In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation
x
=
f
+
a
1
g
1
(
x
)
+
⋯
+
a
n
g
n
(
x
)
x = f + {a_1}{g_1}(x) + \cdots + {a_n}{g_n}(x)
, where the
a
i
{a_i}
are continuous linear operators mapping a Fréchet space
F
\mathcal {F}
into itself and the
g
i
{g_i}
are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of
F
\mathcal {F}
having a stronger topology. The equations are studied first when the
g
i
{g_i}
are of the form
g
i
(
x
)
=
x
+
h
i
(
x
)
{g_i}(x) = x + {h_i}(x)
where
h
i
(
x
)
{h_i}(x)
is “small", and then when the
g
i
{g_i}
are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.