Show that there is a unique continuous function

Question

I have no idea where to even start, i have never dealt with question like this before, any direction you can give me would be greatly appreciated.

Answer 1

Hint: Consider the operator

$$T \colon C([0,1]) \to C([0,1]);\quad T(f) \colon x \mapsto x^2 + \int_0^1 t^2f(tx)\,dt.$$

Show that this operator satisfies the hypotheses of Banach's fixed point theorem.