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The Scientific World Journal
Volume 2014 (2014), Article ID 194346, 28 pages
A Study of Impulsive Multiterm Fractional Differential Equations with Single and Multiple Base Points and Applications
1Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, China
2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 10 October 2013; Accepted 17 November 2013; Published 21 January 2014
Academic Editors: A. M. A. El-Sayed, A. Kılıçman, and S. C. O. Noutchie
Copyright © 2014 Yuji Liu and Bashir Ahmad. 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 discuss the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line. Our study includes the cases for a single base point fractional differential equation as well as multiple base points fractional differential equation. The asymptotic behavior of solutions for the problems is also investigated. We demonstrate the utility of our work by applying the main results to fractional-order logistic models.
The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a characteristic arise naturally and are often, for example, studied in physics, chemical technology, population dynamics, biotechnology, and economics. These processes are modeled by impulsive differential equations. In 1960, Milman and Myshkis introduced the concept of impulsive differential equations . Afterwards, this subject was extensively investigated and several monographs have been published by many authors like Samoilenko and Perestyuk , Lakshmikantham et al. , Baino and Simeonov , Baino and Covachev , and Benchohra et al. .
Fractional differential equations (FDEs for short), regarded as the generalizations of ordinary differential equations to an arbitrary noninteger order, find their genesis in the work of Newton and Leibniz in the seventieth century. Recent investigations indicate that many physical systems can be modeled more accurately with the help of fractional derivatives . Fractional differential equations, therefore, find numerous applications in the field of viscoelasticity, feedback amplifiers, electrical circuits, electroanalytical chemistry, fractional multipoles, and neuron modelling encompassing different branches of physics, chemistry, and biological sciences [8–10].
Some recent work on the existence of solutions for initial value problems of Caputo type impulsive fractional differential equations can be found in a series of papers [11–16], whereas the solvability of boundary value problems of impulsive differential equations involving Caputo fractional derivatives was investigated in [17–26].
In the left and right fractional derivatives and , is called a left base point and right base point. Both and are called base points of fractional derivatives. A fractional differential equation (FDE) containing more than one base points is called a multiple base points FDE while an FDE containing only one base point is called a single base point FDE.
Henderson and Ouahab  studied the solvability of the following initial value problems for impulsive fractional differential equations: where , is a fixed real number, is continuous, are continuous functions, and and . One can see that both fractional differential equations in (1) are multiple base points FDEs with base points , which are in fact the impulse points.
In , the authors used the concept of upper and lower solutions together with Schauder's fixed point theorem to study the impulsive fractional-order differential equation: One can notice that the problem (2) contains a multiple base points FDE with base points (impulse points).
In , the authors studied the existence and uniqueness of solutions of the following initial value problem of fractional order differential equations: where the fractional differential equations are a multiple base points FDE with the base points (impulse points).
Fečkan et al.  studied the existence of solutions of the following initial value problem of impulsive fractional differential equations: where , is a fixed real number, is jointly continuous, are continuous functions, and and . Observe that the fractional differential equation in (4) is a single base point FDE with the base point . So the impulse points are different from the base point.
Liu and Ahmad  studied a problem of multi-term and multiorder quasi-Laplacian singular fractional differential equations: where , , are fixed points, is the Riemann-Liouville fractional derivative, is a sup-multiplicative function, , , are impulsive Caratheodory functions, are continuous functions, and are impulse functions. In (5), the fractional differential equation is a single base point FDE with the base point . Clearly the impulse points are different from the base point.
Remark. It is clear from the abovementioned work that IVPs of impulsive fractional differential equations can be categorized into two classes: (a) IVPs of one base point FDEs [20, 29, 30] and (b) IVPs of multiple base points FDEs [12, 27, 28].
In this paper, we study the following two initial value problems (IVPs for short) of nonlinear multi-term FDEs with impulses on half lines: where , , , with , is the standard Caputo fractional derivative at the base point , satisfies that there exists such that for all , may be singular at , is the standard Caputo fractional derivative at the base points ; that is, for all , and is a Caratheodory function, and is a Caratheodory function sequence, and , .
