Existence and uniqueness results for Liénard́s equation having a dead band

In the first part of the present paper we consider general systems of first-order autonomous differential equations and generalize a uniqueness criterion by Dulac concerning periodic solutions to equations of the formdot{x}=P(x,y), dot{y}=Q(x,y). In the second part we use this result to generalize a uniqueness theorem by de Figueiredo concerning periodic solutions to Liénard's equationddot{x} +f(x)dot{x} + g(x) = 0.

By our method we are able to avoid the hitherto usual conditionxg(x) > 0, x {neq} 0, which excludes the possibility for the equation to have a dead band. Finally, we prove an existence theorem concerning periodic solutions to such equations. The use of the theorems is illustrated by a simple example in the last section.