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Perturbed Second-Order Stochastic Evolution Equations

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Abstract

In this paper, we discuss a class of perturbed second-order stochastic evolution equations. A perturbed second-order stochastic evolution equation for the unperturbed one is proposed. We show the mild solutions of perturbed second-order stochastic evolution equation and the unperturbed one are close on finite time-interval and on interval whose length tends to infinity as small perturbations tend to zero. A class of stochastic partial differential equations with perturbations is proposed as an application to illustrate the theoretical results.

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Acknowledgements

The authors wish to thank the editor and anonymous referees for their valuable comments, correcting errors and improving written language.

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Correspondence to Yong Ren.

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Appendix.

Appendix.

Proof of Theorem 3

Setting

$$\begin{aligned} Z^{\varepsilon }(t)=x^{\varepsilon }(t)-x(t),~\varDelta ^{\varepsilon }(t)=E\mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s)|^{2m},~t\in [0,T]. \end{aligned}$$

By Lemma 1, for \(p=2m\) and \(~\theta \in (0,1),\) we obtain

$$\begin{aligned} \begin{aligned} \varDelta ^{\varepsilon }(t)\le&~\frac{1}{(1-\theta )^{4m-2}}E \left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s)-{\widetilde{G}} (x^{\varepsilon }_{s},\varepsilon )+G(x_{s})|^{2m}\right) \\&+\frac{1}{[\theta (1-\theta )]^{2m-1}}E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|\alpha (x^{\varepsilon }_{s},\varepsilon )|^{2m}\right) +\frac{1}{\theta ^{2m-1}}E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|G(x_{s}^{\varepsilon })-G(x_{s})|^{2m}\right) . \end{aligned} \end{aligned}$$

By using the Assumption 2, we get

$$\begin{aligned} \begin{aligned} \varDelta ^{\varepsilon }(t)\le ~&\frac{1}{(1-\theta )^{4m-2}}E \left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s) -{\widetilde{G}}(x^{\varepsilon }_{s},\varepsilon )+G(x_{s})|^{2m}\right) \\&+\frac{{\overline{\alpha }}^{2m}(\varepsilon )}{[\theta (1-\theta )]^{2m-1}} +\frac{M_{G}^{2m}}{\theta ^{2m-1}}\varDelta ^{\varepsilon }(t). \end{aligned} \end{aligned}$$

For \(1-\frac{M_{G}^{2m}}{\theta ^{2m-1}}>0\), we have

$$\begin{aligned} \varDelta ^{\varepsilon }(t)\le \frac{\frac{{\overline{\alpha }}^{2m} (\varepsilon )}{[\theta (1-\theta )]^{2m-1}} +\frac{1}{(1-\theta )^{4m-2}}E\mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s)-{\widetilde{G}}(x^{\varepsilon }_{s}, \varepsilon )+G(x_{s})|^{2m}}{1-\frac{M_{G}^{2m}}{\theta ^{2m-1}}}. \end{aligned}$$
(22)

(1) Let \(m>1.\) By Itô’s formula, we get

$$\begin{aligned} |Z^{\varepsilon }(t)-{\widetilde{G}}(x^{\varepsilon }_{t},\varepsilon ) +G(x_{t})|^{m}\le ~&|\xi ^{\varepsilon }(0)-\xi (0)-G(x^{\varepsilon }_{0}) +G(x_{0})-\alpha (x^{\varepsilon }_{0},\varepsilon ))|^{m} \nonumber \\&+mI_{1}(t)+mI_{2}(t)+\frac{m(m-1)}{2}I_{3}(t), \end{aligned}$$
(23)

