Theorem 3.1.
Assume that
the functions
are, respectively, lower and upper solutions of (1.1)-(1.2) such that
on 
the function
satisfies a Nagumo condition relative to
and
on
where
is a positive constant depending on
and the Nagumo function
. Further, there exists a function
such that
with
on
where
Then, there exists a monotone sequence
of approximate solutions converging uniformly to a unique solution of the problems (1.1)-(1.2).
Proof.
For
we define
and consider the following modified
-point BVP
We note that
are, respectively, lower and upper solutions of (3.2) and for every
we have
where
As
so
is a Nagumo function. Furthermore, there exists a constant
depending on
, and Nagumo function
such that
where
. Thus, any solution
of (3.2) with
,
satisfies
on
and hence it is a solution of (1.1)-(1.2).
Let us define a function
by
In view of the assumption
it follows that
and satisfies
on
Therefore, by Taylor's theorem, we obtain
We set
and observe that
By the mean value theorem, we can find
and
(
depend on
, resp.), such that
Letting
we note that
Let us define
as
Clearly
is continuous and bounded on
and satisfies a Nagumo condition relative to
. For every
and
, we consider the
-point BVP
Using (3.9), (3.12) and (3.13), we have
Thus,
are lower and upper solutions of (3.14), respectively. Since
satisfies a Nagumo condition, there exists a constant
(depending on
and a Nagumo function) such that any solution
of (3.14) with
satisfies
on 
Now, we choose
and consider the problem
Using
, (3.9), (3.12) and (3.13), we obtain
which imply that
and
are lower and upper solutions of (3.16). Hence by Theorems 2.2 and 2.3, there exists a unique solution
of (3.16) such that
Note that the uniqueness of the solution follows by Theorem 2.3. Using (3.9) and (3.13) together with the fact that
is solution of (3.16), we find that
is a lower solution of (3.2), that is,
In a similar manner, it can be shown by using
, (3.12), (3.13), and (3.19) that
and
are lower and upper solutions of the following
-point BVP
Again, by Theorems 2.2 and 2.3, there exists a unique solution
of (3.20) such that
Continuing this process successively, we obtain a bounded monotone sequence
of solutions satisfying
where
is a solution of the problem
and is given by
Since
is bounded on
,
,
therefore it follows that the sequences
are uniformly bounded and equicontinuous on
Hence, by Ascoli-Arzela theorem, there exist the subsequences and a function
such that
uniformly on
as
Taking the limit
we find that
which consequently yields
This proves that
is a solution of (3.2).
Theorem 3.2.
Assume that
and
hold. Further, one assumes that
the function
satisfies
for
where
and 
Then, the convergence of the sequence
of approximate solutions (obtained in Theorem 3.1) is quadratic.
Proof.
Let us set
so that
satisfies the boundary conditions
In view of the assumption
for every
it follows that
Now, by Taylor's theorem, we have
where
,
,
,
on
,
and
,
with
satisfying
on
Also, in view of (3.13), we have
where
and
, 
Now we show that
By the mean value theorem, for every
and
we obtain
Let
for some
Then
and (3.30) becomes
In particular, taking
and using (3.27), we have
which contradicts that
Similarly, letting
for some
we get a contradiction. Thus, it follows that
for every
, which implies that
and consequently, (3.28) and (3.29) take the form
where
and
Now, by a comparison principle, we can obtain
on
, where
is a solution of the problem
Since
is continuous and bounded on
, there exist
(independent of
) such that
on
Since
on
so we can rewrite (3.35) as
whose solution is given by
where
Introducing the integrating factor
such that
(3.34) takes the form
Integrating (3.39) from
to
and using
we obtain
which can alternatively be written as
where
,
. Using the fact that
together with (3.41) yields
which, on substitutingin (3.37), yields
where
Taking the maximum over
and then solving (3.43) for
we obtain
Also, it follows from (3.33) that
Integrating (3.46) from
to
and using
(from the boundary condition
we obtain
which, in view of the fact
and (3.45), yields
where
As
, there exists
such that
Integrating (3.46) from
to
(
) and using (3.50), we have
Using (3.45) in (3.34), we obtain
where
. Since
is bounded on
,
we can choose
such that
on
,
and
so that (3.52) takes the form
Integrating (3.53) from
to
(
), and using (3.51), we find that
Letting
it follows from (3.51) and (3.54) that
Hence, from (3.48) and (3.56), it follows that
where
From (3.45) and (3.57) with
we obtain
This proves the quadratic convergence in
norm.
Example 3.3.
Consider the boundary value problem
Let
and
be, respectively, lower and upper solutions of (3.60). Clearly
and
are not the solutions of (3.60) and
Also, the assumptions of Theorem 3.1 are satisfied. Thus, the conclusion of Theorem 3.1 applies to the problem (3.60).