The salient features of the present work include the following: (i) to establish sufficient conditions for the existence of solutions for the IVP (6) with a single base point and IVP (7) with multiple base points (same as the impulse points). We emphasize that the conditions for the existence of solutions for the IVPs (6) and (7) are different; (ii) the asymptotic behavior of solutions for the problems is studied and the sufficient criterion for every solution to tend to zero as is established; (iii) the method of proof relies on the Schauder fixed point theorem; (iv) our approach for dealing with impulsive problems at hand is different from the ones employed in earlier work on the topic and thus opens a new avenue for studying impulsive fractional differential equations; (v) as an application, we apply our results to fractional-order logistic models and present sufficient conditions for the existence and asymptotic behavior of solutions of these logistic models.
The paper is organized as follows: the auxiliary material is given in Section 2, the main results are presented in Sections 3 and 4, while the application of the main results is demonstrated in Section 5.
Define the Gamma function and Beta function, respectively, as
Definition 1 (see ). Riemann-Liouville fractional integral of order of a continuous function is given by provided that the right-hand side exists.
Definition 2 (see ). Caputo's derivative of fractional-order for a function is defined by for , . If , then Obviously, Caputo's derivative of a constant is zero.
Lemma 3 (see ). For , the general solution of fractional differential equation is given by , where , .
Definition 4. A function is said to be a solution of the IVP (6) if both and are continuous, satisfies the differential equation a.e. on , and the limits and exist and the following conditions are satisfied:
Definition 5. A function is said to be a solution of the IVP (7) if both and are continuous, satisfies the differential equation on , and the limits and exist and the following conditions are satisfied:
Choose and . LetFor , define the norm on as It is easy to show that is a real Banach space.
Definition 6. is called a Caratheodory function if it satisfies the following assumptions:(i) is continuous on ;(ii)for each , there exists a constant such that implies that
Definition 7. is called a Caratheodory function sequence if it satisfies the following assumptions:(i) is continuous on for each ;(ii)for each , there exist constants such that implies that
If , then we have
Lemma 8. Suppose that is a Caratheodory function and is a Caratheodory function sequence on . Then is a solution of if and only if is a solution of the fractional integral equation
Proof. For and , we have Since is a Caratheodory function and is a Caratheodory function sequence, therefore, there exist and such that Let us assume that satisfies (48). Then, by Lemma 3, the solution of (48) can be written as Observe that From and , we get and This implies that Thus, we have Hence, satisfies (49). Next, we show that . Indeed It is easy to see that Furthermore, for , we have This implies that . Conversely, suppose that satisfies (49). By a direct computation, it follows that the solution given by (49) satisfies the problem (48). This completes the proof.
Choose and and defineFor , we define the norm on as It is easy to show that is a real Banach space.
Lemma 9. Suppose that is a Caratheodory function and is a Caratheodory function sequence, and . Then is a solution of the problem if and only if is a solution of the fractional integral equation
Proof. For , we have that there exists such that
Since is a Caratheodory function and is a Caratheodory function sequence, then there exist and such that
Assume that satisfies the problem (50). Then, in view of Lemma 3, we can write the solution of (50) as
From , we get . Since
and , we get
Hence the solution of the problem (50) is
Next, we need to show that . Clearly,
Furthermore, for , we have Since and , we get for all . Then So Moreover, for , we get So Thus, . Conversely, assume that satisfies (51). Then, by direct computation, it follows that the solution given by (51) satisfies the problem (50). This completes the proof.
3. Existence Results for an IVP with a Single Base Point
In relation to the IVP (6), we define an operator by
Lemma 10. Let be a Caratheodory function and let be a Caratheodory function sequence. Then (i) is well defined; (ii) the fixed point of the operator coincides with the solution of IVP (6); (iii) is completely continuous.
Proof. (i) For , let
Since is a Caratheodory function, is Caratheodory function sequence; there exist positive numbers and such that
It is easy to show that
As in the proof of Lemma 8, it can be shown that both and are bounded on .
Hence, and consequently is well defined.
(ii) It follows from Lemma 8 that the fixed point of the operator coincides with the solution of IVP (6).
(iii) To establish that is completely continuous, we show that (a) is continuous, (b) maps bounded sets of to bounded sets, and (c) maps bounded sets of to relatively compact sets.
(a) In order to show that the operator is continuous, let with as . We will prove that as . It is easy to see that there exists such that Since is a Caratheodory function and is a Caratheodory function sequence, then there exist and such that Notice that From the inequality it follows that there exists for such that Since is uniformly continuous on , there exists such that holds for all with , . From (54), there exists such that Hence, Since therefore, we can find such that holds for all , .
As is a Caratheodory function, there exists such that holds for all and with , . From (54), there exists such that So, for , we have