| Periodic sequences governed by recurrence relations are basic in many branches of mathe-matics. Yet for very simple relations involving nonlinear and discontinuous functions,relatively little is known.Therefore,in this paper,we study a very simple three term recurrence relation involving the discontinuous Heaviside step function, where Z is the set of integers and f:R→R, f is the Heaviside step function defined byBecause the equation (1) involves a discontinuous functions, analytic tools cannot be used to study the property of it.In this paper,the solutions of (1) are investigated by means of combinatorial techniques and recurrence according to the property of symmetry.The author is concerned with traveling waves in polygonal neural networks in [1]. Authors intend to explain the existence and nonexistence of all periodic solutions of (1)with least periods 1 through 8 in [2].They show that there are no solutions which are 2-,3-,7-or 8-periodic,while necessary and sufficient conditions for the existence of 1-,4-,5-and 6-periodic solutions.Some periodic solutions with periods 12,20,and 36 can also be found.On this basis,We intend to explain the existence and nonexistence of all periodic solutions of (1)with least periods 9 through 16.They show that there are no solutions which are 10-,11-,15-or 16-periodic,while necessary and sufficient conditions for the existence of 9-,12-,13-and 14-periodic solutions and their explicit forms can be giren.Some periodic solutions with periods 8n+1-,8n+4-,8n+5-and 8n+6-can also be found. |
