Picard's theorem

Assume that f is a continuous function in the first variable and Lipschitz in the second variable. Then there exists an open interval about x0 on which there is a unique solution y(x). The solution can be constructively found as the limit of functions yn(x), where

As an example, suppose dy/dx =  y and y(0) = 1. Then

So yn(x) is the first n + 1 terms of the exponential series and converges to the solution y(x) = ex.

• Frobenius, (Ferdinand) Georg
• Frobenius normal form
• from above, from the right
• from below, from the left
• frontier
• FRS
• frustum
• Fry, Hannah
• Fubini's Theorem
• fulcrum
• full angle
• full measure
• full rank
• function
• functional
• functional analysis
• functionally separable
• function of a function
• function space
• functor
• fundamental group
• Fundamental Theorem of Algebra
• Fundamental Theorem of Arithmetic
• Fundamental Theorem of Calculus
• Fundamental Theorem of Game Theory