The Existence of Solutions for Impulsive Fractional Partial Neutral Differential Equations

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.


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.ough, 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  = 1, 2, … 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.us, 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][2][3][4][5][6][7][8][9][10][11][12][13][14][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][17][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][20][21][22][23][24][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].e 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][20][21][22][23][28][29][30][31][32][33][34][35][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]),   () ≜  −1   (  ) does not satisfy the semigroup relations, that is,   ( + ) ≠   ()  (), indicating the properties of corresponding resolvent operator.It is difficult to de�ne 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].e 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 de�ned using piecewise functions, and a sufficient condition, which guarantees the existence and uniqueness of solutions to the equations, is obtained.e 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 �rst de�ne the mild solution of fractional neutral functional differential equation using Laplace transform.en, by introducing the operator   (), the mild solution of impulsive fractional differential equation is de�ned.rough 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 0 <   1, is the Caputo fractional derivative, − is the in�nitesimal generator of an analytic semigroup {()} 0 , and (   ) is a Banach space.e rest of this paper is organized as follows.In Section 2, we present some notations, de�nitions, and theorems which will be used in the following sections.In Section 3, the de�nition of mild solution of the system (1) and the relation between analytic semigroup () and some solution operators are given.e main results of our paper are given in Section 4. In Section 5, application of the obtained results is presented.

Preliminaries
In this section, we will introduce some notations, de�nitions, and theorems, which will be used throughout this paper.Let (     ) and (     ) be Banach spaces, and let ℒ( ) be the Banach space of bounded linear operators from  into  equipped with its natural topology.When  = , we use the notation ℒ() to denote the Banach space.We say that a function Moreover, an axiomatic de�nition for the phase space  is employed.�ere, the de�nition of  is similar to that given in [41].In particular,  is a linear space of functions mapping (−∞ 0] into  endowed with a seminorm     and the following axioms hold.where is continuous,  is locally bounded, and    are independent of ().
(II) e space  is complete.
�ext, we give an example to illustrate the above de�nitions.
is said to be sectorial if there exist 0 <  < /2,  > 0, and    such that the resolvent of  exists outside the sector (for short, we say that  is sectorial of type (  )).

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 de�ned.en, the relationship between the analytic semigroup () and some solution operators is given.Lemma 6.Let − be the in�nitesimal generator of an analytic semigroup {()} 0 .If  and  satisfy the uniform Hölder condition with exponent   (0 1], then the solutions of the Cauchy problem (8) are �xed points of the operator e�uation where with  being a suitable path such that     +   for   .
Proof.Applying the Laplace transform to (9), we obtain It follows that Noting that   (  + ) −1 =  − (  + ) −1 , and applying the inverse Laplace transform, we have Since  and  satisfy a uniform Hölder condition, with exponent   (0 1), then the classical solutions of Cauchy problem (8) are �xed points of the following operator equations (see [43]): Remark 8.It is easy to verify that a classical solution of ( 1) is a mild solution of the same system.us, �e�nition 7 is well de�ned (see [42,43]).

Main Results
In this section, we will present the main results of this paper.e mild solution will be understood in the sense of �e�nition 7. To investigate the uniqueness of the mild solution, we require the following assumptions.
Remark 11.Let  ∶ (−∞ ]   be such that  0 =  and |  ∈ , and assume that  1 holds.We have the following estimations: On the other hand, from [44]), we have en, by ( 29) and (30), it is easy to see that It is obvious that the function     ( − )(   ) is integrable on [0 ) for every  > 0. �imilarly, it can be veri�ed that the function     ( − )(   ) is integrable on [0 ) for every  > 0.

Journal of Mathematics 7
Remark 13. e above inequality makes use of which can be obtained directly from (30).erefore, it can be deduced that (ii) when us, erefore, in view of the contraction mapping principle, assumption ( 2 ) implies that Γ has a unique �xed point on ℬ.us, the system (1) has a unique mild solution.

An Example
In this section, we present an example to further illustrate the applications of the results given in Section 4.
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.
() is a normalized piecewise continuous function on [ ] if  is piecewise continuous, and lecontinuous on ( ].e space formed by the normalized piecewise continuous functions from [ ] to  is denoted by ([ ]; ), where the notation  stands for the space formed by all functions   ([0 ]; ) such that () is continuous at  ≠   , and ( −  ) = (  ) and ( +  ) exist for all  = 1 …  .In addition, we use (     ) to denote the space  endowed with the norm   = sup  ().Obviously, (     ) is a Banach space.