I need to show f:[0,1]->[0,1] is a contraction mapping. where g is continous map form [0,1] to itself
The definition of "contraction mapping" is that |f(a)- f(b)|< |a- b|. A "strict contraction" would have |f(a)- f(b)|<= c|a- b| where c< 1. Can you show either of those? Use the fact that |sin(x)|<= 1 of course.
Yes I'm looking for the latter strict contraction as you say. ok so |F(x)-F(y)|=|sinx|||x^{0.5}-g(x)|-|y^{0.5}-g(y)| which is less than or equal to [x^{0.5}-g(x)|+|y^0.5-g(y)| by triangle inequality and |sinx|<=1. Hmm tex doesn't seem to be working. Hope you can read it