Hypothesis 2 Let $A$ be a positive linear operator with discrete spectrum on $(H,\|\cdot\|)$, i.e., there exists an orthonormal basis $\{e_{k}\}_{k\in\mathbb{Z}^{+}}$ of $H$ such that

\begin{align*} Ae_{k}=\lambda_{k}e_{k}, \ \ \ with \ 0 \lt \lambda_{1}\leq\lambda_{2}\leq\cdots, \ \ \ \lim_{k\rightarrow\infty}\lambda_{k}=\infty. \end{align*}

Hypothesis 3 Assume that the nonlinear term $B:C_{\alpha}\times\Omega\rightarrow H, \ (u,\omega)\mapsto B(u,\omega)$ satisfies:

1. (i) the mapping $B$ is $(\mathcal{B}(C_{\alpha})\otimes\mathcal{F},\mathcal{B}(H))$-measurable;

2. (ii) the mapping $B$ is Lipschitz continuous, i.e., there exists some constant $L \gt 0$ such that

   (2.3)\begin{equation} \|B(u_1,\omega)-B(u_2,\omega)\|\leq L|u_1-u_2|_{C_{\alpha}}, \ \ \forall u_1,u_2\in C_{\alpha}, \ \omega\in\Omega; and \end{equation}

3. (iii) $K_0:=\sup_{\omega\in \Omega}\|B(0,\omega)\| \lt \infty$.

Hypothesis 4 $B(u,\omega)$ is $C^{1 }$ in $u$.

Proof. For claim (i), by using the integration by parts formula, we find that

\begin{align*} \eta_0(t,\sigma,\omega)-\eta_\delta(t,\sigma,\omega) & = C\left[W(t)-W(\sigma)-\int_\sigma^t \zeta_{\delta}(\theta_{s}\omega){ds}\right]\\ & \quad-\int_\sigma^t Ae^{-A(t-\tau)}C[W(\tau)-W(\sigma) -\int_\sigma^\tau \zeta_{\delta}(\theta_{s}\omega){ds} ]d\tau. \end{align*}

Along with (2.1), we get that

\begin{align*} & \|\eta_0(t,\sigma,\omega)-\eta_\delta(t,\sigma,\omega)\|_\alpha\\ & \quad\le \|C\|_{L(H,D(A))} \sup_{t\in[\sigma, \sigma+T]}\left\|W(t)-W(\sigma)-\int_\sigma^t \zeta_{\delta}(\theta_{s}\omega){ds}\right\|\\ & \qquad\times \left[\|A^{\alpha-1}\|+\int_\sigma^t K(\alpha) (t-\tau)^{-\alpha} e^{-\lambda (t-\tau)} d\tau\right]. \end{align*}

Consequently, in view of (1.1) and the fact that

\begin{align*} \int_\sigma^t K(\alpha) (t-\tau)^{-\alpha} e^{-\lambda (t-\tau)} d\tau \le K(\alpha) \lambda^{\alpha-1} \Gamma(1-\alpha), \end{align*}

claim (i) is true.

To prove claim (ii), we first consider the case $\delta =0$. Employing the integration by parts formula and (2.1), we find that

\begin{align*} & \|\eta_0(t,\sigma,\omega)-P_{M}\eta_0(t,\sigma,\omega)\|_\alpha\\ & \quad\le \|CW(t)-CP_M W(t)\|_\alpha +\|CP_M W(\sigma)-CW(\sigma)\|_\alpha\\ & \qquad + \left\|\int_\sigma^t Ae^{-A(t-\tau)}C[P_M W(\tau)-P_M W(\sigma)-(W(\tau)- W(\sigma))]d\tau\right\|_\alpha\\ & \quad\le \|C\|_{L(H,D(A))}(\|A^{\alpha-1}\|+K(\alpha) \lambda^{\alpha-1} \Gamma(1-\alpha))\\ & \qquad\times [\sup_{t\in[\sigma, \sigma+T]}\{\|W(t)-P_M W(t)\|\}+\|P_M W(\sigma)-W(\sigma)\|]. \end{align*}

Since $W(t)-P_MW(t), t \in [\sigma-1, \sigma+T+1]$ is a continuous function, for any $t_0 \in [\sigma, \sigma+T]$ and $\epsilon \gt 0$, there exists a small enough $\delta_{t_0}( \gt 0)$ neighbourhood $U(t_0;\delta_{t_0})$ such that

(2.13)\begin{equation} \|W(t)-P_M W(t)-(W(t_0)-P_M W(t_0))\| \lt \frac{\epsilon}{2}, \forall t \in U(t_0;\delta_{t_0}). \end{equation}

Then $\{U(t_0;\delta_{t_0})\}_{t_0 \in [\sigma, \sigma+T]}$ forms an open cover of $[\sigma, \sigma+T]$. Applying the Finite Covering Theorem, there exists a finite sequence $\{t_j\}_{j=1}^{m_1} \subset [\sigma, \sigma+T]$ such that $\cup_{j=1}^{m_1} U(t_j;\delta_{t_j})\supset [\sigma, \sigma+T]$. On the other hand, clearly, for each $t_j, 1\le j \le m_1$, there exists a $M_j \gt 0$ such that for all $M \ge M_j$,

(2.14)\begin{equation} \|W(t_j)-P_M W(t_j)\| \lt \frac{\epsilon}{2}. \end{equation}

Taking $M_0:=\max_{1 \le j \le m_1}\{M_j\}$, for any $M \ge M_0$, by combining (2.13) with (2.14), we get that

\begin{align*} \|W(t)-P_M W(t)\| \lt \epsilon, \forall t \in [\sigma, \sigma+T]. \end{align*}

This implies that

\begin{equation*}\sup_{t\in[\sigma, \sigma+T]}\{\|W(t)-P_M W(t)\|\}\rightarrow 0\end{equation*}

as $M\rightarrow +\infty$. Thus claim (ii) holds for $\delta =0$.

Next, for the case $\delta \neq 0$, using the same argument as $\delta =0$, we get that

\begin{align*} & \|\eta_\delta(t,\sigma,\omega)-P_{M}\eta_\delta(t,\sigma,\omega)\|_\alpha\\ & \quad\le \|C\|_{L(H,D(A))} K(\alpha) \lambda^{\alpha-1} \Gamma(1-\alpha) \times \int_\sigma^{\sigma+T} \|\zeta_{\delta}(\theta_{s}\omega)-P_M\zeta_{\delta}(\theta_{s}\omega)\|{ds}. \end{align*}

For any $s \in [\sigma, \sigma+T]$, we have that

\begin{equation*}\|\zeta_{\delta}(\theta_{s}\omega)-P_M\zeta_{\delta}(\theta_{s}\omega)\|\rightarrow 0\end{equation*}

as $M\rightarrow +\infty$ and

\begin{equation*}\|\zeta_{\delta}(\theta_{s}\omega)-P_M\zeta_{\delta}(\theta_{s}\omega)\| \le 2 \|\zeta_{\delta}(\theta_{s}\omega)\|.\end{equation*}

By the Dominated Convergence Theorem, claim (ii) also holds for any $\delta \neq 0$.

Finally, we observe that for any $\sigma \le t \le \sigma+T$ and $\delta \neq 0$,

\begin{align*} & \|P_{M}\eta_\delta(t,\sigma,\omega)-\eta_0(t,\sigma,\omega)\|_{\alpha}\\ & \quad\le \| \eta_\delta(t,\sigma,\omega)- \eta_0(t,\sigma,\omega)\|_{\alpha}+\|P_{M}\eta_0(t,\sigma,\omega)-\eta_0(t,\sigma,\omega)\|_{\alpha}. \end{align*}

Then claim (iii) follows directly from claims (i)–(ii). Thus this lemma is complete.

Proof. The case $\delta = 0$ has been settled by [Reference Chueshov and Scheutzow9, Proposition 3.1], which proves the existence of $\bar{\eta}_0$ satisfying (i)–(vi). We only consider the case $\delta \neq 0$. For each $\delta\neq 0$, let

\begin{align*} \bar{\eta}_{\delta}(t,s,\omega):= \left\{ \begin{array}{ll} \int_{s}^{t}e^{-A(t-\tau)}C\zeta_{\delta}(\theta_{\tau}\omega){d\tau} , & {\omega \in \tilde{\Omega}}, \\ 0, & {\omega \notin \Omega \setminus \tilde{\Omega}}, \end{array} \right. \end{align*}

where $\tilde{\Omega}$ is the $\theta_t$-invariant set of full measure which is given by [Reference Chueshov and Scheutzow9, Proposition 3.1]. Obviously, $\bar{\eta}_{\delta}(t,s,\omega)$ satisfies (i)–(v). Next we show (vi) for $\bar{\eta}_{\delta}$. By using (1.2) and (2.1), for any $\beta \gt 0$ and $\omega \in \Omega$, we have

\begin{align*} \|\bar{\eta}_{\delta}(t,-\infty,\omega)\|_\alpha e^{-\beta|t|} &\leq \|C\|\int_{-\infty}^{t} K(\alpha)(t-\tau)^{-\alpha}e^{-\lambda(t-\tau)}K_{\delta}C_{\omega}(|\tau|+1) e^{-\beta|t|} {d\tau}\\ & = \|C\|K(\alpha)K_{\delta}C_{\omega}\left(\frac{1}{\lambda}\right)^{1-\alpha}\int_{0}^{+\infty}u^{-\alpha}e^{-u}(|t-\frac{1}{\lambda}u|+1) e^{-\beta|t|} {du} \\ &\leq \|C\| K(\alpha)K_{\delta}C_{\omega}\left(\frac{1}{\lambda}\right)^{1-\alpha}(|t|+1)e^{-\beta|t|}\Gamma(1-\alpha) \\ & \quad +\|C\| K(\alpha)K_{\delta}C_{\omega}\left(\frac{1}{\lambda}\right)^{2-\alpha}e^{-\beta|t|}\Gamma(2-\alpha) \lt \infty, \end{align*}

where the first equality follows from the integral transformation $u=\lambda(t-\tau)$. Then the proof of Proposition 2.1 is complete.

Proof. The case $\delta=0$ has been proved in [Reference Chueshov and Scheutzow9, Theorem 3.1]. Here we focus on the case $\delta\neq 0$. The global existence of solutions in this case is established by applying the method of [Reference Travis and Webb27]. Let $K(\alpha)$ and $\lambda$ be given by (2.1). We arbitrarily fix a constant $\rho \gt 0$ such that

\begin{equation*} K(\alpha)L(\lambda+\rho)^{\alpha-1}\Gamma(1-\alpha) \lt \frac{1}{2} \ \mbox{and} \ e^{-\rho \sigma} \lt \frac{1}{2}. \end{equation*}

Define a Banach space

\begin{align*} \mathcal{C}_{\rho,\sigma}:= \{u\in C([\sigma-r,\infty);D(A^{\alpha})):\sup_{t\geq\sigma-r} \{e^{-\rho (t-\sigma)}\|u(t)\|_\alpha\} \lt \infty\} \end{align*}

with the norm

\begin{align*} \|u\|_{\mathcal{C}_{\rho,\sigma}}:=\sup_{t\geq\sigma-r}\{e^{-\rho (t-\sigma)}\|u(t)\|_\alpha\}. \end{align*}

Fix $\omega\in\Omega$, $\sigma\in\mathbb{R}$ and $u^{\delta}_{\sigma}\in C_{\alpha}$. For any $u^{\delta} \in \mathcal{C}_{\rho,\sigma}$, define the mapping $\mathcal{P}_\delta^\omega$ on $\mathcal{C}_{\rho,\sigma}$ by

\begin{align*} (\mathcal{P}_\delta^\omega (u^{\delta}))(t)= \left\{ \begin{array}{ll} e^{-A(t-\sigma)}u^{\delta}_{\sigma}(0)+\int_{\sigma}^{t}e^{-A(t-\tau)}B(u^{\delta}_{\tau},\theta_{\tau}\omega){d\tau} +\eta_{\delta}(t,\sigma,\omega) , & {t \gt \sigma },\\ u^{\delta}_{\sigma}(t-\sigma), & {\sigma-r\leq\! t\!\leq \sigma}. \end{array} \right. \end{align*}

We claim that $\mathcal{P}_\delta^\omega$ is a contraction on $\mathcal{C}_{\rho,\sigma}$. Indeed, for $t \gt \sigma$, by using (2.1), we obtain that

\begin{align*} \|(\mathcal{P}_\delta^\omega (u^{\delta}))(t)\|_\alpha &\leq K(0)e^{-\lambda (t-\sigma)} \|u^{\delta}_{\sigma}(0)\|_\alpha + K(\alpha)\int_{\sigma}^{t}(t-\tau)^{-\alpha}e^{-\lambda (t-\tau)}\|B(u^{\delta}_{\tau},\theta_{\tau}\omega)\|d\tau \\ &\quad+\|\eta_{\delta}(t,\sigma,\omega)\|_\alpha. \end{align*}

By Remark 2.1, there exists a constant $M_1 \gt 0$ such that $\|\eta_{\delta}(t,\sigma,\omega)\|_\alpha \leq M_{1}e^{\frac{\rho}{2}|t|}$, and by Hypothesis 3, we find that

\begin{align*} \|B(u^{\delta}_{\tau},\theta_{\tau}\omega)\| &\leq \|B(u^{\delta}_{\tau},\theta_{\tau}\omega)-B(0,\theta_{\tau}\omega)\|+\|B(0,\theta_{\tau}\omega))\|\\ &\leq L|u^{\delta}_{\tau}|_{C_{\alpha}}+K_0 \lt \infty. \end{align*}

Thus we have that

(3.2)\begin{align} & e^{-\rho (t-\sigma)}\|(\mathcal{P}_\delta^\omega (u^{\delta}))(t)\|_\alpha \leq K(0)|u^{\delta}_{\sigma}|_{C_\alpha}+e^{-\rho (t-\sigma)}M_{1}e^{\frac{\rho}{2}|t|}\nonumber\\ & \quad +K(\alpha)\int_{\sigma}^{t}(t-\tau)^{-\alpha}e^{-\lambda(t-\tau)}e^{-\rho(t-\sigma)}(L|u^{\delta}_{\tau}|_{C_{\alpha}}+K_0)d\tau.\end{align}

For the second term in (3.2), we observe that

\begin{align*} e^{-\rho (t-\sigma)}M_{1}e^{\frac{\rho}{2}|t|} &\leq e^{-\rho (t-\sigma)}M_{1}e^{\frac{\rho}{2}|t-\sigma|}e^{\frac{\rho}{2}|\sigma|} \\ &= e^{(-\rho+\frac{\rho}{2}) (t-\sigma)}M_{1}e^{\frac{\rho}{2}|\sigma|} \le M_{1}e^{\frac{\rho}{2}|\sigma|} \lt \infty. \end{align*}

For the third term in (3.2), we find that

(3.3)\begin{align} & K(\alpha)\int_{\sigma}^{t}(t-\tau)^{-\alpha}e^{-\lambda(t-\tau)}e^{-\rho(t-\sigma)}L|u^{\delta}_{\tau}|_{C_{\alpha}}d\tau\nonumber\\ & \quad = \ K(\alpha)L\int_{\sigma}^{t}(t-\tau)^{-\alpha} e^{-\lambda(t-\tau)}e^{-\rho(t-\sigma)}\sup_{\xi \in [-r,0]}\|u^{\delta}(\tau+ \xi)\|_\alpha d\tau\nonumber\\ & \quad \leq \ K(\alpha)L\|u^{\delta}\|_{\mathcal{C}_{\rho, \sigma}}\int_{\sigma}^{t}(t-\tau)^{-\alpha}e^{-(\lambda+\rho)(t-\tau)} \sup_{\xi \in [-r,0]}\{e^{\rho \xi}\}d\tau\nonumber\\ & \quad \le K(\alpha)L\|u^{\delta}\|_{\mathcal{C}_{\rho,\sigma}} (\lambda+\rho)^{\alpha-1}\Gamma(1-\alpha) \lt \frac{1}{2}\|u^{\delta}\|_{\mathcal{C}_{\rho,\sigma}} \lt \infty,\end{align}

and

\begin{align*} K(\alpha)K_0\int_{\sigma}^{t}(t-\tau)^{-\alpha}e^{-(\lambda+\rho)(t-\tau)}d\tau & \le K(\alpha)K_0(\lambda+\rho)^{\alpha-1}\Gamma(1-\alpha) \lt \infty. \end{align*}

It then follows that

\begin{align*} \sup_{t \gt \sigma}\{e^{-\rho (t-\sigma)}\|(\mathcal{P}_\delta^\omega (u^{\delta}))(t)\|_\alpha\} \lt \infty. \end{align*}

Additionally, by the definition of $\mathcal{P}_\delta^{\omega}$ and $u^{\delta}_{\sigma}\in C_{\alpha}$, we get that

\begin{align*} \sup_{\sigma\geq t\geq\sigma-r} \{e^{-\rho (t-\sigma)}\|(\mathcal{P}_\delta^\omega (u^{\delta}))(t)\|_\alpha\} & \le \sup_{\sigma\geq t\geq\sigma-r}\{\|u^{\delta}_{\sigma}(t-\sigma,\omega)\|_{\alpha}\}=|u^{\delta}_{\sigma}|_{C_\alpha} \lt \infty. \end{align*}

Thus $\mathcal{P}_\delta^{\omega}$ is a self-mapping from $\mathcal{C}_{\rho,\sigma}$ to $\mathcal{C}_{\rho,\sigma}$. Then considering any $u^{\delta}, \hat{u}^{\delta}\in \mathcal{C}_{\rho,\sigma}$ with the same initial data $u^{\delta}_{\sigma}$, for $t \gt \sigma$, by using the same argument as (3.3), we find that

\begin{align*} & e^{-\rho (t-\sigma)} \|(\mathcal{P}_\delta^\omega (u^{\delta}))(t)-(\mathcal{P}_\delta^{\omega}(\hat{u}^{\delta}))(t)\|_\alpha\\ & \quad\leq e^{-\rho (t-\sigma)} \int_{\sigma}^{t} (t-\tau)^{-\alpha}e^{-\lambda (t-\tau)} \|B(u^{\delta}_{\tau},\theta_{\tau}\omega)-B(\hat{u}^{\delta}_{\tau},\theta_{\tau}\omega)\|_\alpha d\tau\\ & \quad\leq K(\alpha)L\|u^{\delta}-\hat{u}^{\delta}\|_{\mathcal{C}_{\rho,\sigma}}(\lambda+\rho)^{\alpha-1}\Gamma(1-\alpha) \le \frac{1}{2}\|u^{\delta}-\hat{u}^{\delta}\|_{\mathcal{C}_{\rho,\sigma}}. \end{align*}

Besides, for $\sigma-r\leq t\leq \sigma$, by using the definition of $\mathcal{P}_\delta^{\omega}$, we obtain that

\begin{align*} & e^{-\rho (t-\sigma)} \|(\mathcal{P}_\delta^\omega (u^{\delta}))(t)-(\mathcal{P}_\delta^{\omega}(\hat{u}^{\delta}))(t)\|_\alpha\\ & \quad= e^{-\rho (t-\sigma)} \| u^{\delta}_{\sigma}(t-\sigma)-\hat{u}^{\delta}_{\sigma}(t-\sigma) \|_{\alpha} \leq \frac{1}{2}\|u^{\delta}-\hat{u}^{\delta}\|_{\mathcal{C}_{\rho,\sigma}}. \end{align*}

These facts yield that

\begin{align*} \|\mathcal{P}_\delta^\omega (u^{\delta})-\mathcal{P}_\delta^{\omega}(\hat{u}^{\delta})\|_{\mathcal{C}_{\rho,\sigma}} \leq \frac{1}{2}\|u^{\delta}-\hat{u}^{\delta}\|_{\mathcal{C}_{\rho,\sigma}}. \end{align*}

Therefore, by using the Contraction Mapping Principle, we find that $\mathcal{P}_\delta^{\omega}$ has a unique fixed point $u^{\delta} (\cdot,\sigma,\omega,u^{\delta}_{\sigma})\in \mathcal{C}_{\rho,\sigma}$ for any $u^{\delta}_{\sigma}\in C_{\alpha}$.

