# Existence of positive solutions of third-order boundary value problems with integral boundary conditions in Banach spaces

- Dan Fu
^{1}and - Wei Ding
^{1}Email author

**2013**:65

https://doi.org/10.1186/1687-1847-2013-65

© Fu and Ding; licensee Springer 2013

**Received: **13 September 2012

**Accepted: **18 January 2013

**Published: **20 March 2013

## Abstract

This paper deals with positive solutions of a third-order differential equation in ordered Banach spaces,

subject to the following integral boundary conditions:

where *θ* is the zero element of *E*, $g\in L[0,1]$ is nonnegative, $\phi :R\to R$ is an increasing and positive homomorphism, and $\phi (0)={\theta}_{1}$. The arguments are based upon the fixed-point principle in cone for strict set contraction operators. Meanwhile, as an application, we also give an example to illustrate our results.

## Keywords

## 1 Introduction

The theory of boundary value problems is experiencing a rapid development. Many methods are used to study this kind of problems such as fixed point theorems, shooting method, iterative method with upper and lower solutions, *etc.* We refer the readers to the papers [1–14]. Among them, the fixed-point principle in cone has become an important tool used in the study of existence and multiplicity of positive solutions. Many papers that use this method have been published in recent years (see [15–20]).

Recently, scientists have noticed that the boundary conditions in many areas of applied mathematics and physics come down to integral boundary conditions. For instance, the models on chemical engineering, heat conduction, thermo-elasticity, plasma physics, and underground water flow can be reduced to the nonlocal problems with integral boundary conditions. For more information about this subject, we refer the readers to the excellent survey by Gallardo [21–23], Corduneanu [24], and Agarwal and O’Regan [25].

where $f:[a,b]\times {R}^{2}\to R$ satisfies the local Carathéodory conditions and $\mu :[a,b]\to R$ is the function of bounded variation. These criteria apply to the case where the function *f* has nonintegrable singularities in the first argument at the points *a* and *b*.

where the kernel $K(t,s)$ satisfies a continuity assumption in the ${L}^{1}$-sense and it is monotone and concave. The main method is the Krasnosel’skii fixed point theorem on a suitable cone, then the above equation has at least one positive solution.

where $J=[0,1]$, $f\in C([0,1]\times P,P)$, *θ* is the zero element of *E*, *E* is a real Banach space with the norm $\parallel x\parallel $, and $g\in L[0,1]$ is nonnegative; $\phi :R\to R$ is an increasing and positive homomorphism (see Definition 1.2) and $\phi (0)={\theta}_{1}$.

According to Definition 1.2, we know that many problems, such as the problems with *p*-Laplacian operator, three-order boundary-value problems and so on, are special cases of (1). To the best of our knowledge, there have been few results on the positive solutions for odd-order boundary-value problems (or *p*-Laplacian problems) with integral boundary conditions in Banach spaces (see [26–31]).

The plan of this paper is as follows. We introduce some notations and lemmas in the rest of this section. In Section 2, we provide some necessary backgrounds. In particular, we state some properties of the Green’s function associated with BVP (1). In Section 3, we establish the main results of the paper. Finally, one example is also included to illustrate the main results.

**Definition 1.1**Let $(E,\parallel \cdot \parallel )$ be a real Banach space. A nonempty, closed, and convex set $P\subset E$ is said to be a cone provided that the following conditions are satisfied:

- (a)
If $y\in P$ and $\lambda \ge 0$, then $\lambda y\in P$;

- (b)
If $y\in P$ and $-y\in P$, then $y=0$.

If $P\subset E$ is a cone, we denote the order induced by *P* on *E* by ≤, that is, $x\le y$ if and only if $y-x\in P$. *P* is said to be normal if there exists a positive constant *N* such that $\theta \le x\le y$ implies $\parallel x\parallel \le N\parallel y\parallel $. *N* is called the normal constant of *P*.

