# Schauder fixed point theorem based existence of periodic solution for the response of Duffing’s oscillator

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## Abstract

An initial-boundary value problem that is Duffing’s oscillator with time varying coefficients will be studied. Using Banach’s fixed-point theorem, the existence of periodic solution of the equation will be predicted. The method applied in this paper is the Schauder second fixed point theorem, which includes the response of structures under vibratory force systems. As an example, the dynamics of nonlinear simply supported rectangular thin plate under influence of a relatively moving mass is studied. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development vibration amplitude, which is Duffing’s oscillator. Finally, a parametric study is developed, after that some numerical examples are solved, and the validity of the present analysis is clearly shown.

## Keywords

Banach’s theorem Large deformation of thin plates Moving loads Non-linear vibration Schauder fixed point theorem## Preview

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