Existence of Subharmonic Periodic Solutions to a Class of Second-Order Non-Autonomous Neutral Functional Differential Equations

Abstract

By introducing subdifferentiability of lower semicontinuous convex function $\phi \left(x\right(t),x(t-\tau \left)\right)$ 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 $x^{″}\left(t\right)+x^{″}(t-2\tau )+f(t,x(t),x(t-\tau ),x(t-2\tau \left)\right)=0$, $x\left(0\right)=0.$