To prove claim (ii), for any $u^{\delta}_{\sigma}, \hat{u}^{\delta}_{\sigma} \in C_\alpha$, with the same analysis as above, we observe that

\begin{align*} & \sup_{t \gt \sigma} \{e^{-\rho (t-\sigma)} \|u^{\delta} (t,\sigma,\omega,u^{\delta}_{\sigma})-u^{\delta} (t,\sigma,\omega,\hat{u}^{\delta}_{\sigma})\|_\alpha \} \\ &\leq K(0) |u^{\delta}_{\sigma}-\hat{u}^{\delta}_{\sigma}|_{C_\alpha}+\frac{1}{2}\|u^{\delta} (\cdot,\sigma,\omega,u^{\delta}_{\sigma})-u^{\delta} (\cdot,\sigma,\omega,\hat{u}^{\delta}_{\sigma})\|_{\mathcal{C}_{\rho,\sigma}}, \end{align*}

and

\begin{align*} \sup_{\sigma\geq t\geq\sigma-r} \{e^{-\rho (t-\sigma)} \|u^{\delta} (t,\sigma,\omega,u^{\delta}_{\sigma})-u^{\delta} (t,\sigma,\omega,\hat{u}^{\delta}_{\sigma})\|_\alpha \} \le |u^{\delta}_{\sigma}-\hat{u}^{\delta}_{\sigma}|_{C_\alpha}, \end{align*}

which suggests that

\begin{align*} \|u^{\delta} (\cdot,\sigma,\omega,u^{\delta}_{\sigma})-u^{\delta} (\cdot,\sigma,\omega,\hat{u}^{\delta}_{\sigma})\|_{\mathcal{C}_{\rho,\alpha}} \le (2K(0)+1)|u^{\delta}_{\sigma}-\hat{u}^{\delta}_{\sigma}|_{C_\alpha}. \end{align*}

Thus $u^\delta(t,\sigma, \omega,u^\delta_\sigma)$ is Lipschitz continuous in $u^\delta_\sigma$. On the other hand, since $u^\delta(t, \sigma, \omega,u^\delta_\sigma)$ can be regarded as an $\omega$-wise limit of the iteration of contraction mapping $\mathcal{P}_\delta^{\omega}$, which maps an $\mathcal{F}$-measurable function to an $\mathcal{F}$-measurable function, $u^\delta(t,\sigma,\omega,u^\delta_\sigma)$ is $\mathcal{F}$-measurable in $\omega$ as well. Employing [Reference Castaing and Valadier4, Lemma III.14], $u^\delta(t,\sigma, \omega,u^\delta_\sigma)$ is measurable with respect to $(\omega,u^\delta_\sigma)$. Recall that $u^\delta(t,\sigma,\omega,u_\sigma^\delta)$ is continuous in $t$ for each $(\omega,u_\sigma^\delta)$. Applying [Reference Castaing and Valadier4, Lemma III.14] again, $u^\delta(t,\sigma,\omega,u_\sigma^\delta)$ is measurable with respect to $(t,\omega,u_\sigma^\delta)$. Claim (ii) is established.

Finally, we prove claim (iii). To this end, we first notice that $\Phi^{\delta}(0,\omega,\cdot)=Id_{C_{\alpha}}$ for each $\omega \in \Omega$ due to $\Phi^{\delta}(0,\omega,\tilde{u})(\xi)=u^{\delta}(0+\xi,0,\omega,\tilde{u})=\tilde{u}(\xi)$ for $\tilde{u} \in C_\alpha$ and $\xi \in [-r,0]$.

Then for given $\tilde{u} \in C_{\alpha}$ and $t \ge s$, define $V^{\delta}(t,s,\omega,\tilde{u}) \in C_\alpha$ by $V^{\delta}(t,s,\omega,\tilde{u})(\xi):=u^{\delta}(t+\xi,s,\omega,\tilde{u})$ for $\xi\in[-r,0]$. By using the uniqueness of solutions to (2.8), we get that

(3.4)\begin{equation} V^{\delta}(t,s,\omega,\tilde{u})=V^{\delta}(t+\tau,s+\tau,\theta_{-\tau}\omega,\tilde{u}),\forall t \ge s, \tau \in \mathbb{R}, \tilde{u} \in C_\alpha, \omega\in\Omega, \end{equation}

and

(3.5)\begin{equation} V^\delta(t,s,\omega, V^\delta(s,0,\omega, \tilde{u}))=V^\delta(t,0,\omega,\tilde{u}),\ \forall t\ge s, \tilde{u} \in C_\alpha, \omega\in\Omega. \end{equation}

Therefore, for all $t,s \ge 0$, $\omega \in \Omega$ and $\tilde{u} \in C_\alpha$ we get that

\begin{align*} \Phi^{\delta}(t+s,\omega,\tilde{u}) &= V^{\delta}(t+s,0,\omega,\tilde{u}) =V^{\delta}(t+s,s,\omega,V^\delta(s,0,\omega, \tilde{u})) \\ &= V^{\delta}(t,0,\theta_{s}\omega,V^\delta(s,0,\omega, \tilde{u}))=\Phi^{\delta}(t,\theta_{s}\omega,\Phi^{\delta}(s,\omega,\tilde{u})), \end{align*}

where the second and third equalities follow from (3.5) and (3.4), respectively. Thus $\Phi^\delta$ forms an $\textrm{RDS}$ in $C_\alpha$ over the $\textrm{MDS}$ $\theta$.

Moreover, by using (3.4), we observe that for all $t\geq\sigma$,

\begin{align*} \Phi^{\delta}(t-\sigma,\theta_{\sigma}\omega,u^{\delta}_{\sigma})(0) &= u^{\delta}(t-\sigma,0,\theta_{\sigma}\omega,u^{\delta}_{\sigma}) =V^{\delta}(t-\sigma,0,\theta_{\sigma}\omega,u^{\delta}_{\sigma})(0) \\ &= V^{\delta}(t,\sigma,\omega,u^{\delta}_{\sigma})(0)=u^{\delta}(t,\sigma,\omega,u^{\delta}_{\sigma}). \end{align*}

Then we complete the proof of Theorem 3.1.

Proof. We denote by $U(t,\omega):=u^{0}(t,\sigma,\omega,u_{\sigma})-u^{\delta}(t,\sigma,\omega,u_{\sigma})$. By using (2.8), we find that

\begin{align*} U(t,\omega)= \left\{\begin{array}{ll} \int_{\sigma}^{t}e^{-A(t-\tau)}[B(u^{0}_{\tau},\theta_{\tau}\omega)- B(u^{\delta}_{\tau},\theta_{\tau}\omega)]d\tau\\ \qquad +\eta_0(t,\sigma,\omega)-\eta_\delta(t,\sigma,\omega), & {t\in (\sigma,\sigma+T] },\\ 0 , & {t\in [\sigma-r,\sigma]}. \end{array} \right. \end{align*}

Next we show that $ \sup_{t \in [\sigma-r,\sigma+T]}\{\|U(\cdot,\omega)\|_\alpha\} \rightarrow 0$ as $\delta \rightarrow 0$. Set

\begin{equation*}\Lambda_\delta:=\sup_{t\in [\sigma,\sigma+T]}\{\|\eta_0(t,\sigma,\omega)-\eta_\delta(t,\sigma,\omega)\|_{\alpha}\}.\end{equation*}

For any $t\in (\sigma,\sigma+T]$, observe that

\begin{align*} \|U(t,\omega)\|_{\alpha} &\leq \int_{\sigma}^{t}\|A^{\alpha}e^{-A(t-\tau)}[B(u^{0}_{\tau},\theta_{\tau}\omega)- B(u^{\delta}_{\tau},\theta_{\tau}\omega)]\| {d\tau}+\Lambda_\delta. \end{align*}

Let $\rho$ be given in Theorem 3.1. Along with (2.1) and (2.3), the above inequality implies that

\begin{align*} & e^{-\rho(t-\sigma)}\|U(t,\omega)\|_{\alpha}\\ & \quad\leq \int_{\sigma}^{t}K(\alpha)(t-\tau)^{-\alpha}e^{-\lambda(t-\tau)}e^{-\rho(t-\sigma)}L \sup_{\xi \in [-r,0]}\{\|U(\tau+\xi,\omega)\|_\alpha\}{d\tau}+\Lambda_\delta\\ & \quad\leq \int_{\sigma}^{t}K(\alpha)(t-\tau)^{-\alpha}e^{-(\lambda+\rho)(t-\tau)}L \sup_{t \in [\sigma-r,\sigma+T]}\{\|U(t,\omega)\|_\alpha\}{d\tau}+\Lambda_\delta\\ & \quad\leq K(\alpha)L (\lambda+\rho)^{\alpha-1} \Gamma(1-\alpha)\sup_{t \in [\sigma-r,\sigma+T]}\{\|U(t,\omega)\|_\alpha\}+\Lambda_\delta\\ & \quad\leq \frac{1}{2} \sup_{t \in [\sigma-r,\sigma+T]}\{\|U(t,\omega)\|_\alpha\}+\Lambda_\delta \end{align*}

for all $t\in (\sigma,\sigma+T]$. It then follows that

\begin{align*} \sup_{t \in [\sigma ,\sigma+T]}\{e^{-\rho(t-\sigma)} \|U(t,\omega)\|_{\alpha}\} &\leq \frac{1}{2} \sup_{t \in [\sigma-r,\sigma+T]}\{\|U(t,\omega)\|_\alpha\}+\Lambda_\delta. \end{align*}

Also, we note that

\begin{align*} \sup_{t \in [\sigma-r ,\sigma ]}\{e^{-\rho(t-\sigma)} \|U(t,\omega)\|_{\alpha}\}=0 \leq \frac{1}{2} \sup_{t \in [\sigma-r,\sigma+T]}\{\|U(t,\omega)\|_\alpha\}+\Lambda_\delta. \end{align*}

Thus by Lemma 2.1, we obtain that

\begin{align*} \sup_{t \in [\sigma-r,\sigma+T]}\{e^{-\rho(t-\sigma)} \|U(t,\omega)\|_\alpha\} \le 2 \Lambda_\delta \rightarrow 0,\ \mbox{as} \ \delta\rightarrow 0. \end{align*}

Finally, employing the following fact that for all $t \in [\sigma-r,\sigma+T]$,

\begin{align*} \|U(t,\omega)\|_{\alpha} &\le e^{\rho T} \sup_{t \in [\sigma-r,\sigma+T]}\{e^{-\rho(t-\sigma)}\|U(t,\omega)\|_{\alpha}\}, \end{align*}

we establish Theorem 3.2.

Proof. Hypothesis 3 means that

(4.7)\begin{equation} \|B(v_{\tau},\theta_{\tau}\omega)\| \leq L|v_{\tau}|_{C_{\alpha}}+K_0. \end{equation}

By a simple calculation, for any $t\in (-\infty,s]$ we observe that

(4.8)\begin{align} \int_{t}^{s} \lambda_N^\alpha e^{(\gamma-\lambda_N)(t-\tau)}d\tau\leq & \frac{\lambda_{N}^{\alpha} }{\gamma-\lambda_{N}}, \nonumber\\ \lambda_N^\alpha e^{\gamma(t-s)} \int_{t}^{s}e^{-\lambda_{N}(t-\tau)} d\tau \le & \lambda_N^{\alpha-1},\nonumber\\ \int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right] e^{(\gamma-\lambda_{N+1})(t-\tau)}d\tau\leq & \frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-\gamma)^{\alpha}}{\lambda_{N+1}-\gamma},\nonumber\\ \int_{-\infty}^{t} \left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{-\lambda_{N+1}(t-\tau)} d\tau \leq & \lambda_{N+1}^{\alpha-1}(k+1).\end{align}

We first claim that $\mathcal{J}^{s,\omega}_{\delta,p}$ is a self-mapping on $C_{\gamma,s}$. By using (2.2) and (4.7), we find that for $t\leq s$,

(4.9)\begin{align} e^{\gamma(t-s)}\|(\mathcal{J}^{s,\omega}_{\delta,p}(v))(t)\|_{\alpha} \leq&\ e^{(\gamma-\lambda_{N}) (t-s)} \|p\|_\alpha +I_A+I_B+e^{\gamma(t-s)}\|e^{-A(t-s)}P\eta_{\delta}(s,t,\omega)\|_\alpha \nonumber\\ &\ +e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_\alpha,\end{align}

where

\begin{align*} I_{A} &:= \lambda_{N}^{\alpha} \int_{t}^{s}e^{(\gamma-\lambda_N)(t-\tau)} e^{\gamma(\tau-s)} (L|v_{\tau}|_{C_{\alpha}}+K_0)d\tau, \\ I_{B} &:= \int_{-\infty}^{t} \left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{(\gamma-\lambda_{N+1})(t-\tau)} e^{\gamma(\tau-s)} (L|v_{\tau}|_{C_{\alpha}}+K_0) d\tau. \end{align*}

For $I_A$, by using (4.8), we find that

(4.10)\begin{align} I_A \le & \ \ L\lambda_{N}^{\alpha}\|v\|_{C_{\gamma,s}}\int_{t}^{s}e^{(\gamma-\lambda_{N})(t-\tau)} \sup_{\xi \in [-r,0]}\{e^{- \gamma \xi}\}d\tau \nonumber\\ &+K_0 \lambda_{N}^{\alpha} e^{\gamma(t-s)} \int_{t}^{s}e^{-\lambda_{N}(t-\tau)} d\tau \nonumber\\ \le & \ Le^{\gamma r}\frac{\lambda_{N}^{\alpha}}{(\gamma-\lambda_{N})}\|v\|_{C_{\gamma,s}} + K_0\lambda_N^{\alpha-1} \lt \infty. \end{align}

For $I_B$, by using (4.8), we have that

(4.11)\begin{equation} I_B \leq Le^{\gamma r}\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-\gamma)^{\alpha}}{\lambda_{N+1}-\gamma}\|v\|_{C_{\gamma,s}} +K_0 \lambda_{N+1}^{\alpha-1}(k+1) \lt \infty. \end{equation}

Moreover, it follows from (2.2), Remark 2.1(ii) and Proposition 2.1(vi) that

\begin{align*} e^{\gamma(t-s)}\|e^{-A(t-s)}P\eta_{\delta}(s,t,\omega)\|_\alpha \le &\ e^{(\gamma -\lambda_N)(t-s)} e^{\lambda_N |s|} \sup_{t \le s}\{\|\eta_{\delta}(s,t,\omega)\|_\alpha e^{-\lambda_N |s|}\} \\ \le &\ e^{\lambda_N |s|} \sup_{t \le s}\{\|\eta_{\delta}(s,t,\omega)\|_\alpha e^{-\lambda_N |s|}\} \lt \infty, \\ e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_\alpha \le &\ e^{\gamma(t-s)} e^{\lambda_{N+1}|t|} \sup_{t \in \mathbb{R}}\{\|\eta_{\delta}(t,-\infty,\omega)\|_\alpha e^{-\lambda_{N+1}|t|}\} \\ \le &\ e^{\lambda_{N+1}|s|} \sup_{t \in \mathbb{R}}\{\|\eta_{\delta}(t,-\infty,\omega)\|_\alpha e^{-\lambda_{N+1}|t|}\} \lt \infty, \end{align*}

which imply that for the above given $s \in \mathbb{R}$,

(4.12)\begin{align} c_{P,\eta,s} :=&\; \sup_{t \le s}\{e^{\gamma(t-s)}\|e^{-A(t-s)}P\eta_{\delta}(s,t,\omega)\|_\alpha\} \lt +\infty, \nonumber\\ d_{Q,\eta,s} :=&\; \sup_{t \le s}\{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_\alpha\} \lt +\infty.\end{align}

Combining (4.9)–(4.12), we get that

\begin{align*} \sup_{t\leq s}\{e^{\gamma (t-s)}\|(\mathcal{J}^{s,\omega}_{\delta,p}(v))(t)\|_{\alpha}\} \lt \infty. \end{align*}

Therefore, $\mathcal{J}^{s,\omega}_{\delta,p}$ is a self-mapping on $C_{\gamma,s}$.

