This paper consists of the results about \(\omega\)-order preserving partial contraction mapping using perturbation theory to generate a one-parameter semigroup. We show that adding a bounded linear operator \(B\) to an infinitesimal generator \(A\) of a semigroup of the linear operator does not destroy A’s property. Furthermore, \(A\) is the generator of a one-parameter semigroup, and \(B\) is a small perturbation so that \(A+B\) is also the generator of a one-parameter semigroup.
Perturbation theory comprises methods for finding an approximate solution to a problem; in perturbation theory, the solution is expressed as a power series in a small parameter \(\varepsilon \). The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \(\varepsilon\) usually become smaller. Assume \(X\) is a Banach space, \(X_n\subseteq X\) is a finite set, \(T(t)\) the \(C_{0}\)-semigroup, \(\omega-OCP_n\) the \(\omega\)-order preserving partial contraction mapping, \(M_{m}\) be a matrix, \(L(X)\) be a bounded linear operator on \(X\), \(P_n\) a partial transformation semigroup, \(\rho(A)\) a resolvent set, \(\sigma(A)\) a spectrum of \(A\) and \(A\in \omega-OCP_n\) is a generator of \(C_{0}\)-semigroup. This paper consists of results of \(\omega\)-order preserving partial contraction mapping generating a one-parameter semigroup.
Akinyele et al., [1] introduced perturbation of the infinitesimal generator in the semigroup of the linear operator. Batty [2] established some spectral conditions for stability of one-parameter semigroup and also in [3] Batty et al., revealed some asymptotic behavior of semigroup of the operator. Balakrishnan [4] obtained an operator calculus for infinitesimal generators of the semigroup. Banach [5] established and introduced the concept of Banach spaces. Chill and Tomilov [6] deduced some resolvent approaches to stability operator semigroup. Davies [7] obtained linear operators and their spectra. Engel and Nagel [8] introduced a one-parameter semigroup for linear evolution equations. R\(\ddot{a}\)biger and Wolf [9] deduced some spectral and asymptotic properties of the dominated operator. Rauf and Akinyele [10] introduced \(\omega\)-order preserving partial contraction mapping and established its properties, also in [11], Rauf et al., deduced some results of stability and spectra properties on semigroup of a linear operator. Vrabie [12] proved some results of \(C_{0}\)-semigroup and its applications. Yosida [13] established and proved some results on differentiability and representation of one-parameter semigroup of linear operators.
In this paper, we show that adding a bounded linear operator \(B\) to an infinitesimal generator \(A\) of a semigroup of the linear operator does not destroy A’s property. Furthermore, \(A\) is the generator of a one-parameter semigroup, and \(B\) is a small perturbation so that \(A+B\) is also the generator of a one-parameter semigroup.
Definition 1.(\(C_0\)-Semigroup) [8] A \(C_0\)-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.
Definition 2.(\(\omega\)-\(OCP_n\))[11] A transformation \(\alpha\in P_{n}\) is called \(\omega\)-order preserving partial contraction mapping if \(\forall x,y \in~ \) Dom \(\alpha:x\le y~~\implies~~ \alpha x\le \alpha y\) and at least one of its transformation must satisfy \(\alpha y=y\) such that \(T(t+s)=T(t)T(s)\) whenever \(t,s>0\) and otherwise for \(T(0)=I\).
Definition 3.(Perturbation) [1] Let \(A : D(A) \subseteq X \to X\) be the generator of a strongly continuous semigroup \((T(t))_{t\geq 0}\) and consider a second operator \(B : D(B) \subseteq X \to X\) such that the sum \(A + B\) generates a strongly continuous semigroup \((S(t))_{t\geq 0}\). We say that \(A\) is perturbed by operator \(B\) or that \(B\) is a perturbation of \(A\).
