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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 498781, 21 pages
Existence Results for a Coupled System of Nonlinear Singular Fractional Differential Equations with Impulse Effects
1Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, China
2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 2 October 2012; Accepted 15 February 2013
Academic Editor: Jocelyn Sabatier
Copyright © 2013 Yuji Liu et al. 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.
A boundary value problem for the singular fractional differential system with impulse effects is presented. By applying Schauder's fixed point theorem in a suitably Banach space, we obtain the existence of at least one solution for this problem. Two examples are presented to illustrate the main theorem.
Fractional differential equations have received increasing attention during recent years since the behavior of many physical, chemical, and engineering processes can be properly described by using fractional differential equations theory; see the books [1–3], papers [4, 5] and references therein. For details on the geometric and physical interpretation of the derivatives of noninteger order, see, for example, [6–11]. For some recent works with applications to engineering we refer the reader to [12–15].
For an introduction of the basic theory of impulsive differential equation, we refer the reader to . Among previous research, little is concerned with differential equations with fractional order with impulses . Ahmad and Sivasundaram [18, 19] gave some existence results for two-point boundary value problems involving nonlinear impulsive hybrid differential equations of fractional order . Ahmad and Nieto in  establish sufficient conditions for the existence of solutions of the antiperiodic boundary value problem for impulsive differential equations with the Caputo derivative of order . Some recent results on impulsive initial value problems or boundary value problems for fractional differential equations on a finite interval can be found in [21–23] and references therein. The memory property of fractional calculus makes studies more complicated.
This paper is motivated by  in which the following boundary value problem for the fractional differential equation was studied, where , , and , , , and are continuous functions, and is the Riemann-Liouville fractional derivative. An existence result was proved for BVP (1) in . The growth assumptions imposed on and are sublinear cases (see [24, Theorem 2.1]); that is, there exist functions , nonnegative constants , and such that
In , the following boundary value problem for the fractional differential equation was studied, where , , and , and are continuous functions, and is the Riemann-Liouville fractional derivative. The growth assumptions imposed on and are sublinear cases (see [25, Theorem 3.1]), that is, there exist functions , nonnegative constants , , and such that or sublinear cases, that is, there exist nonnegative constants , and such that We find that in the superlinear cases, BVP (3) has a pair of solutions without needing any other assumptions. Hence, these cases are trivial ones discussed in .
In this paper we consider the following nonlinear boundary value problem for the singular multiterm fractional differential equation with impulse effects whose boundary conditions are of integral form where(a), , and , is the Riemann-Liouville fractional derivative,(b) defined on ,(c) with , defined on ,(d),(e).
The novelty of this paper is as follows: first, the fractional differential equations in (6) are multiterm ones and their nonlinearities depend on the lower fractional derivatives; second, both and may be singular at and , that is, and may be not continuous functions on , the boundary conditions are integral boundary conditions, and we obtain the results on the existence of at least one solution of BVP (6)-(7); third, and are supposed; the growth assumptions imposed on and are allowed to be sublinear cases. Finally, two examples are given to illustrate the efficiency of the main theorem.
The remainder of this paper is as follows: in Section 2, we present preliminary results. In Section 3, the main theorem and its proof are given. In Section 4, two examples are given to illustrate the main results.
In this section, we present some background definitions and preliminary results.
Definition 1 (see ). The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side exists.
Definition 2 (see ). The Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided that the right-hand side is pointwise defined on .
Definition 3. is called a -Caratheodory function if satisfies that(i) is continuous on for every ;(ii) is continuous on for every ;(iii)for each there exists a constant such that , , .
Definition 4. is called a -Caratheodory function if satisfies that(i) is continuous on for every ;(ii) is continuous on for every ;(iii)for each there exists a constant such that , , .
Lemma 5 (the Leray-Schauder nonlinear alternative ). Let be a Banach space and be a completely continuous operator. Suppose is a nonempty open subset of centered at zero. Then either there exists and such that or there exists such that .
Let the gamma and beta functions and be defined by Choose For , define the norm by It is easy to show that is a real Banach space. For , define the norm by It is easy to show that is a real Banach space. Thus, is a Banach space with the norm defined by .
In this paper, we suppose the following:(A) satisfies that there exist constants , , such that , , and for all ; satisfies that there exist constants , , such that , , and for all .(B) are -Caratheodory functions and are -Caratheodory functions.
Remark 6. Suppose that is a -Caratheodory function. For example, , , choose , and , then , such that , , and for all . It is easy to see that is singular at and .
Lemma 7. Suppose that , and (a)–(e), (A)-(B) hold. Then is a solution of if and only if satisfies the integral equation where
Proof. If is a solution of BVP (15), then
and satisfies all equations in (31) From (B), is a -Caratheodory function, then there exists such that
Similarly we get that there exist constants such that
It follows from (15) that, for , there exist constants such that
From , we get
From , we get
From , we get
From , we get
It follows that Then So
Hence, for , we have And for , we have Hence, satisfies (16).
On the other hand, if and is a solution of (16), then we can prove that is a solution of BVP (6)-(7). The proof is completed.
Lemma 8. Suppose that , and (a)–(e), (A)-(B) hold. Then is a solution of if and only if satisfies the integral equation where
Proof. The proof is similar to that of the proof of Lemma 7 and is omitted.
Now, we define the operator on by with
Lemma 10. Suppose that (a)–(e) and (A)-(B) hold. Then is well defined and is completely continuous.
Proof. The proof is very long, so we list the steps. First, we prove that is well defined; second, we prove that is continuous, and, finally, we prove that is compact. So is completely continuous. Thus, the proof is divided into three steps.
Step 1. Prove that is well defined.
For , we have . Then From (B), are -Caratheodory functions, then there exist constants such that Hence, From (34), (37), and (38), we see that is defined on , continuous on and , respectively. One sees that and there exits the limit .
On the other hand, we have