I was trying to figure out how to solve the following problem concerning a retraction theorem:
Let j:C->D be an inclusion of the circle into the disk. Suppose that we have two continuous maps f:D->D and g:D->D and that g satisfies goj=j. Use the retraction theorem for the disk to show that there must be a point x in the disk at which f(x)=g(x).
Supposedly, this can be proven by constructing a retraction for j by assuming there is no `fixed point' (or something like this), but I am not sure how to do this. Any general suggestion would be greatly appreciated even if they are suggestions of another way of constructing a proof to answer the question.