1. Introduction

Suppose X is a Banach space, $X_n\subseteq X$ a finite set, ${\left(T\left(t\right)\right)}_{t\ge 0}$ the C₀-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 C₀-semigroup. Furthermore, let $Mm\left(\mathbb{N}\right)$ be a matrix, $L\left(X\right)$ a bounded linear operator on X, $P_n$ a partial transformation semigroup, $\rho \left(A\right)$ a resolvent set, $F\left(x\right)$ 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-OCP_n 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-OCP_n) [11]

Transformation $\alpha \in P_n$ is called ω-order-preserving partial contraction mapping if $\forall x\mathrm{,}y\in \text{Dom}\alpha \mathrm{<mo>\; :\; </mo>}x\le y$ $\Rightarrow \alpha x\le \alpha y$ and at least one of its transformation must satisfy $\alpha y=y$ such that $T\left(t+s\right)=T\left(t\right)T\left(s\right)$ whenever $t,s>0$ and otherwise for $T\left(0\right)=I$.

Definition 2.3 (Subspace Semigroup) [8]

A subspace semigroup is the part of A in Y which is the operator $A_{\mathrm{<mo>\; *\; </mo>}}$ defined by $A_{\mathrm{<mo>\; *\; </mo>}}y=Ay$ with domain $D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)=\left\{y\in D\left(A\right)\cap Y\mathrm{<mo>\; :\; </mo>}Ay\in Y\right\}$.

Definition 2.4 (Duality set)

Let X be a Banach space, for every $x\in X$, a nonempty set defined by $F\left(x\right)=\left\{x^{\mathrm{<mo>\; *\; </mo>}}\in X^{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}{x}^{\mathrm{<mo>\; *\; </mo>}}\right)={\Vert x\Vert }^2={\Vert x^{\mathrm{<mo>\; *\; </mo>}}\Vert }^2\right\}$ is called the duality set.

Definition 2.5 (Dissipative) [9]

A linear operator $\left(A\mathrm{,}D\left(A\right)\right)$ is dissipative if each $x\in X$, there exists $x^{\mathrm{<mo>\; *\; </mo>}}\in F\left(x\right)$ such that $Re\left(Ax\mathrm{,}{x}^{\mathrm{<mo>\; *\; </mo>}}\right)\le 0$.

2.1. Properties of Dissipative Operator

For dissipative operator $A\mathrm{<mo>\; :\; </mo>}D\left(A\right)\subseteq X\to X$, the following properties hold:

a) $\lambda -A$ is injective for all $\lambda >0$ and

$\Vert {\left(\lambda -A\right)}^{-1}\Vert \le 1/\lambda \Vert y\Vert$ (2.1)

for all y in the range $\text{rg}\left(\lambda -A\right)=\left(\lambda -A\right)D\left(A\right)$.

b) $\lambda -A$ is surjective for some $\lambda >0$ if and only if it is surjective for each $\lambda >0$. In that case, we have $\left(\mathrm{0,}\infty \right)\subset \rho \left(A\right)$, where $\rho \left(A\right)$ is the resolvent of the generator A.

c) A is closed if and only if the range $\text{rg}\left(\lambda -A\right)$ is closed for some $\lambda >0$.

d) If $\text{rg}\left(A\right)\subseteq D\left(A\right)$, that is if A is densely defined, then A is closable. its closure A is again dissipative and satisfies $\text{rg}\left(\lambda -A\right)=\text{rg}\left(\lambda -A\right)$ for all $\lambda >0$.

