On singular functional-differential equations (Q942000)

The paper deals with the existence of solutions to the singular retarded differential equation \[ t^rx'(t)=F(t,x(t),x(g(t)),x'(t),x'(h(t))),\quad t\in(0,\tau), \] with the initial value \(x(0)=0,\) where \(F\) is linear, or it is a perturbation of a linear function, or it is nonlinear and satisfies a kind of Lipschitz condition with nonconstant coefficients. The problem in all three cases is solved by applying Schauder's fixed point theorem to an implicitly defined operator.