Notes on Fixed Point Theorems 4 Basic Definitions

A fixed point of a map is a point in its domain which satisfies the equation F (x) = x. It is clear that, for such a point to exist, the domain and range must have common points. We point out two things, first, that the problem of existence of fixed points is equivalent to the problem of the solution of equations of the form f(x) = 0. Indeed, given such an equation, we can add the variable x to each side to obtain the equation F (x) = f(x) + x = x so that x is a fixed point of F if and only if it is a zero of f . Conversely, if we wish to solve the fixted point problem, then if f(x) = F (x)− x then x is a fixed point of F if and only if f(x) = 0.