Solutions of a system of functional differential equations in a right neighborhood of zero

We give some sufficient conditions for the system of nonlinear functional differential equationsx′(qt)=x′(t)+F(t,x(t),x(φ1(qt,x(φ2(qt,…x(φk(qt,x(qt)))…)))),x′(g(t))),where q∈(0,1) and functions F:[0,T]×Rn×Rn×Rn→Rn, φj:[0,T]×Rn→[0,T],j=1,k¯ and g:[0,T]→[0,T] are continuous, for the unique existence of continuously differentiable solutions x(t) on the interval [0,T],0