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Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator

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DOI: 10.4236/apm.2019.96026    83 Downloads   136 Views  

ABSTRACT

This paper consists of dissipative properties and results of dissipation on infinitesimal generator of a C0-semigroup of ω-order preserving partial contraction mapping (ω-OCPn) in semigroup of linear operator. The purpose of this paper is to establish some dissipative properties on ω-OCPn which have been obtained in the various theorems (research results) and were proved.

1. Introduction

Suppose X is a Banach space, X n X a finite set, ( T ( t ) ) t 0 the C0-semigroup that is strongly continuous one-parameter semigroup of bounded linear operator in X. Let ω-OCPn be ω-order-preserving partial contraction mapping (semigroup of linear operator) which is an example of C0-semigroup. Furthermore, let M m ( ) be a matrix, L ( X ) a bounded linear operator on X, P n a partial transformation semigroup, ρ ( A ) a resolvent set, F ( x ) a duality mapping on X and A is a generator of C0-semigroup. Taking the importance of the dissipative operator in a semigroup of linear operators into cognizance, dissipative properties characterized the generator of a semigroup of linear operator which does not require the explicit knowledge of the resolvent.

This paper will focus on results of dissipative operator on ω-OCPn on Banach space as an example of a semigroup of linear operator called C0-semigroup.

Yosida [1] proved some results on differentiability and representation of one-parameter semigroup of linear operators. Miyadera [2] , generated some strongly continuous semigroups of operators. Feller [3] , also obtained an unbounded semigroup of bounded linear operators. Balakrishnan [4] introduced fractional powers of closed operators and semigroups generated by them. Lumer and Phillips [5] , established dissipative operators in a Banach space and Hille & Philips [6] emphasized the theory required in the inclusion of an elaborate introduction to modern functional analysis with special emphasis on functional theory in Banach spaces and algebras. Batty [7] obtained asymptotic behaviour of semigroup of operator in Banach space. More relevant work and results on dissipative properties of ω-Order preserving partial contraction mapping in semigroup of linear operator could be seen in Engel and Nagel [8] , Vrabie [9] , Laradji and Umar [10] , Rauf and Akinyele [11] and Rauf et al. [12] .

2. Preliminaries

Definition 2.1 (C0-Semigroup) [9]

C0-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.

Definition 2.2 (ω-OCPn) [11]

Transformation α P n is called ω-order-preserving partial contraction mapping if x , y Dom α : x y α x α y and at least one of its transformation must satisfy α y = y such that T ( t + s ) = T ( t ) T ( s ) whenever t , s > 0 and otherwise for T ( 0 ) = I .

Definition 2.3 (Subspace Semigroup) [8]

A subspace semigroup is the part of A in Y which is the operator A * defined by A * y = A y with domain D ( A * ) = { y D ( A ) Y : A y Y } .

Definition 2.4 (Duality set)

Let X be a Banach space, for every x X , a nonempty set defined by F ( x ) = { x * X * : ( x , x * ) = x 2 = x * 2 } is called the duality set.

Definition 2.5 (Dissipative) [9]

A linear operator ( A , D ( A ) ) is dissipative if each x X , there exists x * F ( x ) such that R e ( A x , x * ) 0 .

2.1. Properties of Dissipative Operator

For dissipative operator A : D ( A ) X X , the following properties hold:

a) λ A is injective for all λ > 0 and

( λ A ) 1 1 / λ y (2.1)

for all y in the range rg ( λ A ) = ( λ A ) D ( A ) .

b) λ A is surjective for some λ > 0 if and only if it is surjective for each λ > 0 . In that case, we have ( 0, ) ρ ( A ) , where ρ ( A ) is the resolvent of the generator A.

c) A is closed if and only if the range rg ( λ A ) is closed for some λ > 0 .

d) If rg ( A ) D ( A ) , that is if A is densely defined, then A is closable. its closure A is again dissipative and satisfies rg ( λ A ) = rg ( λ A ) for all λ > 0 .

Example 1

2 × 2 matrix [ M m ( { 0 } ) ]

Suppose

A = ( 1 2 2 2 )

and let T ( t ) = e t A , then

e t A = ( e t e 2 t e 2 t e 2 t )

3 × 3 matrix [ M m ( { 0 } ) ]

Suppose

A = ( 1 2 3 1 2 2 2 3 )

and let T ( t ) = e t A , then

e t A = ( e t e 2 t e 3 t e t e 2 t e 2 t I e 2 t e 3 t )

Example 2

In any 2 × 2 matrix [ M m ( ) ] , and for each λ > 0 such that λ ρ ( A ) where ρ ( A ) is a resolvent set on X.

