\[localstability\], we obtain $$|N_{0}(t,m)| \leq \varepsilon e^{-I(t-\overline{\tau})} \leq \varepsilon e^{-\rho(t-\overline{\tau})} \leq\varepsilon.$$ The estimate $$|N_{1}(t,m)-N_{0}(t,m)| \leq |G(N_{0})(t,m)|+|J(N_{0})(t,m)|,$$ with $$\begin{aligned}
|J(N_{0})(t,m)| & \leq & L\int_{\overline{\tau}}^{t}|N_{0}(s,\pi_{-(t-s)}(m))|e^{-I(t-s)} ds, \\
        & \leq & \varepsilon L\int_{\overline{\tau}}^{t}e^{-I(t-s)}e^{-\rho(s-\overline{\tau})}ds, \\
        & \leq & \varepsilon Le^{-It}e^{\rho\overline{\tau}}\int_{\overline{\tau}}^{t}e^{(I-\rho)s}ds,\end{aligned}$$ and $$\begin{aligned}
|G(N_{0})(t,m)| & \leq & 2L\int_{\overline{\tau}}^{t}\bigg(\int_{\underline{\tau}}^{\overline{\tau}} \hat{\zeta}(a) |N_{0}(s-a,\Delta(a,\pi_{-(t-s)}(m)))| da \bigg) e^{-I(t-s)}ds,\\
     & \leq & 2\varepsilon L\int_{\overline{\tau}}^{t} e^{-I(t-s)}\bigg(\int_{\underline{\tau}}^{\overline{\tau}} \hat{\zeta}(a) e^{-\rho(s-a-\overline{\tau})} da \bigg) ds, \\
        & \leq & 2\tilde{\zeta}\varepsilon L\int_{\overline{\tau}}^{t} e^{-I(t-s)}e^{-\rho(s-2\overline{\tau})}ds, \\
        & \leq & 2\tilde{\zeta}\varepsilon Le^{-It}e^{2\rho\overline{\tau}}\int_{\overline{\tau}}^{t}e^{(I-\rho)s}ds,\end{aligned}$$ yields to $$|N_{1}(t,m)-N_{0}(t,m)| \leq  \varepsilon Le^{-It}e^{\rho\overline{\tau}}\Big[
1+2\tilde{\zeta}e^{\rho\overline{\tau}} \Big]\int_{\overline{\tau}}^{t}e^{(I-\rho)s}ds.$$ Note that $$\begin{aligned}
e^{-It}e^{\rho\overline{\tau}}\int_{\overline{\tau}}^{t} e^{(I-\rho)s} ds & = & e^{-It}e^{\rho\overline{\tau}}\frac{e^{(I-\rho)t}-e^{(I-\rho)\overline{\tau}}}{I-\rho}, \\
    & = & e^{(\rho-I)t}e^{\rho(\overline{\tau}-t)}\frac{e^{(I-\rho)t}-e^{(I-\rho)\overline{\tau}}}{I-\rho}, \\
    & = & e^{-\rho(t-\overline{\tau})}\frac{1-e^{(I-\rho)(\overline{\tau}-t)}}{I-\rho}, \\
    & \leq & \frac{1}{I-\rho}e^{-\rho(t-\overline{\tau})}.\end{aligned}$$ Consequently $$|N_{1}(t,m)-N_{0}(t,m)| \leq  \varepsilon L\theta e^{-\rho(t-\overline{\tau})},$$ where $$\theta=\frac{1+2\tilde{\zeta}e^{\rho\overline{\tau}}}{I-\rho}.$$ We show by induction that, for all $n\geq 1$ and all $(t,m)\in[\overline{\tau},T]\times[0,1]$, $$\label{hyprecurrence2}
|N_{n}(t,m)-N_{n-1}(t,m)| \leq  \varepsilon (L\theta)^{n} e^{-\rho(t-\overline{\tau})}.$$ Suppose that (\[hyprecurrence2\]) is true for $n\geq1$ ; then, for all $(t,m)\in[\overline{\tau},T]\times[0,1]$, $$|N_{n+1}(t,m)-N_{n}(t,m)| \leq  |(G(N_{n})-G(N_{n-1}))(t,m)|+|(J(N_{n})-J(N_{n-1}))(t,m)|,$$ with $$\begin{aligned}
|(G(N_{n})-G(N_{n-1}))(t,m)| & \leq & 2L\int_{\overline{\tau}}^{t} \bigg(\int_{\underline{\tau}}^{\overline{\tau}}\hat{\zeta}(a) |N_{n}(s-a,\Delta(a,\pi_{-(t-s)}(m))) \\
            &   & -N_{n-1}(s-a,\Delta(a,\pi_{-(t-s)}(m)))| da \bigg) e^{-I(t-s)} ds, \\
            & \leq & 2\varepsilon L^{n+1}\theta^{n}\!\!\int_{\overline{\tau}}^{t}\!\! e^{-I(t-s)}\bigg(\int_{\underline{\tau}}^{\overline{\tau}}\!\! \hat{\zeta}(a) e^{-\rho(s-a-\overline{\tau})}da\bigg) ds, \\
            & \leq & 2\tilde{\zeta}\varepsilon L^{n+1}\theta^{n}\int_{\overline{\tau}}^{t} e^{-I(t-s)} e^{-\rho(s-2\overline{\tau})} ds, \\
            & \leq & 2\tilde{\zeta}\varepsilon L^{n+1}\theta^{n}e^{2\rho\overline{\tau}}e^{-It}\int_{\overline{\tau}}^{t} e^{(I-\rho)s}ds,\end{aligned}$$ and $$\begin{aligned}
|(J(N_{n})-J(N_{n-1}))(t,m)| & \leq & L\int_{\overline{\tau}}^{t}|N_{n}(s,\pi_{-(t-s)}(m)) \\
            &   & \ 