### Abstract:

In the study of differential equations there are two fundamental questions: Is there a solution? And what is it? One of the most elegant was to prove that an equation has a solution is to pose it as fixed point problem that is to find a function such that x is a solution if and only if ( ) . Results from fixed point theory can then be employed to show that has a fixed point. However the results of fixed point theory are often non-constructive: they guarantee that a fixed point exists but do not help in finding the fixed point. Thus these methods tend to answer the first questions 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.
In this thesis we present a selection of fixed point theorems with applications in semilinear elliptic equation. We begin with the Banach fixed point theorem. Then prove in succession the fixed point theorems of Brouwer Schauder and Schaeffer after which we conclude with applications for semilinear elliptic equation.