**Definition 1.2**A projection $\phi :R\to R$ is called an increasing and positive homomorphism if the following conditions are satisfied:

- (1)
If $x\le y$, then $\phi (x)\le \phi (y)$ for all $x,y\in R$;

- (2)
*φ*is a continuous bijection and its inverse is also continuous; - (3)
$\phi (xy)=\phi (x)\phi (y)$ for all $x,y\in {R}_{+}=[0,+\mathrm{\infty})$.

- (4)
$\phi (xy)=\phi (x)\phi (y)$ for all $x,y\in R$, where $R=(-\mathrm{\infty},+\mathrm{\infty})$.

**Definition 1.3** Let *E* be a real Banach space and $P\subset E$ be a cone in *E*. If ${P}^{\ast}=\{\psi \in {E}^{\ast}|\psi (x)\ge 0,\mathrm{\forall}x\in P\}$, then ${P}^{\ast}$ is a dual cone of the cone *P*.

**Definition 1.4** Let *S* be a bounded set in a real Banach space *E*. Let $\alpha (S)$ = inf{$\delta >0:S$ be expressed as the union of a finite number of sets such that the diameter of each set does not exceed *δ*, *i.e.*, $S={\bigcup}_{i=1}^{m}{S}_{i}$ with $diam({S}_{i})\le \delta $, $i=1,2,\dots ,m$}. Clearly, $0\le \alpha (S)<\mathrm{\infty}$. $\alpha (S)$ is called Kuratowski’s measure of noncompactness.

**Definition 1.5** Let *E* be an ordered Banach space, *D* be a bounded set of *E*. The operator $A:D\to E$ is said to be a *k*-set contraction if $A:D\to E$ is continuous and bounded, and there is a constant $k\ge 0$ such that $\alpha (A(S))\le k\alpha (S)$ for any bounded $S\subset D$; a *k*-set contraction with $k<1$ is called a strict set contraction.

*E*can be found in [15–18]. For a bounded set

*C*in the Banach space

*E*, we denote by $\alpha (C)$ the Kuratowski’s measure of noncompactness. In the following, we denote by $\alpha (\cdot )$, ${\alpha}_{C}(\cdot )$ the Kuratowski’s measure of noncompactness of a bounded subset in

*E*and in $C(J,E)$, respectively. And we set

where $x\in P$, *β* denotes 0 or ∞, $\psi \in {P}^{\ast}$, $\parallel \psi \parallel =1$, and $h(t,x)={\phi}^{-1}({\int}_{0}^{t}f(s,x(s))\phantom{\rule{0.2em}{0ex}}ds)$.

**Lemma 1.1** [1]

*If* $H\in C(J,E)$ *is bounded and equicontinuous*, *then* ${\alpha}_{C}(H)=\alpha (H(J))={max}_{t\in J}\alpha (H(t))$, *where* $H(J)=\{x(t):t\in J,x\in H\}$, $H(t)=\{x(t):x\in H\}$.

**Lemma 1.2** [1]

*Let*

*D*

*be a bounded set of*

*E*;

*if*

*f*

*is uniformly continuous and bounded from*$J\times S$

*into*

*E*,

*then*

*where* ${\eta}_{l}$ *is a nonnegative constant*.

**Lemma 1.3**

*Let*

*K*

*be a cone of the Banach space*

*E*

*and*${K}_{r}=\{x\in K:\parallel x\parallel \le r\}$, ${K}_{r,{r}^{\prime}}=\{x\in K,r\le \parallel x\parallel \le {r}^{\prime}\}$

*with*${r}^{\prime}>r>0$.

*Suppose that*$A:{K}_{{r}^{\prime}}\to K$

*is a strict set contraction such that one of the following two conditions is satisfied*:

- (a)
$\parallel Ax\parallel \ge \parallel x\parallel $, $\mathrm{\forall}x\in K$, $\parallel x\parallel =r$; $\parallel Ax\parallel \le \parallel x\parallel $, $\mathrm{\forall}x\in K$, $\parallel x\parallel ={r}^{\prime}$.

