ABSTRACT

This paper consists of dissipative properties and results of dissipation on infinitesimal generator of a C₀-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.

Keywords:

Semigroup, Linear Operator, Dissipative Operator, Contraction Mapping and Resolvent

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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Akinyele, A.Y, Rauf, K., Adebowale, A.M. and Babatunde, O.J. (2019) Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator. Advances in Pure Mathematics, 9, 544-550. https://doi.org/10.4236/apm.2019.96026

References