- Open Access
Multiple positive solutions of boundary value problems for fractional order integro-differential equations in a Banach space
© Liu et al.; licensee Springer. 2013
- Received: 25 December 2012
- Accepted: 18 March 2013
- Published: 8 April 2013
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.
- 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 .
where , .
provided the right-hand side is pointwise defined on .
where , provided the right-hand side is pointwise defined on .
for some , , .
, , , denotes the set of all nonnegative real numbers.
To establish the existence of multiple positive solutions in of (1), let us list the following assumptions.
() , , , as ().
() For any and , is relatively compact in E, where .
where , .
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 ().
In our main results, we make use of the following lemmas.
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 .
where λ is defined in the operator A.
Thus, we have .
It follows from (14), (17) and (18) that (), and the continuity of A is proved. □
i.e., u is a fixed point of the operator A defined by (3) in .
Now, substituting (22) and (23) into (21), we see that satisfies integral equation (19).
Consequently, , and by (19), (24) and (25), it is easy to see that satisfies BVP (1). □
By simple calculation, we can prove the rest of the lemma. □
uniformly with respect to , as .
This, together with (9) and (10), implies that are equicontinuous on any finite subinterval of J.
for all and .
Consequently, the proof is complete. □
First, we claim that .
uniformly with respect to and .
where denote the diameters of bounded subsets of .
Consequently, the proof is complete. □
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.
where , , , .
Thus, we can conclude that AU is relatively compact in , i.e., A is compact.
Thus, we have proved that is open in .
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 .
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.
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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