where

$$\begin{aligned} I_{1}(t)&=\int _{0}^{t}|Z^{\varepsilon }(s)-{\widetilde{G}}(x^{\varepsilon }_{s}, \varepsilon )+G(x_{s})|^{m-1}|S(t-s)||{\widetilde{f}}(s,x^{\varepsilon }(s), \mu ^{\varepsilon }(s),\varepsilon )\\&\quad -f(s,x(s),\mu (s))|{\mathrm{d}}s,\\ I_{2}(t)&=\int _{0}^{t}|Z^{\varepsilon }(s)-{\widetilde{G}}(x^{\varepsilon }_{s}, \varepsilon )+G(x_{s})|^{m-2}(Z^{\varepsilon }(s)-{\widetilde{G}}(x^{\varepsilon }_{s}, \varepsilon )+G(x_{t}))^{T}\\&\quad \times |S(t-s)||{\widetilde{g}}(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )-g(s,x(s),\mu (s))|{\mathrm{d}}w(s),\\ I_{3}(t)&=\int _{0}^{t}|Z^{\varepsilon }(s)-{\widetilde{G}}(x^{\varepsilon }_{s}, \varepsilon )+G(x_{s})|^{m-2}|S(t-s)|^{2}|{\widetilde{g}}(s,x^{\varepsilon }(s), \mu ^{\varepsilon }(s),\varepsilon )\\&\quad -g(s,x(s),\mu (s))|_{BL}^{2}{\mathrm{d}}s. \end{aligned}$$

Since \(|\xi ^{\varepsilon }(0)-\xi (0)-G(x^{\varepsilon }_{0})+G(x_{0}) -\alpha (x^{\varepsilon }_{0},\varepsilon )|^{2m}\le 3^{2m-1}[\delta _{0} (\varepsilon )+{\overline{\alpha }}^{2m}(\varepsilon )],\) we have

$$\begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s)-{\widetilde{G}}(x^{\varepsilon }_{s}, \varepsilon )+G(x_{s})|^{2m}\right) \\&\quad = E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s) -{\widetilde{G}}(x^{\varepsilon }_{s},\varepsilon )+G(x_{s})|^{m})^{2}\right) \\&\quad \le 4\Bigg [3^{2m-1}[\delta _{0}(\varepsilon )+{\overline{\alpha }}^{2m} (\varepsilon )]+m^{2}E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{1}(s)|^{2}\right) \\&\qquad +m^{2}E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{2}(s)|^{2}\right) +\frac{[m(m-1)]^{2}}{4}E \left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{3}(s)|^{2}\right) \Bigg ]. \end{aligned}$$

It follows from Hölder inequality and (4) that

$$\begin{aligned}&~E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{1}(s)|^{2}\right) \nonumber \\&\quad \le 2\cdot 3^{2m-3}tM_{S}^{2} E\int _{0}^{t}\left[ |f(s,x^{\varepsilon }(s), \mu ^{\varepsilon }(s))-f(s,x(s),\mu (s))|^{2}+|\beta (s,x^{\varepsilon }(s), \mu ^{\varepsilon }(s),\varepsilon )|^{2}\right] \nonumber \\&\qquad \times \left[ |z^{\varepsilon }(s)|^{2m-2}+|G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-2} +|\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-2}\right] {\mathrm{d}}s. \end{aligned}$$
(24)

By expanding the right side of the inequality and using the Assumption 2 and (4), we obtain

$$\begin{aligned}&~E\int _{0}^{t}|f(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-f(s,x(s),\mu (s))|^{2} \cdot |z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\\&\quad \le E\int _{0}^{t}M_{f}^{2}(|x^{\varepsilon }(s)-x(s)|+\rho (\mu ^{\varepsilon }(s), \mu (s)))^{2}|z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\\&\quad \le 2M_{f}^{2}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+2M_{f}^{2}\rho ^{2} (\mu ^{\varepsilon }(s),\mu (s))\int _{0}^{t}(\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}} {\mathrm{d}}s,\\&E\int _{0}^{t}|f(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-f(s,x(s),\mu (s))|^{2} \cdot |G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-2}{\mathrm{d}}s\\&\quad \le E\int _{0}^{t}M_{f}^{2}(|x^{\varepsilon }(s)-x(s)|+\rho (\mu ^{\varepsilon }(s), \mu (s)))^{2}M_{G}^{2m-2}|z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\\&\quad \le 2M_{f}^{2}M_{G}^{2m-2}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+2M_{f}^{2}M_{G}^{2m-2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&E\int _{0}^{t}|f(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-f(s,x(s), \mu (s))|^{2}\cdot |\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-2}{\mathrm{d}}s\\&\quad \le 2M_{f}^{2}{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t}(\varDelta ^{\varepsilon } (s))^{\frac{1}{m}}{\mathrm{d}}s+2tM_{f}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s)) {\overline{\alpha }}^{2m-2}(\varepsilon ).\\ \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \begin{aligned}&E\int _{0}^{t}|\beta (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{2}\cdot |z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\le \int _{0}^{t}{\overline{\beta }}^{2}(s,\varepsilon ) \cdot (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s,\\&E\int _{0}^{t}|\beta (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{2}\cdot |G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-2}{\mathrm{d}}s\\&\quad \le ~ M_{G}^{2m-2}\int _{0}^{t}{\overline{\beta }}^{2}(s,\varepsilon ) \cdot (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s\\ \end{aligned} \end{aligned}$$

