Results of a perturbation theory generating a one-parameter semigroup

1. Introduction

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.

2. Preliminaries

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:

1. \(T(t) = S(t)\) for each \(t \geq 0\);
2. \(S(z_{1} + z_{2}) = S(z_{1})S(z_{2})\) for \(z_{1},z_{2} \in \bar{\mathbb{C}}_{\theta}\);
3. \(\lim_{z_{1} \in \bar{\mathbb{C}}_{\theta},z_{1} \to 0} S(z_{1})x = x\) for \(x \in X\); and
4. the mapping \(z_{1} \to S(z_{1})\) is analytic from \(\bar{\mathbb{C}}_{\theta}\) to \(L(X)\). In addition, for each \(0 < \delta < \theta \), the mapping \(z_{1} \to S(z_{1})\) is bounded from \( \mathbb{C}_{\delta}\) to \(L(X)\), then the \(C_{0}\)-Semigroup \(\{T(t);t \geq 0\}\) is called analytic and uniformly bounded.

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:

Note that \(BT(t)\) is bounded for all \(t > 0\) under conditions (1)\(_1\) and (1)\(_2\) by the closed graph theorem.

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 \[ e^{t A_\lambda}=\begin{pmatrix} e^{2t\lambda}&e^{2t\lambda}&e^{3t\lambda}\\ e^{2t\lambda}&e^{2t\lambda}&e^{2t\lambda}\\ e^{t\lambda}&e^{2t\lambda}&e^{2t\lambda}\end{pmatrix} .\]

3. Main results

This section present results of one-parameter semigroup generated by \(\omega\)-\(OCP_{n}\) using perturbation theory.

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

The \(nth\) term is an \(n\)-fold integral whose integrand is a norm continuous function of the variables. It is easy to verify that the series is norm convergent and that for all \(f\in X\), \(t\geqslant 0\) and \(B\in\omega-OCP_n\).

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 \(B\in\omega-OCP_n\), then It follows that \(f\) lies in the domain of the generator \(Y\) of \(S(t)\) if and only if it lies in the domain of \(A\), and that for such \(f\).

As well as being illuminating in its own right, (2) easily leads to the identities

Hence the proof is complete.

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

for all large enough \(\lambda>0.\) Hence \(B\) has relative bound \(0\) with respect to \(A\).

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 \[ \int_{0}^{\infty}\|BT(t)\|e^{-\lambda t}dt< \infty\,,\] for all \(\lambda>a\). 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

for all large enough \(\lambda>0\). This implies the last statement of the theorem and hence the proof is complete.

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

We may define \(S(t)\) by the convergent series (2) for \(0\leqslant t\leqslant 2a\), and verify as in the proof of Theorem 1 that \(S(s)S(t)=S(s+t)\) for all \(s,t\geqslant 0\) such that \(s+t\leqslant 2a\). We now extend the definition of \(S(t)\) inductively for \(t\geqslant 2a\) by putting if \(n\in\mathbb{N}\) and \(na< t\leqslant(n+1)a\). It is straight forward to verify that \(S(t)\) is a semigroup. Now suppose that \(\|T(t)\|\leqslant N\) for \(0\leqslant t\leqslant a\).

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 that and \(S(t)\) is a one-parameter semigroup on \(X\). It is an immediate consequence of the definition that for all \(f\in X\), \(B\in\omega-OCP_n\) and all \(0\leqslant t\leqslant a\). Suppose that this holds for all \(t\) such that \(0\leqslant t\leqslant na\). If \(na\leqslant u\leqslant(n+1)a\), then By induction, (12) holds for all \(t\geqslant 0\).

We 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, Therefore, \(Dom(Y)\) contains \(D\) and \(Yf(B+A)\) for all \(f\in D\) and \(A,B\in\omega-OCP_n\). If \(f\in Dom(A)\), then there exists a sequence \(f_n\in D\) such that \(\|f_n-f\|\rightarrow 0\) and \(\|Af_n-Af\|\rightarrow 0\) as \(n\rightarrow\infty\). It follows by Theorem 2 that \(\|Bf_n-Bf\|\rightarrow 0\) and hence that \(Yf_n\) converges. Since \(Y\) is a generator that is closed, then we deduce that \[ Yf=(B+A)f \,,\] for all \(f\in Dom(A)\) and \(A,B\in\omega-OCP_n\). Multiplying (12) by \(e^{-\lambda t}\) and integrating over \((0,\infty)\), we see as in the proof of Theorem 2 that if \(\lambda>0\) is large enough, then \[ R(\lambda,Y)f=R(\lambda,A)f+R(\lambda,Y)BR(\lambda,A)f \,,\] for all \(f\in Y\) and \(A,B\in\omega-OCP_n\).

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 \[ (Bf)(x):=\sum_{|\alpha|< 2n}a_\alpha(x)(D^\alpha f)(x). \] If \(a_\alpha\in L^{P_\alpha}(\mathbb{R}^N)+L^\infty(\mathbb{R}^N)\) for each \(\alpha\), where \(P_\alpha\geqslant 2\) and \(P_\alpha>N/(2n-|\alpha|)\), 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

for all \(f\in Dom(a)\) and \(0< \varepsilon\leqslant 1\). Then for all \(t\in(0,1)\) so that \(B\) is a class \(P\) perturbation of \(A\) and by Theorem 3 under the stated conditions on \(t\) and \(\varepsilon\), we have \begin{align*} \|BT(t)f\|&\leqslant\varepsilon\|AT(t)f\|+c_3\varepsilon^{-\alpha/(1-\alpha)}\|T(t)f\|\\ &\leqslant(\varepsilon c_2t^{-1}+c_1c_3\varepsilon^{-\alpha/(1-\alpha)})\|f\|. \end{align*} By putting \(\varepsilon=t^{1-\alpha}\), then we obtain (16).

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 \(|\alpha|/2n< \beta< 1\). On the other hand, if \(a_\alpha\in L^P(\mathbb{R}^N)\) where \(P\geqslant 2\) and \(P>N/(2n-|\alpha|)\), then there exists \(\beta\) such that \[ \frac{N+|\alpha|P}{2np}< \beta< 1. \] This implies that \((|\alpha|-2n\beta)p+N< 0\) and hence \(b_\alpha\in L^p(\mathbb{R}^N)\).

4. Conclusion

In this paper, it has been established that \(\omega\)-order preserving partial contraction mapping generates a one-parameter semigroup using a perturbation theory on Banach space by showing that the semigroup of a linear operator is bounded, that \(B\) has a relative bound \(0\) with respect to \(A\), and also that \(B + A\) is a generator of the one-parameter semigroup.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''

Data Availability:

All data required for this research is included within this paper.

Funding Information:

No funding is available for this research.