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 Open Access
The method of lower and upper solutions for the cantilever beam equations with fully nonlinear terms
 Yongxiang Li^{1}Email authorView ORCID ID profile and
 Yabing Gao^{1}
https://doi.org/10.1186/s1366001920885
© The Author(s) 2019
 Received: 21 August 2018
 Accepted: 2 May 2019
 Published: 14 May 2019
Abstract
Keywords
 Fully fourthorder boundary value problem
 Cantilever beam equation
 Lower and upper solution
 Existence
MSC
 34B15
 34B18
 47N20
1 Introduction
We will use the method of lower and upper solutions to discuss BVP (1.1). For BVP (1.1), since the boundary conditions are different from BVP (1.5), the definitions of lower and upper solutions are different from those in [16] and the argument methods in [16] are not applicable to BVP (1.1). In Sect. 2, under \(f(t, x_{0}, x_{1}, x_{2}, x _{3})\) increasing on \(x_{0}\), \(x_{1}\), \(x_{2}\) and decreasing on \(x_{3}\) in the domain surrounded by lower and upper solutions, we use a monotone iterative technique to obtain the existence of a solution between lower and upper solutions. In Sect. 3, under \(f(t, x_{0}, x_{1}, x_{2}, x_{3})\) without monotonicity on \(x_{3}\), we use a truncating technique to prove the existence of a solution between lower and upper solutions. In Sect. 4, we use the lower and upper theorem built in Sect. 3 to obtain a new existence result of positive solution.
2 Monotone iterative method
The monotone iterative method is an important method for solving nonlinear BVPs. For the special BVP (1.3), a monotone iterative method has been built, see [8]. In this section, we will develop the monotone iterative method of lower and upper solutions for BVP (1.1).
Let \(I=[0, 1]\) and \(C(I)\) denote the Banach space of all continuous functions \(u(t)\) on I with norm \(\u\_{C}=\max_{t\in I}u(t)\). Generally, for \(n\in \mathbb{N}\), we use \(C^{n}(I)\) to denote the Banach space of all nthorder continuous differentiable functions on I with the norm \(\u\_{C^{n}}=\max \{ \u\_{C}, \u'\_{C}, \ldots , \u^{(n)}\_{C}\}\). Let \(C^{+}(I)\) denote the cone of all nonnegative functions in \(C(I)\).
Lemma 2.1
Proof
Lemma 2.2
For every \(h\in C(I)\), LBVP (2.4) has a unique solution \(u:=S h\in C ^{4}(I)\). Moreover, the solution operator \(S: C(I)\to C^{3}(I)\) is a completely continuous linear operator.
Proof
Lemma 2.3
Proof
Theorem 2.1
 (F1):

for every \(t\in I\) and \(x_{3}\in [w_{0}'''(t), v_{0}'''(t)]\), \(f(t, x_{0}, x_{1}, x_{2}, x_{3})\) is increasing on \(x_{0}\), \(x _{1}\), and \(x_{2}\) in \([v_{0}(t), w_{0}(t)]\times [v_{0}'(t), w_{0}'(t)] \times [v_{0}''(t), w_{0}''(t)]\);
 (F2):

for every \(t\in I\) and \((x_{0},x_{1},x_{2})\in [v_{0}(t),w_{0}(t)] \times [v_{0}'(t),w_{0}'(t)]\times [v_{0}''(t),w_{0}''(t)]\), \(f(t, x_{0}, x_{1}, x_{2}, x_{3})\) is decreasing on \(x_{3}\) in \([w_{0}'''(t), v_{0}'''(t)]\).
Proof
Example 2.1
3 A theorem of lower and upper solutions
In this section, we discuss the existence of a solution between a lower solution and an upper solution for BVP (1.1) under the case of nonlinearity \(f(t,x_{0},x_{1},x_{2},x_{3})\) without monotonicity on \(x_{3}\). In [15], an existence result between a lower solution and an upper solution was established for BVP (1.5), in which the authors requested nonlinearity \(f(t,x_{0},x_{1},x_{2},x_{3})\) to satisfy a Nagumotype condition on \(x_{3}\), see [15, Theorem 3.1]. Since the boundary conditions and the definitions of lower and upper solutions of BVP (1.5) are different from those of BVP (1.1), the results presented in [15] are not applicable to BVP (1.1). We will use a directly truncating function technique to establish a similar existence result. A remarkable difference is that our existence result does not need the Nagumotype condition. Our result is as follows:
Theorem 3.1
 (F3):

