TITLE: ﬁnding a ﬁxed-point ˆ x∈R for the function
QUESTION [0 upvotes]: I don't know how to verify this function if I can't use the Banach-Caccioppoli Contraction Principle. Please help

REPLY [0 votes]: A fixed point for $g(x)$ should be a point $\hat{x}$ with the property that $g(\hat{x})=x$. I.e., we want to solve $$1+x-\frac{1}{8}x^3=x$$ for $x$. We rewrite this to $$1-\frac{1}{8}x^3=0,$$ and finally to $$x^3=-8.$$ It is clear that this last equation has $-2$ as its only solution.
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