Open Access

Triple positive solutions for a second order m-point boundary value problem with a delayed argument

Boundary Value Problems20152015:178

https://doi.org/10.1186/s13661-015-0436-z

Received: 23 June 2015

Accepted: 10 September 2015

Published: 30 September 2015

Abstract

In this paper, we establish the new expression and properties of Green’s function for an m-point boundary value problem with a delayed argument. Furthermore, using Hölder’s inequality and a fixed point theorem due to Leggett and Williams, the existence of at least three positive solutions is also given. We discuss our problem with a delayed argument. In this case, our results cover m-point boundary value problems without delayed arguments and are compared with some recent results. An example is included to illustrate our main results.

Keywords

triple positive solutionsdelayed argumentLeggett-Williams’ fixed point theoremHölder’s inequality

1 Introduction

It is well known that multi-point boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics. For example, the vibrations of a guy wire of uniform cross-section composed of N parts of different densities can be set up as a multi-point boundary value problem. Many problems in the theory of elastic stability can be handled as multi-point boundary value problems too. Recently, the existence and multiplicity of positive solutions for multi-point boundary value problems of ordinary differential equations have received a great deal of attention. To identify a few, we refer the reader to [118] and the references therein. Recently, Sun and Liu [19] applied the Leray-Schauder nonlinear alternative to study the existence of a nontrivial solution for the problem given by
$$\left \{ \textstyle\begin{array}{@{}l} u''(t)+f(t,u)=0, \quad 0< t < 1, \\ u(0)=0, \qquad u(1)=\alpha u(\eta), \end{array}\displaystyle \right . $$
where \(\eta\in(0,1)\), \(\alpha\in R\) and \(\alpha\neq1\).
At the same time, a type of boundary value problems with deviating arguments has also received much attention. For example, in [20], Yang et al. studied the existence and multiplicity of positive solutions to a three-point boundary value problem with an advanced argument
$$ \left \{ \textstyle\begin{array}{@{}l} u^{\prime\prime}(t)+a(t)f(u(\alpha(t)))=0, \quad t\in(0,1),\\ u(0)=0,\qquad b u(\eta)=u(1), \end{array}\displaystyle \right . $$
(1.1)
where \(0<\eta<1\), \(b>0\) and \(1-b\eta>0\). The main tool is the fixed point index theory.

It is easy to see that the solution of problem (1.1) is concave when \(a(t)\geq0\) on \([0,1]\) and \(f(u)\geq0\) on \([0,\infty)\). However, few papers have reported the same problems where the solution is without concavity; for example, see some recent excellent results and applications of the case of ordinary differential equations with deviating arguments to a variety of problems from Jankowski [2123], Jiang and Wei [24], Wang [25], Wang et al. [26] and Hu et al. [27].

In the present paper, we shall investigate the existence of triple positive solutions for the following m-point boundary value problem with a delayed argument:
$$ \left \{ \textstyle\begin{array}{@{}l} Lx=\omega(t)f(t,x(\alpha(t))), \quad 0< t < 1 , \\ x'(0)=0, \qquad x(1)=\sum_{i=1}^{m-2}\beta_{i}x(\xi_{i}), \end{array}\displaystyle \right . $$
(1.2)
where \(\xi_{i}\in(0,1)\), \(\beta_{i}\in(0,+\infty)\) (\(i=1,2,\ldots, m-2\)) are given constants and L denotes the linear operator
$$Lx:=-x^{\prime\prime}-ax^{\prime}+bx, $$
here \(a\in C([0,1],[0,+\infty))\) and \(b\in C([0,1],(0,+\infty))\).
Throughout this paper, we assume that \(\alpha(t) \not\equiv t\) on \(J=[0,1] \). In addition, ω, f and \(\beta_{i}\) (\(i=1,2,\ldots, m-2\)) satisfy:
(H1): 

\(\omega\in L^{p}[0,1]\) for some \(p \in[1,+\infty)\), and there exists \(n>0\) such that \(\omega(t) \geq n\) a.e. on J;

(H2): 

\(f\in C([0,1]\times[0,+\infty),[0,+\infty))\), \(\alpha\in C(J,J)\) with \(\alpha(t)\leq t\) on J;

(H3): 
\(\sum_{i=1}^{m-2}\beta_{i}\phi(\xi_{i})<1\), where ϕ satisfies
$$ L\phi=0, \qquad \phi'(0)=0, \qquad \phi(1)=1. $$
(1.3)

Remark 1.1

By a positive solution of problem (1.2) we mean a function \(x\in C^{2}(0,1)\cap C[0,1]\) with \(x(t)>0\) on \((0,1)\) that satisfies (1.2).

Remark 1.2

Generally, when \(y(t)\geq0\) on J, the solution x is not concave for the linear equation
$$Lx-y(t)=0. $$
This means that the method depending on concavity is no longer valid, and we need to introduce a new method to study this kind of problems.

For the case \(\alpha(t)\equiv t\) on J, problem (1.2) reduces to the problem studied by Feng and Ge in [11]. By using the fixed point theorem in a cone, the authors obtained some sufficient conditions for the existence, nonexistence and multiplicity of positive solutions for problem (1.2) when \(\alpha(t)\equiv t\) on J. However, Feng and Ge did not obtain any results of triple solutions on problem (1.2). This paper will resolve this problem.

