Complementary Lidstone Interpolation and Boundary Value Problems

  • Ravi P. Agarwal
  • Sandra Pinelas
  • Patricia J. Y. Wong
Open Access
Research Article

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]).

where Open image in new window is the Lidstone interpolating polynomial of degree Open image in new window

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]).

The following hold:

Theorem 2.3.

where Open image in new window is the complementary Lidstone interpolating polynomial of degree Open image in new window
and Open image in new window is the residue term

Remark 2.4.

Proof.

In (2.1), we let Open image in new window and integrate both sides from Open image in new window to Open image in new window to obtain
Now, since
and, similarly
it follows that
Next since
and similarly, for Open image in new window we have
Finally, since (2.12) is exact for any polynomial of degree up to Open image in new window we find
and hence, for Open image in new window it follows that
Combining (2.23) and (2.25), we obtain

Theorem 2.5.

Let Open image in new window 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

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

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 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.

From the identity (see [47, equation (1.2.21)])
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 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.

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 Open image in new window 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 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).

Example 2.11.

In Table 2, 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 2

Open image in new window

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.

Suppose that Open image in new window , Open image in new window are given real numbers and let Open image in new window be the maximum of Open image in new window on the compact set Open image in new window where
Further, suppose that

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

Proof.

is a closed convex subset of the Banach space Open image in new window We define an operator Open image in new window 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 Open image in new window Then, from (1.5), (3.2), and (3.4), we find

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.

Assume that the function Open image in new window on Open image in new window satisfies the following condition:

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.

Suppose that the function Open image in new window on Open image in new window satisfies the following condition:

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

Theorem 3.4.

Suppose that the differential equation (1.1) together with the homogeneous boundary conditions
has a nontrivial solution Open image in new window and the condition (3.7) with Open image in new window is satisfied on Open image in new window where

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.

Suppose that for all Open image in new window the function Open image in new window satisfies the Lipschitz condition

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.

Consider the complementary Lidstone boundary value problem
where Open image in new window is fixed. Here, Open image in new window and the interpolating polynomial satisfying (1.4) is computed as Open image in new window with

We illustrate Theorem 3.1 by the following two cases.

Case 1.

Suppose Open image in new window and Open image in new window then, Theorem 3.1 states that (3.14), (3.15) has a solution in the set Open image in new window provided
We will look for a constant Open image in new window that satisfies (3.17). Since
the condition Open image in new window simplifies to Open image in new window Coupled with another condition Open image in new window we see that Open image in new window fulfills (3.17). Therefore, we conclude that the differential equation

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

Case 2.

Suppose Open image in new window and Open image in new window then, Theorem 3.1 states that (3.14), (3.15) has a solution in the set Open image in new window , Open image in new window , Open image in new window provided
Pick Open image in new window , Open image in new window , Open image in new window which satisfy (3.22) and also Open image in new window , Open image in new window It follows from Theorem 3.1 that the differential equation

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.

Consider the complementary Lidstone boundary value problem
with the boundary conditions (3.15). Here, Open image in new window , Open image in new window and the interpolating polynomial Open image in new window satisfying (1.4) is given in Example 3.7. To illustrate Theorem 3.3, we note that for Open image in new window and any Open image in new window
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 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.

A function Open image in new window is called an 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
where Open image in new window and Open image in new window are polynomials of degree Open image in new window satisfying (1.2), and

respectively.

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

Thus, from Theorem 2.3 the approximate solution Open image in new window can be expressed as

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

(i)the function Open image in new window satisfies the Lipschitz condition (3.13) on Open image in new window where

(ii) Open image in new window

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

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

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.

For Open image in new window obtained from (4.10), the following inequality holds:

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,

(ii) Open image in new window

(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

and the following error estimate holds

In our next result we will assume the following.

Condition C2.

For Open image in new window obtained from (4.10), the following inequality is satisfied:

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,

(ii) 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 and the following error estimate holds:

Example 4.5.

Consider the complementary Lidstone boundary value problem
with the boundary conditions (3.15). Pick Open image in new window to be an approximate solution of (4.16), (3.15), that is, let Open image in new window Then, from (4.2) we get Open image in new window Further, from (4.1) we have
To illustrate Theorem 4.2, we note that the Lipschitz condition (3.13) is satisfied 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

Suppose that we require the accuracy Open image in new window then from above we just set

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

Next, integrating (4.23) from Open image in new window to Open image in new window as well as from Open image in new window to Open image in new window respectively, gives
Adding (4.24) and (4.25) yields Open image in new window Now, integrate (4.24) (or (4.25)) from Open image in new window to Open image in new window gives

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

(ii)there exist nonnegative constants Open image in new window , Open image in new window such that for all 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

(iv) 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.

