Correction to: J Nonlinear Sci https://doi.org/10.1007/s00332-023-09921-1

There are some small but annoying errors in Ihara and Yagasaki (2023). The following corrections are necessary.

  • Line 12 from below on Page 13: “\(\Vert \bar{\textbf{u}}(T)-\bar{\textbf{u}}_n (T)-\varvec{\theta }\Vert \)” should be read as “\(\Vert \bar{\textbf{u}}(T)-\bar{\textbf{u}}_n (T)\Vert .\)”

  • Line 4 on Page 18: “some” should be inserted before “\(i\in [n]\).”

In Appendix A, the dependence of the graphons \(W_k\) on \(k\in [m]\) is dropped out by mistake. It makes several wrong expressions there. They should be read as follows:

  • For Eq. (A1):

    $$\begin{aligned} T=\left( 2(L_f+mL_D(C_2+\max _{k\in [m]}\Vert W_k\Vert _{L^2(I^2)}))\right) ^{-1}. \end{aligned}$$
    (A.1)

  • For the equation below (A.1):

    $$\begin{aligned} K(\textbf{u})(t) =g(\cdot )+\int _0^t\biggl (f(u(s,\cdot ),s)+\sum _{k=1}^m\int _I W_k(\cdot ,y)D_k(u(s,y)-u(s,\cdot ))dy\biggr )ds \end{aligned}$$

  • For the second and subsequent lines of the last equation on Page 36:

    $$\begin{aligned}&\le \max _{t\in [0,T]}\int _0^t\biggl \Vert f(u(s,\cdot ),s)-f(v(s,\cdot ),s)\\&\quad +\sum _{k=1}^m\int _I W_k(\cdot ,y)(D_k(u(s,y)-u(s,\cdot ))-D_k(v(s,y)-v(s,\cdot ))dy\biggr \Vert ds\\&\le T\max _{t\in [0,T]}\biggl \Vert L_f|u(t,\cdot )-v(t,\cdot )|\\&\quad +L_D\sum _{k=1}^m\int _I W_k(\cdot ,y)|u(t,y)-u(t,\cdot )-v(t,y)+v(t,\cdot )|dy\biggr \Vert \\&\le T\max _{t\in [0,T]}\biggl (L_f\Vert \textbf{u}(t)-\textbf{v}(t)\Vert +L_D\sum _{k=1}^m\biggl (\biggl \Vert \int _I W_k(\cdot ,y)|u(t,\cdot )-v(t,\cdot )|dy\biggr \Vert \\&\quad +\biggl \Vert \int _I W_k(\cdot ,y)|u(t,y)-v(t,y)|dy\biggr \Vert \biggr ). \end{aligned}$$

  • For the first equation on Page 37:

    $$\begin{aligned} \biggl \Vert \int _I W_k(\cdot ,y)|u(t,\cdot )-v(t,\cdot )|dy\biggr \Vert \le C_2\Vert \textbf{u}(t)-\textbf{v}(t)\Vert \end{aligned}$$

  • For the second equation on Page 37:

    $$\begin{aligned} \biggl \Vert \int _I W_k(\cdot ,y)|u(t,y)-v(t,y)|dy\biggr \Vert \le \Vert W_k\Vert _{L^2(I^2)}\,\Vert \textbf{u}(t)-\textbf{v}(t)\Vert , \end{aligned}$$

  • For the third equation on Page 37:

    $$\begin{aligned}&\Vert K(\textbf{u})-K(\textbf{v})\Vert \\&\quad \le T(L_f+mL_D(C_2+\max _{k\in [m]}\Vert W_k\Vert _{L^2(I^2)}))\Vert \textbf{u}(t)-\textbf{v}(t)\Vert =\frac{1}{2}\Vert \textbf{u}(t)-\textbf{v}(t)\Vert . \end{aligned}$$

These corrections do not affect the main conclusions of the paper.