Periodic Solution for a Kind of Third-Order Neutral-Type Differential Equation

Abstract

In this paper, we investigate a class of a third-order neutral-type differential equation with time-varying delays. Some sufficient conditions on the existence of a periodic solution are established for the considered system. Different from the previously reported research results, by utilizing the properties of neutral operators and a special variable substitution, we transform a high-order neutral equation into a first-order three-dimensional nonneutral system. The existence of a periodic solution such as the high-order neutral equation has not been given much attention in past papers due to the difficulty of estimation of prior bounds of solutions. This paper is devoted to the use of properties of neutral-type operators with a variable parameter and Mawhin’s continuation theorem for overcoming the above difficulties. The neutral term in the third-order neutral differential equation in this paper contains a variable parameter which is different from third-order neutral-type equations that have been studied. The third-order neutral-type equation studied in this paper is more general, and similar equations studied in the past are special cases of the equations studied in this paper. Finally, an example is given to elucidate the effectiveness and values of the present results. Our results are new and complement the related results of third-order functional differential equations.