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

Abstract

We give some sufficient conditions for the system of nonlinear functional differential equations x' (qt)= x' (t)+F(t,x(t),x(φ 1(qt, x(φ 2(qt,⋯x(φ k(qt,x(qt)))⋯)))), x'(g(t))),where q ∈(0,1) and functions F:[0,T]×ℝ n×ℝ n×ℝ n→ℝ n, φ j:[0,T]×ℝ n→[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<T<∞, satisfying the condition x(0)=0.