Definition 4.(Analytic Semigroup) [12] We say that a \(C_0\)-semigroup \(\{T(t); t \geq 0\}\) is analytic if there exists \(0 < \theta \leq \pi\), and a mapping \(S : \bar{\mathbb{C}}_{\theta} \to L(X)\) such that:
Definition 5.(Perturbation class) [7] We say that operator \(B\) is a class \(P\) perturbation of the generator \(A\) of the one-parameter semigroup \(T(t)\) if:
Example 1(\(2\times 2\) matrix \({M_m(\mathbb{N} \cup\{0\})}\)). Suppose \[ A=\begin{pmatrix} 2&0\\ 1&2 \end{pmatrix} \] and let \(T(t)=e^{t A}\), then \[ e^{t A}=\begin{pmatrix} e^{2t}&e^{I}\\ e^{t}& e^{2t} \end{pmatrix} .\]
Example 2(\(3\times 3\) matrix \({M_m(\mathbb{N} \cup \{0\})}\)). Suppose \[ A=\begin{pmatrix} 2&2&3\\2& 2&2\\1& 2&2 \end{pmatrix} \] and let \(T(t)=e^{t A}\), then \[ e^{t A}=\begin{pmatrix} e^{2t}&e^{2t}&e^{3t}\\e^{2t}&e^{2t}&e^{2t}\\e^{t}& e^{2t}&e^{2t} \end{pmatrix}. \]
Example 3(\(3\times 3\) matrix \({M_m(\mathbb{C})}\)). Since we have for each \(\lambda>0\) such that \(\lambda\in \rho(A)\) where \(\rho(A)\) is a resolvent set on \(X\). Suppose we have \[ A=\begin{pmatrix} 2&2&3\\ 2&2&2\\ 1&2&2 \end{pmatrix} \] and let \(T(t)=e^{t A_\lambda}\), then \[ e^{t A_\lambda}=\begin{pmatrix} e^{2t\lambda}&e^{2t\lambda}&e^{3t\lambda}\\ e^{2t\lambda}&e^{2t\lambda}&e^{2t\lambda}\\ e^{t\lambda}&e^{2t\lambda}&e^{2t\lambda}\end{pmatrix} .\]
Theorem 1. Let \(A\in\omega-OCP_n\) be the generator of a one-parameter semigroup \(T(t)_{z\geqslant 0}\) on the Banach space \(X\) and suppose that \[ \|T(t)\|\leqslant Me^{at} \] for all \(t\geqslant 0\). If \(B\) is a bounded operator on \(X\), then \((A+B)\) is the generator of a one-parameter semigroup \(S(t)_{t\geqslant 0}\) on \(X\) such that \[ \|S(t)\|\leqslant Me^{(a+M\|B\|)t} \] for all \(t\geqslant 0\) and \(B\in\omega-OCP_n\).
Proof. We define the operators \(S(t)\) by
Since \(S(s)S(t)=S(s+t)\) and if \(f\in X\), then
\[ \lim\limits_{t\rightarrow 0}\|s(t)f-f\|\leqslant\lim\limits_{t\rightarrow 0}\left\{\|T(t)f-f\|+\sum_{n=1}^{\infty}Me^{at}\|f\|(tM\|B\|)^n/n!\right\}\geqslant 0 \,,\] so that \(s(t)\) is a one-parameter semigroup. If \(f\in X\) and \(B\in\omega-OCP_n\), thenAs well as being illuminating in its own right, (2) easily leads to the identities
Theorem 2. Suppose \(B\) is a class \(P\) perturbation of the generator \(A\), then \[ Dom(B)\supseteq Dom(A). \] If \(\varepsilon>0\) and \(A,B\in\omega-OCP_n\), then
Proof. Combining (1) with the bound \[ \|BT(t)\|\leqslant\|BT(t)\|Me^{a(t-1)}\,, \] valid for all \(t\geqslant 1\), we then see that \[ \int_{0}^{\infty}\|BT(t)\|e^{-\lambda t}dta\). Suppose \(\varepsilon>0\) and \(A,B\in\omega-OCP_n\), then for all large enough \(\lambda\) we have \[ \int_{0}^{\infty}\|BT(t)\|e^{-\lambda t}dt\leqslant\varepsilon. \] Now, \[ \int_{0}^{\infty}T(t)e^{-\lambda t}fdt = R(\lambda,A)f \,,\] for all \(f\in X\), so by the closedness of \(B\), we see that \(R(\lambda,A)f\in Dom(B)\) and \[ \|BR(\lambda,A)f\|\leqslant\varepsilon\|f\| \,,\] as required to prove (7).