Example 1

$2\times 2$ matrix $\left[M_m\left(\mathbb{N}\cup \left\{0\right\}\right)\right]$

Suppose

$A=\left(\begin{array}{cc}1& 2\\ 2& 2\end{array}\right)$

and let $T\left(t\right)=e^{tA}$, then

$e^{tA}=\left(\begin{array}{cc}{e}^t& e^{2t}\\ e^{2t}& e^{2t}\end{array}\right)$

$3\times 3$ matrix $\left[M_m\left(\mathbb{N}\cup \left\{0\right\}\right)\right]$

Suppose

$A=\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 2\\ -& 2& 3\end{array}\right)$

and let $T\left(t\right)=e^{tA}$, then

$e^{tA}=\left(\begin{array}{ccc}{e}^t& e^{2t}& e^{3t}\\ e^t& e^{2t}& e^{2t}\\ I& e^{2t}& e^{3t}\end{array}\right)$

Example 2

In any $2\times 2$ matrix $\left[M_m\left(ℂ\right)\right]$, and for each $\lambda >0$ such that $\lambda \in \rho \left(A\right)$ where $\rho \left(A\right)$ is a resolvent set on X.

Also, suppose

$A=\left(\begin{array}{cc}1& 2\\ -& 2\end{array}\right)$

and let $T\left(t\right)=e^{t{A}_{\lambda }}$, then

$e^{t{A}_{\lambda }}=\left(\begin{array}{cc}{e}^{t\lambda }& e^{2t\lambda }\\ I& e^{2t\lambda }\end{array}\right)$

Example 3

Let $X=C_{ub}\left(\mathbb{N}\cup \left\{0\right\}\right)$ be the space of all bounded and uniformly continuous function from $\mathbb{N}\cup \left\{0\right\}$ to $ℝ$, endowed with the sup-norm ${\Vert \cdot \Vert }_{\infty }$ and let $\left\{T\left(t\right)\mathrm{<mo>\; ;\; </mo>}t\ge 0\right\}\subseteq L\left(X\right)$ be defined by

$\left[T\left(t\right)f\right]\left(s\right)=f\left(t+s\right)$

For each $f\in X$ and each $t\mathrm{,}s\in {ℝ}_{+}$, it is easily verified that $\left\{T\left(t\right)\mathrm{<mo>\; ;\; </mo>}t\ge 0\right\}$ satisfies Examples 1 and 2 above.

Example 4

Let $X=C\left[0,1\right]$ and consider the operator $Af=-f^{\prime }$ with domain $D\left(A\right)=\left\{f\in C^{\prime }\left[0,1\right]:f\left(0\right)=0\right\}$. It is a closed operator whose domain is not dense. However, it is dissipative, since its resolvent can be computed explicitly as

$R\left(\lambda \mathrm{,}A\right)f\left(t\right)={\displaystyle {\int }_0^t{e}^{-\lambda \left(t-s\right)}f\left(s\right)\text{d}s}$

for $t\in \left[\mathrm{0,1}\right]$, $f\in C\left[\mathrm{0,1}\right]$. Moreover, $\Vert R\left(\lambda \mathrm{,}A\right)\Vert \le \frac{1}{\lambda }$ for all $\lambda >0$. Therefore $\left(A\mathrm{,}D\left(A\right)\right)$ is dissipative.

2.2. Theorem (Hille-Yoshida [9] )

A linear operator $A\mathrm{<mo>\; :\; </mo>}D\left(A\right)\subseteq X\to X$ is the infinitesimal generator for a C₀-semigroup of contraction if and only if

1) A is densely defined and closed,

2) $\left(\mathrm{0,}+\infty \right)\subseteq \rho \left(A\right)$ and for each $\lambda >0$

${\Vert R\left(\lambda \mathrm{,}A\right)\Vert }_{L\left(X\right)}\le \frac{1}{\lambda }$ (2.2)

2.3. Theorem (Lumer-Phillips [5] )

Let X be a real, or complex Banach space with norm $\Vert \cdot \Vert$, and let us recall that the duality mapping $F\mathrm{<mo>\; :\; </mo>}X\to 2^x$ is defined by

$F\left(x\right)=\left\{x^{*}\in X^{*};\left(x,x^{*}\right)={\Vert x\Vert }^2={\Vert x^{*}\Vert }^2\right\}$ (2.3)

for each $x\in X$. In view of Hahn-Banach theorem, it follows that, for each $x\in X$, $F\left(x\right)$ is nonempty.