In this paper, we present several new and more general results for the existence of triple positive solutions for problem (1.2) by using Leggett-Williams’ fixed point theorem. Another contribution of this paper is to study the expression and properties of Green’s function associated with problem (1.2). The expression of the integral equation is simpler than that of [11].

The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.2). In Section 3, we present some definitions and lemmas which are useful to obtain our main results. In Section 4, we formulate sufficient conditions under which delayed problem (1.2) has at least three positive solutions. In Section 5, we provide an example to illustrate our main results.

2 Expression and properties of Green’s function

Lemma 2.1

Assume that \(\sum_{i=1}^{m-2}\beta_{i}\phi(\xi _{i})\neq1\). Then, for any \(y\in C[0,1]\), the boundary value problem
$$ \left \{ \textstyle\begin{array}{@{}l} Lx=y(t), \quad 0< t < 1 , \\ x'(0)=0, \qquad x(1)=\sum_{i=1}^{m-2}\beta_{i}x(\xi_{i}) \end{array}\displaystyle \right . $$
(2.1)
has a unique solution
$$ x(t)=\int_{0}^{1}H(t,s)q(s)y(s)\,ds, $$
(2.2)
where
$$\begin{aligned} &q(t)=\exp \biggl(\int_{0}^{t}a(s)\,ds \biggr),\qquad H(t,s)=G(t,s)+G_{1}(t,s), \\ &G(t,s)=\frac{1}{\Delta} \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \phi(s)\psi(t), &\textit{if } 0\leq s\leq t\leq1,\\ \phi(t)\psi(s), &\textit{if } 0\leq t\leq s\leq1, \end{array}\displaystyle \right . \\ &G_{1}(t,s)=\frac{\phi(t)\sum_{i=1}^{m-2}\beta_{i}G(\xi _{i},s)}{1-\sum_{i=1}^{m-2}\beta_{i}\phi(\xi_{i})}, \\ &\Delta:=-\phi(0)\psi'(0), \end{aligned}$$
(2.3)
here ϕ and ψ satisfy (1.3) and
$$ L\psi=0, \qquad \psi(0)=1, \qquad \psi(1)=0, $$
(2.4)
respectively.

Proof

First suppose that x is a solution of problem (2.1). Similar to the proof of Lemma 2.3 in [10], we can get
$$x(t)=\int_{0}^{1}G(t,s)q(s)y(s)\,ds+A\phi(t), $$
where
$$A=\frac{\sum_{i=1}^{m-2}\beta_{i}\int_{0}^{1}G(\xi_{i},s)q(s)y(s)\,ds}{ 1-\sum_{i=1}^{m-2}\beta_{i}\phi(\xi_{i})}. $$
So
$$\begin{aligned} x(t) =&\int_{0}^{1}G(t,s)q(s)y(s)\,ds+A\phi(t) \\ =&\int_{0}^{1}G(t,s)q(s)y(s)\,ds+\phi(t)\int _{0}^{1}\frac {\sum_{i=1}^{m-2}\beta_{i}G(\xi_{i},s)}{1-\sum_{i=1}^{m-2}\beta_{i}\phi(\xi_{i})}q(s)y(s)\,ds \\ =& \int_{0}^{1} \biggl[G(t,s)+\phi(t) \frac{\sum_{i=1}^{m-2}\beta_{i}G(\xi_{i},s)}{1-\sum_{i=1}^{m-2}\beta _{i}\phi(\xi_{i})} \biggr]q(s)y(s)\,ds \\ =&\int_{0}^{1} \bigl[G(t,s)+G_{1}(t,s) \bigr]q(s)y(s)\,ds \\ =&\int_{0}^{1}H(t,s)q(s)y(s)\,ds. \end{aligned}$$
Then the proof is completed. □

Remark 2.1

The proof of Lemma 2.1 is supplementary of Theorem 3 in [28], which helps the readers to understand that (2.2) holds.

Remark 2.2

The expression of the integral equation (2.2) is different from that of (2.10) in [10] and that of (2.9) in [11], which shows that we can use a completely different technique from that of [10] and [11] to study problem (1.2).

Remark 2.3

It is not difficult from [10, 11] to show that \(\Delta >0\) and that (i) ϕ is nondecreasing on J and \(\phi>0\) on J; (ii) ψ is strictly decreasing on J.

Remark 2.4

Noticing \(a(t)\in C([0,1],[0,+\infty))\), it follows from the definition of \(q(t)\) that
$$ 1\leq q(t)\leq e^{M} \quad\mbox{for } t\in J, $$
(2.5)
where
$$M=\max_{t\in J}a(t). $$

Lemma 2.2

(See [28])

