Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

<p style='text-indent:20px;'>Consider the second order nonautonomous lattice systemswith singular perturbations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \epsilon \ddot{u}_{m}+\dot{u}_{m}+(Au)_{m}+\lambda_{m}u_{m}+f_{m}(u_{j}|j\in I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k},\; \epsilon>0 \tag{*} \label{0} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and first systems</p><p id="FE2"> \dot{u}_{m}+(Au)_{m}+\lambda _{m}u_{m}+f_{m}(u_{j}|j∈I_{mq}) \mathbb{Z}^{k}. \tag{**} \label{00} style='text-indent:20px;'>Under certain conditions, there are pullback attractors <inline-formula><tex-math id="M1">\begin{document}$ \{\mathcal{A}_{\epsilon }(t)\subset \ell ^{2}\times ^{2}\}_{t\in \mathbb{R}} $\end{document}</tex-math></inline-formula> and id="M2">\begin{document}$ \{\mathcal{A}(t)\subset for systems (*)and (**), respectively. In this paper, we mainly consider uppersemicontinuity of id="M3">\begin{document}$ \mathcal{A}_{\epsilon \ell^{2}\times ^{2} $\end{document}</tex-math></inline-formula>, id="M4">\begin{document}$ t\in \mathbb{R} with respect to coefficient id="M5">\begin{document}$ derivative term under Hausdorff semidistance. First, studythe relationship between id="M6">\begin{document}$ }(t) id="M7">\begin{document}$ \mathcal{A}(t) when id="M8">\begin{document}$ \rightarrow 0^{+} $\end{document}</tex-math></inline-formula>. We construct a family compact sets id="M9">\begin{document}$ \mathcal{A}_{0}(t)\subset id="M10">\begin{document}$ such that id="M11">\begin{document}$ is naturally embedded into id="M12">\begin{document}$ \mathcal{A}_{0}(t) as firstcomponent, prove id="M13">\begin{document}$ can enter anyneighborhood id="M14">\begin{document}$ id="M15">\begin{document}$ small enough. Thenfor id="M16">\begin{document}$ _{0}>0 id="M17">\begin{document}$ enterany neighborhood id="M18">\begin{document}$ _{0}}(t) id="M19">\begin{document}$ \epsilon\rightarrow _{0} Finally, existence andexponentially attraction singleton (*)-(**).</p>