2.4. Theorem (Hahn-Banach Theorem [2] )

Let V be a real vector space. Suppose $p\mathrm{<mo>\; :\; </mo>}V\in \left[\mathrm{0,}+\infty \right]$ is mapping satisfying the following conditions:

1) $p\left(0\right)=0$;

2) $p\left(tx\right)=tp\left(x\right)$ for all $x\in V$ and real of $t\ge 0$; and

3) $p\left(x+y\right)\le p\left(x\right)+p\left(y\right)$ for every $x\mathrm{,}y\in v$.

Assume, furthermore that for each $x\in V$, either both $p\left(x\right)$ and $p\left(-x\right)$ are $\infty$ or that both are finite.

3. 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:

Theorem 3.1

Let $A\in w\text{-}OC{P}_n$ where $A\mathrm{<mo>\; :\; </mo>}D\left(A\right)\subseteq X\to X$ is a dissipative operator on a Banach space X such that $\lambda -A$ is surjective for some $\lambda >0$. Then

1) the part A, of A in the subspace $X_0=\overline{D\left(A\right)}$ 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

$A_{*}x=Ax$ (3.1)

for

$x\in D\left(A_{*}\right)=\left\{x\in x\in D\left(A\right):Ax\in X_0\right\}=R\left(\lambda ,A\right)X_0$ (3.2)

Since $R\left(\lambda ,A\right)$ exists for $\lambda >0$, this implies that $R{\left(\lambda ,A\right)}_{*}=R\left(\lambda ,A_{*}\right)$, hence

$\left(\mathrm{0,}\infty \right)\subset \rho (A*)$

we need to show that $D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)$ is dense in $X_0$.

Take $x\in D\left(A\right)$ and set $x_n=nR\left(n,A\right)x$. Then $x_n\in D\left(A\right)$ and

$\underset{n\to \infty }{\lim }{x}_n=\underset{n\to \infty }{\lim }R\left(n,A\right)Ax+x=x,$

since $\Vert R\left(n\mathrm{,}A\right)\Vert \le \frac{1}{n}$. Therefore the operators $nR\left(n\mathrm{,}A\right)$ converge pointwise on

$D\left(A\right)$ to the identity. Since $\Vert nR\left(n\mathrm{,}A\right)\Vert \le 1$ for all $n\in \mathbb{N}$, we obtain the convergence of $y_n=nR\left(n,A\right)y\to y$ for all $y\in X_0$. If for each $y_n$ in $D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)$, the density of $D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)$ in $X_0$ is shown which proved (i).

To prove (ii), we need to obtain the density of $D\left(A\right)$.

Let $x\in X$ and define $x_n=nR\left(n,A\right)x\in D\left(A\right)$. The element $y=nR\left(1,A\right)x$, also belongs to $D\left(A\right)$. Moreover, by the proof of (i) the operators $nR\left(n\mathrm{,}A\right)$ converges towards the identity pointwise on $X_0=\overline{D\left(A\right)}$. It follows that

$y_n=R\left(1,A\right)x_n=nR\left(n,A\right)R\left(1,A\right)x\to y\text{for}n\to \infty$

Since X is reflexive and $\left\{x_n\mathrm{<mo>\; :\; </mo>}n\in \mathbb{N}\right\}$ is bounded, there exists a subsequence, still denoted by ${\left(x_n\right)}_{\left(n\in \mathbb{N}\right)}$, that converges weakly to some $z\in X$. Since $x_n\in D\left(A\right)$, implies that $z\in \overline{D\left(A\right)}$.

On the other hand, the elements $x_n=\left(1-A\right)y_n$ converges weakly to z, so the weak closedness of A implies that $y\in D\left(A\right)$ and $x=\left(1-A\right)y=z\in \overline{D\left(A\right)}$ which proved (ii).

