New results for positive solutions of singular fourthorder fourpoint pLaplacian problem
 Minghe Pei^{1} and
 Libo Wang^{1}Email author
Received: 27 January 2016
Accepted: 27 April 2016
Published: 3 May 2016
Abstract
The existence and uniqueness of positive solutions are obtained for singular fourthorder fourpoint boundary value problem with pLaplace operator \([\varphi_{p}(u''(t))]''=f(t,u(t))\), \(0< t<1\), \(u(0)=0\), \(u(1)=au(\xi)\), \(u''(0)=0\), \(u''(1)=bu''(\eta)\), where \(f(t,u)\) is singular at \(t=0,1\) and \(u=0\). A fixed point theorem for mappings that are decreasing with respect to a cone in a Banach space plays a key role in the proof.
Keywords
MSC
1 Introduction
It is well known that the bending of elastic beam can be described by some fourthorder boundary value problems. There are extensive studies on fourthorder boundary value problems with diverse boundary conditions by using different methods, for instance, [1–28] and the references therein.
 (H_{1}):

\(f\in C((0,1)\times(0,\infty),[0,\infty))\), and \(f(t,x)\) is nonincreasing in x;
 (H_{2}):

For any constant \(\lambda>0\), \(0<\int_{0}^{1}H(s,s)f(s,\lambda s(1s))\,\mathrm{d}s<\infty\);
 (H_{3}):

There exist a continuous function \(a(t)\) in \([0,1]\) and a fixed positive number k such that \(a(t)\geq kt(1t)\), \(t \in [0,1]\), andwhere \(G(t,s)\), \(H(t,s)\) will be given in Section 2.$$\begin{aligned}& \int_{0}^{1}G(t,r)\varphi_{p}^{1} \biggl( \int_{0}^{1}H(r,s)f\bigl(s,a(s)\bigr)\,\mathrm{d}s \biggr)\,\mathrm{d}r:=b(t)\geq a(t), \quad t\in[0,1], \\& \int_{0}^{1}G(t,r)\varphi_{p}^{1} \biggl( \int_{0}^{1}H(r,s)f\bigl(s,b(s)\bigr)\,\mathrm{d}s \biggr)\,\mathrm{d}r\geq a(t),\quad t\in[0,1], \end{aligned}$$
The purpose of this paper is to improve the existence results of [25]. Using a fixed point theorem for mappings that are decreasing with respect to a cone in a Banach space, we obtain the existence and uniqueness of positive solutions of SBVP (1.1)(1.2). We note that, in our proofs, we just assume that (H_{1}) and (H_{2}) of [25] with (H_{3}) of [25] removed. Our study is motivated by the papers [11, 29].
In addition, we note that we also obtained the uniqueness of a positive solution for SBVP (1.1)(1.2).
The rest of the paper is organized as follows. The fixed point theorem of Gatica et al. [29] and some definitions and lemmas are given in Section 2. The main results on the existence of positive solutions for SBVP (1.1)(1.2) are presented in Section 3.
2 Preliminary
 (i)
\(a u+b v \in K\) for all \(u,v\in K\) and all \(a, b \geq0\);
 (ii)
\(u, u \in K\) imply \(u=0\).
Given a cone K, a partial order ⪯ is induced on B as follows; \(u\preceq v\) for \(u,v\in B\) iff \(vu\in K\) (for clarity, we sometimes write \(u \preceq v\) (w.r.t. K)). For \(u,v\in B\) with \(u\preceq v\), we denote by \(\langle u,v\rangle\) the closed order interval between u and v, that is, \(\langle u,v\rangle=\{w\in B: u\preceq w \preceq v\}\). A cone K is normal in B if there exists \(\delta>0\) such that \(\e_{1}+e_{2}\\geq\delta\) for all \(e_{1}, e_{2} \in K\) with \(\e_{1}\=\e_{2}\=1\).
Lemma 2.1
 (I)
\(Tu_{0}\preceq u_{0}\) and \(T^{2}u_{0}\preceq u_{0}\) or \(u_{0}\preceq Tu_{0}\) and \(u_{0}\preceq T^{2}u_{0}\), or
 (II)
the complete sequence of iterates \(\{T^{n}u_{0}\} _{n=0}^{\infty}\) is defined and there exists \(v_{0}\in D\) such that \(Tv_{0}\in D\) and \(v_{0}\preceq T^{n}u_{0}\) for all \(n\geq0\).
Lemma 2.2
([25])
3 Main results
In this section, we first establish an existence theorem of positive solutions for SBVP (1.1)(1.2) by applying Lemma 2.1.
We now state and prove our existence result for SBVP (1.1)(1.2).
Theorem 3.1
Assume that conditions (H_{1}) and (H_{2}) are satisfied. Then SBVP (1.1)(1.2) has at least one positive solution \(u^{*}\in D\).
Proof
In addition, observe that for all n and \(u\in K\), \(T_{n}u\) satisfies the boundary conditions (1.2). Furthermore, for each n, since \(T_{n}\) satisfies (H_{1}), it follows that \(T_{n}\) is nonincreasing relative to the cone K. Also, it is clear that \(0\preceq T_{n}(0)\) and \(0\preceq T_{n}^{2}(0)\) for each n. Thus, by Lemma 2.1, for each n, there exists \(u_{n}\in K\) such that \(T_{n}u_{n}=u_{n}\). Hence, for each n, \(u_{n}(t)\) satisfies the boundary conditions (1.2).
Theorem 3.2
Assume that conditions (H_{1}) and (H_{2}) are satisfied. Then SBVP (1.1)(1.2) has exactly one positive solution \(u^{*}\in D\).
Proof
The existence of positive solution to SBVP (1.1)(1.2) immediately follows from Theorem 3.1. Thus, we only need to show the uniqueness.
Let \(w(t)=u_{1}(t)u_{2}(t)\) on \([0,1]\). Without loss of generality, we may assume that \(w(1)\geq0\). Now we show that \(w(t)\equiv0\) on \([0,1]\). There are two cases to consider.
In summary, \(w(t)\equiv0\) on \([0,1]\). This completes the proof of the theorem. □
Declarations
Acknowledgements
The authors thank the referee for valuable suggestions, which led to improvement of the original manuscript. This work was supported by the National Natural Science Foundation of China (11201008) and the Education Department of JiLin Province ([2016]45).
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
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