We develop the approximation technique and show that, under suitable conditions on , there exists a bounded monotone sequence of solutions of linear problems that converges uniformly and quadratically to a solution of the nonlinear original problem. If and is bounded on , where

there always exists a function such that

where , and it is such that . For example, let : , then we choose . Clearly,

Define by . Note that and

Theorem 3.1.

Assume that

are lower and upper solutions of the BVP (1.1) such that on ,

and is increasing in for each .

Then, there exists a monotone sequence of solutions of linear problems converging uniformly and quadratically to a unique solution of the BVP (1.1).

Proof.

Conditions and ensure the existence of a unique solution of the BVP (1.1) such that

In view of (3.4), we have

The mean value theorem and the fact that is nonincreasing in on for each yield

where such that . Substituting in (3.6), we have

on . Define by

We note that is continuous for each and is rd-continuous for each . Moreover, satisfies the following relations on :

Now, we develop the iterative scheme to approximate the solution. As an initial approximation, we choose and consider the linear problem

Using (3.11) and the definition of lower and upper solutions, we get

which imply that and are lower and upper solutions of (3.12), respectively. Hence by Theorem 2.4 and Corollary 2.3, there exists a unique solution of (3.12) such that

Using (3.11) and the fact that is a solution of (3.12), we obtain

which implies that is a lower solution of the problem (1.1). Similarly, in view of (3.11), and (3.15), we can show that and are lower and upper solutions of the problem

Hence by Theorem 2.4 and Corollary 2.3, there exists a unique solution of the problem (3.16) such that

Continuing in the above fashion, we obtain a bounded monotone sequence of solutions of linear problems satisfying

where the element of the sequence is a solution of the linear problem

and is given by

By standard arguments as in [19], the sequence converges to a solution of (1.1).

Now, we show that the convergence is quadratic. Set , where is a solution of (1.1). Then, on and the boundary conditions imply that

Now, in view of the definitions of and , we obtain

Using the mean value theorem repeatedly and the fact that on , we obtain

where , : , and : . Hence (3.22) can be rewritten as

where , : , and we used the fact that on . Choose such that

Therefore, we obtain

where .

By comparison result, , where is the unique solution of the linear BVP

Hence,

where . Taking the maximum over , we obtain

which shows the quadratic convergence.