Skip to content

Advertisement

Boundary Value Problems
Boundary Value Problems Cover Image
  • Research
  • Open Access

Triple positive solutions of fourth-order impulsive differential equations with integral boundary conditions

Boundary Value Problems20152015:2

https://doi.org/10.1186/s13661-014-0263-7

©  Zhou and Zhang; licensee Springer 2015

  • Received: 2 June 2014
  • Accepted: 5 December 2014
  • Published:

Abstract

By using Leggett-Williams’ fixed point theorem and Hölder’s inequality, the existence of three positive solutions for the fourth-order impulsive differential equations with integral boundary conditions \(x^{(4)}(t)=\omega(t)f(t,x(t))\), \(0< t<1\), \(t\neq t_{k}\), \(\Delta x|_{t=t_{k}}=I_{k}(t_{k},x(t_{k}))\), \(\Delta x^{\prime}|_{t=t_{k}}=0\), \(k=1,2,\ldots,m\), \(x(0)=\int_{0}^{1}g(s)x(s)\,ds\), \(x'(1)=0\), \(x^{\prime\prime}(0)=\int_{0}^{1}h(s)x^{\prime\prime}(s)\,ds\), \(x^{\prime\prime\prime}(1)=0 \) is considered, where \(\omega(t)\) is \(L^{p}\)-integrable. Our results cover a fourth-order boundary value problem without impulsive effects and are compared with some recent results.

Keywords

  • triple positive solutions
  • impulsive differential equations
  • integral boundary conditions
  • Leggett-Williams’ fixed point theorem
  • Hölder’s inequality

1 Introduction

Impulsive differential equations occur in many applications. Various mathematical models, such as population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc., can be expressed by differential equations with impulses. Therefore, the study of impulsive differential equations has gained prominence and it is a rapidly growing field; see [1–22] and the references therein. We note that the difficulties dealing with such problems are that theirs states are discontinuous. Therefore, the results of impulsive differential equations, especially for higher-order impulsive differential equations, are fewer in number than those of differential equations without impulses.

At the same time, owing to its importance in modeling the stationary states of the deflection of an elastic beam, fourth-order boundary value problems have attracted much attention from many authors; see, for example [23–53] and the references therein. In particular, we would like to mention some results of Yang [28], Anderson and Avery [31], and Zhang et al. [36]. In [28], Yang considered the following fourth-order two-point boundary value problem:
$$\left \{ \begin{array}{@{}l} x^{(4)}(t)=g(t)f(x(t)), \quad 0\leq t \leq1,\\ x(0)=x^{\prime}(0)=x^{\prime\prime}(1)=x^{\prime\prime\prime}(1)=0. \end{array} \right . $$
By using Krasnoselskii’s fixed point theorem, the author established some new estimates to the positive solutions to the above problem and obtained some sufficient conditions for the existence of at least one positive solution.
In [31], Anderson and Avery considered the following fourth-order four-point right focal boundary value problem:
$$\left \{ \begin{array}{@{}l} -x^{(4)}(t)=f(x(t)),\quad t\in[0,1] , \\ x(0)=x^{\prime}(q)=x^{\prime\prime}(r)=x^{\prime\prime\prime}(1)=0, \end{array} \right . $$
where \(0< q<r<1\) are two constants, \(f: R\rightarrow R\) is continuous and \(f(x)\geq0\) for \(x\geq0\). By using the five functionals fixed point theorem, the authors gave sufficient conditions for the existence of three positive solutions of above problem.
Recently, Zhang et al. [36] studied the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:
$$\left \{ \begin{array}{@{}l} x^{(4)}(t)-\lambda f(t,x(t))=\theta,\quad 0< t < 1, \\ x(0)=x(1)=\int_{0}^{1}g(s)x(s)\,ds,\\ x^{\prime\prime}(0)=x^{\prime\prime}(1)=\int_{0}^{1}h(s)x(s)\,ds, \end{array} \right . $$
where θ is the zero element of E.

However, to the best of our knowledge, no paper has considered the existence results of triple positive solutions for fourth-order impulsive differential equations with integral boundary conditions till now; for example, see [54–58] and the references therein.

In this paper, we investigate the existence of three positive solutions for the following fourth-order impulsive differential equations with integral boundary conditions:
$$ \left \{ \begin{array}{@{}l} x^{(4)}(t)=\omega(t)f(t,x(t)),\quad 0< t<1, t\neq t_{k},\\ \Delta x|_{t=t_{k}}=I_{k}(t_{k},x(t_{k})),\\ \Delta x^{\prime}|_{t=t_{k}}=0, \quad k=1,2,\ldots,m,\\ x(0)=\int_{0}^{1}g(s)x(s)\,ds, \qquad x'(1)=0,\\ x^{\prime\prime}(0)=\int_{0}^{1}h(s)x^{\prime\prime}(s)\,ds,\qquad x^{\prime\prime\prime}(1)=0. \end{array} \right . $$
(1.1)
Here \(\omega\in L^{p}[0,1]\) for some \(1\leq p \leq+\infty\), \(t_{k}\) (\(k=1,2,\ldots,m\)) (where m is fixed positive integer) are fixed points with \(0=t_{0}< t_{1}<t_{2}<\cdots <t_{k}<\cdots <t_{m}<t_{m+1}=1\), \(\Delta x |_{t=t_{k}}\) denotes the jump of \(x(t)\) at \(t=t_{k}\), i.e. \(\Delta x |_{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), where \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) represent the right-hand limit and left-hand limit of \(x(t)\) at \({t=t_{k}}\), respectively. In addition, ω, f, \(I_{k}\), g, and h satisfy
(H1): 

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

(H2): 

\(f\in C([0,1]\times[0,+\infty), [0,+\infty))\), \(I_{k}\in C([0,1]\times[0,+\infty),[0,+\infty))\);

(H3): 
\(g,h\in L^{1}[0,1]\) are nonnegative and \(\mu\in[0,1)\), \(\nu \in[0,1)\), where
$$ \nu=\int_{0}^{1}g(t)\,dt, \qquad\mu=\int _{0}^{1}h(t)\,dt. $$
(1.2)

Remark 1.1

The idea of impulsive effect for problem (1.1) is from Ding and O’Regan [59].

