let X=C[0,1/2] with standard sup-norm and F:X---->X defined by
F(f)(x)=1+integral of f(x-t)dt from 0 to x
how to prove F is a contraction with factor k=1/2?
and how to find the unique fixed point?
let X=C[0,1/2] with standard sup-norm and F:X---->X defined by
F(f)(x)=1+integral of f(x-t)dt from 0 to x
how to prove F is a contraction with factor k=1/2?
and how to find the unique fixed point?
What is ? a constant? if it is then which is clearly a contraction. The fixed point of this function however is not so easy to find (to me at least).
If however you meant then, arguing as in your last question, one gets it's indeed a contraction and using induction we prove that (where means composition -times)