### Abstract

In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first-order impulsive superlinear singular integro-differential equations on the half line in a Banach space by means of the fixed-point theorem of cone expansion and compression with norm type.

**MSC: **
45J05, 34G20, 47H10.

##### Keywords:

impulsive singular integro-differential equation in a Banach space; infinite three-point boundary value problem; fixed-point theorem of cone expansion and compression with norm type### 1 Introduction

In recent years, multiple solutions of boundary value problems for impulsive differential
equations in scalar spaces had been extensively studied (see, for example, [1-3]). In recent papers [4] and [5], Professor D. Guo discussed two infinite boundary value problems for *n*th-order impulsive nonlinear singular integro-differential equations of mixed type
on the half line in a Banach space. By constructing a bounded closed convex set, apart
from the singularities, and using the Schauder fixed-point theorem, he obtained the
existence of positive solutions for the infinite boundary value problems. But such
equations are sublinear, and there are no results on existence of two positive solutions.
Now, in this paper, we shall discuss the existence of two positive solutions for first-order
superlinear singular equations by means of a different method, *i.e.*, by using the fixed-point theorem of cone expansion and compression with norm type
(see [6,7]), and the key point is to introduce a new cone *Q*.

Let *E* be a real Banach space and *P* be a cone in *E*, which defines a partial ordering in *E* by
*P* is said to be normal if there exists a positive constant *N* such that
*θ* denotes the zero element of *E*, and the smallest *N* is called the normal constant of *P*. If
*i.e.*,

Consider the infinite three-point boundary value problem for a first-order impulsive
nonlinear singular integro-differential equation of mixed type on the half line in
*E*:

where
*m*),

*i.e.*,

where

and

(where
*i.e.*,

*i.e.*,

Let
*J* into *E* such that

Let
*Q* are two cones in space

### 2 Several lemmas

Let us list some conditions.

(

In this case, let

(

and

where

(

and

(
*E*, where

**Remark** Obviously, condition (
*E* is finite dimensional.

**Remark** It is clear: If condition (
*T* and *S* defined by (2) are bounded linear operators from

We shall reduce the infinite three-point boundary value problem (1) to an impulsive
integral equation. To this end, we consider the operator *A* defined by

In what follows, we write

**Lemma 1***Let cone**P**be normal and conditions* (
*be satisfied*. *Then operator**A**defined by* (6) *is a continuous operator from*
*into**Q*; *moreover*, *for any*
*is relatively compact*.

*Proof* Let

so,

where *N* denotes the normal constant of cone *P*, and consequently,

By condition (

where

which implies the convergence of the infinite integral

and

On the other hand, by condition (

where

which implies the convergence of the infinite series

and

It follows from (6), (11), and (14) that

which implies that

Moreover, by (6), we have

and

It is clear,

so, (17) and (18) imply

It follows from (16) and (19) that

Hence,
*i.e.*, *A* maps
*Q*.

Now, we are going to show that *A* is continuous. Let

So, by (7),

and

Similar to (15), it is easy to get

It is clear that

and, similar to (9) and observing (21) and (22), we have

where

It follows from (24), (25), and the dominated convergence theorem that

On the other hand, for any
*j* such that

where

And then, choose an positive integer

From (27), (28), and observing condition (

hence,

It follows from (23), (26), and (29) that
*A* is proved.

Finally, we prove that

Similar to (9), (12), (15), and observing (30), we have

and

where

and

Consider
*i*. By (6) and (31), we have

which implies that the functions

(

We have, by (35), (6), (31), (32), and (36),

and

It follows from (37), (38), (39), and ([8], Theorem 1.2.3) that

where

and

It follows from (40) to (42) that

which implies by virtue of the arbitrariness of *ϵ* that

By the Ascoli-Arzela theorem (see [[8], Theorem 1.2.5]), we conclude that
*i* may be any positive integer, so, by diagonal method, we can choose a subsequence

It is clear that

which implies that

Let
*τ* such that

For any

which implies by virtue of (31), (32), and (43) that

Letting

On the other hand, since

It follows from (44) to (46) that

By (46) and (47), we have

hence,

**Lemma 2***Let cone**P**be normal and conditions* (
*be satisfied*. *Then*
*is a positive solution of the infinite three*-*point boundary value problem* (1) *if and only if*
*is a solution of the following impulsive integral equation*:

*i*.*e*., *u**is a fixed point of operator**A**defined by* (6) *in*

*Proof* For

Let

We have shown in the proof of Lemma 1 that the infinite integral (10) and the infinite
series (13) are convergent, so, by taking limits as

On the other hand, by (1) and (50), we have

and

It follows from (51) to (53) that

and, substituting it into (50), we see that
*i.e.*,

Conversely, assume that

and

Moreover, by taking limits as

It follows from (54) to (56) that

On the other hand, direct differentiation of (48) gives

and, it is clear, by (48),

Hence,

**Lemma 3** (The fixed-point theorem of cone expansion and compression with norm type; see [[6], Theorem 3] or [[7], Theorem 2.3.4])

*Let**P**be a cone in real Banach space**E**and*
*be two bounded open sets in**E**such that*
*and operator*
*be completely continuous*, *where**θ**denotes the zero element of**E**and*
*denotes the closure of*
*Suppose that one of the following two conditions is satisfied*:

*where*
*denotes the boundary of*

*Then**A**has at least one fixed point in*

**Remark 1** Lemma 3 is different from the Krasnoselskii fixed-point theorem of cone expansion
and compression (see [[9], Theorem 44.1]). In Krasnoselskii’s theorem, the condition corresponding to (a) is

It is clear, conditions (a) and (a′) are independent each other. On the other hand,
in Krasnoselskii’s theorem,
*θ*.

### 3 Main theorems

Let us list more conditions.

(

and

and

**Remark 2** Condition (
*u*.

(

and

and

**Theorem 1***Let cone**P**be normal and conditions* (
*be satisfied*. *Assume that there exists a*
*such that*

*where**N**denotes the normal constant of**P*, *and*

(*for*
*and*
*see conditions* (
*and* (
*Then the infinite three*-*point boundary value problem* (1) *has at least two positive solutions*
*such that*

*Proof* By Lemma 1 and Lemma 2, operator *A* defined by (6) is continuous from
*Q* and we need to prove that *A* has two fixed points

By condition (

so,

Choose

For

so, (6), (63), (61), and (7) imply

and consequently,

By condition (

so,

Choose

For

so, we get by (6), (69), and (67),

which implies

and consequently,

On the other hand, for

and

It is clear, by (17),

It follows from (71) to (73) that

Thus, (74) and (57) imply

From (62) and (68), we know
*A* has two fixed points

**Theorem 2***Let cone**P**be normal and conditions* (
*and* (
*be satisfied*. *Assume that*

*uniformly for*
*and*

(*for*
*and*
*see conditions* (
*and* (
*Then the infinite three*-*point boundary value problem* (1) *has at least one positive solution*

*Proof* As in the proof of Theorem 1, we can choose

and

where

Choose

For

so, (78) and (79) imply

and

It follows from (73), conditions (

and consequently,

Since
*A* has a fixed point

**Example 1**

Consider the infinite system of scalar first-order impulsive singular integro-differential equations of mixed type on the half line:

**Conclusion** Infinite system (85) has at least two positive solutions

*Proof* Let
*P* is a normal cone in *E* with normal constant

and

It is easy to see that

so, observing the inequality

which implies that condition (

and

with

By (87), we have

so,

which implies that condition (

On the other hand, (86) implies

and

so, we see that condition (

which implies that (3) and (4) hold, *i.e.*,