Abstract
We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial 
of degree 
, which involves interpolating data at the odd-order derivatives. For 
we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a 
th order differential equation and the complementary Lidstone boundary conditions.
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1. Introduction
In our earlier work [1, 2] we have shown that the interpolating polynomial theory and the qualitative as well as quantitative study of boundary value problems such as existence and uniqueness of solutions, and convergence of various iterative methods are directly connected. In this paper we will extend this technique to the following complementary Lidstone boundary value problem involving an odd order differential equation
and the boundary data at the odd order derivatives
Here 
, 
but fixed, and 
is continuous at least in the interior of the domain of interest. Problem (1.1), (1.2) complements Lidstone boundary value problem (nomenclature comes from the expansion introduced by Lidstone [3] in 1929, and thoroughly characterized in terms of completely continuous functions in the works of Boas [4], Poritsky [5], Schoenberg [6–8], Whittaker [9, 10], Widder [11, 12], and others) which consists of an even-order differential equation and the boundary data at the even-order derivatives
Problem (1.3) has been a subject matter of numerous studies in the recent years [13–45], and others.
In Section 2, we will show that for a given function 
explicit representations of the interpolation polynomial 
of degree 
satisfying the conditions
and the corresponding residue term 
can be deduced rather easily from our earlier work on Lidstone polynomials [46–48]. Our method will avoid unnecessarily long procedure followed in [49] to obtain the same representations of 
and 
We will also obtain error inequalities
where the constants 
are the best possible in the sense that in (1.5) equalities hold if and only if 
is a certain polynomial. The best possible constant 
was also obtained in [49]; whereas they left the cases 
without any mention. In Section 2, we will also provide best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound.
If 
then the complementary Lidstone boundary value problem (1.1), (1.2) obviously has a unique solution 
if 
is linear, that is, 
then (1.1), (1.2) gives the possibility of interpolation by the solutions of the differential equation (1.1). In Sections 3–5, we will use inequalities (1.5) to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problem (1.1), (1.2). In Section 6, we will show the monotone convergence of Picard's iterative method. Since the proofs of most of the results in Sections 3–6 are similar to those of our previous work [1, 2] the details are omitted; however, through some simple examples it is shown how easily these results can be applied in practice.
2. Interpolating Polynomial
We begin with the following well-known results.
Lemma 2.1 (see [47]).
Let 
Then,
where 
is the Lidstone interpolating polynomial of degree 
and 
is the residue term
here
Recursively, it follows that
(
is the Bernoulli polynomial of degree 
and 
is the 
th Bernoulli number 
, 
; 
, 
, 
, 
, 
, 
, 
, 
, 
).
Lemma 2.2 (see [47]).
The following hold:
(
is the Euler polynomial of degree 
and 
is the 
th Euler number 
, 
; 
, 
, 
, 
)
Theorem 2.3.
Let 
Then,
where 
is the complementary Lidstone interpolating polynomial of degree 
and 
is the residue term
here
Remark 2.4.
From (2.4) and (2.15) it is clear that 
; 
, 
; 
, 
; 
, 
; 
, 
; 
, 
Proof.
In (2.1), we let 
and integrate both sides from 
to 
to obtain
Now, since
and, similarly
it follows that
Next since
for 
from (2.7), we get
and similarly, for 
we have
Finally, since (2.12) is exact for any polynomial of degree up to 
we find
and hence, for 
it follows that
Combining (2.23) and (2.25), we obtain
Theorem 2.5.
Let 
Then, inequalities (1.5) hold with
.
Proof.
From (2.14) and (2.8) it follows that
Now, from (2.11) and (2.13), we find
However, from (2.9), we have
Thus, from 
, 
, 
and 
we obtain
Using the above estimate in (2.29), the inequality (1.5) for 
follows.
Next, from (2.11), (2.13) and (2.14), we have
and hence in view of (2.5) and (2.9) it follows that
and similarly, by (2.5) and (2.10), we get
Remark 2.6.
