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# Solutions for nth-order boundary value problems of impulsive singular nonlinear integro-differential equations in Banach spaces

Yanlai Chen1*, Tingqiu Cao1 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, 250013, People’s Republic of China

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Boundary Value Problems 2014, 2014:128  doi:10.1186/1687-2770-2014-128

 Received: 2 January 2014 Accepted: 6 May 2014 Published: 20 May 2014

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

Not considering the Green’s function, the present study starts to construct a cone formed by a nonlinear term in Banach spaces, and through the cone creates a convex closed set. We obtain the existence of solutions for the boundary values problems of nth-order impulsive singular nonlinear integro-differential equations in Banach spaces by applying the Mönch fixed point theorem. An example is given to illustrate the main results.

MSC: 45J05, 34G20, 47H10.

##### Keywords:
impulsive singular integro-differential equation; Banach spaces; boundary value problem; Mönch fixed point theorem; measure of noncompactness

### 1 Introduction and preliminaries

By using the Schauder fixed point theorem, Guo [1] obtained the existence of solutions of initial value problems for nth-order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with infinite number of impulsive times in a Banach space. In [2], by using the fixed point theorem in a cone, Chen and Qin investigated the existence of multiple solutions for a class of boundary value problems of singular nonlinear integro-differential equations of mixed type in Banach spaces. For singular differential equations in Banach spaces please see [3-9]. Generally based on Green’s function to construct a cone, but using the cone to study different nonlinear terms, we encountered difficulties, especially in infinite dimensional Banach spaces. In this paper, informed by the characteristics of the nonlinear term we construct a new cone, and through this cone create a convex closed set. On the new convex closed set, we apply the Mönch fixed point theorem to investigate the existence of solutions for the boundary value problems of nth-order impulsive singular nonlinear integro-differential equations in Banach spaces. Finally, an example of scalar second-order impulsive integro-differential equations for an infinite system is offered. Because of difficulties of compactness arising from impulsiveness and the use of nth-order integro-differential equations, a space is introduced. Let E be a real Banach space and . Let := { continuous at , left continuous at , and exists, }. Obviously is a Banach space with norm

Let := { exists and let it be continuous at , let and exist, }, where and represent the right and the left limits of at , respectively. For , we have

So observing the existence of and taking limits as in the above equality, we see that exists and

Similarly, we can show that exists. In the same way, we get the existence of . Define (, ). Then (), and, as is natural, in the following, is understood as . It is easy to see that is a Banach space with norm

Let 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 the smallest N is called the normal constant of P. For convenience, let . Let , in which and . For , we write . We consider the following singular boundary value problem (SBVP for short) for an nth-order impulsive nonlinear integro-differential equation in E:

(1)

where ,

(; ), and

(2)

with (), . denotes the jump of at , i.e.,

and θ denotes the zero element of E.

is singular at (), and/or if

, (), (), and

(), ().

Remark Obviously, , and is a normal cone of E if P is a normal cone of E. and P has the same normal constant N.

In the following, we assume that P is a normal cone. Let . A map is called a solution of SBVP (1) if it satisfies (1).

### 2 Several lemmas

To continue, let us formulate some lemmas.

Lemma 2.1Ifis bounded and the elements ofare equicontinuous on each (), then

in whichαdenotes the Kuratowski measure of noncompactness, ().

Proof For , it is easy to prove that

Since (), we know (). Hence

(3)

Next, we check that

In fact, for any , there is a division () such that

(4)

By hypothesis, the elements of are equicontinuous on each and there is a division:

such that

(5)

and

(6)

Let , (). By virtue of (5) and (6), we know that

(7)

Let . There is a division such that

(8)

Let F be the finite set of all maps into (). For , let . It is clear that . For any , , we have for some , and so

(9)

Consequently,

which implies . Since is arbitrary, we get

(10)

Finally, the conclusion follows from (3) and (10). For details of the Kuratowski measure of noncompactness, please see [10]. □

Lemma 2.2 (see [11])

Let us take a countable set (). For all, there issuch that, a.e, . Then, and

Lemma 2.3Supposeis bounded and equicontinuous on each (). Then, and

Proof By Theorem 1.2.2 of [10], the conclusion is obvious. □

Lemma 2.4Letbe two countable sets. Supposeand. Then

Proof The conclusion is obvious by Lemma 6 of [12]. □

Lemma 2.5 (see [13]) (The Mönch fixed point theorem)

LetEbe a Banach space. Assume thatis close and convex. Assume also thatis continuous with the further property that for some, we havecountable, implies thatCis relatively compact. ThenAhas a fixed-point inD.