Some special cases of problem (1.1) have been investigated. For example, Zhang and Ge [45] studied the existence and multiplicity of symmetric positive solutions for problem (1.1) with \(I_{k}\equiv0\) (\(k=1, 2, \ldots, m\)) and \(\omega\in C(0,1)\), not \(\omega\in L^{p}[0,1]\).

Motivated by the results mentioned above, in this paper we study the existence of three positive solutions for problem (1.1) by new technique (different from the proof of Theorems 3.1-3.4 of [45]) to overcome difficulties arising from the appearances of \(I_{k}\neq0\) (\(k=1, 2, \ldots, m\)) and \(\omega(t)\) is \(L^{p}\)-integrable. The arguments are based upon a fixed point theorem due to Leggett and Williams which deals with fixed points of a cone-preserving operator defined on an ordered Banach space.

The rest of the paper is organized as follows: In Section 2, we provide some necessary background. In particular, we state some properties of the Green’s function associated with problem (1.1). In Section 3, the main results of problem (1.1) will be stated and proved.

2 Preliminaries

Let \(J^{\prime}=J\backslash\{t_{1},t_{2},\ldots,t_{m}\}\), and
$$PC[0,1]= \bigl\{ x:x\mbox{ is continuous at }t\neq t_{k}, x \bigl(t_{k}^{-}\bigr)=x(t_{k})\mbox{ and }x \bigl(t_{k}^{+}\bigr)\mbox{ exists}, k=1,2,\ldots ,m \bigr\} . $$
Then \(PC[0,1]\) is a real Banach space with norm
$$\|x\|=\max_{t\in J} \bigl|x(t) \bigr|. $$

Definition 2.1

(See [60])

Let E be a real Banach space over R. A nonempty closed set \(P \subset E\) is said to be a cone provided that
  1. (i)

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

     
  2. (ii)

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

     

Every cone \(P \subset E\) induces an ordering in E given by \(x\leq y\) if and only if \(y-x \in P\).

Definition 2.2

The map β is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that \(\beta:P \rightarrow[0,\infty)\) is continuous and
$$\beta\bigl(tx+(1-t)y\bigr)\geq t\beta(x)+(1-t)\beta(y) $$
for all \(x,y \in P\) and \(0 \leq t \leq1\).

Definition 2.3

A function \(x\in PC[0,1]\cap C^{4}(J^{\prime})\) is called a solution of problem (1.1) if it satisfies (1.1).

We shall reduce problem (1.1) to an integral equation. With this goal, firstly by means of the transformation
$$ x''(t)=-y(t), $$
(2.1)
we convert problem (1.1) into
$$ \left \{ \begin{array}{@{}l} y^{\prime\prime}(t)+\omega(t)f(t,x(t))=0, \quad t\in J,\\ y(0)=\int_{0}^{1}h(t)y(t)\,dt, \qquad y'(1)=0, \end{array} \right . $$
(2.2)
and
$$ \left \{ \begin{array}{@{}l} -x^{\prime\prime}(t)=y(t), \quad t\in J, t\neq t_{k},\\ \Delta x|_{t=t_{k}}=I_{k}(t_{k},x(t_{k})),\\ \Delta x^{\prime}|_{t=t_{k}}=0, \quad k=1,2,\ldots,m,\\ x(0)=\int_{0}^{1}g(t)x(t)\,dt,\qquad x'(1)=0. \end{array} \right . $$
(2.3)

Lemma 2.1

Assume that (H1)-(H3) hold. Then problem (2.2) has a unique solution y given by
$$ y(t)=\int_{0}^{1}H(t,s)\omega(s)f\bigl(s,x(s) \bigr)\,ds, $$
(2.4)
where
$$\begin{aligned}& H(t,s)=G(t,s)+\frac{1}{1-\mu}\int_{0}^{1}G(s, \tau)h(\tau)\,d\tau, \end{aligned}$$
(2.5)
$$\begin{aligned}& G(t,s)=\left \{ \begin{array}{@{}l@{\quad}l} t, & 0\leq t \leq s\leq1,\\ s, & 0\leq s \leq t \leq1. \end{array} \right . \end{aligned}$$
(2.6)

Proof

The proof of Lemma 2.1 is similar to that of Lemma 2.1 in [61]. □

Write \(e(t)=t\). Then from (2.5) and (2.6), we can prove that \(H(t,s)\) and \(G(t,s)\) have the following properties.

Proposition 2.1

Let \(\delta\in(0,\frac{1}{2})\), \(J_{\delta }=[\delta,1-\delta]\). If \(\mu\in[0,1)\), then we have
$$\begin{aligned}& H(t,s)>0, \qquad G(t,s)>0, \quad\forall t,s\in(0,1), \end{aligned}$$
(2.7)
$$\begin{aligned}& H(t,s)\geq0, \qquad G(t,s)\geq0, \quad\forall t,s\in J, \end{aligned}$$
(2.8)
$$\begin{aligned}& e(t)e(s)\leq G(t,s)\leq G(t,t)= t=e(t)\leq1,\quad \forall t,s\in J, \end{aligned}$$
(2.9)
$$\begin{aligned}& \rho e(t)e(s)\leq H(t,s)\leq\gamma s=\gamma e(s)\leq\gamma,\quad \forall t,s\in J, \end{aligned}$$
(2.10)
$$\begin{aligned}& G(t,s)\geq\delta G(s,s), \qquad H(t,s)\geq\delta H(s,s),\quad \forall t\in J_{\delta}, s\in J, \end{aligned}$$
(2.11)
where
$$ \gamma=\frac{1}{1-\mu},\qquad \rho=1+\frac{\int_{0}^{1}sh(s)\,ds}{1-\mu}. $$
(2.12)

