# Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space

Ruijuan Liu12*, Chunhai Kou3 and Ran Jin1

Author Affiliations

1 College of Information Science and Technology, Donghua University, Shanghai, 201620, China

2 College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, China

3 Department of Applied mathematics, Donghua University, Shanghai, 201620, China

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Boundary Value Problems 2013, 2013:79  doi:10.1186/1687-2770-2013-79

 Received: 25 December 2012 Accepted: 18 March 2013 Published: 8 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, we obtain the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space by means of fixed point index theory of completely continuous operators.

MSC: 26A33, 34B15.

##### Keywords:
fractional order; integro-differential equation; measure of noncompactness; fixed point index; boundary value problem

### 1 Introduction

Fractional differential equations (FDEs) have been of great interest for the last three decades [1-11]. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity [12], electrochemistry [13], control, porous media [14], etc. Therefore, the theory of FDEs has been developed very quickly. Many qualitative theories of FDEs have been obtained. Many important results can be found in [15-19] and references cited therein.

In this paper, we shall use the fixed point index theory of completely continuous operators to investigate the multiple positive solutions of a boundary value problem for a class of α order nonlinear integro-differential equations in a Banach space.

Let E be a real Banach space, P be a cone in E and denote the interior points of P. A partial ordering in E is introduced 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 constant N is called the normal constant of P. P is called solid if is nonempty. If and , we write . If P is solid and , we write . For details on cone theory, see [1].

For the application in the sequel, we first state the following lemmas and definitions which can be found in [1,10,20].

Lemma 1.1LetPbe a cone in a real Banach spaceE, and let Ω be a nonempty bounded open convex subset ofP. Suppose thatis completely continuous and, wheredenotes the closure of Ω inP. Then the fixed point index

Lemma 1.2LetPbe a cone in a real Banach spaceE, and let, where () are nonempty bounded open convex subsets ofPand. Suppose thatis a strict set contraction and. Then

Lemma 1.3Ifis bounded and equicontinuous, thenis continuous onI, and set

where, .

Definition 1.1 The fractional integral of order of a function is given by

provided the right-hand side is pointwise defined on .

Definition 1.2 The fractional derivative of order of a function is given by

where , provided the right-hand side is pointwise defined on .

Lemma 1.4Let, then

for some, , .

In this article, let , . It is easy to see that is a Banach space with the norm

Consider the boundary value problem (BVP) for a fractional nonlinear integro-differential equation of mixed type in E:

(1)

where is the standard Riemann-Liouville fractional derivative of order , , (), , and

(2)

, , , denotes the set of all nonnegative real numbers.

### 2 Several lemmas

To establish the existence of multiple positive solutions in of (1), let us list the following assumptions.

() , , , as ().

() There exist and such that

() There exists such that

uniformly for , and

() There exists such that

uniformly for , and

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

() P is normal and solid, and there exist , and such that

and

where , .

() There exist , and such that

and

where , .

Remark 2.1 It is clear that () is satisfied automatically when E is finite dimensional.

Remark 2.2 It is clear that assumption () is weaker than assumption ().

We shall reduce BVP (1) to an integral equation in E. To this end, we first consider the operator A defined by

(3)

where .

In our main results, we make use of the following lemmas.

Lemma 2.1Let assumption () be satisfied, then the operatorsTandSdefined by (2) are bounded linear operators frominto, and

(4)

Moreover,

(5)

Proof Inequalities (4) follow from two simple inequalities:

and (5) is obvious. □

Lemma 2.2Let assumptions (), () and () be satisfied, then the operatorAdefined by (3) is a continuous operator frominto.

Proof

Let

where λ is defined in the operator A.

By virtue of assumptions () and (), there exists an such that

(6)

and

(7)

where

It follows from (6) and (7) that for , , we have

(8)

Let , we have, by (8) and Lemma 2.1,

(9)

which implies the convergence of the infinite integral

and

(10)

Thus, we have, by (3), (9) and (10),

(11)

It follows from (11) that

(12)

Thus, we have .

Finally, we show that A is continuous. Let , (). Then and . By (3), we have

(13)

(14)

It is clear that

(15)

and by (9),

(16)

It follows from (15) and (16) and the dominated convergence theorem that

(17)

and

(18)

It follows from (14), (17) and (18) that (), and the continuity of A is proved. □

Lemma 2.3Let assumptions (), () and () be satisfied, thenis a solution of BVP (1) if and only ifis a solution of the following integral equation:

(19)

i.e., uis a fixed point of the operatorAdefined by (3) in.

Proof If is a solution of BVP (1), then by applying Lemma 1.4 we reduce to an equivalent integral equation

(20)

for some , , . (20) can be rewritten

(21)

By , we have

(22)

By , we obtain

(23)

Now, substituting (22) and (23) into (21), we see that satisfies integral equation (19).

Conversely, if u is a solution of (19), the direct differentiation of (19) gives

(24)

and

(25)

Consequently, , and by (19), (24) and (25), it is easy to see that satisfies BVP (1). □

Lemma 2.4Integral equation (19) can be expressed as

(26)

andfor any, where

(27)

Proof Let . For , one has

,

By simple calculation, we can prove the rest of the lemma. □

Lemma 2.5Let assumptions (), () and () be satisfied, and letUbe a bounded subset of. Thenis equicontinuous on any finite subinterval ofJ, and for any given, there existssuch that

uniformly with respect to, as.

