λx) ≤ λψ(x) for 0 ≤ λ ≤ 1 such that, for some positive numbers h and d, for all x ∈ P (γ, d).Suppose that A : P (γ, d) → P (γ, d) is a completely continuous operator and there exist positive real numbers a, b and c with a < b such that the following conditions are satisfied: Then A has at least three fixed points Let E = R.For the sake of convenience, we list the following hypotheses.
Assume there exist the function for some given positive number λ 0 and any t ∈ [0, t 1 ].Moreover, assume that there exist positive constants We assume ulteriorly there exist constants a, b, d, l, k 1 , k 2 and m satisfying and any t ∈ J, we have EJQTDE, 2011 No. 57, p. 4 (H4) There exists q 0 ∈ (l, ∞) such that, for all t ∈ J, u , f and I ik satisfy, respectively, and .

Main Results
Throughout this section we will work in the Banach space DP C n−1 [J, R] and our considerations are placed in the Banach space DP C n−1 [J, R] considered previously.Let us denote For any x, y ∈ DP C n−1 [J, R], define x ≤ y if and only if x(t) ≤ y(t) for each t ∈ J, x < y if and only if x ≤ y and there exists some t ∈ J such that x(t) = y(t).
Let h = L −1 .For x ∈ P and the positive real number l given in (5), define Remark 2. Distinctly, γ and θ are nonnegative continuous convex functions, α is the nonnegative continuous concave function and ψ is nonnegative continuous function on the cone P .Furthermore, from the fact that x (i) ≥ 0, we see that x (i) is increasing on is satisfied.We also have that ψ(λx) = λψ(x) for λ ∈ [0, 1] and x ∈ P .
EJQTDE, 2011 No. 57, p. 5 Let us define that a function u 1) if it is a nonnegative solution and u(t) ≡ 0.
Theorem 1.If the conditions (H1)-(H4) hold, then BVP(1) has at least three positive solutions x 1 , x 2 and x 3 satisfying Proof.Define a operator A as follows: [4, Lemma 3] In what follows, we write We are now in a position to prove that the operator A has three fixed points by means of Lemma 1.To verify that all conditions of Lemma 1 hold, we shall divide this proof into three steps.
Similar to the proof of [4, Lemma 2] , we can get that A is continuous.As a consequence of Arzela-Ascoli theorem we get that A is a completely continuous operator.