Abdi Tadesse (PhD)Demissie Tsedeke2018-07-192023-11-042018-07-192023-11-042014-03http://etd.aau.edu.et/handle/123456789/9333In this project we will see the existence of periodic solution(s) to the second order ODE of the form: x00(t) + a(t)x0(t) = g(t; x) f(t; x(t); x0(t)) by means of Schauders Fixed Point Theorem where a is a continuous !- periodic function , g(t; u), f(t; u; v) are !-periodic functions in t for u = x(t), v = x0(t) and ! > 0. The method of proof is composed of two steps, the _rst step is to transform the original equation into integrodi erential equation through a linear integral operator and the second step is an application of the Schauder's Fixed Point Theorem. Keywords: Periodic solution; Schauder's _xed point theorem; Fundamental matrixenPeriodic solutionSchauder's _xed point theoremFundamental matrixThesis