I need to show whether $f:X \rightarrow X$ is a contraction mapping where $$f(x,y)=(-y,x^3), X = \{(x,y) \in \mathbb R^2 : x+y\geq 1 \}$$
I know we have to show that $d(f(x),f(y)) \leq c.d(x,y)$ for some $0 \leq c < 1$ but this is my attempt at it: $$d((-y_1,x_1^3),(-y_2,x_2^3)) = \sqrt{(-y_1+y_2)^2+(x_1^3-x_2^3)^2}=\sqrt{(y_1-y_2)^2+[(x_1-x_2)(x_1^2+x_1x_2+x_2^2)]^2} \leq d((x_1,y_1),(x_2,y_2)).(x_1^2+x_1x_2+x_2^2)$$
which implies that $f$ is a contraction mapping if and only if $0 \leq x_1^2+x_1x_2+x_2^2 < 1$. So then would this mean that $f$ isn't actually a contraction since the above is not true $\forall (x,y) \in X$. Thank you in advance for the help.
As said in my comment $f$ is not a well defined (So it is not even a function). And a contraction is a function F defined from a metric space $(M,d)$ to itself with a constant $0 \leq c <1$ such that $$d(F(x),F(y)) \leq cd(x,y)$$ for all $x,y \in M$. So if it is not a function there is no point in trying to show it is a contraction. Take a look at these links if you want more information about what it means to be well defined. What are well-defined functions? "Well defined" function - What does it mean?