Let X be a normed linear space and let K be a convex subset of X. The inward set,
I
K
(
x
)
{I_K}(x)
, of x relative to K is defined as follows:
I
K
(
x
)
=
{
x
+
c
(
u
−
x
)
:
c
⩾
1
,
u
∈
K
}
{I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}
. A mapping
T
:
K
→
X
T:K \to X
is said to be inward if
T
x
∈
I
K
(
x
)
Tx \in {I_K}(x)
for each
x
∈
K
x \in K
, and weakly inward if Tx belongs to the closure of
I
K
(
x
)
{I_K}(x)
for each
x
∈
K
x \in K
. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.