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When designing dynamic neural networks, it is important to know whether steady state solutions can be guaranteed. By means of critical point theory, several easy-to-use fixed point theorems in R n are given for such purposes. Examples are also provided for illustration.

Existence criteria are proved for the periodic solutions of a neutral differential equation with piecewise constant arguments.

In this paper, we study a very simple three term recurrence relation involving the discontinuous Heaviside step function. One reason for studying such an relation is that solutions of our recurrence relation are steady state distributions in some basic neural network models. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by combining combinatorial elimination technique as well as existence arguments for linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 8. Some periodic solutions with periods 12, 20 and 36 can also be found, but exhaustive results are not yet available.

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