Let in Theorem 5.1 the function Open image in new window Further, let Open image in new window be twice continuously differentiable with respect to all Open image in new window , Open image in new window on Open image in new window and
where Open image in new window Thus, the convergence is quadratic if

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,

(ii) Open image in new window

(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

(3)the sequence Open image in new window converges to Open image in new window the unique solution of (1.1), (1.2) if and only if Open image in new window and the following error estimate holds:

Theorem 5.4.

Let the conditions of Theorem 5.3 be satisfied. Further, let Open image in new window for all Open image in new window and Open image in new window be twice continuously differentiable with respect to all Open image in new window , Open image in new window on Open image in new window and

where Open image in new window 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 Open image in new window and Open image in new window (so Open image in new window ). Clearly, Open image in new window is continuously differentiable with respect to Open image in new window for all Open image in new window For Open image in new window we have
Let Open image in new window so that Open image in new window Next, from (4.1) we have Open image in new window Also, from (4.2) we find
The corresponding range of Open image in new window will then be

The conditions of Theorem 5.1 are satisfied and so

(1)the sequence Open image in new window generated by

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

the sequence Open image in new window converges to the unique solution Open image in new window of (5.13), (3.15) with

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.

A function Open image in new window is said to be a lower solution of (1.1), (1.2) with Open image in new window provided
Similarly, a function Open image in new window is said to be an upper solution of (1.1), (1.2) with Open image in new window if

Lemma 6.2.

Let Open image in new window and Open image in new window be lower and upper solutions of (1.1), (1.2) with Open image in new window and let Open image in new window and Open image in new window be the polynomials of degree Open image in new window satisfying

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

Proof.

it follows that

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.

With respect to the boundary value problem (1.1), (1.2) with Open image in new window one assumes that Open image in new window is nondecreasing in Open image in new window and Open image in new window Further, let there exist lower and upper solutions Open image in new window such that Open image in new window Then, the sequences Open image in new window where Open image in new window and Open image in new window are defined by the iterative schemes

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.

Consider the complementary Lidstone boundary value problem
Here, Open image in new window and the function Open image in new window is nondecreasing in Open image in new window and Open image in new window We find that (6.8) has a lower solution
and an upper solution
such that
Hence, Open image in new window and the conditions of Theorem 6.3 are satisfied. The iterative schemes
will converge respectively to some Open image in new window and Open image in new window Moreover,
and Open image in new window are solutions of (6.8). Any solution Open image in new window of (6.8) which is such that Open image in new window fulfills Open image in new window As an illustration, by direct computation (as in Example 4.5), we find

