Research

# Triple solutions of complementary Lidstone boundary value problems via fixed point theorems

Patricia JY Wong

Author Affiliations

School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

Boundary Value Problems 2014, 2014:125  doi:10.1186/1687-2770-2014-125

 Received: 25 June 2013 Accepted: 15 August 2013 Published: 20 May 2014

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We consider the following complementary Lidstone boundary value problem:

By using fixed point theorems of Leggett-Williams and Avery, we offer several criteria for the existence of three positive solutions of the boundary value problem. Examples are also included to illustrate the results obtained. We note that the nonlinear term F depends on and this derivative dependence is seldom investigated in the literature and a new technique is required to tackle the problem.

MSC: 34B15, 34B18.

##### Keywords:
positive solutions; complementary Lidstone boundary value problems; derivative-dependent nonlinearity; fixed point theorems

### 1 Introduction

In this paper we shall consider the complementary Lidstone boundary value problem

(1.1)

where and F is continuous at least in the interior of the domain of interest. It is noted that the nonlinear term F involves , a derivative of the dependent variable. Most research papers on boundary value problems consider nonlinear terms that involve y only, and derivative-dependent nonlinearities are seldom tackled as special techniques are required.

The complementary Lidstone interpolation and boundary value problems have been very recently introduced in [1], and drawn on by Agarwal et al. in [2,3] where they consider an th order differential equation together with boundary data at the odd order derivatives

(1.2)

The boundary conditions (1.2) are known as complementary Lidstone boundary conditions, they naturally complement the Lidstone boundary conditions [4-7] which involve even order derivatives. To be precise, the Lidstone boundary value problem comprises an 2mth order differential equation and the Lidstone boundary conditions

(1.3)

There is a vast literature on Lidstone interpolation and boundary value problems. In fact, the Lidstone interpolation was first introduced by Lidstone [8] in 1929 and further characterized in the work of [9-16]. More recent research on Lidstone interpolation as well as Lidstone spline can be found in [1,17-23]. Meanwhile, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4,18,24-37] and the references cited therein. In most of these works the nonlinear terms considered do not involve derivatives of the dependent variable, only a handful of papers [30,31,34,35] tackle nonlinear terms that involve even order derivatives. In the present work, our study of the complementary Lidstone boundary value problem (1.1) where F depends on a derivative certainly extends and complements the rich literature on boundary value problems and notably on Lidstone boundary value problems. The literature on complementary Lidstone boundary value problems pales in comparison with that on Lidstone boundary value problems - after the first work [2] on complementary Lidstone boundary value problems, the recent paper [38] discusses the eigenvalue problem, while in [39] the existence of at least one or two positive solutions of the complementary Lidstone boundary value problem is derived by Leray-Schauder alternative and Krasnosel’skii’s fixed point theorem in a cone.

In the present work, we shall establish the existence of at least three positive solutions using fixed point theorems of Leggett and Williams [40] as well as of Avery [41]. Estimates on the norms of these solutions will also be provided. Besides achieving new results, we also compare the results in terms of generality and illustrate the importance of the results through some examples. As remarked earlier, the presence of the derivative in the nonlinear term F requires a special technique to tackle the problem.

The paper is organized as follows. Section 2 contains the necessary definitions and fixed point theorems. The existence criteria are developed and discussed in Section 3. Finally, examples are presented in Section 4 to illustrate the importance of the results obtained.

### 2 Preliminaries

In this section we shall state some necessary definitions, the relevant fixed point theorems and properties of certain Green’s function. Let B be a Banach space equipped with the norm .

Definition 2.1 Let C (⊂B) be a nonempty closed convex set. We say that C is a cone provided the following conditions are satisfied:

(a) If and , then ;

(b) If and , then .

Definition 2.2 Let C (⊂B) be a cone. A map ψ is a nonnegative continuous concave functional on C if the following conditions are satisfied:

(a) is continuous;

(b) for all and .

Definition 2.3 Let C (⊂B) be a cone. A map β is a nonnegative continuous convex functional on C if the following conditions are satisfied:

(a) is continuous;

(b) for all and .

Let γ, β, Θ be nonnegative continuous convex functionals on C and α, ψ be nonnegative continuous concave functionals on C. For nonnegative numbers , , we shall introduce the following notations:

The following fixed point theorems are our main tools, the first is usually called Leggett-Williamsfixed point theorem, and the second is known as the five-functional fixed point theorem.

