Skip to main content
  • Research Article
  • Open access
  • Published:

Triple Positive Solutions for a Type of Second-Order Singular Boundary Problems

Abstract

This paper deals with the existence of triple positive solutions for a type of second-order singular boundary problems with general differential operators. By using the Leggett-Williams fixed point theorem, we establish an existence criterion for at least three positive solutions with suitable growth conditions imposed on the nonlinear term.

1. Introduction

In this paper, we study the existence of triple positive solutions for the following second-order singular boundary value problems with general differential operators:

(1.1)

where , , and with

(1.2)

It is easy to see that and may be singular at and/or

When or and , the two kinds of singular boundary value problems have been discussed extensively in the literature; see [1–10] and the references therein. Hence, the problem that we consider is more general and is different from those in previous work.

Furthermore, we will see in the later that the presence of brings us three main difficulties:

(1) the Green's function cannot be explicitly expressed;

(2) the equivalence between BVP (1.1) and its associated integral equation has to be proved;

(3) the compactness of associated integral operator has to be verified.

We will overcome the above mentioned difficulties in Section 2. Also, although the Leggett-William fixed point theorem is used extensively in the study of triple positive differential equations, the method has not been used to study this type of second-order singular boundary value problem with general differential operators. We are concerned with solving these problems in this paper.

To state our main tool used in this paper, we give some definitions and notations.

Let be a real Banach space with a cone . A map is said to be a nonnegative continuous concave functional on if is a continuous and

(1.3)

for all and . Let be two numbers such that and a nonnegative continuous concave functional on . We define the following convex sets:

(1.4)

Theorem 1.1 (Leggett-Williams fixed point theorem).

Let be completely continuous, and let be a nonnegative continuous concave functional on such that for all . Suppose that there exist such that

(i) and , for ;

(ii) , for ;

(iii) , for with .

Then has at least three fixed points in satisfying and .

Remark 1.2.

We note the existence of triple positive solutions of other kind of boundary value problems; see He and Ge [11], Zhao et al. [12], Zhang and Liu [13], Graef et al. [14], and the references therein.

The rest of the paper is organized as follows. In Section 2, we overcome the above-mentioned difficulties in this work. The main results are formulated and proved in Section 3. Finally, an example is presented to demonstrate the application of the main theorems in Section 4.

2. Preliminaries and Lemmas

Throughout this paper, we assume the following:

(H1) ;

(H2) , and ;

(H3) is continuous and does not vanish identically on any subinterval of , and ;

(H4) is continuous.

Lemma 2.1.

Suppose that (H1) and (H2) hold. Then

(i) the initial value problem

(2.1)

has a unique solution and ;

(ii) the initial value problem

(2.2)

has a unique solution and .

Proof.

We only prove (i). (ii) can be treated in the same way.

Suppose that and is a solution of (2.1), that is,

(2.3)

Let

(2.4)

Multiplying both sides of (2.3) by , then

(2.5)

Since and , integrating (2.5) on , we have

(2.6)

Moreover, integrating (2.6) on , , we have

(2.7)

Let

(2.8)

Clearly, , and (2.7) reduces to

(2.9)

By using Fubini's theorem, we have

(2.10)

Therefore,

(2.11)

which implies that is a solution of integral equation (2.11).

Conversely, if is a solution of (2.11) with , by reversing the above argument we could deduce that the function is a solution of (2.1) and satisfy and . Therefore, to prove that (2.1) has a unique solution, , and is equivalent to prove that (2.11) has a unique solution .

To do this, we endow the following norm in :

(2.12)

Let be operator defined by

(2.13)

Since

(2.14)

then, is well defined. Set

(2.15)

Then, for any ,

(2.16)

and subsequently,

(2.17)

Thus,

(2.18)

Since , has a unique fixed point by Banach contraction principle. That is, (2.11) has a unique solution .

Remark 2.2.

Lemma 2.1 generalizes Theorem of [1], where .

Lemma 2.3.

Suppose that (H1) and (H2) hold. Then

(i) is nondecreasing in ;

(ii) is nonincreasing in .

Proof.

We only prove (i). (ii) can be treated in the same way.

