A discontinuous functional differential equation (Q1868999)

The authors give sufficient conditions for the existence of maximal and minimal solutions to discontinuous differential-functional initial value problems of the form \(x'(t)=f(t,x(t), x(\cdot))\), \(x(0)=0\). Apart from measurability and boundedness conditions, \(f\) is assumed to be quasimonotone increasing (with respect to the coordinate cone) in the second and increasing in the third variable. The result is applied to the Darboux problem for a PDE.