Existence of positive solutions for singular impulsive differential equations with integral boundary conditions on an infinite interval in Banach spaces ∗

Existence of positive solutions for singular impulsive differential equations with integral boundary conditions on an infinite interval in Banach spaces∗ Xu Chen, Xingqiu Zhang1,2† 1 School of Mathematics, Liaocheng University, Liaocheng, 252059, Shandong, China 2 School of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, China Email: woshchxu@163.com, zhxq197508@163.com


Introduction
The theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics.For example, heat conduction, chemical engineering,underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions.In recent years, the theory of ordinary differential equations in Banach space has become a new important branch of investigation (see, for example, [1][2][3][4] and references therein).In a recent paper [7], using the cone theory and monotone iterative technique, Zhang et al investigated the existence of minimal nonnegative solution of the following nonlocal boundary value problems for second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times Very recently, by using Schauder fixed point theorem, Guo [6] obtained the existence of positive solutions for a class of nth-order nonlinear impulsive singular integro-differential equations in a Banach space.Motivated by Guo's work, in this paper, we shall use the cone theory and the Mönch fixed point theorem to investigate the positive solutions for a class of second-order nonlinear impulsive integro-differential equations in a Banach space.
The main features of the present paper are as follows: Firstly, compared with [7], the secondorder boundary value problem we discussed here is in Banach spaces and nonlinear term permits singularity not only at t = 0 but also at x, x ′ = θ.Secondly, compared with [6], the relative compact conditions we used are weaker.

Preliminaries and several lemmas
Obviously, F P C[J, E] is a Banach space with norm and DP C 1 [J, E] is also a Banach space with norm where The basic space using in this paper is DP C 1 [J, E].
Let P be a normal cone in E with normal constant N which defines a partial ordering in E by x ≤ y.If x ≤ y and x = y, we write x < y.Let P + = P \{θ}.So, x ∈ P + if and only if x > θ.For details on cone theory, see [4].
Let α, α F , α D denote the Kuratowski measure of non-compactness in E, F P C[J, E], DP C 1 [J, E].For details on the definition and properties of the measure of non-compactness, the reader is referred to references [1][2][3][4]. Denote Let us list the following assumptions, which will stand throughout this paper.
(H 1 ) f ∈ C[J + ×P 0λ ×P 0λ , P ] for any λ > 0 and there exist a, b, c ∈ L[J + , J] and z ∈ C[J + ×J + , J] such that and f (t, x, y) c(t)( x + y ) → 0, as x ∈ P 0λ * , y ∈ P 0λ * , x + y → ∞, uniformly for t ∈ J + , and (H 2 ) I ik ∈ C[P 0λ × P 0λ , P ] for any λ > 0 and there exist and EJQTDE, 2011 No. 28, p. 3 (H 3 ) For any t ∈ J + , R > 0 and countable bounded set V i ⊂ DP C 1 [J, P 0λ * R ] (i = 0, 1), there exist h i (t) ∈ L[J, J] (i = 0, 1) and positive constants m ikj (i, where In what follows, we write We shall reduce BVP (1) to an impulsive integral equations in E. To this end, we first consider operator A defined by where and By (H 1 ), there exists a R > r such that Hence On the other hand, let We see that, by condition (H 2 ), there exists a R 1 > r such that and Hence Let x ∈ Q, by (5), we can get which together with condition (H 1 ) implies the convergence of the infinite integral On the other hand, by (7), we have EJQTDE, 2011 No. 28, p. 5 which together with (2), (H 1 ) and (H 2 ) implies that Therefore, Differentiating (2), we get Hence, It follows from ( 12) and ( 14) that Hence, Ax ∈ Q.Thus, we have proved that A maps Q into Q and (15) holds.Finally, we show that Then {x m } is a bounded subset of Q.Thus, there exists r > 0 such that x m D < r for m ≥ 1 and x D ≤ r.Similar to (12) and ( 14), it is easy to get It is clear that, By ( 8), we get It follows from (17), (18) and the dominated convergence theorem that It is clear that, It follows from ( 16), ( 19) and ( 21) that Ax m − Ax D → 0 as m → ∞.Therefore, the continuity of A is proved.
] is a solution of BVP (1) if and only if x ∈ Q is a solution of the following impulsive integral equation: EJQTDE, 2011 No. 28, p. 7 ] is a solution of BVP (1).For t ∈ J, integrating (1) from 0 to t, we have Taking limit for t → ∞, we get Thus, x x Integrating (27) from 0 to t, we obtain which together with the boundary value condition implies that EJQTDE, 2011 No. 28, p. 8 Thus, Substituting (30) into (28), we have Obviously, integral Conversely, if x a solution of integral equation, then direct differentiation gives the proof.
Lemma 3 Let (H 1 ) be satisfied, V ⊂ Q be a bounded set.Then (AV )(t) 1+t and (A ′ V )(t) are equicontinuous on any finite subinterval J k of J and for any ε > 0, there exists N > 0 such that uniformly with respect to x ∈ V as t ′ , t ′′ ≥ N.
Proof.For x ∈ V, t ′′ > t ′ , t ′′ , t ′ ∈ J k , we have EJQTDE, 2011 No. 28, p. 9 which implies that { AV (t) 1+t : x ∈ V } is equicontinuous on any finite subinterval J k of J. Since V ⊂ Q is bounded, there exists r > 0 such that for any x ∈ V, x D ≤ r.By (13), t ′′ , t ′ ∈ J k , we get In the following, we are in position to show that for any ε > 0, there exists N > 0 such that uniformly with respect to x ∈ V as t ′ , t ′′ ≥ N.
Combining with (33), we need only to show that for any ε > 0, there exists sufficiently large N > 0 such that The rest part of the proof is very similar to Lemma 2.3 in [5], we omit the details.
Proof.The proof is similar to that of Lemma 2.4 in [5], we omit it.

Main results
Theorem 1 Assume conditions (H 1 ), (H 2 ) and Proof.By Lemma 1, operator A defined by ( 2) is a continuous operator from Q into Q, and by Lemma 2, we need only to show that A has a fixed point and let and By Lemma 4, we have where (AV By (9), we know that the infinite integral By ( 38), ( 40) and (41) that On the other hand, α D (V ) ≤ α D {co({u} ∪ (AV ))} = α D (AV ).Then, (42) implies α D (V ) = 0, i.e., V is relatively compact in DP C 1 [J, E].Hence, the Mönch fixed point theorem guarantees that A has a fixed point x in Q 1 .Thus, Theorem 1 is proved.