From (2.13), (2.28), and the above considerations it is clear that
Remark 2.7.
Inequality (1.5) with the constants 
given in (2.27) is the best possible, as equalities hold for the function 
(polynomial of degree 
) whose complementary Lidstone interpolating polynomial 
and only for this function up to a constant factor.
Remark 2.8.
From the identity (see [47, equation (1.2.21)])
we have
and hence
We also have the estimate (see [47, equation (1.2.41)])
Thus, from (2.27), (2.38), and (2.39), we obtain
Therefore, it follows that
Combining (1.5) and (2.41), we get
Hence, if 
for a fixed 
as 
, 
converges absolutely and uniformly to 
in 
provided that there exists a constant 
and an integer 
such that 
for all 
, 
In particular, the function 
, 
satisfies the above conditions. Thus, for each fixed 
expansions
converge absolutely and uniformly in 
provided 
For 
(2.43) and (2.44), respectively, reduce to absurdities, 
and 
Thus, the condition 
is the best possible.
Remark 2.9.
If 
then
Thus, in view of 
, 
we have
Now, since 
, 
, 
from (2.6), we find
and hence by (2.15) it follows that
Using these relations in (2.47), we obtain an approximate quadrature formula
It is to be remarked that (2.50) is different from the Euler-MacLaurin formula, but the same as in [49] obtained by using different arguments. To find the error 
in (2.50), from (2.28) and (2.46) we have
Thus, it immediately follows that
From (2.52) it is clear that (2.50) is exact for any polynomial of degree at most 
Further, in (2.52) equality holds for the function 
and only for this function up to a constant factor.
We will now present two examples to illustrate the importance of (2.50) and (2.52).
Example 2.10.
Consider integrating 
over 
Here, 
, 
and 
The exact value of the integral is
In Table 1, we list the approximates of the integral using (2.50) with different values of 
the actual errors incurred, and the error bounds deduced from (2.52).
Note that 
hence the error 
when 
or 
Although the errors for other values of 
are large, ultimately the approximates tend to the exact value as 
Example 2.11.
Consider integrating 
over 
Here, 
, 
and 
The exact value of the integral is
In Table 2, we list the approximates of the integral using (2.50) with different values of 
the actual errors incurred, and the error bounds deduced from (2.52).
Unlike Example 2.10, here the error decreases as 
increases. In both examples, the approximates tend to the exact value as 
Of course, for increasing accuracy, instead of taking large values of 
one must use composite form of formula (2.50).
3. Existence and Uniqueness
The equalities and inequalities established in Section 2 will be used here to provide necessary and sufficient conditions for the existence and uniqueness of solutions of the complementary Lidstone boundary value problem (1.1), (1.2).
Theorem 3.1.
Suppose that 
, 
are given real numbers and let 
be the maximum of 
on the compact set 
where
Further, suppose that
then, the boundary value problem (1.1), (1.2) has a solution in 
Proof.
The set
is a closed convex subset of the Banach space 
We define an operator 
as follows:
In view of Theorem 2.3 and (2.28) it is clear that any fixed point of (3.4) is a solution of the boundary value problem (1.1), (1.2). Let 
Then, from (1.5), (3.2), and (3.4), we find
Thus, 
Inequalities (3.5) imply that the sets 
, 
are uniformly bounded and equicontinuous in 
Hence, 
that is compact follows from the Ascoli-Arzela theorem. The Schauder fixed point theorem is applicable and a fixed point of 
in 
exists.
Corollary 3.2.
Assume that the function 
on 
satisfies the following condition:
where 
, 
are nonnegative constants, and 
, 
then, the boundary value problem (1.1), (1.2) has a solution.
Theorem 3.3.
Suppose that the function 
on 
satisfies the following condition:
where
then, the boundary value problem (1.1), (1.2) has a solution in 
Theorem 3.4.
