Existence of Subharmonic Periodic Solutions to a Class ofSecond‐Order Non‐Autonomous Neutral Functional Differential Equations

Xiao-Bao Shu,Fei Xu,Yongzeng Lai

doi:10.1155/2012/404928

Abstract

By introducing subdifferentiability of lower semicontinuous convex function φ(x(t), x(t − τ)) and its conjugate function, as well as critical point theory and operator equation theory, we obtain the existence of multiple subharmonic periodic solutions to the following second‐order nonlinear nonautonomous neutral nonlinear functional differential equation x″(t) + x″(t − 2τ) + f(t, x(t), x(t − τ), x(t − 2τ)) = 0, x(0) = 0.

Highlights

The existence of periodic solutions for differential system has received a great deal of attention in the last few decades
We show the validity of the second part of the inequality
For assumptions A1 ∼ A3, function F is a solution to the following partial differential equation: F2 t, x1, x2 F1 t, x2, x3 f t, x1, x2, x3

Summary

Introduction

The existence of periodic solutions for differential system has received a great deal of attention in the last few decades. Different from ordinary differential equations and partial differential equations that do not contain delay variate, it is very difficult to study the existence of periodic solutions for functional differential equations. For this reason, many mathematicians developed different approaches such as the averaging method 1 , the Massera-Yoshizawa theory 2, 3 , the Kaplan-York 4 method of coupled systems, the Grafton cone mapping method 5 , the Nussbaum method of fixed point theory 6 and Mawhin 7 coincidence degree theory. By using critical point and operator equation theories, we study the existence of the following second-order nonlinear and nonautonomous mixed-type functional differential equation:. F2 t, x1, x2 F1 t, x2, x3 f t, x1, x2, x3 , 1.2 where F2 t, x1, x2 and F1 t, x2, x3 denote ∂F t, x1, x2 /∂x2 and ∂F t, x2, x3 /∂x2, respectively; A3 F t τ, x1, x2 F t, x1, x2 for all x1, x2 ∈ R

Variational Structure

Example