Multiple periodic solutions to a class of second-order nonlinear mixed-type functional differential equations (Q2368386)

This paper deals with the existence of periodic solutions for two classes of second-order mixed-type functional-differential equations of the following form \[ x''(t-\tau)+f(t,x(t),x(t-\tau),x(t-2\tau))=0, \] \[ x''(t-\tau)+\lambda(t)f_1(t,x(t),x(t-\tau),x(t-2\tau))=x(t-\tau). \] Under some assumptions on the functions \(f,f_1\in C(\mathbb{R}^4,\mathbb{R})\) and applying the critical point theory and \(Z_2\) group index theory, the existence of nontrivial periodic solutions for such equations is proved without reducing them to ordinary differential equations.