& \varepsilon \big( (1-\alpha) + \alpha \big).\end{aligned}$$ That is, finally, $$|N_{n+1}(t,m)| \leq  \varepsilon.$$ Then, (\[hyprecurrence\]) is true for every $n\in\mathbb{N}$.\
Remarking that the sequence $(N_{n})_{n\in\mathbb{N}}$ converges uniformly to $N^{\varphi}$ for $(t,m)\in[\overline{\tau},T]\times[0,1]$ (this can be proved using the method of steps), we conclude that $$|N^{\varphi}(t,m)|\leq \varepsilon.$$ for all $t\in[\overline{\tau},T]$ and $m\in[0,1]$. This result is true for every $T>\overline{\tau}$. So, it is true for $t\geq\overline{\tau}$ and the proof of the proposition is ended.

The result of invariance proved in Proposition \[localstability\] is fundamental to obtain the local exponential stability of the trivial solution of Problem (\[mildformulation\]). This result is presented in the following theorem.

\[localexpstability\] If we suppose that $L(2\tilde{\zeta}+1) < I$, then the trivial solution $N\equiv 0$ of Problem (\[mildformulation\]) is locally exponentially stable.

Let $\varepsilon>0$ be given by (\[hypbeta\]) and $\varphi\in C([0,\overline{\tau}]\times[0,1])$ such that $||\varphi||\leq\varepsilon$, and let $T>\overline{\tau}$ be given.\
We consider the sequence defined by (\[N0def\])-(\[Nndef\]). From the proof of Proposition \[localstability\], it follows that for all $n\in\mathbb{N}$ and $(t,m)\in[\overline{\tau},T]\times[0,1]$, $$|N_{n}(t,m)|\leq \varepsilon.$$ Since the mapping $$\rho\in[0,I]\mapsto \frac{I-\rho}{1+2\tilde{\zeta}e^{\rho\overline{\tau}}}$$ is continuous on $[0,I]$ and reaches its maximum for $\rho = 0$, there exists $\rho\in(0,I)$ such that $$L < \frac{I-\rho}{1+2\tilde{\zeta}e^{\rho\overline{\tau}}} < \frac{I}{2\tilde{\zeta}+1}.$$ Using the same arguments as in the proof of Proposition 