DELAY - DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS IN A BANACH SPACE

Abstract

Sufficient conditions are given for the existence, uniqueness and continuous dependence of solutions on the initial data of delay - differential equations with infinite delays in a Banach space. These conditions are also shown to be sufficient for the uniform stability of the solutions of these equations. Two kinds of delay - differential equations are: (A):x^' (t)=f(t,x(t),x_t )+g(t,x(t),x_t ),t∈R (B):[x(t)-f(t,x(t-r))]^'= g(t,x(t),x_t),t∈R