Also, suppose

A = ( 1 2 2 )

and let T ( t ) = e t A λ , then

e t A λ = ( e t λ e 2 t λ I e 2 t λ )

Example 3

Let X = C u b ( { 0 } ) be the space of all bounded and uniformly continuous function from { 0 } to , endowed with the sup-norm and let { T ( t ) ; t 0 } L ( X ) be defined by

[ T ( t ) f ] ( s ) = f ( t + s )

For each f X and each t , s + , it is easily verified that { T ( t ) ; t 0 } satisfies Examples 1 and 2 above.

Example 4

Let X = C [ 0 , 1 ] and consider the operator A f = f with domain D ( A ) = { f C [ 0 , 1 ] : f ( 0 ) = 0 } . It is a closed operator whose domain is not dense. However, it is dissipative, since its resolvent can be computed explicitly as

R ( λ , A ) f ( t ) = 0 t e λ ( t s ) f ( s ) d s

for t [ 0,1 ] , f C [ 0,1 ] . Moreover, R ( λ , A ) 1 λ for all λ > 0 . Therefore ( A , D ( A ) ) is dissipative.

2.2. Theorem (Hille-Yoshida [9] )

A linear operator A : D ( A ) X X is the infinitesimal generator for a C0-semigroup of contraction if and only if

1) A is densely defined and closed,

2) ( 0, + ) ρ ( A ) and for each λ > 0

R ( λ , A ) L ( X ) 1 λ (2.2)

2.3. Theorem (Lumer-Phillips [5] )

Let X be a real, or complex Banach space with norm , and let us recall that the duality mapping F : X 2 x is defined by

F ( x ) = { x * X * ; ( x , x * ) = x 2 = x * 2 } (2.3)

for each x X . In view of Hahn-Banach theorem, it follows that, for each x X , F ( x ) is nonempty.

2.4. Theorem (Hahn-Banach Theorem [2] )

Let V be a real vector space. Suppose p : V [ 0, + ] is mapping satisfying the following conditions:

1) p ( 0 ) = 0 ;

2) p ( t x ) = t p ( x ) for all x V and real of t 0 ; and

3) p ( x + y ) p ( x ) + p ( y ) for every x , y v .

Assume, furthermore that for each x V , either both p ( x ) and p ( x ) are or that both are finite.

3. Main Results

In this section, dissipative results on ω-OCPn as a semigroup of linear operator were established and the research results(Theorems) were given and proved appropriately:

Theorem 3.1

Let A w - O C P n where A : D ( A ) X X is a dissipative operator on a Banach space X such that λ A is surjective for some λ > 0 . Then

1) the part A, of A in the subspace X 0 = D ( A ) ¯ is densely defined and generates a constrain semigroup in X 0 , and

2) considering X to be a reflexive, A is densely defined and generates a contraction semigroup.

Proof

We recall from Definition 2.3 that

A * x = A x (3.1)

for

x D ( A * ) = { x x D ( A ) : A x X 0 } = R ( λ , A ) X 0 (3.2)

Since R ( λ , A ) exists for λ > 0 , this implies that R ( λ , A ) * = R ( λ , A * ) , hence

( 0, ) ρ (A*)

we need to show that D ( A * ) is dense in X 0 .

Take x D ( A ) and set x n = n R ( n , A ) x . Then x n D ( A ) and

lim n x n = lim n R ( n , A ) A x + x = x ,

since R ( n , A ) 1 n . Therefore the operators n R ( n , A ) converge pointwise on

D ( A ) to the identity. Since n R ( n , A ) 1 for all n , we obtain the convergence of y n = n R ( n , A ) y y for all y X 0 . If for each y n in D ( A * ) , the density of D ( A * ) in X 0 is shown which proved (i).

To prove (ii), we need to obtain the density of D ( A ) .

