Existence and Uniqueness of a Periodic Solution to a Certain Third-Order Neutral Functional Differential Equation

Abstract

In this paper, by applying Mawhin's continuation theorem of the coincidence
degree theory, some sufficient conditions for the existence and uniqueness of an $\omega$-periodic solution for the following third-order neutral functional differential equation are established

\dfrac{d^{3}}{dt^{3}}\bigg ( x(t)-d(t)x\big (t-\delta(t)\big ) \bigg )+a(t)\ddot{x}(t)+b(t)f\big (t,\dot{x}(t)\big )+\sum_{i=1}^{n}c_{i}(t)g\big (t,x(t-\tau_{i}(t))\big )=e(t).

Moreover, we present an example and a graph to demonstrate the validity of analytical conclusion.