 Research
 Open access
 Published:
Positive solutions of singular beam equations with the bending term
Boundary Value Problems volume 2015, Article number: 84 (2015)
Abstract
Using a new technique for dealing with the bending term of beam equations, we consider the existence and multiplicity of positive solutions for a beam equation. Besides achieving new results, upper and lower bounds for these positive solutions will also be provided. The results are shown by using a novel technique and fixed point theories.
1 Introduction
In this paper, we shall investigate the existence and multiplicity of positive solutions for the fourth order differential equation with integral boundary conditions
where \(a,b>0\), \(F:[0,1]\times R\times R\rightarrow R\) is continuous, and the \(y''\) in F is the bending moment term which represents bending effect. Problem (1.1) is often referred to as the deformation of an elastic beam under a variety of boundary conditions; for details, see [1–18]. Most research papers on beam equations consider nonlinear terms that F in (1.1) involves y only, and derivativedependent nonlinearities are seldom tackled; see [3–7, 9–11] to name a few.
Some classical tools such as fixed point theorems in cones [5, 8–11, 17, 18], the method of lower and upper solutions [8, 16], and the monotone iterative method [9, 12] and the theory of critical point theory and variational methods [6, 13, 14] have been widely used to study beam equations.
Some new techniques via appropriate transformation are proved to be very effective in studying the solvability of differential equations. Such techniques have attracted the attention of Zhang et al. [19] and Wong [20], etc. In [19], Zhang et al. considered the existence of positive solutions of the following problems:
where \(J=[0,1]\).
Firstly, by means of the transformation
the authors converted (1.2) into
and
Then it follows from Lemma 2.3 in [19] that they converted the results obtained for problem (1.2) to the counterpart for problem (1.3).
In [20], Wong transformed the following problems:
into
and
by using \(y^{\prime}(t)=x(t)\). So the existence of a solution of the complementary Lidstone boundary value problem (1.4) follows from the existence of a solution of the Lidstone boundary value problem (1.5). We notice that the above paper requires that F satisfies some assumptions of monotonicity which are essential for the technique used.
Being directly inspired by [19, 20], in the present paper, by using transformation techniques and fixed point theories, the authors shall prove some new and more general results of the existence of at least one or two positive solutions for problem (1.1). The main features of this paper are as follows. Firstly, comparing with [1–18], besides achieving new results, estimates on the norms of these solutions will also be provided. Secondly, we transform problem (1.1) into a differential system without bending term, i.e., the technique to deal with bending term is completely different from that of [8, 16–18]. Finally, it is pointed out that we do not need any monotone assumption on F, which is weaker than the corresponding assumptions on F in [20].
On the other hand, boundary value problems with integral boundary conditions arise naturally in thermal conduction problems [21], semiconductor problems [22], hydrodynamic problems [23] and so on. It is interesting to point out that such problems include two, three, multipoint and nonlocal boundary value problems as special cases and have been extensively studied in the last ten years (see, for example, [24–30]). Therefore, it is important to study fourth order elasticity problems with integral boundary conditions.
The rest of the paper is organized as follows. In Section 2, we provide some preliminaries and lemmas. In particular, we transform problem (1.1) into a differential system without the bending moment term. In Section 3, the main results will be stated and proved.
2 Preliminaries
To establish the existence of positive solutions for problem (1.1), let us list the following assumptions, which will hold throughout this paper:
 (H_{1}):

\(\omega\in C((0,1), [0,+\infty))\) with \(0<\int_{0}^{1}\omega (s)\,ds<\infty\) and ω does not vanish on any subinterval of \((0,1)\);
 (H_{2}):

\(F\in C([0,1]\times[0,+\infty)\times(\infty,0],[0,+\infty ))\);
 (H_{3}):

