Duffing Equation With Singular Periodic Solution Of The Existence And Multiplicity Of Solutions

In this paper, we consider the existence and multiplicity of periodic solutions of Duffing equation x" + g(x) = p(t) with singularity at origin, where g : R+ (?) R is local lipschitz continuous function and has singularity at origin, p(t) is continuous and 2π- periodic.The first section of this paper discusses the existence and multiplicity of periodic solutions of equation x"+g(x) = p(t) with singularity. When the time-map has oscillating properties, we obtain the existence of harmonic solutions and the multiplicity of subharmonic solutions of the given equations by using the phase-plane analysis methods and generalized Poincaré-Birkhoff twist theorem and Poincaré-Bohl fixed point theorem .The second part deals with the existence of periodic solutions of x" + g(x) = p(t) with singularity. When the time-map satisfies limit conditionwe prove that the equation has at least one 2π- periodic solution by using the phase-plane analysis methods and Poincare-Bohl fixed point theorem .