Let \(\xi\in(0,1)\), \(G(t,s)\), \(G_{1}(t,s)\) and \(H(t,s)\) be given as in Lemma  2.1. Then we have the following results:
$$\begin{aligned}& G(t,s)\geq0,\qquad G_{1}(t,s)\geq0, \qquad H(t,s)\geq0, \quad \forall t,s\in J, \end{aligned}$$
(2.6)
$$\begin{aligned}& G(t,s)\leq G(s,s),\qquad G_{1}(t,s)\leq G_{1}(1,s), \qquad H(t,s)\leq H(s)\leq H^{0},\quad \forall t,s\in J, \end{aligned}$$
(2.7)
$$\begin{aligned}& \begin{aligned} &G(t,s)\geq\sigma G(s,s),\qquad G_{1}(t,s)\geq\phi(0)G_{1}(1,s), \\ &H(t,s)\geq\sigma H(s)\geq\sigma H_{0}, \quad\forall t\in[0, \xi], s\in J, \end{aligned} \end{aligned}$$
(2.8)
where
$$ \begin{aligned} &H(s)=G(s,s)+G_{1}(1,s),\qquad H^{0}=\max _{s\in J}H(s),\\ & H_{0}=\min_{s\in J}H(s), \qquad\sigma=\min\bigl\{ \psi(\xi),\phi(0)\bigr\} . \end{aligned} $$
(2.9)

Remark 2.5

From (2.8) it follows that
$$H(t,s)>\frac{1}{2}\sigma H_{0}. $$

Remark 2.6

By (1.3), (2.4) and the definition of σ, we obtain
$$0< \sigma< 1. $$

3 Preliminaries

In this section, we provide some background material from the theory of cones in Banach spaces, and then we state Hölder’s inequality and Legget-Williams’ fixed point theorem. The following definitions can be found in the book by Deimling [29] as well as in the book by Guo and Lakshmikantham [30].

Definition 3.1

Let E be a real Banach space over R. A nonempty closed set \(K\subset E\) is said to be a cone provided that the following two conditions are satisfied:
  1. (i)

    \(au+bv\in K\) for all \(u, v\in K\) and all \(a\geq0\), \(b\geq0\);

     
  2. (ii)

    \(u, -u\in K\) implies \(u=0\).

     

Note that every cone \(K\subset E\) induces an ordering in E given by \(x\leq y\) if and only if \(y-x\in K\).

Definition 3.2

A map Λ is said to be a nonnegative continuous concave functional on a cone K of a real Banach space E if \(\Lambda:K\rightarrow R_{+}\) is continuous and
$$\Lambda\bigl(tx+(1-t)y\bigr)\geq t\Lambda(x)+(1-t)\Lambda(y) $$
for all \(x, y\in K\) and \(t\in J\).

Definition 3.3

An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.

Lemma 3.1

(Arzelà-Ascoli)

A set \(M\subset C(J,R)\) is said to be a pre-compact set provided that the following two conditions are satisfied:
  1. (i)

    All the functions in the set M are uniformly bounded. It means that there exists a constant \(r>0\) such that \(|u(t)|\leq r\), \(\forall t\in J\), \(u\in M\);

     
  2. (ii)
    All the functions in the set M are equicontinuous. It means that for every \(\varepsilon>0\), there is \(\delta=\delta(\varepsilon )>0\), which is independent of the function u, such that
    $$\bigl|u(t_{1})-u(t_{2})\bigr|< \varepsilon $$
    whenever \(|t_{1}-t_{2}|<\delta\), \(t_{1}, t_{2}\in J\).
     

Lemma 3.2

(Hölder)

Let \(u\in L^{p}[a,b]\) and \(v\in L^{q}[a,b]\), where \(p,q\in(0,+\infty)\) and \(\frac{1}{p}+\frac {1}{q}=1\). Then \(uv\in L^{1}[a,b]\) and
$$\|uv\|_{1}\leq\|u\|_{p}\|v\|_{q}. $$
Let \(u\in L^{1}[a,b]\), \(v\in L^{\infty}[a,b]\). Then \(uv\in L^{1}[a,b]\) and
$$\|uv\|_{1}\leq\|u\|_{1}\|v\|_{\infty}. $$
The basic space used in this paper is \(E=C[0,1]\). It is well known that E is a real Banach space with the norm \(\|\cdot\|\) defined by
$$\|x\|=\max_{t\in J}\bigl|x(t)\bigr|. $$
Define a cone K in E by
$$ K= \Bigl\{ x\in E:x(t)\geq0,\min_{t\in[0,\xi]}x(t)\geq\sigma\|x\| \Bigr\} . $$
(3.1)
Define an operator \(T:K \rightarrow K\) by
$$ (Tx) (t)=\int_{0}^{1}H(t,s)q(s)\omega(s)f \bigl(s,x\bigl(\alpha(s)\bigr)\bigr)\,ds. $$
(3.2)

Lemma 3.3

Assume that (H1)-(H3) hold. Then \(T(K)\subset K\) and \(T:K\rightarrow K\) is completely continuous.

Proof

For \(x\in K\), it follows from (2.7) and (3.2) that
$$\begin{aligned} \|Tx\|&=\max_{t\in J}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ &\leq\int_{0}^{1}H(s)q(s)\omega(s)f\bigl(s,x \bigl(\alpha(s)\bigr)\bigr)\,ds. \end{aligned}$$
(3.3)
It follows from (2.8), (3.2) and (3.3) that
$$\begin{aligned} \min_{t\in[0,\xi]}(Tx) (t)&=\min_{t\in[0,\xi]}\int _{0}^{1}H(t,s)q(s)\omega(s)f\bigl(s,x\bigl( \alpha(s)\bigr)\bigr)\,ds \\ &\geq\sigma\int_{0}^{1}H(s)q(s)\omega(s)f \bigl(s,x\bigl(\alpha(s)\bigr)\bigr)\,ds \\ &\geq\sigma\|Tx\|. \end{aligned}$$
Thus, \(T(K)\subset K\).

