1. Introduction

In this paper we studied some new classes of impulsive multi-integro fractional order and multi-fractional order with Caputo and Riemann level derivative respectively and nonlocal conditions and natural infinite delay.

We consider the new classes of impulsive multi fractional order differential nonlinear equations with nonlocal conditions and natural infinite delay with cosin semigroup generated by unbounded linear operator and uniformly bounded as well as its derivative and presented some properties of the Riemann-Leovel derivatives and integration that need in proving of locally existence and a new definition of function. The necessary and sufficient conditions of existence have been presented and given in details. The uniqueness solution of this classes have been presented without delay with some necessary conditions. The results in [6], [15], [21], [20], was developed with this classes in system (3.1) with multi-fractional derivatives which defined on , all above result have been given the mild solution in piecewise continuous space .

The existence conditions for the fractional order impulsive integro-differential non-quasilinear equations with neutral infinite delay and nonlocal integral condition have been considered and developed of the works in [11], [8], [12], [5], [10], [13], [3], [4], [18], [17], [2]. The class of equations considered in the paper constitutes an important subclass of control systems since such models appear as a natural description of several real processes, and extend many classical initial value problems. The paper studies the existence-uniqueness of the mild solution, which constitutes a crucial step in the study of any control problem.

2. Preliminaries

Let X and U be a pair of real Banach spaces, with norms and , respectively. Which is use it in next section consider the existance of the fractional impulsive mixed-type integro-differential partial functional equation with neutral infinite delay and nonlocal conditions.

Definition (1.2), [18]:

The piecewise continuous space which defined on a Banach space X is denoted by is continuous at left continuous at and the right-hand limit exists for .

Note that the equipped with the norm

is a Banach space.

Definition (2.2) "Compact Operator", [25]:

Let X and Y be normed spaces, the operator is called a compact if:

1. A is continuous, and

2. A transforms bounded subset M of X into relatively compact subset in is compact).

Definitions (2.3), [24]:

A subset U of is said to be equicontinuous if for each there is a such that:

and , imply .

Lemma (2.1), "Completely Continuous Operator", [19]:

Let X be a normed space, the mapping is called completely continuous if it is compact.

Remark (2.1), [9]:

Compact operator on Banach space are always completely continuous.

Definition (2.4), [16]:

Let X be a metric space equipped with a distance d. A map is said to be Lipschitz continuous if there is such that

The smallest for which the inequality holds is the Lipschitz constant of f. If f is said to be nonexpansive, if f is said to be a contraction.

Theorem (2.1) "Arezola-Ascoli's theorem", [19]:

Let satisfy:

i. For any is relatively compact on X.

ii. is equicontinuous on that is, for any and any there exists such that:

for any satisfying and all .

Then is a relatively compact.

Theorem (2.2) "Schauder's Fixed-Point theorem", [19]:

Let be a nonempty bounded convex closed subset in if is continuous and is relatively compact, then has at least one fixed point.

Theorem (2.3) "Leray-Schauder Principle", [25]:

Suppose and any solution of satisfies the priori bound for then T has a fixed point.

Definition (2.5) "Strongly Continuous Semigroup", [7]:

A semigroup , of bounded linear operators on a Banach space X is a -semigroup of bounded linear operators if:

, for every .

Definition(2.6), [14]:

If is a strongly continuous cosine family in X,

i. associated to the given strongly continuous cosine family, is defined by

ii. The infinitesimal generator of a cosine family is defined by

where

Remark (2.2), [22]:

Let cosine family of bounded linear operators. Then

is once continuously differentiable in .

Lemma (2.2), [22]:

Let A be a generator of strongly continuous cosine family of bounded linear operators. Then

Definition (2.7) "mild solution", [23]:

Let A be infinitesimal generator of a C_o-semigroup T(t). Let The function given by:

is the mild solution of the above non homgenous initial value problem.

Definition (2.8), [1]:

The Riemann-Liouvill fractional derivative of order for a function is defined by

, provided the right hand side is pointwise defined on .

Definition (2.9), [1]: The Riemann-Liouvill fractional integral of order for a function is defined as provided the right hand side is pointwise defined on .

Lemma (2.3), [20]:

If is a linear operator such that then, for we have

Lemma (2.4), [20]:

If is a continuous function such that and is continuous, then, for we have

Corollary (2.1), [20]:

For the sine family associated with the cosine family and we have

Hypotheses (2.1):

The following hypotheses of constriction needed in description of piecewise continuous space .

1. is a continuous function satisfy The Banach space induced by the function is defined as follows for any is a bounded and measurable function on and endowed with the norm

2. Let and there exist and with , where is the restriction of to , Denote by a seminorm in space as follows where

with norm .