Remark 2.1

From (2.5) and (2.11), we can obtain
$$H(t,s)\geq\delta s= \delta G(s,s), \quad \forall t\in J_{\delta}, s\in J. $$

Lemma 2.2

If (H2) and (H3) hold, then problem (2.3) has a unique solution x and x can be expressed in the form
$$ x(t) =\int_{0}^{1}H_{1}(t,s)y(s)\,ds + \sum_{k=1}^{m}H_{1s}'(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr), $$
(2.13)
where
$$\begin{aligned}& H_{1}(t,s)=G(t,s)+\frac{1}{1-\nu}\int_{0}^{1}G(s, \tau)g(\tau)\,d\tau, \end{aligned}$$
(2.14)
$$\begin{aligned}& H_{1s}'(t,s)=G'_{s}(t,s)+ \frac{1}{1-\nu}\int_{0}^{1}G'_{s}( \tau,s)g(\tau )\,d\tau, \end{aligned}$$
(2.15)
$$\begin{aligned}& G'_{s}(t,s)=\left \{ \begin{array}{@{}l@{\quad}l} 0, & 0\leq t \leq s\leq1,\\ 1, & 0\leq s \leq t \leq1. \end{array} \right . \end{aligned}$$
(2.16)

Proof

The proof of Lemma 2.2 is similar to that of Lemma 2.6 in [53]. □

From (2.14)-(2.16), we can prove that \(H_{1}(t,s)\), \(H_{1s}'(t,s)\), and \(G'_{s}(t,s)\) have the following properties.

Proposition 2.2

If \(\nu\in[0,1)\), then we have
$$\begin{aligned}& H_{1}(t,s)\geq0,\quad \forall t,s\in J; \end{aligned}$$
(2.17)
$$\begin{aligned}& \rho_{1} e(t)e(s)\leq H_{1}(t,s)\leq\gamma_{1} s= \gamma_{1} e(s)\leq \gamma_{1},\quad \forall t,s\in J, \end{aligned}$$
(2.18)
$$\begin{aligned}& H_{1}(t,s)\geq\delta H_{1}(s,s),\quad \forall t\in J_{\delta}, s\in J. \end{aligned}$$
(2.19)
$$\begin{aligned}& G'_{s}(t,s)\leq1, \qquad 0\leq H_{1s}'(t,s) \leq\frac{1}{1-\nu}, \end{aligned}$$
(2.20)
where
$$ \gamma_{1}=\frac{1}{1-\nu}, \qquad \rho_{1}=1+ \frac{\int_{0}^{1}sg(s)\,ds}{1-\nu}. $$
(2.21)

Remark 2.2

From (2.14) and (2.19), we can obtain
$$H_{1}(t,s)\geq\delta s= \delta G(s,s), \quad \forall t\in J_{\delta}, s\in J. $$

Remark 2.3

From (2.20), one can prove that
$$ 0< H_{1s}'(t,s) (1-\nu)\leq1, \quad\forall t\in J_{\delta},s\in[0,1). $$
(2.22)
Suppose that x is a solution of problem (1.1). Then from Lemma 2.1 and Lemma 2.2, we have
$$x(t)=\int_{0}^{1}\int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum _{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr). $$
Define a cone in \(PC[0,1]\) by
$$ K= \bigl\{ x\in PC[0,1]:x\geq0 \bigr\} . $$
(2.23)
It is easy to see K is a closed convex cone of \(PC[0,1]\).
Define an operator \(T:K\rightarrow PC[0,1]\) by
$$ (Tx) (t) =\int_{0}^{1}\int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum _{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr). $$
(2.24)

From (2.24), we know that \(x\in PC[0,1]\) is a solution of problem (1.1) if and only if x is a fixed point of operator T.

Lemma 2.3

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

Proof

The proof of Lemma 2.3 is similar to that of Lemma 2.4 in [53]. □

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

Finally we state Leggett-Williams’ fixed point theorem [62].

Lemma 2.4

Let K be a cone in a real Banach space E, \(A:\bar{K}_{a}\rightarrow\bar{K}_{a}\) be completely continuous and β be a nonnegative continuous concave functional on K with \(\beta(x)\leq\|x\|\) for all \(x\in K_{a}\). Suppose 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 positive solutions \(x_{1}\), \(x_{2}\), \(x_{3}\) satisfying
$$\|x_{1}\|< d,\qquad a<\beta(x_{2}),\qquad \|x_{3} \|>d\quad \textit{and}\quad \beta(x_{3})<a. $$

To obtain some of the norm inequalities in Theorem 3.1 and Corollary 3.1, we employ Hölder’s inequality.

Lemma 2.5

(Hölder)

Let \(f\in L^{p}[a,b]\) with \(p>1\), \(g\in L^{q}[a,b]\) with \(q>1\), and \(\frac{1}{p}+\frac{1}{q}=1\). Then \(fg\in L^{1}[a,b]\) and
$$\|fg\|_{1}\leq\|f\|_{p}\|g\|_{q}. $$
Let \(f\in L^{1}[a,b]\), \(g\in L^{\infty}[a,b]\). Then \(fg\in L^{1}[a,b]\) and
$$\|fg\|_{1}\leq\|f\|_{1}\|g\|_{\infty}. $$

3 Existence of triple positive solutions to (1.1)

In this section, we apply Lemma 2.4 and Lemma 2.5 to establish the existence of triple positive solutions for problem (1.1). 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 introduce the following notation:
$$\begin{aligned}& D=\gamma\gamma_{1}\|e\|_{q}\|\omega\|_{p}, \qquad D_{1}=\frac{m}{1-\nu},\\& \delta_{1}=\min _{t\in J_{\delta},s\in(0,1)}H_{1s}'(t,s) (1-\nu), \qquad \delta^{*}=\min\biggl\{ \frac{\delta}{\gamma_{1}},\delta_{1}\biggr\} , \\& f^{\infty}=\limsup_{x \rightarrow\infty}\max_{t\in J} \frac{f(t,x)}{x},\qquad I^{\infty}(k)=\limsup_{x \rightarrow\infty}\max _{t\in J} \frac{I_{k}(t,x)}{x},\quad k=1,2,\ldots,m. \end{aligned}$$