Next we shall show that operator T is completely continuous. We break the proof into several steps.

Step 1. Operator T is continuous. Since the function \(f(t,x)\) is continuous on \(J\times[0,+\infty)\), this conclusion can be easily obtained.

Step 2. For each constant \(l>0\), let \(B_{l}=\{x\in K:\|x\|\leq l\}\). Then \(B_{l}\) is a bounded closed convex set in K. \(\forall x\in B_{l}\), from (3.2), we have
$$\begin{aligned} \|Tx\|&=\max_{t\in J}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ &\leq\int_{0}^{1}H(s)q(s)\omega(s)f\bigl(s,x \bigl(\alpha(s)\bigr)\bigr)\,ds \\ &\leq e^{M}\int_{0}^{1}H(s)\omega(s)f \bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ &\leq e^{M}\|H\|_{q}\|\omega\|_{p}L, \end{aligned}$$
where \(L= \mathop{\sup} _{t \in J, \|x\| \leq l } f(s,x(\alpha(s)))\). This proves that \(T(B_{r})\) is uniformly bounded.
Step 3. The family \(\{Tx: x\in B_{l}\}\) is a family of equicontinuous functions. Since \(H(t,s)\) is continuous on \(J\times J\), and noticing \(J=[0,1]\), \(H(t,s)\) is uniformly continuous on \(J\times J\). Therefore, for all \(\varepsilon>0\), there exists \(l>0\), when \(|t_{1}-t_{2}|< l\), such that
$$\bigl|H(t_{1},s)-H(t_{2},s)\bigr|< \frac{\varepsilon}{e^{M}\| \omega\|_{1}|L}. $$
Then, for all \(x\in B_{l}\), when \(|t_{1}-t_{2}|<\delta\), we get
$$\begin{aligned} \bigl|{Tx({t_{1}}) - Tx({t_{2}})}\bigr| ={}& \biggl| \int _{0}^{1} H(t_{1},s)q(s)\omega (s)f \bigl({s},x\bigl(\alpha({s})\bigr)\bigr)\,ds \\ &{}- \int_{0}^{1} {H(t_{2},s)q(s)\omega (s)f\bigl({s},x\bigl(\alpha({s})\bigr)\bigr)\,ds} \biggr| \\ ={}&\biggl|{\int_{0}^{1} \bigl[{H(t_{1},s)-H(t_{2},s) \bigr]q(s)\omega(s)f\bigl({s},x\bigl(\alpha({s})\bigr)\bigr)\,ds} } \biggr| \\ \le{}& {e^{M}}\|\omega\| _{1} L \int_{0}^{1} \bigl|H(t_{1},s)-H(t_{2},s)\bigr|\,ds \\ < {}& \varepsilon. \end{aligned}$$
Thus, the set \(\{Tx: x\in B_{l}\}\) is equicontinuous.

As a consequence of Step 1 to Step 3 together with Lemma 3.1, we can prove that \(T: K\rightarrow K\) is completely continuous. □

Remark 3.1

From Lemma 2.1 and (3.1), we know that \(x\in E\) is a solution of problem (1.2) if and only if x is a fixed point of operator T.

Let \(0< r_{1}< r_{2}\) be given and let β be a nonnegative continuous concave functional on the cone K. Define the convex sets \(K_{r_{1}}\), \(K(\beta,r_{1},r_{2})\) by
$$\begin{aligned}& K_{r_{1}}=\bigl\{ x\in K:\|x\|< r_{1}\bigr\} , \\& K(\beta,r_{1},r_{2})=\bigl\{ x\in K: r_{1}\leq \beta(x), \|x\|\leq r_{2}\bigr\} . \end{aligned}$$

Finally we state Leggett-Williams’ fixed point theorem [31].

Lemma 3.4

Let K be a cone in a real Banach space E, \(A:\bar{K}_{c}\rightarrow\bar{K}_{c}\) be completely continuous and β be a nonnegative continuous concave functional on K with \(\beta(x)\leq\|x\|\) (\(\forall x\in\bar{K}_{c}\)). Suppose that there exist \(0< d < a < b \leq c\) such that
  1. (i)

    \(\{x\in K(\beta,a,b): \beta(x)>a\} \neq\emptyset \) and \(\beta(Ax)>a\) for \(x\in K(\beta,a,b)\);

     
  2. (ii)

    \(\|Ax\|< d\) for \(\|x\|\leq d\);

     
  3. (iii)

    \(\beta(Ax)>a\) for \(x\in K(\beta,a,c)\) with \(\|Ax\|>b\).

     
Then A has at least three fixed points \(x_{1}\), \(x_{2}\), \(x_{3}\) satisfying
$$\|x_{1}\|< d,\qquad a< \beta(x_{2}),\qquad \|x_{3}\|>d\quad \textit{and}\quad \beta(x_{3})< a. $$

4 Existence of triple positive solutions

In this section, we apply Lemma 3.2 and Lemma 3.4 to establish the existence of three positive solutions for problem (1.2). We consider the following three cases for \(\omega\in L^{p}[0,1]:p> 1\), \(p=1\), and \(p=\infty\). Case \(p>1\) is treated in the following theorem.