Suppose on the contrary that is not nondecreasing in . Then there exists such that

(2.19)

This together with the equation implies that

(2.20)

which is a contradiction!

Remark 2.4.

From Lemmas 2.1 and 2.3, there exist positive constants , , , and such that

(2.21)

In fact, since

(2.22)

we have that and . Then, there exist constants and , such that

(2.23)

that is

(2.24)

In the following, we will show that . Suppose on the contrary, if there exist , such that

(2.25)

then, , which is a contradiction!

The other inequality can be treated in the same manner.

Lemma 2.5.

Suppose that (H1), (H2), and (H3) hold. Then

(2.26)

Proof.

We only prove the first equality; the other can be treated in the same way. From Remark 2.4 and (H3), we have

(2.27)

Lemma of [2] together with the facts that and (H3) implies that

(2.28)

Combining (2.27) and (2.28), we have

(2.29)

Lemma 2.6.

Suppose that (H1), (H2), and (H3) hold. Then the problem

(2.30)

has a unique solution

(2.31)

where

(2.32)

Moreover, on .

Proof.

By Lemma 2.3 and (2.32), we have

(2.33)

This together with Remark 2.4 implies that the right side of (2.31) is well defined.

Now we check that the function

(2.34)

satisfies (2.30). In fact,

(2.35)

Therefore,

(2.36)

Equation (2.34) and Lemma 2.5 imply that

(2.37)

Since for , then

(2.38)

Let with the norm

(2.39)

and let be a cone in defined by

(2.40)

Lemma 2.7.

Suppose that (H1)–(H3) hold and is a positive solution of (2.30). Then

(2.41)

where

(2.42)

Furthermore, for any , there exists corresponding such that

(2.43)

Proof.

In fact, if , then

(2.44)

and if , then

(2.45)

Combining this and , we have

(2.46)

Take

(2.47)

Then Lemma 2.3 guarantees that , and Lemma 2.7 guarantees that (2.43) holds.

Remark 2.8.

From Lemma 2.7 and Remark 2.4, we have

(2.48)

Now, for any , we can define the operator by

(2.49)

Lemma 2.9.

Let (H1)–(H4) hold. Then is a completely continuous operator.

Proof.

From (H3) and (H4) and Lemma 2.6, it is easy to see that , and is continuous by the Lebesgues dominated convergence theorem.

Let be any bounded set. Then (H3) and (H4) imply that is a bounded set in .

Since

(2.50)

then this together with the similar proof of Lemma of [2] yields

(2.51)

From this fact, it is easy to verify that is equicontinuous. Therefore, by the Arzela-Ascoli theorem, is a completely continuous operator.

3. Main Result

Let be nonnegative continuous concave functional defined by

(3.1)

We notice that, for each , , and also that by Lemma 2.6, is a solution of (1.1) if and only if is a fixed point of the operator .

For convenience we introduce the following notations. Let

(3.2)

Theorem 3.1.

Assume that (H1)–(H4) hold. Let , and suppose that satisfies the following conditions:

(S1) , for ;

(S2) , for ;

(S3) , for .

Then the boundary value problem (1.1) has at least three positive solutions in satisfying , and .

Proof.

From Lemma 2.9, is a completely continuous operator. If , then , and assumption (S3) implies that . Therefore

(3.3)

Hence, . In the same way, if , then . Therefore, condition (ii) of Leggett-williams fixed-point theorem holds.

To check condition (i) of Leggett-Williams fixed-point theorem, choose . It is easy to see that and . so,

(3.4)

Hence, if , then . From assumption (S2) and Remark 2.8, we have

(3.5)

Finally, we assert that if and , then . To see this, suppose that and , then

(3.6)

To sum up, all the conditions of Leggett-williams fixed-point theorem are satisfied. Therefore, has at least three fixed points, that is, problem (1.1) has at least three positive solutions in satisfying and .

Theorem 3.2.

Assume that (H1)–(H4) hold. Let , and suppose that satisfies the following conditions:

(A1) , for ;

(A2) , for .

Then the boundary value problem (1.1) has at least positive solutions.

Proof.

When , it follows from condition (A1) that , which means that has at least one fixed point by the Schauder fixed-point Theorem. When , it is clear that Theorem 3.1 holds (with ). Then we can obtain at least three positive solutions , and satisfying and with . Following this way, we finish the proof by the induction method.

