Open Access

Complementary Lidstone Interpolation and Boundary Value Problems

Journal of Inequalities and Applications20092009:624631

https://doi.org/10.1155/2009/624631

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 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.

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

(1.1)

and the boundary data at the odd order derivatives

(1.2)

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 [68], 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

(1.3)

Problem (1.3) has been a subject matter of numerous studies in the recent years [1345], and others.

In Section 2, we will show that for a given function explicit representations of the interpolation polynomial of degree satisfying the conditions

(1.4)

and the corresponding residue term can be deduced rather easily from our earlier work on Lidstone polynomials [4648]. Our method will avoid unnecessarily long procedure followed in [49] to obtain the same representations of and We will also obtain error inequalities

(1.5)

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,
(2.1)
where is the Lidstone interpolating polynomial of degree
(2.2)
and is the residue term
(2.3)
here
(2.4)
(2.5)

Recursively, it follows that

(2.6)

( is the Bernoulli polynomial of degree and is the th Bernoulli number , ; , , , , , , , , ).

Lemma 2.2 (see [47]).

The following hold:
(2.7)
(2.8)
(2.9)
( is the Euler polynomial of degree and is the th Euler number , ; , , , )
(2.10)

Theorem 2.3.

Let Then,
(2.11)
where is the complementary Lidstone interpolating polynomial of degree
(2.12)
and is the residue term
(2.13)
here
(2.14)
(2.15)

Remark 2.4.

From (2.4) and (2.15) it is clear that ; , ; , ; , ; , ; ,
(2.16)

Proof.

In (2.1), we let and integrate both sides from to to obtain
(2.17)
Now, since
(2.18)
and, similarly
(2.19)
it follows that
(2.20)
Next since
(2.21)
for from (2.7), we get
(2.22)
and similarly, for we have
(2.23)
Finally, since (2.12) is exact for any polynomial of degree up to we find
(2.24)
and hence, for it follows that
(2.25)
Combining (2.23) and (2.25), we obtain
(2.26)

Theorem 2.5.

Let Then, inequalities (1.5) hold with
(2.27)
.

Proof.

From (2.14) and (2.8) it follows that
(2.28)
Now, from (2.11) and (2.13), we find
(2.29)
However, from (2.9), we have
(2.30)
Thus, from , , and we obtain
(2.31)

Using the above estimate in (2.29), the inequality (1.5) for follows.

Next, from (2.11), (2.13) and (2.14), we have

(2.32)
and hence in view of (2.5) and (2.9) it follows that
(2.33)
and similarly, by (2.5) and (2.10), we get
(2.34)

Remark 2.6.

From (2.13), (2.28), and the above considerations it is clear that
(2.35)

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)])
(2.36)
we have
(2.37)
and hence
(2.38)
We also have the estimate (see [47, equation (1.2.41)])
(2.39)
Thus, from (2.27), (2.38), and (2.39), we obtain
(2.40)
Therefore, it follows that
(2.41)
Combining (1.5) and (2.41), we get
(2.42)

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

(2.43)
(2.44)

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
(2.45)
(2.46)
Thus, in view of , we have
(2.47)
Now, since , , from (2.6), we find
(2.48)
and hence by (2.15) it follows that
(2.49)
Using these relations in (2.47), we obtain an approximate quadrature formula
(2.50)
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
(2.51)
Thus, it immediately follows that
(2.52)

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
(2.53)

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

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.

Consider integrating over Here, , and The exact value of the integral is
(2.54)
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).

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.

Suppose that , are given real numbers and let be the maximum of on the compact set where
(3.1)
Further, suppose that
(3.2)

then, the boundary value problem (1.1), (1.2) has a solution in

Proof.

The set
(3.3)
is a closed convex subset of the Banach space We define an operator as follows:
(3.4)
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
(3.5)

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:
(3.6)

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:
(3.7)
where
(3.8)
(3.9)
(3.10)

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
(3.11)
has a nontrivial solution and the condition (3.7) with is satisfied on where
(3.12)

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
(3.13)

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
(3.14)
(3.15)
where is fixed. Here, and the interpolating polynomial satisfying (1.4) is computed as with
(3.16)

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
(3.17)
We will look for a constant that satisfies (3.17). Since
(3.18)
the condition simplifies to Coupled with another condition we see that fulfills (3.17). Therefore, we conclude that the differential equation
(3.19)

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
(3.20)
Here
(3.21)
and the conditions , reduce to
(3.22)
Pick , , which satisfy (3.22) and also , It follows from Theorem 3.1 that the differential equation
(3.23)

with the boundary conditions (3.15) has a solution in , ,

Example 3.8.