Theorem 3.1

Assume that (H1)-(H3) hold. Furthermore, suppose that there exist constants \(0< d<a<\frac{a}{\delta^{*}}\leq c\) such that
(H4): 

\(f^{\infty}<\frac{1}{2D}\), \(I^{\infty}(k)<\frac {1}{2D_{1}}\), \(k=1,2,\ldots,m\);

(H5): 

\(f(t,x)> \frac{3a}{\delta^{2}(1-2\delta)n}\) for \((t,x)\in J_{\delta}\times[a,\frac{a}{\delta^{*}}]\);

(H6): 

\(f(t,x)<\frac{d}{2D}\), \(I_{k}(t,x)<\frac{d}{2D_{1}}\) for \((t,x)\in J\times[0,d]\), \(k=1,2,\ldots,m\).

Then problem (1.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
$$\|x_{1}\|< d, \qquad a<\beta(x_{2}), \qquad \textit{and} \quad x_{3}>d \quad\textit{with }\beta(x_{3})<a. $$

Proof

By the definition of operator T and its properties, it suffices to show that the conditions of Lemma 2.4 hold with respect to T.

Let \(\beta(x)=\min_{t\in J_{\delta}}x(t)\). Then \(\beta(x)\) is a nonnegative continuous concave functional on the cone K satisfying \(\beta(x)\leq\|x\|\) for all \(x\in K\).

For convenience, we denote \(b=\frac{a}{\delta^{*}}\).

Considering (H4), there exist \(0<\sigma<\frac{1}{2D}\), \(0<\sigma _{1}<\frac{1}{2D_{1}}\), and \(l>0\) such that
$$f(t,x)\leq\sigma x,\qquad I_{k}(t,x)\leq\sigma_{1} x,\quad k=1,2,\ldots,m, \forall t\in J, x\geq l. $$
Let
$$\eta=\max_{0\leq x\leq l, t\in J}f(t,x),\qquad \eta_{1}=\max _{0\leq x\leq l, t\in J}I_{k}(t,x),\quad k=1,2,\ldots,m. $$
Then
$$ f(t,x)\leq\sigma x+\eta, \qquad I_{k}(t,x)\leq\sigma_{1} x+ \eta_{1}, \quad \forall t\in J, 0\leq x\leq+\infty. $$
(3.1)
Set
$$c>\max \biggl\{ \frac{2D\eta}{1-2D\sigma},\frac{2D_{1}\eta _{1}}{1-2D_{1}\sigma_{1}},\frac{a}{\delta^{*}} \biggr\} . $$
Then, for \(x\in\bar{K}_{c}\), it follows from (2.19), (2.22), and (3.1) that
$$\begin{aligned} (Tx) (t)&=\int_{0}^{1}\int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum _{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau) (\sigma x+\eta)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}( \sigma_{1} x+\eta_{1}) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau) \bigl(\sigma\|x\|+\eta\bigr)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}\bigl( \sigma_{1} \|x\|+\eta_{1}\bigr) \\ &\leq\gamma_{1}\gamma(\sigma c+\eta)\int_{0}^{1} e(\tau)\omega(\tau)\,d\tau+\frac{m}{1-\nu}(\sigma_{1} c+ \eta_{1}) \\ &\leq\gamma_{1}\gamma(\sigma c+\eta)\|e\|_{q}\| \omega \|_{p} +\frac{m}{1-\nu}(\sigma_{1} c+\eta_{1}) \\ &<\frac{c}{2}+\frac{c}{2} = c, \end{aligned}$$
which shows that \(Tx\in K_{c}\).

Hence, we have shown that if (H4) holds, then T maps \(\bar{K}_{c}\) into \(K_{c}\).

Next, we verify that \(\{x\in K(\beta,a,b):\beta(x)>a\}\neq\emptyset\) and \(\beta(Tu)>a\) for all \(x\in K(\beta,a,b)\).

Take \(\varphi_{0}(t)\equiv\frac{\delta^{*}+1}{2\delta^{*}}a\), for \(t\in J\). Then
$$\varphi_{0}\in \biggl\{ x\Bigm| x\in K\biggl(\beta, a, \frac{a}{\delta^{*}}\biggr), \beta(x)>a \biggr\} . $$
This shows that
$$\bigl\{ x\in K(\beta,a,b):\beta(x)>a\bigr\} \neq\emptyset. $$
Therefore, it follows from (H5) that
$$\begin{aligned} \beta(Tx)&=\min_{t\in J_{\delta}}(Tx) (t) \\ &=\min_{t\in J_{\delta}}\int_{0}^{1}\int _{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f \bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum_{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\geq\min_{t\in J_{\delta}}\int_{0}^{1}\int _{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f \bigl(\tau,x(\tau)\bigr)\,d\tau \,ds \\ &\geq\delta\int_{0}^{1}\int_{0}^{1}e(s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds \\ &\geq\delta\int_{0}^{1}\int_{\delta}^{1-\delta}e(s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds \\ &\geq\delta^{2}\int_{0}^{1}s^{2}\,ds \int_{\delta}^{1-\delta }\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \\ &>\frac{1}{3}\delta^{2}n(1-2\delta)\frac{3a}{\delta ^{2}(1-2\delta)n} \\ &=a. \end{aligned}$$
If \(x\in\bar{K}_{d}\), then it follows from (H6) that
$$\begin{aligned} (Tx) (t)&=\int_{0}^{1}\int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum _{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &< \int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau )\omega(\tau)\frac{d}{2D}\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m} \frac {d}{2D_{1}} \\ &= d. \end{aligned}$$

Finally, we assert that if \(x\in K(\beta,a,c)\) and \(\|Tx\|>b\), then \(\beta(Tx)>a\).

