Dissipative Properties of Ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator
Advances in Pure Mathematics, 2019, 9, 544-550 http://www.scirp.org/journal/apm ISSN Online: 2160-0384 ISSN Print: 2160-0368
Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator
Akinola Yussuff Akinyele, Kamilu Rauf*, Aasa Moses Adebowale, Omosowon Jude Babatunde
Department of Mathematics, University of Ilorin, Ilorin, Nigeria
How to cite this paper: Akinyele, A.Y, Abstract Rauf, K., Adebowale, A.M. and Babatunde, O.J. (2019) Dissipative Properties of This paper consists of dissipative properties and results of dissipation on infi-
ω-Order Preserving Partial Contraction nitesimal generator of a C0-semigroup of ω-order preserving partial contrac- Mapping in Semigroup of Linear Operator. tion mapping (ω-OCPn) in semigroup of linear operator. The purpose of this Advances in Pure Mathematics, 9, 544-550. https://doi.org/10.4236/apm.2019.96026 paper is to establish some dissipative properties on ω-OCPn which have been obtained in the various theorems (research results) and were proved. Received: March 5, 2019 Accepted: June 25, 2019 Keywords Published: June 28, 2019 Semigroup, Linear Operator, Dissipative Operator, Contraction Mapping and Copyright © 2019 by author(s) and Resolvent Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). 1. Introduction http://creativecommons.org/licenses/by/4.0/ Open Access XX⊆ Tt Suppose X is a Banach space, n a finite set, ( ( ))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 Mm() be a matrix, LX( ) a bounded linear operator on X,
Pn a partial transformation semigroup, ρ ( A) a resolvent set, Fx( ) 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, dis- sipative 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
DOI: 10.4236/apm.2019.96026 Jun. 28, 2019 544 Advances in Pure Mathematics
A. Y. Akinyele et al.
strongly continuous semigroups of operators. Feller [3], also obtained an un- bounded 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 in- troduction 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 se- migroup of linear operator could be seen in Engel and Nagel [8], Vrabie [9], La- radji 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 ∈ Pn is called ω-order-preserving partial contraction mapping if ∀∈xy, Domα : x ≤ y ⇒≤ααxy and at least one of its transfor- mation must satisfy α yy= such that Tt( += s) TtTs( ) ( ) whenever ts,0> and otherwise for TI(0) = . 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= Ay with domain DA( * ) =∈∈{ y DA( ) Y: AyY} . Definition 2.4 (Duality set) Let X be a Banach space, for every xX∈ , a nonempty set defined by 2 2 F( x) =∈=={ x** X:,( xx *) x x *} is called the duality set. Definition 2.5 (Dissipative) [9] A linear operator ( AD, ( A)) is dissipative if each xX∈ , there exists x* ∈ Fx( ) such that Re( Ax,0 x* ) ≤ .
2.1. Properties of Dissipative Operator
For dissipative operator ADA: ( ) ⊆→ X X, the following properties hold: a) λ − A is injective for all λ > 0 and
−1 (λλ−≤Ay) 1 (2.1)
for all y in the range rg (λλ−=−A) ( ADA) ( ) . 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) ⊆ DA( ) , that is if A is densely defined, then A is closable. its clo- sure A is again dissipative and satisfies rg(λλ−=AA) rg ( −) for all λ > 0 .
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Example 1 22× matrix M m ( {0}) Suppose 12 A = 22
and let Tt( ) = etA , then eett2 tA = e 22tt ee 33× matrix M m ( {0}) Suppose 123 A = 122 − 23
and let Tt( ) = etA , then eettt23 e tAttt22 e= ee e 23tt Ie e
Example 2
In any 22× matrix M m () , and for each λ > 0 such that λρ∈ ( A) where ρ ( A) is a resolvent set on X. Also, suppose 12 A = − 2
and let Tt( ) = etAλ , then eettλλ2 tAλ = e 2tλ Ie
Example 3
Let XC= ub ( {0}) be the space of all bounded and uniformly continuous ⋅ function from {0} to , endowed with the sup-norm ∞ and let {Tt( );0 t≥⊆} LX( ) be defined by
Tt( ) f( s) = ft( + s)
For each fX∈ and each ts, ∈ + , it is easily verified that {Tt( );0 t≥ } satisfies Examples 1 and 2 above. Example 4 Let XC= [0,1] and consider the operator Af= − f ′ with domain DA( ) =∈={ 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
t −−λ(ts) R(λ,d Aft) ( ) = e fs( ) s ∫0
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1 for t ∈[0,1] , fC∈ [0,1] . Moreover, RA(λ, ) ≤ for all λ > 0 . Therefore λ ( AD, ( A)) is dissipative.
2.2. Theorem (Hille-Yoshida [9])
A linear operator ADA: ( ) ⊆→ 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 1 RA(λ, ) ≤ (2.2) LX( ) λ
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 FX:2→ x is defined by
2 2 F( x) =∈=={ x** X;,( xx *) x x *} (2.3)
for each xX∈ . In view of Hahn-Banach theorem, it follows that, for each xX∈ , Fx( ) is nonempty.
