Let be a closed subset of which contains the entire line segment between any two of its points and let be a continuously differentiable map from an open subset of containing into . Suppose that and that there is a real number such that
for all . Prove that the restriction of to is a contraction map, so that the fixed point theorem is applicable.
I have my idea, but I could not go through. I need to show that for any and
by the Mean Value Theorem for several variables, we have
where is the point on the line segment between and which yields
From here, I have no idea to proceed to make the desired inequality. There is a hint said that I could use the preceding problem in the book as follows:
If , its length is
but I could not relate how I can use it. The assumption that should be the hint that and the fact that is closed ensures the existence of cauchy sequences converging to a fixed point.
The Schwarz inequality seems to be the way to go, but I could not utilize it.
so that I could somehow use to relate that
Anyone has an idea on how to proceed ? Thanks.
EDITED: FORGET WHAT I WROTE ABOVE. I was careless. The Mean Value Theorem cannot be applied here. I wrongly wrote which is not correct. In face