New Trends on Fractional and Functional Differential EquationsView this Special Issue
Research Article | Open Access
Fractional Differential Equations with Fractional Impulsive and Nonseparated Boundary Conditions
This paper studies the existence results for nonseparated boundary value problems of fractional differential equations with fractional impulsive conditions. By means of 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.
The subject of fractional differential equations has recently evolved as an interesting and popular field of research. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. More and more researchers have found that fractional differential equations play important roles in many research areas, such as physics, chemical technology, population dynamics, biotechnology, and economics (see [1–4]). Hence, there are many literatures devoted to solving fractional differential equations through theoretical analysis or numerical methods (e.g., see [5–9]).
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 [10–15] and so forth. However, impulsive differential equations of fractional order 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 [16–24] and the references therein.
In this paper, we investigate the existence and uniqueness of solutions for fractional differential equations with fractional impulsive conditions and nonseparated boundary conditions where is the Caputo fractional derivative of order , , , , , with , representing the right and left limits of at , has a similar meaning for , and , are real constants such that for (2) and for (3).
We note that, as pointed out in the papers [22–24], the concept of piecewise continuous solutions used in some already published works to handle the impulsive fractional differential equations is not appropriate (see counterexamples given in Lemma 3.1 of , Section 1 in , and Section 3 in ). 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 nonseparated boundary conditions [25–27].
The rest of the paper is organized as follows. In Section 2 we introduce some notations and definitions needed in the following sections and give the appropriate formula of solutions for our problems. In Section 3 we present the existence results for the problems (1), (2) and (1), (3). Two examples are presented in Section 4 to illustrate the results. Concluding remarks are provided in Section 5.
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 function is defined as provided the integral exists.
Definition 2 (see ). For a 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 where , , .
Definition 4. A function with its -derivative existing on is said to be a solution of the problems (1), (2) (or the problems (1), (3)), if satisfies the equation on , the impulsive conditions and the nonseparated boundary conditions (2) (or (3)).
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 boundary value problem
Proof. For , by Lemma 3, we know that a general solution of the equation on each interval () is given by
where and are arbitrary constants. Then we have
Since ( is a constant), , and (see ), then from (13), we have
Applying the boundary conditions of (12), we get (since and )
Next using the impulsive conditions in (12), we obtain that, for ,
That is to say, we have, for ,
Now the above two equations together with (16) imply that where
Hence, for , we have
Substituting the value of and in (21), and by (13), we get the fractional integral equation (9).
Conversely, assume that satisfies the fractional integral equation (9); 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 boundary value problem (12). Next we will verify that satisfies the impulsive conditions. Taking fractional derivative of (22), we have, for ,
From (22), we obtain
The two equations above imply that, for ,
Similarly, from (25), we have
Therefore, for , we have
Finally, it follows from (22) and (23) that (since , )
Hence, we get
Therefore, given by (9) satisfies the impulsive fractional boundary value problem (12). The proof is complete.
Lemma 6. Let . A function is a solution of the fractional integral equation where with and defined by (11) and if and only if is a solution of the impulsive fractional boundary value problem
Proof. The proof is similar to that of Lemma 5. Let the notations be given as in the proof of Lemma 5. Applying the boundary conditions in (35), from the relations (13), (15), and (17), we have
After a direct computation, we obtain that with and defined by (11) and
The remaining part of proof is the same as that of Lemma 5.
Remark 7. We note that the solution expression (32) of the problem (35) does not depend on the parameter appearing in the boundary conditions. Thus, by Lemma 6, we conclude that the parameter is of arbitrary nature of the problem (35).
Remark 8. Our approach for the construction of solutions for impulsive fractional differential equations is general, which provides an effective way to deal with such problems. Taking the problem (1) for example, on each interval , we use
From the counterexamples given in Lemma 3.1 of , Section 1 in , and Section 3 in , we know that if the case was chosen to construct solutions, the solution formula obtained in the fractional integral equation form is not equivalent to the original impulsive fractional differential equation. This is the main difference between impulsive fractional differential equations and impulsive ordinary differential equations. Observe that fractional calculus has memory property.
Theorem 9 (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 10 (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
First we study the existence results for the problem (1), (2). In view of Lemma 5, we define an operator by where Here , , , and mean that , , , and defined in Lemma 5 are related to . It is obvious that is well defined because of the continuity of , , and and that the problem (1), (2) has solutions if and only if the operator equation has fixed points.
For the sake of convenience, we set
Lemma 11. The operator defined by (39) is completely continuous.
Proof. Since , , and are continuous, it is easy to check that is continuous on .
Let be bounded; then there exist positive constants , , such that , , and , for all , . Thus, for and , we have Therefore, we can deduce that, for all and , which implies that the operator is uniformly bounded on .
On the other hand, let and, for any , , with , we have From (43) and the above inequality, we deduce that tends to zero as . This implies that is equicontinuous on the interval . Hence, by PC-type Arzela-Ascoli Theorem (see Theorem 2.1 ), the operator is completely continuous.(H1): there exist and continuous nondecreasing such that for ; there exist and continuous nondecreasing such that and for all and .
Proof. We will show that the operator defined by (39) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.
From Lemma 11, the operator is continuous and completely continuous.
Let be such that for some . Then using the computations in the proof that maps bounded sets into bounded sets in Lemma 11, we have Consequently, we have Then, in view of condition (46), there exists such that . 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 9), we deduce that has a fixed point in which is a solution of the impulsive fractional boundary value problem (1), (2). The proof is complete.(H2):there exist and positive constants and such that, for , , ,
Proof. Lemma 11 tells us that the operator defined by (39) is continuous and compact on each bounded subset of .
Now let us show that the set is bounded. Let ; then for some . For each , using the similar estimations given in Theorem 12, we have This implies that there exists some such that for all ; that is, is bounded. Thus, by Theorem 10, the operator has at least one fixed point. Hence the problem (1), (2) has at least one solution. The proof is completed.(H3):there exist and positive constants and such that, for , , ,
Proof. Let . Then, for each , we have Since then combining these two estimations with (55), we obtain Therefore, by (54), the operator is a contraction mapping on . Then it follows from Banach’s fixed point theorem that the problems (1), (2) has a unique solution on . This completes the proof.
Remark 18. The results obtained in this paper can easily be generalized to the boundary value problems of impulsive fractional differential equations (1) with nonseparated integral boundary conditions where and are given continuous functions.
In this section, we give two simple examples to show the applicability of our results.
Example 1. Consider the following impulsive fractional BVP:
Here , , , , , , , , , and . Clearly, we can take , , and such that the relations (53) hold. Moreover,