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
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 , electrochemistry , control, porous media , 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 .
Lemma 1.1LetPbe a cone in a real Banach spaceE, and let Ω be a nonempty bounded open convex subset ofP. Suppose that is completely continuous and , where denotes 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 that is a strict set contraction and . Then
Lemma 1.3If is bounded and equicontinuous, then is 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:
where is the standard Riemann-Liouville fractional derivative of order , , ( ), , and
, , , 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
where , .
( ) There exist , and such that
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
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 from into , and
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 from into .
where λ is defined in the operator A.
By virtue of assumptions ( ) and ( ), there exists an such that
It follows from (6) and (7) that for , , we have
Let , we have, by (8) and Lemma 2.1,
which implies the convergence of the infinite integral
Thus, we have, by (3), (9) and (10),
It follows from (11) that
Thus, we have .
Finally, we show that A is continuous. Let , ( ). Then and . By (3), we have
It is clear that
and by (9),
It follows from (15) and (16) and the dominated convergence theorem that
It follows from (14), (17) and (18) that ( ), and the continuity of A is proved. □
Lemma 2.3Let assumptions ( ), ( ) and ( ) be satisfied, then is a solution of BVP (1) if and only if is a solution of the following integral equation:
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
for some , , . (20) can be rewritten
By , we have
By , we obtain
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
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
and for any , where
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 . Then is equicontinuous on any finite subinterval ofJ, and for any given , there exists such that
uniformly with respect to , as .
Proof For , , by using (3), we have
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
and there exists such that
On the other hand, let , , , then we have
Thus, there exists such that for ,
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
uniformly with respect to and .
Since is equicontinuous on , by Lemma 1.3, we know
that is, is the restriction of AU on . Therefore, there exists such that
where denote the diameters of bounded subsets of .
At the same time, for any , by (32) and (33), we obtain
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
Since together with (35), we get
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 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
It follows from Lemma 2.5 that
is equicontinuous on . Thus, by (3), (36) and (37), we have
where , , , .
Since for , where , we see that, by virtue of assumption ( ),
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
where is given in assumption ( ), and let . Then and, by (12) and (40), we have
By virtue of ( ), there exists an such that
Then, for with , we have by (42)
It follows from (3), (43) and (44) that
Let . Then , and we have, by (45) and (46),
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
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,
where . Since , there exists an such that
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
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
On the other hand, for , we have , and so
By (53), (54) and the additivity of the fixed point index (Lemma 1.2), we can obtain
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 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
Hence, , and therefore . Thus, the Schauder fixed point theorem implies that A has a fixed point , and by (56) for . The proof is complete. □
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.
The authors declare that they have no competing interests.
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.
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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