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
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).
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 , 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
By comparison result, , where is the unique solution of the linear BVP
where . Taking the maximum over , we obtain
which shows the quadratic convergence.