3. Existence of a Multi-Fractional Order Impulsive Nonlinear Control System Via Cosine Semigroup

Consider the following multi-fractional order impulsive control system with nonlocal initial value conditions.

(3.1)

The operator A is a linear operator with where defined in hypotheses (2.1). is a nonlinear function from , the control function u(.) lies in a Banach space of admissible control functions. B is a bounded linear from a Banach space U into X, for all

which defined as

We shall make the following hypotheses:

1. is the infinitesimal generator of a strongly continuous cosine family of bounded linear operator in the Banach space . The associated sine family is defined in definition (2.6).

2. and on depend on .

To define and find the mild solution of problem (3.1), let by condition (1), C(t), is a strongly cosine semigroup generated by the linear operator A, and let x(t) be the solution of (3.1). Then by Remark (2.2) we have C(t)x is differentiable that implies -valued function

is differentiable for and:

Take the integration from 0 to t of both sides and shchas the technique that are using to find the mild in theorem (3.4), we used in the following:

Hence,

Therefore,

(3.2)

Definition (3.10):

A function is called a mild solution of the problem (3.1) if , the impulsive conditions , is verified, the restriction of to the interval is continuous and the following integral equation holds for .

(3.3)

To investigate the existence of the mild solution to the system (3.1), we assume the following conditions:

3.

i. is uniformly continuous for a.e For every the function is strongly measurable.

ii. There exists a nonnegative continuous function and a continuous nondecreasing positive function such that:

iii. For each the set is relatively compact in

4.

i.

ii. There exist a nonnegative continuous function and continuous nondecreasing positive function such that , for .

iii. The family of functions is equicontinuous on

iv. For each the set is relatively compact in

5.

6.

7. The linear operator B satisfies the following , , for positive constant .

8. Let

Remark (3.3):

From conditions (3) (iii) and (4) (ii) above, we have the following:

(a) , where is positive constant.

(b) , where is positive constant.

Remark (3.4):

We need the following inequalities for complete the prove of existence theorem. , thus

We define as follows:

(3.4)

Definition (3.11):

Let and for , we define function by

Theorem (3.4):

Assume the Hypotheses (1-7) and

(3.5)

are hold. Then for every , the Multi-fractional order impulsive nonlocal initial value control problem (3.1) has a mild solution for every control

Proof:

From definition(3.10) and for the maps define as:

(3.6)

(3.7)

By using lemma (2.4), and corollary (2.1), we obtain and , and from lemma(2.3), we get .

To satisfies theorem (2.2) for finding a fixed point need to do the following:

(3.8)

From (3.6) and assumption (1-7) above with remark (3.3) and remark (3.4),

thus

Also, from equation (3.7), we obtain:

From condition (8)(i), we have that

(3.9)

From condition (8), (ii), (iii), we get

(3.10)

Now if we put,

Then and

, thus

, therefore

, By taking the integral in both sides we get:

let ,

(3.11)

From (3.5), we get:

Hence is bounded and then is bounded.

Thus the set of solution of (3.8) is bounded in . To prove that are completely continuous, from the mild solution (3.2), we get

Since and are bounded on [0,T] and closed, then are compact.

and also are bounded, then the multiplication of bounded and compact set is compact, hence compact on Banach Space is completely continuous (see lemma(2.1)). Now:

By (1) and (4), for and , there exists such that:

for and where

This together with (3), (4) and the fact that satisfies Lipschitz condition, imply that:

Thus is equicontinuous on

By is continuous from to and from (4) (iv) that is relative compact in .

The set .

Since is continuous and defined on relative compact set then the above set is relative compact in

By remark (2.1), we get is completely continuous. Let thus

and select the partition of for

For then for

Since is bounded from (3), (iii) and

also is uniformly Lipschitz on we obtain

where

Therefore is relative compact in

By theorem (2.1), we have that is relatively compact in then is completely continuous (see lemma (2.1)). Similarly we can prove that is completely continuous. Hence has a fixed point in .

3.1. Uniqueness of the Multi-Fractional Order impulsive Differential Nonlinear control System via Cosine Semigroup

In this part, the Uniqueness of the solution to the system (3.1) without Neutral infinite delay has been developed on. Via cosine semigroup operators.

Theorem (3.5):

Assume the hypotheses (1-3), (4)(i), (5)(i), (7) and

i. and

ii. satisfies Lipschitz condition such that

iii. .

Then for every , the Multi-fractional order impulsive nonlocal initial value control problem(3.1) without Neutral infinite delay has a unique mild solution for every control that is

Proof: Let be two local mild solution of the nonlocal initial value impulsive control problem by equation (3.1) on the interval we must prove: , Assume that

Set . Now,

by (3.5), hence .