Third Order Opial Inequalities

The purpose of the thesis was to find extremals y and constants K for a basic problem and for several variations of the problem. The basic problem was the third order opial inequality. We proved the existence of an extremal of this problem and other related problems. We conjectured that the extremal was a quintic spline with at most one knot, then found the quintic splines and constants for the basic problem and several variations. Maple was an important tool in finding these extremals and constants due to the complexity of the equations. We also proved that these extremals can be approximated by polynomials. We discussed an application involving a bound for the least eigenvalue for y^(vi) = ^_λρ(x)y with certain boundary conditions. Finally, we discussed the related problem in L^p space.