# Complementary Lidstone Interpolation and Boundary Value Problems

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## Abstract

We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial Open image in new window of degree Open image in new window , which involves interpolating data at the odd-order derivatives. For Open image in new window 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 Open image in new window 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 Open image in new window , Open image in new window but fixed, and Open image in new window 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, 7, 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, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45], and others.

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

and the corresponding residue term Open image in new window can be deduced rather easily from our earlier work on Lidstone polynomials [46, 47, 48]. Our method will avoid unnecessarily long procedure followed in [49] to obtain the same representations of Open image in new window and Open image in new window We will also obtain error inequalities

where the constants Open image in new window are the best possible in the sense that in (1.5) equalities hold if and only if Open image in new window is a certain polynomial. The best possible constant Open image in new window was also obtained in [49]; whereas they left the cases Open image in new window 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 Open image in new window then the complementary Lidstone boundary value problem (1.1), (1.2) obviously has a unique solution Open image in new window if Open image in new window is linear, that is, Open image in new window 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]).

Recursively, it follows that

( Open image in new window is the Bernoulli polynomial of degree Open image in new window and Open image in new window is the Open image in new window th Bernoulli number Open image in new window , Open image in new window ; Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window ).

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 Open image in new window follows.

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

Remark 2.6.

Remark 2.7.

Inequality (1.5) with the constants Open image in new window given in (2.27) is the best possible, as equalities hold for the function Open image in new window (polynomial of degree Open image in new window ) whose complementary Lidstone interpolating polynomial Open image in new window and only for this function up to a constant factor.

Remark 2.8.

Hence, if Open image in new window for a fixed Open image in new window as Open image in new window , Open image in new window converges absolutely and uniformly to Open image in new window in Open image in new window provided that there exists a constant Open image in new window and an integer Open image in new window such that Open image in new window for all Open image in new window , Open image in new window

In particular, the function Open image in new window , Open image in new window satisfies the above conditions. Thus, for each fixed Open image in new window expansions

converge absolutely and uniformly in Open image in new window provided Open image in new window For Open image in new window (2.43) and (2.44), respectively, reduce to absurdities, Open image in new window and Open image in new window Thus, the condition Open image in new window 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 Open image in new window Further, in (2.52) equality holds for the function Open image in new window 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 Open image in new window the actual errors incurred, and the error bounds deduced from (2.52).

Table 1

Approximate (2.50) | Actual error Open image in new window | 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 Open image in new window | 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 Open image in new window increases. In both examples, the approximates tend to the exact value as Open image in new window Of course, for increasing accuracy, instead of taking large values of Open image in new window 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 Open image in new window

Proof.

Thus, Open image in new window Inequalities (3.5) imply that the sets Open image in new window , Open image in new window are uniformly bounded and equicontinuous in Open image in new window Hence, Open image in new window that is compact follows from the Ascoli-Arzela theorem. The Schauder fixed point theorem is applicable and a fixed point of Open image in new window in Open image in new window exists.

Corollary 3.2.

where Open image in new window , Open image in new window are nonnegative constants, and Open image in new window , Open image in new window 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 Open image in new window

Theorem 3.4.

and Open image in new window then, it is necessary that Open image in new window

Remark 3.5.

Conditions of Theorem 3.4 ensure that in (3.7) at least one of the Open image in new window , Open image in new window will not be zero; otherwise the solution Open image in new window will be a polynomial of degree at most Open image in new window and will not be a nontrivial solution of (1.1), (1.2). Further, Open image in new window is obviously a solution of (1.1), (1.2), and if Open image in new window then it is also unique.

Theorem 3.6.

where Open image in new window then, the boundary value problem (1.1), (1.2) has a unique solution in Open image in new window

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 Open image in new window where Open image in new window

Case 2.

with the boundary conditions (3.15) has a solution in Open image in new window , Open image in new window , Open image in new window

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 Open image in new window converges to the unique solution Open image in new window of the problem (1.1), (1.2).

Definition 4.1.

*approximate solution*of (1.1), (1.2) if there exist nonnegative constants Open image in new window and Open image in new window such that

respectively.

Inequality (4.1) means that there exists a continuous function Open image in new window such that

In what follows, we will consider the Banach space Open image in new window and for Open image in new window

Theorem 4.2.

With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution Open image in new window and

Then, the following hold:

(1)there exists a solution Open image in new window of (1.1), (1.2) in Open image in new window

(2) Open image in new window is the unique solution of (1.1), (1.2) in Open image in new window

- (4)

In Theorem 4.2 conclusion (3) ensures that the sequence Open image in new window obtained from (4.8) converges to the solution Open image in new window of the boundary value problem (1.1), (1.2). However, in practical evaluation this sequence is approximated by the computed sequence, say, Open image in new window To find Open image in new window the function Open image in new window is approximated by Open image in new window Therefore, the computed sequence Open image in new window satisfies the recurrence relation

where Open image in new window

With respect to Open image in new window we will assume the following condition.

