Triple positive solutions for a second order mpoint boundary value problem with a delayed argument
 Jie Zhou^{1} and
 Meiqiang Feng^{1}Email author
Received: 23 June 2015
Accepted: 10 September 2015
Published: 30 September 2015
Abstract
In this paper, we establish the new expression and properties of Green’s function for an mpoint boundary value problem with a delayed argument. Furthermore, using Hölder’s inequality and a fixed point theorem due to Leggett and Williams, the existence of at least three positive solutions is also given. We discuss our problem with a delayed argument. In this case, our results cover mpoint boundary value problems without delayed arguments and are compared with some recent results. An example is included to illustrate our main results.
Keywords
1 Introduction
It is easy to see that the solution of problem (1.1) is concave when \(a(t)\geq0\) on \([0,1]\) and \(f(u)\geq0\) on \([0,\infty)\). However, few papers have reported the same problems where the solution is without concavity; for example, see some recent excellent results and applications of the case of ordinary differential equations with deviating arguments to a variety of problems from Jankowski [21–23], Jiang and Wei [24], Wang [25], Wang et al. [26] and Hu et al. [27].
 (H_{1}):

\(\omega\in L^{p}[0,1]\) for some \(p \in[1,+\infty)\), and there exists \(n>0\) such that \(\omega(t) \geq n\) a.e. on J;
 (H_{2}):

\(f\in C([0,1]\times[0,+\infty),[0,+\infty))\), \(\alpha\in C(J,J)\) with \(\alpha(t)\leq t\) on J;
 (H_{3}):

\(\sum_{i=1}^{m2}\beta_{i}\phi(\xi_{i})<1\), where ϕ satisfies$$ L\phi=0, \qquad \phi'(0)=0, \qquad \phi(1)=1. $$(1.3)
Remark 1.1
By a positive solution of problem (1.2) we mean a function \(x\in C^{2}(0,1)\cap C[0,1]\) with \(x(t)>0\) on \((0,1)\) that satisfies (1.2).
Remark 1.2
For the case \(\alpha(t)\equiv t\) on J, problem (1.2) reduces to the problem studied by Feng and Ge in [11]. By using the fixed point theorem in a cone, the authors obtained some sufficient conditions for the existence, nonexistence and multiplicity of positive solutions for problem (1.2) when \(\alpha(t)\equiv t\) on J. However, Feng and Ge did not obtain any results of triple solutions on problem (1.2). This paper will resolve this problem.
In this paper, we present several new and more general results for the existence of triple positive solutions for problem (1.2) by using LeggettWilliams’ fixed point theorem. Another contribution of this paper is to study the expression and properties of Green’s function associated with problem (1.2). The expression of the integral equation is simpler than that of [11].
The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.2). In Section 3, we present some definitions and lemmas which are useful to obtain our main results. In Section 4, we formulate sufficient conditions under which delayed problem (1.2) has at least three positive solutions. In Section 5, we provide an example to illustrate our main results.
2 Expression and properties of Green’s function
Lemma 2.1
Proof
Remark 2.1
The proof of Lemma 2.1 is supplementary of Theorem 3 in [28], which helps the readers to understand that (2.2) holds.
Remark 2.2
The expression of the integral equation (2.2) is different from that of (2.10) in [10] and that of (2.9) in [11], which shows that we can use a completely different technique from that of [10] and [11] to study problem (1.2).
Remark 2.3
It is not difficult from [10, 11] to show that \(\Delta >0\) and that (i) ϕ is nondecreasing on J and \(\phi>0\) on J; (ii) ψ is strictly decreasing on J.
Remark 2.4
Lemma 2.2
(See [28])
Remark 2.5
3 Preliminaries
In this section, we provide some background material from the theory of cones in Banach spaces, and then we state Hölder’s inequality and LeggetWilliams’ fixed point theorem. The following definitions can be found in the book by Deimling [29] as well as in the book by Guo and Lakshmikantham [30].
Definition 3.1
 (i)
\(au+bv\in K\) for all \(u, v\in K\) and all \(a\geq0\), \(b\geq0\);
 (ii)
\(u, u\in K\) implies \(u=0\).
Note that every cone \(K\subset E\) induces an ordering in E given by \(x\leq y\) if and only if \(yx\in K\).
Definition 3.2
Definition 3.3
An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Lemma 3.1
(ArzelàAscoli)
 (i)
All the functions in the set M are uniformly bounded. It means that there exists a constant \(r>0\) such that \(u(t)\leq r\), \(\forall t\in J\), \(u\in M\);
 (ii)All the functions in the set M are equicontinuous. It means that for every \(\varepsilon>0\), there is \(\delta=\delta(\varepsilon )>0\), which is independent of the function u, such thatwhenever \(t_{1}t_{2}<\delta\), \(t_{1}, t_{2}\in J\).$$\biglu(t_{1})u(t_{2})\bigr< \varepsilon $$
Lemma 3.2
(Hölder)
Lemma 3.3
Assume that (H_{1})(H_{3}) hold. Then \(T(K)\subset K\) and \(T:K\rightarrow K\) is completely continuous.
Proof
Next we shall show that operator T is completely continuous. We break the proof into several steps.
Step 1. Operator T is continuous. Since the function \(f(t,x)\) is continuous on \(J\times[0,+\infty)\), this conclusion can be easily obtained.
As a consequence of Step 1 to Step 3 together with Lemma 3.1, we can prove that \(T: K\rightarrow K\) is completely continuous. □
Remark 3.1
From Lemma 2.1 and (3.1), we know that \(x\in E\) is a solution of problem (1.2) if and only if x is a fixed point of operator T.
Finally we state LeggettWilliams’ fixed point theorem [31].
Lemma 3.4
 (i)
\(\{x\in K(\beta,a,b): \beta(x)>a\} \neq\emptyset \) and \(\beta(Ax)>a\) for \(x\in K(\beta,a,b)\);
 (ii)
\(\Ax\< d\) for \(\x\\leq d\);
 (iii)
\(\beta(Ax)>a\) for \(x\in K(\beta,a,c)\) with \(\Ax\>b\).
4 Existence of triple positive solutions
In this section, we apply Lemma 3.2 and Lemma 3.4 to establish the existence of three positive solutions for problem (1.2). We consider the following three cases for \(\omega\in L^{p}[0,1]:p> 1\), \(p=1\), and \(p=\infty\). Case \(p>1\) is treated in the following theorem.
Theorem 4.1
 (H_{4}):