Suppose \(x\in K(\beta,a,c)\) and \(\|Tx\|>b\), then it follows from (2.18), (2.20), and (2.23) that
$$\begin{aligned} \beta(Tx)={}&\min_{t\in J_{\delta}}(Tx) (t) \\ ={}&\min_{t\in J_{\delta}} \Biggl[\int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega( \tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum_{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr] \\ \geq{}&\delta\int_{0}^{1}\int _{0}^{1}e(s)H(s, \tau)\omega(\tau)f\bigl(\tau,x( \tau)\bigr)\,d\tau \,ds \\ &{}+\min_{t\in J_{\delta}}\sum_{k=1}^{m}H'_{1s}(t,t_{k}) (1-\nu)\frac{1}{1-\nu}I_{k}\bigl(t_{k},x(t_{k}) \bigr) \\ \geq&{}\frac{\delta}{\gamma_{1}}\int_{0}^{1}\int _{0}^{1}\gamma_{1}e(s)H(s, \tau)\omega( \tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\delta_{1}\sum _{k=1}^{m}\frac {1}{1-\nu}I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ \geq{}&\min\biggl\{ \frac{\delta}{\gamma_{1}},\delta_{1}\biggr\} \Biggl[\int _{0}^{1}\int_{0}^{1} \gamma_{1}e(s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\! \sum_{k=1}^{m}\frac{1}{1-\nu }I_{k} \bigl(t_{k},x(t_{k})\bigr)\! \Biggr] \\ \geq&{}\delta^{*}\|Tx\| \\ >{}&a. \end{aligned}$$
To sum up, the hypotheses of Lemma 2.5 hold. Therefore, an application of Lemma 2.5 implies problem (1.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
$$\|x_{1}\|< d,\qquad a<\beta(x_{2}), \quad \mbox{and} \quad x_{3}>d \quad\mbox{with }\beta(x_{3})<a. $$
 □

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

Corollary 3.1

Assume that (H1)-(H6) hold. Then problem (1.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
$$\|x_{1}\|< d, \qquad a<\beta(x_{2}), \quad \textit{and}\quad x_{3}>d \quad\textit{with }\beta(x_{3})<a. $$

Proof

Let \(\|e\|_{1}\|\omega\|_{\infty}\) replace \(\|e\|_{p}\| \omega\|_{q}\) and repeat the argument above. □

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

\(f^{\infty}<\frac{1}{D'}\), \(I^{\infty}(k)<\frac {1}{D_{1}}\), \(k=1,2,\ldots,m\);

\((\mathrm{H}_{6})'\) : 

\(f(t,x)\leq\frac{d}{2D'}\), \(I_{k}(t,x)\leq\frac {d}{2D_{1}}\) (\(k=1,2,\ldots,m\)) for \((t,x)\in J\times[0,d]\),

where
$$D'=\gamma\gamma_{1}\|\omega\|_{1}. $$

Corollary 3.2

Assume that (H1)-(H3), \((\mathrm{H}_{4})'\), (H5), and \((\mathrm{H}_{6})'\) hold. Then problem (1.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
$$\|x_{1}\|< d, \qquad a<\beta(x_{2}), \quad\textit{and}\quad x_{3}>d \quad\textit{with }\beta(x_{3})<a. $$

Proof

Set
$$c'>\max \biggl\{ \frac{2D'\eta}{1-2D'\sigma'},\frac{2D_{1}\eta _{1}}{1-2D_{1}\sigma_{1}}, \frac{a}{\delta^{*}} \biggr\} , $$
where \(0<\sigma'<\frac{1}{2D'}\). Then, for \(x\in\bar{K}_{c'}\), it follows from (2.19), (2.22), and (3.1) that
$$\begin{aligned} (Tx) (t)&=\int_{0}^{1}\int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum _{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau) (\sigma x+\eta)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}( \sigma_{1} x+\eta_{1}) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau) \bigl(\sigma\|x\|+\eta\bigr)\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m}\bigl( \sigma_{1} \|x\|+\eta_{1}\bigr) \\ &\leq\gamma_{1}\gamma\bigl(\sigma c'+\eta\bigr)\int _{0}^{1}\omega(\tau)\,d\tau+\frac{m}{1-\nu}\bigl( \sigma_{1} c'+\eta_{1}\bigr) \\ &\leq\gamma_{1}\gamma\bigl(\sigma c'+\eta\bigr)\|\omega\| _{1} +\frac{m}{1-\nu}\bigl(\sigma_{1} c'+ \eta_{1}\bigr) \\ &<\frac{c'}{2}+\frac{c'}{2} =c', \end{aligned}$$
which shows that \(Tx\in K_{c'}\).

Hence, we have shown that if \((\mathrm{H}_{4})'\) holds, then T maps \(\bar{K}_{c'}\) into \(K_{c'}\).

If \(x\in\bar{K}_{d}\), then it follows from \((\mathrm{H}_{6})'\) that
$$\begin{aligned} (Tx) (t)&=\int_{0}^{1}\int_{0}^{1}H_{1}(t,s)H(s, \tau)\omega(\tau)f\bigl(\tau,x(\tau)\bigr)\,d\tau \,ds+\sum _{k=1}^{m}H'_{1s}(t,t_{k})I_{k} \bigl(t_{k},x(t_{k})\bigr) \\ &\leq\int_{0}^{1}\int_{0}^{1} \gamma_{1}\gamma e(\tau)\omega(\tau)\frac{d}{2D'}\,d\tau \,ds+ \frac{1}{1-\nu}\sum_{k=1}^{m} \frac{d}{2D_{1}} \\ &\leq\gamma_{1}\gamma\frac{d}{2D'}\int_{0}^{1} \omega(\tau)\,d\tau+\frac{1}{1-\nu}\sum_{k=1}^{m} \frac {d}{2D_{1}} \\ &= d. \end{aligned}$$