Theorem 3.2

The linear operator $A\mathrm{<mo>\; :\; </mo>}D\left(A\right)\subseteq X\to X$ is a dissipative if and only if for each $x\in D\left(A\right)$ and $\lambda >0$, where $A\in \omega \text{-}OC{P}_n$, then we have

$\Vert \left({\lambda }_1-A\right)x\Vert \ge \lambda \Vert x\Vert$ (3.3)

Proof

Suppose A is dissipative, then, for each $x\in D\left(A\right)$ and $\lambda >0$, there exists $x^{\mathrm{<mo>\; *\; </mo>}}\in F\left(x\right)$ such that $Re\left(\lambda x-Ax\mathrm{,}{x}^{\mathrm{<mo>\; *\; </mo>}}\right)\le 0$. Therefore

$\Vert x\Vert \Vert \lambda x-Ax\Vert \ge \left|\left(\lambda x-Ax\mathrm{,}x\right)\right|\ge Re\left(\lambda x-Ax\mathrm{,}x\right)\ge \lambda {\Vert x\Vert }^2$

and this completes the proof. Next, let $x\in D\left(A\right)$ and $\lambda >0$.

Let $y_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\in F\left(\lambda x-Ax\right)$ and let us observe that, by virtue of (3.3), $\lambda x-Ax=0$ $\Rightarrow$ $x=0$.

So, in this case, we clearly have $Re\left(x^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\lambda x-Ax\right)=0.$ Therefore, by assuming that $\lambda x-Ax\ne 0$. As a consequence, $y_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\ne 0$, and thus

$z_{\lambda }^{*}=\frac{y_{\lambda }^{*}}{\Vert y_{\lambda }^{*}\Vert }$

lies on the unit ball, i.e. $\Vert z_{\lambda }^{*}\Vert =1$. We have $\left(\lambda x-Ax\mathrm{,}{z}_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\right)=\Vert \lambda x-Ax\Vert \ge \lambda \Vert x\Vert$ $\Rightarrow$ $Re\left(x\mathrm{,}{z}_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\right)-Re\left(Ax\mathrm{,}{z}_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\right)\le \lambda \Vert x\Vert -Re\left(Ax\mathrm{,}{z}_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\right)$ hence

$Re\left(Ax\mathrm{,}{z}_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\right)\le 0$

and $Re\left(z_{\lambda }^{\mathrm{<mo>\; *\; </mo>}},x\right)\ge \Vert x\Vert -\frac{1}{\lambda }\Vert Ax\Vert$. Now, let us recall that the closed unit ball in $X^{\mathrm{<mo>\; *\; </mo>}}$

is weakly-star compact. Thus, the net ${\left(z_{\lambda }^{\mathrm{<mo>\; *\; </mo>}}\right)}_{\lambda >0}$ has at least one weak-star cluster point $z^{\mathrm{<mo>\; *\; </mo>}}\in X^{\mathrm{<mo>\; *\; </mo>}}$ with

$\Vert z^{\mathrm{<mo>\; *\; </mo>}}\Vert \le 1$ (3.4)

From (3.4), it follows that $Re\left(Ax\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)\le 0$ and $Re\left(x\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)\ge \Vert x\Vert$. Since $Re\left(x\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)\le \left|\left(x\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)\right|\le \Vert x\Vert$, it follows that $\left(x\mathrm{,}{z}^{\mathrm{<mo>\; *\; </mo>}}\right)=\Vert x\Vert$. Hence $x^{\mathrm{<mo>\; *\; </mo>}}=\Vert x\Vert z^{\mathrm{<mo>\; *\; </mo>}}\in F\left(x\right)$ and $Re\left(Ax\mathrm{,}{x}^{\mathrm{<mo>\; *\; </mo>}}\right)\le 0$ and this completes the proof.