2.4. Theorem (Hahn-Banach Theorem [2])
Let V be a real vector space. Suppose pV:∈[ 0, +∞] is mapping satisfying the following conditions: 1) p(00) = ; 2) p( tx) = tp( x) for all xV∈ and real of t ≥ 0 ; and 3) px( +≤ y) px( ) + py( ) for every xy, ∈ v. Assume, furthermore that for each xV∈ , either both px( ) and px(− ) 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 ap- propriately: Theorem 3.1
Let A∈ w- OCPn where ADA: ( ) ⊆→ 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 X0 = DA( ) is densely defined and gene-
rates a constrain semigroup in X 0 , and 2) considering X to be a reflexive, A is densely defined and generates a con- traction semigroup. Proof We recall from Definition 2.3 that
A* x= Ax (3.1)
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for
x∈ DA( *) ={ x ∈∈ x DA( ) :, AxX ∈00} = R(λ AX) (3.2)
λ λ > λλ= Since RA( , ) exists for 0 , this implies that RA( ,,)* RA( * ) , hence
(0,∞⊂) ρ ( A* )
we need to show that DA( * ) is dense in X 0 .
Take x∈ DA( ) and set xn = nRnAx( , ) . Then xn ∈ DA( ) and
limxn = lim R( n , A) Ax+= x x, nn→∞ →∞ 1 since RnA( , ) ≤ . Therefore the operators nR( n, A) converge pointwise on n DA( ) to the identity. Since nR( n,1 A) ≤ for all n ∈ , we obtain the con-
vergence of yn = nR( n, A) y → y for all yX∈ 0 . If for each yn in DA( * ) ,
the density of DA( * ) in X 0 is shown which proved (i). To prove (ii), we need to obtain the density of DA( ) .
Let xX∈ and define xn = nRnAx( , ) ∈ DA( ) . The element y= nR(1, A) x , also belongs to DA( ) . Moreover, by the proof of (i) the operators nR( n, A)
converges towards the identity pointwise on X0 = DA( ) . It follows that
ynn= R(1, Ax) = nRnAR( ,) ( 1, Ax) → y for n →∞
Since X is reflexive and {xnn : ∈ } is bounded, there exists a subsequence, still denoted by ( x ) , that converges weakly to some zX∈ . Since n (n∈) xn ∈ DA( ) , implies that z∈ DA( ) .
On the other hand, the elements xnn=(1 − Ay) converges weakly to z, so the weak closedness of A implies that y∈ DA( ) and x=−=∈(1 Ay) z DA( ) which proved (ii). Theorem 3.2 The linear operator ADA: ( ) ⊆→ X X is a dissipative if and only if for
each x∈ DA( ) and λ > 0 , where A∈ω- OCPn , then we have
(λλ1 −≥Ax) x (3.3)
Proof Suppose A is dissipative, then, for each x∈ DA( ) and λ > 0 , there exists x* ∈ Fx( ) such that Re(λ x−≤ Ax,0 x* ) . Therefore 2 xλλ xAx−≥( xAxx −,,) ≥ RexAxx( λ −) ≥ λ x
and this completes the proof. Next, let x∈ DA( ) and λ > 0 . * Let yλ ∈− F(λ x Ax) and let us observe that, by virtue of (3.3), λx−= Ax 0 ⇒ x = 0 . So, in this case, we clearly have Re( x* ,λ x−= Ax) 0. Therefore, by assuming * that λx−≠ Ax 0 . As a consequence, yλ ≠ 0 , and thus y* z* = λ λ * yλ
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* * lies on the unit ball, i.e. zλ = 1 . We have (λx− Ax, zλ ) =−≥ λλ x Ax x ⇒ ** * Re( x,, zλλ) − Re( Ax z) ≤−λ x Re( Ax , z λ) hence * Re( Ax,0 zλ ) ≤
* 1 * and Re( zλ , x) ≥− x Ax . Now, let us recall that the closed unit ball in X λ * is weakly-star compact. Thus, the net zλ has at least one weak-star cluster ( )λ >0 point zX**∈ with z* ≤ 1 (3.4)
From (3.4), it follows that Re( Ax,0 z* ) ≤ and Re( x, z* ) ≥ x . Since * Rexz( ,,**) ≤≤( xz) x, it follows that ( xz, ) = x. Hence x**= xz ∈ F( x) and Re( Ax,0 x* ) ≤ and this completes the proof. Proposition 3.3
Let ADA: ( ) ⊆→ X X be infinitesimal generator of a C0-semigroup of
contraction and A∈ω- OCPn . Suppose X* = DA( ) is endowed with the ⋅→ = − ∈ graph-norm DA( ) :0X* { } defined by uDA( ) u Au for uX* .
Then operator A*: DA( *) ⊆→ X ** X defined by
DA( *) =∈∈{ xXAxX **; } A** x= Ax, for x ∈ D( X )
is the infinitesimal generator of a C0-semigroup of contractions on X* . Proof
Let λ > 0 and fX∈ * and let us consider the equation λu−= Au F Since
A generates a C0-semigroup of contraction [6], it follows that this equation has a unique solution u∈ DA( ) .
Since fX∈ * , we conclude that Au∈ D( A) and thus u∈ DA( * ) .
Thus λu−= Au* f . On the other hand, we have
−−11 (λλIA−* ) f =−−( IA)( IA) f DA( ) (3.5) −1 11 =(λIA −) ( IAf −) ≤ fAf −= f λλDA( )
which shows that A* satisfies condition (ii) in Theorem 2.2. Moreover, it fol-
lows that A* is closed in X* . −1 Indeed, as (λI−∈ A) LX( * ) , it is closed, and consequently λIA− * enjoys
the same property which proves that A* is closed.
Now, let xX∈ * , λ > 0 , A∈ω- OCPn and let xλ =λ x − Ax* . Clearly x∈ DA −= DA λ ( * ) , and in addition limλλ→∞ xxDA( ) 0 Thus, ( * ) 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 dis-
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sipative results on ω-OCPn.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this pa- per.
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