Initial value problem for mixed-type differential equations (Q1366664)

The authors obtain various existence and uniqueness results for the following initial value problem: \[ {d\over dt} [x(t)- D(t,\widehat x_t)- g(t)]= L(t,\widehat x_t)+ f(t);\quad t\in(t_0;+\infty),\tag{1} \] \[ x(t_0+ \theta)=\varphi(\theta),\quad \theta\in(-r; 0];\tag{2} \] where \(\widehat x_t(\theta)= x(t+\theta)\), \(\theta\in[-r;\rho]\); \(f\), \(g\) are functions from \([t_0;+\infty)\) in \(\mathbb{C}^N\), \(D(t,\cdot)\), \(L(t,\cdot)\) are bounded linear operators from \(C([-r; \rho];\mathbb{C}^N)\) to \(\mathbb{C}^N\) and \(\varphi\in C([-r; 0];\mathbb{C}^N)\).