A Study of  Impulsive Multiterm Fractional Differential Equations with Single and Multiple Base Points and Applications

Liu, Yuji; Ahmad, Bashir

doi:https://doi.org/10.1155/2014/194346

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Theory, Methods, and Applications of Fractional Calculus

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Volume 2014 | Article ID 194346 | https://doi.org/10.1155/2014/194346

A Study of Impulsive Multiterm Fractional Differential Equations with Single and Multiple Base Points and Applications

Yuji Liu¹and Bashir Ahmad²

Academic Editor: A. Kılıçman, S. C. O. Noutchie, A. M. A. El-Sayed

Received10 Oct 2013

Accepted17 Nov 2013

Published21 Jan 2014

Abstract

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.

1. Introduction

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 [1]. Afterwards, this subject was extensively investigated and several monographs have been published by many authors like Samoilenko and Perestyuk [2], Lakshmikantham et al. [3], Baino and Simeonov [4], Baino and Covachev [5], and Benchohra et al. [6].

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 [7]. 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 [12] 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 [27], 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 [28], 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. [29] 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 [30] 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.

2. Preliminaries

We recall some basic concepts of fractional calculus [9, 10] and show auxiliary results.

Define the Gamma function and Beta function, respectively, as

Definition 1 (see [9]). Riemann-Liouville fractional integral of order of a continuous function is given by provided that the right-hand side exists.

Definition 2 (see [9]). 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 [9]). 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 which gives 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 this section, we discuss the existence and uniqueness of solutions for the single base point IVP (6). The asymptotic behaviour of solutions of IVP (6) is also investigated.

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 Consequently, for all , , we get
In particular, for , we find that Thus, it follows that Similarly, it can be shown that From (69) and (70), we conclude that . This implies that is continuous.
(b) Let us recall that is relatively compact if it is bounded, both and are equicontinuous on any closed subinterval of and equiconvergent at , and .
Let be a nonempty bounded set. To prove that is completely continuous, we need to prove that is bounded, is equicontinuous on finite closed sub-interval on , is equiconvergent at , and is equiconvergent at .
Since is bounded, therefore, (49), (50), and (51) hold for . Following the method of proof for Lemma 8, it can easily be shown that is bounded.
Next we show that is equicontinuous on finite closed sub-interval on .
For with with and , we have So Thus, uniformly as with . Similarly, we have From (73) and (74), we conclude that is equicontinuous on finite closed interval on .
Now wee prove that is equiconvergent as . For , we find that It follows that From (75), it follows that is equiconvergent as .
For , we have which imply that is equiconvergent as .
Our next task is to show that is equiconvergent as . Observe that Hence, is equiconvergent as .
From the above steps, it follows that is completely continuous. This completes the proof.

In the sequel, we need the following assumption: is a Caratheodory function such that where and are real numbers; is a Caratheodory sequence and there exist numbers , , such that

Furthermore, we set where

Theorem 11. Suppose that and hold. Then IVP (6) has at least one solution if

Proof. Let be the Banach space as defined in Section 2 and let be an operator given by (98). In view of Lemma 8, it follows from the assumptions and that is well defined and is completely continuous. Thus, we seek solutions of IVP (6) by finding fixed points of in .
Let us introduce It is easy to show that . For , we define . Then, for , we have Using the assumptions and , we find that Thus, by (81), it follows that Next, we have the following cases.
(i) For , we can choose sufficiently large such that . Let . It is easy to see that . Then, by Schauder's fixed point theorem, the operator has a fixed point , which is a bounded solution of IVP (6).
(ii) In case , we choose Let . Then it can easily be shown that . Thus, Schauder's fixed point theorem applies and the operator has a fixed point , which is a bounded solution of IVP (6).
(iii) For , we choose such that Let . As before, it is easy to show that . Then, it follows from Schauder's fixed point theorem that has a fixed point , which corresponds to a solution of IVP (6). This completes the proof.

Theorem 12. Suppose that and hold with . Then IVP (6) has a unique solution if .

Proof. By Theorem 11, IVP (6) has at least one solution. Let and be two different solutions of IVP (6). Then , , and . Employing the method used in the proof of Theorem 11, we find that Thus, . On the other hand, by (51), we get which is a contradiction. Hence, IVP (6) has a unique solution if . This completes the proof.

Next, consider the following IVP: where are constants, is convergent, and is a Caratheodory function; there exists such that for all .

Theorem 13. Assume that the conditions and hold. Then every solution of (93) tends to as provided that (84) is satisfied.