- (b)
$\parallel Ax\parallel \le \parallel x\parallel $, $\mathrm{\forall}x\in K$, $\parallel x\parallel =r$; $\parallel Ax\parallel \ge \parallel x\parallel $, $\mathrm{\forall}x\in K$, $\parallel x\parallel ={r}^{\prime}$.

*Then* *A* *has a fixed point* $x\in {K}_{r,{r}^{\prime}}$.

## 2 Preliminaries

To establish the existence of positive solutions in ${C}^{3}(J,P)$ of (1), let us list the following assumptions:

(H_{0}) $f\in C(J\times P,P)$, and for any $l>0$, *f* is uniformly continuous on $J\times {P}_{l}$. Further suppose that $g\in L[0,1]$ is nonnegative, $\sigma ={\int}_{0}^{1}sg(s)\phantom{\rule{0.2em}{0ex}}ds$, and $\sigma \in [0,1)$, $\gamma =\frac{1+{\int}_{0}^{1}(1-s)g(s)\phantom{\rule{0.2em}{0ex}}ds}{1-\sigma}$, $h(s,x(s))={\phi}^{-1}({\int}_{0}^{s}f(t,x(t))\phantom{\rule{0.2em}{0ex}}dt)$, where ${P}_{l}=\{x\in P,\parallel x\parallel \le l\}$.

_{1}) There exists a nonnegative constant ${\eta}_{l}$ with $\gamma {\eta}_{l}<1$ such that

Evidently, $(C(J,E),{\parallel \cdot \parallel}_{c})$ is a Banach space, and the norm is defined as ${\parallel x\parallel}_{C}={max}_{t\in J}\parallel x(t)\parallel $.

In the following, we construct a cone $K=\{x\in Q:x(t)\ge \delta x(v),t\in {J}_{\delta},v\in [0,1]\}$, where $Q=\{x\in {C}^{3}(J,P):x(t)\ge \theta ,t\in J\}$, and let ${B}_{l}=\{x\in C(J,P):{\parallel x\parallel}_{c}\le l\}$, $l>0$. It is easy to see that *K* is a cone of ${C}^{3}(J,E)$ and ${K}_{r,{r}^{\prime}}=\{x\in K:r\le \parallel x\parallel \le {r}^{\prime}\}\subset K$, $K\subset Q$.

In our main results, we will make use of the following lemmas.

**Lemma 2.1**

*Assume that*(H

_{0})

*is satisfied*.

*Then*$x(t)$

*is a solution of problem*(1)

*if and only if*$x\in K$

*is a solution of the integral equation*

*Here*,

*we define an operator*

*A*

*by*

*That is*, *x* *is a fixed point of the operator* *A* *in* *K*.

**Lemma 2.2** *If condition* (H_{0}) *is satisfied*, *then the operator* *A* *defined by* (5) *is a continuous operator*.

*Proof* It can be verified easily by the definition of $(Ax)(t)$, we omit it here. □

**Lemma 2.3** *For* $t,s\in [0,1]$, $0\le G(t,s)\le \frac{1}{4}$.

*Proof* It is obvious that $G(t,s)\ge 0$ for any $s,t\in [0,1]$. When $0\le s\le t\le 0$, $G(t,s)=s(1-t)\le t(1-t)=-{(t-\frac{1}{2})}^{2}+\frac{1}{4}$, thus when $t=\frac{1}{2}$, ${max}_{0\le t,s\le 1}G(t,s)=\frac{1}{4}$, therefore $G(t,s)\le \frac{1}{4}$, which implies the proof is complete. □

**Lemma 2.4**

*Assume that*(H

_{0})

*holds*,

*choose*$\delta \in (0,\frac{1}{2})$

*and let*${J}_{\delta}=[\delta ,1-\delta ]$,

*then for all*$v,s\in [0,1]$,

*where* *γ* *is as defined in* (H_{0}).