Consider the complementary Lidstone boundary value problem
(3.24)
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
(3.25)
Thus, condition (3.7) is satisfied with , , , The constants and are then computed as
(3.26)
By Theorem 3.3, problem (3.24), (3.15) has a solution in
(3.27)

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
(4.1)
(4.2)
where and are polynomials of degree satisfying (1.2), and
(4.3)

respectively.

Inequality (4.1) means that there exists a continuous function such that

(4.4)
Thus, from Theorem 2.3 the approximate solution can be expressed as
(4.5)

In what follows, we will consider the Banach space and for

(4.6)

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
(4.7)

(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
(4.8)
where converges to with and
(4.9)
  1. (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

(4.10)

where

With respect to we will assume the following condition.

Condition C1.

For obtained from (4.10), the following inequality holds:
(4.11)

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

(4.12)
and the following error estimate holds
(4.13)

In our next result we will assume the following.

Condition C2.

For obtained from (4.10), the following inequality is satisfied:
(4.14)

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:
(4.15)

Example 4.5.

Consider the complementary Lidstone boundary value problem
(4.16)
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
(4.17)
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
(4.18)

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

(4.19)
where converges to with
(4.20)
Suppose that we require the accuracy then from above we just set
(4.21)

to get Thus, will fulfill the required accuracy.

Finally, we will illustrate how to obtain from (4.19). First, we integrate

(4.22)
from to to get
(4.23)
Next, integrating (4.23) from to as well as from to respectively, gives
(4.24)
(4.25)
Adding (4.24) and (4.25) yields Now, integrate (4.24) (or (4.25)) from to gives
(4.26)

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

(5.1)
(5.2)

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
(5.3)

(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

(5.4)

Theorem 5.2.

Let in Theorem 5.1 the function Further, let be twice continuously differentiable with respect to all , on and
(5.5)
Then,
(5.6)
where Thus, the convergence is quadratic if
(5.7)

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

(5.8)

where

With respect to we will assume the following condition.

Condition C3.

is continuously differentiable with respect to all , on with
(5.9)

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:
(5.10)

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
(5.11)
Then,
(5.12)

where is the same as in Theorem 5.2.

Example 5.5.

Consider the complementary Lidstone boundary value problem
(5.13)
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
(5.14)
Thus,
(5.15)
Let so that Next, from (4.1) we have Also, from (4.2) we find
(5.16)
and so we take Now,
(5.17)
yields Coupled with (so that ), we should impose
(5.18)
The corresponding range of will then be
(5.19)

The conditions of Theorem 5.1 are satisfied and so

(1)the sequence generated by
(5.20)

where remains in that is,

the sequence converges to the unique solution of (5.13), (3.15) with
(5.21)

Next, we will illustrate Theorem 5.2. For we have

(5.22)

Hence, we may take From Theorem 5.2, we have

(5.23)

The convergence is quadratic if

(5.24)

which is the same as

(5.25)

and is satisfied if or Combining with (5.18), we conclude that the convergence of the scheme (5.20) is quadratic if

(5.26)

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
(6.1)
Similarly, a function is said to be an upper solution of (1.1), (1.2) with if
(6.2)

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
(6.3)
and
(6.4)

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
(6.5)
it follows that
(6.6)

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
(6.7)

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
(6.8)
Here, and the function is nondecreasing in and We find that (6.8) has a lower solution
(6.9)
and an upper solution
(6.10)
such that
(6.11)
Hence, and the conditions of Theorem 6.3 are satisfied. The iterative schemes
(6.12)
(6.13)
will converge respectively to some and Moreover,
(6.14)
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
(6.15)

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Department of Mathematics, Azores University
(3)
School of ELectrical & Electronic Engineering, Nanyang Technological University
(4)
Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals

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