Total solvability of the initial value problem for Pfaff differential equations with constant delay of one variable (Q2768777)

The author obtains necessary and sufficient conditions of total solvability of the initial value problem NEWLINE\[NEWLINE\begin{multlined} dx(t_1,t_2)=a_1(t_1,t_2,x(t_1,t_2),x(t_1-\Delta,t_2)) dt_1+ a_2(t_1,t_2,x(t_1,t_2)) dt_2,\\ (t_1,t_2)\in T=[0,T_1]\times[0,T_2],\end{multlined}NEWLINE\]NEWLINE \(x(t_1,t_2)=\phi(t_1,t_2)\), \(t_1\in[-\Delta,0]\), \(t_2\in [0,T_2]\)NEWLINENEWLINENEWLINEand the initial value problem NEWLINE\[NEWLINEdx(t_1,t_2)=a_1(t_1,t_2,x(t_1,t_2),x(t_1-\Delta_1,t_2),\ldots,x(t_1-\Delta_{m},t_2)) dt_1+ a_2(t_1,t_2,x(t_1,t_2)) dt_2,NEWLINE\]NEWLINE \((t_1,t_2)\in T\), \(x(t_1,t_2)=\phi(t_1,t_2)\), \(t_1\in[-\Delta,0]\), \(t_2\in [0,T_2]\), where \(x\in D\subset \mathbb{R}^{n}\), \(\Delta>0\), \(\Delta_1>\Delta_2>\ldots>\Delta_{m}>0\).