Computer-assisteed proof of a periodic solution in a non-linear feedback DDE

In this paper we rigorously prove the existance of a non-trivial periodic orbit for the non-linear delay differential equation: $x'(t) = K \sin(x(t-1))$ for $K=1.6$. We show that the equations on the Fourier equations have a solution by computing the local Brower degree. This degree can be computed by using a homotopy which validity can be checked by checking a finite number of inequalities. Checking these inequalities is done by a computer program.