For convenience, we write
$$\Gamma= e^{M}\|H\|_{q}\|\omega\|_{p},\qquad \Psi= \frac{1}{2}n\sigma H_{0}. $$
Let the nonnegative continuous concave functional Λ on the cone K be defined by
$$\Lambda(x)=\min_{[0,\xi]}\bigl|x(t)\bigr|. $$
Note that for \(x\in K\), \(\Lambda(x)\leq\|x\|\).

Theorem 4.1

Assume that (H1)-(H3) hold. In addition, there exist constants \(0< d< l<\frac{l}{\sigma}\leq c\) such that
(H4): 

\(f(t,x)\leq\frac{c}{\Gamma}\) for \((t,x)\in J\times[0,c]\);

(H5): 

\(f(t,x)\geq\frac{l}{\Psi}\) for \((t,x)\in[0,\xi ]\times[l,\frac{l}{\sigma}]\);

(H6): 

\(f(t,x)\leq\frac{d}{\Gamma}\) for \((t,x)\in J\times \in[0,d]\).

Then problem (1.2) has at least three positive solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\) satisfying
$$ \|x_{1}\|< d, \qquad l< \Lambda(x_{2}), \qquad d< \|x_{3}\| \quad \textit{and} \quad \Lambda(x_{3})< l. $$
(4.1)
For details, see Figure  1.
Figure 1

The relationship of \(\pmb{x_{1}}\) , \(\pmb{x_{2}}\) and \(\pmb{x_{3}}\) .

Proof

By the definition of operator T and its properties, it suffices to show that the conditions of Lemma 3.4 hold with respect to T.

Let \(x\in\bar{P}_{c}\). Then \(0\leq x(t) \leq c\) on J. Since \(0\leq\alpha(t)\leq t\leq1\) on J, it follows from \(0\leq x(t) \leq c\) on J that \(0\leq x(\alpha(t)) \leq c\) on J.

Consequently, for \(t\in J\) and \(x\in P_{c} \), it follows from (H4), (2.8) and (3.2) that
$$\begin{aligned} \|Tx\|&=\max_{t\in J}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq\int_{0}^{1}H(s)q(s)\omega(s)f\bigl(s,x \bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}\int_{0}^{1}H(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}\|H\|_{q}\|\omega\|_{p}\int _{0}^{1}f\bigl(s,x\bigl(\alpha(s)\bigr)\bigr)\,ds \\ & \leq e^{M}\|H\|_{q}\|\omega\|_{p} \frac{c}{\Gamma} \\ & \leq c, \end{aligned}$$
(4.2)
which implies \(Tx\in P_{c}\). This proves that \(T: \bar{P}_{c}\rightarrow\bar{P}_{c}\) is completely continuous.

We first show that the condition (i) of Lemma 3.4 holds.

Take \(x(t)=\frac{1}{2} (l+\frac{l}{\sigma} )\), \(\forall t\in J\). Then
$$\|x\|=\frac{1}{2} \biggl(l+\frac{l}{\sigma} \biggr)< \frac{l}{\sigma}, \qquad \Lambda(x)=\min_{t\in[0,\xi]}x(t)=\frac{1}{2} \biggl(l+ \frac {l}{\sigma} \biggr)>l. $$
This shows that
$$\biggl\{ x\in K \biggl(\Lambda,l,\frac{l}{\sigma} \biggr): \Lambda(x) > l \biggr\} \neq\emptyset. $$
Therefore, for all \(\{x\in K (\Lambda,l,\frac{l}{\sigma} ): \Lambda(x) > l \}\) and \(t\in J\), we have
$$l\leq x(t)\leq\frac{l}{\sigma}. $$

Since \(0\leq\alpha(t)\leq t\leq\xi\) on \([0,\xi]\), it follows from \(l\leq x(t) \leq\frac{l}{\sigma}\) on \([0,\xi]\) that \(l\leq x(\alpha (t))\leq\frac{l}{\sigma}\) for \(t\in[0,\xi]\).

Therefore, it follows from Remark 2.1 and (H5) that
$$\begin{aligned} \Lambda(Tx)&=\min_{t\in[0,\xi]}\bigl|(Tx) (t)\bigr| \\ &\geq\sigma\int_{0}^{1}H(s)q(s)\omega(s)f \bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ &\geq n\sigma\int_{0}^{1}H(s)f\bigl(s,x\bigl( \alpha(s)\bigr)\bigr)\,ds \\ &\geq n \sigma\int_{0}^{\xi}H(s)f\bigl(s,x\bigl( \alpha(s)\bigr)\bigr)\,ds \\ &>\frac{1}{2}n\sigma H_{0}\frac{l}{\Psi} \\ &=l. \end{aligned}$$
(4.3)
Therefore, we have
$$\Lambda(Tx)>l, \quad \forall x\in K \biggl(\Lambda,l,\frac{l}{\sigma } \biggr). $$
This implies that condition (i) of Lemma 3.4 is satisfied.

Secondly, we prove that condition (ii) of Lemma 3.4 is satisfied. If \(x\in K_{d}\), then \(0\leq x(t)\leq d\) on J.

Since \(0\leq\alpha(t)\leq t \leq1\) on J, it follows from \(0\leq x(t)\leq d\) on J that \(0\leq x(\alpha(t))\leq d\) on J.

