Journal of Mathematics

Volume 2013 (2013), Article ID 147193, 9 pages

http://dx.doi.org/10.1155/2013/147193

Research Article

The Existence of Solutions for Impulsive Fractional Partial Neutral Differential Equations

Xiao-Bao Shu¹ and Fei Xu²

¹Department of Mathematics, Hunan University, Changsha 410082, China

²Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada, N2L 3C5

Received 16 August 2012; Revised 27 October 2012; Accepted 27 October 2012

Academic Editor: Beny Neta

Copyright © 2013 Xiao-Bao Shu and Fei Xu. 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.

Abstract.

This paper deals with the existence of mild solutions of a class of impulsive fractional partial neutral semilinear differential equations. A series of analytical results about the mild solutions are obtained by using fixed-point methods. Then, we present an example to further illustrate the applications of these results.

1. Introduction

Great progress in fractional differential equations has been achieved in recent years. Due to their broad applications in science and engineering such as physics, economics, biology, and mechanical engineering, fractional differential equations attract attention of researchers from different areas. Compared with ordinary differential equation systems and partial differential equation systems, fractional differential equation systems have the potential to model real-world problems with more accuracy. In order to further investigate these models, it is essential to study the fractional differential equations analytically. Though, mathematically, a fractional differential equation is closely related to its corresponding ordinary differential equation or partial differential equation, that is, the ordinary differential equation or the partial differential equation can be obtained by letting in its corresponding fractional differential equation, many mathematical methods which can be used in investigating ordinary differential equations or partial differential equations fail in analyzing fractional differential equations since fractional differential equations usually have more complicated structures and different properties. Thus, it is essential to develop novel methods to study fractional differential equations analytically. With the aim of analyzing fractional differential equations, extensive investigations had been carried out [1–15].

In applied mathematics, another two types of differential equations, impulsive differential equations and functional differential equations with different conditions have wide applications in modeling particular phenomena and dynamical processes in physics, automatics, robotics, biology, medicine, and so forth. For example, impulsive differential equations can be used to investigate natural phenomena or dynamical processes which are subject to great changes in a short period of time [16–18]. To further understand the mathematical models with impulsive differential equations or functional differential equations with different conditions, analytical investigation of these equations is required. During the last ten years, progress in studying these two types of differential equations has been made [19–25].

One of the important techniques for analyzing impulsive differential equations is the semigroup theory, which has been successfully used in investigating the existence, uniqueness, and continuous dependence of the solutions of impulsive differential equations. It also has wide applications in the study of periodic and almost periodic solutions of different kinds of differential equations. For example, Fan and Li studied the existence results for semilinear differential equations with nonlocal and impulsive conditions using semigroup theory [26]. The theory has also been used to investigate a mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces [27]. With the aid of the semigroup theory, a series of conclusions about the existence of the solutions of functional differential equations and functional integral equations have been obtained [19–23, 28–36].

Since the fractional derivative does not satisfy the Leibniz rule and the corresponding fractional operator equation does not satisfy the variation of constants formula [37], which implies that the corresponding operator does not satisfy the properties of semigroups (see [38]), does not satisfy the semigroup relations, that is, , indicating the properties of corresponding resolvent operator. It is difficult to define the mild solutions of fractional partial differential evolution equations. In particular, semigroup theory was used inappropriately to study the existence and uniqueness of mild solutions to impulsive fractional differential equations [13] and impulsive partial neutral functional differential equation [39]. The existence of mild solutions for a class of impulsive fractional partial semilinear differential equations was investigated in [15] with errors in [13] corrected.

A solution to this problem has been given in [15], in which the mild solutions of impulsive fractional differential equations are defined using piecewise functions, and a sufficient condition, which guarantees the existence and uniqueness of solutions to the equations, is obtained. The existence of solutions to a fractional neutral integrodifferential equation with unbounded delay was studied in [40].

In this paper, with the aim of investigating the existence theorem for the solutions of impulsive fractional neutral functional differential equations, we first define the mild solution of fractional neutral functional differential equation using Laplace transform. Then, by introducing the operator , the mild solution of impulsive fractional differential equation is defined. Through analyzing operator , , and the corresponding semigroup , a sufficient condition, which guarantees the existence and uniqueness of solutions to the following system, is obtained:

where , for , is the Caputo fractional derivative, is the infinitesimal generator of an analytic semigroup , and is a Banach space.

The rest of this paper is organized as follows. In Section 2, we present some notations, definitions, and theorems which will be used in the following sections. In Section 3, the definition of mild solution of the system (1) and the relation between analytic semigroup and some solution operators are given. The main results of our paper are given in Section 4. In Section 5, application of the obtained results is presented.

2. Preliminaries

In this section, we will introduce some notations, definitions, and theorems, which will be used throughout this paper. Let and be Banach spaces, and let be the Banach space of bounded linear operators from into equipped with its natural topology. When , we use the notation to denote the Banach space. We say that a function is a normalized piecewise continuous function on if is piecewise continuous, and left-continuous on . The space formed by the normalized piecewise continuous functions from to is denoted by , where the notation stands for the space formed by all functions such that is continuous at , and and exist for all . In addition, we use to denote the space endowed with the norm . Obviously, is a Banach space. Moreover, an axiomatic definition for the phase space is employed. Here, the definition of is similar to that given in [41]. In particular, is a linear space of functions mapping into endowed with a seminorm and the following axioms hold.

