X=C[0,1/2] , F:X---->X defined by
F(f)(x)=1+f(x)/3+integral of f(s)ds from 0 to x
how to prove F is a contraction?
how to find the unique fixed point?
To prove it's a contraction you proceed as follows:
now taking the supremum over we get that this last is
To find the fixed point you could follow th proof of the fixed point theorem and iterate given a simple initial function (say or some such) or you could try by inspection see if you can find it.
Edit: For simplicity you could assume is differentiable, derive the expression and obtain an easy diff. equation, solve, substitute and determine the constant.