 Research
 Open Access
The method of lower and upper solutions for fourth order equations with the Navier condition
 Ruyun Ma^{1}Email author,
 Jinxiang Wang^{1} and
 Dongliang Yan^{1}
 Received: 30 May 2017
 Accepted: 2 October 2017
 Published: 19 October 2017
Abstract
Keywords
 elastic beam
 fourth order equations
 lower and upper solutions
 Green function
MSC
 34B10
 34B18
1 Introduction
 (H1):

\(k_{1}\) and \(k_{2}\) are two constants with$$ k_{2}< k_{1}< 0. $$(1.3)
Definition 1.1
([4])
Theorem A
([4, Theorem 7])
 (H2):

\(k_{1}<0<k_{2}\).
Roughly speaking, for some kind of second order boundary value problems, it is well known that the existence of a lower solution α and an upper solution β, which are well ordered, that is, \(\alpha\leq\beta\), implies the existence of a solution between them (see [5]). However, the use of lower and upper solutions in boundary value problems of the fourth order, even for the simple boundary conditions (1.2), is heavily dependent on the positiveness properties for the corresponding linear operators, see the counterexample in [5, Remark 3.1].
For the related results on the existence and multiplicity of positive solutions or signchanging solutions for fourth order problems, see Bai and Wang [6], Chu and O’Regan [7], Cid et al. [8], Drábek and Holubová [9, 10], Hernandez and Manasevich [11], Korman [12], Liu and Li [13], Ma et al. [14–18], Rynne [19, 20], Schröder [3], Webb et al. [21], Yang [22] and Yao [23] and the references therein.
The rest of the paper is arranged as follows. In Section 2, we show that the Green function of (1.4) possesses the positiveness properties under the condition \(k_{1}<0<k_{2}<\pi^{2}\). Finally, in Section 3, we develop the method of lower and upper solutions for (1.1), (1.2) under some monotonic condition on the nonlinearity f, and give some applications of our main results.
2 Green function in the case \(k_{1}<0<k_{2}\)
Firstly, we construct the Green function \(G(x,s)\) for \(Ly=0\).
Theorem 2.1
Proof
3 Method of lower and upper solutions
In this section, we will establish the method of lower and upper solutions for (1.1), (1.2) in the case \(k_{1}<0<k_{2}\).
Lemma 3.1
 (1)
Let \(0 < r < \infty\) and \(0< m<\pi\). Then \(w(x)> 0\) for \(x\in(0,1)\).
 (2)
Let \(0 < r < \infty\) and \(0< m<\pi\). Then \(\chi(x)< 0\) for \(x\in(0,1)\).
Proof
(1) Since \(r(1x)>0\) and \(\infty< m(1x)<\pi\) for \(x\in (0,1)\), it follows from (3.6) that \(w(x)> 0\) for \(x\in(0,1)\).
As a direct consequence of the Schauder fixed point theorem [4, Theorem 5], we have the following lemma.
Lemma 3.2
Theorem 3.1
Proof
Therefore, \(\alpha(x)\leq y(x)\leq\beta(x)\) for \(x\in[0,1]\), and accordingly, y is a solution of (1.1), (1.2). □
Remark 3.1
It is worth remarking that if (3.13) is not valid, then the existence of a lower solution α and an upper solution β with \(\alpha (x)\leq\beta(x)\) in \([0,1]\) cannot guarantee the existence of solutions in the order interval \([\alpha(x), \beta(x)]\). Let us see the counterexample in Cabada et al. [5, Remark 3.1].
Remark 3.2
In the case \(\vert k_{1} \vert > k_{2}\), the assertions of Theorem 3.1 can be deduced from Habets and Sanchez [24, Theorem 4.1].
Remark 3.3
Declarations
Acknowledgements
This work was supported by NSFC (No. 11671322) and NSFC (No. 11361054). The authors are very grateful to the anonymous referees for their valuable suggestions.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Authors’ contributions
RM and JW completed the main study together. RM wrote the manuscript, DY checked the proofs process and verified the calculation. Moreover, all the authors read and approved the last version of the manuscript.
Competing interests
All of the authors of this article claim that together they have no competing interests.
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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