Theorem 2.1[40]

LetC (⊂B) be a cone, andbe given. Assume thatψis a nonnegative continuous concave functional onCsuch thatfor all, and letbe a continuous and completely continuous operator. Suppose that there exist numbers, , , where, such that

(a) , andfor all;

(b) for all;

(c) for allwith.

ThenShas (at least) three fixed points, andin. Furthermore, we have

(2.1)

Theorem 2.2[41]

LetC (⊂B) be a cone. Assume that there exist positive numbers, M, nonnegative continuous convex functionalsγ, β, Θ onC, and nonnegative continuous concave functionalsα, ψonC, with

for all. Letbe a continuous and completely continuous operator. Suppose that there exist nonnegative numbers, , withsuch that

(a) , andfor all;

(b) , andfor all;

(c) for allwith;

(d) for allwith.

ThenShas (at least) three fixed points, andin. Furthermore, we have

(2.2)

We also require the definition of an -Carathéodory function.

Definition 2.4[42]

A function is an -Carathéodory function if the following conditions hold:

(a) The map is measurable for all .

(b) The map is continuous for almost all .

(c) For any , there exists such that implies that for almost all .

To tackle the complementary Lidstone boundary value problem (1.1), let us review certain attributes of the Lidstone boundary value problem. Let be the Green’s function of the Lidstone boundary value problem

(2.3)

The Green’s function can be expressed as [4,5]

(2.4)

where

Further, it is known that

(2.5)

The following two lemmas give the upper and lower bounds of , they play an important role in subsequent development. We remark that the bounds in the two lemmas are sharper than those given in the literature [4,5,35,37].

Lemma 2.1[38]

For, we have

Lemma 2.2[38]

Letbe given. For, we have

### 3 Triple positive solutions

In this section, we shall use the fixed point theorems stated in Section 2 to obtain the existence of at least three positive solutions of the complementary Lidstone boundary value problem (1.1). By a positive solutiony of (1.1), we mean a nontrivial satisfying (1.1) and for .

To tackle (1.1), we first consider the initial value problem

(3.1)

whose solution is simply

(3.2)

Taking into account (3.1) and (3.2), the complementary Lidstone boundary value problem (1.1) reduces to the Lidstone boundary value problem

(3.3)

If (3.3) has a solution , then by virtue of (3.2), the boundary value problem (1.1) has a solution given by

(3.4)

So the existence of a solution of the complementary Lidstone boundary value problem (1.1) follows from the existence of a solution of the Lidstone boundary value problem (3.3). It is clear from (3.4) that ; moreover if is positive, so is . With the tools in Section 2 and a technique to handle the nonlinear term F, we shall study the boundary value problem (1.1) via (3.3).

Let the Banach space be equipped with the norm for . Define the operator by

(3.5)

where is the Green’s function given in (2.4). A fixed point of the operator S is clearly a solution of the boundary value problem (3.3), and as seen earlier is a solution of (1.1).

For easy reference, we shall list the conditions that are needed later. In these conditions the sets K and are defined by

(3.6)

(C1) is an -Carathéodory function.

(C2) We have

(C3) There exist continuous functions f, ν, μ with and such that

(C4) There exists a number such that

If (C2) and (C3) hold, then it follows from (3.5) that for and ,

(3.7)

Let be fixed. We define a cone C in B as

(3.8)

where θ is given in (C4). Clearly, we have .

Lemma 3.1Let (C1)-(C4) hold. Then the operatorSdefined in (3.5) is continuous and completely continuous, andSmapsCintoC.

Proof From (2.4) we have , and the map is continuous from to . This together with is an -Carathéodory function ensures (as in [[42], Theorem 4.2.2]) that S is continuous and completely continuous.

Let . From (3.7) we have for . Next, using (3.7) and Lemma 2.1 gives for ,

(3.9)

Hence, we have

(3.10)

Now, employing (3.7), Lemma 2.2, (C4) and (3.10), we find for ,

We have shown that . □

For subsequent results, we define the following constants for fixed and :

(3.11)

Lemma 3.2Let (C1)-(C4) hold, and assume

(C5) the functionon a subset ofof positive measure.

Suppose that there exists a numbersuch that for,

(3.12)

Then

(3.13)

Proof Let . So , which implies immediately that

Then, using (3.9), (C5) and (3.12), we find for ,

This implies . Together with the fact that (Lemma 3.1), we have shown that . Conclusion (3.13) is now immediate. □

Using a similar argument as Lemma 3.2, we have the following lemma.