Similar to the proof of Theorem 3.1, one can find the results of Corollary 3.2. □

We remark that the condition (H6) in Theorem 3.1 can be replaced by the following condition:
\((\mathrm{H}_{6})''\) : 
\(f_{0}^{d}\leq\frac{1}{2D}\), \(I_{0}^{d}(k)\leq\frac {1}{2D_{1}}\), \(k=1,2,\ldots,m\), where
$$f_{0}^{d}=\max \biggl\{ \max_{t\in J} \frac{f(t,x)}{d}:x\in [0,d] \biggr\} ,\qquad I_{0}^{d}(k)= \max \biggl\{ \max_{t\in J}\frac {I_{k}(t,x)}{d}:x\in [0,d] \biggr\} . $$
\((\mathrm{H}_{6})'''\) : 

\(f^{0}\leq\frac{1}{2D}\), \(I^{0}(k)\leq\frac {1}{2D_{1}}\), \(k=1,2,\ldots,m\).

Corollary 3.3

If the condition (H6) in Theorem 3.1 is replaced by \((\mathrm{H}_{6})''\) or \((\mathrm{H}_{6})'''\), respectively, then the conclusion of Theorem 3.1 also holds.

Proof

It follows from the proof of Theorem 3.1 that Corollary 3.3 holds. □

Remark 3.1

Comparing with Zhang and Ge [45], the main features of this paper are as follows.
  1. (i)

    Triple positive solutions are available.

     
  2. (ii)

    \(I_{k}\neq0\) (\(k=1, 2, \ldots, m\)) is considered.

     
  3. (iii)

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

     

4 Example

To illustrate how our main results can be used in practice, we present an example.

Example 4.1

Let \(\delta=\frac{1}{4}\), \(m=1\), \(t_{1}=\frac{1}{2}\), \(p=1\). It follows from \(p=1\) that \(q=\infty\). Consider the following boundary value problem:
$$ \left \{ \begin{array}{@{}l} x^{(4)}(t)=\omega(t)f(t,x(t)),\quad 0< t<1, t\neq\frac{1}{2},\\ \Delta x|_{t=\frac{1}{2}}=I_{1}(\frac{1}{2},x(\frac{1}{2})),\\ \Delta x^{\prime}|_{t=\frac{1}{2}}=0,\\ x(0)=\int_{0}^{1}g(s)x(s)\,ds,\qquad x'(1)=0,\\ x^{\prime\prime}(0)=\int_{0}^{1}h(s)x^{\prime\prime}(s)\,ds, \qquad x^{\prime\prime\prime}(1)=0, \end{array} \right . $$
(4.1)
where \(\omega(t)=2t+3\in L^{1}[0,1]\), \(g(t)=h(t)=t\), \(I_{1}(t,x)=\frac {tx}{20\delta}\),
$$f(t,x)=\left \{ \begin{array}{@{}l@{\quad}l} \frac{d}{48}, & t\in J, x\in[0,d],\\ \frac{d}{48}\times\frac{a-x}{a-d}+64a\frac{x-d}{a-d}, & t\in J, x\in[d,a],\\ 64a,& t\in J, x\in[a,\frac{a}{\delta^{*}}],\\ 64a+t\sqrt{x-\frac{a}{\delta^{*}}},& t\in J, x\in[\frac{a}{\delta ^{*}}, \infty). \end{array} \right . $$
Thus it is easy to see by calculating that \(\omega(t)\geq n=3\) for a.e. \(t\in J\), and
$$\begin{aligned}& \mu=\int_{0}^{1}h(t)\,dt=\frac{1}{2}, \qquad \nu=\int_{0}^{1}g(t)\,dt=\frac {1}{2}, \\& \gamma= \frac{1}{1-\mu}=2, \qquad\gamma_{1}=\frac{1}{1-\nu}=2,\qquad \delta_{1}=\frac{3}{4}, \qquad\delta^{*}= \frac{1}{8}. \end{aligned}$$

Therefore, it follows from the definitions ω, f, \(I_{1}\), g, and h that (H1)-(H3) hold.

On the other hand, it follows from \(\omega(t)=2t+3\) and \(e(t)=t\) that
$$\|\omega\|_{1}=\int_{0}^{1}(2t+3)\,dt=4, \qquad \|e\|_{q}=\|e\|_{\infty}=\lim_{q\rightarrow\infty} \biggl(\int_{0}^{1}t^{q} \,dt\biggr)^{\frac{1}{q}}=\lim_{q\rightarrow\infty}\biggl(\frac{1}{q+1} \biggr)^{\frac{1}{q}}=1. $$
Thus, we have
$$D=\gamma\gamma_{1}\|\omega\|_{1}\|e\|_{\infty}=16, \qquad D_{1}=\frac{m}{1-\mu }=2, \qquad\frac{1}{2D}= \frac{1}{32},\qquad \frac{1}{2D_{1}}=\frac{1}{4}. $$
Choosing \(0< d<a<8a\leq c\), we have
$$\begin{aligned}& f^{\infty}=0<\frac{1}{32}=\frac{1}{2D}, \qquad I^{\infty}(1)=\frac {1}{5}<\frac{1}{4}=\frac{1}{2D_{1}}, \\& f(t,x)=64a>32a=\frac{3a}{\delta^{2}(1-2\delta)n}, \quad\forall(t,x)\in\biggl[\frac {1}{4}, \frac{3}{4}\biggr]\times[a,8a], \\& f(t,x)=\frac{d}{48}<\frac{d}{32}=\frac{1}{2D},\qquad I_{1}(t,x)\leq\frac {d}{5}<\frac{d}{4}=\frac{d}{2D_{1}}, \quad\forall(t,x)\in J\times[0,d], \end{aligned}$$
which shows that (H4)-(H6) hold.
By Corollary 3.2, problem (4.1) has at least three positive solutions \(x_{1}\), \(x_{2}\), and \(x_{3}\) such that
$$\|x_{1}\|< d, \qquad a<\beta(x_{2}),\quad \mbox{and}\quad x_{3}>d \quad \mbox{with }\beta(x_{3})<a. $$

Remark 4.1

In Example 4.1, we consider the norm of \(L^{\infty }[0,1]\), which is different from that used in [28, 31, 36, 45].

