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

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.


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][9][10].
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.
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 0, 1 , 2 , . . . , (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 0, 1 , 2 , . . . , (impulse points).
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].
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.

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 > 0 of a continuous function : (0, ∞) → is given by provided that the right-hand side exists.
Proof. For ∈ and > 0, we have Since is a Caratheodory function and { } is a Caratheodory function sequence, therefore, there exist > 0 and > 0 such that Let us assume that satisfies (48). Then, by Lemma 3, the solution of (48) can be written as Observe that The Scientific World Journal This implies that Thus, we have Hence, satisfies (49). Next, we show that ∈ . Indeed It is easy to see that Furthermore, for ∈ ( , +1 ], 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 > max{0, + } and > max{ , − − } and define 6 The Scientific World Journal For ∈ , 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 0 =: is a solution of the problem * ( ) = ( ) ( , ( ) , * ( )) , ∈ (0, ∞) , if and only if ∈ is a solution of the fractional integral equation Proof. For ∈ , we have that there exists > 0 such that Since is a Caratheodory function and { } is a Caratheodory function sequence, then there exist > 0 and > 0 such that Assume that satisfies the problem (50). Then, in view of Lemma 3, we can write the solution of (50) as The Scientific World Journal 7 From (0) = 0 , we get 0 = 0 . Since and Δ ( ) = ( , ( )), we get which gives Hence the solution of the problem (50) is Next, we need to show that ∈ . Clearly, Furthermore, for ∈ ( , +1 ], we have The Scientific World Journal Since − −1 ≥ 0 and 0 = 0, we get ≥ 0 for all = 0, 1, 2, . . .. Then Moreover, for ∈ ( , +1 ], 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.

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 ∈ ( , +1 ] , = 0, 1, 2, . . . . Since is a Caratheodory function, { } is Caratheodory function sequence; there exist positive numbers > 0 and > 0 ( = 1, 2, . . .) such that It is easy to show that As in the proof of Lemma 8, it can be shown that both 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 → 0 as → ∞. We will prove that → 0 as → ∞. It is easy to see that there exists > 0 such that sup Since is a Caratheodory function and { } is a Caratheodory function sequence, then there exist > 0 and > 0 such that Notice that From the inequality it follows that there exists > 0 for > 0 such that ∈ (0, ∞) , > 1 , 10 The Scientific World Journal Hence, therefore, we can find > 0 such that holds for all > , = 1, 2, . . .. As is a Caratheodory function, there exists 1 > 0 such that holds for all ∈ [0, ] and 1 , 2 , So, for ∈ [0, ], we have Consequently, for all > 2 , ∈ [0, ∞), we get In particular, for ∈ ( , +1 ], we find that Thus, it follows that Similarly, it can be shown that sup ∈(0,∞) From (69) and (70), we conclude that lim → ∞ = 0 . This implies that is continuous.
Our next task is to show that is equiconvergent as Hence, is equiconvergent as → ∞. From the above steps, it follows that is completely continuous. This completes the proof.

15
In the sequel, we need the following assumption: ( 1 ) is a Caratheodory function such that where 0 < 1 < 2 < ⋅ ⋅ ⋅ < and , ( = 1, 2, . . . , ), ≥ 0 are real numbers; Furthermore, we set Theorem 11. Suppose that ( 1 ) and ( 2 ) 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 ( 1 ) and ( 2 ) that is well defined and is completely continuous. Thus, we seek solutions of IVP (6) by finding fixed points of in .
It is easy to show that Ψ ∈ . For > 0, we define Ω = { ∈ : ‖ − Ψ‖ ≤ }. Then, for ∈ Ω , we have Using the assumptions ( 1 ) and ( 2 ), we find that The Scientific World Journal Thus, by (81), it follows that Next, we have the following cases.
Proof. By Theorem 11, IVP (6) has at least one solution. Let 1 and 2 be two different solutions of IVP (6). Then ‖ 1 − 2 ‖ > 0, 1 = 1 , and 2 = 2 . Employing the method used in the proof of Theorem 11, we find that On the other hand, by (51), we get which is a contradiction. Hence, IVP (6) has a unique solution ∈ if 0 < 1. This completes the proof.
Since is a Caratheodory function by ( 1 ), therefore, there exists > 0 such that So This completes the proof.

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 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 2. As in the proof of Lemma 10, it is easy to show that is bounded.
Step 5. is equiconvergent as → ∞. Notice that The Scientific World Journal 23 Hence, is equiconvergent as → ∞. This completes the proof in which is completely continuous. ( 1 ) and ( 2 ) hold. Then IVP (7) has at least one solution ∈ if

Theorem 15. Assume that
where 0 = max{ 2 , 3 }, 2 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 Φ ∈ . Let > 0 and define For ∈ Ω , we have ‖ − Φ‖ ≤ . Then Using the assumptions ( 1 ) and ( 2 ), we find that The Scientific World Journal Furthermore, we have The Scientific World Journal Thus, it follows that Now we discuss the cases for different values of . (i) For < 1, we can choose 0 > 0 sufficiently large so that [ 0 + ‖Φ‖] 0 < 0 . Let Ω 0 = { ∈ : ‖ ‖ < 0 }. It is easy to show that Ω 0 ⊂ Ω 0 . Then, the Schauder fixed point theorem implies that the operator has a fixed point ∈ Ω 0 , which is a bounded solution of IVP (7). (ii) For = 1, we select Let Ω 0 = { ∈ : ‖ ‖ < 0 }. It can easily be shown that Ω 0 ⊂ Ω 0 . Then, the Schauder fixed point theorem applies and the operator has a fixed point ∈ Ω 0 , which is a bounded solution of IVP (7).
Proof. The proof is similar to that of Theorem 12, so we omit it. In [33], the authors presented the following logistic model with fractional order:  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: