A discontinuous functional Cauchy problem

The existence of extremal solutions of the following functional Cauchy problem [FCP]:(1)x′=f(t,x,x(.)),x(0)=0, is investigated here. The main results established for non-functional Cauchy problem [N-FCP] in R [Dariusz Walachowski, Witold Rzymowski, One-dimensional differential equations under weak assumptions, J. Math. Anal. Appl. 198 (1996) 657-670] are generalized here in R, as well as in Rn.