Then we show that $\mathcal{J}^{s,\omega}_{\delta,p}$ is a contraction on $C_{\gamma,s}$. For any $v,\tilde{v}\in C_{\gamma,s}$, by Hypothesis 3, we find that for all $\tau\leq s$,

(4.13)\begin{align} & \|B(v_{\tau},\theta_{\tau}\omega)-B(\tilde{v}_{\tau},\theta_{\tau}\omega)\|\nonumber \\ & \quad\leq L \max_{\xi \in [-r,0]}\|v(\tau+\xi)-\tilde{v}(\tau+\xi)\|_{\alpha}\nonumber \\ & \quad\leq L\sup_{\xi \in [-r,0]}\{e^{-\gamma (\tau+\xi-s)}\} \times \sup_{\tau+\xi \leq s}\{\|v(\tau+\xi)-\tilde{v}(\tau+\xi))\|_{\alpha}e^{\gamma (\tau+\xi-s)}\}\nonumber \\ & \quad\leq L\|v-\tilde{v}\|_{C_{s,\gamma}}e^{-\gamma (\tau-s)} e^{\gamma r}.\end{align}

Employing (2.2), (4.13) and (4.8), we have that

\begin{align*} & e^{\gamma (t-s)}\| (\mathcal{J}^{s,\omega}_{\delta,p}(v))(t)-(\mathcal{J}^{s,\omega}_{\delta,p}(\tilde{v}))(t))\|_{\alpha}\\ & \quad\leq L e^{\gamma r} \|v-\tilde{v}\|_{C_{\gamma,s}} \left\{\int_{t}^{s} e^{(\gamma-\lambda_N)(t-\tau)} d\tau+ \int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right] e^{(\gamma-\lambda_{N+1})(t-\tau)}d\tau\!\right\}\\ & \quad\leq L e^{\gamma r}\left(\frac{\lambda_{N}^{\alpha} }{\gamma-\lambda_{N} } +\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-\gamma)^{\alpha}}{\lambda_{N+1}-\gamma}\right)\|v-\tilde{v}\|_{C_{\gamma,s}}\\ & \quad=\left(\frac{\zeta}{2}+\frac{\zeta \mu}{2} \frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-\lambda_{N}-\mu\lambda_{N}^{\alpha})^{\alpha}}{\lambda_{N+1}-\lambda_{N}-\mu\lambda_{N}^{\alpha}}\right)\|v-\tilde{v}\|_{C_{\gamma,s}} \lt \zeta \|v-\tilde{v}\|_{C_{\gamma,s}}, \end{align*}

which leads to

(4.14)\begin{equation} \| \mathcal{J}^{s,\omega}_{\delta,p}(v)- \mathcal{J}^{s,\omega}_{\delta,p}(\tilde{v})\|_{C_{\gamma,s}} \lt \zeta \|v-\tilde{v}\|_{C_{\gamma,s}}. \end{equation}

Therefore, by using the Contraction Mapping Principle, $\mathcal{J}^{s,\omega}_{\delta,p}$ has a unique fixed point in $C_{\gamma,s}$, which is denoted by $u^{\delta}(\cdot;s,\omega,p)$.

Now we prove the aforementioned three conclusions, respectively.

For claim (i), it follows from (4.9)–(4.12) that the bounded estimate (4.3) holds. With the same arguments as in (4.14), for any $p_{1}, p_{2} \in PD(A^{\alpha})$, we obtain that

\begin{align*} & \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma,s}}= \|\mathcal{J}^{s,\omega}_{\delta,p_{1}}(u^{\delta}(\cdot;s,\omega,p_1)) -\mathcal{J}^{s,\omega}_{\delta,p_{2}}(u^{\delta}(\cdot;s,\omega,p_2))\|_{C_{\gamma,s}}\\ & \quad \lt \zeta \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma,s}}+\sup_{t\leq s}\{e^{\gamma(t-s)}\| e^{-A(t-s)}(p_{1}-p_{2})\|_{\alpha}\}\\ & \quad \lt \zeta \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma,s}}+\| p_{1}-p_{2} \|_\alpha, \end{align*}

which implies that for any $p_1, p_2 \in PD(A^\alpha)$,

\begin{align*} \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma,s}} \lt \frac{1}{1-\zeta } \| p_{1}-p_{2} \|_{\alpha}. \end{align*}

Thus claim (i) holds.

Observe that $u^\delta(t;s,\omega, p)$ is $\mathcal{F}$-measurable in $\omega$ because it can be regarded as an $\omega$-wise limit of the iteration of contraction mapping $\mathcal{J}^{s,\omega}_{\delta,p}$, which maps an $\mathcal{F}$-measurable function to a measurable function. And recall that $u^\delta(t;s,\omega,p)$ is Lipschitz continuous in $p$. Applying [Reference Castaing and Valadier4, Lemma III.14], $u^\delta(t;s,\omega,p)$ is measurable with respect to $(\omega,p)$. To prove (4.5), we notice that

\begin{align*} u^{\delta}(t;s,\omega,p) &= e^{-A(t-s)}p -\int_{t}^{s}e^{-A(t-\tau)}PB(u^{\delta}_{\tau}(\cdot; s,\omega,p),\theta_{\tau}\omega)d\tau\\ & \quad+\int_{-\infty}^{t}e^{-A(t-\tau)}QB(u^{\delta}_{\tau}(\cdot; s,\omega,p),\theta_{\tau}\omega)d\tau\\ & \quad-e^{-A(t-s)}P\eta_{\delta}(s,t,\omega)+Q\eta_{\delta}(t,-\infty,\omega). \end{align*}

Taking the transformation $r=\tau-s$ in the integral above and using Proposition 2.1(ii), we get that

\begin{align*} u^{\delta}(t;s,\omega,p) &= e^{-A((t-s)-0)}p -\int_{t-s}^{0}e^{-A((t-s)-r)}PB(u^{\delta}_{r+s}(\cdot;s,\omega,p),\theta_{r}\theta_{s}\omega)dr\\ & \quad + \int_{-\infty}^{t-s}e^{-A((t-s)-r)}QB(u^{\delta}_{r+s}(\cdot;s,\omega,p),\theta_{r}\theta_{s}\omega)dr\\ & \quad-e^{-A(t-s)}P\eta_{\delta}(0,t-s,\theta_s\omega)+ Q\eta_{\delta}(t-s,-\infty,\theta_s\omega) =u^{\delta}(t-s;0,\theta_{s}\omega,p), \end{align*}

where the last equality follows from the uniqueness of the fixed point of $\mathcal{J}^{0,\theta_{s}\omega}_{\delta,p}$. Then claim (ii) is established.

For claim (iii), the conditions (4.1), $\gamma=\lambda_{N}+\mu\lambda_{N}^{\alpha}$ and $\zeta=\frac{2}{\mu}Le^{\gamma r} \lt 1$ suggest that there exist two sufficiently small positive constants $\varepsilon_0$ and $\varepsilon_1$ such that for every $ 0 \lt \eta\leq 2\varepsilon_0$,

(4.15)\begin{align} & \gamma-\eta \in (\lambda_{N},\lambda_{N+1}),\;\;\; \frac{\lambda_N^\alpha}{\gamma-\eta-\lambda_N} =\frac{\lambda_N^\alpha}{\mu\lambda_N^\alpha-\eta} \le \frac{1}{\mu}+\varepsilon_1, \nonumber\\ & \lambda_{N+1}-(\gamma-\eta) \gt \mu\{\lambda_{N+1}^{\alpha}+k[\lambda_{N+1}-(\gamma-\eta)]^{\alpha}\},\;\;\; \tilde{\zeta}(\eta):=\left(\frac{2}{\mu}+\varepsilon_1\right)Le^{(\gamma-\eta) r} \lt 1.\end{align}

With the similar arguments as the previous one, we find that $\mathcal{J}^{s,\omega}_{\delta,p}(\cdot)$ has a unique fixed point in $C_{\gamma-\eta,s}$. In view of $C_{\gamma-\eta,s}\subset C_{\gamma ,s}$, we conclude that $u^{\delta}(\cdot;s,\omega,p) \in C_{\gamma-\eta,s}$. Exactly like (4.14), we have that for any $p_1, p_2 \in PD(A^\alpha)$,

\begin{align*} \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma-\eta,s}} \lt \frac{1}{1-\tilde{\zeta}(\eta)} \| p_{1}-p_{2} \|_{\alpha}. \end{align*}

From now on, we restrict $0 \lt \eta \le \frac{3}{2}\varepsilon_0$. We prove that for any $\delta\in \mathbb{R}$, $u^{\delta}(\cdot;s,\omega,p)$ is differentiable in $p \in PD(A^{\alpha})$. For any $t\leq s$ and $p \in PD(A^{\alpha})$, we define the mapping

\begin{align*} (Sp)(t):=e^{-A(t-s)}p. \end{align*}

By a simple calculation, we observe that

\begin{align*} e^{(\gamma-\eta)(t-s)}\|(Sp)(t)\|_{\alpha} \leq e^{(\gamma-\eta-\lambda_{N})(t-s)}\| p\|_{\alpha} \lt \|p\|_\alpha. \end{align*}

This implies that $S \in L (PD(A^{\alpha}),C_{\gamma-\eta,s})$ with $\Arrowvert S\Arrowvert_{L(PD(A^\alpha),C_{\gamma-\eta,s})}\leq1.$ Meanwhile, for $t\leq s$, $p_{0} \in PD(A^{\alpha})$ and $v \in C_{\gamma-\eta, s}$, we also define the mapping

\begin{align*} (\mathcal{T}v)(t) & := -\int_{t}^{s}e^{-A(t-\tau)}PD_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,p_{0} ),\theta_{\tau}\omega)v_{\tau}d\tau\\ &\quad+\int_{-\infty}^{t}e^{-A(t-\tau)}QD_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,p_{0}),\theta_{\tau}\omega)v_{\tau}d\tau, \end{align*}

where $v_{\tau} \in C_{\alpha}$. In view of (2.3), we find that $\|D_{u}B(u,\omega)\|_{L(C_\alpha, H)}\leq L$ for all $u \in C_{\alpha}$ and $\omega\in\Omega$. Along with (2.2) and (4.8), we get that

\begin{align*} & e^{(\gamma-\eta)(t-s)}\|(\mathcal{T}v)(t)\|_{\alpha}\\ & \quad\leq e^{(\gamma-\eta)(t-s)}L\left\{\int_{t}^{s} \lambda_{N}^{\alpha} e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\|v(\tau+\xi )\|_{\alpha}d\tau\right.\\ & \qquad\left.+ \int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{-\lambda_{N+1} (t-\tau)} \sup_{\xi \in [-r,0]}\|v(\tau+\xi )\|_{\alpha}d\tau\right\}\\ & \quad\leq Le^{(\gamma-\eta )r}\left(\frac{\lambda_N^\alpha}{\gamma-\eta-\lambda_N}+\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-(\gamma-\eta))^{\alpha}}{\lambda_{N+1}-(\gamma-\eta) }\right)\|v\|_{C_{\gamma-\eta,s }}\\ & \quad \lt \tilde{\zeta}(\eta) \|v\|_{C_{\gamma-\eta,s }}, \end{align*}

which suggests that $\mathcal{T}\in L (C_{\gamma-\eta,s},C_{\gamma-\eta,s})$ with $\|\mathcal{T}\|_{L (C_{\gamma-\eta,s},C_{\gamma-\eta,s})} \lt \tilde{\zeta}(\eta) \lt 1.$ Thus $Id-\mathcal{T}$ is invertible on $C_{\gamma-\eta,s}$.

For any given $ p_{0} \in PD(A^{\alpha}) $, we denote by

\begin{align*} I_\delta(t) &:= -\int_{t}^{s}e^{-A(t-\tau)}P[B(u_{\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega) -B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}),\theta_{\tau}\omega)\\ & \quad-D_{u }B(u_{\tau}^{\delta}(\cdot; s,\omega,p_{0}),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega,p)-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}))]d\tau\\ & \quad+\int_{-\infty}^{t}e^{-A(t-\tau)}Q[B(u_{\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega)-B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}),\theta_{\tau}\omega)\\ & \quad-D_{u }B(u_{\tau}^{\delta}(\cdot; s,\omega,p_{0}),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega,p)-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}))]d\tau. \end{align*}

Then we have that

(4.16)\begin{equation} u^{\delta}(t;s,\omega, p)-u^{\delta}(t;s,\omega, p_{0})-\mathcal{T}(u^{\delta}(t;s,\omega, p)-u^{\delta}(t;s,\omega, p_{0})) = (S(p-p_{0}))(t)+I_\delta(t). \end{equation}

Next we claim that $\|I_\delta\|_{C_{\gamma-\eta,s}}=o(\|p-p_{0}\|_\alpha )$ as $p\rightarrow p_{0}$. To this end, we divide $ I_\delta(t)$ into a sum of $I_{\delta, P}^1(t)$, $I_{\delta, P}^2(t)$, $I_{\delta, Q}^1(t)$ and $I_{\delta, Q}^2(t)$:

\begin{align*} I_\delta(t)=I^{1}_{\delta,P}(t)+I^{2}_{\delta,P}(t)+I^{1}_{\delta,Q}(t)+I^{2}_{\delta,Q}(t), \end{align*}

where

\begin{align*} I^{1}_{\delta,P}(t):= \left\{ \begin{array}{ll} -\int_{t}^{N_{P}} e^{-A(t-\tau)}P[ B(u_{\tau}^{\delta}(\cdot;s,\omega, p ),\theta_{\tau}\omega)-B(u_{\tau}^{\delta}(\cdot; s,\omega, p_{0}),\theta_{\tau}\omega) \\ \ \ \ \ \ -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega, p )-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}))]d\tau , & {t \lt N_{P} },\\ 0, & {t \geq N_{P}}, \end{array} \right. \end{align*}

\begin{align*} I^{2}_{\delta, P}(t):= \left\{ \begin{array}{ll} -\int_{N_{P}}^s e^{-A(t-\tau)}P[B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega) -B(u_{\tau}^{\delta}(\cdot; s,\omega, p_{0}),\theta_{\tau}\omega) \\ \ \ \ \ \ -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega, p )-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ))]d\tau , & {t \lt N_{P} },\\ -\int_{t}^{s}e^{-A(t-\tau)}P[B(u_{\tau}^{\delta}(\cdot;s,\omega, p ),\theta_{\tau}\omega)-B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega) \\ \ \ \ \ \ -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega, p )-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ))]d\tau , & {t \geq N_{P}}, \end{array} \right. \end{align*}

\begin{align*} I^{1}_{\delta,Q}(t):= \left\{ \begin{array}{ll} \int_{N_{Q}}^{t}e^{-A(t-\tau)}Q[B(u_{\tau}^{\delta}(\cdot;s,\omega, p ),\theta_{\tau}\omega)-B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega) \\ \ \ \ \ \ -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot; s,\omega, p )-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ))]d\tau, & {t \gt N_{Q} },\\ 0, & {t \leq N_{Q}}, \end{array} \right. \end{align*}

\begin{align*} I^{2}_{\delta, Q}(t):= \left\{ \begin{array}{ll} \int_{-\infty}^{N_{Q}}e^{-A(t-\tau)}Q[B(u_{\tau}^{\delta}(\cdot;s,\omega, p ),\theta_{\tau}\omega)-B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega) \\ \ \ \ \ \ -D_{u }B(u_{\tau}^{\delta}(\cdot ;s,\omega, p_{0} ),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega, p )-u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ))]d\tau, & {t \gt N_{Q} },\\ \int_{-\infty}^{t}e^{-A(t-\tau)}Q[B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega) \\ \ \ \ \ \ -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)(u_{\tau}^{\delta}(\cdot;s,\omega, p)-u_{\tau}^{\delta}(\cdot; s,\omega, p_{0} ))]d\tau, & {t\leq N_{Q}}. \end{array} \right. \end{align*}

Here $N_P ( \lt s)$ and $N_Q ( \lt s)$ are constants to be determined.

For $I^{1}_{\delta, P}(t)$, applying (4.4), we obtain that for $ t \lt N_{P} \lt s$,

\begin{align*} & e^{(\gamma-\eta)(t-s)} \|I^{1}_{\delta,P}(t)\|_{\alpha}\\ & \quad\leq e^{(\gamma-\eta)(t-s)} \int_{t}^{N_{P}} \lambda_{N}^{\alpha} e^{-\lambda_{N}(t-\tau)} 2L \sup_{\xi \in [-r,0]}\|u ^{\delta}(\tau+\xi;s,\omega, p)\\ & \qquad -u ^{\delta}(\tau+\xi;s,\omega, p_{0} )\|_{\alpha} d\tau\\ & \quad\leq \int_{t}^{N_{P}} e^{(\gamma-2\varepsilon_0-\lambda_{N})(t-\tau) } e^{( 2\varepsilon_0-\eta)(t-s) } \lambda_{N}^{\alpha}2 Le^{(\gamma-2\varepsilon_0) r}\|u^{\delta}(\cdot;s,\omega, p)\\ & \qquad-u^{\delta}(\cdot; s,\omega, p_{0} ) \|_{C_{\gamma-2\varepsilon_0,s}}d\tau\\ & \quad\leq \frac{e^{\frac{\varepsilon}{2}(t-s) }}{(1-\tilde{\zeta}(2\varepsilon_0))} \|p-p_{0}\|_\alpha \lambda_{N}^{\alpha} 2 Le^{(\gamma-2\varepsilon_0) r} \int_{t}^{N_{P}} e^{(\gamma-2\varepsilon_0-\lambda_{N})(t-\tau) } d\tau\\ & \quad\leq \frac{2\tilde{\zeta}(2\varepsilon_0) e^{\frac{\varepsilon}{2}(N_{P} -s) }}{(1-\tilde{\zeta}(2\varepsilon_0))} \|p-p_{0}\|_\alpha. \end{align*}

For any given $\epsilon \gt 0$, choose $N_{P}$ to be sufficiently negative such that

\begin{align*} \sup_{t\leq s}\{e^{(\gamma-\eta)(t-s)} \|I^{1}_{\delta, P}(t)\|_{\alpha}\} \leq \frac{\epsilon}{4}\|p-p_{0}\|_\alpha. \end{align*}

For such a chosen $N_{P}$, we observe that

\begin{align*} & e^{(\gamma-\eta)(t-s)} \|I^{2}_{\delta,P}(t)\|_{\alpha}\\ & \quad\leq \int_{\max\{N_{P},t\}}^{s} e^{(\gamma-\eta-\lambda_{N})(t-\tau) } \lambda_{N}^{\alpha} 2 Le^{(\gamma-\eta) r}\frac{1 }{1-\tilde{\zeta}(\eta) } \|p-p_{0}\|_\alpha\\ & \qquad\times \left\{\int_{0}^{1} \|D_{u }B(\mu u_{\tau}^{\delta}(\cdot;s,\omega,p ) +(1-\mu)u_{\tau}^{\delta}(\cdot; s,\omega,p_{0} ),\theta_{\tau}\omega)\right.\\ & \qquad\left.\quad -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)\| d\mu \vphantom{\int_{0}^{1}}\right\} d\tau. \end{align*}

Notice that the continuity of $D_{u }B(u, \omega)$ in $u$ and the fact that for all $\tau\in[N_{P},s]$,

\begin{align*} \|u_{\tau}^{\delta}(\cdot;s,\omega,p)-u_{\tau}^{\delta}(\cdot;s,\omega,p_0)\|_{C_\alpha} &\le e^{\gamma r} e^{-\gamma(\tau-s)} \|u^{\delta}(\cdot;s,\omega,p)-u^{\delta}(\cdot;s,\omega,p_0)\|_{C_{\gamma,s}} \\ &\le e^{\gamma r} e^{-\gamma(N_p-s)} \frac{1}{1-\zeta} \|p-p_0\|_\alpha. \end{align*}

