This paper consists of dissipative properties and results of dissipation on infinitesimal generator of a C 0 -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.
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 ( ℕ ) 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 [
Definition 2.1 (C0-Semigroup) [
C0-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.
Definition 2.2 (ω-OCPn) [
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) [
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) [
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 .
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 ( ℕ ∪ { 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 ( ℕ ∪ { 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 ( ℂ ) ] , 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 ( ℕ ∪ { 0 } ) be the space of all bounded and uniformly continuous function from ℕ ∪ { 0 } to ℝ , 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 ∈ ℝ + , 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.
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)
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.
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.
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 ∈ 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 ∈ ℕ } is bounded, there exists a subsequence, still denoted by ( x n ) ( 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 * → ℕ ∪ { 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 [
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.
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.
The authors declare no conflicts of interest regarding the publication of this paper.
Akinyele, A.Y, Rauf, K., Adebowale, A.M. and Babatunde, O.J. (2019) Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator. Advances in Pure Mathematics, 9, 544-550. https://doi.org/10.4236/apm.2019.96026