Proposition 3.3

Let $A\mathrm{<mo>\; :\; </mo>}D\left(A\right)\subseteq X\to X$ be infinitesimal generator of a C₀-semigroup of contraction and $A\in \omega \text{-}OC{P}_n$. Suppose $X_{*}=D\left(A\right)$ is endowed with the graph-norm ${\left|\cdot \right|}_{D\left(A\right)}\mathrm{<mo>\; :\; </mo>}{X}_{\mathrm{<mo>\; *\; </mo>}}\to \mathbb{N}\cup \left\{0\right\}$ defined by ${\left|u\right|}_{D\left(A\right)}=\Vert u-Au\Vert$ for $u\in X_{\mathrm{<mo>\; *\; </mo>}}$. Then operator $A_{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo>\; :\; </mo>}D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)\subseteq X_{\mathrm{<mo>\; *\; </mo>}}\to X_{\mathrm{<mo>\; *\; </mo>}}$ defined by

$\{\begin{array}{l}D\left(A_{*}\right)=\left\{x\in X_{*};\mathrm{<mi>\; </mi>\; <mi>\; </mi>}Ax\in X_{*}\right\}\\ A_{*}x=Ax,\mathrm{<mi>\; </mi>\; <mi>\; </mi>\; <mi>\; </mi>}\text{for}x\in D(X*)\end{array}$

is the infinitesimal generator of a C₀-semigroup of contractions on $X_{\mathrm{<mo>\; *\; </mo>}}$.

Proof

Let $\lambda >0$ and $f\in X_{\mathrm{<mo>\; *\; </mo>}}$ and let us consider the equation $\lambda u-Au=F$ Since A generates a C₀-semigroup of contraction [6] , it follows that this equation has a unique solution $u\in D\left(A\right)$.

Since $f\in X_{\mathrm{<mo>\; *\; </mo>}}$, we conclude that $Au\in D\left(A\right)$ and thus $u\in D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)$.

Thus $\lambda u-A_{*}u=f$. On the other hand, we have

$\begin{array}{l}{\left|{\left(\lambda I-A_{\mathrm{<mo>\; *\; </mo>}}\right)}^{-1}f\right|}_{D\left(A\right)}=\Vert \left(I-A\right){\left(\lambda I-A\right)}^{-1}f\Vert \\ =\Vert {\left(\lambda I-A\right)}^{-1}\left(I-A\right)f\Vert \le \frac{1}{\lambda }\Vert f-Af\Vert =\frac{1}{\lambda }{\left|f\right|}_{D\left(A\right)}\end{array}$ (3.5)

which shows that $A_{\mathrm{<mo>\; *\; </mo>}}$ satisfies condition (ii) in Theorem 2.2. Moreover, it follows that $A_{\mathrm{<mo>\; *\; </mo>}}$ is closed in $X_{\mathrm{<mo>\; *\; </mo>}}$.

Indeed, as ${\left(\lambda I-A\right)}^{-1}\in L\left(X_{\mathrm{<mo>\; *\; </mo>}}\right)$, it is closed, and consequently $\lambda I-A_{\mathrm{<mo>\; *\; </mo>}}$ enjoys the same property which proves that $A_{\mathrm{<mo>\; *\; </mo>}}$ is closed.

Now, let $x\in X_{\mathrm{<mo>\; *\; </mo>}}$, $\lambda >0$, $A\in \omega \text{-}OC{P}_n$ and let $x_{\lambda }=\lambda x-A_{*}x$. Clearly $x_{\lambda }\in D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)$, and in addition ${\lim }_{\lambda \to \infty }{\left|x_{\lambda }-x\right|}_{D\left(A\right)}=0$ Thus, $D\left(A_{\mathrm{<mo>\; *\; </mo>}}\right)$ is dense in $X_{\mathrm{<mo>\; *\; </mo>}}$ by virtue of Theorem 2.2, $A_{\mathrm{<mo>\; *\; </mo>}}$ generates a C₀-semigroup of contraction on $X_{\mathrm{<mo>\; *\; </mo>}}$. Hence the proof.

4. Conclusion

In this paper, it has been established that ω-OCP_n possesses the properties of dissipative operators as a semigroup of linear operator, and obtaining some dissipative results on ω-OCP_n.