It then follows that for all $\tau\in[N_{P},s]$,

\begin{align*} \lim_{p\rightarrow p_{0}} \|D_{u }B(\mu u_{\tau}^{\delta}(\cdot;s,\omega,p )+(1-\mu)u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0} ),\theta_{\tau}\omega)\|=0. \end{align*}

Then for given $\epsilon$ as above, there exists a $\varsigma_{P} \gt 0$ so that if $\|p-p_{0}\|_\alpha \leq \varsigma_{P}$, then

\begin{align*} \sup_{t\le s}\{e^{(\gamma-\eta)(t-s)} \|I^{2}_{\delta, P}(t)\|_{\alpha}\} \leq \frac{\epsilon}{4}\|p-p_{0}\|_\alpha. \end{align*}

By using the similar arguments as above, we choose $N_{Q}$ to be sufficiently negative such that

\begin{align*} \sup_{t\leq s}\{e^{(\gamma-\eta)(t-s)} \|I^{1}_{\delta,Q}(t)\|_{\alpha}\} \leq \frac{\epsilon}{4}\|p-p_{0}\|_\alpha, \end{align*}

while for such chosen $N_{Q}$ and given $\epsilon$, there exists a $\varsigma_{Q} \gt 0$ so that if $\|p-p_{0}\|_\alpha \leq \varsigma_{Q}$, then

\begin{align*} \sup_{t\leq s}\{e^{(\gamma-\eta)(t-s)} \|I^{2}_{\delta,Q}(t)\|_{\alpha}\} \leq \frac{\epsilon}{4}\|p-p_{0}\|_\alpha. \end{align*}

Combining the above analysis, there exists a $\varsigma:=\min\{\varsigma_{P},\varsigma_{Q}\} \gt 0$ such that if $\|p-p_{0}\|_\alpha \leq \varsigma $,

\begin{align*} \sup_{t\leq s}\{e^{(\gamma-\eta)(t-s)} \|I_{\delta}(t)\|_{\alpha}\} \leq \epsilon\|p-p_{0}\|_\alpha. \end{align*}

Thus we complete this claim. Along with (4.16), we get that

\begin{align*} u^{\delta}(t;s,\omega, p)-u^{\delta}(t;s,\omega, p_{0})= (Id-\mathcal{T})^{-1} (S(p-p_{0}))(t)+o(\|p-p_{0}\|_\alpha ), \end{align*}

which suggests that $u^{\delta}(\cdot; s,\omega,p)\in C_{\gamma-\eta,s} $ is differentiable in $p \in PD(A^{\alpha})$. Besides, for any $p \in PD(A^{\alpha})$, its derivative $D_{p}u^{\delta}(\cdot;s,\omega, p) \in L(PD(A^{\alpha}), C_{\gamma-\eta,s})$ and satisfies that

(4.17)\begin{align} D_{p}u^{\delta}(t;s,\omega,p) =& \ e^{-A(t-s)}P-\int_{t}^{s}e^{-A(t-\tau)}PD_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega)D_{p}u_{\tau}^{\delta}(\cdot;s,\omega,p)d\tau \nonumber \\ & \ +\int_{-\infty}^{t}e^{-A(t-\tau)}QD_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega)D_{p}u_{\tau}^{\delta}(\cdot;s,\omega,p)d\tau.\end{align}

In what follows, to simplify the notation, sometimes the operator norm is directly denoted as $\|\cdot\|$ where no confusion arises. Clearly, (4.6) is a consequence of (2.2) and (4.8). Indeed, for any $p\in PD(A^{\alpha})$ and $t\leq s$, we have that

\begin{align*} & e^{(\gamma-\eta)(t-s)}\|D_{p}u^{\delta}(t;s,\omega,p)\|_{L(PD(A^\alpha), D(A^\alpha))}\\ & \quad\leq \lambda_{N}^{\alpha}e^{(\gamma-\eta- \lambda_{N})(t-s)}\\ & \qquad +e^{(\gamma-\eta)(t-s)}L\left\{\int_{t}^{s} \lambda_{N}^{\alpha} e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\|D_{p}u^{\delta}(\tau+\xi ;s,\omega,p)\|_{L(PD(A^\alpha), D(A^\alpha))} d\tau\right.\\ & \qquad \left.+ \int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{-\lambda_{N+1} (t-\tau)} \sup_{\xi \in [-r,0]}\right.\\ &\qquad\quad\left.\|D_{p}u^{\delta}(\tau+\xi ;s,\omega,p) )\|_{L(PD(A^\alpha), D(A^\alpha))} d\tau\vphantom{\int_{t}^{s}}\right\}\\ & \quad\leq \lambda_{N}^{\alpha}+ Le^{(\gamma-\eta )r}\left(\frac{\lambda_N^\alpha}{\gamma-\eta-\lambda_N}+\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-(\gamma-\eta))^{\alpha}}{\lambda_{N+1}-(\gamma-\eta) }\right)\\ & \qquad\quad\|D_{p}u^{\delta}(\cdot;s,\omega,p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})}\\ & \quad \lt \lambda_{N}^{\alpha}+ \tilde{\zeta}(\eta)\|D_{p}u^{\delta}(\cdot;s,\omega,p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})}, \end{align*}

which implies that (4.6) is established.

Finally, we show that $D_{p}u^{\delta}(\cdot;s,\omega, p)$ is continuous in $p$. In fact, by using (4.17), we find that

\begin{align*} & D_{p}u^{\delta}(t;s,\omega, p)-D_{p}u^{\delta}(t;s,\omega, p_{0})\\ &\quad= -\int_{t}^{s}e^{-A(t-\tau)}PD_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega) (D_{p}u_{\tau}^{\delta}(\cdot; s,\omega,p) -D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}))d\tau\\ &\qquad+\int_{-\infty}^{t}e^{-A(t-\tau)}QD_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega) (D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p)\\ & \qquad -D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}))d\tau+II_\delta(t), \end{align*}

where

\begin{align*} II_\delta(t) & := -\int_{t}^{s}e^{-A(t-\tau)}P[D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega) -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}),\theta_{\tau}\omega)]\\ &\qquad D_{p}u_{\tau}^{\delta}(\cdot; s,\omega,p_{0})d\tau\\ & \quad +\int_{-\infty}^{t}e^{-A(t-\tau)}Q[D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega) -D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p_{0}),\theta_{\tau}\omega)]\\ & \qquad D_{p}u_{\tau}^{\delta}(\cdot; s,\omega,p_{0})d\tau. \end{align*}

With the similar arguments as above, we have that $\|II_\delta\|_{L(PD(A^{\alpha}),C_{\gamma-\eta ,s})}=o(1)$ as $p \rightarrow p_{0}$, and

\begin{align*} \|D_{p}u^{\delta}(\cdot;s,\omega, p)-D_{p}u^{\delta}(\cdot;s,\omega, p_{0})\|_{L(PD(A^{\alpha}),C_{\gamma-\eta,s })} \leq \frac{\|II_\delta\|_{L(PD(A^{\alpha}),C_{\gamma-\eta ,s})}}{1-\tilde{\zeta}(\eta)}. \end{align*}

Thus the continuity holds. Then claim (iii) is established and the proof of Lemma 4.1 is complete.

Proof. For claim (i), by using (2.2), (2.3), (4.4), (4.8) and (4.18), for any $p_{1},p_{2}\in PD(A^{\alpha})$ and $\xi \in [-r,0]$, we have that

\begin{align*} & \| \Psi^{\delta}_{s}(\omega,p_{1},\xi)-\Psi^{\delta}_{s}(\omega,p_{2},\xi) \|_{\alpha}\\ & \quad\le \int_{s+\xi}^{s} \lambda_N^\alpha e^{-\lambda_N (s+\xi-\tau)} L e^{\gamma ( s-\tau)}\sup_{\xi \in [-r,0]}\{e^{- \gamma \xi}\} \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma,s}} d\tau\\ &\qquad+\int_{-\infty}^{s+\xi}\left[\left(\frac{\alpha}{s+\xi-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right] e^{-\lambda_{N+1} (s+\xi-\tau)} L e^{\gamma ( s-\tau)}\\ & \qquad \times\sup_{\xi \in [-r,0]}\{e^{- \gamma \xi}\} \|u^{\delta}(\cdot;s,\omega,p_1)-u^{\delta}(\cdot;s,\omega,p_2)\|_{C_{\gamma,s}} d\tau\\ & \quad \lt \frac{1}{1-\zeta }\| p_{1}-p_{2} \|_\alpha \left\{\int_{s+\xi}^{s} \lambda_N^\alpha e^{(\gamma-\lambda_N)(s+\xi-\tau)} e^{- \gamma \xi}L e^{\gamma r} d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{s+\xi}\left[\left(\frac{\alpha}{s+\xi-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right] e^{(\gamma-\lambda_{N+1}) (s+\xi-\tau)} e^{- \gamma \xi} L e^{\gamma r}d\tau\right\} d\tau\\ & \quad \lt \frac{Le^{\gamma r}}{1-\zeta }\left(\frac{\lambda_{N}^{\alpha} }{\gamma-\lambda_{N}}+ \frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-\gamma)^{\alpha}}{\lambda_{N+1}-\gamma}\right)\| p_{1}-p_{2} \|_{\alpha} e^{- \gamma \xi}\\ & \quad \lt \frac{ \zeta e^{\gamma r}}{1-\zeta }\| p_{1}-p_{2} \|_{\alpha}, \end{align*}

which yields that (4.19) holds.

Obviously, by Lemma 4.1(iii), we have that $\Psi^{\delta}_{s}(\omega,p,\xi)$ is $C^{1}$ in $p$. Moreover, applying (4.5) and (4.18), for any $\xi \in [-r,0]$ and $p\in PD(A^{\alpha})$, we find that

\begin{align*} \Psi^{\delta}_{s}(\omega,p,\xi)= u^{\delta}(s+\xi;s,\omega,p)-e^{-A\xi}p u^{\delta}(\xi;0,\theta_{s}\omega,p)-e^{-A\xi}p=\Psi^{\delta}_{0}(\theta_{s}\omega,p,\xi), \end{align*}

which suggests that $\Psi^{\delta}_{s}(\omega,p,\cdot)$ is a stationary process. Thus claim (i) holds.

To obtain claim (ii), we first prove that $\mathcal{M}^{\delta}_{s}(\omega)$ is a random set. Clearly, for each $s \in \mathbb{R}$, $\delta \in \mathbb{R}$, $\xi \in [-r,0]$ and $\hat{p}\in \hat{P}C_{\alpha}$, $\Psi^{\delta}_{s}(\cdot,\hat{p}(0),\xi)$ is measurable by (4.18). For any $v \in C_{\alpha},$ applying the continuity of $v$, $\hat{p}$ and $\Psi^{\delta}_{s}(\omega,\hat{p}(0),\cdot)$, we find the mapping

\begin{align*} \omega & \mapsto dist_{C_{\alpha}}(v(\cdot),\hat{p}(\cdot)+\Psi^{\delta}_{s}(\omega,\hat{p}(0),\cdot)) = \sup_{\xi\in[-r,0]\cap \mathbb{Q}}\{\| v(\xi)-\hat{p}(\xi)-\Psi^{\delta}_{s}(\omega,\hat{p}(0),\xi)\|_{\alpha}\}. \end{align*}

It then follows from [Reference Castaing and Valadier4, Lemma III.14] that $\mathcal{M}^{\delta}_{s}(\omega)$ is a random set.

Next we prove the invariance of $\mathcal{M}^{\delta}_{s}(\omega)$ under the cocycle $\Phi^{\delta}$. For any fixed $s\in \mathbb{R}$ and $\hat{p}\in \hat{P}C_{\alpha}$, let $u^{\delta}(t )$ be a solution of (2.8) for $t\geq s$ with the initial data $u^{\delta}_{s}(\cdot):=\hat{p}(\cdot)+\Psi^{\delta}_{s}(\omega,p,\cdot) \in \mathcal{M}^{\delta}_{s}(\omega)$, where $\hat{p}(0):=p$. According to Theorem 3.1(iii), we are to show that $u^{\delta}(t )=\Phi^{\delta}(t-s,\theta_{s}\omega, u^{\delta}_{s} ) \in \mathcal{M}^{\delta}_{t}(\omega)$. Define the mapping

(4.21)\begin{equation} \varpi^{\delta}(\sigma):= \begin{cases} u^{\delta}(\sigma; s, \omega, p), & \sigma \leq s, \\ u^{\delta}(\sigma), & \sigma \geq s. \end{cases} \end{equation}

For $s-r \leq \sigma\leq s$, there exists $\xi \in [-r,0]$ such that $\sigma=s+\xi$. Then by using (4.18), we observe that

\begin{align*} u^{\delta}(\sigma;s,\omega,p) =&\, u^{\delta}(s+\xi;s,\omega,p)=\Psi^{\delta}_{s}(\omega,p,\xi)+e^{-A\xi}p \\ =&\, \Psi^{\delta}_{s}(\omega,p,\xi)+\hat{p}(\xi) = \Psi^{\delta}_{s}(\omega,p,\sigma-s)+\hat{p}(\sigma-s) \\ =&\, u^{\delta}_{s}(\sigma-s)=u^{\delta}(\sigma), \end{align*}

which together with (4.21) implies that for $\sigma \ge s-r$, we have that

(4.22)\begin{equation} \varpi^{\delta}(\sigma)=u^{\delta}(\sigma). \end{equation}

Subsequently, we claim that for every $t\geq s$ and $\sigma \leq t$,

(4.23)\begin{equation} \varpi^{\delta}(\sigma) =u^{\delta}(\sigma;t,\omega,Pu^{\delta}(t)). \end{equation}

Indeed, using (2.8) and (4.18), for $t\geq s$, we have that

\begin{align*} u^{\delta}(t) & = e^{-A(t-s)}u^{\delta}_{s}(0)+\int_{s}^{t}e^{-A(t-\tau)}B(u_{\tau}^\delta,\theta_{\tau}\omega){d\tau} +\eta_{\delta}(t,s,\omega)\\ &= e^{-A(t-s)}\left\{p+\int_{-\infty}^{s }e^{-A(s -\tau)}QB(u^{\delta}_{\tau}(\cdot; s,\omega, p),\theta_{\tau}\omega)d\tau +Q\eta_{\delta}(s ,-\infty,\omega)\right\} \\ & \quad+ \int_{s}^{t}e^{-A(t-\tau)}B(u_{\tau}^\delta,\theta_{\tau}\omega){d\tau} +\eta_{\delta}(t,s,\omega). \end{align*}

Thus we get that

\begin{align*} Pu^{\delta}(t) &= e^{-A(t-s)}p +\int_{s}^{t}e^{-A(t-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau +P\eta_{\delta}(t,s ,\omega). \end{align*}

Then for $\sigma \leq t$, we have that

\begin{align*} & u^{\delta}(\sigma;t,\omega,Pu^{\delta}(t))\\ & \quad= e^{-A(\sigma-t)}Pu^{\delta}(t) -\int_{\sigma}^{t}e^{-A(\sigma-\tau)}PB(\varpi^{\delta}_{\tau},\theta_{\tau}\omega)d\tau -e^{-A(\sigma-t)}P\eta_{\delta}(t,\sigma,\omega)\\ & \qquad+\int_{-\infty}^{\sigma}e^{-A(\sigma-\tau)}QB(\varpi^{\delta}_{\tau},\theta_{\tau}\omega)d\tau +Q\eta_{\delta}(\sigma,-\infty,\omega)\\ & \quad= e^{-A(\sigma-s)}p+\int_{s}^{t}e^{-A(\sigma-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau+e^{-A(\sigma-t)}P\eta_{\delta}(t,s ,\omega)\\ & \qquad-\int_{\sigma}^{t}e^{-A(\sigma-\tau)}PB(\varpi^{\delta}_{\tau},\theta_{\tau}\omega)d\tau -e^{-A(\sigma-t)}P\eta_{\delta}(t,\sigma,\omega)\\ & \qquad+\int_{-\infty}^{\sigma}e^{-A(\sigma-\tau)}QB(\varpi^{\delta}_{\tau},\theta_{\tau}\omega)d\tau +Q\eta_{\delta}(\sigma,-\infty,\omega)\\ & \quad= e^{-A(\sigma-s)}p+\int_{s}^{t}e^{-A(\sigma-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau +P\eta_{\delta}(\sigma,s ,\omega) +Q\eta_{\delta}(\sigma,-\infty,\omega)\\ & \qquad-\int_{\sigma}^{t}e^{-A(\sigma-\tau)}PB(\varpi^{\delta}_{\tau},\theta_{\tau}\omega)d\tau +\int_{-\infty}^{\sigma}e^{-A(\sigma-\tau)}QB(\varpi^{\delta}_{\tau},\theta_{\tau}\omega)d\tau=:L_R, \end{align*}

where the third equality follows from the fact that $e^{-A(\sigma-t)}P(\eta_{\delta}(t,s ,\omega)-\eta_{\delta}(t,\sigma ,\omega))=P\eta_{\delta}(\sigma,s ,\omega)$. If $s \leq \sigma \leq t$, by (4.21) and (4.22), we find that

\begin{align*} L_R &= e^{-A(\sigma-s)}p+\int_{s}^{t}e^{-A(\sigma-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau + P\eta_{\delta}(\sigma,s ,\omega)\\ & \qquad-\int_{\sigma}^{t}e^{-A(\sigma-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau +\int_{s}^{\sigma}e^{-A(\sigma-\tau)}QB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau\\ & \qquad+\int_{-\infty}^{s}e^{-A(\sigma-\tau)}QB(u^{\delta}_\tau(\cdot;s,\omega,p),\theta_{\tau}\omega)d\tau+Q\eta_{\delta}(\sigma,-\infty,\omega)\\ &= e^{-A(\sigma-s)}p +\int_{s}^{\sigma}e^{-A(\sigma-\tau)}B(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau + \eta_{\delta}(\sigma,s ,\omega)\\ & \qquad+\int_{-\infty}^{s}e^{-A(\sigma-\tau)}QB(u^{\delta}_\tau(\cdot;s,\omega,p),\theta_{\tau}\omega)d\tau+e^{-A(\sigma-s)}Q\eta_{\delta}(s,-\infty,\omega)\\ & = e^{-A(\sigma-s)}\left\{p+\int_{-\infty}^{s }e^{-A(s -\tau)}QB(u^{\delta}_{\tau}(\cdot; s,\omega, p),\theta_{\tau}\omega)d\tau +Q\eta_{\delta}(s ,-\infty,\omega)\right\}\\ & \qquad+\int_{s}^{\sigma}e^{-A(\sigma-\tau)}B(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau + \eta_{\delta}(\sigma,s ,\omega) =u^\delta(\sigma)=\varpi^{\delta}(\sigma). \end{align*}