*Proof* First, we prove that $G(t,s)\ge \delta G(v,s)$. Obviously, for $t\in {J}_{\delta}$, $v,s\in \{0,1\}$, $G(t,s)\ge \delta G(v,s)$ hold. And for $v,s\in (0,1)$, we have the following four cases.

To sum up, we get that $G(t,s)\ge \delta G(v,s)$.

So, we complete the proof. □

*Proof of Lemma 2.1*Necessity. First, we suppose that

*x*is a solution of equation (1). By taking the integral of (1) on $[0,t]$, we have

*t*, we have

which implies $x\in K$. To sum up, we know that *x* is a solution of the integral equation (4) in *K*.

*x*be as in (4). Taking the derivative of (4), it implies that

hold, which implies $x(t)$ is a solution of (1). The proof is complete. □

**Lemma 2.5** *Suppose that* (H_{0}) *and* (H_{1}) *hold*. *Then*, *for each* $l>0$, *A* *is a strict set contraction on* $Q\cap {B}_{l}$, *i*.*e*., *there exists a constant* $0\le {k}_{l}<1$ *such that* ${\alpha}_{C}(A(S))\le {k}_{l}{\alpha}_{C}(S)$ *for any* $S\subset Q\cap {B}_{l}$.

*Proof*By Lemmas 2.1 and 2.3, we know that $A:Q\to Q$ is continuous and bounded. Now, let

*S*be a bounded set in

*Q*. Then by (H

_{1}), we get that

where ${k}_{l}=\gamma {\eta}_{l}$, $0\le {k}_{l}<1$. The proof is complete. □

## 3 Main results

*f*which allow us to apply Lemma 1.3 to establish the existence of positive solutions of (1). At the beginning, we introduce the notation

**Theorem 3.1** *Suppose* (H_{0}) *and* (H_{1}) *hold and* *P* *is normal*. *If* $\gamma {h}^{0}<1<\mathrm{\Lambda}{(\psi h)}_{\mathrm{\infty}}$, *then problem* (1) *has at least one positive solution*.

*Proof* *A* is defined as (5). Considering $\gamma {h}^{0}<1$, there exists ${\overline{r}}_{1}$ such that $\parallel h(t,x)\parallel \le ({h}^{0}+{\epsilon}_{1})\parallel x\parallel $ for $t\in J$, $x\in K$, $\parallel x\parallel \le {\overline{r}}_{1}$, where ${\epsilon}_{1}>0$ satisfies $\gamma ({h}^{0}+{\epsilon}_{1})\le 1$.

*i.e.*, for $x\in K$, ${\parallel x\parallel}_{C}={r}_{1}$, ${\parallel Ax\parallel}_{C}\le {\parallel x\parallel}_{C}$ holds.

Next, turning to $1<\mathrm{\Lambda}{(\psi h)}_{\mathrm{\infty}}$, there exists ${\overline{r}}_{2}>0$ such that $\psi (h(t,x(t)))>({(\psi h)}_{\mathrm{\infty}}-{\epsilon}_{2})\parallel x\parallel $, $t\in J$, $x\in P$, $\parallel x\parallel \ge {\overline{r}}_{2}$, where ${\epsilon}_{2}>0$ satisfies $({(\psi h)}_{\mathrm{\infty}}-{\epsilon}_{2})\parallel x\parallel \mathrm{\Lambda}\ge 1$.

*i.e.*, for $x\in K$, ${\parallel x\parallel}_{C}={r}_{2}$, ${\parallel Ax\parallel}_{C}\ge {\parallel x\parallel}_{C}$ holds.