(I) If , such that and , then for every the following conditions hold:

(i) is in ;(ii);(iii),

where is a constant, , is continuous, is locally bounded, and are independent of .

(II)The space is complete.

Next, we give an example to illustrate the above definitions.

Example 1 (the phase space ). Let and be a nonnegative, locally Lebesgue integrable function. Assume that there is a non-negative measurable, locally bounded function on such that for all and , where is a set with Lebesgue measure zero. denotes the set of all functions such that and . In , we consider the seminorm defined by

The preceding conditions indicate that the space satisfies the axioms (I) and (II). Moreover, when , we can take , and for .

Definition 2. Let ,. A function is said to be in the space if there exist a real number and a function such that . Moreover, is said to be in the space for some positive integer if .

Definition 3. If the function ,  where , the fractional derivative of order of in the Caputo sense is defined as

Definition 4. Let be a closed linear operator. is said to be sectorial if there exist , , and such that the resolvent of exists outside the sector

(for short, we say that is sectorial of type ).

Let denote the infinitesimal generator of an analytic semigroup in a Banach space and , where is the resolvent set of . We define the fractional power by

For is a closed linear operator whose domain is dense in . For analytic semigroup , we have the following theorem.

Theorem 5 ([42, Theorem 6.13]). Let be the infinitesimal generator of an analytic semigroup . If  , then the following statements hold.

(a) for every and .(b)For every , we have .(c)For every , the operator is bounded and

(d) and , then

3. Mild Solutions

In this section, we investigate the classical solution of the following equation:

Based on the classical solution, the mild solution of system (1) is defined. Then, the relationship between the analytic semigroup and some solution operators is given.

Lemma 6. Let be the infinitesimal generator of an analytic semigroup . If and satisfy the uniform Hölder condition with exponent , then the solutions of the Cauchy problem (8) are fixed points of the operator equation

where

with being a suitable path such that for .

Proof. Applying the Laplace transform to (9), we obtain

It follows that

Noting that , and applying the inverse Laplace transform, we have

Since and satisfy a uniform Hölder condition, with exponent , then the classical solutions of Cauchy problem (8) are fixed points of the following operator equations (see [43]):

In what follows, we use the notations , and .

Definition 7. A function is called a mild solution of (1) if it satisfies the operator equation

Remark 8. It is easy to verify that a classical solution of (1) is a mild solution of the same system. Thus, Definition 7 is well defined (see [42, 43]).

Lemma 9. If is the infinitesimal generator of an analytic semigroup and , then we have

where is the probability density function defined on such that its Laplace transform is given by

and satisfies

Proof. For all , we have

Let

where (see [44]). Thus, we obtain

where is the probability density function defined on and satisfies (see [44]). Then, from (21), we get

On the other hand, for all , we notice that

Thus,

Remark 10. If , then from Lemma (9), we have

4. Main Results

In this section, we will present the main results of this paper. The mild solution will be understood in the sense of Definition 7. To investigate the uniqueness of the mild solution, we require the following assumptions.

  The functions and , are continuous and satisfy the following conditions

(i) For every such that and , the function is strongly measurable and the function belongs to .(ii) There exist a positive constant , and a function as well as , such that is -valued, is continuous and

  The function is defined by

(i)

when , or

(ii)

when . Here, and satisfies , for all .

Remark 11. Let be such that and , and assume that holds. We have the following estimations:

On the other hand, from (see [44]), we have

Then, by (29) and (30), it is easy to see that

It is obvious that the function is integrable on for every . Similarly, it can be verified that the function is integrable on for every .

Theorem 12. Under the assumptions and , the system (1) has a unique mild solution.

Proof. On the metric space endowed with the metric , we define the operator by

From Remark 11, we know that and are integrable on for every . Thus, is well defined with values in . Let . Using , we get

for all .

Remark 13. The above inequality makes use of

which can be obtained directly from (30).

Remark 14. If in the above inequality, we replace by .

 Therefore, it can be deduced that (i) when ,

 (ii) when ,

Thus,

Therefore, in view of the contraction mapping principle, assumption implies that has a unique fixed point on . Thus, the system (1) has a unique mild solution.

5. An Example

In this section, we present an example to further illustrate the applications of the results given in Section 4.

Let be the space of the square integrable functions and . We investigate the operator defined by , where

We note that is the infinitesimal generator of an analytic compact semigroup on . In particular, we have that is a uniformly stable semigroup with for all . Moreover,

It follows from the above properties that analytic semigroup satisfies

Consider the impulsive fractional partial neutral differential equation system:

where is a strictly increasing sequence of positive real numbers. To treat this system, we assume that the following conditions hold.

The function is continuous and there is a continuous integrable function such that

The function is integrable, , and

The function is continuous and there exists a positive constant such that

We now define the functions and by

respectively. It is easy to see that problem (41) can be modeled as the abstract impulsive Cauchy problem (1).

Theorem 15. Assume that the conditions , and hold and

If

for all , then there is a mild solution of the impulsive system (41).

Proof. It follows from that is -valued and is continuous. Moreover, is a bounded linear operator, and as such for all . By , one can easily conclude that the function is continuous for all , and is a strongly measurable function for each . In addition, for all and , we have

It follows from condition that

for all . This implies that . The proof is complete since the existence of a mild solution is a consequence of Theorem 15.

Acknowledgments

This research was partially supported by the NNSF of China (11271115), the Postdoctoral Science Foundation of Hunan province (2006FJ4249), and the Doctoral Fund of Ministry of Education of China (200805321017).

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