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.
Similar content being viewed by others
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:
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:
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:
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:
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
Then \(PC[0,1]\) is a real Banach space with norm
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
-
(i)
\(au+bv \in P\) for all \(u, v \in P\) and all \(a\geq0\), \(b\geq0\) and
-
(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
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
we convert problem (1.1) into
and
Lemma 2.1
Assume that (H1)-(H3) hold. Then problem (2.2) has a unique solution y given by
where
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
where
Remark 2.1
From (2.5) and (2.11), we can obtain
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
where
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
where
Remark 2.2
From (2.14) and (2.19), we can obtain
Remark 2.3
From (2.20), one can prove that
Suppose that x is a solution of problem (1.1). Then from Lemma 2.1 and Lemma 2.2, we have
Define a cone in \(PC[0,1]\) by
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
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
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
-
(i)
\(\{x\in K(\beta,a,b): \beta(x)>a\} \neq\emptyset\) and \(\beta(Ax)>a\) for \(x\in K(\beta,a,b)\);
-
(ii)
\(\|Ax\|< d\) for \(\|x\|\leq d\);
-
(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
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
Let \(f\in L^{1}[a,b]\), \(g\in L^{\infty}[a,b]\). Then \(fg\in L^{1}[a,b]\) and
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:
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
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
Let
Then
Set
Then, for \(x\in\bar{K}_{c}\), it follows from (2.19), (2.22), and (3.1) that
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
This shows that
Therefore, it follows from (H5) that
If \(x\in\bar{K}_{d}\), then it follows from (H6) that
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
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
□
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
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
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
Proof
Set
where \(0<\sigma'<\frac{1}{2D'}\). Then, for \(x\in\bar{K}_{c'}\), it follows from (2.19), (2.22), and (3.1) that
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
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.
-
(i)
Triple positive solutions are available.
-
(ii)
\(I_{k}\neq0\) (\(k=1, 2, \ldots, m\)) is considered.
-
(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:
where \(\omega(t)=2t+3\in L^{1}[0,1]\), \(g(t)=h(t)=t\), \(I_{1}(t,x)=\frac {tx}{20\delta}\),
Thus it is easy to see by calculating that \(\omega(t)\geq n=3\) for a.e. \(t\in J\), and
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
Thus, we have
Choosing \(0< d<a<8a\leq c\), we have
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
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].
References
Yan, J: Existence and global attractivity of positive periodic solution for an impulsive Lasota-Wazewska model. J. Math. Anal. Appl. 279, 111-120 (2003)
Yan, J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters. J. Math. Anal. Appl. 327, 854-868 (2007)
Agarwal, RP, O’Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114, 51-59 (2000)
Nieto, JJ, López, RR: Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl. 55, 2715-2731 (2008)
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)
Ding, W, Han, M: Periodic boundary value problem for the second order impulsive functional differential equations. Appl. Math. Comput. 155, 709-726 (2004)
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)
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)
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
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)
Ma, R, Yang, B, Wang, Z: Positive periodic solutions of first-order delay differential equations with impulses. Appl. Math. Comput. 219, 6074-6083 (2013)
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)
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)
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)
Jankowski, T: Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. Nonlinear Anal. TMA 74, 3775-3785 (2011)
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)
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)
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)
Shen, J, Wang, W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. TMA 69, 4055-4062 (2008)
Bai, L, Dai, B: Three solutions for a p-Laplacian boundary value problem with impulsive effects. Appl. Math. Comput. 217, 9895-9904 (2011)
Xu, J, Kang, P, Wei, Z: Singular multipoint impulsive boundary value problem with p-Laplacian operator. J. Appl. Math. Comput. 30, 105-120 (2009)
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
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
Yao, Q: Positive solutions of nonlinear beam equations with time and space singularities. J. Math. Anal. Appl. 374, 681-692 (2011)
Yao, Q: Local existence of multiple positive solutions to a singular cantilever beam equation. J. Math. Anal. Appl. 363, 138-154 (2010)
O’Regan, D: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 161, 78-116 (1991)
Wei, Z: A class of fourth order singular boundary value problems. Appl. Math. Comput. 153, 865-884 (2004)
Yang, B: Positive solutions for the beam equation under certain boundary conditions. Electron. J. Differ. Equ. 2005, 78 (2005)
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)
Gupta, GP: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26, 289-304 (1988)
Anderson, DR, Avery, RI: A fourth-order four-point right focal boundary value problem. Rocky Mt. J. Math. 36(2), 367-380 (2006)
Graef, JR, Yang, B: On a nonlinear boundary value problem for fourth order equations. Appl. Anal. 72, 439-448 (1999)
Agarwal, RP: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ. 2, 91-110 (1989)
Davis, J, Henderson, J: Uniqueness implies existence for fourth-order Lidstone boundary value problems. Panam. Math. J. 8, 23-35 (1998)
Kosmatov, N: Countably many solutions of a fourth order boundary value problem. Electron. J. Qual. Theory Differ. Equ. 2004, 12 (2004)
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)
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)
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)
Bonanno, G, Bella, B: A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. 343, 1166-1176 (2008)
Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. Anal. 59, 225-231 (1995)
Han, G, Xu, Z: Multiple solutions of some nonlinear fourth-order beam equations. Nonlinear Anal. TMA 68, 3646-3656 (2008)
Zhang, X, Ge, W: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 3553-3564 (2012)
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)
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)
Zhang, X, Ge, W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput. Math. Appl. 58, 203-215 (2009)
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)
Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl. 116, 415-426 (1986)
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)
Xu, J, Yang, Z: Positive solutions for a fourth order p-Laplacian boundary value problem. Nonlinear Anal. TMA 74, 2612-2623 (2011)
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)
Ma, H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal. TMA 68, 645-651 (2008)
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)
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
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)
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
Sun, J, Chen, H, Yang, L: Variational methods to fourth-order impulsive differential equations. J. Appl. Math. Comput. 35, 323-340 (2011)
Xie, J, Luo, Z: Solutions to a boundary value problem of a fourth-order impulsive differential equation. Bound. Value Probl. 2013, 154 (2013)
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)
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
Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
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)
Leggett, R, Williams, L: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ. Math. J. 28, 673-688 (1979)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All results belong to YZ and XZ. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Zhou, Y., Zhang, X. Triple positive solutions of fourth-order impulsive differential equations with integral boundary conditions. Bound Value Probl 2015, 2 (2015). https://doi.org/10.1186/s13661-014-0263-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-014-0263-7