Solvability of Impulsive Nonlinear Partial Differential Equations with Nonlocal Conditions

The aim of this paper is to investigate the theoretical approach for solvability of impulsive abstract Cauchy problem for impulsive nonlinear fractional order partial differential equations with nonlocal conditions, where the nonlinear extensible beam equation is a particular application case of this problem.


Introduction
The fractional differential equations are models of many applications, such as medicine, engineering, physics and other sciences. They are becoming more important tools among researchers and have been attracting many authors in recent years [1,2,3,4,5,6,7,8,9,10]. Furthermore, fractional impulsive differential equations played a main role in the modeling phenomena, for example in describing population dynamics which are subject to abrupt changes, as well as other phenomena such as diseases, harvesting, and so forth. From this reason, there are many researchers who discussed the existence of a mild solution of impulsive fractional differential equations [11,12,13,14,15,16,17]. There are some authors who also studied impulsive fractional differential equations with delay [18,19,20]. However, there are few authors who discussed the existence of mild solutions of the impulsive fractional integro-differential equations of order . In this work, we will study the existence of mild solutions for considered an abstract Cauchy problem, which is entitled the impulsive nonlinear fractional order partial differential equations with nonlocal conditions, as follows: where ̂ . 0‖ ⁄ ‖ 1/ is a linear operator in the Banach space , , are the Caputo fractional derivative and fractional integration of , respectively, ( ) belongs to Banach space X, and ( )| is defined by ( ) ( ) representing the right and left limits of at respectively, ( ) ( ) representing the right and left limits of at respectively, and is a given function. Let ( ) denoted by ( ) ( ) ( -Many scientific problem of the impulsive nonlinear fractional order partial differential equations with nonlocal conditions can be expressed as in the above systems (1.1) and (1.2). Some of them do not have easy solvability. They are even difficult to study sometimes. Also their behaviors for their solutions is not appearing in general and, thus, we need more effort and practices. Therefore, these problems have taken our interest. The aim of this article is to study and present all the results of the solvability for the proposal class impulsive nonlinear fractional order partial differential equations with nonlocal conditions. Some of special types of cases of the proposed problem have been approached, using nonlinear functional analysis theorems and abstract Cauchy problem involving semigroup operators with Krasnoselskii's fixed point theorem, to show the solvability with some recent sufficient and necessary conditions.

Preliminaries
In this section, we present some assumptions, notation and results needed in our proofs later.

Assumptions (2.1)
The following Assumptions of are needed in the description of piecewise continuous space (( -).

1), [21]
The fractional integral of order α with the lower limit 0 for a function h is defined as: where ( ) ∫ , (gamma function).

Definition (2.2), [21]
The Caputo fractional derivative of order α with lower limit 0 for a function can be written as: , .

Definition (2.3), [21]
The Laplace transform, of the Caputo fractional derivative of order α given as:

Lemma (2.2), [10]
Let be a closed convex subset of a Banach space . Let and be two operators which satisfy 1. whenever ; 2.
is continuous and compact; 3. is a contraction. Then there exists such that .

Lemma (2.4)
If ̂ is a densely defined sectorial operator of type ( ) , then ̂ is the infinitesimal generator of a -resolvent family

Proof
Since ̂ is sectorial operator then, form definition (2.4), we get ̂ is closed linear operator and satisfies (i) For some , (ii) There exists a constant such that ‖( ) ‖ | | , .

Lemma (2.5), [10]
If the sectorial operator ̂ is of type ( ) in definition (2.4), then we have the following: (i) If and ( ), then we have that

Main results
To investigate the existence of the mild solution of the impulsive abstract Cauchy problem (1.2), we assume the following conditions: (A 1 ) The semigroups ( ), ( ), ( ) and ( ) generated by the operator ̂ are compact in ( ̂ ) ̅̅̅̅̅̅̅̅ when and (A 2 ) The and there exist constants , ̅ and there exist constants and , satisfying

Lemma (3.6)
If the sectorial operator ̂ of type ( ) and is a map that satisfies (A 3 ), then the impulsive abstract Cauchy problem (1.2) is equivalent to the integral equation given by the following: Such that for . Proof By integrating the both sides of impulsive abstract Cauchy problem (1.2) of order , we get the equivalent equation Now, by taking a Laplace transform of both sides of (3.3), we have We can rewrite the equation (3.5) as follows: ∫ ∫ ( ) ( ( ) ) / ( ̂ ) ̅ ( ( )) (3.8) By (3.6), (3.7) and (3.8), we get that ( -Now, we define the mild solution of impulsive abstract Cauchy problem (1.2) for every .

Definition (3.6):
Let ( ) ( ) ( ) and for . To introduce the following main theorem, we define the function as follows:

Proof
For , if and , then we have Also, if , then we get From Assumption (A 1 ), the compactness of ( ), ( ), ( ) and ( ) for satisfy the continuity in the uniform operator topology. Then the right-hand sides of (3.10) and (3.11) tend to zero as . Therefore, the family *( ̅ )( ) + satisfies the equicontinuous functions. Since the proof of equicontinuities to the case and is simple, then the proof is omitted.

Theorem (3.1)
If the assumptions (A 1 )-(A 7 ) hold, then the impulsive abstract Cauchy problem (1.2) has at least one mild solution that belongs to . Proof It suffices to prove that the operator defined as follows has a fixed point ( ) ( -We consider the operators and ̅ on defined as follows: ) .
Then the set *( ̅ )( ) + is uniformly bounded. Therefore, from lemma (3.7), we have that the ̅ ( ) is equicontinuous, and from lemma (3.8), we have that the ( ) *( ̅ )( ) + is relatively compact for any * +. Then by the Arzela-Ascoli theorem, we get that the closure of *( ̅ )( ) + is compact . Next, we shall prove that ̅ is continuous: Let * + be a sequence in and for . From assumptions (A 2 )-(A 3 ), we get and are continuous, i.e. for all there is a positive integer such that for ‖ . ( )( ) Hence , ̅ is continuous. By the analysis above, we can see that ̅ implies the condition (2) of lemma (2.2) which means that ̅ is completely continuous. For ending the proof, we will show that is a contraction. Let Then is contraction. Therefore, the three conditions of lemma (2.2) are satisfied. Therefore, for this the operator ̅ has a fixed point in . Then the impulsive abstract Cauchy problem (1.2) has a mild solution on J.

Conclusions:
We conclude that the solvability of impulsive nonlinear fractional order partial differential equations with nonlocal conditions needed to define an approach to use the fractional Laplace transforms which make important roles for computing the formula of the semigroup family operators. Hence, we thought that it is an important and basic issue for solvability. The assumptions presented in this work are needed to prove that our problem, with the specific main results, was never assumed before for solving other problems. We used this approach to simulate the theoretical approach and gain solvability results of impulsive nonlinear fractional order partial differential equations with nonlocal conditions by transforming it to an impulsive abstract Cauchy problem. We also used the nonlinearity functional analysis as a suitable analytic tool for specific spaces and domains of operators to generalize the problem of nonlinear extensible beam equations and other problems.