Proof. By Theorem 11, there exist solutions for IVP (93) satisfying the integral equation Clearly, Since is a Caratheodory function by , therefore, there exists such that So This completes the proof.

4. Existence of Solutions for an IVP with Multiple Base Points

In this section, we show the existence for solutions for IVP (7) with multiple base points. Let us introduce an operator on as

Lemma 14. Suppose that is a Caratheodory function and is a Caratheodory function sequence and . Then(i) is well defined;(ii)the fixed point of the operator coincides with the solution of IVP (7);(iii) is completely continuous.

Proof. (i) For , we set Since is a Caratheodory function, is Caratheodory function sequence; there exist positive numbers and such that It is easy to show that As in Lemma 9, we can show that Hence, . This implies that is well defined.
(ii) It follows from Lemma 9 that the fixed point of the operator coincides with the solution of IVP (7).
(iii) To show that is completely continuous, we split the proof into several steps.
Step 1. is continuous.
Let with as . We will prove that as . It is easy to see that there exists such that As in the proof of Lemma 10, From , we get for all .
Since is convergent, there exists such that Then
Since is a Caratheodory function, there exists such that holds for all and with , . From (104), there exists such that So, for , we have Thus, for with , we have In consequence, Similarly, we can show that From (112) and (113), it follows that which implies that is continuous.
Let be a nonempty bounded set. To prove that is completely continuous, we need to prove that is bounded, is equicontinuous on finite closed sub-interval on , is equiconvergent at , and is equiconvergent at .
Step 2. As in the proof of Lemma 10, it is easy to show that is bounded.
Step 3. We prove that is equicontinuous on finite closed sub-interval on . For with with and , we have So It follows that uniformly as with .
In a similar manner, one can find that From (116) and (117), we deduce that is equicontinuous on finite closed interval on .
Step 4. We prove that is equiconvergent as .
As in Lemma 10, is equiconvergent as . For , we have Hence, is equiconvergent as .
Step 5. is equiconvergent as . Notice that Hence, is equiconvergent as . This completes the proof in which is completely continuous.

Theorem 15. Assume that and hold. Then IVP (7) has at least one solution if where , is given by (83) and

Proof. Let denote the Banach space equipped with the norm (introduced in Section 2). Let be an operator defined by (98). In view of Lemma 8, we need to show that the operator has a fixed point in which will be a solution of IVP (7). By Lemma 14, is well defined and completely continuous. Lets us introduce It is easy to show that . Let and define
For , we have . Then Using the assumptions and , we find that Furthermore, we have Thus, it follows that Now we discuss the cases for different values of .
(i) For , we can choose sufficiently large so that . Let . It is easy to show that . Then, the Schauder fixed point theorem implies that the operator has a fixed point , which is a bounded solution of IVP (7).
(ii) For , we select
Let . It can easily be shown that . Then, the Schauder fixed point theorem applies and the operator has a fixed point , which is a bounded solution of IVP (7).
(iii) For , we set so that
Let . Then we can show that . Thus, by the Schauder fixed point theorem, the operator has a fixed point , which is a solution of IVP (7). This completes the proof.

Theorem 16. Suppose that and hold with . Then IVP (7) has a unique solution if .

Proof. The proof is similar to that of Theorem 12, so we omit it.

5. Applications

Malthusian geometrical law is expressed as , where is the population at time and is the proportionality constant. When the growth of the population in any environment is stopped due to the density of the population, this model modifies to a nonlinear logistic model of the form . The generalization of the nonlinear logistic model is represented by . For , the model is known as the Gompertz model and can be found in the literature on actuarial science and mortality analysis of elderly person [31].

In [32], Das et al. presented the following fractional-order logistic model (Das Model): In [33], the authors presented the following logistic model with fractional order: where is a constant, are impulse functions, , and with , .

As an application of the main results established in the paper, we discuss the sufficient conditions for the existence and asymptotic behavior of solutions for the logistic models: where , , , are continuous functions, and are constants.

Theorem 17. Suppose that and there exists such that Then IVP (133) has at least one solution if where

Proof. Let . Then In association with Theorem 11, we choose , , , , , . Then the conditions and hold. By Theorem 11, IVP (133) has at least one solution. This completes the proof.

Theorem 18. Suppose that and there exists such that Then IVP (134) has at least one solution if where

Proof. The proof immediately follows from Theorem 15.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Bashir Ahmad is partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. This work was supported by the Natural Science Foundation of Guangdong province (no. S2011010001900) and the Guangdong Higher Education Foundation for High-Level Talents.

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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.

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