consider the following system on $\Bbb{R}^n$
$\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $
assume that f(0,t)=g(0,t) = 0 and
1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$
2.$|g(x,t)| \leq \mu|x| $ $\forall x \in \Bbb{R}^n$
show that 0 is is an exponentiolly stable equilibrium of $(*)$
for small enough $\mu$
all I know for now is that x=0 is critical point of g(x,t), so if I define $F(x,t) = f(x,t) +g(x,t)$ and I prove that $DF_0 (=Df_0 + Dg_0 )$ have only negative eigenvalues, then 0 is exponentiolly stable of the system but how can I show that for small enough $\mu$ .
thanks ahead