Let $L : B_1 \rightarrow B_1$ be an isomorphism of Banach spaces
(i.e., L is a bijective bounded linear operator) and
$A: B_1 \rightarrow B_2$ a non linear operator. Consider the following equation
$$ Lu_t + Au_t = \xi_t$$
where $u_t$ is unknown and $\xi_t$ is known.
Suppose we know that $u_0$ is a solution to the above equation, i.e.
$$ Lu_0 + Au_0 = \xi_0.$$
Furthermore, suppose that the linearization of the operator $A$ at
the point $u_0$ is the zero operator (said in a different way, the
linearization of $L+A$ at $u_0$ is the operator $L$). Is it correct to
say that if $\xi_t$ is sufficiently close to $\xi_0$, then there
exists a unique solution $u_t$, sufficiently close to $u_0$ to the
equation
$$ Lu_t + Au_t = \xi_t$$
?
If $B_1$ and $B_2$ were finite dimensional spaces, I think this is correct. Does one have to check some additional hypothesis in infinte dimensions?