 Research
 Open Access
New existence and uniqueness results for an elastic beam equation with nonlinear boundary conditions
 Shunyong Li^{1, 2} and
 Chengbo Zhai^{1}Email author
 Received: 30 March 2015
 Accepted: 29 May 2015
 Published: 19 June 2015
Abstract
In this article, we study the existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions, and some sufficient conditions which guarantee the existence of unique monotone positive solution are established. The methods employed are two fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of unique monotone positive solution, but also be applied to construct an iterative scheme for approximating it. Two examples are given to illustrate our main results.
Keywords
 existence and uniqueness
 monotone positive solution
 elastic beam equation
 fixed point theorem for mixed monotone operators
MSC
 34B18
 34B15
1 Introduction
Motivated by the work [17, 21], we will discuss the existence and uniqueness of monotone positive solutions for problem (1.1) by using two fixed point theorems for mixed monotone operators with perturbation. As we know, there are still very few works to utilize fixed point theorems of mixed monotone operators to study fourthorder boundary value problems. So it is worthwhile to investigate problem (1.1) and the methods used here are relatively new to the literature. The main features of this article are as follows. First, we consider the monotone positive solutions for fourthorder boundary value problems. Second, comparing with [16, 17, 19], we establish the existence and uniqueness of monotone positive solutions via different methods. Third, our results can not only guarantee the existence of a unique monotone positive solution, but also be applied to construct an iterative scheme for approximating it. In addition, few papers can be found in the literature on the existence and uniqueness of monotone positive solutions for fourthorder boundary value problems. Hence we improve the results of [17] to some degree, and so it is important to study the existence and uniqueness of monotone positive solutions for problem (1.1).
2 Preliminaries
In the following, for completeness we list some basic concepts in ordered Banach spaces and two fixed point theorems for mixed monotone operators which will be used later. For the convenience of readers, we refer them to [21–23] for details.
Let \((E,\\cdot\)\) be a real Banach space which is partially ordered by a cone \(P \subset E\), i.e., \(x \leq y\) if and only if \(yx \in P\). If \(x \leq y\) and \(x \neq y\), then we denote \(x < y\) or \(y > x\). By θ we denote the zero element of E. A nonempty closed convex set \(P \subset E\) is a cone if it satisfies (i) \(x \in P\), \(\lambda\geq0 \Rightarrow\lambda x \in P\); (ii) \(x \in P\), \(x \in P \Rightarrow x=\theta\).
P is called normal if there is a constant \(N > 0\) such that, for all \(x,y \in E\), \(\theta\leq x \leq y\) implies \(\ x \\leq N \ y \\); in this case N is the infimum of such constants, it is called the normality constant of P. If \(x_{1},x_{2} \in E\), the set \([x_{1},x_{2}]=\{x \in E \mid x_{1}\leq x \leq x_{2}\}\) is called the order interval between \(x_{1}\) and \(x_{2}\). We say that an operator \(A:E \rightarrow E\) is increasing (decreasing) if \(x \leq y\) implies \(Ax \leq Ay \) (\(Ax \geq Ay\)).