\(g, h\in L^{1}[0,1]\) are nonnegative and \(\mu\in[0,a)\), \(\nu\in[0,1)\), where
$$ \mu=\int_{0}^{1}g(s)\,ds,\qquad \nu=\int _{0}^{1}h(s)\,ds. $$(2.1)
Lemma 2.1
[11]
Assume that (H_{3}) holds. Then, for any \(x\in C[0,1]\), the boundary value problem
has a unique solution y given by
where
Taking into account (2.2) and (2.3), problem (1.1) reduces to the following problem:
Lemma 2.2
[11]
Assume that (H_{1})(H_{3}) hold. Then boundary value problem (2.6) has a unique solution x given by
where
If problem (2.6) has a solution \(x^{*}\), then by (2.3) problem (1.1) has a solution given by
So the existence of a solution of problem (1.1) follows from the existence of a solution of problem (2.6).
Lemma 2.3
[11]
For \(t,s\in J\), we have the following results:
Lemma 2.4
[11]
Let (H_{3}) hold. Then we obtain the following results:
where
It is clear from (2.3) that \(\y^{*}\\leq\frac{\gamma}{4}\x^{*}\\); moreover, if \(x^{*}\) is positive, so is \(y^{*}\).
Let \(E=C[0,1]\). It is well known that E is a real Banach space with the norm \(\\cdot\\) defined by \(\x\=\max_{t\in J}x(t)\).
Define an operator \(T:E\rightarrow E\) as follows:
Let C be a cone in E which is defined as
where
It is easy to see that E is a closed convex cone of E.
Lemma 2.5
Let (H_{1})(H_{3}) hold. Then we have \(T(C)\subset C\), and \(T:C\rightarrow C\) is completely continuous.
Proof
In view of condition (2.13), we see that
Moreover, it follows from (2.13) and (2.14) that
This proves that \(T(C)\subset C\).
Next, by standard methods and the AscoliArzelà theorem, one can prove that \(T:C\rightarrow C\) is completely continuous. □
To obtain positive solutions of problem (1.1), the following fixed point theorem in cones, which can be found in [31], p.94, is fundamental.
Lemma 2.6
Let P be a cone in a real Banach space E. Assume \(\Omega_{1}\), \(\Omega_{2}\) are bounded open sets in E with \(0 \in\Omega_{1}\), \(\bar{\Omega}_{1}\subset\Omega_{2}\). If
is completely continuous such that either

(a)
\(\Ax\\leq\x\\), \(\forall x\in P\cap\partial \Omega_{1}\) and \(\Ax\\geq\x\\), \(\forall x\in P\cap\partial\Omega_{2}\), or

(b)
\(\Ax\\geq\x\\), \(\forall x\in P\cap\partial \Omega_{1}\) and \(\Ax\\leq\x\\), \(\forall x\in P\cap\partial\Omega_{2}\),
then A has at least one fixed point in \(P\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\).
3 Main results
In this section, we apply Lemma 2.6 to establish the existence of positive solutions for problem (1.1). We begin by introducing the following conditions on \(F(t,u,v)\):
 (H_{4}):

There exist two positive constants r, R with \((\frac{\gamma }{4}+1)r<(\sigma+\delta)R\) such that:
$$\begin{aligned}& F(t,u,v)\leq\frac{1}{\rho_{2}\eta}r,\quad \forall t\in J, u+v\leq\biggl( \frac{\gamma}{4}+1\biggr)r, \end{aligned}$$(3.1)$$\begin{aligned}& F(t,u,v)\geq\frac{1}{\rho_{1}\eta\delta} R,\quad \forall t\in J, u+v\geq(\sigma+ \delta)R, \end{aligned}$$(3.2)
where
Theorem 3.1
Assume that (H_{1})(H_{4}) hold. Then we have the following conclusions:

(i)
Problem (2.6) has (at least) one positive solution \(x\in C\) such that
$$ \delta r\leq x(t)\leq\frac{1}{\delta} R, \quad t\in J. $$(3.3) 
(ii)
Problem (1.1) has (at least) one positive solution y such that
$$ \left \{ \begin{array}{@{}l} y(t)=\int_{0}^{1}H_{1}(t,s)x(s)\,ds, \quad t\in J;\\ \y\\leq\frac{\gamma}{4}\x\;\\ y(t)\geq\sigma\x\,\quad t\in J. \end{array} \right . $$(3.4)
We further have
Proof
Let T be the cone preserving, completely continuous operator that was defined by (2.14).
Let \(x\in C\) with \(\x\=r\). Then \(0\leq x(t)\leq r\), \(t\in J\), and \(0\leq\int_{0}^{1}H_{1}(s,\tau)x(\tau)\,d\tau\leq\frac{\gamma}{4}r\). And hence, for \(x\in C\) with \(\x\=r\), we have
Then it follows from (3.1) that
Now if we let \(\Omega_{1}=\{x\in C:\x\< r\}\), then (3.6) shows that
Further, let
and
Then \(x\in C\) and \(\x\=R_{1}\) imply
that is,
Hence, \(x(t)\geq R\) for all \(t\in J\), and
So
Using condition (3.2), it follows from \(x\in C\) and \(\x\=R_{1}\) that
that is, \(x\in\partial\Omega_{2}\) implies
It now follows from Lemma 2.6 that problem (2.6) has (at least) one positive solution \(x\in\bar{\Omega}_{2}\setminus\Omega_{1}\) satisfying (3.3).
It is observed from (2.3) that problem (1.1) has (at least) one positive solution y such that
Moreover, since \(x\in C\), we get for \(t\in J\)
Then we get (3.4).
Further, it follows from (3.3) and (3.4) that (3.5) holds. □
In Theorem 3.2 we assume the following condition on \(f(t,u,v)\):
 (H_{5}):

