Open Access Senior Thesis
Bachelor of Science
Jon T. Jacobsen
Francis Edward Su
In the study of differential equations there are two fundamental questions: is there a solution? and what is it? One of the most elegant ways to prove that an equation has a solution is to pose it as a fixed point problem, that is, to find a function f such that x is a solution if and only if f (x) = x. Results from fixed point theory can then be employed to show that f has a fixed point. However, the results of fixed point theory are often nonconstructive: they guarantee that a fixed point exists but do not help in finding the fixed point. Thus these methods tend to answer the first question, but not the second. One such result is Schauder’s fixed point theorem. This theorem is broadly applicable in proving the existence of solutions to differential equations, including the Navier-Stokes equations under certain conditions. Recently a semi-constructive proof of Schauder’s theorem was developed in Rizzolo and Su (2007). In this thesis we go through the construction in detail and show how it can be used to search for multiple solutions. We then apply the method to a selection of differential equations.
Rizzolo, Douglas, "Approximating Solutions to Differential Equations via Fixed Point Theory" (2008). HMC Senior Theses. 213.