Existence and Uniqueness of Positive Solutions to Three-point Boundary Value Problems for Second Order Impulsive Differential Equations

Using a fixed point theorem of generalized concave operators, we present in this paper criteria which guarantee the existence and uniqueness of positive solutions to three-point boundary value problems for second order impulsive differential equations.


Introduction
In this paper, we study the existence and uniqueness of positive solutions to the following three-point boundary value problems for second order impulsive differential equations: x ′ (0) = 0, βx(η) = x(1), (1.1) where x ′ (t − k ) denote the right limit(left limit) of x ′ (t) at t = t k respectively.
Impulsive differential equations have been studied extensively in recent years.The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments.Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics.For an introduction of the basic theory of impulsive differential equations in R n , see EJQTDE, 2011 No. 70, p. 1 [3,17,26] and the references therein.The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see for instance [1,[4][5][6][7][8][9][10][12][13][14][15][18][19][20][21][23][24][25] and their references).Second-order impulsive differential equations have been studied by many authors with much of the attention given to positive solutions.For a small sample of such work, we refer the reader to works by Agarwal, O'Regan [1], Feng, Xie [9], Hu et al. [12], Jankowski [14,15], Lee [18], Lin, Jiang [19], Liu et al. [20], Wang et al. [27] and Zhang [30].The results of these papers are based on Schauder fixed point theorem, Leggett-Williams theorem, fixed point index theorems in cones, Krasnoselskii's fixed point theorem, the method of upper-lower solutions, fixed point theorems in cones and so on.However, few papers can be found in the literature on the existence of positive solutions to three-point boundary value problems for second-order impulsive differential equations.Three-point boundary value problems for differential equations or difference equations have been studied by many authors with much of the attention given to positive solutions.Here we mention only a few of them, see for example papers by Ahmad, Nieto [2], Gupta and Trofimchuk [11], Karaca [16], Ma [22] and Yang, Zhai and Yan [28].
To the best of our knowledge, no paper can be found in the literature on the existence and uniqueness of positive solutions to three-point boundary value problems for second-order impulsive differential equations.In this paper, we shall study the problem (1.1) and not suppose the existence of upper-lower solutions.Different from the above works mentioned, in this paper we will use a fixed point theorem of generalized concave operators to show the existence and uniqueness of positive solutions for the problem (1.1).

Preliminaries
In this section, we state some definitions, notations and known results.For convenience of readers, we suggest that one refer to [29]  x ≤ y implies x ≤ N y ; in this case N is called the normality constant of P .We say that an operator A : E → E is increasing(decreasing) if x ≤ y implies Ax ≤ Ay(Ax ≥ Ay).For all x, y ∈ E, the notation x ∼ y means that there exist λ > 0 and µ > 0 such that λx ≤ y ≤ µx.Clearly, ∼ is an equivalence relation.Given h > θ(i.e., h ≥ θ and h = θ), we denote by P h the set P h = {x ∈ E| x ∼ h}.Clearly, P h ⊂ P is convex and aP h = P h for all a > 0.
We now present a fixed point theorem of generalized concave operators which will be used in the latter proof.See [29] for further information.
Theorem 2.1(from the Lemma 2.1 and Theorem 2.1 in [29]).Let h > θ and P be a normal cone.Assume that: (D 1 ) A : P → P is increasing and Ah ∈ P h ; (D 2 ) For any x ∈ P and t ∈ (0, 1), there exists α(t) ∈ (t, 1) such that A(tx) ≥ α(t)Ax.Then (i) there are u 0 , v 0 ∈ P h and r ∈ (0, 1) such In what follows, for the sake of convenience, let is called a solution of the problem (1.1), if it satisfies the problem (1.1).
is the solution of the following integral equation: (2.1) It is easy to see by integration of (1.1) that EJQTDE, 2011 No. 70, p. 3 Integrate again, we can get Letting t = 1 and t = η in (2.2), we find From the boundary condition x(1) = βx(η), we have 2), we have Thus, the proof of sufficient is complete.
Conversely, if x is a solution of (2.1).Direct differentiation of (2.1) implies, for Further R] and it is easy to verify that x ′ (0) = 0, x(1) = βx(η), and the lemma is proved.2In this section, we apply Theorem 2.1 to study the problem (1.1) and we obtain a new result on the existence and uniqueness of positive solutions.The method used in this paper is new to the literature and so is the existence and uniqueness result to the second-order impulsive differential equations.This is also the main motivation for the study of (1.1) in the present work.

Define an operator
Set P = {x ∈ C[J, R]|x(t) ≥ 0, t ∈ J}, the standard cone.It is clear that P is a normal cone in C[J, R] and the normality constant is 1.Our main result is summarized in the following theorem.
Secondly, we prove Ah ∈ Ph .Set Then from (H 1 ), we have r 2 ≥ r 1 > 0. Further, from (H 1 ), (H 2 ), the above two cases (i),(ii) and (3.2), we have From (H 2 ), we have Further, from (H 1 ), (H 2 ), (H 4 ), the above two cases (i),(ii) and (3.1)-(3.3),we have Hence, EJQTDE, 2011 No. 70, p. 7 That is, Ah ∈ Ph .Finally, an application of Theorem 2.1 implies that (i) there are u 0 , v 0 ∈ Ph such that u 0 ≤ Au 0 , Av 0 ≤ v 0 ; (ii) operator equation x = Ax has a unique solution in Ph .That is, and the problem (1.1) has a unique solution x * in Ph .Moreover, from Lemmas 2.4 and 2.5 we know that x * ∈ P C 1 [J, R].Evidently, x * is a positive solution of the problem (1.1).2 Remark 3.2.For the case of I k = 0, k = 1, 2, . . ., m, the problem (1.1) reduces to the following three-point boundary value problem for ordinary differential equations: We can establish the existence and uniqueness of positive solutions for the problem (3.1) by the same method used in this paper, which is new to the literature.So the method employed in this paper is different from previous ones in literature and the result obtained in this paper is new.

4 3
is a solution of the problem (1.1) if and only if x ∈ P C 1 [J, R] is a fixed point of the operator A. EJQTDE, 2011 No. 70, p. Existence and uniqueness of positive solutions for problem (1.1)