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