for any \(t\in I\) and \((x_{0}, x_{1}, x_{2})\in [v_{0}(t), w _{0}(t)]\times [v_{0}'(t), w_{0}'(t)]\times [v_{0}''(t), w_{0}''(t)]\),$$\begin{aligned}& f\bigl(t, x_{0}, x_{1}, x_{2}, v_{0}'''(t)\bigr) \ge f \bigl(t, v_{0}(t), v _{0}'(t), v_{0}''(t), v_{0}'''(t) \bigr), \\& f\bigl(t, x_{0}, x_{1}, x_{2}, w_{0}'''(t)\bigr) \le f \bigl(t, w_{0}(t), w _{0}'(t), w_{0}''(t), w_{0}'''(t) \bigr); \end{aligned}$$
In Theorem 3.1, condition (F3) is weaker than condition (F1) of Theorem 2.1, and Theorem 3.1 does not need the monotonicity of \(f(t, x_{0}, x_{1}, x_{2}, x_{3})\) on \(x_{3}\). For the existence, Theorem 3.1 is more applicable than Theorem 2.1, but it has no monotone iterative procedure of seeking solutions. The proof of Theorem 3.1 needs the following lemma.
Lemma 3.1
Let \(f: I\times \mathbb{R}^{4}\to \mathbb{R}\) be continuous and bounded. Then BVP (1.1) has at least one solution \(u\in C^{4}(I)\).
Proof
Proof of Theorem 3.1
Example 3.1
4 Existence of positive solutions
In [17], the present first author have discussed the existence of positive solution of BVP (1.1) by using fixed point theory in cones. In this section we present a different existence result of positive for BVP (1.1) by Theorem 3.1.
Theorem 4.1
 (F4):

for every \(t\in I\) and \(x_{3}\in (\infty , 0]\), \(f(t, x_{0}, x_{1}, x_{2}, x_{3})\) is increasing on \(x_{0}\), \(x_{1}\), and \(x_{2}\) in \([0, +\infty )\);
 (F5):

there exists a positive constant \(\delta >0\) such that$$ f(t, x_{0}, x_{1}, x_{2}, x_{3}) \ge 21 x_{0} \quad \textit{for all }(t, x_{0}, x_{1}, x_{2}, x_{3})\in I\times [0, \delta ]^{3}\times [ \delta , 0]; $$
 (F6):

there exist nonnegative constants \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a _{3}\) satisfying \(a_{0}+a_{1}+a_{2}+a_{3}<1\) and a positive constant \(C_{0}>0\) such thatfor all \((t, x_{0}, x_{1}, x_{2}, x_{3})\in I\times [0, + \infty )^{3}\times (\infty , 0]\).$$ f(t, x_{0}, x_{1}, x_{2}, x_{3}) \le a_{0} x_{0}+a_{1} x_{1}+a _{2} x_{2}+a_{3} \vert x_{3} \vert +C_{0} $$
The proof of Theorem 4.1 needs the following existence and uniqueness result of a general fourthorder linear boundary value problem.
Lemma 4.1
Proof
Set \(K_{3}=\{ u\in C^{3}: u\succeq 0\}\). Then \(K_{3}\) is a closed convex cone in \(C^{3}(I)\). For every \(v\in K_{3}\), by the definition (4.7) of B, \(Bv\in C^{+}(I)\). By Lemma 2.3, \(SBv=S(Bv)\in K_{3}\). Hence, \(SB(K_{3})\subset K_{3}\). Let \(h\in C^{+}(I)\). By Lemma 2.3, \(v=Sh\in K_{3}\). Hence, for every \(n\in \mathbb{N}\), \((S B)^{n} S h=(SB)^{n} v\in K_{3}\). By (4.10) and the completeness of \(K_{3}\), \(u\in K_{3}\), that is, u satisfies (4.2). □
Proof of Theorem 4.1
Example 4.1
Consequently, the nonlinearity f of BVP (4.16) satisfies conditions (F4)–(F6). By Theorem 4.1, BVP (4.16) has at least one positive solution.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The work is supported by NNSFs of China (11661071, 11761063).
Authors’ contributions
YL and YG carried out the first draft of this manuscript, YL prepared the final version of the manuscript. All authors read and approved the final version of the manuscript.
Competing interests
The authors declare that 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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