On positive solutions for singular boundary value problems of differential and difference equations / Noor Halimatus Sa'diah Ismail - Ismail, Noor Halimatus Sa'diah (2016)
This thesis is concerned with the existence and multiplicity of positive solutions to singular boundary value problems (BVPs) of differential and difference equations. By using the Krasnoselskii fixed point theorem on compression and expectation in cone, sufficient conditions for the existence of positive solutions are established for a singular system of first-order differential equations and singular second-order BVPs of difference equations. Our results give an almost complete structure of the existence of positive solutions for the problems studied with an appropriately chosen parameter. By choosing appropriate cone, the singularity of the equations is essentially removed and the associated positive operator becomes well defined for certain ranges of functions even when et is negative. By employing the Krasnoselskii fixed point theorem in cone, the existence and multiplicity of positive periodic solutions for a singular system of first-order ordinary differential equations is established. As an extension, the discrete analogue of singular differential problems of second-order BVPs with a parameter is derived. The existence of positive solutions is obtained by applying the Krasnoselskii fixed point theorem in cone. The result is then extended to a singular discrete system of second-order two point BVPs. Also the existence of positive solutions is investigated for a singular discrete system of second-order multi-point BVPs.