Lemma 3.3Let (C1)-(C4) hold. Suppose that there exists a numbersuch that for,

Then

We are now ready to establish the existence of three positive solutions for the complementary Lidstone boundary value problem (1.1). The first result below uses Leggett-Williams’ fixed point theorem (Theorem 2.1).

Theorem 3.1Letbe fixed. Let (C1)-(C5) hold, and assume

(C6) for each, the functionon a subset ofof positive measure.

Suppose that there exist numbers, , with

such that the following hold:

(P) for;

(Q) one of the following holds:

(Q1) or;

(Q2) there exists a numberd () such thatfor;

(R) forand.

Then we have the following conclusions:

(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions (whereCis defined in (3.8)) such that

(3.14)

(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions, , such that for,

(3.15)

(wheres are those in conclusion (a)). We further have

(3.16)

Proof We shall employ Theorem 2.1 with the cone C defined in (3.8). First, we shall prove that condition (Q) implies the existence of a number , where , such that

(3.17)

Suppose that (Q2) holds. Then by Lemma 3.3 we immediately have (3.17) where we pick . Suppose now that of (Q1) is satisfied. Then there exist and such that

(3.18)

Let

Noting (3.18), it is then clear that for ,

(3.19)

Now, pick the number so that

(3.20)

Let . Using (3.9), (3.19) and (3.20) yields for ,

Hence, and so . Thus, (3.17) follows immediately. Note that the argument is similar if we assume that of (Q1) is satisfied.

Let be defined by

Clearly, ψ is a nonnegative continuous concave functional on C and for all .

We shall verify that condition (a) of Theorem 2.1 is satisfied. It is obvious that

and so . Next, let . Then and which imply

(3.21)

Using (3.7), (3.21), (C6) and (R), it follows that

Therefore, we have shown that for all .

Next, by condition (P) and Lemma 3.2 (with ), we have . Hence, condition (b) of Theorem 2.1 is fulfilled.

Finally, we shall show that condition (c) of Theorem 2.1 holds. Let with . Using (3.7), Lemma 2.2, (C4), (3.10) and the inequality , we find

Hence, we have proved that for all with .

It now follows from Theorem 2.1 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions satisfying (2.1). It is easy to see that here (2.1) reduces to (3.14). This completes the proof of conclusion (a).

Finally, it is observed from (3.4) that the complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions , , such that for ,

(3.22)

Moreover, since , we get for ,

(3.23)

Combining (3.22) and (3.23) gives (3.15) immediately.

Further, since for , we have for ,

(3.24)

Hence, noting (3.14), (3.15) and (3.24), we get (3.16). This completes the proof of conclusion (b). □

We shall now employ the five-functional fixed point theorem (Theorem 2.2) to give other existence criteria. In applying Theorem 2.2 it is possible to choose the functionals and constants in different ways, indeed we shall do so and derive two results. Our first result below turns out to be a generalization of Theorem 3.1.

Theorem 3.2Letbe fixed. Let (C1)-(C4) hold. Assume that there exist numbers, , with

such that

(C7) for each, the functionon a subset ofof positive measure;

(C8) the functionon a subset ofof positive measure.

Suppose that there exist numbers, , with

such that the following hold:

(P) forand;

(Q) for;

(R) forand.

Then we have the following conclusions:

(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions (whereCis defined in (3.8)) such that

(3.25)

(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions, , such that (3.15) holds for. We further have

(3.26)

Proof We shall apply Theorem 2.2 with the cone C defined in (3.8). We define the following five functionals on the cone C:

(3.27)

First, we shall show that the operator S maps into . Note that . By (Q) and Lemma 3.3 (with ), we immediately have .

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we note that

since it has an element . Let . Then by definition we have and , which imply

(3.28)

Noting (3.7), (3.28), (C7) and (R), we find

Hence, for all .

We shall now verify that condition (b) of Theorem 2.2 is satisfied. Let be such that . Note that

because it has an element . Let . Then we have and , i.e.,

(3.29)

(3.30)

Using (3.9), (3.29), (3.30), (C8), (P) and (Q) successively, we find

Therefore, for all .

Next, we shall show that condition (c) of Theorem 2.2 is met. Let . Clearly, we have

(3.31)

Moreover, using the fact that S maps C into C, we find

(3.32)

Combining (3.31) and (3.32) yields

(3.33)

Now, let with . Then it follows from (3.33) and the inequality that

(3.34)

Thus, for all with .

