<p>In this paper, a class of discontinuous Cohen-Grossberg neural networks with time-varying delays is considered. Firstly, under the extended Filippov differential inclusions framework, the problem of periodic solutions of the considered neural networks with more relaxed conditions imposed on the amplification functions is analyzed by using set-valued mapping and Kakutani’s fixed point theorem, which has rarely been used to study such problem. Secondly, the fixed-time synchronization of the error system of the considered neural networks is also investigated by designing a novel control strategy, which can improve not only the previous ones with sign function greatly, but also can reduce the chattering phenomenon. Finally, two numerical examples are presented to further illustrate the validity of the obtained results.</p>