Research Article | Open Access

# Triple Positive Solutions of a Nonlocal Boundary Value Problem for Singular Differential Equations with *p*-Laplacian

**Academic Editor:**Bashir Ahmad

#### Abstract

We establish the existence of triple positive solutions of an *m*-point boundary value problem for the nonlinear singular second-order differential equations of mixed type with a *p*-Laplacian operator by Leggett-William fixed point theorem. At last, we give an example to demonstrate the use of the main result of this paper. The conclusions in this paper essentially extend and improve the known results.

#### 1. Introduction

The existence and multiplicity of positive solutions for differential equations boundary value problems (BVPs) with the *p*-Laplacian operator subject to Dirichlet, Sturm-Liouville, or nonlinear boundary value conditions have been extensively investigated in recent years; see [1–10] and the references therein. Particularly, the following differential equations with one-dimensional *p*-Laplacian
have been studied subject to different kinds of boundary conditions; see [1–4] and the references therein. The methods mainly depend on Kransnosel’skii fixed point theorem, upper and lower solution technique, Leggett-Williams fixed point theorem, and some new fixed point theorems in cones, and so forth.

Recently, in [9], Kong et al. have studied the existence of triple positive solutions for the following BVP:

More recently, in [10], Hu and Ma have pointed out that the equivalent integral equation of BVP (2) is wrong in [9] and studied the existence of triple positive solutions for the following BVP:

Firstly, we confirm that the mistakes which have been pointed out in [10] exist. At the same time, we think that the value of designed in Theorem 3.1 in [10] is not suitable, since the proof needs the condition , but in fact this condition does not always hold.

Motivated by the work above, in this paper, we will study the following more extensive second-order *m*-point BVP:
where is an increasing function, , ; , , , and ; and are two linear operators defined by
in which , , , , , , , and .

Obviously, when , BVP (4) reduces to BVP (3), and when , , BVP (4) reduces to BVP (2), so BVP (2) and BVP (3) are special cases of BVPs (4).

Throughout this paper, we always suppose the following conditions hold:; may be singular at and , so it is easy to see that there exists a constant such that ; is nonincreasing and continuous, and for .

#### 2. Preliminary Results

In this section, we firstly present some definitions, theorems, and lemmas, which will be needed in the proof of the main result.

*Definition 1. *Let be a real Banach space. A nonempty closed convex set is called a cone if it satisfies the following two conditions:(i), implies ;(ii), implies .

*Definition 2. *Given a cone in a real Banach space , a continuous map is called a concave (resp., convex) functional on if and only if, for all and , it holds: , (resp., .

We consider the Banach space equipped with norm , where .

We denote, for any fixed constants , , = is concave and nonincreasing on , , . It’s easy to see that is a cone in .

Theorem 3 (Leggett-William). * Let be a completely continuous map and let be a nonnegative continuous concave functional on with for any . Suppose there exist constants , and with such that*(i)* and for all ;*(ii)* for all ;*(iii)* for all with .**
Then has at least three fixed points , , and satisfying
*

Lemma 4. *Suppose with satisfies
**
Then, is concave and nonincreasing on , that is, .*

*Proof. *Since , we know that is nonincreasing, that is, is nonincreasing, which means is concave. At the same time, we have , so , namely . Then, ; that is to say, is nonincreasing. So . Above all, . This completes the proof.

Lemma 5. *Let and , then, BVP
**
has a unique solution
**
where .**Define the operator by
**
Obviously, is well defined and is a solution of BVP (4) if and only if is a fixed point of .*

Lemma 6. * is completely continuous.*

*Proof. *It is similar to the proof of Lemma 2.2 in [9].

Lemma 7. *For any , one has , .*

*Proof. *From (10), we obtain

Since , so we have , which completes the proof.

#### 3. Main Results

For any , we define a nonnegative continuous concave function by . Obviously, the following two conclusions hold:

The main result of this paper is following.

