Funkcialaj Ekvacioj, 30 (1987) 9-17 Existence and Uniqueness of Solutions of Neutral Delay-Differential Equations with State Dependent Delays* By Zdzislaw JACKIEWICZ (University of Arkansas, U. S. A.) 1. Introduction Let there be given real numbers $ gamma leq a<b$ , a function $f:[a, b] times R^{3} rightarrow R$ , initial function $g$ : $[ gamma, a] rightarrow R$ , and delay functions $ alpha$ , $ beta:[a, b] times R rightarrow R$ such that $ gamma leq alpha(t, y) leq t$ , $ gamma leq beta(t, y) leq t$ . Here, $R$ denotes the set of real numbers. We consider the initial-value problem for delay-differential equations of neutral type $y^{ prime}(t)=F(t, y, y^{ prime})$ , $t$ $ in[a, b]$ , (1) $y(t)=g(t)$ , $t$ $ in[ gamma, a]$ , where for any functions $y$ , $z$ : $[ gamma, b] rightarrow R$ , $F$ is defined by $F(t, y, z):=f(t, y(t), y( alpha(t, y(t))), z( beta(t, y(t))))$ . Denote by $ mathrm{L} mathrm{i} mathrm{p}_{1}$ $[t_{1}, t_{2}]$ the space of real-valued Lipschitz continuous functions on $[t_{1}, t_{2}]$ and by $C^{1,1}[t_{1}, t_{2}]$ the space of functions whose first derivative belongs to $ mathrm{L} mathrm{i} mathrm{p}_{1}$ $[t_{1}, t_{2}]$ . We impose the following conditions on the functions /, $g$ , $ alpha$ , and $ beta$ which define problem (1): (i) $g$ $ in C^{1,1}[ gamma, a]$ ; (ii) $F(a, g, g^{ prime})=g_{-}^{ prime}(a)$ ( $g_{-}^{ prime}(a)$ denotes the left hand derivative of $g$ at $a$ ); (iii) $|f(t_{1}, y_{1}, u_{1}, z_{1})-f(t_{2}, y_{2}, u_{2}, z_{2})| leq L_{1}(|t_{1}-t_{2}|+|y_{1}-y_{2}|+|u_{1}-u_{2}|)$ $+L_{2}|z_{1}-z_{2}|$ , $L_{1}$ , $L_{2} geq 0$ , $t_{1}$ , $t_{2} in[a, b]$ , $y_{1}$ , $y_{2}$ , $u_{1}$ , $u_{2}$ , $z_{1}$ , $z_{2} in R$ ; (iv) $| alpha(t_{1}, y_{1})- alpha(t_{2}, y_{2})| leq A_{1}|t_{1}-t_{2}|+A_{2}|y_{1}-y_{2}|$ , $A_{1}$ , $A_{2} geq 0$ , $t_{1}$ , $t_{2} in[a, b]$ , $y_{1}$ , $y_{2} in R$ . (v) $| beta(t_{1}, y_{1})- beta(t_{2}, y_{2})| leq B_{1}|t_{1}-t_{2}|+B_{2}|y_{1}-y_{2}|$ , $B_{1}$ , $B_{2} geq 0$ , $t_{1}$ , $t_{2} in[a, b]$ , $y_{1}$ , $y_{2} in R$ . Additional conditions on some Lipschitz constants appearing above will be given in the formulation of our theorems. Equations of type (1) arise as a model for a two-body problem of classical electrodynamics and were studied extensively by Driver [2?4]. He proved the $*)$ This research was supported in part by the National Science Foundation under grant DMS-8401013.