In this paper, we discuss the expression and properties of Green’s function for boundary value problems of nonlinear Sturm-Liouville-type fractional order impulsive differential equations. Its applications are also given. Our results are compared with some recent results by Bai and Lü.
Keywords:fractional differential equation; impulse; Green’s function; existence; fixed point theorem
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [1-5]. For the basic theory and recent development of the subject, we refer to a text by Lakshmikantham et al.. For more details and examples, see [7-23] and the references therein.
Integer-order impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see, for instance, [24-26]. Recently, the boundary value problems of impulsive differential equations of integer order have been studied extensively in the literature (see [27-36]).
On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see [37-46] and the references therein. However, as pointed out in [38,39], the fractional impulsive differential equations have not been addressed so extensively and many aspects of these problems are yet to be explored. For example, the theory using Green’s function to express the solution of fractional impulsive differential equations has not been investigated till now. Now, in this paper, we shall study the expression of the solution of fractional impulsive differential equations by using Green’s function.
Consider the following nonlinear boundary value problem of fractional impulsive differential equations:
where is the Caputo fractional derivative, , is a continuous function, is a continuous function, are continuous functions, and . with , , , . has a similar meaning for .
Some special cases of (1.1) have been investigated. For example, Bai and Lü  considered problem (1.1) with and . By using the fixed point theorem in cones, they proved some existence and multiplicity results of positive solutions of problem (1.1).
At the end of this section, it is worth mentioning that it is an important method to express the solution of differential equations by Green’s function. According to the previous work, we find that the solution of impulsive differential equations with integer order can be expressed by Green’s function of the case without impulse. For example, Green’s function of the following boundary value problem
can be expressed by
where . The solution of problem (1.2) can be expressed by
If we consider impulsive differential equations
then the solution of problem (1.3) can be expressed by
where , .
Naturally, one wishes to know whether or not the same result holds for the fractional order case. We first study the fractional order differential equations with Caputo derivatives
where is the Caputo fractional derivative. Then
Then we study whether the solution of fractional order impulsive differential equations with Caputo derivatives
can be expressed by
Thus, it is an interesting problem, and so it is worthwhile to study. We will give the answers in the following sections.
The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.1). In Section 3, we give some preliminaries about the operator and the fixed point theorem. In Section 4, we get some existence results for problem (1.1) by means of some standard fixed point theorems. The final section of the paper contains two examples to illustrate our main results.
2 Expression and properties of Green’s function
Consider the following fractional impulsive boundary value problem:
where is the Caputo fractional derivative, , are continuous functions, and . with , , , . has a similar meaning for .
Theorem 2.1The solution of problem (2.1) can be expressed by
Proof Suppose that x is a solution of (2.1). Then, for some constants , we have
It follows from (2.5) that
If , then, for some constants , we can write
Using the impulse conditions and , we find that
If , repeating the above procedure, we obtain
It follows that , and
By the boundary conditions, we have
where η is defined in (2.4).
Substituting (2.7), (2.8) into (2.6), we obtain (2.2). This completes the proof. □
Remark 2.1 It is clear that is Green’s function of the boundary value problem
From (2.3), we can prove the following results.
Proposition 2.1For all , we have
Proposition 2.2For all , we have
For the sake of convenience, let
Then it follows from (2.9) and (2.10) that
From the proof of Theorem 2.1 we have the following results.
Proposition 2.3The solution of fractional impulsive differential equations can be expressed by Green’s function, and it is not Green’s function of the corresponding fractional differential equations, but Green’s function of the corresponding integer order differential equations.
In this section, we give some preliminaries for discussing the solvability of problem (1.1) as follows.
Then is a Banach space with the norm
is a Banach space with the norm
Definition 3.1 A function with its Caputo derivative of order q existing on J is a solution of problem (1.1) if it satisfies (1.1).
We give the following hypotheses:
(H1) is a continuous function, and there exists such that ;
(H2) is a continuous function;
(H3) are continuous functions.
It follows from Theorem 2.1 that:
Lemma 3.1If (H1)-(H3) hold, then a function is a solution of problem (1.1) if and only if is a solution of the impulsive fractional integral equation
Using Lemma 3.1, problem (1.1) reduces to a fixed point problem , where T is given by (3.2). Thus problem (1.1) has a solution if and only if the operator T has a fixed point.
From (3.2) and Lemma 3.1, it is easy to obtain the following result.
Lemma 3.2Assume that (H1)-(H3) hold. Then is completely continuous.
Proof Note that the continuity of f, ω, and together with and ensures the continuity of T.
Let be bounded. Then there exist positive constants , and such that , and , . Thus, , we have
Furthermore, for any , , we obtain
On the other hand, from (3.5), for with , we have
It follows from (3.3), (3.5) and (3.6) that T is equicontinuous on all subintervals , . Thus, by the Arzela-Ascoli theorem, the operator is completely continuous. □
To prove our main results, we also need the following two lemmas.
LetDbe a nonempty, closed, bounded, convex subset of aB-spaceX, and suppose that is a completely continuous operator. ThenThas a fixed point .
LetXbe a real Banach space and be a completely continuous operator. If
is bounded, thenThas a fixed point , where
4 Existence of solutions
In this section, we apply Lemma 3.3, Lemma 3.4 and the contraction mapping principle to establish the existence of solutions of problem (1.1). Let us begin by introducing some notation. Define
Theorem 4.1Assume that (H1)-(H3) hold. Suppose further that
hereγis defined in (3.4). Then problem (1.1) has at least one solution .
Proof We shall use Schauder’s fixed point theorem to prove that T has a fixed point. First, recall that the operator is completely continuous (see the proof of Lemma 3.2).
On account of (4.1), we can choose , and such that
By the definition of ξ, there exists such that
Similarly, we have
where , are positive constants.
It follows from (3.2) and (4.4)-(4.6) that
where is defined by (4.2) and is defined by
Similarly, from (3.2) and (4.4)-(4.6), we get
where is defined by (4.3) and is defined by
It follows from (4.7) and (4.8) that
Hence, we can choose a sufficiently large such that , where
Consequently, Lemma 3.3 implies that T has a fixed point in , and the proof is complete. □
Remark 4.1 Condition (4.1) is certainly satisfied if uniformly in as , as and as ( ).
Theorem 4.2Assume that (H1)-(H3) hold. In addition, letf, and satisfy the following conditions:
(H4) There exists a nonnegative function with on a subinterval of such that
(H5) There exist constants such that
for all , .
Then problem (1.1) has at least one solution.
Proof It follows from Lemma 3.2 that the operator is completely continuous.
Next, we show that the set
Let . Then for . For any , we have
It follows from (H4), (H5), (3.2) and (4.9) that
It follows from (4.10) that
Furthermore, for any , , we obtain
which, for any , yields
It follows from (4.11) and (4.12) that
where , here
So it follows from (4.13) that the set V is bounded. Thus, as a consequence of Lemma 3.4, the operator T has at least one fixed point. Consequently, the problem (1.1) has at least one solution. This finishes the proof. □
Finally we consider the existence of a unique solution for problem (1.1) by applying the contraction mapping principle.
Theorem 4.3Assume that (H1)-(H3) hold. In addition, letf, and satisfy the following conditions:
(H6) There exists a constant such that
for each and all .
(H7) There exist constants such that
for all , .
here and are defined in (2.10), then problem (1.1) has a unique solution.
Proof Let . Then, for each , it follows from (H6), (H7) and (3.2) that