Theorem 8. *Let and . Suppose , , and hold. Suppose further that there exist numbers and such that , and**, for × × ;**, for × × ;**, for × × , where ;** ≥ , where × .**
Then, BVP (4) has at least three positive solutions , and such that
*

*Proof. *We divide the proof into three steps.*Step 1.* We prove , ; that is, (ii) of Theorem 3. By Lemma 6, we have , so , we get , , , . For and by
Hence, and . Similarly, we obtain .*Step 2.* We show
that is, (i) of Theorem 3.

Let , then . Hence, (15) holds. For any , we have , so by , we have
Hence (16) holds.*Step 3.* We show that for all with , that is, (iii) of Theorem 3.

If with , we obtain , , , , for any , and so by , we have
Furthermore, we have
Therefore, by Lemma 7, we have
Hence, by Theorem 3, the results of Theorem 8 hold. This completes the proof of Theorem 8.

#### 4. Example

Consider the following BVP: where

*Proof. *Since and , , , , , , , and , then we can obtain , and
Next, we show that – are satisfied.

If , then
So is satisfied.

If , then
So is satisfied.

If , then
So is satisfied.

For any , we have
Hence, it’s easy to know that is satisfied.

So by Theorem 8, we conclude that the BVP (21) has three positive solutions , , and satisfying

#### Acknowledgments

This paper is supported by the Natural Science Foundation of China (10901045) and (11201112), the Natural Science Foundation of Hebei Province (A2009000664) and (A2011208012) and the Foundation of Hebei University of Science and Technology (XL200757).

#### References

- J. Wang, “The existence of positive solutions for the one-dimensional $p$-Laplacian,”
*Proceedings of the American Mathematical Society*, vol. 125, no. 8, pp. 2275–2283, 1997. View at: Publisher Site | Google Scholar | MathSciNet - L. Kong and J. Wang, “Multiple positive solutions for the one-dimensional $p$-Laplacian,”
*Nonlinear Analysis*, vol. 42, no. 8, pp. 1327–1333, 2000. View at: Publisher Site | Google Scholar | MathSciNet - X. He and W. Ge, “Twin positive solutions for the one-dimensional $p$-Laplacian boundary value problems,”
*Nonlinear Analysis*, vol. 56, no. 7, pp. 975–984, 2004. View at: Publisher Site | Google Scholar | MathSciNet - D. Zhao, H. Wang, and W. Ge, “Existence of triple positive solutions to a class of $p$-Laplacian boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 328, no. 2, pp. 972–983, 2007. View at: Publisher Site | Google Scholar | MathSciNet - A. Lakmeche and A. Hammoudi, “Multiple positive solutions of the one-dimensional $p$-Laplacian,”
*Journal of Mathematical Analysis and Applications*, vol. 317, no. 1, pp. 43–49, 2006. View at: Publisher Site | Google Scholar | MathSciNet - D.-X. Ma, “Existence and iteration of positive solution for a three-point boundary value problem with a $p$-Laplacian operator,”
*Journal of Applied Mathematics & Computing*, vol. 25, no. 1-2, pp. 329–337, 2007. View at: Publisher Site | Google Scholar | MathSciNet - H. Feng and W. Ge, “Existence of three positive solutions for $m$-point boundary-value problems with one-dimensional $p$-Laplacian,”
*Nonlinear Analysis*, vol. 68, no. 7, pp. 2017–2026, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Y. Guo, C. Yu, and J. Wang, “Existence of three positive solutions for $m$-point boundary value problems on infinite intervals,”
*Nonlinear Analysis*, vol. 71, no. 3-4, pp. 717–722, 2009. View at: Publisher Site | Google Scholar | MathSciNet - D. Kong, L. Liu, and Y. Wu, “Triple positive solutions of a boundary value problem for nonlinear singular second-order differential equations of mixed type with $p$-Laplacian,”
*Computers & Mathematics with Applications*, vol. 58, no. 7, pp. 1425–1432, 2009. View at: Publisher Site | Google Scholar | MathSciNet - J.-X. Hu and D.-X. Ma, “Triple positive solutions of a boundary value problem for second order three-point differential equations with $p$-Laplacian operator,”
*Journal of Applied Mathematics and Computing*, vol. 36, no. 1-2, pp. 251–261, 2011. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2013 Jufang Wang et al. 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.