Periodic solutions of a singular differential delay equation with the Farey-type nonlinearity

We address the problem of existence of periodic solutions for the differential delay equation ε x ˙ ( t ) + x ( t ) = f ( x ( t - 1 ) ) , 0 < ε ⪡ 1 , with the Farey nonlinearity f ( x ) of the form f ( x ) = mx + A if x ⩽ 0 , mx - B if x > 0 , where | m | < 1 , A > 0 , B > 0 . We show that when the map x ↦ f ( x ) has an attracting 2-cycle then the delay differential equation has a periodic solution, which is close to the square wave corresponding to the limit (as ε → 0 + ) difference equation x ( t ) = f ( x ( t - 1 ) ) .