References

  1. 1.
    Agarwal RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Teaneck, NJ, USA; 1986:xii+307.CrossRefGoogle Scholar
  2. 2.
    Agarwal RP: Focal Boundary Value Problems for Differential and Difference Equations, Mathematics and Its Applications. Volume 436. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:x+289.CrossRefGoogle Scholar
  3. 3.
    Lidstone GJ: Notes on the extension of Aitken's theorem (for polynomial interpolation) to the Everett types. Proceedings of the Edinburgh Mathematical Society 1929, 2: 16–19.CrossRefMATHGoogle Scholar
  4. 4.
    Boas RP Jr.: Representation of functions by Lidstone series. Duke Mathematical Journal 1943, 10: 239–245. 10.1215/S0012-7094-43-01021-XMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Poritsky H: On certain polynomial and other approximations to analytic functions. Transactions of the American Mathematical Society 1932,34(2):274–331. 10.1090/S0002-9947-1932-1501639-4MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Schoenberg IJ: On certain two-point expansions of integral functions of exponential type. Bulletin of the American Mathematical Society 1936,42(4):284–288. 10.1090/S0002-9904-1936-06293-2MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Schoenberg IJ: Contributions to the problem of approximation of equidistant data by analytic functions—part A. Quarterly of Applied Mathematics 1946, 4: 45–99.MathSciNetGoogle Scholar
  8. 8.
    Schoenberg IJ: Contributions to the problem of approximation of equidistant data by analytic functions—part B. Quarterly of Applied Mathematics 1946, 4: 112–141.MathSciNetGoogle Scholar
  9. 9.
    Whittaker JM: On Lidstone's series and two-point expansions of analytic functions. Proceedings of the London Mathematical Society 1934,36(1):451–469. 10.1112/plms/s2-36.1.451MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Whittaker JM: Interpolatory Function Theory. Cambridge University Press, Cambridge, UK; 1935.MATHGoogle Scholar
  11. 11.
    Widder DV: Functions whose even derivatives have a prescribed sign. Proceedings of the National Academy of Sciences of the United States of America 1940, 26: 657–659. 10.1073/pnas.26.11.657MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Widder DV: Completely convex functions and Lidstone series. Transactions of the American Mathematical Society 1942, 51: 387–398.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Agarwal RP, Akrivis G: Boundary value problems occurring in plate deflection theory. Journal of Computational and Applied Mathematics 1982,8(3):145–154. 10.1016/0771-050X(82)90035-3MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Agarwal RP, Wong PJY: Lidstone polynomials and boundary value problems. Computers & Mathematics with Applications 1989,17(10):1397–1421. 10.1016/0898-1221(89)90023-0MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Agarwal RP, Wong PJY: Quasilinearization and approximate quasilinearization for Lidstone boundary value problems. International Journal of Computer Mathematics 1992,42(1–2):99–116. 10.1080/00207169208804054CrossRefMATHGoogle Scholar
  16. 16.
    Agarwal RP, O'Regan D, Wong PJY: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1999:xii+417.CrossRefMATHGoogle Scholar
  17. 17.
    Agarwal RP, O'Regan D: Lidstone continuous and discrete boundary value problems. Memoirs on Differential Equations and Mathematical Physics 2000, 19: 107–125.MathSciNetMATHGoogle Scholar
  18. 18.
    Agarwal RP, O'Regan D, Staněk S: Singular Lidstone boundary value problem with given maximal values for solutions. Nonlinear Analysis: Theory, Methods & Applications 2003,55(7–8):859–881. 10.1016/j.na.2003.06.001CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Avery RI, Davis JM, Henderson J: Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem. Electronic Journal of Differential Equations 2000,2000(40):1–15.MathSciNetMATHGoogle Scholar
  20. 20.
    Bai Z, Ge W: Solutions of 2Open image in new windowth Lidstone boundary value problems and dependence on higher order derivatives. Journal of Mathematical Analysis and Applications 2003,279(2):442–450. 10.1016/S0022-247X(03)00011-8MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Baldwin P: Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methods. Philosophical Transactions of the Royal Society of London. Series A 1987,322(1566):281–305. 10.1098/rsta.1987.0051MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Baldwin P: Localised instability in a Bénard layer. Applicable Analysis 1987,24(1–2):117–156. 10.1080/00036818708839658MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Boutayeb A, Twizell EH: Finite-difference methods for twelfth-order boundary-value problems. Journal of Computational and Applied Mathematics 1991,35(1–3):133–138.MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Chyan CJ, Henderson J: Positive solutions of 2Open image in new windowth-order boundary value problems. Applied Mathematics Letters 2002,15(6):767–774. 10.1016/S0893-9659(02)00040-XMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Davis JM, Eloe PW, Henderson J: Triple positive solutions and dependence on higher order derivatives. Journal of Mathematical Analysis and Applications 1999,237(2):710–720. 10.1006/jmaa.1999.6500MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Davis JM, Henderson J, Wong PJY: General Lidstone problems: multiplicity and symmetry of solutions. Journal of Mathematical Analysis and Applications 2000,251(2):527–548. 10.1006/jmaa.2000.7028MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Eloe PW, Henderson J, Thompson HB: Extremal points for impulsive Lidstone boundary value problems. Mathematical and Computer Modelling 2000,32(5–6):687–698. 10.1016/S0895-7177(00)00165-5MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Eloe PW, Islam MN: Monotone methods and fourth order Lidstone boundary value problems with impulse effects. Communications in Applied Analysis 2001,5(1):113–120.MathSciNetMATHGoogle Scholar