Declarations

Acknowledgements

This work is sponsored by the project NSFC (11301178) and the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58). The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, Republic of China

References

  1. Yan, J: Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model. J. Math. Anal. Appl. 279, 111-120 (2003) View ArticleMATHMathSciNetGoogle Scholar
  2. Yan, J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters. J. Math. Anal. Appl. 327, 854-868 (2007) View ArticleMATHMathSciNetGoogle Scholar
  3. Agarwal, RP, O’Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114, 51-59 (2000) View ArticleMATHMathSciNetGoogle Scholar
  4. Nieto, JJ, López, RR: Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl. 55, 2715-2731 (2008) View ArticleMATHMathSciNetGoogle Scholar
  5. Zhang, X, Feng, M: Transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations. J. Comput. Appl. Math. 271, 117-129 (2014) View ArticleMathSciNetMATHGoogle Scholar
  6. Ding, W, Han, M: Periodic boundary value problem for the second order impulsive functional differential equations. Appl. Math. Comput. 155, 709-726 (2004) View ArticleMATHMathSciNetGoogle Scholar
  7. Lin, X, Jiang, D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501-514 (2006) View ArticleMATHMathSciNetGoogle Scholar
  8. Liu, B, Yu, J: Existence of solution of m-point boundary value problems of second-order differential systems with impulses. Appl. Math. Comput. 125, 155-175 (2002) View ArticleMATHMathSciNetGoogle Scholar
  9. Feng, M: Positive solutions for a second-order p-Laplacian boundary value problem with impulsive effects and two parameters. Abstr. Appl. Anal. (2014). doi:10.1155/2014/534787 MathSciNetGoogle Scholar
  10. Feng, M, Du, B, Ge, W: Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian. Nonlinear Anal. TMA 70, 3119-3126 (2009) View ArticleMATHMathSciNetGoogle Scholar
  11. Ma, R, Yang, B, Wang, Z: Positive periodic solutions of first-order delay differential equations with impulses. Appl. Math. Comput. 219, 6074-6083 (2013) View ArticleMATHMathSciNetGoogle Scholar
  12. Zhang, X, Feng, M, Ge, W: 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. 233, 1915-1926 (2010) View ArticleMATHMathSciNetGoogle Scholar
  13. Ding, W, Wang, Y: New result for a class of impulsive differential equation with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 18, 1095-1105 (2013) View ArticleMATHMathSciNetGoogle Scholar
  14. Infante, G, Pietramala, P, Zima, M: Positive solutions for a class of nonlocal impulsive BVPs via fixed point index. Topol. Methods Nonlinear Anal. 36, 263-284 (2010) MATHMathSciNetGoogle Scholar
  15. Jankowski, T: Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. Nonlinear Anal. TMA 74, 3775-3785 (2011) View ArticleMATHMathSciNetGoogle Scholar
  16. Liu, Y, O’Regan, D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 1769-1775 (2011) View ArticleMATHMathSciNetGoogle Scholar
  17. Hao, X, Liu, L, Wu, Y: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 16, 101-111 (2011) View ArticleMATHMathSciNetGoogle Scholar
  18. Sun, J, Chen, H, Yang, L: The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method. Nonlinear Anal. TMA 73, 440-449 (2010) View ArticleMATHMathSciNetGoogle Scholar
  19. Shen, J, Wang, W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. TMA 69, 4055-4062 (2008) View ArticleMATHMathSciNetGoogle Scholar
  20. Bai, L, Dai, B: Three solutions for a p-Laplacian boundary value problem with impulsive effects. Appl. Math. Comput. 217, 9895-9904 (2011) View ArticleMATHMathSciNetGoogle Scholar
  21. Xu, J, Kang, P, Wei, Z: Singular multipoint impulsive boundary value problem with p-Laplacian operator. J. Appl. Math. Comput. 30, 105-120 (2009) View ArticleMATHMathSciNetGoogle Scholar
  22. Ning, P, Huan, Q, Ding, W: Existence result for impulsive differential equations with integral boundary conditions. Abstr. Appl. Anal. (2013). doi:10.1155/2013/134691 MathSciNetMATHGoogle Scholar
  23. Sun, J, Wang, X: Monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. Math. Probl. Eng. (2011). doi:10.1155/2011/609189 MathSciNetMATHGoogle Scholar
  24. Yao, Q: Positive solutions of nonlinear beam equations with time and space singularities. J. Math. Anal. Appl. 374, 681-692 (2011) View ArticleMATHMathSciNetGoogle Scholar
  25. Yao, Q: Local existence of multiple positive solutions to a singular cantilever beam equation. J. Math. Anal. Appl. 363, 138-154 (2010) View ArticleMATHMathSciNetGoogle Scholar
  26. O’Regan, D: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 161, 78-116 (1991) View ArticleMATHMathSciNetGoogle Scholar
  27. Wei, Z: A class of fourth order singular boundary value problems. Appl. Math. Comput. 153, 865-884 (2004) View ArticleMATHMathSciNetGoogle Scholar
  28. Yang, B: Positive solutions for the beam equation under certain boundary conditions. Electron. J. Differ. Equ. 2005, 78 (2005) MathSciNetMATHGoogle Scholar
  29. Zhang, X: Existence and iteration of monotone positive solutions for an elastic beam equation with a corner. Nonlinear Anal., Real World Appl. 10, 2097-2103 (2009) View ArticleMATHMathSciNetGoogle Scholar
  30. Gupta, GP: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26, 289-304 (1988) View ArticleMATHMathSciNetGoogle Scholar
  31. Anderson, DR, Avery, RI: A fourth-order four-point right focal boundary value problem. Rocky Mt. J. Math. 36(2), 367-380 (2006) View ArticleMATHMathSciNetGoogle Scholar
