A note on the shooting method and its applications in the Stieltjes integral boundary value problems
 Huilan Wang^{1},
 Zigen Ouyang^{1}Email author and
 Hengsheng Tang^{1}
Received: 15 October 2014
Accepted: 25 May 2015
Published: 18 June 2015
Abstract
In this paper, the existence results of positive solutions for threepoint RiemannStieltjes integral BVPs (boundary value problems) is considered. By applying shooting method and comparison principle, we obtain some new results which extend the known ones. At the same time, the theorems in one of our published articles are corrected by another theorem in this paper.
Keywords
integral boundary value problem existence positive solution shooting method comparison principleMSC
34B10 34B18 34C101 Introduction
 (i)
\(\bar{f}_{0}<\mu\) and \(\underline{f}_{\infty}>\mu\);
 (ii)
\(\bar{f}_{0}>\mu\) and \(\underline{f}_{\infty}<\mu\),
When \(\theta= \pi/2\) and \(0\leq\sum_{i=1}^{m2} \alpha_{i} \eta _{i}<1\), Ma [8] has studied BVP (1.1) with (1.4) by using Krasnoselskii’s fixed point theorem in a cone. The sufficient condition for the existence of positive solutions is also the superlinear case or the sublinear case.
However, Theorem 1.1 and some proofs in [9] need to be corrected, which is one of the reasons why we write this paper. Furthermore, more general existence criteria are presented in this article as well as the application of the shooting method in the study of BVPs. For simplicity and without loss of generality, we start from BVP (1.1)(1.2).
2 Preliminaries: some notation and lemmas
 (H_{1}):

\(f\in C([0,\infty);[0,\infty))\), \(a\in C([0,1];[0,\infty))\).
Furthermore, we assume that f is strong continuous enough to guarantee that \(u(t, m)\) is uniquely defined and that it depends continuously on both t and m. As for the discussion of this problem, see [6].
Next, we present some comparison theorems which help us to establish the main results.
Lemma 2.1
(Sturm comparison theorem)
Lemma 2.2
Proof
Since \(0\leq g(t)\leq f(y(t))/y(t)\leq G(t)\) and \(Z(t)\) does not vanish in \((0,1]\), from Lemma 2.1, it follows that \(y(t)\) and \(z(t)\) will not vanish in \((0,1]\). The proof for (2.3) can be seen in [6]. The continuity of the integrands implies the existence of the Riemann integral. In view of the definition of Riemann integral, by using the inequality of the limit, we have (2.4). □
Lemma 2.3
 (i)
If \(A=\pi^{2}\), then \(y(t)\) vanishes at \(t=1\) for the first time on interval \((0,1]\) and \(b=0\);
 (ii)
if \(0< A<\pi^{2}\), then \(y(t)\) does not vanish on the interval \((0,1]\) and \(b>0\);
 (iii)
if \(A>\pi^{2}\), then \(y(t)\) vanishes before \(t=1\) on interval \((0,1]\).
Proof
Obviously, \(y(t)=\sin(\pi^{2}t)\) satisfies the conditions \(y(0)=0\), \(y(1)=0\), and \(y(t)>0\) for \(t\in(0,1)\), hence (i) is established. According to the Sturm comparison theorem, we can draw the conclusions (ii) and (iii). □
Lemma 2.4
([1])
Assume that (H_{1}) holds and \(\alpha \eta^{2}> 2\), then BVP (1.1)(1.2) has no positive solution.
 (H_{2}):