Thus it follows from (H6) that
$$\begin{aligned} \|Tx\|&=\max_{t\in J}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq\int_{0}^{1}H(s)q(s)\omega(s)f\bigl(s,x \bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}\int_{0}^{1}H(s) \omega(s)\,ds\bigl|f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\bigr| \\ & \leq e^{M}\|H\|_{q}\|\omega\|_{p} \frac{d}{\Gamma} \\ & = d. \end{aligned}$$
(4.4)
Hence, the condition (ii) of Lemma 3.4 is satisfied.

Finally, we prove that the condition (iii) of Lemma 3.4 is satisfied.

In fact, for all \(x\in K(\Lambda,l,c)\) and \(\|Tx\|>\frac{l}{\sigma}\), it follows from (2.5), (2.9), (3.2) and (3.3) that
$$\begin{aligned} \Lambda(Tx)&=\min_{t\in[0,\xi]}\bigl|(Tx) (t)\bigr| \\ & = \min_{t\in[0,\xi]}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha(s)\bigr)\bigr)\,ds \\ & \geq\sigma\int_{0}^{1}H(s)q(s)\omega (s)f \bigl(s,x\bigl(\alpha(s)\bigr)\bigr)\,ds \\ & \geq\sigma\|Tx\| \\ & > \sigma\frac{l}{\sigma} \\ & =l. \end{aligned}$$
(4.5)
This gives the proof of the condition (iii) of Lemma 3.4.
To sum up, the hypotheses of Lemma 3.4 hold. Therefore, an application of Lemma 3.4 implies that problem (1.2) has at least three positive solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\) such that
$$\|x_{1}\|< d, \qquad l< \Lambda(x_{2}),\quad \mbox{and} \quad x_{3}>d \quad\mbox{with }\Lambda (x_{3})< l. $$
 □

The following corollary deals with the case \(p=\infty\).

Corollary 4.1

Assume that (H1)-(H6) hold. Then problem (1.2) has at least three positive solutions \(x_{1}\), \(x_{2} \) and \(x_{3}\) satisfying (4.1).

Proof

Let \(\|H\|_{1}\|\omega\|_{\infty}\) replace \(\|H\|_{q}\| \omega\|_{p}\) and repeat the argument above, we can get the corollary. □

Finally we consider the case of \(p=1\). Let
\((\mathrm{H}_{4})^{\prime}\)

\(f(t,x)\leq\frac{c}{\Gamma_{1}}\) for \((t,x)\in J\times[0,c]\);

\((\mathrm{H}_{6})^{\prime}\)

\(f(t,x)\leq\frac{d}{\Gamma_{1}}\) for \((t,x)\in J\times\in[0,d]\),

where
$$\Gamma_{1}=e^{M}H^{0}\|\omega\|_{1}. $$

Corollary 4.2

Assume that (H1)-(H3), \((\mathrm{H}_{4})^{\prime}\), (H5) and \((\mathrm{H}_{6})^{\prime}\) hold. Then problem (1.2) has at least three positive solutions \(x_{1}\), \(x_{2} \) and \(x_{3}\) satisfying (4.1).

Proof

Similar to the proof of (4.1), it follows from (2.3) and \((\mathrm{H}_{4})^{\prime}\) that
$$\begin{aligned} \|Tx\|&=\max_{t\in J}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq\int_{0}^{1}H(s)q(s)\omega(s)f\bigl(s,x \bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}\int_{0}^{1}H(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}H^{0}\int_{0}^{1} \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}H^{0}\|\omega\|_{1}\int _{0}^{1}f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}H^{0}\|\omega\|_{1} \frac{c}{\Gamma_{1}} \\ & \leq c, \end{aligned}$$
(4.6)
which shows that \(Tx\in\bar{K}_{c}\), \(\forall x\in\bar{K}_{c}\).
Next turning to \((\mathrm{H}_{6})^{\prime}\), we have
$$\begin{aligned} \|Tx\|&=\max_{t\in J}\int_{0}^{1}H(t,s)q(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq\int_{0}^{1}H(s)q(s)\omega(s)f\bigl(s,x \bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}\int_{0}^{1}H(s) \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}H^{0}\int_{0}^{1} \omega(s)f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}H^{0}\|\omega\|_{1}\int _{0}^{1}f\bigl(s,x\bigl(\alpha (s)\bigr)\bigr)\,ds \\ & \leq e^{M}H^{0}\|\omega\|_{1} \frac{d}{\Gamma_{1}} \\ & \leq c, \quad\forall x\in\bar{K}_{d}. \end{aligned}$$
(4.7)

Similar to the proof of Theorem 4.1, we can get Corollary 4.2. □

Remark 4.1

Comparing with Feng and Ge [11], the main features of this paper are as follows.
  1. (i)

    Three positive solutions are available.

     
  2. (ii)

    \(\alpha(t) \not\equiv t\) is considered throughout this paper.

     
  3. (iii)

    \(\omega(t)\) is \(L^{p}\)-integrable, not only \(\omega(t)\in C(0,1)\) on \(t\in J\).

