I have a following question on the fixed point argument.
Assume that we have some space $X$ with a norm $\|\cdot\|_X$. Suppose also that there is an equation of type $u = Au$, were $A$ is an operator. We also know that $$\|Au\|_X \leq \|u\|_X^{1+a}, \quad a>0$$
Is it possible to show that there exists an unique solution to $u=Au$ in the space $\|u\|_X \lesssim \delta$ for sufficiently small $\delta$? If so, what type of fixed point theorem one can use?
I tried the Banach fixed point theorem, and it could work if $A$ is some polynomial function of $u$, for example. But what if $A$ is something more complicated? Is there anything other than Banach fixed point theorem?
Thanks!