Existence and continuous dependence for a class of nonlinear neutral-differential equations

Abstract:

This paper presents existence, uniqueness, and continuous dependence theorems for solutions of initial-value problems for neutral-differential equations of the form \[ x’(t) = f(t,x(t),x(g(t,x)),x’(h(t,x))),\quad x(0) = {x_0},\] where f, g, and h are continuous functions with $g(0,{x_0}) = h(0,{x_0}) = 0$. The existence of a continuous solution of the functional equation $z(t) = f(t,z(h(t)))$ is proved as a corollary.