 Research Article
 Open Access
Complementary Lidstone Interpolation and Boundary Value Problems
 Ravi P. Agarwal^{1, 4}Email author,
 Sandra Pinelas^{2} and
 Patricia J. Y. Wong^{3}
https://doi.org/10.1155/2009/624631
© Ravi P. Agarwal et al. 2009
 Received: 21 August 2009
 Accepted: 6 November 2009
 Published: 29 December 2009
Abstract
We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial of degree , which involves interpolating data at the oddorder 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.
Keywords
 Iterative Method
 Boundary Data
 Iterative Scheme
 Quadrature Formula
 Lipschitz Condition
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 evenorder differential equation and the boundary data at the evenorder 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 wellknown results.
Lemma 2.1 (see [47]).
Recursively, it follows that
( is the Bernoulli polynomial of degree and is the th Bernoulli number , ; , , , , , , , , ).
Lemma 2.2 (see [47]).
Theorem 2.3.
Remark 2.4.
Proof.
Theorem 2.5.
Proof.
Using the above estimate in (2.29), the inequality (1.5) for follows.
Next, from (2.11), (2.13) and (2.14), we have
Remark 2.6.
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.
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.
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.
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).
Table 1
 Approximate (2.50)  Actual error  Error bound (2.52) 

1 

 91 
2 

 1001 
3 

 7293 
4 

 31031 
5 

 62881 
6 

 38227 
7 
 0  0 
Example 2.11.
Table 2
 Approximate (2.50)  Actual error  Error bound (2.52) 

1  0.822467  0.177533  2.029356 
2  0.957757  0.042243  2.002894 
3  0.989549  0.010451  2.000310 
4  0.997394  0.002606  2.000034 
5  0.999349  0.000651  2.0000038 
6  0.999837  0.000163  2.00000042 
7  0.999959  0.000041  2.000000046 
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.
then, the boundary value problem (1.1), (1.2) has a solution in
Proof.
Thus, Inequalities (3.5) imply that the sets , are uniformly bounded and equicontinuous in Hence, that is compact follows from the AscoliArzela theorem. The Schauder fixed point theorem is applicable and a fixed point of in exists.
Corollary 3.2.
where , are nonnegative constants, and , then, the boundary value problem (1.1), (1.2) has a solution.
Theorem 3.3.
then, the boundary value problem (1.1), (1.2) has a solution in
Theorem 3.4.
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.
where then, the boundary value problem (1.1), (1.2) has a unique solution in
Example 3.7.
We illustrate Theorem 3.1 by the following two cases.
Case 1.
with the boundary conditions (3.15) has a solution in where
Case 2.
with the boundary conditions (3.15) has a solution in , ,
Example 3.8.
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.
respectively.
Inequality (4.1) means that there exists a continuous function such that
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
(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
 (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.
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
In our next result we will assume the following.
Condition C2.
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
Example 4.5.
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
to get Thus, will fulfill the required accuracy.
Finally, we will illustrate how to obtain from (4.19). First, we integrate
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
(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.
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.
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
Theorem 5.4.
where is the same as in Theorem 5.2.
Example 5.5.
The conditions of Theorem 5.1 are satisfied and so
where remains in that is,
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.
Lemma 6.2.
respectively. Then, for all , ,
Proof.
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.
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.
Authors’ Affiliations
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