Fixed Point Theory for Study the Controllability of Boundary Control Problems in Reflexive Banach Spaces

In this paper, we extend the work of our proplem in uniformly convex Banach spaces using Kirk fixed point theorem. Thus the existence and sufficient conditions for the controllability to general formulation of nonlinear boundary control problems in reflexive Banach spaces are introduced. The results are obtained by using fixed point theorem that deals with nonexpanisive mapping defined on a set has normal structure and strongly continuous semigroup theory. An application is given to illustrate the importance of the results.


Introduction
Many engineering and scientific systems in the control theory in infinite dimensional spaces can be formulated by partial differential equations, integral equations, or fractional differential equations. We can characterize these systems as differential equations by using semigroup theory,and then study the solution of these problems. Controllability is one of most significant properties of the control system, it means that the ability to transmit the system from an arbitrary initial

ISSN: 0067-2904
Bassem and Al-Jawary Iraqi Journal of Science, 2022, Vol. 63, No. 1, pp: 222-232 223 state to an arbitrary final state of a given set in a finite time by a convenient option of the control function, one can refer to the references [1][2][3]. In this paper, we introduce the sufficient conditions for controllability of the following boundary control problem in arbitrary reflexive Banach spaces (rBs). where ( ) takes values in (rBs) with norm ‖ ‖, the control function ( ) ( ) be a (rBs) of admissible control functions, with is a Banach space (Bs). Let a closed linear and densely defined operator, with the domain of B, ( ) , ‖ ‖ where is a positive constant, and be a linear operator such that ( ) and the range of , ( ) , where is a (Bs), be a linear continuous operator. The nonlinear operators , and are continuous from into and all of them satisfy Lipschitz condition on the second argument. Here be a linear operator generates a strongly continuous semigroup (semigroup) ( ), , on (rBs) Z and be a bounded linear operator with ‖ ‖ , where is a positive constant. Fixed point theorems (FPTs) are basic mathematical tools which are used in studying the controllability results of nonlinear equations. Controllability of the system (1.1) with different geometric conditions on the spaces and has been studied by using Banach contraction theorem, Schauder (FPT) and Kirk (FPT), see [4][5][6][7]. Nonexpansive mappings on a space has normal structure, these mappings play an important role in fixed point theory, see [8,11]. Since every uniformly convex Banach space (ucBs) is (rBs) ,howeverthe converse is not ture in general, [7], as well as a nonexpansive mapping on a (Bs) has no fixed point (FP) in general. . Then we extend the work of our problem by using Kirk (FPT) [5]. Thus, the aim of this article is to study the controllability of the system (1.1) in arbitrary (rBs) by using (FPT) that deals with nonexpansive mapping defined on a set has normal structure.

Preliminaries
In this section some well known definitions, theorems and examples that will be used in the proof of the main results.

Definition 2.5 [7]:
Let be a normed space, a subset of is called weakly compact, if every sequence * + contains a subsequence which converges weakly in . Remark 2.6: Every nonempty, bounded, closed and convex (bcc) subset of (rBs) is weakly compact [8,9]. Definition 2.7 [8]: Let be a Banach space, and be nonempty, (bcc). A point is said to be diametral if * ‖ ‖ + . A subset of has normal structure, if for each nonempty, convex with diam , there exist a point which is not diametral . Example 2.8 [8]: In (Bs) compact convex set has normal structure, so nonempty, (bcc) subset of (ucBs) has normal structure, Opial's condition also implies normal structure, see [10].
To have an extension of Kirk (FPT), on (ucBs), we need some geometric conditions on the spaces in the domain of the nonexpansive maps in (rBs). Theorem 2.9 [8,11]: Let be nonexpansive mapping from into , where is a nonempty weakly compact convex subset having normal structure in a (Bs) , then has a (FP) in . Remark 2.10 [11]: In previous theorem the convexity can t be dispense one can see the following simple example Let , -,and is a self mapping on defined by , , therefore is nonexpansive, but has no (FP) in Note that, the nonexpansive map on a non convex set in (Bs) has no fixed point . Definition 2.11 [1]: Let be a (Bs). A one parameter family ( ) of linear bounded operators from a (Bs) into itself , is called a strongly continuous semigroup (semigroup) , if it's satisfied the following conditions: Definition 2.12 [11]: The infinitesimal generator of the semigroup ( ) on a (Bs) is defined by: ( ( ) ) , for x ( ) whenever the limit exists .

Controllability Of Nonlinear Control Problems
The main objective of this section, is to study the controllability of mild solution to the boundary value control problem (1.1) in (rBs) by using semigroup and Theorem 2. is a (rBs) [7]. Throughout this paper, we also suppose the basic hypothesis as follows: (C 1 ) ( ) ( ) and the restriction of to ( ) is continuous relative to such graph norm of ( ). There is positive constant. Also, let ‖ ( ( ))‖ .

(C 6 )
The linear operator from ( ) into , defined by : This leads to a bounded inverse operator ̃ defined on ( ) ( ) ⁄ , and hence ‖ ̃ ‖ , where is a positive constant. For more details about the existence of bounded inverse operator of , see [5,12].

Result of Controllability to Problem (1.1):
Throughout this subsection, we want to define, and to find the mild solution to the problem (1.1).
Suppose that ( ) be a solution of problem (1.1), then we can define a function : From assumptions, we obtain that ( ) ( ). Therefore the problem (1.1) can be written in term of B 1 and A 2 , as follows :

By condition (C 3 ), we have
( ) is continuously differentiable, if is continuously differentiable on , then by definition of the mild solution ( ) ( ) ( ) it can be defined as a mild solution to Cauchy problem [1], , which is the C 0 -semigroup generated by the linear operator B 1 , and ( ) is a solution of (3.3), hence the function ( ) ( ) ( ) is differentiable for for more dailies see [1].

Main Results
In this section, we will prove the theorem that deals with the controllability of the problem (1.1). Theorem 3.1: Let be a (rBs) which satisfying Opial's condition and the hypothesis (C 1 )_(C 6 ) are satisfied for the nonlinear boundary control problem (1.1) Further, suppose that (C 7 ) There exists a constant 3 , such that ∫ ( ) 3 .

Bassem and Al-Jawary
Iraqi Journal of Science, 2022, Vol. 63, No. 1, pp: 222-232 227 (C 9 ) ,, -, ‖ ‖ -, --, such that . Then the system (1.1) is controllable on . Proof: By using definition (3.1) and mild solution equation (3.8) we obtain that ) From constriction the operator in (C 6 ) since ( ) ̃ ( ( )), thus We can explain implication when using this control, the operator defined by has a (FP), this (FP) is then a solution of (1.1). Note that, ( )( ) , it know that the control transmit the control system from the initial to in time , provided we can obtain a (FP) of the nonlinear operator . Let be a (rBs) that satisfies the Opial's condition and let =* ( ) ( ) ‖ ( )‖ , for }, it's clear that is (bcc) subset of [7]. We have also is weakly compact and has normal structure in (rBs) , see Remark 2.6 and Example 2.8. Here, we will define a mapping : by : ∫ ( ) ( ( ( ))) -( ) ( ) Now we want to show that the operator is continuous and maps into itself. Thus, we take the norm of both sides of (3.10)