On the other hand, if $\sigma \leq s(\le t)$, by using the similar arguments as above, we get that

\begin{align*} L_R &= e^{-A(\sigma-s)}p+\int_{s}^{t}e^{-A(\sigma-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau + P\eta_{\delta}(\sigma,s ,\omega)\\ & \qquad-\int_{s}^{t}e^{-A(\sigma-\tau)}PB(u_{\tau}^\delta,\theta_{\tau}\omega)d\tau -\int_{\sigma}^{s}e^{-A(\sigma-\tau)}PB(u^{\delta}_\tau(\cdot;s,\omega,p),\theta_{\tau}\omega)d\tau\\ & \qquad+\int_{-\infty}^{\sigma}e^{-A(\sigma-\tau)}QB(u^{\delta}_\tau(\cdot;s,\omega,p),\theta_{\tau}\omega)d\tau+Q\eta_{\delta}(\sigma,-\infty,\omega) \\ &= e^{-A(\sigma-s)}p-\int_{\sigma}^{s}e^{-A(\sigma-\tau)}PB(u^{\delta}_\tau(\cdot;s,\omega,p),\theta_{\tau}\omega)d\tau -e^{-A(\sigma-s)} P\eta_{\delta}(s,\sigma ,\omega) \\ & \qquad+\int_{-\infty}^{\sigma}e^{-A(\sigma-\tau)}QB(u^{\delta}_\tau(\cdot;s,\omega,p),\theta_{\tau}\omega)d\tau+Q\eta_{\delta}(\sigma,-\infty,\omega)\\ & =u^{\delta}(\sigma;s,\omega,p)=\varpi^{\delta}(\sigma). \end{align*}

Thus claim (4.23) holds. This yields that for any $ t \geq s$ and $\xi \in [-r,0]$, by using (3.1), (3.4), (4.22), (4.23) and (4.18), we obtain that

\begin{align*} \Phi^{\delta}(t-s,\theta_{s}\omega, u^{\delta}_{s} )(\xi) &= u^{\delta}(t-s+\xi,0,\theta_s\omega,u^{\delta}_{s} ) \\ &= u^{\delta}(t +\xi,s,\omega,u^{\delta}_{s} ) \\ &= u^{\delta}(t+\xi) \\ &= \varpi^{\delta}(t+\xi) \\ &= u^{\delta}(t+\xi;t,\omega,Pu^{\delta}(t)) \\ &= \Psi^{\delta}_{t}(\omega,Pu^{\delta}(t),\xi)+e^{-A\xi}Pu^{\delta}(t), \end{align*}

which suggests that $\Phi^{\delta}(t-s,\theta_{s}\omega, u^{\delta}_{s} ) \in \mathcal{M}^{\delta}_{t}(\omega)$. It then indicates the invariance property of $\mathcal{M}^{\delta}_{t}(\omega)$ under $\Phi^{\delta}$. Moreover, recalling that $\Psi^{\delta}_{s}(\omega,p,\cdot)$ is a stationary process and using (4.20), we get that

\begin{align*} \mathcal{M}^{\delta}_{s}(\omega) &= \{\hat{p}(\cdot)+\Psi^{\delta}_{s}(\omega,\hat{p}(0),\cdot):\hat{p}\in \hat{P}C_{\alpha}\}\\ & = \{\hat{p}(\cdot)+\Psi^{\delta}_{0}(\theta_{s}\omega,\hat{p}(0),\cdot):\hat{p}\in \hat{P}C_{\alpha}\}\\ & =\mathcal{M}^{\delta}_{0}(\theta_{s}\omega),\ \ \ \forall s\in \mathbb{R}, \ \omega \in \Omega. \end{align*}

Then claim (ii) is true.

Finally, we prove claim (iii). To this end, we denote by $C_{+}$ the Banach space

\begin{align*} C_{+} :=\{x\in C([-r,+\infty);D(A^{\alpha})):\sup_{t\geq -r}\{e^{\gamma t}\|x(t)\|_\alpha\} \lt +\infty\}, \end{align*}

with the norm

\begin{align*} \|x\|_{+}:=\sup_{t\geq -r}\{e^{\gamma t}\|x(t)\|_\alpha\}. \end{align*}

Employing the method used in Chueshov and Scheutzow [Reference Chueshov and Scheutzow9, Theorem 4.2], for any $u_0 \in C_\alpha$, we need to show that there exist $u_0^\ast \in \mathcal{M}_0^\delta(\omega)$ and $x^\delta(\cdot,\omega) \in C_+$ such that the induced trajectory $u^{\delta}(t,0,\omega,u^\ast_{0} )$ of (2.8) can be represented in the form

\begin{align*} u^{\delta}(t,0,\omega,u^\ast_{0} ):= \left\{ \begin{array}{ll} u^{\delta}(t,0,\omega,u_{0} )+x^\delta(t,\omega), & {t \geq 0}, \\ u_{0}(t)+x^\delta(t,\omega), & {0 \geq t \geq -r}. \end{array} \right. \end{align*}

Let $\hat{q}(\xi):=(Id_{C_\alpha}-\hat{P})(u^\ast_{0}(\xi)-u_{0}(\xi))$. By a simple observation, we find that $x^\delta$ satisfies the following equations in $C_{+}$:

(4.24)\begin{align} x (t,\omega ) & =e^{-At}\hat{q}(0)+\int_{0}^{t}e^{-A(t-\tau)}Q(B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_\tau(\cdot,\omega) ,\theta_{\tau}\omega)\nonumber\\ &\qquad-B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} ),\theta_{\tau}\omega))d\tau \nonumber\\ & \qquad-\int_{t}^{+\infty}e^{-A(t-\tau)}P(B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_\tau(\cdot,\omega) ,\theta_{\tau}\omega)\nonumber\\ &\qquad-B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} ),\theta_{\tau}\omega))d\tau \nonumber\\ & =:e^{-At}\hat{q}(0)+ R_x, t \ge 0, \end{align}

and for $0 \geq \xi \geq -r$,

(4.25)\begin{align} x(\xi,\omega ) & = \hat{q}(\xi) -e^{-A\xi}\int_{0}^{+\infty}e^{A\tau}P(B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_\tau(\cdot,\omega) ,\theta_{\tau}\omega)\nonumber\\ & \qquad-B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} ),\theta_{\tau}\omega))d\tau. \end{align}

Since $u^{\delta}(t,0,\omega,u_{0} )=\Phi^{\delta}(t,\omega, u^\ast_{0} ) \in \mathcal{M}^{\delta}_{t}(\omega)$ holds for all $t \geq 0$ if and only if $u_0^\ast \in \mathcal{M}^{\delta}_{0}(\omega)$, i.e., $u_0^\ast=\hat{P}u_0^\ast+\Psi^{\delta}_{0}(\omega,Pu_0^\ast(0),\cdot)$. Notice that $x^\delta(\xi,\omega )=u^{\delta}(\xi,0,\omega,u^\ast_{0} )-u_{0}(\xi)=u^\ast_{0}(\xi)-u_{0}(\xi)$. Consequently,

(4.26)\begin{equation} \hat{q}(\xi)=-((Id-\hat{P})u_0)(\xi) +\Psi^{\delta}_{0}(\omega,p^\ast,\xi) \end{equation}

with

(4.27)\begin{align} p^\ast & := Pu_0^\ast(0)=P( u_0(0)+x^\delta(0,\omega ))\nonumber\\ & = P u_0(0)-\int_{0}^{+\infty}e^{A \tau }P(B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_\tau(\cdot,\omega) ,\theta_{\tau}\omega)\nonumber\\ &\quad-B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} ),\theta_{\tau}\omega))d\tau.\end{align}

In what follows, by applying the Contraction Mapping Principle, we prove that Equations (4.24)–(4.26) have a unique fixed point in $C_{+}$, which is denoted by $x^\delta(\cdot,\omega ) \in C_{+}$ as desirable and satisfies the estimate:

(4.28)\begin{equation} |x^\delta(\cdot,\omega ) |_{+} \leq\; C(u_0,\delta,\omega). \end{equation}

Actually, for any $x(\cdot,\omega ) \in C_{+}$, we define a mapping $\mathcal{X}[x(\cdot,\omega )](t)$ by the right-hand sides of (4.24) and (4.25).

We next verify that $\mathcal{X} $ is a self-mapping on $C_{+}$. Using (4.18) and (4.26), we find that

\begin{align*} \hat{q}(0)&=-((Id-\hat{P})u_0)(0)+\Psi^{\delta}_{0}(\omega, p^\ast ,0)\\ & = -((Id-\hat{P})u_0)(0) +\int_{-\infty}^{0}e^{A\tau}QB(u^{\delta}_{\tau}(\cdot; 0,\omega,p^\ast),\theta_{\tau}\omega)d\tau\\ &\qquad +Q\eta_{\delta}(0,-\infty,\omega). \end{align*}

Thus for $t \ge 0$, we get that

\begin{align*} e^{\gamma t}\|e^{-At} \hat{q}(0)\|_{\alpha} &\le e^{-(\lambda_{N+1}-\gamma )t}\| \hat{q}(0)\|_{\alpha}\\ & \leq |u_0|_{C_\alpha}+ e^{\gamma t} \|\int_{-\infty}^{0}e^{A\tau}QB(u^{\delta}_{\tau}(\cdot; 0,\omega,p^\ast),\theta_{\tau}\omega)d\tau\|_\alpha\\ & \quad+\|Q\eta_{\delta}(0,-\infty,\omega)\|_\alpha \\ & \leq |u_0|_{C_\alpha}+\frac{\zeta}{2}\|u^{\delta}(\cdot; 0,\omega,p^\ast)\|_{C_{\gamma,0}} +K_0 \lambda_{N+1}^{\alpha-1}(k+1)+\|Q\eta_{\delta}(0,-\infty,\omega)\|_\alpha\\ & \leq |u_0|_{C_\alpha} +K_0 \lambda_{N+1}^{\alpha-1}(k+1)+\|Q\eta_{\delta}(0,-\infty,\omega)\|_\alpha\\ & \qquad+\frac{\zeta}{2(1-\zeta)}\| p^\ast- Pu_0(0) \|_{\alpha}+ \frac{\zeta}{2}\| u^{\delta}(\cdot;0,\omega,Pu_0(0)) \|_{C_{\gamma,0}}, \end{align*}

where the third and fourth inequalities follow from (4.11) and (4.4), respectively. By (4.27), (2.2) and (4.8), we notice that

\begin{align*} \| p^\ast- Pu_0(0) \|_{\alpha} &\le L \lambda_N^\alpha \int_{0}^{+\infty} e^{\lambda_N \tau} |x_\tau(\cdot,\omega)|_{C_\alpha}d\tau \\ &\le L \lambda_N^\alpha e^{\gamma r} \int_{0}^{+\infty} e^{(\lambda_N-\gamma)\tau} \|x(\cdot,\omega)\|_+d\tau \le \frac{\zeta}{2}\|x(\cdot,\omega)\|_+. \end{align*}

Similarly, for $R_x$, we get that

\begin{align*} e^{\gamma t}\|R_x\|_{\alpha} & \le Le^{\gamma r} \|x(\cdot,\omega)\|_{+} \times \left[\int_{0}^{t} \left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right] e^{(\gamma-\lambda_{N+1})(t-\tau)}d\tau\right.\\ & \qquad\left.+ \lambda_N^{\alpha } \int_{t}^{+\infty}e^{(\gamma-\lambda_{N })(t-\tau)}d\tau\right] \\ & \le \zeta\|x(\cdot,\omega)\|_{+}. \end{align*}

Then, combining the analyses above and noting that the assumption $\zeta \le \frac{1}{2}$, we observe for $t \ge 0$ that

(4.29)\begin{align} e^{\gamma t}\|\mathcal{X}[x(\cdot,\omega )(t)]\|_{\alpha} \leq&\; |u_0|_{C_\alpha}+ \frac{1}{4}\| u^{\delta}(\cdot;0,\omega,Pu_0(0)) \|_{C_{\gamma,0}}+ \frac{5}{8}\|x(\cdot,\omega)\|_{+} \nonumber\\ &\; +K_0 \lambda_{N+1}^{\alpha-1}(k+1)+\|Q\eta_{\delta}(0,-\infty,\omega)\|_\alpha.\end{align}

While for $ \xi \in [-r,0]$, using the same arguments as the case $t \geq 0$, we find that

(4.30)\begin{align} & e^{\gamma \xi}\|\mathcal{X}[x(\cdot,\omega )(\xi)]\|_{\alpha}\leq e^{\gamma \xi}|u_0|_{C_\alpha}+ e^{\gamma \xi}\|\Psi^{\delta}_{0}(\omega, p^\ast ,\xi) \|_{\alpha}\nonumber\\ & \qquad+e^{\gamma \xi}\|\int_{0}^{+\infty}e^{-A(\xi-\tau)}P(B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_\tau(\cdot,\omega) ,\theta_{\tau}\omega)\nonumber\\ & \qquad-B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} ),\theta_{\tau}\omega))d\tau\|_{\alpha}\nonumber\\ & \quad \leq |u_0|_{C_\alpha}+ \frac{1}{2}\| u^{\delta}(\cdot;0,\omega,Pu_0(0)) \|_{C_{\gamma,0}}+ \frac{1}{2}\|x(\cdot,\omega)\|_{+}+K_0 (\lambda_{N+1}^{\alpha-1}(k+1)+\lambda_N^{\alpha-1})\nonumber\\ & \qquad+c_{P,\eta,0}+d_{Q,\eta,0}.\end{align}

Then we get that $\|\mathcal{X}[x(\cdot,\omega ) ]\|_{+} \lt \infty$ by (4.29) and (4.30), which means that $\mathcal{X}$ is a self-mapping on $C_{+}$. Next we show that $\mathcal{X}$ is a contraction on $C_{+}$. For any $x_1(\cdot,\omega ),x_2(\cdot,\omega )\in C_{+}$ with corresponding to $p^\ast_1, p^\ast_2$ defined by (4.27), we find that for $t \geq 0$,

\begin{align*} & e^{\gamma t} \|\mathcal{X}[x_1(\cdot,\omega )](t)-\mathcal{X}[x_2(\cdot,\omega )](t)]\|_{\alpha}\\ & \quad\leq Le^{\gamma r} \int_{-\infty}^{0}\left[\left(\frac{\alpha}{-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{(\lambda_{N+1}-\gamma)( \tau-t)}\\ & \qquad\|u^{\delta}_{\tau}(\cdot; 0,\omega,p_1^\ast)-u^{\delta}_{\tau}(\cdot; 0,\omega,p_2^\ast)\|_{C_{\gamma,0}}d\tau\\ & \qquad+Le^{\gamma r} \int_{0}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{(\lambda_{N+1}-\gamma)( \tau-t)} \|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+}d\tau \\ & \qquad+Le^{\gamma r} \int_{t}^{+\infty} \lambda_{N }^{\alpha} e^{( \gamma-\lambda_{N })( t-\tau)} \|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+}d\tau \\ & \quad\leq \frac{\zeta}{2(1-\zeta)}\|p_1^\ast -p_2^\ast \|_{\alpha} +\zeta\|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+} \\ & \quad\leq \frac{1}{2}Le^{\gamma r} \int_{0}^{+\infty}\lambda_{N }^{\alpha} e^{( \gamma-\lambda_{N }) \tau}\|x_{1 }(\cdot,\omega) -x_{2 }(\cdot,\omega)\|_{+} d\tau + \frac{1}{2}\|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+} \\ & \quad\leq \frac{3}{4}\|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+}. \end{align*}

While for $ \xi \in [-r,0]$, we have that

\begin{align*} & e^{\gamma \xi} \|\mathcal{X}[x_1(\cdot,\omega )](\xi)-\mathcal{X}[x_2(\cdot,\omega )](\xi)]\|_{\alpha} \\ & \quad\leq e^{\gamma \xi} ( \|\Psi^{\delta}_{0}(\omega, p_1^\ast ,\xi)-\Psi^{\delta}_{0}(\omega, p_2^\ast ,\xi)\|_{\alpha} \\ & \qquad+\| \int_{+\infty}^{0}e^{-A(\xi-\tau)}P (B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_{1,\tau}(\cdot,\omega) ,\theta_{\tau}\omega) \\ & \qquad-B(u^{\delta}_{\tau}(\cdot,0,\omega,u_{0} )+x_{2,\tau}(\cdot,\omega) ,\theta_{\tau}\omega))d\tau \|_{\alpha}) \\ & \quad\leq Le^{\gamma r} \int_{\xi}^{0}\lambda_{N }^{\alpha}e^{( \gamma-\lambda_{N } )( \xi-\tau )} \|u^{\delta}_{\tau}(\cdot; 0,\omega,p_1^\ast)-u^{\delta}_{\tau}(\cdot; 0,\omega,p_2^\ast)\|_{C_{\gamma,0}}d\tau \\ & \qquad+Le^{\gamma r} \int_{-\infty}^{\xi}\left[\left(\frac{\alpha}{\xi-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{(\lambda_{N+1}-\gamma)( \tau -\xi )}\|u^{\delta}_{\tau}(\cdot; 0,\omega,p_1^\ast)\\ &\qquad-u^{\delta}_{\tau}(\cdot; 0,\omega,p_2^\ast)\|_{C_{\gamma,0}}d\tau \\ & \qquad+Le^{\gamma r} \int_{0}^{+\infty}\lambda_{N }^{\alpha}e^{( \gamma-\lambda_{N } )( \xi-\tau )}\|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+}d\tau \\ & \quad\leq \frac{\zeta}{1-\zeta }\|p_1^\ast -p_2^\ast \|_{\alpha} +\frac{\zeta}{2}\|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+} \\ & \quad\leq \frac{3}{4}\|x_{1 }(\cdot,\omega) -x_{2}(\cdot,\omega)\|_{+} , \end{align*}

which leads to

\begin{align*} \| \mathcal{X}[x_1(\cdot,\omega )] -\mathcal{X}[x_2(\cdot,\omega )]\|_{+} \le \frac{3}{4}\|x_1(\cdot,\omega ) - x_2(\cdot,\omega )\|_{+}. \end{align*}

Therefore, $\mathcal{X}$ has a unique fixed point $x^\delta(\cdot,\omega ) \in C_{+}$ and it satisfies the estimate (4.28) according to (4.29) and (4.30). Recall that

\begin{align*} u^{\delta}(t,0,\omega,u^\ast_{0} )=u^{\delta}(t,0,\omega,u_{0} )+x^\delta(t,\omega), \forall t \ge 0. \end{align*}

Consequently, by using (4.28), we have for $t \geq 0$ that

\begin{align*} |\Phi^{\delta}(t , \omega,u_0)-\Phi^{\delta}(t , \omega,u_0^\ast)|_{C_\alpha} &= \sup_{\xi \in [-r,0]}\|x^\delta(t+\xi,\omega)\|_\alpha \\ &\leq \sup_{\xi \in [-r,0]}\{e^{-\gamma(t+\xi)}\}|x^\delta(\cdot,\omega ) |_{+} \le e^{-\gamma(t-r)} C(u_0,\delta,\omega). \end{align*}

Thus claim (iii) holds and the proof is complete.

