Fractional Halanay inequality of order between one and two and application to neural network systems

We extend the (integer-order) Halanay inequality with distributed delay to the fractional-order case between one and two. The main feature is the passage from integer order to noninteger order between one and two. This case of order between one and two is more delicate than the case between zero and one because of several difficulties explained in this paper. These difficulties are encountered, in fact, in general differential equations. Here we show that solutions decay to zero as a power function in case the delay kernel satisfies a general (integral) condition. We provide a large class of admissible functions fulfilling this condition. The even more complicated nonlinear case is also addressed, and we obtain a local stability result of power type. Finally, we give an application to a problem arising in neural network theory and an explicit example.