Moletsane, Boitumelo2019-09-112019-09-112019https://hdl.handle.net/10539/28082A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy School of Mathematics University of the Witwatersrand Johannesburg, South Africa, January 2019In this thesis,wegivefirstorderasymptoticsofeigenvaluesofquadraticpencils presenting a fourth order differential equation together a mixture of boundary conditions that depend on the eigenvalue parameter and are periodic or antiperiodic. The non-self-adjoint quadratic pencils have the two constant coefficient operators and the differential operator all self-adjoint. For the same differential equationandthesamesetofboundaryconditionswheretheonlydifferenceisthat the boundary conditions which are periodic are replaced with anti-periodic one, thezerosoftheircharacterisiticdeterminantsareinterlaced. Thus,theeigenvalues of their quadratic pencils with periodic and anti-periodic boundary conditions, respectively, are interlaced and lie in the first and third quadrant of the complex plane. In both cases the periodic and anti-periodic boundary conditions do not depend on the eigenvalue parameterenThesis