Consider the following statement of the fiber contraction principle. Theorem[ Fi
ID: 2987447 • Letter: C
Question
Consider the following statement of the fiber contraction principle.
Theorem[Fiber Contraction Principle] Let (X, dX) be a metric space and let (Y, dY) be a complete metric space. Consider the metric space X x Y equipped with the metric dXxY
= dX + dY.
Let S : X x Y -> X x Y given by
S(x, y) = (T(x), A(x, y)).
Assume:
1. T : X -> X has an attracting fixed point x0. (X knot)
2. For each y in Y is fixed, the map X->Y : x -> A(x, y) is continuous.
3. There exists lambda in [0, 1) such that dY(A(x, y1), A(x, y2)) ? lambda dY(y1, y2) for all x in X and for all y1, y2 in Y. (This last requirement means that S is a fiber contraction).
Under this assumptions, S : X x Y -> X x Y has an attracting fixed point.
Prove this theorem.
[Hint/Remark: The attracting fixed point for S will be given by (x0, y0) where y0 is the unique fixed point of the contraction y -> A(x0, Y) in the fiber {x0} x Y over x0.
Notice that in this form of the fiber contraction, we DO NOT REQUIRE S to be continuous,
we just require the continuity of the map X -> Y : x -> A(x, y) for each fixed y in Y]
Explanation / Answer
please check the link http://www.math.wisc.edu/~robbin/angelic/hadper.pdf, it might help u