and

$$\begin{aligned}&E\int _{0}^{t}|\beta (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{2}\cdot |\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-2}{\mathrm{d}}s\le {\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t} {\overline{\beta }}^{2}(s,\varepsilon ){\mathrm{d}}s. \end{aligned}$$

Combining these inequalities and (24), we have

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{1}(s)|^{2}\right) \\&\quad \le 2\cdot 3^{2m-3}\cdot t\cdot M_{S}^{2}\bigg [2M_{f}^{2}(1+M_{G}^{2m-2})\int _{0}^{t}\varDelta ^{\varepsilon }(s) {\mathrm{d}}s\\&\qquad +2M_{f}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))(1+M_{G}^{2m-2})\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s\\&\qquad +(1+M_{G}^{2m-2})\int _{0}^{t}{\overline{\beta }}^{2}(s,\varepsilon ) (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s+2M_{f}^{2} {\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}}{\mathrm{d}}s\\&\qquad +{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t}{\overline{\beta }}^{2} (s,\varepsilon ){\mathrm{d}}s+2tM_{f}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-2} (\varepsilon )\bigg ]. \end{aligned} \end{aligned}$$

Take \(~|a|^{r_{2}}\le |a|^{r_{1}}+|a|\), such that

$$\begin{aligned} ~a=\varDelta ^{\varepsilon }(s),~r_{1}=\frac{1}{m},~r_{2}=\frac{m-1}{m}, ~(\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}\le (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}} +\varDelta ^{\varepsilon }(s).~ \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{1}(s)|^{2}\right) \\ \le ~&~2\cdot 3^{2m-3}\cdot t\cdot M_{S}^{2}\Bigg [(1+M_{G}^{2m-2})\int _{0}^{t}[2M_{f}^{2}(1+\rho ^{2} (\mu ^{\varepsilon }(s),\mu (s)))+{\overline{\beta }}^{2}(s,\varepsilon )] \varDelta ^{\varepsilon }(s){\mathrm{d}}s\\&+\int _{0}^{t}\Big [2M_{f}^{2}((1+M_{G}^{2m-2})\cdot \rho ^{2}(\mu ^{\varepsilon }(s), \mu (s))+{\overline{\alpha }}^{2m-2}(\varepsilon ))\\&\quad +(1+M_{G}^{2m-2}) {\overline{\beta }}^{2}(s,\varepsilon )\Big ] (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}}{\mathrm{d}}s\\&+2tM_{f}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-2} (\varepsilon )+{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t} {\overline{\beta }}^{2}(s,\varepsilon ){\mathrm{d}}s\Bigg ]. \end{aligned} \end{aligned}$$

Using (4) and Lemma 3, we obtain

$$\begin{aligned} \begin{aligned}&~ E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{2}(s)|^{2}\right) \\ \le&~L_{g}E\int _{0}^{t}|S(t-s)|^{2}|{\widetilde{g}}(s,x^{\varepsilon }(s), \mu ^{\varepsilon }(s),\varepsilon )-g(s,x(s),\mu (s))|_{BL}^{2}\cdot |Z^{\varepsilon }(s)\\&\quad -{\widetilde{G}}(x^{\varepsilon }_{s},\varepsilon )+G(x_{t})|^{2m-2}{\mathrm{d}}s\\ \le&~2\cdot 3^{2m-3}\cdot L_{g}\cdot M_{S}^{2}\cdot E\int _{0}^{t}\left[ |g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s)) -g(s,x(s),\mu (s))|_{BL}^{2}\right. \\&\quad \left. +|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{2}\right] \\&\times \left[ |z^{\varepsilon }(s)|^{2m-2}+|G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-2} +|\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-2}\right] {\mathrm{d}}s. \end{aligned} \end{aligned}$$

