## Boundary Value Problems

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

Boundary Value Problems20132013:79

DOI: 10.1186/1687-2770-2013-79

Accepted: 18 March 2013

Published: 8 April 2013

## 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 [111]. 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 [1519] 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 1.1 Let P be a cone in a real Banach space E, and let Ω be a nonempty bounded open convex subset of P. Suppose that is completely continuous and , where denotes the closure of Ω in P. Then the fixed point index
Lemma 1.2 Let P be a cone in a real Banach space E, and let , where () are nonempty bounded open convex subsets of P and . Suppose that is a strict set contraction and . Then
Lemma 1.3 If is bounded and equicontinuous, then is continuous on I, 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 1.4 Let , 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 2.1 Let assumption () be satisfied, then the operators T and S defined by (2) are bounded linear operators from into , and
(4)
Moreover,
(5)
Proof Inequalities (4) follow from two simple inequalities:

and (5) is obvious. □

Lemma 2.2 Let assumptions (), () and () be satisfied, then the operator A defined by (3) is a continuous operator from into .

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 2.3 Let assumptions (), () and () be satisfied, then is a solution of BVP (1) if and only if is a solution of the following integral equation:
(19)

i.e., u is a fixed point of the operator A defined 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 2.4 Integral equation (19) can be expressed as
(26)
and for any , where
(27)
Proof Let . For , one has
,

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

Lemma 2.5 Let assumptions (), () and () be satisfied, and let U be a bounded subset of . Then is equicontinuous on any finite subinterval of J, and for any given , there exists such 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 2.6 Let assumptions (), () and () be satisfied, and let U be 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 3.1 Let ()-() be satisfied. Then BVP (1) has at least two positive solutions such that for .

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 3.2 Let ()-() and () be satisfied. Then BVP (1) has at least one positive solution such that for .

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.

## Declarations

### 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.

## Authors’ Affiliations

(1)
College of Information Science and Technology, Donghua University
(2)
College of Fundamental Studies, Shanghai University of Engineering Science
(3)
Department of Applied mathematics, Donghua University

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