Lemma 1.3 yields that *A* has at least one fixed point ${x}^{\ast}\in {\overline{K}}_{{r}_{1},{r}_{2}}$, ${r}_{1}\le \parallel {x}^{\ast}\parallel \le {r}_{2}$ and ${x}^{\ast}(t)\ge \delta \parallel {x}^{\ast}\parallel >0$, $t\in {J}_{\delta}$. Thus, BVP (1) has at least one positive solution ${x}^{\ast}$. The proof is complete. □

**Remark 3.1**If $\phi =I$, where

*I*denotes a unit operator, then the differential equation can change into the general differential equation

This system has been studied in Ref. [1]. By Theorem 3.1, we can easily obtain the main result (Theorem 3.1 in Ref. [1]).

**Corollary 3.1** *Assume that* (H_{0}) *holds and* *P* *is normal*. *If* $\frac{1}{2}\gamma {f}^{0}<1<\mathrm{\Lambda}{(\psi f)}_{\mathrm{\infty}}$, *then problem* (14) *has at least one positive solution*.

**Remark 3.2**If $\phi (x)={\mathrm{\Phi}}_{p}(x)={|x|}^{p-2}x$ for some $p>1$, where ${\mathrm{\Phi}}_{P}^{-1}={\mathrm{\Phi}}_{q}$, then (1) can be written as a BVP with a

*p*-Laplace operator,

First, we reduce (H_{0}) as ${({\mathrm{H}}_{0})}^{\ast}$.

${({\mathrm{H}}_{0})}^{\ast}$ $f\in C(J\times P,P)$, and for any $l>0$, *f* is uniformly continuous on $J\times {P}_{l}$. Further suppose that $g\in L[0,1]$ is nonnegative, $\sigma ={\int}_{0}^{1}sg(s)\phantom{\rule{0.2em}{0ex}}ds$, and $\sigma \in [0,1)$, $\gamma =\frac{1+{\int}_{0}^{1}(1-s)g(s)\phantom{\rule{0.2em}{0ex}}ds}{1-\sigma}$, ${h}_{1}(s,x(s))={\mathrm{\Phi}}_{q}({\int}_{0}^{s}f(t,x(t))\phantom{\rule{0.2em}{0ex}}dt)$, where ${P}_{l}=\{x\in P,\parallel x\parallel \le l\}$.

Then we can get the following similar conclusion.

**Corollary 3.2** *Suppose* ${({\mathrm{H}}_{0})}^{\ast}$ *and* (H_{1}) *hold and* *P* *is normal*. *If* $\gamma {h}_{1}^{0}<1<\mathrm{\Lambda}{(\psi {h}_{1})}_{\mathrm{\infty}}$, *then problem* (15) *has at least one positive solution*.

## 4 Example

Next, we will give an example to illustrate our results.

**Example 4.1**

Let $E={R}^{n}=\{x=({x}_{1},{x}_{2},\dots ,{x}_{n}):{x}_{i}\in R,i=1,2,\dots ,n\}$ with the norm $\parallel x\parallel ={max}_{1\le i\le n}|{x}_{i}|$, and $P=\{x=({x}_{1},{x}_{2},\dots ,{x}_{n}):{x}_{i}\ge 0,i=1,2,\dots ,n\}$. Then we have the following result.

**Theorem 4.1** *For any* $t\in J$, *system* (16) *has at least one positive solution* $x(t)$.

*Proof* First, according to (16), (17), we have $g=1$ and ${f}_{i}(t,{x}_{i})=8t{x}_{i}^{3}$. Then, when ${x}_{i}={t}^{2}$, we have $f(t,x)=8{t}^{7}$. Moreover, we can get that ${P}^{\ast}=P$. Choose $\psi =(1,1,\dots ,1)$, then it is clear that (H_{0}) and (H_{1}) are satisfied.

Choose $\delta =\frac{1}{4}\in (0,\frac{1}{2})$, then $\mathrm{\Lambda}=\delta {\int}_{\delta}^{1-\delta}H(\frac{1}{2},s)\phantom{\rule{0.2em}{0ex}}ds=\frac{1}{4}{\int}_{\frac{1}{4}}^{\frac{3}{4}}H(\frac{1}{2},s)\phantom{\rule{0.2em}{0ex}}ds=\frac{1}{16}$.