Proof For , , by using (3), we have

(28)

This, together with (9) and (10), implies that are equicontinuous on any finite subinterval of J.

Now, we are going to prove that for any given , there exists sufficiently large , which satisfies

for all and .

Together with (28), we need only to show that for any given , there exists sufficiently large such that

It follows from (10) that for any given , there exists a sufficiently large such that

(29)

and there exists such that

(30)

On the other hand, let , , , then we have

Thus, there exists such that for ,

(31)

Therefore, from (29), (30) and (31) we have

Consequently, the proof is complete. □

Lemma 2.6Let assumptions (), () and () be satisfied, and letUbe a bounded subset of. Then

Proof By Lemma 2.2, we know AU is a bounded subset of . Thus,

First, we claim that .

In fact, by Lemma 2.5, we know that for any given , there exists a such that

(32)

uniformly with respect to and .

Since is equicontinuous on , by Lemma 1.3, we know

where

that is, is the restriction of AU on . Therefore, there exists such that

satisfying

(33)

where denote the diameters of bounded subsets of .

At the same time, for any , by (32) and (33), we obtain

(34)

It follows from (33) and (34) that

Then, by using , we have

On the other hand, for any given , there exist , , such that

Hence, for , , , we have

(35)

Since together with (35), we get

that is,

Because ε is arbitrary, we obtain

Consequently, the proof is complete. □

### 3 Main results

In this section, we give and prove our main results.

Theorem 3.1Let ()-() be satisfied. Then BVP (1) has at least two positive solutionssuch thatfor.

Proof By Lemma 2.2 and Lemma 2.4, the operator A defined by (3) is continuous from into , and by Lemma 2.3, we need only to show that A has two positive fixed points such that for .

First, we shall prove A is compact.

Let be bounded and (). From (9), we can choose a sufficiently large such that for all

(36)

It follows from Lemma 2.5 that

(37)

is equicontinuous on . Thus, by (3), (36) and (37), we have

(38)

where , , , .

Since for , where , we see that, by virtue of assumption (),

(39)

It follows from (38) and (39) that

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

Using Lemma 2.6, we have

Thus, we can conclude that AU is relatively compact in , i.e., A is compact.

As in the proof of Lemma 2.2, (12) holds. Choose

(40)

where is given in assumption (), and let . Then and, by (12) and (40), we have

(41)

By virtue of (), there exists an such that

(42)

where

(43)

Let

Then, for with , we have by (42)

(44)

It follows from (3), (43) and (44) that

which implies

(45)

Choose

(46)

Let . Then , and we have, by (45) and (46),

(47)

Let , and we are going to show that is an open set of . It is clear that we need only to show the following: for any , there exists such that , implies that for . We have for . So, for any , there exists a such that

(48)

Since and is continuous on J, we can find an open interval () such that

which implies by virtue of (48) that

Since I is compact, there is a finite collection of such intervals () which cover I, and

where (). Consequently,

(49)

where . Since , there exists an such that

(50)

whenever satisfying , which implies by virtue of (49) and (50) that

Thus, we have proved that is open in .

On the other hand, Lemma 2.4 and assumption () imply

(51)

Hence

(52)

Since , and are nonempty bounded convex open subsets of , we see that (41), (47) and (52) imply by virtue of Lemma 1.1 the fixed point indices

(53)

On the other hand, for , we have , and so

Consequently,

(54)

By (53), (54) and the additivity of the fixed point index (Lemma 1.2), we can obtain

(55)

Finally, (53), (54) and (55) imply that A has two fixed points and . We have, by (51), for . The proof is complete. □

Remark 3.1 Assumption () and the continuity of f imply that for . Hence, under the assumptions of the theorem, BVP (1) has the trivial solution besides two positive solutions and .

Theorem 3.2Let ()-() and () be satisfied. Then BVP (1) has at least one positive solutionsuch thatfor.

Proof By Lemma 2.2, Lemma 2.4 and the proof of Theorem 3.1, the operator A defined by (3) is completely continuous from into , and by Lemma 2.3, we need only to show that A has one positive fixed point such that for .

As in the proof of Lemma 2.2, (12) holds. Choose R satisfying (40) and let , where is given by assumption (). It is clear that U is a nonempty bounded closed convex subset in ( because ). Let , by (40), we have . On the other hand, as in the proof of Theorem 3.1, Lemma 2.4 and assumption () imply

(56)

Hence, , and therefore . Thus, the Schauder fixed point theorem implies that A has a fixed point , and by (56) for . The proof is complete. □

### 4 Conclusion

In this paper, the issue on the existence of multiple positive solutions of a boundary value problem for α-order nonlinear integro-differential equations in a Banach space has been addressed for the first time. Taking advantage of the fixed point index theory of completely continuous operators, the existence conditions for such boundary value problems have been established.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

RL completed the proof and wrote the initial draft. CK provided the problem and gave some suggestions for amendment. RL then finalized the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

This work was supported by the Natural Science Foundation of China under grant No. 11271248 and the Science and Technology Research Program of Zhejiang Province under grant No. 2011C21036.

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