We deduce from (b) and (4

$$\begin{aligned}&~ E\int _{0}^{t}|g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-g(s,x(s), \mu (s))|_{BL}^{2}\cdot |z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\\ \le&~ E\int _{0}^{t}M_{g}^{2}(|x^{\varepsilon }(s)-x(s)|+\rho (\mu ^{\varepsilon }(s), \mu (s)))^{2}|z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\\ \le&~ 2M_{g}^{2}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+2M_{g}^{2}\rho ^{2} (\mu ^{\varepsilon }(s),\mu (s))\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s,\\&~ E\int _{0}^{t}|g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-g(s,x(s), \mu (s))|_{BL}^{2}\cdot |G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-2}{\mathrm{d}}s\\ \le&~ E\int _{0}^{t}M_{g}^{2}(|x^{\varepsilon }(s)-x(s)|+\rho (\mu ^{\varepsilon }(s), \mu (s)))^{2}M_{G}^{2m-2}|z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\\ \le&~ 2M_{g}^{2}M_{G}^{2m-2}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+2M_{g}^{2}M_{G}^{2m-2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&E\int _{0}^{t}|g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-g(s,x(s), \mu (s))|_{BL}^{2}\cdot |\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-2}{\mathrm{d}}s\\ \le ~&2M_{g}^{2}{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}}{\mathrm{d}}s+2tM_{g}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-2} (\varepsilon ). \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \begin{aligned}&~ E\int _{0}^{t}|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s),\varepsilon )|^{2} \cdot |z^{\varepsilon }(s)|^{2m-2}{\mathrm{d}}s\le \int _{0}^{t}{\overline{\gamma }}^{2} (s,\varepsilon )\cdot (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s,\\&~ E\int _{0}^{t}|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s),\varepsilon )|^{2} \cdot |G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-2}{\mathrm{d}}s\\&\quad \le M_{G}^{2m-2}\int _{0}^{t}{\overline{\gamma }}^{2}(s,\varepsilon ) \cdot (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s \end{aligned} \end{aligned}$$

and

$$\begin{aligned}&E\int _{0}^{t}|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s),\varepsilon )|^{2} \cdot |\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-2}{\mathrm{d}}s\le {\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t} {\overline{\gamma }}^{2}(s,\varepsilon ){\mathrm{d}}s. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{2}(s)|^{2}\right) \\&\quad \le 2\cdot 3^{2m-3}\cdot L_{g}\cdot M_{S}^{2}\Bigg [2M_{g}^{2}((1+M_{G}^{2m-2})\int _{0}^{t}\varDelta ^{\varepsilon } (s){\mathrm{d}}s\\&+2M_{g}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))(1+M_{G}^{2m-2})\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s\\&+(1+M_{G}^{2m-2})\int _{0}^{t}{\overline{\gamma }}^{2}(s,\varepsilon ) (\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}}{\mathrm{d}}s+2M_{g}^{2}{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t}(\varDelta ^{\varepsilon }(s))^{\frac{1}{m}} {\mathrm{d}}s\\&+{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t}{\overline{\gamma }}^{2} (s,\varepsilon ){\mathrm{d}}s +2tM_{g}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-2} (\varepsilon )\Bigg ]. \end{aligned} \end{aligned}$$

By applying the inequality \((\varDelta ^{\varepsilon }(s))^{\frac{m-1}{m}} \le (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}}+\varDelta ^{\varepsilon }(s),\) we get

