# Multiple positive solutions for first-order impulsive singular integro-differential equations on the half line in a Banach space

Yanlai Chen1* and Baoxia Qin2

Author Affiliations

1 School of Economics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

2 Department of Mathematics, Qilu Normal University, Jinan, Shandong, 250100, People’s Republic of China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:69  doi:10.1186/1687-2770-2013-69

 Received: 22 December 2012 Accepted: 18 March 2013 Published: 2 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 nth-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 if and only if . P is said to be normal if there exists a positive constant N such that implies , where θ denotes the zero element of E, and the smallest N is called the normal constant of P. If and , we write . Let , i.e., . For details on cone theory, see [7].

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:

(1)

where , ,  , , , , (), , , (for some m), and

(2)

, , , denotes the set of all nonnegative numbers. denotes the jump of at , i.e.,

where and represent the right and left limits of at , respectively. In the following, we always assume that

(3)

and

(4)

(where means , ), i.e., is singular at and . We also assume that

(5)

i.e., () are singular at .

Let = { is a map from J into E such that is continuous at , left continuous at , and exists, } and . It is clear that is a Banach space with norm

Let and . Obviously, and Q are two cones in space and . is called a positive solution of the infinite three-point boundary value problem (1) if for and satisfies (1). Let and for .

### 2 Several lemmas

Let us list some conditions.

() , and

In this case, let

() There exist and such that

and

where .

() There exist () and such that

and

() For any and , and () are relatively compact in E, where and .

Remark Obviously, condition () is satisfied automatically when E is finite dimensional.

Remark It is clear: If condition () is satisfied, then the operators T and S defined by (2) are bounded linear operators from into and , ; moreover, we have and .

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

(6)

In what follows, we write , ().

Lemma 1Let conePbe normal and conditions ()-() be satisfied. Then operatorAdefined by (6) is a continuous operator fromintoQ; moreover, for any, is relatively compact.

Proof Let and . Then and

so,

(7)

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

(8)

By condition () and (8), we have

(9)

where

which implies the convergence of the infinite integral

(10)

and

(11)

On the other hand, by condition () and (8), we have

(12)

where

which implies the convergence of the infinite series

(13)

and

(14)

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

which implies that and

(15)

Moreover, by (6), we have

(16)

and

(17)

It is clear,

(18)

so, (17) and (18) imply

(19)

It follows from (16) and (19) that

(20)

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

Now, we are going to show that A is continuous. Let , (). Write () and we may assume that

So, by (7),

(21)

and

(22)

Similar to (15), it is easy to get

(23)

It is clear that

(24)

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

(25)

where

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

(26)

On the other hand, for any , we can choose a positive integer j such that

(27)

where

And then, choose an positive integer such that

(28)

From (27), (28), and observing condition () and (21), (22), we get

hence,

(29)

It follows from (23), (26), and (29) that as , and the continuity of A is proved.

Finally, we prove that is relatively compact, where are arbitrarily given. Let (). Then, by (7),

(30)

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

(31)

(32)

and

(33)

where

and

Consider for any fixed i. By (6) and (31), we have

(34)

which implies that the functions () defined by

(35)

( denotes the right limit of at ) are equicontinuous on . On the other hand, for any , choose a sufficiently large and a sufficiently large positive integer such that

(36)

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

(37)

and

(38)

(39)

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

(40)

where , , , and denotes the Kuratowski measure of noncompactness of bounded set (see [[8], Section 1.2]). Since and for , where and , we see that, by condition (),

(41)

and

(42)

It follows from (40) to (42) that

which implies by virtue of the arbitrariness of ϵ that for .

By the Ascoli-Arzela theorem (see [[8], Theorem 1.2.5]), we conclude that is relatively compact in , hence, has a subsequence which is convergent uniformly on , so, has a subsequence which is convergent uniformly on . Since i may be any positive integer, so, by diagonal method, we can choose a subsequence of such that is convergent uniformly on each (). Let

It is clear that . By (33), we have

which implies that and

Let be arbitrarily given and choose a sufficiently large positive number τ such that

(43)

For any , we have, by (6),

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

(44)

Letting in (44), we get

(45)

On the other hand, since converges uniformly to on as , there exists a positive integer such that

(46)

It follows from (44) to (46) that

(47)

By (46) and (47), we have

hence, as , and the relative compactness of is proved. □

Lemma 2Let conePbe normal and conditions ()-() be satisfied. Thenis a positive solution of the infinite three-point boundary value problem (1) if and only ifis a solution of the following impulsive integral equation:

(48)

i.e., uis a fixed point of operatorAdefined by (6) in.

Proof For , it is easy to get the following formula:

(49)

Let be a positive solution of the infinite three-point boundary value problem (1). By (1) and (49), we have

(50)

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 in both sides of (50), we get

(51)

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

(52)

and

(53)

It follows from (51) to (53) that

and, substituting it into (50), we see that satisfies equation (48), i.e., .

Conversely, assume that is a solution of Equation (48). We have, by (48),

(54)

and

(55)

Moreover, by taking limits as in (48), we see that exists and

(56)

It follows from (54) to (56) that

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

and, it is clear, by (48),

Hence, and satisfies (1). Since , so (7) holds and , hence for . Consequently, is a positive solution of the infinite three-point boundary value problem (1). □

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

LetPbe a cone in real Banach spaceEand, be two bounded open sets inEsuch that, , and operatorbe completely continuous, whereθdenotes the zero element ofEanddenotes the closure of (). Suppose that one of the following two conditions is satisfied:

wheredenotes the boundary of ().

ThenAhas 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, and are balls with center θ.

