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).