There exist two positive constants r, R with \((\frac{\gamma }{4}+1)r< R\) such that:
$$\begin{aligned}& F(t,u,v)\geq\frac{1}{\rho_{1}\eta(\sigma+\delta)}\bigl(u+v\bigr),\quad \forall t\in J, u+v\leq\biggl( \frac{\gamma}{4}+1\biggr)r, \end{aligned}$$(3.10)$$\begin{aligned}& F(t,u,v)\leq\frac{1}{2\rho_{2}\eta(\frac{\gamma}{4}+1)} \bigl(u+v\bigr),\quad \forall t\in J, u+v\geq R, \end{aligned}$$(3.11)
and write
Theorem 3.2
Assume that (H_{1})(H_{3}) and (H_{5}) hold. Then we have the following conclusions:

(i)
Problem (2.6) has (at least) one positive solution \(x\in C\) such that
$$ \delta r\leq x(t)\leq\max\{2R,2\rho_{2}\eta M\}, \quad t\in J. $$(3.13) 
(ii)
Problem (1.1) has (at least) one positive solution y such that (3.4) holds. We further have
$$ \sigma\delta r\leq y(t) \leq\max\biggl\{ \frac{\gamma R}{4(\sigma+\delta )},\frac{\rho_{2}\eta M\gamma}{2} \biggr\} , \quad t\in J. $$(3.14)
Proof
Let \(x\in C\) with \(\x\=r\). Then \(0\leq x(t)\leq r\), \(t\in J\), and \(0\leq\int_{0}^{1}H_{1}(s,\tau)x(\tau)\,d\tau\leq\frac{\gamma }{4}r\). And hence for \(x\in C\) with \(\x\=r\), we have
and it follows from condition (H_{5}) that
that is, \(x\in\partial\Omega_{1}\) implies that
Next, we turn to (3.11) and (3.12). From (3.11) and (3.12), we have
Further, let
and
Notice that for \(x\in\partial\Omega_{3}\) we have
Thus, for \(x\in\partial\Omega_{3}\), it follows from (3.17) that
that is, \(x\in\partial\Omega_{3}\) implies
It now follows from Lemma 2.6 that problem (2.6) has (at least) one positive solution \(x\in\bar{\Omega}_{3}\setminus\Omega_{1}\) satisfying (3.13).
It follows from (2.3) that problem (1.1) has (at least) one positive solution y. Similar to the proof of (3.5), one can show that y satisfies (3.14). □
Theorem 3.3
Assume that (H_{1})(H_{3}), (3.2) of (H_{4}) and (3.10) of (H_{5}) hold. In addition, letting f satisfies the following condition:
 (H_{6}):

Let l, ζ and L satisfy
$$0< l< \biggl(\frac{\gamma}{4}+1\biggr)l< \zeta< \biggl(\frac{\gamma}{4}+1\biggr) \zeta< \delta L< L. $$
If
then we have the following conclusions:

(i)
Problem (2.6) has (at least) two positive solutions \(x_{1},x_{2}\in C\) such that
$$ \delta l\leq x_{1}(t)< \zeta< \biggl(\frac{\gamma}{4}+1\biggr) \zeta< x_{2}(t)\leq L,\quad t\in J. $$(3.20) 
(ii)
Problem (1.1) has (at least) two positive solutions \(y_{1}\), \(y_{2}\) such that for \(i=1,2\),
$$ \left \{ \begin{array}{@{}l} y_{i}(t)=\int_{0}^{1}H_{1}(t,s)x_{i}(s)\,ds, \quad t\in J;\\ \y_{i}\\leq\frac{\gamma}{4}\x_{i}\;\\ y_{i}(t)\geq\sigma\x_{i}\, \quad t\in J. \end{array} \right . $$(3.21)
We further have
Proof
If (3.10) of (H_{5}) holds, similar to the proof of (3.16), we can prove that
If (3.2) of (H_{4}) holds, similar to the proof of (3.9), we have
Finally, we show that
In fact, for \(x\in C\) with \(\x\=\zeta\), we have
and
Therefore,
and hence it follows from (H_{6}) that
which shows that (3.25) holds.
Applying Lemma 2.6 to (3.23), (3.24) and (3.25) yields that problem (2.6) has (at least) two positive solutions \(x_{1}\), \(x_{2}\) with \(x_{1}\in C_{\bar{l},\zeta}=\{x \in C, l \leq\x\ < \zeta\}\), \(x_{2}\in C_{\zeta,\bar{L}}=\{x \in C, (\frac{\gamma}{4}+1)\zeta< \ x\ \leq L \}\). Hence, since for \(x_{1}\in C\) we have \(x_{1}(t)\geq \delta\x_{1}\\), \(t \in J\), it follows that (3.20) holds.
Similar to the proof of (3.4) and (3.5), one can show that (3.21) and (3.22) hold. □
Remark 3.1
In Theorems 3.13.3, we generalize the results of [8, 16–18] in three main directions as follows:

(1)
Upper and lower bounds for these positive solutions are given.