Finally, we shall prove that condition (d) of Theorem 2.2 is fulfilled. Let with . Then we have and which give (3.29) and (3.30). As in proving condition (b), we get . Hence, condition (d) of Theorem 2.2 is satisfied.

It now follows from Theorem 2.2 that the Lidstone boundary value problem (3.3) has (at least) three positive solutions satisfying (2.2). Furthermore, (2.2) reduces to (3.25) immediately. This completes the proof of conclusion (a).

Finally, as in the proof of Theorem 3.1, we see that (3.15) holds for the positive solutions , , of the complementary Lidstone boundary value problem (1.1). Moreover, noting that for , we find for ,

Next, noting for , we get for ,

Lastly, using (3.15) and , we find for ,

The proof of conclusion (b) is complete. □

We shall now consider the special case of Theorem 3.2 when

Then, from definitions (3.11), we see that

In this case Theorem 3.2 yields the following corollary.

Corollary 3.1Letbe fixed. Let (C1)-(C4) hold, and assume

(C7)′ for each, the functionon a subset ofof positive measure;

(C8)′ the functionon a subset ofof positive measure.

Suppose that there exist numbers, , with

such that the following hold:

(P) for;

(Q) for;

(R) forand.

Then we have the following conclusions:

(a) The Lidstone boundary value problem (3.3) has (at least) three positive solutions (whereCis defined in (3.8)) such that

(3.35)

(b) The complementary Lidstone boundary value problem (1.1) has (at least) three positive solutions, , such that (3.15) holds for. We further have

(3.36)

Remark 3.1 Corollary 3.1 is actually Theorem 3.1. Since Corollary 3.1 is a special case of Theorem 3.2, this shows that Theorem 3.2 is more general than Theorem 3.1.

The next theorem illustrates another application of Theorem 2.2. Compared to the conditions in Theorem 3.2, here the numbers , and have different ranges and condition (P) is also different. Note that in the proof of Theorem 3.3 the functionals ψ and Θ are chosen differently from those in Theorem 3.2.

Theorem 3.3Letbe fixed. Let (C1)-(C4) hold. Assume that there exist numbers, , with

such that (C7) and (C8) hold. Suppose that there exist numbers, , with

such that the following hold:

(P) forand;

(Q) for;

(R) forand.

Then we have conclusions (a) and (b) of Theorem 3.2.

Proof To apply Theorem 2.2, we shall define the following functionals on the cone C (see (3.8)):

(3.37)

As in the proof of Theorem 3.2, using (Q) and Lemma 3.3 we can show that .

Next, to see that condition (a) of Theorem 2.2 is fulfilled, we use (R) and a similar argument as in the proof of Theorem 3.2.

We shall now prove that condition (b) of Theorem 2.2 is satisfied. Note that

Let . Then we have , and which imply

(3.38)

and also (3.30). In view of (3.9), (3.38), (3.30), (C8), (P) and (Q), we obtain, as in the proof of Theorem 3.2, that . Therefore, condition (b) of Theorem 2.2 is fulfilled.

Next, using a similar argument as in the proof of Theorem 3.2, we see that condition (c) of Theorem 2.2 is met.

Finally, we shall verify that condition (d) of Theorem 2.2 is fulfilled. Let . It is clear that

(3.39)

Noting that S maps C into C, we find

(3.40)

A combination of (3.39) and (3.40) gives

(3.41)

Let with . Then (3.41) and the inequality lead to

Thus, for all with .

Conclusion (a) now follows from Theorem 2.2 immediately, while conclusion (b) is similarly obtained as in Theorem 3.2. □

### 4 Examples

In this section, we shall present examples to illustrate the usefulness as well as to compare the generality of the results obtained in Section 3.

Example 4.1 Consider the complementary Lidstone boundary value problem (1.1) with and the nonlinear term F given by

(4.1)

where is continuous in each argument and satisfies

(4.2)

Here, is fixed and the ’s and d are in the context of Theorem 3.1 satisfying

(4.3)

Let the functions (which implies ). Then it is clear that (C1)-(C6) are fulfilled. Moreover, by direct computation we get , and on using Lemma 2.2 we find

For convenience, we take although this will lead to more stringent conditions.

Hence, (4.3) reduces to

(4.4)

and clearly we can easily find numbers ’s and d that satisfy (4.4).