Then $h(s,x(s))={\phi}^{-1}({\int}_{0}^{s}f(t,x(t))\phantom{\rule{0.2em}{0ex}}dt)=\sqrt{{\int}_{0}^{s}f(t,x(t))\phantom{\rule{0.2em}{0ex}}dt}={t}^{4}={x}^{2}$.

thus we have $\gamma {h}^{0}=0<1$.

which implies that $\mathrm{\Lambda}{(\psi h)}_{\mathrm{\infty}}\to \mathrm{\infty}>1$. So, all the conditions in Theorem 3.1 are satisfied, then the conclusion follows, and the proof is complete. □

## Declarations

### Acknowledgements

This work was supported by the NNSF of China under Grant (No. 11271261), Natural Science Foundation of Shanghai (No. 12ZR1421600), Shanghai municipal education commission (No. 10YZ74), and Shanghai Normal University Leading Academic Discipline project (No. DZW912).

## Authors’ Affiliations

## References

- Zhao JF, Wang PG, Ge WG: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces.
*Commun. Nonlinear Sci. Numer. Simul.*2011, 16: 402-413. 10.1016/j.cnsns.2009.10.011MathSciNetView ArticleGoogle Scholar - Zhang XM, Feng MQ, Ge WG: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces.
*J. Comput. Appl. Math.*2010, 233: 1915-1926. 10.1016/j.cam.2009.07.060MathSciNetView ArticleGoogle Scholar - Liang SH, Zhang JH: Existence of three positive solutions of three-order with
*m*-point impulsive boundary value problems.*Acta Appl. Math.*2010, 110: 1265-1292.MathSciNetView ArticleGoogle Scholar - Gupta CP: A note on a second order three-point boundary value problem.
*J. Math. Anal. Appl.*1994, 186: 277-281. 10.1006/jmaa.1994.1299MathSciNetView ArticleGoogle Scholar - Ma RY: Existence theorems for a second order three-point boundary value problem.
*J. Math. Anal. Appl.*1997, 212: 430-442. 10.1006/jmaa.1997.5515MathSciNetView ArticleGoogle Scholar - Ma RY, Castaneda N: Existence of solutions of nonlinear
*m*-point boundary-value problems.*J. Math. Anal. Appl.*2001, 256: 556-567. 10.1006/jmaa.2000.7320MathSciNetView ArticleGoogle Scholar - Guo DJ, Lakshmikanntham V: Multiple solutions of two-point boundary-value problems of ordinary differential equations in Banach spaces.
*J. Math. Anal. Appl.*1988, 129: 211-222. 10.1016/0022-247X(88)90243-0MathSciNetView ArticleGoogle Scholar - Guo DJ: Multiple positive solutions of impulsive nonlinear Fredholm integral equations and application.
*J. Math. Anal. Appl.*1993, 173: 318-324. 10.1006/jmaa.1993.1069MathSciNetView ArticleGoogle Scholar - Guo DJ: Existence of solutions of boundary value problems for second order impulsive differential equations in Banach spaces.
*J. Math. Anal. Appl.*1994, 181: 407-421. 10.1006/jmaa.1994.1031MathSciNetView ArticleGoogle Scholar - Guo DJ, Liu XZ: Multiple positive solutions of boundary value problems for impulsive differential equations.
*Nonlinear Anal.*1995, 25: 327-337. 10.1016/0362-546X(94)00175-HMathSciNetView ArticleGoogle Scholar - Guo DJ: Periodic boundary value problem for second order impulsive integro-differential equations in Banach spaces.
*Nonlinear Anal.*1997, 28: 938-997.View ArticleGoogle Scholar - Guo DJ: Second order impulsive integro-differential equations on unbounded domains in Banach spaces.
*Nonlinear Anal.*1999, 35: 413-423. 10.1016/S0362-546X(97)00564-6MathSciNetView ArticleGoogle Scholar - Wei ZL, Pang CC: Positive solutions of some singular
