Impulsive Multiorders Riemann-Liouville Fractional Differential Equations

Abstract

Impulsive multiorders fractional differential equations are studied. Existence and uniqueness results are obtained for first- and second-order impulsive initial value problems by using Banach’s fixed point theorem in an appropriate weighted space. Examples illustrating the main results are presented.

1. Introduction

Fractional calculus has become very useful over the last years because of its many applications in almost all applied sciences. By now, almost all fields of research in science and engineering use fractional calculus to better describe them.

Fractional differential equations have been of great interest and are caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various science such as physics, mechanics, chemistry, and engineering. For details and some recent results on the subject we refer to the papers [1–3], books [4–7], and references cited therein.

Recently in [8], Wang et al. studied existence and uniqueness results for the following impulsive multipoint fractional integral boundary value problem involving multiorders fractional derivatives and deviating argument: where is the Caputo fractional derivative of order , is fractional Riemann-Liouville integral of order ,  ,  , , , , , and where , and , denote the right and left hand limits of and at .

We notice that there are some discussions on the concept of solution for impulsive fractional differential equations for both Riemann-Liouville and Caputo fractional derivatives. We refer the interested reader to some recent papers [9–11] and the references cited therein. However, we can point out the problems caused by using the definition of Caputo fractional derivatives of order with the lower limit for a function as If there are impulse points such that for some , then the does not exist, which leads to non-integrability of the right-hand side of (2). The key idea for solving this problem is to apply the definition of fractional derivative only on an interval and combining all intervals by using impulsive conditions.

In this paper, we study impulsive multiorders Riemann-Liouville fractional differential equations. More precisely, in Section 3 we study the existence and uniqueness of solutions for the following initial value problem for impulsive multiorders Riemann-Liouville fractional differential equations of order of the form where is the Riemann-Liouville fractional derivative of order on intervals , , , is a continuous function, and . The notation is defined by where is the Riemann-Liouville fractional integral of order 1  −   on interval . It should be noticed that if in (4), then = = for .

In Section 4, we investigate the initial value problem of impulsive Riemann-Liouville fractional differential equations of the form where , , is a continuous function, for , and is the Riemann-Liouville fractional derivative of order on intervals for . The notation is defined by (4) and is defined by where is the Riemann-Liouville fractional integral of order on . It should be noticed that if in (6), then and for .

By using Banach’s fixed point theorem we prove existence and uniqueness results for the problem (3) and (5) in an appropriate weighted space.

The paper is organized as follows: Section 2 contains some preliminary notations, definitions, and lemmas that we need in the sequel. In Section 3 we present the main results for problem (3), while in Section 4 we present the main results for problem (5). Examples illustrating the obtained results are also presented.

2. Preliminaries

In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later.

Definition 1. The Riemann-Liouville fractional derivative of order of a continuous function is defined by where , denotes the integer part of a real number , provided the right-hand side is pointwise defined on , where is the gamma function defined by

Definition 2. For at least -times differentiable function , the Caputo derivative of fractional order is defined as where .

Definition 3. The Riemann-Liouville fractional integral of order of a continuous function is defined by provided the right-hand side is pointwise defined on

Lemma 4 (see [5]). Let and . Then the fractional differential equation has a unique solution where , , and

Lemma 5 (see [5]). Let . Then for it holds where

3. Impulsive Riemann-Liouville Fractional Differential Equations of Orders

Let , , for . Let , is continuous everywhere except for some at which and exist, and , . For , we introduce the space with the norm and for each and , with the norm Clearly is a Banach space.

In this section we study problem (3).

Lemma 6. If is a solution of (3), then, for any , , with . The converse is also true.

Proof. For , taking the Riemann-Liouville fractional integral of order to the first equation of (3) and using Lemma 5, we have where . The initial condition implies . Then for , we get Applying the Riemann-Liouville fractional integral of order from to , we get For , taking the Riemann-Liouville fractional integral of order to the first equation of (3) and using Lemma 5, we have Since , it follows for that Applying the Riemann-Liouville fractional integral of order to the above equation and substituting , one has For , using the Riemann-Liouville fractional integral of order for (3), we have Repeating the above procession for each , we obtain (13).
On the other hand, assume that is a solution of (3). Applying the Riemann-Liouville fractional derivative of order for (13) on , , and using , it follows that It is easy to verify that , , and . The proof is complete.

Next we will prove that problem (3) has a unique solution by using Banach’s fixed point theorem.

Theorem 7. Assume that
   is a continuous function and satisfies   , , are continuous functions and satisfy If where , , and for , then the initial value problem (3) has a unique solution on .

Proof. In view of Lemma 6, we define the operator as In addition, we define a ball , . To show that , we suppose and then As , we get for each . Therefore, we have . Next we will show that . Suppose that , . Setting we choose such that For any and for each , we have Multiplying both sides of the above inequality by for each , we obtain This implies that .
Finally we will show that is a contraction mapping on . For and for each we have Multiplying both sides of the above inequality by for each , we have It follows that Since , is a contraction mapping on . Therefore (3) has a unique solution on .

Example 8. Consider the following impulsive multiorders Riemann-Liouville fractional initial value problem: Here , , , , , and . Since and , for , then and are satisfied with , .
By choosing , we find that , , and Hence, by Theorem 7, the initial value problem (34) has a unique solution on .

4. Impulsive Riemann-Liouville Fractional Differential Equations of Orders

Problem (5) is studied in this section.

Lemma 9. The unique solution of problem (5) is given by for , with for  .

Proof. For , taking the Riemann-Liouville fractional integral of order for the first equation of (5) and applying Lemma 5, we obtain where and . The initial condition implies which leads to Using the Riemann-Liouville fractional derivative of order for (38) on , we get From the second initial condition of (5), we get Taking the Riemann-Liouville fractional integral of order and for (40) and substituting , we have For , taking the Riemann-Liouville fractional integral of order for (5), we have Since and , it follows that, for , The Riemann-Liouville integrating of the above equation of order and for leads to For , applying the Riemann-Liouville fractional integral of order for (5) and substituting values and , we get Repeating the above process, for , we obtain (36) as requested.

Next, we will prove the existence and uniqueness of a solution to the initial value problem (5) by using Banach’s fixed point theorem.

Theorem 10. Assume that and hold. In addition we suppose that    , , are continuous functions and satisfy  If  where  , , and , for , then problem (5) has a unique solution on .

Proof. We define the operator as follows: for , with for .
It is straightforward to show that ; see Theorem 7. Next we will show that , where a ball is defined by . Assume that , and . Setting we choose a constant such that Let For any , , we have Multiplying both sides of the above inequality by for , we have This implies that .
Finally we will show that is a contraction mapping on . For and for each we have Multiplying both sides of the above inequality by , we have It follows that Since , is a contraction mapping on . Therefore (5) has a unique solution on .

Example 11. Consider the following impulsive multiorders Riemann-Liouville fractional initial value problem: Here , , , , , , and . Since , , and , then , , and are satisfied with , , and , respectively. By choosing , we can find that , , , and Hence, by Theorem 10, the initial value problem (57) has a unique solution on .

Conflict of Interests

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

Acknowledgment

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.