Proof. By using Hypothesis 1(iii), we find that for any $\epsilon \gt 0$, there exists a $T_1=T_1(\omega) \gt 0$ such that for any $t \le -T_1$ and $|\delta| \lt \tilde{\delta}$,

\begin{align*} \frac{1}{s-t}\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\| \lt \epsilon, \end{align*}

which means that there exists $T_2 = T_2(\omega)( \gt T_1(\omega))$ such that for any $t \le -T_2$ and $|\delta| \lt \tilde{\delta}$,

\begin{align*} & e^{\gamma(t-s)}\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\| \\ & \quad= (s-t) e^{\gamma(t-s)} \times \frac{1}{s-t}\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\| \lt \frac{1}{4}\epsilon. \end{align*}

Meanwhile, by using Hypothesis 1(ii), there exists $\delta_0 \in (0,\tilde{\delta})$ such that for any $t \in [-T_2,s]$ and $|\delta| \le \delta_0$,

\begin{align*} & e^{\gamma(t-s)}\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\| \lt \frac{3}{4}\epsilon. \end{align*}

Thus, combining the above analysis, for all $|\delta| \le \delta_0$,

\begin{align*} \sup_{t \le s}\left\{e^{\gamma(t-s)}\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\|\right\} \le \epsilon. \end{align*}

Then this lemma is established.

Proof. First, consider the convergence of inertial manifolds. For any $t \le s$ and $\hat{p} \in \hat{P}C_{\alpha}$, we observe that

\begin{align*} & e^{\gamma(t-s)}\|u^{\delta}(t; s,\omega,\hat{p}(0))-u^{0}(t;s,\omega, \hat{p}(0))\|_{\alpha}\\ & \quad\leq e^{\gamma(t-s)}\left\{\int_{t}^{s}\|e^{-A(t-\tau)}P(B(u^{\delta}_{\tau}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-B(u^{0}_{\tau}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega))\|_{\alpha}d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{t}\|e^{-A(t-\tau)}Q(B(u^{\delta}_{\tau}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-B(u^{0}_{\tau}(\cdot; s,\omega, \hat{p}(0)),\theta_{\tau}\omega))\|_{\alpha}d\tau\right.\\ & \qquad\left.+\left\|\int_{t}^{s} e^{-A(t-\tau)}P C {d_\tau\left[\int_\tau^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right]}\right\|_{\alpha}\right.\\ & \qquad\left.+\left\|\int_{-\infty}^{t} e^{-A(t-\tau)}Q C{d_\tau\left[\int_\tau^s q\zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right]}\right\|_{\alpha}\right\}\\ & \quad \lt \zeta \|u^{\delta}(\cdot; s,\omega,\hat{p}(0))-u^{0}(\cdot; s,\omega,\hat{p}(0))\|_{C_{\gamma,s}} + \mathcal{S}_{\delta,P}(t)+\mathcal{S}_{\delta,Q}(t), \end{align*}

where the last inequality follows from (4.14) and

\begin{align*} \mathcal{S}_{\delta,P}(t) &:= e^{\gamma (t-s)} \left\|\int_{t}^{s} e^{-A(t-\tau)}P C {d_\tau\left[\int_\tau^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right]}\right\|_{\alpha},\\ \mathcal{S}_{\delta,Q}(t) &:= e^{\gamma (t-s)} \left\|\int_{-\infty}^{t} e^{-A(t-\tau)}Q C{d_\tau\left[\int_\tau^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right]}\right\|_{\alpha}. \end{align*}

For $\mathcal{S}_{\delta, P}(t)$, using the integration by parts formula and (2.2), for any $t\leq \tau \leq s$ and $\omega \in \Omega $, we care about that

\begin{align*} \mathcal{S}_{\delta,P}(t) &\leq e^{\gamma (t-s)}\left\|PC \left(\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\right)\right\|_{\alpha}\\ & \quad+e^{\gamma (t-s)} \left\| \int_{t}^{s} Ae^{-A(t-\tau)} PC \left[\int_\tau^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right]d\tau\right\|_{\alpha}\\ &\leq e^{\gamma (t-s)}\|A^{\alpha-1} \|\|C\|_{L(H,D(A))} \left\| \int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\right\|\\ &\quad+ e^{\gamma (t-s)} \lambda_{N}^{\alpha}\|C\|_{L(H,D(A))} \int_{t}^{s} e^{-\lambda_{N}(t-\tau)} \left\|\int_\tau^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right\|d\tau\\ & \le \sup_{t \le s} \left\{e^{\gamma(t-s)}\left\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\right\|\right\} \|C\|_{L(H,D(A))}\\ & \qquad\left(\|A^{\alpha-1} \|+\frac{\lambda_{N}^{\alpha}}{\gamma-\lambda_N}\right). \end{align*}

It then follows from Lemma 4.2 that $\sup_{t \le s} \{\mathcal{S}_{\delta,P}(t)\}\rightarrow 0$ as $\delta\rightarrow 0$.

For $\mathcal{S}_{\delta, Q}(t)$, we first restrict $|\delta| \le \delta_0$ (see Lemma 4.2). Similarly, by applying the integration by parts formula, (2.2) and (4.8), for any $\tau\leq t \leq s$ and $\omega \in \Omega $, we find that

\begin{align*} \mathcal{S}_{\delta,Q}(t) &\leq e^{\gamma (t-s)}\left\|QC \left(\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t))\right)\right\|_{\alpha}\\ & \quad +e^{\gamma (t-s)} \left\|\int_{-\infty}^{t} Ae^{-A(t-\tau)}QC\left(\int_\tau^s\zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(\tau))\right)d\tau\right\|_{\alpha}\\ & \le \sup_{t \le s} \left\{e^{\gamma(t-s)}\left\|\int_t^s \zeta_{\delta}(\theta_{r}\omega){dr}-(W(s)-W(t)) \right\|\right\}\|C\|_{L(H,D(A))}\\ & \quad\times \left(\|A^{\alpha-1} \|+\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-\gamma)^{\alpha}}{\lambda_{N+1}-\gamma}\right). \end{align*}

Lemma 4.2 implies that $\sup_{t \le s} \{\mathcal{S}_{\delta,Q}(t)\}\rightarrow 0 $ as $\delta \rightarrow 0$. Consequently,

\begin{align*} \|u^{\delta}(\cdot; s,\omega,\hat{p}(0))-u^{0}(\cdot;s,\omega, \hat{p}(0))\|_{C_{\gamma,s}} \lt \zeta \|u^{\delta}(\cdot; s,\omega,\hat{p}(0))-u^{0}(\cdot; s,\omega,\hat{p}(0))\|_{C_{\gamma,s}}+o(1), \end{align*}

as $\delta \rightarrow 0$. Then we get that

(4.33)\begin{equation} \|u^{\delta}(\cdot;s,\omega, \hat{p}(0))-u^{0}(\cdot;s,\omega, \hat{p}(0))\|_{C_{\gamma,s}} \rightarrow 0 \end{equation}

as $\delta \rightarrow 0$. Notice that for any $\xi \in [-r,0]$,

\begin{align*} \|\Psi^{\delta}_{s}(\omega,\hat{p}(0),\xi)-\Psi^{0}_{s}(\omega,\hat{p}(0),\xi)\|_\alpha = &\; \|u^{\delta}(s+\xi;s,\omega, \hat{p}(0))-u^{0}(s+\xi;s,\omega, \hat{p}(0))\|_{\alpha}\\ \le &\; e^{\gamma r} \|u^{\delta}(\cdot;s,\omega, \hat{p}(0))-u^{0}(\cdot;s,\omega, \hat{p}(0))\|_{C_{\gamma,s}}. \end{align*}

Therefore, we complete the proof of (4.31).

We next prove the convergence of $C^1$-smooth inertial manifolds. Notice that $D_{p}u^{\delta}(\cdot;s,\omega, p) \in L(PD(A^{\alpha}), C_{\gamma-\eta,s})$ with $0 \lt \eta \le \frac{3}{2}\varepsilon_0$. For any $t \leq s$ and $\hat{p} \in \hat{P}C_{\alpha}$, along with (4.17) and (2.2), we get that

\begin{align*} & e^{( \gamma-\eta) (t-s)}\|D_{p}u^{\delta}(t; s,\omega,\hat{p}(0))-D_{p}u^{0}(t; s,\omega,\hat{p}(0))\|_{L(PD(A^\alpha), D(A^\alpha))}\\ & \quad\leq e^{( \gamma-\eta)(t-s)}\left\{\int_{t}^{s} \lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \times\|D_{u }B(u_{\tau}^{\delta}(\cdot; s,\omega,\hat{p}(0)),\theta_{\tau}\omega)D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0))\right.\\ & \qquad\left.-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)D_{p}u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0))\|d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{-\lambda_{N+1}(t-\tau)} \times\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)\right.\\ & \qquad \left.\times D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0))\right.\\ & \qquad\left.-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)D_{p}u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0))\|d\tau\vphantom{\int_{t}^{s}}\right\}=:\mathcal{I}_{\delta,P}(t)+\mathcal{I}_{\delta,Q}(t). \end{align*}

For $\mathcal{I}_{\delta,P}(t)$, by using (2.3) and (4.8), we find that for any $t\le \tau\leq s$,

\begin{align*} \mathcal{I}_{\delta,P}(t) &\leq e^{(\gamma-\eta)(t-s)} \int_{t}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)}\\ & \quad\times\{\|D_{u }B(u_{\tau}^{\delta}(\cdot; s,\omega,\hat{p}(0)),\theta_{\tau}\omega) (D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0))-D_{p}u_{\tau}^{0}(\cdot; s,\omega,\hat{p}(0)))\| \\ & \quad+ \|(D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)\\ &\qquad-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)) D_{p}u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)) \|\}d\tau\\ &\le Le^{(\gamma-\eta )r}\frac{\lambda_N^\alpha}{\gamma-\eta-\lambda_N} \|D_{p}u ^{\delta}(\cdot; s,\omega,\hat{p}(0))-D_{p}u ^{0}(\cdot;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})}\\ & \quad+\mathcal{\tilde{I}}_{\delta,P}(t), \end{align*}

where

\begin{align*} \mathcal{\tilde{I}}_{\delta,P}(t) &:= e^{(\gamma-\eta)(t-s)} \int_{t}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\|D_{p}u^0(\tau+\xi;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}),D(A^\alpha))} \\ & \times\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega) \|d\tau. \end{align*}

Next for $0 \lt \eta \le \varepsilon_0$, we claim that

(4.34)\begin{equation} \|\mathcal{\tilde{I}}_{\delta,P}(t)\|_{L(DP(A^\alpha),C_{\gamma-\eta,s})}=o(1) \end{equation}

as $\delta \rightarrow 0$. Indeed, with the analogous arguments as $I_{\delta}(t)$ in Lemma 4.1, we split $\mathcal{\tilde{I}}_{\delta,P}(t)$ into a sum of two parts as $\mathcal{\tilde{I}}_{\delta,P}(t)=\mathcal{\tilde{I}}^{1}_{\delta,P}(t)+\mathcal{\tilde{I}}^{2}_{\delta,P}(t)$, where

\begin{align*} & \mathcal{\tilde{I}}^{1}_{\delta,P}(t)\\ & \quad:= \left\{ \begin{array}{ll} e^{(\gamma-\eta)(t-s)} \int_{t}^{\tilde{T}_1}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\\ \quad\times\|D_{p}u^0(\tau+\xi;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}),D(A^\alpha))} \\ \quad\times\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega) \|d\tau, & {t \lt \tilde{T}_1},\\ 0, & {t \ge \tilde{T}_1}, \end{array} \right. \end{align*}

\begin{align*} \mathcal{\tilde{I}}^{2}_{\delta,P}(t):= \left\{ \begin{array}{ll} e^{(\gamma-\eta)(t-s)} \int_{\tilde{T}_1}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\\ \quad\times\|D_{p}u^0(\tau+\xi;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}),D(A^\alpha))}\\ \quad\times\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega) \|d\tau, & {t \lt \tilde{T}_1},\\ e^{(\gamma-\eta)(t-s)} \int_{t}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\\ \quad\times\|D_{p}u^0(\tau+\xi;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}),D(A^\alpha))} \\ \quad\times\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega) \|d\tau, & {t \ge \tilde{T}_1}, \end{array} \right. \end{align*}

where for any given $\epsilon \gt 0$, $\tilde{T}_1$ is a sufficiently negative constant such that

\begin{equation*}\frac{\lambda_{N}^{\alpha}\tilde{\zeta}(\frac{3\varepsilon_0}{2}) e^{\frac{\varepsilon_0}{2}(\tilde{T}_1 -s)}}{1-\tilde{\zeta}(\frac{3\varepsilon_0}{2})} \lt \frac{\epsilon}{4}.\end{equation*}

For $\mathcal{\tilde{I}}^{1}_{\delta,P}(t)$, applying (4.6) and the fact that $\|D_{u}B(u,\omega)\|\leq L$, we have that for $t \lt \tilde{T}_1$,

\begin{align*} \mathcal{\tilde{I}}^{1}_{\delta,P}(t) &\leq 2 L\lambda_{N}^{\alpha}\int_{t}^{\tilde{T}_1}e^{(\gamma-\frac{3}{2}\varepsilon_0-\lambda_{N})(t-\tau) } e^{( \frac{3}{2}\varepsilon_0-\eta)(t-s) } e^{(\gamma-\frac{3}{2}\varepsilon_0) r} \\ & \quad\times\|D_{p}u ^{0}(\cdot;s,\omega,\hat{p}(0))\|_{L(PD(A^{\alpha}), C_{\gamma-\frac{3}{2}\varepsilon_0,s})}d\tau \\ &\le \frac{2 L\lambda_{N}^{2\alpha}e^{\frac{\varepsilon_0}{2}(t-s)}}{(1-\tilde{\zeta}(\frac{3\varepsilon_0}{2}))} e^{(\gamma-\frac{3}{2}\varepsilon_0) r} \int_{t}^{\tilde{T}_1} e^{(\gamma-\frac{3}{2}\varepsilon_0-\lambda_{N})(t-\tau) } d\tau \le \frac{2\lambda_{N}^{\alpha}\tilde{\zeta}(\frac{3\varepsilon_0}{2}) e^{\frac{\varepsilon_0}{2}(\tilde{T}_1 -s)}}{1-\tilde{\zeta}(\frac{3\varepsilon_0}{2})} \lt \frac{\epsilon}{2}. \end{align*}

For $\mathcal{\tilde{I}}^{2}_{\delta,P}(t)$, using (4.6) again, for such fixed $\tilde{T}_1$ and any $t \le s$, we get that

\begin{align*} \mathcal{\tilde{I}}^{2}_{\delta,P}(t) &\leq \int_{\tilde{T}_1}^{s} e^{(\gamma-\eta-\lambda_{N})(t-\tau) } e^{(\gamma-\eta) r}\frac{\lambda_{N}^{2\alpha}}{1-\tilde{\zeta}(\eta) } \\ & \times\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega) \|d\tau. \end{align*}

Following (4.33), notice that for each $\tau \le s$,

\begin{align*} & |u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0))-u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0))|_{C_\alpha} \\ & \le e^{(\gamma-\eta)(s-\tau+r)}\|u^{\delta}(\cdot;s,\omega, \hat{p}(0))-u^{0}(\cdot;s,\omega, \hat{p}(0))\|_{C_{\gamma-\eta,s}} \rightarrow 0 \end{align*}

as $\delta \rightarrow 0$ and

\begin{align*} & \|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega) -D_{u }B(u_{\tau}^{0}(\cdot;s,\omega, \hat{p}(0)),\theta_{\tau}\omega))\| \le 2 L. \end{align*}

Then by the Dominated Convergence Theorem, for the given $\epsilon$ as above, there exists $\delta_{P} \gt 0$ such that for all $ |\delta| \le \delta_{P}$,

\begin{align*} \sup_{t\le s}\{\mathcal{\tilde{I}}^{2}_{\delta,P}(t)\} \lt \frac{\epsilon}{2}. \end{align*}

Thus claim (4.34) holds.