Suppose that the differential equation (1.1) together with the homogeneous boundary conditions
has a nontrivial solution 
and the condition (3.7) with 
is satisfied on 
where
and 
then, it is necessary that 
Remark 3.5.
Conditions of Theorem 3.4 ensure that in (3.7) at least one of the 
, 
will not be zero; otherwise the solution 
will be a polynomial of degree at most 
and will not be a nontrivial solution of (1.1), (1.2). Further, 
is obviously a solution of (1.1), (1.2), and if 
then it is also unique.
Theorem 3.6.
Suppose that for all 
the function 
satisfies the Lipschitz condition
where 
then, the boundary value problem (1.1), (1.2) has a unique solution in 
Example 3.7.
Consider the complementary Lidstone boundary value problem
where 
is fixed. Here, 
and the interpolating polynomial satisfying (1.4) is computed as 
with
We illustrate Theorem 3.1 by the following two cases.
Case 1.
Suppose 
and 
then, Theorem 3.1 states that (3.14), (3.15) has a solution in the set 
provided
We will look for a constant 
that satisfies (3.17). Since
the condition 
simplifies to 
Coupled with another condition 
we see that 
fulfills (3.17). Therefore, we conclude that the differential equation
with the boundary conditions (3.15) has a solution in 
where 
Case 2.
Suppose 
and 
then, Theorem 3.1 states that (3.14), (3.15) has a solution in the set 
, 
, 
provided
Here
and the conditions 
, 
reduce to
Pick 
, 
, 
which satisfy (3.22) and also 
, 
It follows from Theorem 3.1 that the differential equation
with the boundary conditions (3.15) has a solution in 
, 
, 
Example 3.8.
Consider the complementary Lidstone boundary value problem
with the boundary conditions (3.15). Here, 
, 
and the interpolating polynomial 
satisfying (1.4) is given in Example 3.7. To illustrate Theorem 3.3, we note that for 
and any 
Thus, condition (3.7) is satisfied with 
, 
, 
, 
The constants 
and 
are then computed as
By Theorem 3.3, problem (3.24), (3.15) has a solution in
4. Picard's and Approximate Picard's Methods
Picard's method of successive approximations has an important characteristic, namely, it is constructive; moreover, bounds of the difference between iterates and the solution are easily available. In this section, we will provide a priori as well as posteriori estimates on the Lipschitz constants so that Picard's iterative sequence 
converges to the unique solution 
of the problem (1.1), (1.2).
Definition 4.1.
A function 
is called an approximate solution of (1.1), (1.2) if there exist nonnegative constants 
and 
such that
where 
and 
are polynomials of degree 
satisfying (1.2), and
respectively.
Inequality (4.1) means that there exists a continuous function 
such that
Thus, from Theorem 2.3 the approximate solution 
can be expressed as
In what follows, we will consider the Banach space 
and for 
Theorem 4.2.
With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution 
and
(i)the function 
satisfies the Lipschitz condition (3.13) on 
where
(ii)
Then, the following hold:
(1)there exists a solution 
of (1.1), (1.2) in 
(2)
is the unique solution of (1.1), (1.2) in 
(3)the Picard iterative sequence 
defined by
where 
converges to 
with 
and
-
(4)
for any
, 
, 
In Theorem 4.2 conclusion (3) ensures that the sequence 
obtained from (4.8) converges to the solution 
of the boundary value problem (1.1), (1.2). However, in practical evaluation this sequence is approximated by the computed sequence, say, 
To find 
the function 
is approximated by 
Therefore, the computed sequence 
satisfies the recurrence relation
where 
With respect to 
we will assume the following condition.
Condition C1.
For 
obtained from (4.10), the following inequality holds:
where 
is a nonnegative constant.
Inequality (4.11) corresponds to the relative error in approximating the function 
by 
for the 
th iteration.
Theorem 4.3.
