If you are interested in applications of fixed point theorems, you'll find entire journals dedicated to them. For example, you see fixed point techniques pop up in approximation theory where one is interested in finding the best approximation.

For a specific problem, consider the following:

You decide to take a break from the fast paced world of academia to climb Mt. Fuji. You begin your ascent at sunrise along a narrow path. Along the way, you stop a few times to take in the scenery and eat, maybe even work out a math puzzle or two. You reach the top at sunset. The next day you begin your descent at sunrise, again making leisurely stops along the way. It's reasonable to assume that going downhill is easier than uphill so let's assume your average downhill speed is greater than your average uphill speed. Show that there must be a place along the path that you occupy at the exact same time of day during your uphill and downhill trips.