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 form . In the second part we use this result to generalize a uniqueness theorem by de Figueiredo concerning periodic solutions to Liénard\´s equation . By our method we are able to avoid the hitherto usual condition , 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.