1. Introduction
Suppose X is a Banach space, a finite set, 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 be a matrix, a bounded linear operator on X, a partial transformation semigroup, a resolvent set, 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 is called ω-order-preserving partial contraction mapping if and at least one of its transformation must satisfy such that whenever and otherwise for .
Definition 2.3 (Subspace Semigroup) [8]
A subspace semigroup is the part of A in Y which is the operator defined by with domain .
Definition 2.4 (Duality set)
Let X be a Banach space, for every , a nonempty set defined by is called the duality set.
Definition 2.5 (Dissipative) [9]
A linear operator is dissipative if each , there exists such that .
2.1. Properties of Dissipative Operator
For dissipative operator , the following properties hold:
a) is injective for all and
(2.1)
for all y in the range .
b) is surjective for some if and only if it is surjective for each . In that case, we have , where is the resolvent of the generator A.
c) A is closed if and only if the range is closed for some .
d) If , that is if A is densely defined, then A is closable. its closure A is again dissipative and satisfies for all .
Example 1
matrix
Suppose
and let , then
matrix
Suppose
and let , then
Example 2
In any matrix , and for each such that where is a resolvent set on X.
Also, suppose
and let , then
Example 3
Let be the space of all bounded and uniformly continuous function from to , endowed with the sup-norm and let be defined by
For each and each , it is easily verified that satisfies Examples 1 and 2 above.
Example 4
Let and consider the operator with domain . It is a closed operator whose domain is not dense. However, it is dissipative, since its resolvent can be computed explicitly as
for , . Moreover, for all . Therefore is dissipative.
2.2. Theorem (Hille-Yoshida [9] )
A linear operator is the infinitesimal generator for a C0-semigroup of contraction if and only if
1) A is densely defined and closed,
2) and for each
(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 is defined by
(2.3)
for each . In view of Hahn-Banach theorem, it follows that, for each , is nonempty.
2.4. Theorem (Hahn-Banach Theorem [2] )
Let V be a real vector space. Suppose is mapping satisfying the following conditions:
1) ;
2) for all and real of ; and
3) for every .
Assume, furthermore that for each , either both and 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 where is a dissipative operator on a Banach space X such that is surjective for some . Then
1) the part A, of A in the subspace is densely defined and generates a constrain semigroup in , 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
(3.1)
for
(3.2)
Since exists for , this implies that , hence
we need to show that is dense in .
Take and set . Then and
since . Therefore the operators converge pointwise on
to the identity. Since for all , we obtain the convergence of for all . If for each in , the density of in is shown which proved (i).
To prove (ii), we need to obtain the density of .
Let and define . The element , also belongs to . Moreover, by the proof of (i) the operators converges towards the identity pointwise on . It follows that
Since X is reflexive and is bounded, there exists a subsequence, still denoted by , that converges weakly to some . Since , implies that .
On the other hand, the elements converges weakly to z, so the weak closedness of A implies that and which proved (ii).
Theorem 3.2
The linear operator is a dissipative if and only if for each and , where , then we have
(3.3)
Proof
Suppose A is dissipative, then, for each and , there exists such that . Therefore
and this completes the proof. Next, let and .
Let and let us observe that, by virtue of (3.3), .
So, in this case, we clearly have Therefore, by assuming that . As a consequence, , and thus
lies on the unit ball, i.e. . We have hence
and . Now, let us recall that the closed unit ball in
is weakly-star compact. Thus, the net has at least one weak-star cluster point with
(3.4)
From (3.4), it follows that and . Since , it follows that . Hence and and this completes the proof.
Proposition 3.3
Let be infinitesimal generator of a C0-semigroup of contraction and . Suppose is endowed with the graph-norm defined by for . Then operator defined by
is the infinitesimal generator of a C0-semigroup of contractions on .
Proof
Let and and let us consider the equation Since A generates a C0-semigroup of contraction [6] , it follows that this equation has a unique solution .
Since , we conclude that and thus .
Thus . On the other hand, we have
(3.5)
which shows that satisfies condition (ii) in Theorem 2.2. Moreover, it follows that is closed in .
Indeed, as , it is closed, and consequently enjoys the same property which proves that is closed.
Now, let , , and let . Clearly , and in addition Thus, is dense in by virtue of Theorem 2.2, generates a C0-semigroup of contraction on . 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.
[1] Yosida, K. (1948) On the Differentiability and Representation of One-Parameter Semigroups of Linear Operators. Journal of the Mathematical Society of Japan, 1, 15-21.
https://doi.org/10.2969/jmsj/00110015
[2] Miyadera, I. (1952) Generation of Strongly Continuous Semigroups Operators. Tohoku Mathematical Journal, 4, 109-114.
https://doi.org/10.2748/tmj/1178245412
[3] Feller, W. (1953) On the Generation of Unbounded Semigroup of Bounded Linear Operators. Annals of Mathematics, 58, 166-174.
https://doi.org/10.2307/1969826
[4] Balakrishnan, A.V. (1960) Fractional Powers of Closed Operators and Semigroups Generated by Them. Pacific Journal of Mathematics, 10, 419-437.
https://doi.org/10.2140/pjm.1960.10.419
[5] Lumer, G. and Phillips, R.S. (1961) Dissipative Operators in a Banach Space. Pacific Journal of Mathematics, 11, 679-698.
https://doi.org/10.2140/pjm.1961.11.679
[6] Hille, E. and Phillips, R.S. (1981) Functional Analysis and Semigroups. American Mathematical Society, Providence, Colloquium Publications Vol. 31.
[7] Batty, C.J.K. (1994) Asymptotic Behaviour of Semigroup of Operators. Banach Center Publications, 30, 35-52.
https://doi.org/10.4064/-30-1-35-52
[8] Engel, K. and Nagel, R. (1999) One-Parameter Semigroup for Linear Evolution Equations. Graduate Texts in Mathematics Vol. 194, Springer, New York.
[9] Vrabie, I.I. (2003) C0-Semigroup and Application. Mathematics Studies Vol. 191, Elsevier, North-Holland.
[10] Laradji, A. and Umar, A. (2004) Combinatorial Results for Semigroups of Order Preserving Partial Transformations. Journal of Algebra, 278, 342-359.
https://doi.org/10.1016/j.jalgebra.2003.10.023
[11] Rauf, K. and Akinyele, A.Y. (2019) Properties of ω-Order-Preserving Partial Contraction Mapping and Its Relation to C0-Semigroup. International Journal of Mathematics and Computer Science, 14, 61-68.
[12] Rauf, K., Akinyele, A.Y., Etuk, M.O., Zubair, R.O. and Aasa, M.A. (2019) Some Results of Stability and Spectra Properties on Semigroupn of Linear Operator. Advances of Pure Mathematics, 9, 43-51.
https://doi.org/10.4236/apm.2019.91003