Periodic solutions for higher-order neutraldifferential equations with several delays

Neutral differential equations arise in many practical problems and have importantapplications in physics and engineering. This paper introduces Fourier series method and inequality techniques to investigate periodic solutions for a class of higher-order delayed neutral differential equations. This method is different to some traditional methods (such as critical point theory, fixed point method, and topological degree method) which are applied to study periodic solution problem of neutral differential equations. Some new necessary and sufficient conditions are obtained ensuring the existence and uniqueness of periodic solutions. In addition, three examples are given to illustrate the theory.