Periodic Solutions To A Suspension Bridge Torsional Oscillation Equation With Damping

Because of the important mechanical background and realistic significance, many people payattention to the existence of multiple periodic solutions for the suspension bridge equation and domuch study work. In this paper we extend the results about suspension bridge torsional oscillationmodel without damping to have damping case.First of all, in condition of the nonlinear operator only having Ga teaux derivability, withthe aid of the related literature thought, using Leray-Schauder degree theory，the sufficientconditions are given to the suspension bridge torsional oscillations equations with damping havingat least two periodic solutions.Second, we discuss the existence of double periodic positive solutions for the nonlinearsuspension bridge damping equation on the basis of the maximum principle of telegraph equationusing the method of upper and lower solution and thereom of fixed point index.Finally, by using the monotone iterative and the method of upper and lower solution, we usethe regularity of second order damping suspension bridge ordinary differential operator to discussthat, the second order nonlinear suspension bridge ordinary differential equation has periodicsolutions.Parts of our results to some extent answer to the problem proposed in literature about periodicsolutions to suspension bridge torsional oscillation equation with damping.