If \(g\in Dom(A)\) and we put \(f:=(\lambda I-A)g\), then we deduce from (7) that
Theorem 3. Assume \(B\) is a class \(P\) perturbation of the generator \(A\) of the one-parameter semigroup \(T(t)\) on \(X\), then \(B+A\) is the generator of a one-parameter semigroup \(S(t)\) on \(X\) and \(A,B\in\omega-OCP_n\).
Proof. Let \(a\) be small enough that
Assume \(f\in X\) and \(B\in\omega-OCP_n\), then
\[ \|S(t)f-f\|\leqslant\|T(t)f-f\|+\sum_{n=1}^{\infty}N\left(\int_{0}^{t}\|BS(t)\|ds\right)^n\|f\|, \] so thatWe finally have to identify the generator \(Y\) of \(S(t)\). The subspace
\[ D:=\underset{t>0}{\bigcup}T(t)\{Dom(A)\} \,,\] is contained in \(Dom(A)\) and is invariant under \(T(t)\) and so is a core for \(A\). If \(f\in D\), then there exists \(g\in Dom(A)\) where \(A\in\omega-OCP_n\) and \(\varepsilon>0\) such that \(f=T(\varepsilon)g\). Hence,If \(\lambda\) is also large enough that
\[ \|BR(\lambda,A)\|< 1 \,,\] we deduce that \[ R(\lambda,Y)=R(\lambda,A)(I-BR(\lambda,A))^{-1}. \] Hence, \[ Dom(Y)=Ran(R(\lambda,Y))=Ran(R(\lambda,A))=Dom(A) \,,\] and \(Y=A+B\), and this achieve the proof.Theorem 4. Let \(A:=-H\) where \(H=(-\Delta)^n\geqslant 0\) acts in \(L^2(\mathbb{R}^N)\). Also let \(B\) be a lower order perturbation of the form \[ (Bf)(x):=\sum_{|\alpha|N/(2n-|\alpha|)\), the \(A+B\) is the generator of a one-parameter semigroup and \(B\) has relative bound \(0\) with respect to \(A\) where \(A,B\in\omega-OCP_n\).
Proof. Suppose \(A\in\omega-OCP_n\) is the generator of holomorphic semigroup \(T(t)\) such that \[ \|T(t)\|\leqslant c_1,\quad \|AT(t)\leqslant c_2/t\| \,,\] for all \(t\in(0,1)\). And also the operator \(B\in\omega-OCP_n\) has domain containing \(Dom(A)\) and there exists \(\alpha\in(0,1)\), such that
Assume \(\alpha\in(0,1)\), \(H\) is a non-negative self-adjoint operator on \(P\) and \(B\) is a linear operator with \(Dom(B)\geqslant(H)\), we have
\[ \|Bf\|\leqslant\varepsilon\|Af\|+c_3\varepsilon^{-\alpha/(1-\alpha)}\|f\| \,,\] for all \(\varepsilon>0\) if and only if there is a constant \(c_4\) such that \[ \|Bf\|\leqslant c_4\|Af\|^\alpha\|f\|^{1-\alpha} \,,\] for all \(f\in Dom(A)\) and \(A,B\in\omega-OCP_n\).By Theorem 3, it is sufficient to prove that for each \(\alpha\) there exists \(\beta< 1\) for which
\[ X_\alpha:=a_\alpha(\cdot)D^\alpha(H+1)^{-\beta} \] is bounded.Let \(X_\alpha=a_\alpha(Q)b_\alpha(P)\), where
\[ b_\alpha(\varepsilon)=\frac{i^{|\alpha|}\varepsilon^\alpha}{(|\varepsilon|^{2n}+1)^\beta}. \] If \(a_\alpha\in L^\infty(\mathbb{R}^N)\), then \(\|X\|\leqslant\|a_\alpha\|_\infty\|b_\alpha\|_\infty< \infty\) provided \(|\alpha|/2n< \betaN/(2n-|\alpha|)\), then there exists \(\beta\) such that \[ \frac{N+|\alpha|P}{2np}< \beta< 1. \] This implies that \((|\alpha|-2n\beta)p+N< 0\) and hence \(b_\alpha\in L^p(\mathbb{R}^N)\).