Existence of solutions for neutral functional differential equations with causal operators (Q1592710)

The following neutral equation is considered \[ {d\over dt}\Bigl[ x(t)+g\biggl( x\bigl(\alpha (t)\bigr)\biggr) \Bigr]=(Wx)(t),\quad\alpha (t)\leq t,\;\alpha(0)= 0,\quad x(0)=x^0,\;x^0\in\mathbb{R}^n. \] Under standard conditions, in the space of continuous functions, local theorems of existence and uniqueness are proved.