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.