For $\mathcal{I}_{\delta, Q}(t)$, by using the similar arguments as above, we find that for any $\tau\leq t \leq s$,

\begin{align*} \mathcal{I}_{\delta, Q}(t) &\leq e^{(\gamma-\eta)(t-s)}\int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{-\lambda_{N+1}(t-\tau)}\\ & \quad\times\{\|D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,\hat{p}(0)),\theta_{\tau}\omega) (D_{p}u_{\tau}^{\delta}(\cdot;s,\omega,\hat{p}(0))-D_{p}u_{\tau}^{0}(\cdot;s,\omega,\hat{p}(0)))\|\\ & \quad+ \|(D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,\hat{p}(0)),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{0}(\cdot;s,\omega,\hat{p}(0)),\theta_{\tau}\omega))\\ & \quad\times D_{p}u_{\tau}^{0}(\cdot;s,\omega,\hat{p}(0)) \|\}d\tau\\ & \lt Le^{(\gamma-\eta)r}\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-(\gamma-\eta))^{\alpha}}{\lambda_{N+1}-(\gamma-\eta) }\\ & \quad\times \|D_{p}u ^{\delta}(\cdot; s,\omega,\hat{p}(0))-D_{p}u ^{0}(\cdot;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})} + o(1) \end{align*}

as $\delta \rightarrow 0$. Therefore, combining the estimates of $\mathcal{I}_{\delta, P}(t)$ and $\mathcal{I}_{\delta, Q}(t)$, in view of (4.15), we get that

\begin{align*} & \|D_{p}u ^{\delta}(\cdot; s,\omega,\hat{p}(0))-D_{p}u ^{0}(\cdot;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})} \\ & \lt \tilde{\zeta}(\eta) \|D_{p}u ^{\delta}(\cdot; s,\omega,\hat{p}(0))-D_{p}u ^{0}(\cdot;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})} +o(1), \end{align*}

as $\delta \rightarrow 0$. It then shows that

\begin{align*} \|D_{p}u ^{\delta}(\cdot; s,\omega,\hat{p}(0))-D_{p}u ^{0}(\cdot;s,\omega, \hat{p}(0))\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})} \rightarrow 0, \end{align*}

as $\delta \rightarrow 0$. Furthermore for any $\hat{p} \in \hat{P}C_{\alpha}$ and $\xi \in [-r,0]$, we get that

\begin{align*} & \|D_{p}\Psi^{\delta}_{s}(\omega,\hat{p}(0),\xi)-D_{p}\Psi^{0}_{s}(\omega,\hat{p}(0),\xi)\|_{L(PD(A^\alpha),D(A^\alpha))}\\ &\quad=\|D_{p}u^{\delta}(s+\xi;s,\omega,\hat{p}(0))-D_{p}u^{0}(s+\xi;s,\omega,\hat{p}(0))\|_{L(PD(A^\alpha),D(A^\alpha))} \\ & \quad\le e^{(\gamma-\eta)r} \|D_{p}u ^{\delta}(\cdot; s,\omega,\hat{p}(0))-D_{p}u ^{0}(\cdot;s,\omega, \hat{p}(0))\| _{L(PD(A^\alpha),C_{\gamma-\eta,s})}, \end{align*}

which establishes (4.32) and completes the proof.

Proof. (i) Let $\rho \gt 0$ and $\mathcal{C}_{\rho,\sigma}$ be given in Theorem 3.1. Then by Hypothesis 3 and (3.6), for any $\delta \in \mathbb{R}$ and $\tau\in (\sigma,t]$, we find that

(5.5)\begin{equation} \| B(u^\delta_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega)\| \leq\; Le^{\rho(\tau-\sigma)} \Xi_{\sigma,\rho,u_{\sigma}}+K_0. \end{equation}

Define $U_M^\delta(\cdot,\omega):=u^{\delta}_M(\cdot,\sigma,\omega,P_Mu_{\sigma})-u^{\delta}(\cdot,\sigma,\omega,u_{\sigma})$. By (2.8) and (2.11), we get that

\begin{align*} U_M^\delta(t,\omega)= \left\{ \begin{array}{ll} e^{-A(t-\sigma)}[P_M u_{\sigma}(0)-u_\sigma(0)]+P_{M}\eta_\delta(t,\sigma,\omega)-\eta_\delta(t,\sigma,\omega) & \\ +\int_{\sigma}^{t}e^{-A(t-\tau)}[P_{M}B(u^\delta_{M,\tau}(\cdot,\sigma,\omega,P_Mu_{\sigma}),\theta_{\tau}\omega)- B(u^{\delta}_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega)]d\tau, & \\ \qquad \qquad \qquad {t\in (\sigma,\sigma+T] }, & \\ P_M u_{\sigma}(t-\sigma)-u_\sigma(t-\sigma), \quad \qquad \qquad {t\in [\sigma-r,\sigma]}, \end{array} \right. \end{align*}

and set

\begin{equation*}\Upsilon_M^\delta:=\sup_{t\in [\sigma,\sigma+T]}\{\|P_{M}\eta_\delta(t,\sigma,\omega)- \eta_\delta(t,\sigma,\omega)\|_{\alpha}\}.\end{equation*}

According to Lemma 2.1, $\Upsilon_M^\delta\rightarrow 0$ as $M\rightarrow +\infty$. Then for any $t\in (\sigma,\sigma+T]$, by using (2.1), (2.3), (5.4) and (5.5), we observe that

\begin{align*} & e^{-\rho(t-\sigma)} \|U_M^\delta(t,\omega)\|_{\alpha}\\ & \quad\leq K(0)\|P_M u_{\sigma}(0)-u_\sigma(0)\|_\alpha+\Upsilon_M^\delta\\ & \qquad +e^{-\rho(t-\sigma)}\int_{\sigma}^{t}\|A^{\alpha}e^{-A(t-\tau)} Q_{M}(B(u^\delta_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega))\|{d\tau}\\ & \qquad +e^{-\rho(t-\sigma)}\int_{\sigma}^{t}\|A^{\alpha}e^{-A(t-\tau)} P_M[B(u^\delta_{M,\tau}(\cdot,\sigma,\omega,P_Mu_{\sigma}),\theta_{\tau}\omega)\\ & \qquad- B(u^{\delta}_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega)]\|{d\tau}\\ & \quad\leq K(0)\|P_M u_{\sigma}(0)-u_\sigma(0)\|_\alpha+\Upsilon_M^\delta+\frac{1}{2} \sup_{t \in [\sigma-r,\sigma+T]}\|U_M^\delta(t,\omega)\|_{\alpha}\\ & \qquad+\lambda_{M+1}^{-\epsilon_\alpha} e^{-\rho(t-\sigma)} K(\alpha+\epsilon_\alpha) \int_{\sigma}^{t}(t-\tau)^{-(\alpha+\epsilon_\alpha)} e^{-\lambda(t-\tau)} \|B(u^\delta_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega)\|{d\tau}\\ & \quad\le \frac{1}{2} \sup_{t \in [\sigma-r,\sigma+T]}\|U_M^\delta(t,\omega)\|_{\alpha}+\Upsilon_M^\delta+\Theta_M, \end{align*}

where

\begin{align*} \Theta_M &:= K(0)\|P_M u_{\sigma}(0)-u_\sigma(0)\|_\alpha+\lambda_{M+1}^{-\epsilon_\alpha}K(\alpha+\epsilon_\alpha)\Gamma(1-\alpha-\epsilon_\alpha)\\ & \quad\times \{L(\lambda+\rho)^{\alpha+\epsilon_\alpha-1}\Xi_{\sigma,\rho,u_{\sigma}}+\lambda^{\alpha+\epsilon_\alpha-1}K_0\}. \end{align*}

On the other hand, for any $t\in [\sigma-r,\sigma]$, noting that $u_\sigma \in C_\alpha$ and (5.4), we have that

\begin{align*} \|U_M^\delta(t,\omega)\|_{\alpha} & \leq \sup_{t \in [\sigma-r,\sigma]} \{\|P_M u_{\sigma}(t-\sigma)-u_\sigma(t-\sigma)\|_\alpha\} \\ &=\sup_{t \in [-r,0]}\{\|Q_M u_\sigma(t)\|_\alpha\} \le \lambda_{M+1}^{-\epsilon_\alpha} |u_\sigma|_{C_{\alpha+\epsilon_\alpha}}. \end{align*}

Combining the analyses above, we obtain that

\begin{align*} \sup_{t \in [\sigma-r,\sigma+T]}\{e^{-\rho(t-\sigma)}\|U_M^\delta(t,\omega)\|_{\alpha}\} &\le 2 e^{\rho T} \bigl\{e^{\rho r} \lambda_{M+1}^{-\epsilon_\alpha}|u_\sigma|_{C_{\alpha+\epsilon_\alpha}} +\Upsilon_M^\delta+\Theta_M^\delta \bigr\} \rightarrow 0, \end{align*}

as $\ M\rightarrow +\infty$. Finally, following the fact that

\begin{align*} \sup_{t \in [\sigma-r,\sigma+T]}\{\|U_M^\delta(t,\omega)\|_{\alpha}\} &\le e^{\rho T} \sup_{t \in [\sigma-r,\sigma+T]}\{e^{-\rho(t-\sigma)}\|U_M^\delta(t,\omega)\|_{\alpha}\}, \end{align*}

claim (i) is established.

The idea of proving claim (ii) is similar to that of claim (i). For the reader’s convenience, we provide the details below. Let $\tilde{U}^\delta_M(\cdot,\omega):=u^{\delta}_M(\cdot,\sigma,\omega,P_Mu_{\sigma})-u^{0}(\cdot,\sigma,\omega,u_{\sigma})$. Then we have that

\begin{align*} \tilde{U}^\delta_M(t,\omega)= \left\{ \begin{array}{ll} e^{-A(t-\sigma)}[P_M u_{\sigma}(0)-u_\sigma(0)]+P_{M}\eta_\delta(t,\sigma,\omega)-\eta_0(t,\sigma,\omega) & \\ +\int_{\sigma}^{t}e^{-A(t-\tau)}[P_{M}B(u^\delta_{M,\tau}(\cdot,\sigma,\omega,P_Mu_{\sigma}),\theta_{\tau}\omega)- B(u^{0}_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega)]d\tau, & \\ \qquad \qquad \qquad {t\in (\sigma,\sigma+T] }, & \\ P_M u_{\sigma}(t-\sigma)-u_\sigma(t-\sigma), \quad \qquad \qquad {t\in [\sigma-r,\sigma]}. \end{array} \right. \end{align*}

In view of Lemma 2.1, we have that

\begin{equation*}\tilde{\Upsilon}^\delta_M:=\sup_{t\in [\sigma,\sigma+T]}\|P_{M}\eta_\delta(t,\sigma,\omega)- \eta_0(t,\sigma,\omega)\|_{\alpha}\rightarrow0,\end{equation*}

as $M\rightarrow +\infty$ and $\delta\rightarrow0$. Then, along with (2.1), (2.3), (5.4) and (5.5) again, we find that for any $t\in (\sigma,\sigma+T]$,

\begin{align*} & e^{-\rho(t-\sigma)} \|\tilde{U}_M^\delta(t,\omega)\|_{\alpha}\\ & \quad\leq K(0)\|P_M u_{\sigma}(0)-u_\sigma(0)\|_\alpha +\tilde{\Upsilon}_M^\delta \\ & \qquad +e^{-\rho(t-\sigma)}\int_{\sigma}^{t}\|A^{\alpha}e^{-A(t-\tau)} Q_{M}(B(u^0_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega))\|{d\tau}\\ & \qquad +e^{-\rho(t-\sigma)}\int_{\sigma}^{t}\|A^{\alpha}e^{-A(t-\tau)} P_M[B(u^\delta_{M,\tau}(\cdot,\sigma,\omega,P_Mu_{\sigma}),\theta_{\tau}\omega)\\ & \qquad- B(u^{0}_{\tau}(\cdot,\sigma,\omega,u_{\sigma}),\theta_{\tau}\omega)]\|{d\tau}\\ & \quad\leq \frac{1}{2} \sup_{t \in [\sigma-r,\sigma+T]}\|\tilde{U}_M^\delta(t,\omega)\|_{\alpha} +\tilde{\Upsilon}_M^\delta+\Theta_M. \end{align*}

For any $t\in [\sigma-r,\sigma]$, we notice that

\begin{align*} \|\tilde{U}_M^\delta(t,\omega)\|_{\alpha}= \|U_M^\delta(t,\omega)\|_{\alpha} \le \lambda_{M+1}^{-\epsilon_\alpha} |u_\sigma|_{C_{\alpha+\epsilon_\alpha}}. \end{align*}

With all that said, we get the conclusion that

\begin{align*} \sup_{t \in [\sigma-r,\sigma+T]}\{\|\tilde{U}_M^\delta(t,\omega)\|_{\alpha}\} & \le e^{\rho T} \sup_{t \in [\sigma-r,\sigma+T]}\{e^{-\rho(t-\sigma)}\|\tilde{U}_M^\delta(t,\omega)\|_{\alpha}\} \\ & \le 2 e^{\rho T} \bigl\{e^{\rho r} \lambda_{M+1}^{-\epsilon_\alpha}|u_\sigma|_{C_{\alpha+\epsilon_\alpha}} +\tilde{\Upsilon}_M^\delta+\Theta_M\} \rightarrow 0, \end{align*}

as $\delta\rightarrow0$ and $M\rightarrow +\infty$. Then claim (ii) is established and Theorem 5.3 is complete.

Proof. We only prove claim (i) and the proof for claim (ii) is similar. By using (3.1) and (5.1), for any $\xi \in [-r,0]$, we find that

\begin{align*} \|\Phi^{\delta}_M(t,\omega,P_Mu_{0})(\xi)-\Phi^{\delta}(t,\omega,u_{0})(\xi)\|_\alpha \le \sup_{\tilde{t} \in [t-r,t]}\{\|u^{\delta}_M(\tilde{t},0,\omega,P_Mu_{0})-u^{\delta}(\tilde{t},0,\omega,u_{0})\|_\alpha\}, \end{align*}

which, along with Theorem 5.3 (i), suggests that

\begin{align*} |\Phi^{\delta}_M(t,\omega,P_Mu_{0})-\Phi^{\delta}(t,\omega,u_{0})|_{C_{\alpha}} \le \sup_{\tilde{t} \in [t-r,t]}\{\|u^{\delta}_M(\tilde{t},0,\omega,P_Mu_{0})-u^{\delta}(\tilde{t},0,\omega,u_{0})\|_\alpha\} \rightarrow 0, \end{align*}

as $M\rightarrow +\infty$. The proof is complete.

Proof. For claim (i), by Hypothesis 3 and (4.3), we have that for any $\delta \in \mathbb{R}$ and $\tau\le t$,

(5.10)\begin{equation} \| B(u^\delta_{\tau}(\cdot;s,\omega,p),\theta_{\tau}\omega)\| \le Le^{\gamma(s-\tau)+\gamma r} \tilde{\Xi}_{s,\gamma,p}+K_0. \end{equation}

For any $\delta \in \mathbb{R}$, $t \le s$ and $p \in PD(A^{\alpha})$, by applying (2.2), (2.3), (4.8), (5.4) and (5.10), we find that

\begin{align*} & e^{\gamma(t-s)}\|u^{\delta}_M(t; s,\omega,p)-u^{\delta}(t;s,\omega, p)\|_{\alpha}\\ & \quad\leq e^{\gamma(t-s)}\left\{\int_{t}^{s}\|e^{-A(t-\tau)}P(B(u^{\delta}_{M,\tau}(\cdot;s,\omega, p),\theta_{\tau}\omega)-B(u^{\delta}_{\tau}(\cdot;s,\omega, p),\theta_{\tau}\omega))\|_{\alpha}d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{t}\|e^{-A(t-\tau)}QP_M[B(u^{\delta}_{M,\tau}(\cdot;s,\omega,p),\theta_{\tau}\omega)-B(u^{\delta}_{\tau}(\cdot; s,\omega, p),\theta_{\tau}\omega))]\|_{\alpha}d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{t}\|e^{-A(t-\tau)}QQ_MB(u^{\delta}_{\tau}(\cdot; s,\omega, p),\theta_{\tau}\omega))\|_{\alpha}d\tau +\|QQ_M\eta_{\delta}(t,-\infty,\omega)\|_{\alpha}\right\}\\ &\quad \lt \zeta \|u^{\delta}_{M}(\cdot;s,\omega, p)-u^{\delta}(\cdot;s,\omega, p)\|_{C_{\gamma,s}} +\lambda_{M+1}^{-\epsilon_\alpha} \sup_{t\le s}\{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\}\\ & \qquad+\lambda_{M+1}^{-\epsilon_\alpha} \int_{-\infty}^{t}\left[\left(\frac{\alpha+\epsilon_\alpha}{t-\tau}\right)^{\alpha+\epsilon_\alpha}\!+\lambda_{N+1}^{\alpha+\epsilon_\alpha}\right] e^{-\lambda_{N+1}(t-\tau)}e^{\gamma(t-s)}\| B(u^\delta_{\tau}(\cdot;s,\omega,p),\theta_{\tau}\omega)\|d\tau\\ & \quad \lt \zeta \|u^{\delta}_{M}(\cdot;s,\omega, p)-u^{\delta}(\cdot;s,\omega, p)\|_{C_{\gamma,s}}\\ & \qquad+\lambda_{M+1}^{-\epsilon_\alpha}\sup_{t\le s}\{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\}+\tilde{\Theta}_M, \end{align*}

where

\begin{align*} \tilde{\Theta}_M & := \lambda_{M+1}^{-\epsilon_\alpha} \left\{\lambda_{N+1}^{\alpha+\epsilon_\alpha}\left[\frac{K_0}{\lambda_{N+1}}+\frac{Le^{\gamma r} \tilde{\Xi}_{s,\gamma,p}}{\lambda_{N+1}-\gamma}\right]+(\alpha+\epsilon_\alpha)^{\alpha+\epsilon_\alpha}\Gamma(1-\alpha-\epsilon_\alpha)\right.\\ & \quad\left.\times \left[\frac{\lambda_{N+1}^{\alpha+\epsilon_\alpha} K_0}{\lambda_{N+1}}+\frac{(\lambda_{N+1}-\gamma)^{\alpha+\epsilon_\alpha}Le^{\gamma r}\tilde{\Xi}_{s,\gamma,p}}{\lambda_{N+1}-\gamma}\right] \right\}. \end{align*}

In view of (4.12), we observe that $\sup_{t\le s}\{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\} \lt +\infty$. Then we get that

(5.11)\begin{align} & \|u^{\delta}_{M}(\cdot;s,\omega, p)-u^{\delta}(\cdot;s,\omega, p)\|_{C_{\gamma,s}} \nonumber\\ & \quad\le \frac{1}{1-\zeta} \{\lambda_{M+1}^{-\epsilon_\alpha} \sup_{t\le s}\{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\}+\tilde{\Theta}_M\} \rightarrow 0,\end{align}

as $M\rightarrow +\infty$. Following (4.18) and (5.3), notice that for all $\xi \in [-r,0]$,

(5.12)\begin{align} \|\Psi^{\delta}_{s,M}(\omega,p,\xi)- \Psi^{\delta}_{s}(\omega,p,\xi) \|_{\alpha} & = \|u^{\delta}_M(s+\xi;s,\omega, p)-u^{\delta}(s+\xi;s,\omega, p)\|_{\alpha} \nonumber\\ & \le e^{\gamma r} \|u^{\delta}_M(\cdot;s,\omega, p)-u^{\delta}(\cdot;s,\omega, p)\|_{C_{\gamma,s}}.\end{align}

Thus (5.6) holds.

