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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 510808, 13 pages
Impulsive Problems for Fractional Differential Equations with Nonlocal Boundary Value Conditions
Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan 471023, China
Received 7 January 2014; Revised 8 February 2014; Accepted 13 February 2014; Published 22 April 2014
Academic Editor: Yonghuia Xia
Copyright © 2014 Peiluan Li and Youlin Shang. 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.
We investigate the nonlocal boundary value problems of impulsive fractional differential equations. By Banach’s contraction mapping principle, Schaefer’s fixed point theorem, and the nonlinear alternative of Leray-Schauder type, some related new existence results are established via a new special hybrid singular type Gronwall inequality. At last, some examples are also given to illustrate the results.
Fractional differential equations have recently proved to be strong tools in the modeling of many physical phenomena. It draws a great application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media, and fluid dynamic traffic model. For more details on fractional calculus theory, one can see the monographs of Diethelm , Kilbas et al. , Lakshmikantham et al. , Miller and Ross , Podlubny , and Tarasov . Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attentions (see, e.g., [7–13]).
The impulsive differential equations arise from the real world problems to describe the dynamics of processes in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, physics, engineering, and so forth. Due to their significance, many authors have established the solvability of impulsive differential equations. For the general theory and applications of such equations we refer the interested readers to see the papers [14–17] and references therein.
As one of the important topics in the research of differential equations, the boundary value problems have attained a great deal of attention from many researchers; see [18–23] and the references therein. As pointed out in , the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. But there are very few papers (see, e.g., [24–26]) dealing with the nonlocal boundary value problems of fractional differential equations. And even in [24–26], the impulsive effect has not been considered. In , the author considered the following problems: where , is a continuous function, and are continuous functions, , , , , , , , and are two continuous functions, , , , and exist with , .
In , the authors considered the following problem: where is the Caputo fractional derivative of order with the lower limit zero, is jointly continuous, satisfy , and represent the right and left limits of at , , and , , are real constants with .
In , the authors studied the following problem: where is the Caputo fractional derivative of order with the lower limit zero, , is jointly continuous, , and satisfy , and with , representing the right and left limits of at . In , the authors obtained the sufficient condition of the existence of at least one solution for problem (3).
Motivated by the work mentioned above, we consider the following impulsive fractional differential equation with nonlocal boundary value conditions: where , is a continuous function, and are continuous functions; and with , , , , , , and , and are two continuous functions, ; , , and , exist with , . The main methods in our paper are Banach’s contraction mapping principle, Schaefer’s fixed point theorem, and the nonlinear alternative of Leray-Schauder type.
Obviously, the problems in our paper are different from those in , and we generalized the methods and results in . Problems in our paper are more universal than problems in [28, 29]. It should also be noted that the basic space in our paper is , , , exist, and is left continuous at , , which is a Banach space with the norm , where , . The basic space in  is , , and exist with , , with the norm , which is unreasonable for the order because , may not exist, for . So the problem (3) in  are not well defined, and Definition 4.1 is also unreasonable and should be modified.
The rest of this paper is organized as follows. In Section 2, we will give some lemmas which are essential to prove our main results. In Section 3, we give the main results. The first result is based on the Banach contraction principle, the second result is based on Schaefer’s fixed point theorem via a generalized hybrid singular Gronwall inequality, and the third result is based on a nonlinear alternative of Leray-Schauder type. In Section 4, some examples are offered to demonstrate the application of our main results.
At first, we present the necessary definitions for the fractional calculus theory.
Lemma 5 (see [29, Lemma 2.9]). Let satisfy the following inequality: where , ; for some , are constants. Then there exists a constant such that
Lemma 6 (Schaefer’s fixed point theorem). Let be a Banach space and let be a completely continuous operator. If the set is bounded, then has at least a fixed point.
Lemma 7 (nonlinear alternative of Leray-Schauder type). Let be a nonempty convex subset of . Let be a nonempty open subset of with and let be a compact and continuous operator. Then either(i)has fixed points, or(ii)there exist and with .
We define , , and exist with . ; , , , exist and is left continuous at , .
Obviously, is a Banach space with the norm , where , .
Then we can define the solution for the problem (4).
Definition 8. A function with its Caputo derivative of order existing on is a solution of the problem (4) if for and satisfies a.e. on , the restriction of on is just , and the conditions , , , with , .
Lemma 9. For any , a function is a solution of the nonlocal impulsive problem if and only if is a solution of the fractional integral equation with where
Proof. By Lemmas 3 and 4, the solution of (12) can be written as
where . Taking the derivative of gives
If , then we have
where . In view of the impulse conditions
and (16)–(18), we have
Taking (20) into (18), we can get
Repeating the process in this way, the solution for can be written as
By taking the derivative of (22), we have
Taking (16), (17), (22), and (23) to the boundary value conditions
we can get
Then the solution of (12) is (22), where , are given by (25). Taking derivative of (13), we can get
Conversely, taking (13) and (14) into (12), we can easily get the equation and all the impulse conditions and boundary value conditions are satisfied. So we complete the proof of Lemma 9.
Consider the operator defined by where Then we have
Clearly, is well defined.
3. Main Results
This section deals with the existence of solutions for problem (4). Before stating and proving the main results, we make the following hypotheses.(H1) is jointly continuous.(H2) are continuous functions and there exists such that , , for , .(H3)There exist real functions such that ,, for , .(H4) are continuous functions and there exist positive constants , such that , , and .Let
Theorem 10. Assume that hold and ; then problem (4) has a unique solution, where
Step 1. We show that , for all .
For all , , or , , , by (30) and the continuity of , , we have So we know , . It is easy to see that , exist and is left continuous at . So, for , .
Step 2. We show that is a contraction operator on . Consider
Hence , that is, is a contraction operator on . By applying the well-known Banach’s contraction mapping principle, we know that the operator has a unique fixed point on . Therefore, the problem (4) has a unique solution.
In order to get the second main result, we replace with . are continuous functions and there exist positive constants and , such that , , , , for all .
Next, we modify to the following linear growth condition : There exist constants and such that , , .
Theorem 11. Assume that , , and hold; then the problem (4) has at least one solution.
Proof. According to Lemma 6, if we want to get the solution of problem (4), we only need to consider the fixed point of operator , which is defined by (28). We divide the proof into four steps.
Step 1. is continuous.
Let be a sequence such that in . , we have
From and , we know is jointly continuous and are also continuous. Together with the continuity of , , we can also easily draw that , as .
Similarly, we can obtain , as , . Then for , we have , as , which implies that is continuous.
Step 2. maps bounded sets into bounded sets in .
Set . For , by the continuity of , , , , , we know that , , where , are nonnegative constants.
For all , , we have
Then we can obtain , where