4. Example

Consider the following boundary value problem:

(4.1)

where

(4.2)

Then, by computation, we have

(4.3)

Furthermore, for ,

(4.4)

In fact, let , then , and . It is easy to compute that

(4.5)

Then, , that is

(4.6)

The other inequalities in (4.4) can be proved by the same method.

Thus, we can choose that , and . By computation, we have

(4.7)

Let , and . Then, we can compute

(4.8)

Consequently,

(4.9)

Therefore, all the conditions of Theorem 3.1 are satisfied, then problem (4.1) has at least three positive solutions , and satisfying

(4.10)

References

  1. O'Regan D: Singular Dirichlet boundary value problems. I. Superlinear and nonresonant case. Nonlinear Analysis. Theory, Methods & Applications 1997,29(2):221-245. 10.1016/S0362-546X(96)00026-0

    Article  MathSciNet  MATH  Google Scholar 

  2. Asakawa H: Nonresonant singular two-point boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 2001,44(6):791-809. 10.1016/S0362-546X(99)00308-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Dalmasso R: Positive solutions of singular boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 1996,27(6):645-652. 10.1016/0362-546X(95)00073-5

    Article  MathSciNet  MATH  Google Scholar 

  4. Dunninger DR, Wang HY: Multiplicity of positive radial solutions for an elliptic system on an annulus. Nonlinear Analysis. Theory, Methods & Applications 2000,42(5):803-811. 10.1016/S0362-546X(99)00125-X

    Article  MathSciNet  MATH  Google Scholar 

  5. Ha KS, Lee Y-H: Existence of multiple positive solutions of singular boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 1997,28(8):1429-1438. 10.1016/0362-546X(95)00231-J

    Article  MathSciNet  MATH  Google Scholar 

  6. Habets P, Zanolin F: Upper and lower solutions for a generalized Emden-Fowler equation. Journal of Mathematical Analysis and Applications 1994,181(3):684-700. 10.1006/jmaa.1994.1052

    Article  MathSciNet  MATH  Google Scholar 

  7. Kim C-G, Lee Y-H: Existence and multiplicity results for nonlinear boundary value problems. Computers & Mathematics with Applications 2008,55(12):2870-2886. 10.1016/j.camwa.2007.09.007

    Article  MathSciNet  MATH  Google Scholar 

  8. Wong FH: Existence of positive solutions of singular boundary value problems. Nonlinear Analysis. Theory, Methods & Applications 1993,21(5):397-406. 10.1016/0362-546X(93)90083-5

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu X, Ma JP: A note on singular nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2004,293(1):108-124. 10.1016/j.jmaa.2003.12.017

    Article  MathSciNet  MATH  Google Scholar 

  10. Yang XJ: Positive solutions for nonlinear singular boundary value problems. Applied Mathematics and Computation 2002,130(2-3):225-234. 10.1016/S0096-3003(01)00046-7

    Article  MathSciNet  MATH  Google Scholar 

  11. He XM, Ge WG: Triple solutions for second-order three-point boundary value problems. Journal of Mathematical Analysis and Applications 2002,268(1):256-265. 10.1006/jmaa.2001.7824

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhao DX, Wang HZ, Ge WG: Existence of triple positive solutions to a class of p -Laplacian boundary value problems. Journal of Mathematical Analysis and Applications 2007,328(2):972-983. 10.1016/j.jmaa.2006.05.073

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang BG, Liu XY: 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-X

    Article  MathSciNet  MATH  Google Scholar 

  14. Graef JR, Henderson J, Wong PJY, Yang B: Three solutions of an n th order three-point focal type boundary value problem. Nonlinear Analysis. Theory, Methods & Applications 2008,69(10):3386-3404. 10.1016/j.na.2007.09.024

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

The first author was partially supported by NNSF of China (10901075).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jiemei Li.

Rights and permissions

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

Reprints and permissions

About this article

Cite this article

Li, J., Wang, J. Triple Positive Solutions for a Type of Second-Order Singular Boundary Problems. Bound Value Probl 2010, 376471 (2010). https://doi.org/10.1155/2010/376471

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2010/376471

Keywords