### 3 Main theorem and example

For convenience, we list the following conditions:

(H1) There exist , (), () and () such that

where is nonincreasing, () and , are nondecreasing. And there exist , (, ) such that

().

(H2) There exists a ( denotes the dual cone of ) such that . And for any , there exists a such that

(H3) There exists a such that

where b, (), (), φ, (), (), (, ) and are defined as in conditions (H1) and (H2), and , .

(H4) There exist (), such that

(), . There exist (, ) such that

Remark Obviously, condition (H4) is satisfied automatically when E is finite dimensional.

Lemma 3.1Suppose conditions (H1), (H2) and (H3) are satisfied. ThenQdefined by

is a nonempty, convex and closed subset of.

Proof Let

For ,

(11)

It is clear that . Since , for , by (11), one can see that

which implies () for .

By conditions (H1), (H2), (H3), and (11), we have

and . Therefore, and Q is a nonempty set.

Now, we check that Q is a convex subset of . In fact, for any , , we write , which means . It is clear that

(12)

By virtue of the characters of elements of Q and the characters of φ, we have

(13)

In the same way,

(14)

and

Therefore, . Thus, Q is a convex subset of . It is clear that Q is a closed subset of . So the conclusion holds. □

Lemma 3.2Assume that conditions (H1), (H2) and (H3) are satisfied. Then, where the operatorAis defined by

(15)

Proof For any , i.e.,

(16)

and

For any and ,

(17)

and

(18)

Because of , and , , we know

(19)

Analogously, for , it is easy to see

(20)

Hence,

(21)

Differentiating (15) times, we get

(22)

Obviously, () exist and

(23)

where is understood as θ for . Similarly, () exist. Hence,

(24)

Let

(25)

Since

and (; ), it follows from (15), (16), (17) and (18) that

(26)

It is clear that

(27)

Since , by (22), (26) and condition (H2), we have

(28)

Now, we show that

(29)

By (15), (19), (21), conditions (H1) and (H3) imply

(30)

which implies that (29) is true. By (24), (26) to (29), the conclusion holds. □

Lemma 3.3Suppose conditions (H1) to (H4) are satisfied. Let

and

in which, . Then

(31)

and

(32)

in which ().

Proof In order to avoid the singularity, given , let

By conditions (H1), (H2) and (H3), for any , , we have

(33)

By virtue of absolute continuity of the Lebesgue integrable function, we have

(34)

in which, denotes the Hausdorff distance between and . Therefore,

(35)

Now, we show

In fact, by Lemma 2.2, we have

(36)

where , .

On the other hand, for , it follows from (19) and (21) that

(37)

and

(38)

Taking , , by (16), (17) and (18), one can see that

(39)

Therefore, by condition (H4) and (36), for , it is easy to get

(40)

Since B is a bounded set of and is a bounded set, is equicontinuous on each (). By Lemma 2.3, it is easy to get

(41)

Substituting (41) into (40), we get (31).

Similarly, we obtain (32) and our conclusion holds. □

Lemma 3.4Let conditions (H1) to (H3) be satisfied. is a solution of SBVP (1), if and only ifis a fixed point of the operatorAdefined by (15).

Proof First of all, by mathematical induction, for , Taylor’s formula with the integral remainder term holds,

(42)

In fact, as , for , let , it is easy to see that

that is,

(43)

This proves that (42) is true for .

Suppose (42) is true for , i.e., for , the next formula holds:

(44)

Now we check that (42) is also true for n. In fact, suppose . Then , by (43), one can see

(45)

Substituting the above equation into (44), we get

(46)

So, (42) is also true for n. By mathematical induction, (42) holds.

Suppose is a solution of SBVP (1). By (42), we can see that

(47)

Substituting

into (47), by (15), we get . So u is a fixed point of the operator A defined by (15) in Q.

