Abstract
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
R
n
.