$$\begin{aligned} \begin{aligned}&~ E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{2}(s)|^{2}\right) \\&\quad \le 2\cdot 3^{2m-3}\cdot L_{g}\cdot M_{S}^{2}\Bigg [(1+M_{G}^{2m-2})\int _{0}^{t}\left[ 2M_{g}^{2}(1+\rho ^{2} (\mu ^{\varepsilon }(s),\mu (s)))+{\overline{\gamma }}^{2}(s,\varepsilon )\right] \varDelta ^{\varepsilon }(s){\mathrm{d}}s\\&\qquad +\int _{0}^{t}\Big [2M_{g}^{2}((1+M_{G}^{2m-2})\cdot \rho ^{2}(\mu ^{\varepsilon }(s), \mu (s))+{\overline{\alpha }}^{2m-2}(\varepsilon ))+(1+M_{G}^{2m-2}) {\overline{\gamma }}^{2}(s,\varepsilon )\Big ]\\&\qquad (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}} {\mathrm{d}}s\\&\qquad +2tM_{g}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-2} (\varepsilon )+{\overline{\alpha }}^{2m-2}(\varepsilon )\int _{0}^{t} {\overline{\gamma }}^{2}(s,\varepsilon ){\mathrm{d}}s\Bigg ]. \end{aligned} \end{aligned}$$

Using the Hölder inequality and inequality (4), we have

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{3}(s)|^{2}\right) \\&\quad \le t\cdot E\int _{0}^{t}|S(t-s)|^{4}|{\widetilde{g}}(s,x^{\varepsilon }(s), \mu ^{\varepsilon }(s),\varepsilon )-g(s,x(s),\mu (s))|_{BL}^{4}|Z^{\varepsilon }(s)\\&\qquad -{\widetilde{G}}(x^{\varepsilon }_{s},\varepsilon )+G(x_{t})|^{2m-4}{\mathrm{d}}s\\ \le ~&~8\cdot 3^{2m-5}\cdot t\cdot M_{S}^{4}\cdot E\int _{0}^{t}\left[ |g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s)) -g(s,x(s),\mu (s))|_{BL}^{4}\right. \\&\qquad \left. +|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{4}\right] \\&\qquad \times \left[ |z^{\varepsilon }(s)|^{2m-4}+|G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-4} +|\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-4}\right] {\mathrm{d}}s. \end{aligned} \end{aligned}$$

We have

$$\begin{aligned}&~ E\int _{0}^{t}|g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-g(s,x(s), \mu (s))|_{BL}^{4}\cdot |z^{\varepsilon }(s)|^{2m-4}{\mathrm{d}}s\\&\quad \le E\int _{0}^{t}M_{g}^{4}(|x^{\varepsilon }(s)-x(s)|+\rho (\mu ^{\varepsilon }(s), \mu (s)))^{4}|z^{\varepsilon }(s)|^{2m-4}{\mathrm{d}}s\\&\quad \le 8M_{g}^{4}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+8M_{g}^{4}\rho ^{4} (\mu ^{\varepsilon }(s),\mu (s))\int _{0}^{t}(\varDelta ^{\varepsilon }(s))^{\frac{m-2}{m}} {\mathrm{d}}s,\\&~ E\int _{0}^{t}|g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-g(s,x(s),\mu (s))|_{BL}^{4} \cdot |G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-4}{\mathrm{d}}s\\&\quad \le E\int _{0}^{t}M_{g}^{4}(|x^{\varepsilon }(s)-x(s)|+\rho (\mu ^{\varepsilon }(s), \mu (s)))^{4}M_{G}^{2m-4}|z^{\varepsilon }(s)|^{2m-4}{\mathrm{d}}s\\&\quad \le 8M_{g}^{4}M_{G}^{2m-4}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+8M_{g}^{4}M_{G}^{2m-4}\rho ^{4}(\mu ^{\varepsilon }(s),\mu (s))\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{m-2}{m}}{\mathrm{d}}s \end{aligned}$$

and

$$\begin{aligned}&E\int _{0}^{t}|g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s))-g(s,x(s), \mu (s))|_{BL}^{4}\cdot |\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-4}{\mathrm{d}}s\\ \le ~&8M_{g}^{4}{\overline{\alpha }}^{2m-4}(\varepsilon )\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{2}{m}}{\mathrm{d}}s+8tM_{g}^{4}\rho ^{4}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-4} (\varepsilon ). \end{aligned}$$