Conversely, if is a fixed point of the operator A, i.e., u is a solution of the following impulsive integro-differential equation:

Then, by (15), similar to (26), from the derivative of both sides of the above equation one can draw the following conclusions:

(48)

So, we have

(49)

and

(50)

Hence

(51)

It follows from (48) and (50) that . By (48), (49) and (50), we have

(52)

and

(53)

It is easy to see by (48) and (50)

(54)

By (51) to (54), u is a solution of SBVP (1). □

Theorem 3.1Let conditions (H1) to (H4) be satisfied. Assume that

(55)

Then SBVP (1) has at least a solution.

Proof We will use Lemma 2.5 to prove our conclusion. By (H1)-(H3), from Lemma 3.2, we know .

We affirm that is continuous. In fact, let , , (as ). From the continuity of f and (, ) and the definition of A, by virtue of the Lebesgue dominated convergence theorem, we see that

(56)

For (fixed), we have (). We also see that is bounded and is equicontinuous on each . By Lemma 2.1, it is easy to get

(57)

i.e., is a relatively compact set in . The reduction to absurdity is used to prove that A is continuous. Suppose . Then , such that

(58)

On the other hand, since is a relatively compact set in , there exists a subsequence of which converges to . Without loss of generality, we may assume itself converges to y, that is,

(59)

By virtue of (56), we see that . Obviously, this is in contradiction to (58). Hence,

(60)

Consequently, is continuous.

By Lemma 2.4, for any countable , which satisfies , one can see

By virtue of the character of noncompactness, it is easy to get ,

(61)

For any fixed , , by condition (H4) and Lemma 3.3, we have

(62)

Similarly, by Lemma 3.3, for , we have

(63)

Let . It is clear that . By (61) and (62), for , it is easy to see that

(64)

Similarly, for ,

(65)

Since , by (64) and (65), we know that . It is easy to see that is bounded and the elements of are equicontinuous on each (). It follows from (61) and Lemma 2.1 that

(66)

Hence, B is a relatively compact set. By Lemma 2.5 (the Mönch fixed point theorem), A has at least a fixed point , and by Lemma 3.4, is the solution of SBVP (1) which means conclusion holds. □

An application of Theorem 3.1 is as follows.

Example Consider the following infinite system of scalar nonlinear second order impulsive integro-differential equations:

(67)

Conclusion. The infinite system (67) has at least a () solution, , , , .

Proof Let , with norm . We have the cone . Obviously P is a normal cone in E. Taking (), it is easy to see , and . The infinite system (67) can be regarded as a SBVP of the form (1) in E. In this situation,

in which

(68)

and , (), where

(69)

Obviously, for , we have

(70)

which implies

(71)

Since

and

as , we have

That is, . Obviously, . By (71), we can see

(72)

On the other hand, from (68) and (70), we have

(73)

It is easy to get

(74)

It follows from (72), (73) and (74) that

(75)

Hence, . Similarly, we have .

Taking

and

by (69) and (72), condition (H1) holds.

For any , define φ by . It is easy to see , and . For any , let

Therefore, condition (H2) is satisfied. It follows from (68) that

(76)

Now, we check that condition (H3) is true. In fact, it is easy to get

with and . Since

there exists a sufficient large such that

(77)

which implies that condition (H3) is satisfied.

Let

in which

where

(78)

For any

by (71) and (78), it is easy to get

(79)

Hence, the relative compactness of in follows directly from a known result (see [14]): a bounded set X of is relatively compact if and only if

That is,

(80)

For any

by (78), one can get

(81)

in which . Since and , by (81), it is easy to see

(82)

which implies

(83)

Similarly, by (78),

(84)

and

(85)

By (80), (82), (83) and (84), it is easy to get

(86)

In the same way,

(87)

Taking

the condition (H4) follows from (86) and (87). We can calculate and get

(88)

Therefore, by Theorem 3.1, the conclusion holds. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors typed, read and approved the final manuscript.

### Acknowledgements

The authors would like to thank the reviewers for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of P.R. China (71272119), Social Science Foundation of Shandong Province (13CJRJ07) and Teaching and Research Projects of Qi Lu Normal University (201306).

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