Banach Fixed point theorem for ode

Question

I want to prove the existence and unicity of solution of the differential equation $y'=-x^4y^3+\exp(-y^3)$ over $R=\{(x,y), 0\leq x\leq \frac12, |y|\leq 1\}$ using Banach fixed point. My question is how to choose the complete normed space ? I know that the operator is $T: E\to E$ defined by $Ty(x)=y_0+\int_{x_0}^x -s^4y^3(s)+\exp(-y^3(s)) ds$. Can I choose $E=C([0,\frac12],[-1,1]),||.||_{\infty})$?