# fixed point iterations

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!

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 $$\{u\in X:\Vert u\Vert_X<1\}$$.