\(f(t,x)\leq\frac{c}{\Gamma}\) for \((t,x)\in J\times[0,c]\);
 (H_{5}):

\(f(t,x)\geq\frac{l}{\Psi}\) for \((t,x)\in[0,\xi ]\times[l,\frac{l}{\sigma}]\);
 (H_{6}):

\(f(t,x)\leq\frac{d}{\Gamma}\) for \((t,x)\in J\times \in[0,d]\).
Proof
By the definition of operator T and its properties, it suffices to show that the conditions of Lemma 3.4 hold with respect to T.
Let \(x\in\bar{P}_{c}\). Then \(0\leq x(t) \leq c\) on J. Since \(0\leq\alpha(t)\leq t\leq1\) on J, it follows from \(0\leq x(t) \leq c\) on J that \(0\leq x(\alpha(t)) \leq c\) on J.
We first show that the condition (i) of Lemma 3.4 holds.
Since \(0\leq\alpha(t)\leq t\leq\xi\) on \([0,\xi]\), it follows from \(l\leq x(t) \leq\frac{l}{\sigma}\) on \([0,\xi]\) that \(l\leq x(\alpha (t))\leq\frac{l}{\sigma}\) for \(t\in[0,\xi]\).
Secondly, we prove that condition (ii) of Lemma 3.4 is satisfied. If \(x\in K_{d}\), then \(0\leq x(t)\leq d\) on J.
Since \(0\leq\alpha(t)\leq t \leq1\) on J, it follows from \(0\leq x(t)\leq d\) on J that \(0\leq x(\alpha(t))\leq d\) on J.
Finally, we prove that the condition (iii) of Lemma 3.4 is satisfied.
The following corollary deals with the case \(p=\infty\).
Corollary 4.1
Assume that (H_{1})(H_{6}) hold. Then problem (1.2) has at least three positive solutions \(x_{1}\), \(x_{2} \) and \(x_{3}\) satisfying (4.1).
Proof
Let \(\H\_{1}\\omega\_{\infty}\) replace \(\H\_{q}\ \omega\_{p}\) and repeat the argument above, we can get the corollary. □
 \((\mathrm{H}_{4})^{\prime}\) :

\(f(t,x)\leq\frac{c}{\Gamma_{1}}\) for \((t,x)\in J\times[0,c]\);
 \((\mathrm{H}_{6})^{\prime}\) :

\(f(t,x)\leq\frac{d}{\Gamma_{1}}\) for \((t,x)\in J\times\in[0,d]\),
Corollary 4.2
Assume that (H_{1})(H_{3}), \((\mathrm{H}_{4})^{\prime}\), (H_{5}) and \((\mathrm{H}_{6})^{\prime}\) hold. Then problem (1.2) has at least three positive solutions \(x_{1}\), \(x_{2} \) and \(x_{3}\) satisfying (4.1).
Proof
Similar to the proof of Theorem 4.1, we can get Corollary 4.2. □
Remark 4.1
 (i)
Three positive solutions are available.
 (ii)
\(\alpha(t) \not\equiv t\) is considered throughout this paper.
 (iii)
\(\omega(t)\) is \(L^{p}\)integrable, not only \(\omega(t)\in C(0,1)\) on \(t\in J\).
5 An example
Example 5.1
This means that problem (5.1) involves the delayed argument α. For example, we can take \(\alpha(t)=t^{3}\). It is clear that ω is nonnegative and \(\omega\in L^{2}[0,1]\).
Conclusion 5.1
Problem (5.1) has at least three positive solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\) satisfying (4.1).
Proof
By Theorem 4.1, problem (5.1) has least three positive solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\) satisfying (4.1). □
Declarations
Acknowledgements
We wish to express our thanks to Prof. Xuemei Zhang, Department of Mathematics and Physics, North China Electric Power University, Beijing, P.R. China, for her kind help, careful reading, and making useful comments on the earlier version of this paper. The authors are also grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper. This work is sponsored by the project NSFC (11301178), the Fundamental Research Funds for the Central Universities (2014MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201511232018).
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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