Let
I
=
{
a
,
a
+
1
,
…
,
b
}
I = \left \{ {a,a + 1, \ldots ,b} \right \}
be finite, let
n
⩾
1
n \geqslant 1
, and let
I
j
=
{
a
,
a
+
1
,
…
,
b
+
j
}
{I^j} = \left \{ {a,a + 1, \ldots ,b + j} \right \}
,
j
=
1
,
…
,
n
j = 1, \ldots ,n
. Let
B
B
be the set of mappings from
I
n
{I^n}
into the reals and define the linear difference operator
P
P
by (1)
\[
P
u
(
m
)
=
∑
j
=
0
n
α
j
(
m
)
u
(
m
+
j
)
,
where
m
∈
I
,
α
n
(
m
)
≡
1
,
and
α
0
(
m
)
≠
0
on
I
.
Pu(m) = \sum \limits _{j = 0}^n {{\alpha _j}(m)u(m + j),} \quad {\text {where }}m \in I,{\alpha _n}(m) \equiv 1,{\text {and }}{\alpha _0}(m) \ne 0{\text { on }}I.
\]
Existence of solutions theorems and iteration schemes that approximate solutions are given for boundary value problems of the form
P
u
(
m
)
=
f
(
m
,
u
,
E
u
,
…
,
E
n
−
1
u
)
Pu(m) = f(m,u,Eu, \ldots ,{E^{n - 1}}u)
, with boundary conditions
T
u
(
m
)
=
r
Tu(m) = r
, where
P
P
is defined by (1),
E
j
u
(
m
)
=
u
(
m
+
j
)
{E^j}u(m) = u(m + j)
,
j
=
0
,
1
,
…
,
n
−
1
j = 0,1, \ldots ,n - 1
,
f
:
I
×
R
n
→
R
f:I \times {{\mathbf {R}}^n} \to {\mathbf {R}}
is continuous, and
T
:
B
→
R
n
T:B \to {{\mathbf {R}}^n}
is a continuous linear operator. The results are based on solutions of difference inequalities and sign properties of associated Green’s functions.