     

5 An example

In this section, we present an example. Let
$$p=2, \qquad \beta_{1}=e^{-\frac{1}{2}}, \qquad\xi_{1}= \frac{1}{2},\qquad a(t)\equiv 0,\qquad b(t)\equiv1. $$

Example 5.1

Consider the following three-point boundary value problem:
$$ \left \{ \textstyle\begin{array}{@{}l} -x''(t)+x(t)=\omega(t)f(t,x(\alpha(t))), \quad 0< t < 1 , \\ x'(0)=0, \qquad x(1)=e^{-\frac{1}{2}}x(\frac{1}{2}), \end{array}\displaystyle \right . $$
(5.1)
where \(\alpha\in C(J,J)\), \(\alpha(t)\leq t\) on J and
$$\begin{aligned} &\omega(t)=\frac{1}{|t-\frac{1}{2}|^{\frac{1}{3}}}, \\ & f(t,x)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{d}{\Gamma}, & t\in J, x\in[0,d], \\ \frac{d}{\Gamma}\times\frac{l-x}{l-d}+\frac{l}{\Psi}\times\frac {x-d}{l-d}, & t\in J, x\in[d,l], \\ \frac{l}{\Psi}, & t\in J, x\in[l,\frac{l}{\sigma}],\\ \frac{l}{\Psi}\times\frac{c-x}{c-\frac{l}{\sigma}}+\frac{c}{\Gamma }\times\frac{x-\frac{l}{\sigma}}{c-\frac{l}{\sigma}}, & t\in J, x\in[\frac{l}{\sigma},c],\\ \frac{c}{\Gamma}, & t\in J, x\in[c,+\infty). \end{array}\displaystyle \right . \end{aligned}$$
(5.2)

This means that problem (5.1) involves the delayed argument α. For example, we can take \(\alpha(t)=t^{3}\). It is clear that ω is nonnegative and \(\omega\in L^{2}[0,1]\).

Conclusion 5.1

Problem (5.1) has at least three positive solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\) satisfying (4.1).

Proof

It follows from (1.3) and (2.4) that ϕ and ψ satisfy
$$ \begin{aligned} L\phi=0, \qquad \phi'(0)=0, \qquad \phi (1)=1, \\ L\psi=0, \qquad \psi(0)=1, \qquad \phi(1)=0, \end{aligned} $$
(5.3)
where \(Lx=-x''(t)+x(t)\) and
$$\begin{aligned} &\phi(t)=\frac{e^{1-t}+e^{1+t}}{1+e^{2}}, \qquad \phi (0)=\frac{2e}{1+e^{2}}, \\ &\psi(t)=\frac{-e^{2-t}+e^{t}}{1-e^{2}}, \qquad \psi '(0)=\frac{1+e^{2}}{1-e^{2}}, \\ &q(t)=1, \qquad \Delta:=-\phi(0)\psi'(0)=\frac{2e}{e^{2}-1}>0, \qquad \beta_{1}\phi(\xi_{1})=e^{-\frac{1}{2}}\phi\biggl( \frac{1}{2}\biggr)< 1. \end{aligned}$$
(5.4)
On the other hand, it follows from \(a(t)=0\), \(m=3\), \(\alpha _{1}=e^{-\frac{1}{2}}\) and \(\omega(t)=\frac{1}{|t-\frac{1}{2}|^{\frac {1}{3}}}\) that
$$e^{M}=1,\qquad n=\sqrt[3]{2}, \qquad \|\omega\|_{2}= \biggl[\int_{0}^{1} \biggl(t-\frac{1}{2} \biggr)^{-\frac{2}{3}} \,dt \biggr]^{\frac{1}{2}}=\sqrt{ \frac{6}{\sqrt[3]{2}}}. $$
Choosing \(0< d< l<\frac{l}{\sigma}\leq c\), we have
$$\begin{aligned}& f(t,x)\leq\frac{c}{\Gamma} \quad\mbox{for } (t,x)\in J\times[0,c]; \\& f(t,x)\geq\frac{l}{\Psi} \quad \mbox{for } (t,x)\in\biggl[0, \frac{1}{2}\biggr]\times \biggl[l,\frac{2l}{\sigma}\biggr]; \\& f(t,x)=\frac{d}{\Gamma} \quad \mbox{for } (t,x)\in J\times[0,d], \end{aligned}$$
which shows that (H4)-(H6) hold.

By Theorem 4.1, problem (5.1) has least three positive solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\) satisfying (4.1). □

Declarations

Acknowledgements

We wish to express our thanks to Prof. Xuemei Zhang, Department of Mathematics and Physics, North China Electric Power University, Beijing, P.R. China, for her kind help, careful reading, and making useful comments on the earlier version of this paper. The authors are also grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper. This work is sponsored by the project NSFC (11301178), the Fundamental Research Funds for the Central Universities (2014MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201511232018).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Applied Science, Beijing Information Science & Technology University

