fixed point iterations

Question

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!

Answer 1

First, note that $u=0$ is a fixed point of $A$ since we have $$\Vert A(0)\Vert_X\leq\Vert 0\Vert_X^{1+a}=0,$$ thus $A(0)=0$. Now, let $u\in X$ be some arbitrary fixed point. We can then show by induction that $$\Vert u\Vert_X\leq\Vert u\Vert_X^{(1+a)^n}\quad\text{for all }n\in\mathbb{N}$$ which implies $u=0$ if $\Vert u\Vert_X<1$. Hence, $u=0$ is unique solution to $Au=u$ in the set $\{u\in X:\Vert u\Vert_X<1\}$.