For all \(x,y \in E\), the notation \(x\sim y\) means that there exist \(\lambda> 0\) and \(\mu> 0\) such that \(\lambda x \leq y \leq\mu x\). Clearly, ∼ is an equivalence relation. Given \(h > \theta\) (i.e., \(h \geq\theta\) and \(h \neq\theta\)), we denote by \(P_{h}\) the set \(P_{h}=\{ x \in E \mid x \sim h \}\). It is easy to see that \(P_{h} \subset P\).
Definition 2.1
\(A: P\times P\rightarrow P\) is said to be a mixed monotone operator if \(A(x,y)\) is increasing in x and decreasing in y, i.e., \(u_{i},v_{i}\in P\), \(i=1,2\), \(u_{1} \leq u_{2}\), \(v_{1} \geq v_{2}\) imply \(A(u_{1},v_{1}) \leq A(u_{2},v_{2})\). Element \(x \in P\) is called a fixed point of A if \(A(x,x)=x\).
Definition 2.2
Definition 2.3
To prove our results, we need the following fixed point theorems for mixed monotone operators, which were established in [21].
Lemma 2.4
(See Theorem 2.1 in [21])
 (i)
there is \(h_{0} \in P_{h}\) such that \(A(h_{0},h_{0}) \in P_{h}\) and \(Bh_{0} \in P_{h}\);
 (ii)
there exists a constant \(\delta_{0} > 0\) such that \(A(x,y)\geq \delta_{0} Bx\), \(\forall x,y \in P\).
 (1)
\(A: P_{h} \times P_{h} \rightarrow P_{h}\) and \(B: P_{h} \rightarrow P_{h}\);
 (2)there exist \(u_{0},v_{0} \in P_{h}\) and \(r \in(0,1)\) such that$$rv_{0} \leq u_{0} < v_{0},\qquad u_{0} \leq A(u_{0},v_{0})+Bu_{0} \leq A(v_{0},u_{0})+Bv_{0} \leq v_{0}; $$
 (3)
the operator equation \(A(x,x)+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\);
 (4)for any initial values \(x_{0},y_{0} \in P_{h}\), constructing successively the sequences$$x_{n}=A(x_{n1},y_{n1})+Bx_{n1}, \qquad y_{n}=A(y_{n1},x_{n1})+By_{n1}, \quad n=1,2, \ldots, $$
Lemma 2.5
(See Theorem 2.4 in [21])
 (i)
there is \(h_{0} \in P_{h}\) such that \(A(h_{0},h_{0}) \in P_{h}\) and \(Bh_{0} \in P_{h}\);
 (ii)
there exists a constant \(\delta_{0} > 0\) such that \(A(x,y)\leq \delta_{0} Bx\), \(\forall x,y \in P\).
 (1)
\(A: P_{h} \times P_{h} \rightarrow P_{h}\) and \(B: P_{h} \rightarrow P_{h}\);
 (2)there exist \(u_{0},v_{0} \in P_{h}\) and \(r \in(0,1)\) such that$$rv_{0} \leq u_{0} < v_{0},\qquad u_{0} \leq A(u_{0},v_{0})+Bu_{0} \leq A(v_{0},u_{0})+Bv_{0} \leq v_{0}; $$
 (3)
the operator equation \(A(x,x)+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\);
 (4)for any initial values \(x_{0},y_{0} \in P_{h}\), constructing successively the sequences$$x_{n}=A(x_{n1},y_{n1})+Bx_{n1}, \qquad y_{n}=A(y_{n1},x_{n1})+By_{n1},\quad n=1,2, \ldots, $$
3 Main results
In this section, we use Lemmas 2.4, 2.5 to study problem (1.1) and present two new results on the existence and uniqueness of monotone positive solutions. The main results obtained here are relatively new in the literature.
In our considerations we shall consider the Banach space \(E= C^{1}[0,1]\) equipped with the norm \(\u\=\max\{\max_{0\leq t\leq1} u(t),\max_{0\leq t\leq1} u'(t)\}\). In order to find monotone positive solutions, we consider the closed convex cone of nonnegative increasing functions \(P=\{u\in Eu(t)\geq0,u'(t)\geq0,\forall t\in [0,1]\}\). Note that this induces an order relation \(\dot{\leq}\) in E by defining \(u \dot{\leq} v\) if and only if \(vu\in P\). Clearly, this cone is normal. That is, if \(u \dot{\leq} v\), then \(u(t)\leq v(t)\), \(u'(t)\leq v'(t)\), \(t\in[0,1]\). Therefore, \(\u\\leq\v\\) and the normality constant is 1.
Lemma 3.1
Theorem 3.2
 (H_{1}):

