all $\varepsilon>0$, there exists $\mu=\mu(\varepsilon)>0$ such that, if $\varphi\in\Omega$ and $||\varphi-\overline{\varphi}||<\mu$, then, for all $(t,m)\in[\overline{\tau},+\infty)\times[0,1]$, $|N^{\varphi}(t,m)-\overline{N}(t,m)|<\varepsilon$.\
This solution $\overline{N}$ is said to be [*locally exponentially stable on the set $\Omega$*]{} if there exists $\varepsilon>0$ such that, if $\varphi\in\Omega$ and $||\varphi -\overline{\varphi}
||<\varepsilon$, then there exist $c>0$ and $d>0$ such that, for all $(t,m)\in[\overline{\tau},+\infty)\times[0,1]$, $$\label{caracexpstab}
|N^{\varphi}(t,m)-\overline{N}(t,m)|\leq ce^{-d(t-\overline{\tau})}.$$ We will say [*locally stable*]{} (respectively [*locally exponentially stable*]{}) if $\Omega=C([0,\overline{\tau}]\times[0,1])$.

In this section, we make the assumption that the nonlinear function $x\mapsto x\beta(m,x)$, defined for $x\in\mathbb{R}$, is Lipschitz continuous in a neighborhood of zero, for all $m\in[0,1]$, that is, there exist $L\geq 0$ and $\varepsilon >0$ such that, $$\label{hypbeta}
|x\beta(m,x)-y\beta(m,y)| \leq L|x-y|,$$ for all $|x|\leq\varepsilon$, $|y|\leq\varepsilon$ and $m\in[0,1]$. For example, if $\beta$ is defined by (\[betalipschitz\]), then the mapping $x\in\mathbb{R}\mapsto x\beta(m,x)$ is Lipschitz continuous on $\mathbb{R}$ and a Lipschitz constant is given by $L=\sup_{m\in[0,1]}(a(m)/b(m))$.

In the sequel, $I$ and $\tilde{\zeta}$ denote the quantities $$I:=\inf_{m\in[0,1]} (\delta(m)+V'(m)) \quad \textrm{ and } \quad
\tilde{\zeta}:=\int_{\underline{\tau}}^{\overline{\tau}}  \hat{\zeta}(a) da,$$ where $\hat{\zeta}(a):=\sup_{m\in[0,1]}|\zeta(m,a)|$ and $\zeta(m,a)$ is given in the previous section. Then we can prove the following invariance result, which is a first step to prove our local stability result.

\[localstability\] Suppose that $L(2\tilde{\zeta}+1) < I$. If $\varphi\in C([0,\overline{\tau}]\times[0,1])$ satisfies $||\varphi||\leq\varepsilon$, 