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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 785078, 9 pages
Existence Results for Fractional Differential Equations with Separated Boundary Conditions and Fractional Impulsive Conditions
1Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, China
2School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China
Received 6 March 2013; Accepted 14 July 2013
Academic Editor: Stanislaw Migorski
Copyright © 2013 Xi Fu and Xiaoyou Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the fractional separated boundary value problem of fractional differential equations with fractional impulsive conditions. By means of the Schaefer fixed point theorem, Banach fixed point theorem, and nonlinear alternative of Leray-Schauder type, some existence results are obtained. Examples are given to illustrate the results.
Recently, much attention has been paid to study fractional differential equations due to the fact that they have been proven to be valuable tools in the mathematical modeling of many phenomena in physics, biology, mechanics, and so forth, (see [1–3]).
The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation in recent years. For the general theory and applications of impulsive differential equations, see [4–10] and so forth. However, impulsive fractional differential equations have not been much studied, and many aspects of these equations are yet to be explored. For some recent work on impulsive fractional differential equations, we can refer to [11–26] and the references therein.
In this paper, we consider the existence and uniqueness of solutions for the following fractional separated boundary value problem with fractional impulsive conditions: where is the Caputo fractional derivative of order with the lower limit zero, , , , , with , representing the right and left limits of at , has a similar meaning for , and , , are real constants with and .
We note that the papers on this topic cited above except  all deal with the Caputo derivative and the impulsive conditions only involve integer order derivatives. Here we study the fractional differential equations with fractional impulsive conditions and fractional separated boundary conditions.
In , the author considered the following two impulsive problems: where is the Caputo fractional derivative of order with the lower limit zero, , and where is the Riemann-Liouville fractional derivative of order with the lower limit zero and .
In , Fečkan et al. studied the impulsive problem of the following form: where is jointly continuous, and satisfy , , and .
Furthermore, Wang et al.  considered the impulsive fractional differential equations with boundary conditions as follows: where .
The rest of the paper is organized as follows. In Section 2 we introduce some preliminary results needed in the sequel. In Section 3 we present the existence results for the problem (1). Two examples are given in Section 4 to illustrate the results.
Let us set , , , and , and introduce the space , , and there exist and , , with . It is clear that is a Banach space with the norm .
Definition 1 (see ). The Riemann-Liouville fractional integral of order for a continuous function is defined as which provided that the integral exists.
Definition 2 (see ). For times an absolutely continuous function , the Caputo derivative of order is defined as where denotes the integer part of the real number .
Lemma 3 (see ). Let . Then the differential equation has solutions and which hold for almost all points on the interval , here , , .
Definition 4. A function with its -derivative existing on is said to be a solution of the problem (1) if satisfies the equation on and the conditions
By using a similar discussion of , we have the following lemma.
Lemma 5. Let . A function is a solution of the fractional integral equation: where if and only if is a solution of the impulsive fractional BVP:
Proof. For , by Lemma 3, we know that a general solution of the equation on each interval () is given by
where are arbitrary constants. Since ( is a constant), , and (see ), then from (15), we have
for . Applying the boundary conditions of (14), we get
Next, using the impulsive conditions in (14), we obtain that for
Now we can derive the values of , from formulae (17)-(18). That is,
Hence for , we have
Now it is clear that a solution of the problem (14) has the form of (11).
Conversely, assume that satisfies the fractional integral equation (11). That is, for , , we have Since , we have ( is a constant) and . Using the fact that is the left inverse of , we get which means that satisfies the first equation of the impulsive fractional BVP (14). Next we will verify that satisfies the impulsive conditions. Taking fractional derivative of (22), we have, for , From (22), we obtain Hence we have, for , Similarly, from (24), we can obtain that, for , Finally, it follows from (22) and (24) that (since , ) , , and Now we get Therefore given by (11) satisfies the impulsive fractional boundary value problem (14). The proof is complete.
Remark 6. We notice that the expression of (11) does not depend on the parameter appearing in the boundary conditions of the problem (14). Thus by Lemma 5, we conclude that the parameter is of arbitrary nature of the problem (14).
Let be Banach spaces and , and we say that is a compact if the image of each bounded set in under is relatively compact. The following are two fixed point theorems which will be used in the sequel.
Theorem 7 (nonlinear alternative of Leray-Schauder type ). Let be a Banach space, a nonempty convex subset of , and a nonempty open subset of with . Suppose that is a continuous and compact map. Then either (a) has a fixed point in or (b) there exist a (the boundary of ) and with .
Theorem 8 (Schaefer fixed point theorem ). Let be a normed space and a continuous mapping of into which is compact on each bounded subset of . Then either (I) the equation has a solution for or (II) the set of all such solutions , for , is unbounded.
3. Main Results
This section deals with the existence and uniqueness of solutions for the problem (1).
In view of Lemma 5, we define an operator by with Here , mean that , defined in Lemma 5 are related to . It is obvious that is well defined because of the continuity of , , and . Observe that the problem (1) has solutions if and only if the operator equation has fixed points.
Lemma 9. The operator defined by (31) is completely continuous.
Proof. Since , , and are continuous, it is easy to show that is continuous on .
Let be bounded. Then there exist positive constants , , such that , , and for all , , . Thus, for and , we have Now we can obtain that, for all , and , which implies that the operator is uniformly bounded on .
On the other hand, let and for any , , with , we have By (34) and the above inequality, we deduce that This implies that is equicontinuous on the interval . Hence by PC-type Arzela-Ascoli theorem (see Theorem 2.1 ), the operator is completely continuous.
Theorem 10. Assume that (1) there exist and continuous, nondecreasing such that for ; (2) there exist continuous, nondecreasing such that , for all and ; (3) there exists a constant such that where Then, BVP (1) has at least one solution.
Proof. We will show that the operator defined by (31) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.
From Lemma 9, the operator is continuous and completely continuous.
Let such that for some . Then using the computations in proving that maps bounded sets into bounded sets in Lemma 9, we have Consequently, we have Then by condition (38), . Let us set The operator is continuous and compact. From the choice of the set , there is no such that for some . Therefore by the nonlinear alternative of Leray-Schauder type (see Theorem 7), we deduce that has a fixed point in which is a solution of the problem (1). The proof is complete.
Theorem 11. Assume that there exist and positive constants such that, for , , , Then, BVP (1) has at least one solution on .
Proof. Lemma 9 tells us that the operator defined by (31) is continuous and compact on each bounded subset of .
Let . Since, for each , we know that is bounded. Thus, by Theorem 8, the operator has at least one fixed point. Hence the problem (1) has at least one solution. The proof is completed.
Theorem 12. Assume that there exist and positive constants such that, for , , , Moreover Then, BVP (1) has a unique solution on .
Proof. Let . Then for each , we have Since then combining these two estimations with (47), we obtain Therefore, by (46), the operator is a contraction mapping on . Then it follows Banach's fixed point theorem that the problem (1) has a unique solution on . This completes the proof.
Finally we give two simple examples to show the applicability of our results.
Example 1. Consider the following impulsive fractional separated BVP:
Here , , , and . Clearly, we can take , and such that the relations (45) hold. Moreover Thus, all the assumptions of Theorem 12 are satisfied. Hence, by the conclusion of Theorem 12, the impulsive fractional BVP (50) has a unique solution on .
Example 2. Consider the following impulsive fractional separated BVP:
In the context of this problem, we have Put , , and . Then from Theorem 11, the impulsive fractional BVP (52) has at least one solution on .
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