\(0< \alpha\eta^{2} <2 \).
3 Main results
Lemma 3.1
Proof
Theorem 3.1
Proof
Next, we will find a positive number \(m_{2}^{*}\) such that \(\varphi (m_{2}^{*})\geq0\).
Since the solution \(u(t,m)\) is concave, it hits the line \(u=L\) at most two times for the constant L defined in (3.6) and \(t\in(0,1]\). We denote the left intersecting time by \(\underline{\delta}_{m}\) and the right one by \(\overline{\delta}_{m}\) provided they exist. Henceforth, denote \(I_{m}=[\underline{\delta}_{m},\overline{\delta}_{m}]\subseteq (0,1]\). If \(u(1,m)\geq L\), then \(\overline{\delta}_{m}=1\).
The discussion is divided into three steps.
Step 1. We claim that there exists a slope \(m_{0}\) large enough such that \(0\leq u(t,m_{0})\leq L\) for \(t\in[0,\underline{\delta }_{m_{0}}]\) and \(u(t,m_{0})\geq L\) for \(t\in I_{m_{0}}\).
Since \(u(t,m)\) is continuous and concave, there exists a number \(m_{0}\) large enough such that \(u(t,m_{0})\geq L\) for \(t\in I_{m_{0}}\).
By mathematical induction, it is not difficult to show that \(\underline{\delta}_{m_{k}}<\underline{\delta}_{m_{k1}}\), \(k=1,2,\ldots\) .
Step 3. Seek out a slope \(m_{2}^{*}\) and two positive numbers ρ and σ such that \(0<\rho\leq\eta\leq\frac{A_{2}}{A_{2}+\epsilon} \leq\sigma\leq1\) and \(u(t,m_{2}^{*})\geq L\) for \(t\in[\rho,\sigma]\).
Subcase 1. \(\eta\in[\underline{\delta}_{m_{0}},\overline{\delta }_{m_{0}}]\) and \(u(1,m_{0})\geq L\). In this case, we take \(m_{2}^{*}=m_{0}\) and \(\rho=\underline{\delta}_{m_{0}}\), \(\sigma=\overline{\delta}_{m_{0}}=1\).
In the following, we prove that \(k(m_{2}^{*})\geq1\) or \(\varphi(m_{2}^{*})>0\) for the selected \(m_{2}^{*}\) and ρ, σ.
From (3.5) and (3.17), we can find a \(m^{*}\) between \(m_{1}^{*}\) and \(m_{2}^{*}\) such that \(u(t,m^{*})\) is the solution of (1.1)(1.2). The theorem is complete.
The proof for (ii) is similar, so we omit it. □
Now, we present the result for BVP (1.1) with (1.7), which is also the correction of Theorem 3.1 and Theorem 3.2 in [9].
Theorem 3.2
Proof
The remainder of the proof is similar, so we omit it. □
4 Conclusion and discussion
The conditions in [8] and [1] are easy to verify; however, they are not as general as ours, because the suplinear case or the sublinear case is sufficient for the conditions in Theorem 3.1. As an example of [4], where the constant μ is related to the Green’s function and the spectral radius of associated linear operator, our calculation is more direct. The idea of this paper was illuminated by [6, 7]; however, the certain constant \(L_{\theta}\) could not be given explicitly in [7] and η only equals \(1/2\) in [6]. From this point of view, this paper extends the work of [6, 7] and presents another way to find the ‘eigenvalue’ by numerical calculation, though it is related to a transcendental equation which has at least one numerical solution.
 (i):

\(f_{0} = 0\) and \(f_{\infty}= \infty\), or
 (ii):

\(f_{0}= \infty\) and \(f_{\infty}= 0\),
 (i′):

\(0\leq\bar{f}_{0}< A^{2}<\underline{f}_{\infty}\); or
 (ii′):

\(0\leq\bar{f}_{\infty}< A^{2}< \underline{f}_{0}\),
Further, when \(\alpha\eta^{2}=2\), BVP (1.1)(1.2) is at resonant. There may not exist a solution \(x=A_{1}\in(0,\pi)\) and \(x=A_{2}\in(0,\pi)\) to (3.1) and (3.2), respectively. If (3.1) and (3.2) has a solution \(x=A_{1}\in(0,\pi)\) and \(x=A_{2}\in(0,\pi)\), respectively, then we can also obtain the existence result for (1.1)(1.2), similarly for (1.1) with (1.7).
When \(\theta= \pi/2\) and \(\sum_{i=1}^{m2} \alpha_{i} \eta_{i}=1\), BVP (1.1) with (1.4) is resonant. If there exists a number \(A\in (0,\pi)\) such that (4.1), then the existence result for BVP (1.1) with (1.4) can be obtained, similarly for BVP (1.5).
Declarations
Acknowledgements
It was remarked to the authors by Professor Webb that the result is not correct in [9]. The authors would like to express their sincere gratitude to Professor Webb for his helpful comments and suggestion on the manuscript, as well as the anonymous reviewers’ comments. Moreover, the first author HW is sorry for having cited the comparison theorem by mistake in [9]. This project was supported by the Scientific Research Fund of Hunan Provincial Educational Department (No. 13A088), the Scientific Research Foundation of Hengyang City (No. 2012KJ2) and the Construct Program in USC.
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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