Uniqueness Solution Of Abstract Fractional Order Nonlinear Dynamical Control Problems

The aim of this paper is to investigate the Uniqueness solution of Abstract Cauchy Problem represented for fractional order nonlinear dynamical control system involving certain control input and their approach of investigated depended on commutative composite semigroup and some certain conditions in certain space.


Introduction
The semilinear and nonlinear equations appearing in variety of theories and applications ,in particular in the theory of fractional ordinary and fractional partial differential equations as well as integral equations with different types of derivatives have recently been addressed by several researchers for different problems and provided excellent tool for the description of memory and hereditary properties of various materials and processes. In [12], [14], [15], [17], [19], [20], the authors had been studied some classes of nonlinear and semiliner equation without ordinary or fractional derivatives with projectively compact and which among others contains completely continuous , quasi compact and monotone operators with general fixed point theorems as well as the nonlinear and semilinear equation studied with closed linear operator in Hilbert space, self adjoint operator also some time with perturbed operator that has densely defined domain in Banach space , moreover studied with monotonicity and compactness of the linear operator on reflexive Banach space ,the strongly positive operator and maximal monotonicity linear operator with nonlinear functions presented with existence and uniqueness approach. In [7], [1], [11], [21], [16], [5], [6], [13], [2], the authors had been studied the solvability of fractional order nonlinear and semilinear control differential equations by using fractional integral formulation with properties of calculus of fractional derivative and integration and the existence and uniqueness obtained by using classical fixed point theorems with initial values as well as boundary values and integral boundary condition also some of them involving nonlocal initial condition Our intersect in this paper to study the fractional order nonlinear dynamical feedback control system involve sum of N-unbounded operators with feedback perturbation as a generators of N-semigroup with new definitions depended on no expansive prosperity , maximal accretive, maximal monotone, resolvent set , fractional derivative and fixed point theorem also presented some results for solvability without using fractional calculus and equivalent integral formulation . main interest on nonlinear functional analysis and some new properties defined on special space, ]}, T>0. Also appear the role of feedback control operator as a perturbation for the generators still a challenge for many researchers up to our knowledge. Our aim establish necessary and sufficient conditions on sum of nonlinearity operator interacts suitably their system:

Preliminaries
Some necessary mathematical concepts for semigroup theory as well as some non-linear fractional calculus concepts have been presented.
Definition (2.1), [18]: The family of bounded linear operators ( ), 0 T t t    defined on the Banach space X is a semigroup ifq   The Riemann-Liouville fractional integral of order  for a function g is defined as   provided the right hand side is pointwise defined on (0, )  .

Lemma(2.11),[22]:
Let an operator ( ) on the real Hilbert space H . the statements are equivalent:

A is monotone and R(I-A)=H
A is maximal accretive A is maximal monotone. Lemma(2.12), [22]: Let a linear operator ( ) on real Hilbert space H 1. A is the generator of a linear nonexpansive semigroup. 2.maximal accretive and ( ) ̅̅̅̅̅̅̅ Lemma(2.13), [4]: Let A be the generator of 0 C -semigroup of contraction (nonexpansive semigroup) on a Banach space X. A bounded linear operator B is a perturbation of A such that ( ) ( ) and i. Let F denoted the duality on Banach space to defined as Lemma(2.14), [22]: Let the mapping A.B: be maximal monotone on the real reflexive Banach space X,(where is the dual space of X) and let ( ) ( ) . Then the sum A+B:X is also maximal monotone. Lemma(2.15), [9 ]: Let f be a contraction on complete metric space X. Then f has a unique fixed point ̅ . Our problem investigated on the following space that which denoted by ,

2.Main Results:
, for n Then , then by lemma(2.15) we have that ∑ ( ) (∑ ( )) a maximal monotone, then by lemma(2.13) and definition(2.9), we get 1,2,..., i n  , respectively and ( ) satisfies the following condition for every Thus, the operators ( ) are generators of nonexpansive semigroup. Then from theorem (2.12) and lemma (2.13) we have the operators ( ) are maximal monotone for ) (∑ ( ) ) (9) is also a maximal monotone, then by lemma (3.1), we have that Hence, , are nonlinear operators satisfy the following Then by (11), we have that Consider the following semilinear of sum of N-perturbed unbounded operators equations discussed in the following equations. , are nonlinear operators satisfy the following ,, x y H  and some } for all x,y .
Then the following equation , for all ( ) (12) has an unique solution.

Definition (3.5):
Let X be a real separable Banach space a one-parameter family  

S t s S t s S t s S t S t S t S s S s S s
The generator ∑ ( )) of a semigroup of commutative composite perturbed semigroups   , on a real separable Banach space X, defined as the Limit is a domain of ∑ ( ) has a countable subset which is dense in X and defined as follows Proof:   (21) become The Equation (23) can be equivalently written as

S t h S t h S t h xdt e S t S t S t xdt h h
Therefore, ( ∑ ( )) ( ) is a contraction in . Then by theorem (2.16) the Equation (27) and consequently (23) has a unique solution.