Similarly,

$$\begin{aligned}&E\int _{0}^{t}|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s),\varepsilon )|^{4} \cdot |z^{\varepsilon }(s)|^{2m-4}{\mathrm{d}}s\le \int _{0}^{t}{\overline{\gamma }}^{4} (s,\varepsilon )\cdot (\varDelta ^{\varepsilon }(s))^{\frac{m-2}{m}}{\mathrm{d}}s,\\&E\int _{0}^{t}|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s),\varepsilon )|^{4} \cdot |G(x_{s}^{\varepsilon })-G(x_{s})|^{2m-4}{\mathrm{d}}s\\&\le M_{G}^{2m-4}\int _{0}^{t}{\overline{\gamma }}^{4}(s,\varepsilon ) \cdot (\varDelta ^{\varepsilon }(s))^{\frac{m-2}{m}}{\mathrm{d}}s \end{aligned}$$

and

$$\begin{aligned}&E\int _{0}^{t}|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{4}\cdot |\alpha (x_{s}^{\varepsilon },\varepsilon )|^{2m-4}{\mathrm{d}}s\le {\overline{\alpha }}^{2m-4}(\varepsilon )\int _{0}^{t}{\overline{\gamma }}^{4} (s,\varepsilon ){\mathrm{d}}s. \end{aligned}$$

This shows that,

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{3}(s)|^{2}\right) \\&\quad \le 8\cdot 3^{2m-5}\cdot t\cdot M_{S}^{4}\Bigg [8M_{g}^{4}(1+M_{G}^{2m-4})\int _{0}^{t}\varDelta ^{\varepsilon }(s) {\mathrm{d}}s\\&\qquad +\int _{0}^{t}(1+M_{G}^{2m-4})\cdot (8M_{g}^{4}\rho ^{4}(\mu ^{\varepsilon }(s), \mu (s))+{\overline{\gamma }}^{4}(s,\varepsilon )) (\varDelta ^{\varepsilon }(s))^{\frac{m-2}{m}}{\mathrm{d}}s\\&\qquad +8M_{g}^{4}{\overline{\alpha }}^{2m-4}(\varepsilon )\int _{0}^{t} (\varDelta ^{\varepsilon }(s))^{\frac{2}{m}}{\mathrm{d}}s+8tM_{g}^{4}\rho ^{4}(\mu ^{\varepsilon }(s),\mu (s)) {\overline{\alpha }}^{2m-4}(\varepsilon )\\&\qquad +{\overline{\alpha }}^{2m-4}(\varepsilon )\int _{0}^{t} {\overline{\gamma }}^{4}(s,\varepsilon ){\mathrm{d}}s\Bigg ]. \end{aligned} \end{aligned}$$

Since \((\varDelta ^{\varepsilon }(s))^{\frac{2}{m}} \le (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}}+\varDelta ^{\varepsilon }(s)\)  and  \((\varDelta ^{\varepsilon }(s))^{\frac{m-2}{m}} \le (\varDelta ^{\varepsilon }(s))^{\frac{1}{m}}+\varDelta ^{\varepsilon }(s),\)  we get

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{3}(s)|^{2}\right) \\&\quad \le 8\cdot 3^{2m-5}\cdot t\cdot M_{S}^{4}\Bigg [\int _{0}^{t}\Big [(1+M_{G}^{2m-4})(8M_{g}^{4} (1+\rho ^{4}(\mu ^{\varepsilon }(s),\mu (s)))+{\overline{\gamma }}^{4} (s,\varepsilon ))\\&\qquad +8M_{g}^{4}{\overline{\alpha }}^{2m-4}(\varepsilon )\Big ] \varDelta ^{\varepsilon }(s){\mathrm{d}}s\\&\qquad +\int _{0}^{t}\left[ (1+M_{G}^{2m-4})\cdot (8M_{g}^{4}\rho ^{4} (\mu ^{\varepsilon }(s),\mu (s))+{\overline{\gamma }}^{4}(s,\varepsilon )) +8M_{g}^{4}{\overline{\alpha }}^{2m-4}(\varepsilon )\right] (\varDelta ^{\varepsilon } (s))^{\frac{1}{m}}{\mathrm{d}}s\\&\qquad +8tM_{g}^{4}\rho ^{4}(\mu ^{\varepsilon }(s),\mu (s)){\overline{\alpha }}^{2m-4} (\varepsilon )+{\overline{\alpha }}^{2m-4}(\varepsilon )\int _{0}^{t}{\overline{\gamma }}^{4}(s,\varepsilon ){\mathrm{d}}s\Bigg ]. \end{aligned} \end{aligned}$$