  32. Graef, JR, Yang, B: On a nonlinear boundary value problem for fourth order equations. Appl. Anal. 72, 439-448 (1999) View ArticleMATHMathSciNetGoogle Scholar
  33. Agarwal, RP: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ. 2, 91-110 (1989) MATHMathSciNetGoogle Scholar
  34. Davis, J, Henderson, J: Uniqueness implies existence for fourth-order Lidstone boundary value problems. Panam. Math. J. 8, 23-35 (1998) MATHMathSciNetGoogle Scholar
  35. Kosmatov, N: Countably many solutions of a fourth order boundary value problem. Electron. J. Qual. Theory Differ. Equ. 2004, 12 (2004) MathSciNetMATHGoogle Scholar
  36. Zhang, X, Feng, M, Ge, W: Symmetric positive solutions for p-Laplacian fourth order differential equation with integral boundary conditions. J. Comput. Appl. Math. 222, 561-573 (2008) View ArticleMATHMathSciNetGoogle Scholar
  37. Bai, Z, Huang, B, Ge, W: The iterative solutions for some fourth-order p-Laplace equation boundary value problems. Appl. Math. Lett. 19, 8-14 (2006) View ArticleMATHMathSciNetGoogle Scholar
  38. Liu, X, Li, W: Existence and multiplicity of solutions for fourth-order boundary values problems with parameters. J. Math. Anal. Appl. 327, 362-375 (2007) View ArticleMATHMathSciNetGoogle Scholar
  39. Bonanno, G, Bella, B: A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. 343, 1166-1176 (2008) View ArticleMATHMathSciNetGoogle Scholar
  40. Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. Anal. 59, 225-231 (1995) View ArticleMATHMathSciNetGoogle Scholar
  41. Han, G, Xu, Z: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Anal. TMA 68, 3646-3656 (2008) View ArticleMATHMathSciNetGoogle Scholar
  42. Zhang, X, Ge, W: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 3553-3564 (2012) View ArticleMathSciNetMATHGoogle Scholar
  43. Zhai, C, Song, R, Han, Q: The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem. Comput. Math. Appl. 62, 2639-2647 (2011) View ArticleMATHMathSciNetGoogle Scholar
  44. Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. TMA 69, 3310-3321 (2008) View ArticleMATHMathSciNetGoogle Scholar
  45. Zhang, X, Ge, W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput. Math. Appl. 58, 203-215 (2009) View ArticleMATHMathSciNetGoogle Scholar
  46. Zhang, X, Liu, L: A necessary and sufficient condition of positive solutions for fourth order multi-point boundary value problem with p-Laplacian. Nonlinear Anal. TMA 68, 3127-3137 (2008) View ArticleMATHGoogle Scholar
  47. Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116, 415-426 (1986) View ArticleMATHMathSciNetGoogle Scholar
  48. Kang, P, Wei, Z, Xu, J: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Appl. Math. Comput. 206, 245-256 (2008) View ArticleMATHMathSciNetGoogle Scholar
  49. Xu, J, Yang, Z: Positive solutions for a fourth order p-Laplacian boundary value problem. Nonlinear Anal. TMA 74, 2612-2623 (2011) View ArticleMATHMathSciNetGoogle Scholar
  50. Webb, JRL, Infante, G, Franco, D: Positive solutions of nonlinear fourth-order boundary value problems with local and non-local boundary conditions. Proc. R. Soc. Edinb. 138, 427-446 (2008) View ArticleMATHMathSciNetGoogle Scholar
  51. Ma, H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal. TMA 68, 645-651 (2008) View ArticleMATHMathSciNetGoogle Scholar
  52. Zhang, X, Liu, L: Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator. J. Math. Anal. Appl. 336, 1414-1423 (2007) View ArticleMATHMathSciNetGoogle Scholar
  53. Feng, M: Multiple positive solutions of four-order impulsive differential equations with integral boundary conditions and one-dimensional p-Laplacian. Bound. Value Probl. (2011). doi:10.1155/2011/654871 MATHGoogle Scholar
  54. Cabada, A, Tersian, S: Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations. Bound. Value Probl. 2014, 105 (2014) View ArticleMathSciNetMATHGoogle Scholar
  55. Afrouzi, G, Hadjian, A, Radulescu, V: Variational approach to fourth-order impulsive differential equations with two control parameters. Results Math. (2013). doi:10.1007/s00025-013-0351-5 MathSciNetMATHGoogle Scholar
  56. Sun, J, Chen, H, Yang, L: Variational methods to fourth-order impulsive differential equations. J. Appl. Math. Comput. 35, 323-340 (2011) View ArticleMATHMathSciNetGoogle Scholar
  57. Xie, J, Luo, Z: Solutions to a boundary value problem of a fourth-order impulsive differential equation. Bound. Value Probl. 2013, 154 (2013) View ArticleMathSciNetMATHGoogle Scholar
  58. Zhang, X, Feng, M: Positive solutions for classes of multi-parameter fourth-order impulsive differential equations with one-dimensional singular p-Laplacian. Bound. Value Probl. 2014, 112 (2014) View ArticleMathSciNetMATHGoogle Scholar
  59. Ding, Y, O’Regan, D: Positive solutions for a second-order p-Laplacian impulsive boundary value problem. Adv. Differ. Equ. (2012). doi:10.1186/1687-1847-2012-159 MathSciNetGoogle Scholar
  60. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) MATHGoogle Scholar
  61. Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 222, 351-363 (2008) View ArticleMATHMathSciNetGoogle Scholar
  62. Leggett, R, Williams, L: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673-688 (1979) View ArticleMATHMathSciNetGoogle Scholar

Copyright

© Zhou and Zhang; licensee Springer 2015

Advertisement