With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution 
and Condition C1 is satisfied. Further, one assumes that
(i)condition (i) of Theorem 4.2,
(ii)
(iii)
where 
then,
(1)all the conclusions (1)–(4) of Theorem 4.2 hold,
(2)the sequence 
obtained from (4.10) remains in 
(3)the sequence 
converges to 
the solution of (1.1), (1.2) if and only if 
where
and the following error estimate holds
In our next result we will assume the following.
Condition C2.
For 
obtained from (4.10), the following inequality is satisfied:
where 
is a nonnegative constant.
Inequality (4.14) corresponds to the absolute error in approximating the function 
by 
for the 
th iteration.
Theorem 4.4.
With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution 
and Condition C2 is satisfied. Further, one assumes that
(i)condition (i) of Theorem 4.2,
(ii)
then,
(1)all the conclusions (1)–(4) of Theorem 4.2 hold,
(2)the sequence 
obtained from (4.10) remains in 
(3)the sequence 
converges to 
the solution of (1.1), (1.2) if and only if 
and the following error estimate holds:
Example 4.5.
Consider the complementary Lidstone boundary value problem
with the boundary conditions (3.15). Pick 
to be an approximate solution of (4.16), (3.15), that is, let 
Then, from (4.2) we get 
Further, from (4.1) we have
To illustrate Theorem 4.2, we note that the Lipschitz condition (3.13) is satisfied globally with 
, 
, 
and the constants 
and 
are computed directly as
By Theorem 4.2, it follows that
(1)there exists a solution 
of (4.16), (3.15) in 
(2)
is the unique solution of (4.16), (3.15) in 
(3)the Picard iterative sequence 
defined by
where 
converges to 
with
Suppose that we require the accuracy 
then from above we just set
to get 
Thus, 
will fulfill the required accuracy.
Finally, we will illustrate how to obtain 
from (4.19). First, we integrate
from 
to 
to get
Next, integrating (4.23) from 
to 
as well as from 
to 
respectively, gives
Adding (4.24) and (4.25) yields 
Now, integrate (4.24) (or (4.25)) from 
to 
gives
A similar method can be used to obtain 
, 
5. Quasilinearization and Approximate Quasilinearization
Newton's method when applied to differential equations has been labeled as quasilinearization. This quasilinear iterative scheme for (1.1), (1.2) is defined as
where 
is an approximate solution of (1.1), (1.2).
In the following results once again we will consider the Banach space 
and for 
the norm 
is as in (4.6).
Theorem 5.1.
With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution 
and
(i)the function 
is continuously differentiable with respect to all 
, 
on 
(ii)there exist nonnegative constants 
, 
such that for all 
(iii)the function 
is continuous on 
, 
and 
(iv)
Then, the following hold:
(1)the sequence 
generated by the iterative scheme (5.1), (5.2) remains in 
(2)the sequence 
converges to the unique solution 
of the boundary value problem (1.1), (1.2),
(3)a bound on the error is given by
Theorem 5.2.
Let in Theorem 5.1 the function 
Further, let 
be twice continuously differentiable with respect to all 
, 
on 
and
Then,
where 
Thus, the convergence is quadratic if
Conclusion (3) of Theorem 5.1 ensures that the sequence 
generated from the scheme (5.1), (5.2) converges linearly to the unique solution 
of the boundary value problem (1.1), (1.2). Theorem 5.2 provides sufficient conditions for its quadratic convergence. However, in practical evaluation this sequence is approximated by the computed sequence, say, 
which satisfies the recurrence relation
where 
With respect to 
we will assume the following condition.
Condition C3.
is continuously differentiable with respect to all 
, 
on 
with
and Condition C1 is satisfied.
Theorem 5.3.
With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution 
and the Condition C3 is satisfied. Further, one assumes
(i)conditions (i) and (ii) of Theorem 5.1,
(ii)
(iii)
then,
(1)all conclusions (1)–(3) of Theorem 5.1 hold,
(2)the sequence 
generated by the iterative scheme (5.8), remains in 
(3)the sequence 
converges to 
the unique solution of (1.1), (1.2) if and only if 
and the following error estimate holds:
Theorem 5.4.