On the other hand, using the integration by parts formula provided in (1.3), we get that

\begin{align*} \eta_{\delta}(t,-\infty,\omega)-\eta_{0}(t,-\infty,\omega) & = C\left[(W(s)-W(t))-\int_t^s\zeta_{\delta}(\theta_r \omega)dr\right]\\ & \quad -\int_{-\infty}^t Ae^{-A(t-\tau)}C\left[W(s)-W(t)-\int_t^s\zeta_{\delta}(\theta_r \omega)dr\right]d\tau, \end{align*}

which together with (4.8) implies that for all $t \le s$,

\begin{align*} & e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)-Q\eta_{0}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\\ & \quad\le \sup_{t \le s}\left\{e^{\gamma (t-s)}\left\|\int_t^s\zeta_{\delta}(\theta_r \omega)dr-(W(s)-W(t))\right\|\right\}\\ & \qquad\times \left\{\|C\|_{L(H,D(A^{\alpha+\epsilon_\alpha}))}+ \frac{\lambda_{N+1}^{\alpha+\epsilon_\alpha}+(\alpha+\epsilon_\alpha)^{\alpha+\epsilon_\alpha}\Gamma(1-\alpha-\epsilon_\alpha) (\lambda_{N+1}-\gamma)^{\alpha+\epsilon_\alpha}}{\lambda_{N+1}-\gamma}\right.\\ & \qquad\left.\|C\|_{L(H,D(A))}\vphantom{\frac{\lambda_{N+1}^{\alpha+\epsilon_\alpha}+(\alpha+\epsilon_\alpha)^{\alpha+\epsilon_\alpha}\Gamma(1-\alpha-\epsilon_\alpha) (\lambda_{N+1}-\gamma)^{\alpha+\epsilon_\alpha}}{\lambda_{N+1}-\gamma}}\right\}. \end{align*}

It then follows from Lemma 4.2 that

\begin{align*} \lim_{\delta \rightarrow 0}\sup_{t \le s} \{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)-Q\eta_{0}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\}=0. \end{align*}

Hence there exists a $\bar{\bar{\delta}}=\bar{\bar{\delta}}(\omega) \gt 0$ such that for all $|\delta| \lt \bar{\bar{\delta}}$,

\begin{equation*} \sup_{t\le s}\{e^{\gamma(t-s)}\|Q\eta_{\delta}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\} \le 1+ \sup_{t\le s}\{e^{\gamma(t-s)}\|Q\eta_{0}(t,-\infty,\omega)\|_{\alpha+\epsilon_\alpha}\} \lt +\infty. \end{equation*}

Repeating the proof for (5.11) and (5.12), we immediately find that

(5.13)\begin{equation} \|\Psi^{\delta}_{s,M}(\omega,p,\xi)- \Psi^{\delta}_{s}(\omega,p,\xi) \|_{\alpha}\rightarrow 0~\mbox{as}~M\rightarrow +\infty \end{equation}

uniformly for $|\delta| \lt \bar{\bar{\delta}}$.

Furthermore, along with (4.31) and (5.13), we obtain that

\begin{align*} & \|\Psi^{\delta}_{s,M}(\omega,p,\cdot)- \Psi^{0}_{s}(\omega,p,\cdot) \|_{C_\alpha} \\ &\quad\leq \|\Psi^{\delta}_{s,M}(\omega,p,\cdot)- \Psi^{\delta}_{s}(\omega,p,\cdot) \|_{C_\alpha}+ \|\Psi^{\delta}_{s}(\omega,p,\cdot)-\Psi^{0}_{s}(\omega,p,\cdot)\|_{C_\alpha} \rightarrow 0, \end{align*}

as $M\rightarrow +\infty$ and $\delta \rightarrow 0$. Then we have proved (5.7) and claim (i) is established.

Next for claim (ii), in order to show the convergence of $C^1$ inertial manifolds, recall that for any $0 \lt \eta \le \frac{3}{2}\varepsilon_0$, $D_{p}u^{\delta}(\cdot;s,\omega, p) \in L(PD(A^{\alpha}), C_{\gamma-\eta,s})$ and $D_{p}u^{\delta}_M(\cdot;s,\omega, p) \in L(PD(A^{\alpha}), C_{\gamma-\eta,s,M})$. Then for any $\delta \in \mathbb{R}$, $t \le s$ and $p \in PD(A^{\alpha})$, by using (4.17) and (5.2), we have that

\begin{align*} & e^{( \gamma-\eta) (t-s)}\|D_{p}u^{\delta}_M(t; s,\omega,p)-D_{p}u^{\delta}(t; s,\omega,p)\|_{L(PD(A^\alpha),D(A^\alpha))}\\ & \quad\leq e^{( \gamma-\eta)(t-s)}\left\{\int_{t}^{s} \|A^\alpha e^{-A(t-\tau)}P[D_{u }B(u_{M,\tau}^{\delta}(\cdot; s,\omega,p),\theta_{\tau}\omega)D_{p}u_{M,\tau}^{\delta}(\cdot;s,\omega, p)\right.\\ & \qquad \left.-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p)]\| d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{t}\|A^\alpha e^{-A(t-\tau)}QP_M[D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)D_{p}u_{M,\tau}^{\delta}(\cdot;s,\omega, p)\right.\\ & \qquad\left.-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p)]\| d\tau\right.\\ & \qquad\left.+\int_{-\infty}^{t}\|A^\alpha e^{-A(t-\tau)}Q Q_M D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega) D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p)\| d\tau\right\}\\ & \quad =: \mathcal{I}^\delta_{M,P}(t)+\mathcal{I}^\delta_{M,Q}(t)+\hat{\mathcal{I}}^\delta_{M,Q}(t). \end{align*}

The estimates for $\mathcal{I}^\delta_{M,P}(t)$ and $\mathcal{I}^\delta_{M,Q}(t)$ are similar to Theorem 4.2, and for the convenience of the reader, we will briefly prove them as follows.

For $\mathcal{I}^\delta_{M,P}(t)$, according to (2.2), (2.3) and (4.8), we find that for any $t\le \tau\leq s$,

\begin{align*} \mathcal{I}^\delta_{M,P}(t) & \le e^{(\gamma-\eta)(t-s)} \int_{t}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)}\\ & \quad\times\{\|D_{u }B(u_{M,\tau}^{\delta}(\cdot; s,\omega,p),\theta_{\tau}\omega) (D_{p}u_{M,\tau}^{\delta}(\cdot;s,\omega, p)-D_{p}u_{\tau}^{\delta}(\cdot; s,\omega,p))\|\\ & \quad + \|(D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega))D_{p}u_{\tau}^{\delta}(\cdot;s,\omega, p) \|\}d\tau\\ & \lt Le^{(\gamma-\eta )r}\frac{\lambda_N^\alpha}{\gamma-\eta-\lambda_N}\|D_{p}u_M^{\delta}(\cdot; s,\omega,p)-D_{p}u ^{\delta}(\cdot;s,\omega, p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})}\\ & \quad+\mathcal{\tilde{I}}^\delta_{M,P}(t), \end{align*}

where

\begin{align*} \mathcal{\tilde{I}}^\delta_{M,P}(t) &:= e^{(\gamma-\eta)(t-s)} \int_{t}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\|D_{p}u^\delta(\tau+\xi;s,\omega,p)\|_{L(PD(A^{\alpha}),D(A^\alpha))}\\ & \quad\times\|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)\|d\tau. \end{align*}

Next we can claim that for any $0 \lt \eta \le \varepsilon_0$,

(5.14)\begin{equation} \|\mathcal{\tilde{I}}^\delta_{M,P}(t)\|_{L(PD(A^\alpha),C_{\gamma-\eta,s})}=o(1), \end{equation}

as $M \rightarrow +\infty$. To this end, for any given $\epsilon \gt 0$, we first let $\tilde{T}_M ( \lt s)$ be a sufficiently negative constant such that

\begin{equation*}\frac{\lambda_{N}^{\alpha}\tilde{\zeta}(\frac{3\varepsilon_0}{2}) e^{\frac{\varepsilon_0}{2}(\tilde{T}_M -s)}}{1-\tilde{\zeta}(\frac{3\varepsilon_0}{2})} \lt \frac{\epsilon}{4}.\end{equation*}

Then we split $\mathcal{\tilde{I}}^\delta_{M,P}(t)$ into a sum of two parts as $\mathcal{\tilde{I}}^\delta_{M,P}(t)=\mathcal{\tilde{I}}^{\delta,1}_{M,P}(t)+\mathcal{\tilde{I}}^{\delta,2}_{M,P}(t)$, where

\begin{align*} \mathcal{\tilde{I}}^{\delta,1}_{M,P}(t):= \left\{ \begin{array}{ll} e^{(\gamma-\eta)(t-s)} \int_{t}^{\tilde{T}_M}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\\ \quad\|D_{p}u^\delta(\tau+\xi;s,\omega, p)\|_{L(PD(A^{\alpha}),D(A^\alpha))} \\ \times\|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)\|d\tau , & {t \lt \tilde{T}_M},\\ 0, & {t \ge \tilde{T}_M}, \end{array} \right. \end{align*}

\begin{align*} \mathcal{\tilde{I}}^{\delta,2}_{M,P}(t):= \left\{ \begin{array}{ll} e^{(\gamma-\eta)(t-s)} \int_{\tilde{T}_M}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\\ \quad\|D_{p}u^\delta(\tau+\xi;s,\omega, p)\|_{L(PD(A^{\alpha}),D(A^\alpha))} \\\times\|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)\|d\tau , & {t \lt \tilde{T}_M},\\ e^{(\gamma-\eta)(t-s)} \int_{t}^{s}\lambda_{N}^{\alpha}e^{-\lambda_{N}(t-\tau)} \sup_{\xi \in [-r,0]}\\ \quad\|D_{p}u^\delta(\tau+\xi;s,\omega, p)\|_{L(PD(A^{\alpha}),D(A^\alpha))} \\ \times\|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)\|d\tau , & {t \ge \tilde{T}_M}. \end{array} \right. \end{align*}

For $\mathcal{\tilde{I}}^{\delta,1}_{M,P}(t)$, by using (4.6) and the fact that $\|D_{u}B(u,\omega)\|_{L(C_\alpha, H)}\leq L$, we find that for $t \lt \tilde{T}_M$,

\begin{align*} \mathcal{\tilde{I}}^{\delta,1}_{M,P}(t) & \leq 2 L\lambda_{N}^{\alpha}\int_{t}^{\tilde{T}_M}e^{(\gamma-\frac{3}{2}\varepsilon_0-\lambda_{N})(t-\tau) } e^{( \frac{3}{2}\varepsilon_0-\eta)(t-s) } e^{(\gamma-\frac{3}{2}\varepsilon_0) r}\\ & \quad\times\|D_{p}u ^{\delta}(\cdot;s,\omega,p)\|_{L(PD(A^{\alpha}), C_{\gamma-\frac{3}{2}\varepsilon_0,s})}d\tau\\ & \le \frac{2 L\lambda_{N}^{2\alpha}e^{\frac{\varepsilon_0}{2}(t-s)}}{(1-\tilde{\zeta}(\frac{3\varepsilon_0}{2}))} e^{(\gamma-\frac{3}{2}\varepsilon_0) r} \int_{t}^{\tilde{T}_M} e^{(\gamma-\frac{3}{2}\varepsilon_0-\lambda_{N})(t-\tau) } d\tau\\ & \le \frac{2\lambda_{N}^{\alpha}\tilde{\zeta}(\frac{3\varepsilon_0}{2}) e^{\frac{\varepsilon_0}{2}(\tilde{T}_M -s)}}{1-\tilde{\zeta}(\frac{3\varepsilon_0}{2})} \lt \frac{\epsilon}{2}. \end{align*}

For $\mathcal{\tilde{I}}^{\delta,2}_{M,P}(t)$, considering such fixed $\tilde{T}_M$ and any $t \le s$, using (4.6) again, we have that

\begin{align*} \mathcal{\tilde{I}}^{\delta,2}_{M,P}(t) & \leq \int_{\tilde{T}_M}^{s} e^{(\gamma-\eta-\lambda_{N})(t-\tau)} e^{(\gamma-\eta) r}\frac{\lambda_{N}^{2\alpha}}{1-\tilde{\zeta}(\eta) }\\ & \quad\times\|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)\|d\tau. \end{align*}

Together with (5.11), for any $\tilde{T}_M \le \tau \le s$, notice that

\begin{align*} & |u_{M,\tau}^{\delta}(\cdot;s,\omega, p)-u_{\tau}^{\delta}(\cdot;s,\omega, p)|_{C_\alpha}\\ & \quad\le e^{(\gamma-\eta)(s-\tilde{T}_M+r)}\|u^{\delta}_M(\cdot;s,\omega, p)-u^{\delta}(\cdot;s,\omega, p)\|_{C_{\gamma-\eta,s}} \rightarrow 0, \end{align*}

as $M \rightarrow +\infty$ and

\begin{align*} \|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega, p),\theta_{\tau}\omega)\| \le 2 L. \end{align*}

By the Dominated Convergence Theorem, for the given $\epsilon$ as above, there exists an $M_P \gt 0$ such that for any $ M \ge M_P$,

\begin{align*} \sup_{t\le s}\{\mathcal{\tilde{I}}^{\delta,2}_{M,P}(t)\} \lt \frac{\epsilon}{2}. \end{align*}

Then claim (5.14) is true.

For $\mathcal{I}^\delta_{M,Q}(t)$, using the similar methods as for $\mathcal{I}^\delta_{M,P}(t)$, we get that for any $\tau\leq t$ ( $\leq s$),

\begin{align*} \mathcal{I}^\delta_{M,Q}(t) & \le e^{(\gamma-\eta)(t-s)}\int_{-\infty}^{t}\left[\left(\frac{\alpha}{t-\tau}\right)^{\alpha}+\lambda_{N+1}^{\alpha}\right]e^{-\lambda_{N+1}(t-\tau)}\\ & \quad\times\{\|D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega) (D_{p}u_{M,\tau}^{\delta}(\cdot;s,\omega,p)-D_{p}u_{\tau}^{\delta}(\cdot;s,\omega,p)\|\\ & \quad + \|(D_{u }B(u_{M,\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega)-D_{u }B(u_{\tau}^{\delta}(\cdot;s,\omega,p),\theta_{\tau}\omega)) D_{p}u_{\tau}^{\delta}(\cdot;s,\omega,p) \|\}d\tau\\ & \lt Le^{(\gamma-\eta )r}\frac{\lambda_{N+1}^{\alpha}+k(\lambda_{N+1}-(\gamma-\eta))^{\alpha}}{\lambda_{N+1}-(\gamma-\eta) }\\ &\quad\times \|D_{p}u_M^{\delta}(\cdot; s,\omega,p)-D_{p}u ^{\delta}(\cdot;s,\omega, p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})} + o(1), \end{align*}

as $M \rightarrow +\infty$.

For $\hat{\mathcal{I}}^\delta_{M,Q}(t)$, in view of (2.2), (4.6), (4.8) and (5.4), for any $\tau\leq t$ ( $\leq s$), we observe that

\begin{align*} \hat{\mathcal{I}}^\delta_{M,Q}(t) & \le \lambda_{M+1}^{-\epsilon_\alpha}L e^{(\gamma-\eta)(t-s)}\int_{-\infty}^{t}\left[\left(\frac{\alpha+\epsilon_\alpha}{t-\tau}\right)^{\alpha+\epsilon_\alpha}+\lambda_{N+1}^{\alpha+\epsilon_\alpha}\right] e^{-\lambda_{N+1}(t-\tau)}\\ & \quad\times e^{(\gamma-\eta)(s-\tau+r)}\|D_p u^{\delta}(\cdot;s,\omega,p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})}d\tau\\ & \le \lambda_{M+1}^{-\epsilon_\alpha}Le^{(\gamma-\eta)r}\frac{\lambda_N^\alpha}{1-\tilde{\zeta}(\eta)}\times \left\{\frac{\lambda_{N+1}^{\alpha+\epsilon_\alpha}}{\lambda_{N+1}-(\gamma-\eta)}\right.\\ & \quad\left.+(\alpha+\epsilon_\alpha)^{\alpha+\epsilon_\alpha}\Gamma(1-\alpha-\epsilon_\alpha) (\lambda_{N+1}-(\gamma-\eta))^{\alpha+\epsilon_\alpha-1}\right\}=o(1), \end{align*}

as $M \rightarrow +\infty$. Therefore, combining the above estimates of $\mathcal{I}^\delta_{M,P}(t)$, $\mathcal{I}^\delta_{M,Q}(t)$ and $\hat{\mathcal{I}}^\delta_{M,Q}(t)$, and applying (4.15), we get that

\begin{align*} \|D_{p}u_M^{\delta}(\cdot; s,\omega,p)-D_{p}u ^{\delta}(\cdot;s,\omega, p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})} \lt \frac{o(1)}{1-\tilde{\zeta}(\eta)}, \end{align*}

as $M \rightarrow +\infty$. Then for any $\xi \in [-r,0]$, in consideration of the fact that

\begin{align*} &\| D_{p}\Psi^{\delta}_{s,M}(\omega,p,\xi)- D_{p}\Psi^{\delta}_{s}(\omega,p,\xi) \|_{L(PD(A^\alpha),D(A^\alpha))}\\ & \quad\le e^{(\gamma-\eta)r} \|D_{p}u_M^{\delta}(\cdot; s,\omega,p)-D_{p}u ^{\delta}(\cdot;s,\omega, p)\|_{L(PD(A^{\alpha}), C_{\gamma-\eta,s})}, \end{align*}

which yields that (5.8) is established.

Moreover, we obtain that

\begin{align*} & \| D_{p}\Psi^{\delta}_{s,M}(\omega,p,\cdot)- D_{p}\Psi^{0}_{s}(\omega,p,\cdot) \|_{L(PD(A^\alpha),C_\alpha)}\\ & \quad\leq \| D_{p}\Psi^{\delta}_{s,M}(\omega,p,\cdot)- D_{p}\Psi^{\delta}_{s}(\omega,p,\cdot) \|_{L(PD(A^\alpha),C_\alpha)}\\ & \qquad+\| D_{p}\Psi^{\delta}_{s}(\omega,p,\cdot)- D_{p}\Psi^{0}_{s}(\omega,p,\cdot) \|_{L(PD(A^\alpha),C_\alpha)}. \end{align*}

Considering the fact that $DB(\cdot,\omega)$ is uniformly Hölder continuous and (5.11), we basically repeat the proof of (5.8) and find that for all sufficiently small $|\delta|$,

\begin{align*} \| D_{p}\Psi^{\delta}_{s,M}(\omega,p,\cdot)- D_{p}\Psi^{\delta}_{s}(\omega,p,\cdot) \|_{L(PD(A^\alpha),C_\alpha)}\rightarrow 0 \end{align*}

as $M\rightarrow +\infty$ and $\delta\rightarrow 0$. Along with (4.32), (5.9) is obtained and the proof is complete.