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

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-OCP n ) in semigroup of linear operator. The purpose of this paper is to establish some dissipative properties on ω-OCP n which have been obtained in the various theorems (research results) and were proved.


Introduction
Suppose X is a Banach space, n X X ⊆ a finite set, ( ) ( ) 0 t T t ≥ the C 0 -semigroup that is strongly continuous one-parameter semigroup of bounded linear operator in X. Let ω-OCP n be ω-order-preserving partial contraction mapping (semigroup of linear operator) which is an example of C 0 -semigroup.
Furthermore, let ( ) Mm  be a matrix, ( ) L X a bounded linear operator on X, n P a partial transformation semigroup, ( ) A ρ a resolvent set, ( ) F x a duality mapping on X and A is a generator of C 0 -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-OCP n on Banach space as an example of a semigroup of linear operator called C 0 -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].

Preliminaries
Definition 2.1 (C 0 -Semigroup) [9] C 0 -Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space. Definition 2.2 (ω-OCP n ) [11] Transformation n P α ∈ is called ω-order-preserving partial contraction mapping if , Dom : and at least one of its transformation must satisfy y y Definition 2.3 (Subspace Semigroup) [8] A subspace semigroup is the part of A in Y which is the operator

Definition 2.4 (Duality set)
Let X be a Banach space, for every x X ∈ , a nonempty set defined by Definition 2.5 (Dissipative) [9] A linear operator

Properties of Dissipative Operator
For dissipative operator for all y in the range , that is if A is densely defined, then A is closable. its closure A is again dissipative and satisfies e e e I e e is a resolvent set on X.
Also, suppose , then

Theorem (Lumer-Phillips [5])
Let X be a real, or complex Banach space with norm ⋅ , and let us recall that the duality mapping : for each x X ∈ . In view of Hahn-Banach theorem, it follows that, for each x X ∈ , ( ) F x is nonempty.

Theorem (Hahn-Banach Theorem [2])
Let V be a real vector space. Suppose

Main Results
In this section, dissipative results on ω-OCP n as a semigroup of linear operator were established and the research results(Theorems) were given and proved appropriately: Banach space X such that A λ − is surjective for some 0 λ > . Then 1) the part A, of A in the subspace is densely defined and generates a constrain semigroup in 0 X , 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