References

  1. Gupta, CP: A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput. 89, 133-146 (1998) MATHMathSciNetView ArticleGoogle Scholar
  2. Feng, W, Webb, JRL: Solvability of m-point boundary value problems with nonlinear growth. J. Math. Anal. Appl. 212, 467-480 (1997) MATHMathSciNetView ArticleGoogle Scholar
  3. Wei, Z, Pang, C: Positive solutions of some singular m-point boundary value problems at non-resonance. Appl. Math. Comput. 171, 433-449 (2005) MATHMathSciNetView ArticleGoogle Scholar
  4. Liu, B: Positive solutions of a nonlinear four-point boundary value problems in Banach spaces. J. Math. Anal. Appl. 305, 253-276 (2005) MATHMathSciNetView ArticleGoogle Scholar
  5. Zhao, Y, Chen, H: Existence of multiple positive solutions for m-point boundary value problems in Banach spaces. J. Comput. Appl. Math. 215, 79-90 (2008) MATHMathSciNetView ArticleGoogle Scholar
  6. Eloe, PW, Henderson, J: Positive solutions and nonlinear multipoint conjugate eigenvalue problems. Electron. J. Differ. Equ. 1997, 3 (1997) MathSciNetGoogle Scholar
  7. Webb, JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47, 4319-4332 (2001) MATHMathSciNetView ArticleGoogle Scholar
  8. He, X, Ge, W: Triple solutions for second-order three-point boundary value problems. J. Math. Anal. Appl. 268, 256-265 (2002) MATHMathSciNetView ArticleGoogle Scholar
  9. Kosmatov, N: Symmetric solutions of a multi-point boundary value problem. J. Math. Anal. Appl. 309, 25-36 (2005) MATHMathSciNetView ArticleGoogle Scholar
  10. Ma, R: Existence of positive solutions for a nonlinear m-point boundary value problem. Acta Math. Sin. 46, 785-794 (2003) MATHGoogle Scholar
  11. Feng, M, Ge, W: Positive solutions for a class of m-point singular boundary value problems. Math. Comput. Model. 46, 375-383 (2007) MATHMathSciNetView ArticleGoogle Scholar
  12. Xu, X: Positive solutions for singular m-point boundary value problems with positive parameter. J. Math. Anal. Appl. 291, 352-367 (2004) MATHMathSciNetView ArticleGoogle Scholar
  13. Cheung, WS, Ren, JL: Positive solutions for m-point boundary-value problems. J. Math. Anal. Appl. 303, 565-575 (2005) MATHMathSciNetView ArticleGoogle Scholar
  14. Jiang, W, Guo, Y: Multiple positive solutions for second-order m-point boundary value problems. J. Math. Anal. Appl. 327, 415-424 (2007) MATHMathSciNetView ArticleGoogle Scholar
  15. Yang, L, Liu, X, Jia, M: Multiplicity results for second-order m-point boundary value problem. J. Math. Anal. Appl. 324, 532-542 (2006) MATHMathSciNetView ArticleGoogle Scholar
  16. Sun, Y: Positive solutions of nonlinear second-order m-point boundary value problem. Nonlinear Anal. 61, 1283-1294 (2005) MATHMathSciNetView ArticleGoogle Scholar
  17. Liu, YS, O’Regan, D: Multiplicity results using bifurcation techniques for a class of fourth-order m-point boundary value problems. Bound. Value Probl. 2009, 970135 (2009) MathSciNetView ArticleGoogle Scholar
  18. Liu, X, Qiu, J, Guo, Y: Three positive solutions for second-order m-point boundary value problems. Appl. Math. Comput. 156, 733-742 (2004) MATHMathSciNetView ArticleGoogle Scholar
  19. Sun, YP, Liu, LS: Solvability for a nonlinear second-order three-point boundary value problem. J. Math. Anal. Appl. 296, 265-275 (2004) MATHMathSciNetView ArticleGoogle Scholar
  20. Yang, C, Zhai, C, Yan, J: Positive solutions of the three point boundary value problem for second order differential equations with an advanced argument. Nonlinear Anal. 65, 2013-2023 (2006) MATHMathSciNetView ArticleGoogle Scholar
  21. Jankowski, T: Existence of solutions of boundary value problems for differential equations with delayed arguments. J. Comput. Appl. Math. 156, 239-252 (2003) MATHMathSciNetView ArticleGoogle Scholar
  22. Jankowski, T: Advanced differential equations with nonlinear boundary conditions. J. Math. Anal. Appl. 304, 490-503 (2005) MATHMathSciNetView ArticleGoogle Scholar
  23. Jankowski, T: Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives. Nonlinear Anal. 87, 83-101 (2013) MATHMathSciNetView ArticleGoogle Scholar
  24. Jiang, D, Wei, J: Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations. Nonlinear Anal. 50, 885-898 (2002) MATHMathSciNetView ArticleGoogle Scholar
  25. Wang, G: Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments. J. Comput. Appl. Math. 236, 2425-2430 (2012) MATHMathSciNetView ArticleGoogle Scholar
  26. Wang, G, Zhang, L, Song, G: Systems of first order impulsive functional differential equations with deviating arguments and nonlinear boundary conditions. Nonlinear Anal. 74, 974-982 (2011) MATHMathSciNetView ArticleGoogle Scholar
  27. Hu, C, Liu, B, Xie, S: Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments. Appl. Math. Comput. 222, 72-81 (2013) MathSciNetView ArticleGoogle Scholar
  28. Zhang, X, Feng, M: Green’s function and positive solutions for a second-order singular boundary value problem with integral boundary conditions and a delayed argument. Abstr. Appl. Anal. 2014, Article ID 393187 (2014) Google Scholar
  29. Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985) MATHView ArticleGoogle Scholar
  30. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cons. Academic Press, New York (1998) Google Scholar
  31. Leggett, R, Williams, L: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673-688 (1979) MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Zhou and Feng 2015