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

By the mean value theorem, we can find

and

(

depend on

, resp.), such 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

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

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

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

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

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

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).