(2)
Estimates on the norms of positive solutions are considered.

(3)
The method to deal with the bending term of beam equations in this paper is completely different from that of [8, 16–18], which opens a new technique to study the beam equations with the bending term.
References
Gupta, CP: Existence and uniqueness theorems for a bending of an elastic beam equation. Appl. Anal. 26, 289304 (1988)
Agarwal, RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986)
Liu, LS, Zhang, XG, Wu, YH: Positive solutions of fourthorder nonlinear singular SturmLiouville eigenvalue problems. J. Math. Anal. Appl. 326, 12121224 (2007)
Graef, JR, Qian, C, Yang, B: A three point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl. 287, 217233 (2003)
Anderson, DR, Avery, RI: A fourthorder fourpoint right focal boundary value problem. Rocky Mt. J. Math. 36(2), 367380 (2006)
Ma, TF: Positive solutions for a beam equation on a nonlinear elastic foundation. Math. Comput. Model. 39, 11951201 (2004)
Yao, QL: Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right. Nonlinear Anal. 69, 15701580 (2008)
Bai, ZB: The method of lower and upper solutions for a bending of an elastic beam equation. J. Math. Anal. Appl. 248, 95202 (2000)
Zhang, XG, Liu, LS: Positive solutions of fourthorder fourpoint boundary value problems with pLaplacian operator. J. Math. Anal. Appl. 336, 14141423 (2007)
Zhang, XM, Feng, MQ, Ge, WG: Existence results for nonlinear boundaryvalue problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 69, 33103321 (2008)
Zhang, XM, Ge, WG: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 35533564 (2012)
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
Yang, L, Chen, H, Yang, X: The multiplicity of solutions for fourthorder equations generated from a boundary condition. Appl. Math. Lett. 24, 15991603 (2011)
Cabada, A, Tersian, S: Multiplicity of solutions of a two point boundary value problem for a fourthorder equation. Appl. Math. Comput. 219, 52615267 (2013)
Han, G, Xu, Z: Multiple solutions of some nonlinear fourthorder beam equations. Nonlinear Anal. 68, 36463656 (2008)
Ma, R, Zhang, J, Fu, S: The method of lower and upper solutions for fourthorder twopoint boundary value problems. J. Math. Anal. Appl. 215, 415422 (1997)
Li, Y: On the existence of positive solutions for the bending elastic beam equations. Appl. Math. Comput. 189, 821827 (2007)
Zhang, XM, Ge, WG: Positive solutions for a class of boundaryvalue problems with integral boundary conditions. Comput. Math. Appl. 58, 203215 (2009)
Zhang, XM, Yang, XZ, Ge, WG: Positive solutions of nthorder impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 71, 59305945 (2009)
Wong, PJY: Triple solutions of complementary Lidstone boundary value problems via fixed point theorems. Bound. Value Probl. 2014, 125 (2014)
Cannon, JR: The solution of the heat equation subject to the specification of energy. Q. Appl. Math. 21, 155160 (1963)
Ionkin, NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ. 13, 294304 (1977)
Chegis, RY: Numerical solution of a heat conduction problem with an integral boundary condition. Liet. Mat. Rink. 24, 209215 (1984)
Boucherif, A: Secondorder boundary value problems with integral boundary conditions. Nonlinear Anal. 70, 364371 (2009)
Wang, YQ, Liu, LS, Wu, YH: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 35993605 (2011)
Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal., Real World Appl. 9, 17271740 (2008)
Webb, JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47, 43194332 (2001)
Jiang, JQ, Liu, LS, Wu, YH: Secondorder nonlinear singular SturmLiouville problems with integral boundary conditions. Appl. Math. Comput. 215, 15731582 (2009)
Liu, LS, Hao, XA, Wu, YH: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 57, 836847 (2013)
Liu, LS, Liu, BM, Wu, YH: Nontrivial solutions for higherorder mpoint boundary value problem with a signchanging nonlinear term. Appl. Math. Comput. 217, 37923800 (2010)
Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Acknowledgements
This work is sponsored by the project NSFC (11301178), the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201511232018). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has 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
XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Zhang, X., Feng, M. Positive solutions of singular beam equations with the bending term. Bound Value Probl 2015, 84 (2015). https://doi.org/10.1186/s136610150348y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s136610150348y