*m*-point boundary value problems at non-resonance.*Appl. Math. Comput.*2005, 171: 433-449. 10.1016/j.amc.2005.01.043MathSciNetView ArticleGoogle Scholar - Zhang ZX, Wang JY: The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems.
*J. Comput. Appl. Math.*2002, 147: 41-52. 10.1016/S0377-0427(02)00390-4MathSciNetView ArticleGoogle Scholar - Demling K:
*Ordinary Differential Equations in Banach Spaces*. Springer, Berlin; 1977.Google Scholar - Guo DJ: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space.
*Comput. Appl. Math.*2003, 143: 233-249. 10.1016/S0096-3003(02)00356-9View ArticleGoogle Scholar - Guo DJ, Lakshmikanntham V, Liu XZ:
*Nonlinear Integral Equations in Abstract Spaces*. Kluwer Academic, Dordrecht; 1996.View ArticleGoogle Scholar - Lakshmikanntham V, Leela S:
*Nonlinear Differential Equations in Banach Spaces*. Pergamon, Oxford; 1981.Google Scholar - Zhou YM, Su H: Positive solutions of four-point boundary value problem for higher-order with
*p*-Laplacian operator.*Electron. J. Differ. Equ.*2007., 2007: Article ID 5Google Scholar - Liang SH, Zhang JH: The existence of countably many positive solutions for nonlinear singular
*m*-point boundary value problems.*J. Comput. Appl. Math.*2005, 309: 505-516.Google Scholar - Gallardo JM: Second order differential operator with integral boundary conditions and generation of semigroups.
*Rocky Mt. J. Math.*2000, 30: 925-931.Google Scholar - Karakostas GL, Tsamatos PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems.
*Electron. J. Differ. Equ.*2002., 2002: Article ID 30Google Scholar - Lomtatidze A, Malaguti L: On a nonlocal boundary value problems for second order nonlinear singular differential equations.
*Georgian Math. J.*2000, 7: 133-154.MathSciNetGoogle Scholar - Corduneanu C:
*Integral Equations and Applications*. Cambridge University Press, Cambridge; 1991.View ArticleGoogle Scholar - Agarwal RP, O’Regan D:
*Infinite Interval Problems for Differential, Difference and Integral Equations*. Kluwer Academic, Dordrecht; 2001.View ArticleGoogle Scholar - Avery RI, Henderson J: Existence of three positive pseudo-symmetric solutions for a one-dimensional
*p*-Laplacian.*J. Comput. Appl. Math.*2003, 277: 395-404.MathSciNetGoogle Scholar - Feng M, Du B, Ge WG: Impulsive boundary value problems with integral boundary conditions and one-dimensional
*p*-Laplacian.*Nonlinear Anal.*2008. doi:10.1016/j.na.2008.04.015Google Scholar - Guo Y, Ji Y, Liu X: Multiple positive solutions for some multi-point boundary value problems with
*p*-Laplacian.*J. Comput. Appl. Math.*2008, 216: 144-156. 10.1016/j.cam.2007.04.023MathSciNetView ArticleGoogle Scholar - Li J, Shen J: Existence of three positive solutions for boundary value problems with
*p*-Laplacian.*J. Comput. Appl. Math.*2005, 311: 457-465.MathSciNetGoogle Scholar - Zhou CL, Ma DX: Existence iteration of positive solutions for a generalized right-focal boundary value problem with
*p*-Laplacian operator.*J. Comput. Appl. Math.*2008. doi:10.1016/j.cam.2008.07.039Google Scholar - Sang Y, Su H: Positive solutions of nonlinear third-order
*m*-point BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity.*J. Comput. Appl. Math.*2008. doi:10.1016/j.cam.2008.07.039Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.