\(f(t,x,y):[0,1]\times[0,+\infty)\times[0,+\infty)\rightarrow [0,+\infty)\) and \(g:[0,+\infty)\rightarrow(\infty,0]\);
 (H_{2}):

\(f(t,x,y)\) is increasing in \(x \in[0,+\infty)\) for fixed \(t \in [0,1]\) and \(y \in[0,+\infty)\), decreasing in \(y \in[0,+\infty)\) for fixed \(t \in[0,1]\) and \(x\in[0,+\infty)\), and \(g(x)\) is decreasing in \(x \in[0,+\infty)\);
 (H_{3}):

\(g(\lambda x) \leq\lambda g(x)\) for \(\lambda\in(0,1)\), \(x\in [0,+\infty)\), and there exists a constant \(\alpha\in(0,1)\) such that \(f(t,\lambda x,\lambda^{1} y) \geq\lambda^{\alpha} f(t,x,y)\), \(\forall t\in[0,1]\), \(\lambda\in(0,1)\), \(x,y\in[0,+\infty)\);
 (H_{4}):

there exists a constant \(\sigma> 0\) such that \(f(t,x,y) \geq \sigma\geqg(x)>0\), \(t\in[0,1]\), \(x,y\geq0\).
 (1)there exist \(u_{0},v_{0} \in P_{h}\) and \(r \in(0,1)\) such that \(rv_{0} \dot{\leq} u_{0} \dot{<} v_{0}\) andwhere \(h(t)=t^{2}\), \(t\in[0,1]\) and \(G(t,s)\) is given as in (3.1);$$\begin{aligned}& u_{0}(t) \leq\int^{1}_{0} G(t,s)f \bigl(s,u_{0}(s),v_{0}'(s) \bigr)\,dsg \bigl(u_{0}(1) \bigr)\phi (t),\quad t\in[0,1], \\& u_{0}'(t) \leq\int^{1}_{0} G_{t}(t,s)f \bigl(s,u_{0}(s),v_{0}'(s) \bigr)\,dsg \bigl(u_{0}(1) \bigr)\phi'(t),\quad t \in[0,1], \\& v_{0}(t) \geq\int^{1}_{0} G(t,s)f \bigl(s,v_{0}(s),u_{0}'(s) \bigr)\,dsg \bigl(v_{0}(1) \bigr)\phi (t),\quad t\in[0,1], \\& v_{0}'(t) \geq\int^{1}_{0} G_{t}(t,s)f \bigl(s,v_{0}(s),u_{0}'(s) \bigr)\,dsg \bigl(v_{0}(1) \bigr)\phi'(t),\quad t \in[0,1], \end{aligned}$$
 (2)
problem (1.1) has a unique monotone positive solution \(u^{*}\) in \(P_{h}\);
 (3)for any \(x_{0},y_{0}\in P_{h}\), constructing successively the sequences$$\begin{aligned}& x_{n}(t) = \int^{1}_{0} G(t,s)f \bigl(s,x_{n1}(s),y_{n1}'(s) \bigr)\,dsg \bigl(x_{n1}(1) \bigr)\phi(t),\quad n=1,2,\ldots, \\& y_{n}(t) = \int^{1}_{0} G(t,s)f \bigl(s,y_{n1}(s),x_{n1}'(s) \bigr)\,dsg \bigl(y_{n1}(1) \bigr)\phi'(t),\quad n=1,2,\ldots, \end{aligned}$$
Proof
Theorem 3.3
 (H_{5}):

there exists a constant \(\alpha\in(0,1)\) such that \(g(\lambda x) \leq\lambda^{\alpha} g(x)\), \(\forall \lambda\in(0,1)\), \(x\in[0,+\infty)\), and \(f(t,\lambda x,\lambda^{1} y) \geq\lambda f(t,x,y)\) for \(\lambda\in(0,1)\), \(t \in[0,1]\), \(x,y\in [0,+\infty)\);
 (H_{6}):

\(f(t,0,2)\not\equiv0\) for \(t\in[0,1]\) and there exists a constant \(\sigma> 0\) such that \(f(t,x,y) \leq\sigma\leqg(x)\), \(t\in [0,1]\), \(x,y\geq0\).
 (1)there exist \(u_{0},v_{0} \in P_{h}\) and \(r \in(0,1)\) such that \(rv_{0}\dot{ \leq} u_{0} \dot{<} v_{0}\) andwhere \(h(t)=t^{2}\), \(t\in[0,1]\) and \(G(t,s)\) is given as in (3.1);$$\begin{aligned}& u_{0}(t) \leq\int^{1}_{0} G(t,s)f \bigl(s,u_{0}(s),v_{0}'(s) \bigr)\,dsg \bigl(u_{0}(1) \bigr)\phi (t),\quad t\in[0,1], \\& u_{0}'(t) \leq\int^{1}_{0} G_{t}(t,s)f \bigl(s,u_{0}(s),v_{0}'(s) \bigr)\,dsg \bigl(u_{0}(1) \bigr)\phi'(t),\quad t \in[0,1], \\& v_{0}(t) \geq\int^{1}_{0} G(t,s)f \bigl(s,v_{0}(s),u_{0}'(s) \bigr)\,dsg \bigl(v_{0}(1) \bigr)\phi (t),\quad t\in[0,1], \\& v_{0}'(t) \geq\int^{1}_{0} G_{t}(t,s)f \bigl(s,v_{0}(s),u_{0}'(s) \bigr)\,dsg \bigl(v_{0}(1) \bigr)\phi'(t), \quad t \in[0,1], \end{aligned}$$
 (2)
problem (1.1) has a unique monotone positive solution \(u^{*}\) in \(P_{h}\);
 (3)for any \(x_{0},y_{0}\in P_{h}\), constructing successively the sequences$$\begin{aligned}& x_{n}(t) = \int^{1}_{0} G(t,s)f \bigl(s,x_{n1}(s),y_{n1}'(s) \bigr)\,dsg \bigl(x_{n1}(1) \bigr)\phi(t),\quad n=1,2,\ldots, \\& y_{n}(t) = \int^{1}_{0} G(t,s)f \bigl(s,y_{n1}(s),x_{n1}'(s) \bigr)\,dsg \bigl(y_{n1}(1) \bigr)\phi'(t), \quad n=1,2,\ldots, \end{aligned}$$
Sketch of the proof
Remark 3.4
Comparing Theorems 3.23.3 with the main results in [17], we provide some alternative approaches to study the same type of problems under different conditions. Our results can guarantee the existence of a unique monotone positive solution and the existence of upperlower solutions, which are seldom seen in the literature.
4 Examples
To illustrate how our main results can be used in practice we present two examples.
Example 4.1
Example 4.2
Declarations
Acknowledgements
This work was completed when the first author visited College of William and Mary in 2015, and he would like to thank CWM for warm hospitality, and are greatly indebted to Prof. Junping Shi for many helpful suggestions. The first author was partially supported by International Science and Technology Cooperation Projects of Shanxi (2015081020). The second author was partially supported by the Youth Science Foundation of China (11201272) and Shanxi Province Science Foundation (2015011005).
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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