Let kn : [a, b]x[a, b]-------->R be defined inductively by

k1 = k; kn(x, y)= $\displaystyle \int_{y} ^{x} K_{n-1}(t,y)K(x,t)dt$

(n=2,3,4,....)

Show by induction that $\displaystyle T^n f(x)$= $\displaystyle \int_{a} ^{x} f(y)K_{n}(x,y)dy$

(n=2,3,4,....)

where T1 = T; T2 = ToT; : : : ; T^n+1 = ToTn are the iterates of T.

Original question is here http://www.maths.qmul.ac.uk/~cchu/MTH6122/lode095.pdf

I have no idea how to solve this so any help would be greatly appeciated.

Thank you.