For \(m=2\), by (23), Hölder inequality and Lemma 3, we get

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|I_{3}(s)|^{2}\right) \\&\quad =E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}\left| \int _{0}^{t}|S(t-s)|^{2} \cdot |{\widetilde{g}}(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s),\varepsilon ) -g(s,x(s),\mu (s))|_{BL}^{2}{\mathrm{d}}s\right| ^{2}\right) \\&\quad \le 8tM_{S}^{4}E\int _{0}^{t}\left[ |g(s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s)) -g(s,x(s),\mu (s))|_{BL}^{4}+|\gamma (s,x^{\varepsilon }(s),\mu ^{\varepsilon }(s), \varepsilon )|^{4}\right] {\mathrm{d}}s\\&\quad \le 8tM_{S}^{4}E\left[ \int _{0}^{t}(M_{g}(|x^{\varepsilon }(s)-x(s)| +\rho (\mu ^{\varepsilon }(s),\mu (s)))^{4}{\mathrm{d}}s+\int _{0}^{t} {\overline{\gamma }}^{4}(s,\varepsilon ){\mathrm{d}}s\right] \\&\quad \le 64tM_{S}^{4}M_{g}^{4}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s+64M_{S}^{4}M_{g}^{4}t^{2}\rho ^{4}(\mu ^{\varepsilon }(s),\mu (s)) +8tM_{S}^{4}\int _{0}^{t}{\overline{\gamma }}^{4}(s,\varepsilon ){\mathrm{d}}s. \end{aligned} \end{aligned}$$

Substituting the above inequalities into (22) and (23), we get (9) and (10).

(2) For \(m=1\), it follows from \({\mathrm{It{\hat{o}}}}\) integral and (4) that

$$\begin{aligned} \begin{aligned}&E\left( \mathop {\mathrm{sup}}\limits _{s\in [0,t]}|Z^{\varepsilon }(s) -{\widetilde{G}}(x^{\varepsilon }_{s},\varepsilon )+G(x_{s})|^{2}\right) \\&\quad \le 3\Bigg [3((1+M_{G}^{2})\delta _{0}(\varepsilon )+{\overline{\alpha }}^{2} (\varepsilon ))+4tM_{S}^{2}M_{f}^{2}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s +4t^{2}M_{S}^{2}M_{f}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))\\&\qquad +2tM_{S}^{2}\int _{0}^{t}{\overline{\beta }}^{2}(s,\varepsilon ){\mathrm{d}}s+4M_{S}^{2}M_{g}^{2}L_{g}\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s +4tM_{S}^{2}M_{g}^{2}L_{g}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))\\&\qquad +2M_{S}^{2}L_{g}\int _{0}^{t}{\overline{\gamma }}^{2}(s,\varepsilon ){\mathrm{d}}s\Bigg ]\\&\quad = 3\Bigg [3((1+M_{G}^{2})\delta _{0}(\varepsilon )+{\overline{\alpha }}^{2}(\varepsilon )) +4tM_{S}^{2}\rho ^{2}(\mu ^{\varepsilon }(s),\mu (s))(tM_{f}^{2}+L_{g}M_{g}^{2})\\&\qquad +2M_{S}^{2}\int _{0}^{t}(t{\overline{\beta }}^{2}(s,\varepsilon ) +L_{g}{\overline{\gamma }}^{2}(s,\varepsilon )){\mathrm{d}}s +4M_{S}^{2}(tM_{f}^{2}+L_{g}M_{g}^{2})\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s\Bigg ]. \end{aligned} \end{aligned}$$

Putting this inequality in (22), for \(M_{G}^{2}<\theta <1\), we have

$$\begin{aligned} \varDelta ^{\varepsilon }(s)\le a(t,\varepsilon )+b(t)\int _{0}^{t}\varDelta ^{\varepsilon }(s){\mathrm{d}}s. \end{aligned}$$

By Lemma 2, (11)  follows, which completes the proof of Theorem 3. \(\square \)

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Cheng, L., Ren, Y. Perturbed Second-Order Stochastic Evolution Equations. Qual. Theory Dyn. Syst. 20, 37 (2021). https://doi.org/10.1007/s12346-021-00475-9

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