Let $F$ be a bounded linear operator between the Hilbert spaces $H_1$ and $H_2$. Let $F$ satisfies $$\|z- a\|\leq \|F(z)-F(a)\|^t, \ t>0, \ z \in H_1.$$ Please answer the following two questions:
(1) How to show that $F$ is continuously invertible, provided it is?
(2) Is $F$ is continuously invertible even if $F$ is nonlinear.
Please help.
When $F$ is linear put $z=z_1-z_2+a$ to get $\|z_1-z_2\| \leq \|F(z_1)-F(z_2)\|^{t}$ for all $z_1$ and $z_2$. Now you can use the answer for your previous question: Continuously invertible operator.
For a counter-example when $F$ is not linear take $t=1$, $H_1=H_2=\mathbb R$, $a=0$, $F(z)=z$ for $z$ rational and $F(z)=-z$ for $z$ irrational.