Condition C1.

where Open image in new window is a nonnegative constant.

Inequality (4.11) corresponds to the relative error in approximating the function Open image in new window by Open image in new window for the Open image in new window th iteration.

Theorem 4.3.

With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution Open image in new window and Condition C1 is satisfied. Further, one assumes that

(i)condition (i) of Theorem 4.2,

(iii) Open image in new window where Open image in new window

then,

(1)all the conclusions (1)–(4) of Theorem 4.2 hold,

(2)the sequence Open image in new window obtained from (4.10) remains in Open image in new window

(3)the sequence Open image in new window converges to Open image in new window the solution of (1.1), (1.2) if and only if Open image in new window where

In our next result we will assume the following.

Condition C2.

where Open image in new window is a nonnegative constant.

Inequality (4.14) corresponds to the absolute error in approximating the function Open image in new window by Open image in new window for the Open image in new window th iteration.

Theorem 4.4.

With respect to the boundary value problem (1.1), (1.2) one assumes that there exists an approximate solution Open image in new window and Condition C2 is satisfied. Further, one assumes that

(i)condition (i) of Theorem 4.2,

then,

(1)all the conclusions (1)–(4) of Theorem 4.2 hold,

(2)the sequence Open image in new window obtained from (4.10) remains in Open image in new window

Example 4.5.

*globally*with Open image in new window , Open image in new window , Open image in new window and the constants Open image in new window and Open image in new window are computed directly as

By Theorem 4.2, it follows that

(1)there exists a solution Open image in new window of (4.16), (3.15) in Open image in new window

(2) Open image in new window is the unique solution of (4.16), (3.15) in Open image in new window

(3)the Picard iterative sequence Open image in new window defined by

to get Open image in new window Thus, Open image in new window will fulfill the required accuracy.

Finally, we will illustrate how to obtain Open image in new window from (4.19). First, we integrate

A similar method can be used to obtain Open image in new window , Open image in new window

## 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 Open image in new window is an approximate solution of (1.1), (1.2).

In the following results once again we will consider the Banach space Open image in new window and for Open image in new window the norm Open image in new window 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 Open image in new window and

(i)the function Open image in new window is continuously differentiable with respect to all Open image in new window , Open image in new window on Open image in new window

(iii)the function Open image in new window is continuous on Open image in new window , Open image in new window and Open image in new window

Then, the following hold:

(1)the sequence Open image in new window generated by the iterative scheme (5.1), (5.2) remains in Open image in new window

(2)the sequence Open image in new window converges to the unique solution Open image in new window 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 Open image in new window generated from the scheme (5.1), (5.2) converges linearly to the unique solution Open image in new window 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, Open image in new window which satisfies the recurrence relation

where Open image in new window

With respect to Open image in new window 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 Open image in new window and the Condition C3 is satisfied. Further, one assumes

(i)conditions (i) and (ii) of Theorem 5.1,

(iii) Open image in new window

then,

(1)all conclusions (1)–(3) of Theorem 5.1 hold,

(2)the sequence Open image in new window generated by the iterative scheme (5.8), remains in Open image in new window

Theorem 5.4.

where Open image in new window is the same as in Theorem 5.2.

Example 5.5.

The conditions of Theorem 5.1 are satisfied and so

where Open image in new window remains in Open image in new window that is, Open image in new window

Next, we will illustrate Theorem 5.2. For Open image in new window we have

Hence, we may take Open image in new window From Theorem 5.2, we have

The convergence is quadratic if

which is the same as

and is satisfied if Open image in new window or Open image in new window 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 Open image in new window

Definition 6.1.

*lower solution*of (1.1), (1.2) with Open image in new window provided

*upper solution*of (1.1), (1.2) with Open image in new window if

Lemma 6.2.

respectively. Then, for all Open image in new window , Open image in new window , Open image in new window

Proof.

Similarly, we have Open image in new window The proof of Open image in new window , Open image in new window is similar.

In the following result for Open image in new window we will consider the norm Open image in new window Open image in new window and introduce a partial ordering Open image in new window as follows. For Open image in new window we say that Open image in new window if and only if Open image in new window and Open image in new window for all Open image in new window

Theorem 6.3.

are well defined, and Open image in new window converges to an element Open image in new window converges to an element Open image in new window (with the convergence being in the norm of Open image in new window ). Further, Open image in new window , Open image in new window are solutions of (1.1), (1.2) with Open image in new window and each solution Open image in new window of this problem which is such that Open image in new window satisfies Open image in new window

Example 6.4.

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