  29. 29.
    Forster P: Existenzaussagen und Fehlerabschätzungen bei gewissen nichtlinearen Randwertaufgaben mit gewöhnlichen Differentialgleichungen. Numerische Mathematik 1967, 10: 410–422. 10.1007/BF02162874MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Graef JR, Qian C, Yang B: Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations. Proceedings of the American Mathematical Society 2003,131(2):577–585. 10.1090/S0002-9939-02-06579-6MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Guo Y, Ge W: Twin positive symmetric solutions for Lidstone boundary value problems. Taiwanese Journal of Mathematics 2004,8(2):271–283.MathSciNetMATHGoogle Scholar
  32. 32.
    Guo Y, Gao Y: The method of upper and lower solutions for a Lidstone boundary value problem. Czechoslovak Mathematical Journal 2005,55(3):639–652. 10.1007/s10587-005-0051-8MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kiguradze T: On Lidstone boundary value problem for higher order nonlinear hyperbolic equations with two independent variables. Memoirs on Differential Equations and Mathematical Physics 2005, 36: 153–156.MathSciNetMATHGoogle Scholar
  34. 34.
    Liu Y, Ge W: Positive solutions of boundary-value problems for 2Open image in new window-order differential equations. Electronic Journal of Differential Equations 2003,2003(89):1–12.CrossRefMathSciNetGoogle Scholar
  35. 35.
    Liu Y, Ge W: Solutions of Lidstone BVPs for higher-order impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications 2005,61(1–2):191–209. 10.1016/j.na.2004.12.004MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Ma Y: Existence of positive solutions of Lidstone boundary value problems. Journal of Mathematical Analysis and Applications 2006,314(1):97–108. 10.1016/j.jmaa.2005.03.059MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Palamides PK: Positive solutions for higher-order Lidstone boundary value problems. A new approach via Sperner's lemma. Computers & Mathematics with Applications 2001,42(1–2):75–89. 10.1016/S0898-1221(01)00132-8MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Twizell EH, Boutayeb A: Numerical methods for the solution of special and general sixth-order boundary value problems, with applications to Bénard layer eigenvalue problems. Proceedings of the Royal Society of London. Series A 1990,431(1883):433–450. 10.1098/rspa.1990.0142MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wang Y-M: Higher-order Lidstone boundary value problems for elliptic partial differential equations. Journal of Mathematical Analysis and Applications 2005,308(1):314–333. 10.1016/j.jmaa.2005.01.019MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wang Y-M: On 2Open image in new windowth-order Lidstone boundary value problems. Journal of Mathematical Analysis and Applications 2005,312(2):383–400. 10.1016/j.jmaa.2005.03.039MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Wang Y-M, Jiang H-Y, Agarwal RP: A fourth-order compact finite difference method for higher-order Lidstone boundary value problems. Computers and Mathematics with Applications 2008,56(2):499–521. 10.1016/j.camwa.2007.04.051MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Wei Z: Existence of positive solutions for 2Open image in new windowth-order singular sublinear boundary value problems. Journal of Mathematical Analysis and Applications 2005,306(2):619–636. 10.1016/j.jmaa.2004.10.037MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Yao Q: Monotone iterative technique and positive solutions of Lidstone boundary value problems. Applied Mathematics and Computation 2003,138(1):1–9. 10.1016/S0096-3003(01)00316-2MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Zhang B, Liu X: Existence of multiple symmetric positive solutions of higher order Lidstone problems. Journal of Mathematical Analysis and Applications 2003,284(2):672–689. 10.1016/S0022-247X(03)00386-XMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Zhao Z: On the existence of positive solutions for 2Open image in new window-order singular boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2553–2561. 10.1016/j.na.2005.09.003MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Agarwal RP: Sharp inequalities in polynomial interpolation. In General Inequalities 6, International Series of Numerical Mathematics. Volume 103. Edited by: Walter W. Birkhäuser, Basel, Switzerland; 1992:73–92.Google Scholar
  47. 47.
    Agarwal RP, Wong PJY: Error Inequalities in Polynomial Interpolation and Their Applications. Kluwer Academic Publishers, Dodrecht, The Netherlands; 1993.CrossRefMATHGoogle Scholar
  48. 48.
    Agarwal RP, Wong PJY: Error bounds for the derivatives of Lidstone interpolation and applications. In Approximation Theory. In memory of A. K. Varma, Monogr. Textbooks Pure Appl. Math.. Volume 212. Edited by: Govil NKet al.. Marcel Dekker, New York, NY, USA; 1998:1–41.Google Scholar
  49. 49.
    Costabile FA, Dell'Accio F, Luceri R: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. Journal of Computational and Applied Mathematics 2005,176(1):77–90. 10.1016/j.cam.2004.07.004MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Ravi P. Agarwal et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
    • 4
  • Sandra Pinelas
    • 2
  • Patricia J. Y. Wong
    • 3
  1. 1.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  2. 2.Department of MathematicsAzores UniversityPonta DelgadaPortugal
  3. 3.School of ELectrical & Electronic EngineeringNanyang Technological UniversitySingapore
  4. 4.Mathematics and Statistics DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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