Let the conditions of Theorem 5.3 be satisfied. Further, let 
for all 
and 
be twice continuously differentiable with respect to all 
, 
on 
and
Then,
where 
is the same as in Theorem 5.2.
Example 5.5.
Consider the complementary Lidstone boundary value problem
again with the boundary conditions (3.15). First, we will illustrate Theorem 5.1. Pick 
and 
(so 
). Clearly, 
is continuously differentiable with respect to 
for all 
For 
we have
Thus,
Let 
so that 
Next, from (4.1) we have 
Also, from (4.2) we find
and so we take 
Now,
yields 
Coupled with 
(so that 
), we should impose
The corresponding range of 
will then be
The conditions of Theorem 5.1 are satisfied and so
(1)the sequence 
generated by
where 
remains in 
that is, 
the sequence 
converges to the unique solution 
of (5.13), (3.15) with
Next, we will illustrate Theorem 5.2. For 
we have
Hence, we may take 
From Theorem 5.2, we have
The convergence is quadratic if
which is the same as
and is satisfied if 
or 
Combining with (5.18), we conclude that the convergence of the scheme (5.20) is quadratic if
6. Monotone Convergence
It is well recognized that the method of upper and lower solutions, together with uniformly monotone convergent technique offers effective tools in proving and constructing multiple solutions of nonlinear problems. The upper and lower solutions generate an interval in a suitable partially ordered space, and serve as upper and lower bounds for solutions which can be improved by uniformly monotone convergent iterative procedures. Obviously, from the computational point of view monotone convergence has superiority over ordinary convergence. We will discuss this fruitful technique for the boundary value problem (1.1), (1.2) with 
Definition 6.1.
A function 
is said to be a lower solution of (1.1), (1.2) with 
provided
Similarly, a function 
is said to be an upper solution of (1.1), (1.2) with 
if
Lemma 6.2.
Let 
and 
be lower and upper solutions of (1.1), (1.2) with 
and let 
and 
be the polynomials of degree 
satisfying
and
respectively. Then, for all 
, 
, 
Proof.
From (2.5), (2.6), and (2.8) it is clear that 
, 
, 
and this in turn from (2.18) and (2.19) implies that 
, 
, 
, 
, 
Now, since
it follows that
Similarly, we have 
The proof of 
, 
is similar.
In the following result for 
we will consider the norm 
and introduce a partial ordering 
as follows. For 
we say that 
if and only if 
and 
for all 
Theorem 6.3.
With respect to the boundary value problem (1.1), (1.2) with 
one assumes that 
is nondecreasing in 
and 
Further, let there exist lower and upper solutions 
such that 
Then, the sequences 
where 
and 
are defined by the iterative schemes
are well defined, and 
converges to an element 
converges to an element 
(with the convergence being in the norm of 
). Further, 
, 
are solutions of (1.1), (1.2) with 
and each solution 
of this problem which is such that 
satisfies 
Example 6.4.
Consider the complementary Lidstone boundary value problem
Here, 
and the function 
is nondecreasing in 
and 
We find that (6.8) has a lower solution
and an upper solution
such that
Hence, 
and the conditions of Theorem 6.3 are satisfied. The iterative schemes
will converge respectively to some 
and 
Moreover,
and 
are solutions of (6.8). Any solution 
of (6.8) which is such that 
fulfills 
As an illustration, by direct computation (as in Example 4.5), we find
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Agarwal, R.P., Pinelas, S. & Wong, P.J.Y. Complementary Lidstone Interpolation and Boundary Value Problems. J Inequal Appl 2009, 624631 (2009). https://doi.org/10.1155/2009/624631
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DOI: https://doi.org/10.1155/2009/624631