Positive solutions of singular beam equations with the bending term
 Xuemei Zhang^{1}Email author and
 Meiqiang Feng^{2}
Received: 25 November 2014
Accepted: 12 May 2015
Published: 26 May 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.
Keywords
1 Introduction
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.
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
 (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]
Lemma 2.2
[11]
Lemma 2.3
[11]
Lemma 2.4
[11]
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 \(\x\=\max_{t\in J}x(t)\).
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
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
 (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\\), \(\forall x\in P\cap\partial\Omega_{2}\),
3 Main results
 (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)
Theorem 3.1
 (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)
Proof
Let T be the cone preserving, completely continuous operator that was defined by (2.14).
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).
Further, it follows from (3.3) and (3.4) that (3.5) holds. □
 (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)
Theorem 3.2
 (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)
Proof
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
 (H_{6}):

Let l, ζ and L satisfy$$0< l< \biggl(\frac{\gamma}{4}+1\biggr)l< \zeta< \biggl(\frac{\gamma}{4}+1\biggr) \zeta< \delta L< L. $$
 (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)
Proof
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
 (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.
Declarations
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.
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
References
 Gupta, CP: Existence and uniqueness theorems for a bending of an elastic beam equation. Appl. Anal. 26, 289304 (1988) View ArticleMATHMathSciNetGoogle Scholar
 Agarwal, RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Singapore (1986) View ArticleMATHGoogle Scholar
 Liu, LS, Zhang, XG, Wu, YH: Positive solutions of fourthorder nonlinear singular SturmLiouville eigenvalue problems. J. Math. Anal. Appl. 326, 12121224 (2007) View ArticleMATHMathSciNetGoogle Scholar
 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) View ArticleMATHMathSciNetGoogle Scholar
 Anderson, DR, Avery, RI: A fourthorder fourpoint right focal boundary value problem. Rocky Mt. J. Math. 36(2), 367380 (2006) View ArticleMATHMathSciNetGoogle Scholar
 Ma, TF: Positive solutions for a beam equation on a nonlinear elastic foundation. Math. Comput. Model. 39, 11951201 (2004) View ArticleMATHMathSciNetGoogle Scholar
 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) View ArticleMATHMathSciNetGoogle Scholar
 Bai, ZB: The method of lower and upper solutions for a bending of an elastic beam equation. J. Math. Anal. Appl. 248, 95202 (2000) View ArticleMathSciNetGoogle Scholar
 Zhang, XG, Liu, LS: Positive solutions of fourthorder fourpoint boundary value problems with pLaplacian operator. J. Math. Anal. Appl. 336, 14141423 (2007) View ArticleMATHMathSciNetGoogle Scholar
 Zhang, XM, Feng, MQ, Ge, WG: Existence results for nonlinear boundaryvalue problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 69, 33103321 (2008) View ArticleMATHMathSciNetGoogle Scholar
 Zhang, XM, Ge, WG: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 35533564 (2012) View ArticleMathSciNetMATHGoogle Scholar
 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
 Yang, L, Chen, H, Yang, X: The multiplicity of solutions for fourthorder equations generated from a boundary condition. Appl. Math. Lett. 24, 15991603 (2011) View ArticleMATHMathSciNetGoogle Scholar
 Cabada, A, Tersian, S: Multiplicity of solutions of a two point boundary value problem for a fourthorder equation. Appl. Math. Comput. 219, 52615267 (2013) View ArticleMATHMathSciNetGoogle Scholar
 Han, G, Xu, Z: Multiple solutions of some nonlinear fourthorder beam equations. Nonlinear Anal. 68, 36463656 (2008) View ArticleMATHMathSciNetGoogle Scholar
 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) View ArticleMATHMathSciNetGoogle Scholar
 Li, Y: On the existence of positive solutions for the bending elastic beam equations. Appl. Math. Comput. 189, 821827 (2007) View ArticleMATHMathSciNetGoogle Scholar
 Zhang, XM, Ge, WG: Positive solutions for a class of boundaryvalue problems with integral boundary conditions. Comput. Math. Appl. 58, 203215 (2009) View ArticleMATHMathSciNetGoogle Scholar
 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) View ArticleMATHMathSciNetGoogle Scholar
 Wong, PJY: Triple solutions of complementary Lidstone boundary value problems via fixed point theorems. Bound. Value Probl. 2014, 125 (2014) View ArticleMathSciNetMATHGoogle Scholar
 Cannon, JR: The solution of the heat equation subject to the specification of energy. Q. Appl. Math. 21, 155160 (1963) MathSciNetMATHGoogle Scholar
 Ionkin, NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differ. Equ. 13, 294304 (1977) MATHMathSciNetGoogle Scholar
 Chegis, RY: Numerical solution of a heat conduction problem with an integral boundary condition. Liet. Mat. Rink. 24, 209215 (1984) MATHMathSciNetGoogle Scholar
 Boucherif, A: Secondorder boundary value problems with integral boundary conditions. Nonlinear Anal. 70, 364371 (2009) View ArticleMATHMathSciNetGoogle Scholar
 Wang, YQ, Liu, LS, Wu, YH: Positive solutions for a nonlocal fractional differential equation. Nonlinear Anal. 74, 35993605 (2011) View ArticleMATHMathSciNetGoogle Scholar
 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) View ArticleMATHMathSciNetGoogle Scholar
 Webb, JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Anal. 47, 43194332 (2001) View ArticleMATHMathSciNetGoogle Scholar
 Jiang, JQ, Liu, LS, Wu, YH: Secondorder nonlinear singular SturmLiouville problems with integral boundary conditions. Appl. Math. Comput. 215, 15731582 (2009) View ArticleMATHMathSciNetGoogle Scholar
 Liu, LS, Hao, XA, Wu, YH: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 57, 836847 (2013) View ArticleMATHMathSciNetGoogle Scholar
 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) View ArticleMATHMathSciNetGoogle Scholar
 Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) MATHGoogle Scholar