$i=1,$ $\cdots,$ $d$ , under the assumption that $f_{i}$ satisfies the Caratheodory condition, where $x(t)$ stands for $(x_{1}(t), \cdots, x_{d}(t))$ and prime denotes differentiation with respect to $t$ . We assume the existence of a finite number $\alpha$ such that for each $j=2,$ $\cdots,$ $m,$ $\alpha\leqq g_{j}(t, x)\leqq t$ , whenever $g_{j}(t, x)$ is defined; the delays $t-g_{j}(t, x)$ may be unbounded. This type of system arises in studying a two-body problem of classical electrodynamics $[6, 7]$ . Driver [4] developed the basic theory (existence, uniqueness and dependence of solutions, etc.) for the initial value problem for delay differential equations (E) with continuous $f_{i}[5]$ . Since then the theory of delay differential equations (E) has been studied by many authors. Among them Bullock [1] showed the existence theorem and uniqueness theorem for delay differential equations (E’) of Caratheodory type. On the other hand, Strauss and Yorke [13] constructed a fundamental theory for ordinary differential equations by using the convergence theorem which is a generalization of Kamke’s theorem (see [8], Theorem 3.2). Their method proves to be very important in studying the fundamental theory of functional (or delay) differential equations. Costello [3] extended their results to functional differential equations of Caratheodory type with finite delay,