Define the operator
D
:
C
[
0
,
τ
]
→
C
[
0
,
τ
]
D:C[0,\tau ] \to C[0,\tau ]
by
D
[
u
]
=
u
−
a
0
u
′
−
a
1
u
D[u] = u - {a_0}u\prime - {a_1}u
where
a
0
,
a
1
∈
C
[
0
,
τ
]
{a_0},{a_1} \in C[0,\tau ]
and consider the two point boundary value problem
(
BVP
)
D
[
y
]
(
x
)
=
a
2
(
x
)
({\text {BVP}})\;D[y](x) = {a_2}(x)
,
x
∈
[
0
,
τ
]
x \in [0,\tau ]
,
N
0
[
y
]
=
α
0
y
(
0
)
+
α
1
y
′
(
0
)
=
α
2
{N_0}[y] = {\alpha _0}y(0) + {\alpha _1}y\prime (0) = {\alpha _2}
,
N
τ
[
y
]
=
β
0
y
(
τ
)
+
β
1
y
′
(
τ
)
=
β
2
{N_\tau }[y] = {\beta _0}y(\tau ) + {\beta _1}y\prime (\tau ) = {\beta _2}
where
a
2
∈
C
[
0
,
τ
]
{a_2} \in C[0,\tau ]
,
α
0
2
+
α
1
2
≠
0
\alpha _0^2 + \alpha _1^2 \ne 0
and
β
0
2
+
β
1
2
≠
0
\beta _0^2 + \beta _1^2 \ne 0
. Let
Π
k
{\Pi _k}
denote the set of polynomials of degree at most k and define the approximating set
P
k
=
{
p
∈
Π
k
:
N
0
[
p
]
=
α
2
,
N
τ
[
p
]
=
β
2
}
{\mathcal {P}_k} = \{ p \in {\Pi _k}:{N_0}[p] = {\alpha _2},{N_\tau }[p] = {\beta _2}\}
. Then for each
k
⩾
3
k \geqslant 3
there exists
p
k
∈
P
k
{p_k} \in {\mathcal {P}_k}
satisfying
‖
D
[
p
k
]
−
a
2
‖
=
inf
p
∈
P
k
‖
D
[
p
]
−
a
2
‖
=
δ
k
\left \| {D[{p_k}] - {a_2}} \right \| = {\inf _{p \in {\mathcal {P}_k}}}\left \| {D[p] - {a_2}} \right \| = {\delta _k}
, where
‖
⋅
‖
\left \| \cdot \right \|
denotes the uniform norm on
C
[
0
,
τ
]
C[0,\tau ]
. If the homogeneous BVP
D
[
y
]
=
0
D[y] = 0
,
N
0
[
y
]
=
N
τ
[
y
]
=
0
{N_0}[y] = {N_\tau }[y] = 0
has no nontrivial solutions, then the nonhomogeneous BVP has a unique solution y and
lim
k
→
∞
‖
p
k
(
i
)
−
y
(
i
)
‖
=
0
{\lim _{k \to \infty }}\left \| {p_k^{(i)} - {y^{(i)}}} \right \| = 0
for
i
=
0
,
1
,
2
i = 0,1,2
. If X denotes a closed subset of
[
0
,
τ
]
[0,\tau ]
and
\[
δ
k
,
X
=
inf
p
∈
P
k
max
x
∈
X
|
D
[
p
]
(
x
)
−
a
2
(
x
)
|
,
{\delta _{k,X}} = \inf \limits _{p \in {\mathcal {P}_k}} \max \limits _{x \in X} |D[p](x) - {a_2}(x)|,
\]
then for each
ε
>
0
\varepsilon > 0
there exists
δ
>
0
\delta > 0
such that
d
(
x
)
⩽
δ
d(x) \leqslant \delta
implies that
0
⩽
δ
k
−
δ
k
,
X
⩽
ε
0 \leqslant {\delta _k} - {\delta _{k,X}} \leqslant \varepsilon
, where
d
(
X
)
d(X)
denotes the density of X in
[
0
,
τ
]
[0,\tau ]
. Several numerical examples are given.