Time-dependent neural networks with activation functions violating the standard Lipschitz condition

A Hopfield neural network system with discrete (or variable delays) is considered in this paper. The standard assumption of Lipschitz continuity of the activation functions is dropped partially. Having in mind that this condition is needed not only for the uniqueness of solutions but also for the stability of the system, the present work improves the existing ones in the literature. The other feature here, which is in fact the main one, is the treatment of time-dependent activation functions. The time-independent case has been discussed by one of the authors in Tatar (2020). Unfortunately, it is not applicable to the present situation. Indeed, when applied, it will require a uniform boundedness condition. To overcome this difficulty, we provide here a new argument in addition to the introduction of some functionals.