A discontinuous functional Cauchy problem

The existence of extremal solutions of the following functional Cauchy problem [FCP]: x = f t, x, x(.) , x(0) = 0, (1) is investigated here. The main results established for non-functional Cauchy problem [N-FCP] in R [DariuszWalachowski,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. © 2007 Elsevier Inc. All rights reserved.