We shall check the conditions of Theorem 3.1. First, condition (P) is obviously satisfied. Next, from (4.3) we have , therefore for it follows that

Hence, condition (Q2) is met. Finally, (R) is satisfied since for , we have

By Theorem 3.1 (conclusion (b)), the boundary value problem (1.1) with , , (4.1) and (4.2) has (at least) three positive solutions , , such that (from (3.16))

(4.5)

where ’s satisfy (4.4).

Example 4.2 Consider the complementary Lidstone boundary value problem (1.1) with and the nonlinear term F given by

(4.6)

where is continuous in each argument and satisfies

(4.7)

Here, we fix

(4.8)

and the ’s are in the context of Theorem 3.2 satisfying

(4.9)

Let the functions (which implies ). Then it is clear that (C1)-(C4), (C7) and (C8) are fulfilled. Moreover, by direct computation we have

By using Lemma 2.2, we get

For convenience, we take although this will lead to more stringent conditions. Hence, (4.9) reduces to

or equivalently (combining the first two inequalities)

(4.10)

It is clear that we can easily find numbers ’s that fulfill (4.10).

We shall check the conditions of Theorem 3.2. First, condition (P) is obviously satisfied. Next, since

(4.11)

we find for ,

Hence, condition (Q) is met. Finally, (R) is satisfied since for , using (4.11) we get

By Theorem 3.2 (conclusion (b)), the boundary value problem (1.1) with , (4.6), (4.7) and (4.8) has (at least) three positive solutions , , such that (from (3.26))

(4.12)

where ’s satisfy (4.10).

Remark 4.1 In Example 4.2, we see that for ,

Thus, condition (R) of Corollary 3.1 is not satisfied and so Corollary 3.1 cannot be used to establish the existence of triple positive solutions in Example 4.2. Recalling that Corollary 3.1 is actually Theorem 3.1, this illustrates the case when Theorem 3.2 is applicable but not Theorem 3.1. Hence, this example shows that Theorem 3.2 is indeed more general than Theorem 3.1.

### Competing interests

The author declares that she has no competing interests.

### References

1. Costabile, FA, Dell’Accio, F, Luceri, R: Explicit polynomial expansions of regular real functions by means of even order Bernoulli polynomials and boundary values. J. Comput. Appl. Math.. 175, 77–99 (2005). Publisher Full Text

2. Agarwal, RP, Pinelas, S, Wong, PJY: Complementary Lidstone interpolation and boundary value problems. J. Inequal. Appl.. 2009, Article ID 624631 (2009)

3. Agarwal, RP, Wong, PJY: Piecewise complementary Lidstone interpolation and error inequalities. J. Comput. Appl. Math.. 234, 2543–2561 (2010). Publisher Full Text

4. Agarwal, RP, Wong, PJY: Lidstone polynomials and boundary value problems. Comput. Math. Appl.. 17, 1397–1421 (1989). Publisher Full Text

5. Agarwal, RP, Wong, PJY: Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer Academic, Dordrecht (1993)

6. Davis, PJ: Interpolation and Approximation, Blaisdell, Boston (1961)

7. Varma, AK, Howell, G: Best error bounds for derivatives in two point Birkhoff interpolation problem. J. Approx. Theory. 38, 258–268 (1983). Publisher Full Text

8. Lidstone, GJ: Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types. Proc. Edinb. Math. Soc.. 2, 16–19 (1929)

9. Boas, RP: A note on functions of exponential type. Bull. Am. Math. Soc.. 47, 750–754 (1941). Publisher Full Text

10. Boas, RP: Representation of functions by Lidstone series. Duke Math. J.. 10, 239–245 (1943). Publisher Full Text

11. Poritsky, H: On certain polynomial and other approximations to analytic functions. Trans. Am. Math. Soc.. 34, 274–331 (1932). Publisher Full Text

12. Schoenberg, IJ: On certain two-point expansions of integral functions of exponential type. Bull. Am. Math. Soc.. 42, 284–288 (1936). Publisher Full Text

13. Whittaker, JM: On Lidstone’s series and two-point expansions of analytic functions. Proc. Lond. Math. Soc.. 36, 451–469 (1933-1934)

14. Whittaker, JM: Interpolatory Function Theory, Cambridge University Press, Cambridge (1935)

15. Widder, DV: Functions whose even derivatives have a prescribed sign. Proc. Natl. Acad. Sci. USA. 26, 657–659 (1940). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

16. Widder, DV: Completely convex functions and Lidstone series. Trans. Am. Math. Soc.. 51, 387–398 (1942)

17. Agarwal, RP, Wong, PJY: Explicit error bounds for the derivatives of piecewise-Lidstone interpolation. J. Comput. Appl. Math.. 58, 67–81 (1995). Publisher Full Text

