Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Question

Besides the conditions in the title, we have:

1. $0$ is an exponential equilibrium of the system $y'=f(y)$
2. $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$
3. $\mu$ is sufficiently small!

What I have tried so far is that:

1. $||y(0)||<\delta_1\Rightarrow ||y(t)||<\alpha_1||y(0)||e^{-\beta_1 t},t\geq0$
2. We want to prove: $\exists \delta,\alpha,\beta>0,$ such that $||x(0)||<\delta\Rightarrow ||x(t)||<\alpha||x(0)||e^{-\beta t},t\geq0$

Considering the following system:

1. $x_1'(t)=f(x_1)$
2. $x_2'(t)=g(x_2)$

Then we will have:

1. $\exists \alpha_1 ,\beta_1, \delta>0$ such that $||x_1(0)||<\delta_1\Rightarrow ||x_1(t)||<\alpha_1||x_1(0)||e^{-\beta_1 t},t\geq0$
2. $x_2'(t)\leq \mu|x_2(t)|$, and by Gronwall inequality $x_2(t)\leq x_2(0)e^{\mu t}$

I have a sense that it might be somewhat close to what we want to get. And I am stuck at this stage.

I would really appreciate any suggestion to proceed.

Answer 1

If $f(x) = -\frac12\mu x$, then the equation $y' = f(y)$ has $0$ as an exponentially stable fixed point. Now consider $g(x) = \mu x$. Then $x' = f(x) + g(x) = \frac12 \mu x$ has $0$ as an unstable fixed point. What am I missing?