### 3 Main theorems

Let us list more conditions.

() There exist , and such that

and

and

Remark 2 Condition () means that is superlinear with respect to u.

() There exist , and such that

and

and

Theorem 1Let conePbe normal and conditions ()-() be satisfied. Assume that there exists asuch that

(57)

whereNdenotes the normal constant ofP, and

(58)

(59)

(for, , and; see conditions () and ()). Then the infinite three-point boundary value problem (1) has at least two positive solutionssuch that.

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

By condition (), there exists a such that

(60)

so,

(61)

Choose

(62)

For , , we have by (7) and (62),

(63)

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

(64)

and consequently,

(65)

By condition (), there exists such that

(66)

so,

(67)

Choose

(68)

For , , we have by (68) and (7),

(69)

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

which implies

and consequently,

(70)

On the other hand, for , , by condition (), condition (), (58), and (59), we have

(71)

and

(72)

It is clear, by (17),

(73)

It follows from (71) to (73) that

(74)

Thus, (74) and (57) imply

(75)

From (62) and (68), we know , and by Lemma 1, is completely continuous, where , hence, (65), (70), (75), and Lemma 3 imply that A has two fixed points such that . The proof is complete. □

Theorem 2Let conePbe normal and conditions ()-() and () be satisfied. Assume that

(76)

uniformly for, and

(77)

(forand, 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 such that (70) holds (in this case, we put in (66) and (68)). On the other hand, by (76) and (77), there exists such that

(78)

and

(79)

where

(80)

Choose

(81)

For , , we have by (7) and (81),

so, (78) and (79) imply

(82)

and

(83)

It follows from (73), conditions (), condition (), (82), (83), and (80) that

and consequently,

(84)

Since by virtue of (81), we conclude from (70), (84), and Lemma 3 that A has a fixed point such that . The theorem is proved. □

Example 1

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

(85)

Conclusion Infinite system (85) has at least two positive solutions () and () such that

Proof Let with norm and . Then P is a normal cone in E with normal constant , and infinite system (85) can be regarded as an infinite three-point boundary value problem of form (1). In this situation, , , , (), , , , , , and , in which

(86)

and

(87)

It is easy to see that , () and condition () is satisfied and , . We have, by (86),

(88)

so, observing the inequality , we get

which implies that condition () is satisfied for

and

with

By (87), we have

(89)

so,

which implies that condition () is satisfied for and

On the other hand, (86) implies

and

(90)

so, we see that condition () is satisfied for (), and and condition () is satisfied for (), and . In addition, from (90), we have

which implies that (3) and (4) hold, i.e., is singular at and . Moreover, from (87), we get

and so,

which implies that (5) holds, i.e., () are singular at . Now, we check that condition () is satisfied. Let and be fixed, and be any sequence in , where . Then, we have, by (86) and (88),

(91)

So, is bounded, and, by diagonal method, we can choose a subsequence such that

(92)

which implies by virtue of (91) that

(93)

Consequently, . Let be given. Choose a positive integer such that

(94)

By (92), we see that there exists a positive integer such that

(95)

It follows from (91) to (95) that

hence, in E as . Thus, we have proved that is relatively compact in E. Similarly, by using (89), we can prove that is relatively compact in E. Hence, condition () is satisfied. Finally, we check that inequality (57) is satisfied for . In this case,

and

so,

i.e., inequality (57) is satisfied for . Hence, our conclusion follows from Theorem 1. □

Example 2

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

(96)

Conclusion Infinite system (96) has at least one positive solution () such that

Proof Let with norm and . Then P is a normal cone in E with normal constant , and infinite system (96) can be regarded as an infinite three-point boundary value problem of form (1) in E. In this situation, , , , (), , , , , , and , in which

(97)

and

(98)

It is clear that , () and condition () is satisfied and , . We have, by (97) and (98),

and

so, observing

we get

and

which imply that conditions () is satisfied for

and

with

and () is satisfied for () and

By (97), we have

(99)

so, condition () is satisfied for

and . Moreover, (99) implies

so, (3) and (4) are satisfied, i.e., is singular at and . Similarly, (98) implies

so, (5) is satisfied, i.e., () are singular at . Similar to the discussion in Example 1, we can prove that and (for fixed and ; ) are relatively compact in , so, condition () is satisfied. On the other hand, we have

so, (76) is satisfied. Moreover, it is clear that (77) is satisfied. Hence, our conclusion follows from Theorem 2. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors typed, read, and approved the final manuscript.

### Acknowledgements

The author would like to thank Professor D. Guo for his valuable suggestions.

### References

1. Xian, X, O’Regan, D, Agarwal, RP: Multiplicity results via topological degree for impulsive boundary value problems under a non-well ordered upper and lower solution condition. Bound. Value Probl.. 2008, (2008) Article ID 197205

2. Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math.. 69, 83–96 (2005). Publisher Full Text

3. Agarwal, RP, O’Regan, D: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput.. 161, 433–439 (2005). Publisher Full Text

4. Guo, D: Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal.. 68, 2727–2740 (2008). Publisher Full Text

5. Guo, D: Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal.. 70, 2078–2090 (2009). Publisher Full Text

6. Guo, D: Some fixed point theorems of expansion and compression type with applications. In: Lakshmikantham V (ed.) Nonlinear Analysis and Applications, pp. 213–221. Dekker, New York (1987)

7. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, Boston (1988)

8. Guo, D, Lakshmikantham, V, Liu, XZ: Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic, Dordrecht (1996)

9. Krasnoselskii, MA, Zabreiko, PP: Geometrical Methods of Nonlinear Analysis, Springer, New York (1984)