18. Agarwal, RP, Wong, PJY: Error bounds for the derivatives of Lidstone interpolation and applications. In: Govil NK, Mohapatra RN, Nashed Z, Sharma A, Szabados J (eds.) Approximation Theory: In Memory of A.K. Varma, pp. 1–41. Dekker, New York (1998)

19. Costabile, FA, Dell’Accio, F: Lidstone approximation on the triangle. Appl. Numer. Math.. 52, 339–361 (2005). Publisher Full Text

20. Costabile, FA, Serpe, A: An algebraic approach to Lidstone polynomials. Appl. Math. Lett.. 20, 387–390 (2007). Publisher Full Text

21. Wong, PJY: On Lidstone splines and some of their applications. Neural Parallel Sci. Comput.. 1, 472–475 (1995)

22. Wong, PJY, Agarwal, RP: Sharp error bounds for the derivatives of Lidstone-spline interpolation. Comput. Math. Appl.. 28(9), 23–53 (1994). Publisher Full Text

23. Wong, PJY, Agarwal, RP: Sharp error bounds for the derivatives of Lidstone-spline interpolation II. Comput. Math. Appl.. 31(3), 61–90 (1996). Publisher Full Text

24. Agarwal, RP, Akrivis, G: Boundary value problems occurring in plate deflection theory. J. Comput. Appl. Math.. 8, 145–154 (1982). Publisher Full Text

25. Agarwal, RP, O’Regan, D, Wong, PJY: Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht (1999)

26. Agarwal, RP, Wong, PJY: Quasilinearization and approximate quasilinearization for Lidstone boundary value problems. Int. J. Comput. Math.. 42, 99–116 (1992). Publisher Full Text

27. Baldwin, P: Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral method. Philos. Trans. R. Soc. Lond. Ser. A. 322, 281–305 (1987). Publisher Full Text

28. Baldwin, P: A localized instability in a Bénard layer. Appl. Anal.. 24, 117–156 (1987). Publisher Full Text

29. Boutayeb, A, Twizell, EH: Finite-difference methods for twelfth-order boundary value problems. J. Comput. Appl. Math.. 35, 133–138 (1991). Publisher Full Text

30. Davis, JM, Eloe, PW, Henderson, J: Triple positive solutions and dependence on higher order derivatives. J. Math. Anal. Appl.. 237, 710–720 (1999). Publisher Full Text

31. Davis, JM, Henderson, J, Wong, PJY: General Lidstone problems: multiplicity and symmetry of solutions. J. Math. Anal. Appl.. 251, 527–548 (2000). Publisher Full Text

32. Eloe, PW, Henderson, J, Thompson, HB: Extremal points for impulsive Lidstone boundary value problems. Math. Comput. Model.. 32, 687–698 (2000). Publisher Full Text

33. Forster, P: Existenzaussagen und Fehlerabschätzungen bei gewissen nichtlinearen Randwertaufgaben mit gewöhnlichen Differentialgleichungen. Numer. Math.. 10, 410–422 (1967). Publisher Full Text

34. Guo, Y, Ge, W: Twin positive symmetric solutions for Lidstone boundary value problems. Taiwan. J. Math.. 8, 271–283 (2004)

35. Ma, Y: Existence of positive solutions of Lidstone boundary value problems. J. Math. Anal. Appl.. 314, 97–108 (2006). Publisher Full Text

36. 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. Proc. R. Soc. Lond. Ser. A. 431, 433–450 (1990). Publisher Full Text

37. Yao, Q: On the positive solutions of Lidstone boundary value problems. Appl. Math. Comput.. 137, 477–485 (2003). Publisher Full Text

38. Agarwal, RP, Wong, PJY: Eigenvalues of complementary Lidstone boundary value problems. Bound. Value Probl.. 2012, Article ID 49 (2012)

39. Agarwal, RP, Wong, PJY: Positive solutions of complementary Lidstone boundary value problems. Electron. J. Qual. Theory Differ. Equ.. 2012, Article ID 60 (2012)

40. Leggett, RW, Williams, LR: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J.. 28, 673–688 (1979). Publisher Full Text

41. Avery, RI: A generalization of the Leggett-Williams fixed point theorem. Math. Sci. Res. Hot-Line. 2, 9–14 (1998)

42. O’Regan, D, Meehan, M: Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Dordrecht (1998)