Let x X and define x n = n R ( n , A ) x D ( A ) . The element y = n R ( 1 , A ) x , also belongs to D ( A ) . Moreover, by the proof of (i) the operators n R ( n , A ) converges towards the identity pointwise on X 0 = D ( A ) ¯ . It follows that

y n = R ( 1 , A ) x n = n R ( n , A ) R ( 1 , A ) x y for n

Since X is reflexive and { x n : n } is bounded, there exists a subsequence, still denoted by ( x n ) ( n ) , that converges weakly to some z X . Since x n D ( A ) , implies that z D ( A ) ¯ .

On the other hand, the elements x n = ( 1 A ) y n converges weakly to z, so the weak closedness of A implies that y D ( A ) and x = ( 1 A ) y = z D ( A ) ¯ which proved (ii).

Theorem 3.2

The linear operator A : D ( A ) X X is a dissipative if and only if for each x D ( A ) and λ > 0 , where A ω - O C P n , then we have

( λ 1 A ) x λ x (3.3)

Proof

Suppose A is dissipative, then, for each x D ( A ) and λ > 0 , there exists x * F ( x ) such that R e ( λ x A x , x * ) 0 . Therefore

x λ x A x | ( λ x A x , x ) | R e ( λ x A x , x ) λ x 2

and this completes the proof. Next, let x D ( A ) and λ > 0 .

Let y λ * F ( λ x A x ) and let us observe that, by virtue of (3.3), λ x A x = 0 x = 0 .

So, in this case, we clearly have R e ( x * , λ x A x ) = 0. Therefore, by assuming that λ x A x 0 . As a consequence, y λ * 0 , and thus

z λ * = y λ * y λ *

lies on the unit ball, i.e. z λ * = 1 . We have ( λ x A x , z λ * ) = λ x A x λ x R e ( x , z λ * ) R e ( A x , z λ * ) λ x R e ( A x , z λ * ) hence

R e ( A x , z λ * ) 0

and R e ( z λ * , x ) x 1 λ A x . Now, let us recall that the closed unit ball in X *

is weakly-star compact. Thus, the net ( z λ * ) λ > 0 has at least one weak-star cluster point z * X * with

z * 1 (3.4)

From (3.4), it follows that R e ( A x , z * ) 0 and R e ( x , z * ) x . Since R e ( x , z * ) | ( x , z * ) | x , it follows that ( x , z * ) = x . Hence x * = x z * F ( x ) and R e ( A x , x * ) 0 and this completes the proof.

Proposition 3.3

Let A : D ( A ) X X be infinitesimal generator of a C0-semigroup of contraction and A ω - O C P n . Suppose X * = D ( A ) is endowed with the graph-norm | | D ( A ) : X * { 0 } defined by | u | D ( A ) = u A u for u X * . Then operator A * : D ( A * ) X * X * defined by

{ D ( A * ) = { x X * ; A x X * } A * x = A x , for x D (X*)

is the infinitesimal generator of a C0-semigroup of contractions on X * .

Proof

Let λ > 0 and f X * and let us consider the equation λ u A u = F Since A generates a C0-semigroup of contraction [6] , it follows that this equation has a unique solution u D ( A ) .

Since f X * , we conclude that A u D ( A ) and thus u D ( A * ) .

Thus λ u A * u = f . On the other hand, we have

| ( λ I A * ) 1 f | D ( A ) = ( I A ) ( λ I A ) 1 f = ( λ I A ) 1 ( I A ) f 1 λ f A f = 1 λ | f | D ( A ) (3.5)

which shows that A * satisfies condition (ii) in Theorem 2.2. Moreover, it follows that A * is closed in X * .

Indeed, as ( λ I A ) 1 L ( X * ) , it is closed, and consequently λ I A * enjoys the same property which proves that A * is closed.

Now, let x X * , λ > 0 , A ω - O C P n and let x λ = λ x A * x . Clearly x λ D ( A * ) , and in addition lim λ | x λ x | D ( A ) = 0 Thus, D ( A * ) is dense in X * by virtue of Theorem 2.2, A * generates a C0-semigroup of contraction on X * . Hence the proof.

4. Conclusion

In this paper, it has been established that ω-OCPn possesses the properties of dissipative operators as a semigroup of linear operator, and obtaining some dissipative results on ω-OCPn.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Akinyele, A. , Rauf, K. , Adebowale, A. and Babatunde, O. (2019) Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator. Advances in Pure Mathematics, 9, 544-550. doi: 10.4236/apm.2019.96026.

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