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A note on the shooting method and its applications in the Stieltjes integral boundary value problems
Boundary Value Problems volume 2015, Article number: 102 (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.
1 Introduction
By applying the shooting method, we establish the criteria for the existence of positive solutions to the following RiemannStieltjes integral BVPs:
where \(f\in C([0,\infty);[0,\infty))\) and \(0<\eta<1\), \(\alpha\geq0\) are given constants, and \(0<\alpha\eta^{2}<2\).
Set
By Krasnoselskii’s fixed point theorem in a cone, Tariboon and Sitthiwirattham [1] proved that BVP (1.1)(1.2) has a positive solution in the case \(f_{0}=0\) and \(f_{\infty}=\infty\) (superlinear case) or in the case \(f_{0}=\infty\) and \(f_{\infty}=0\) (sublinear case) when \(0<\alpha\eta^{2}<2\).
Some meaningful results of nonlinear secondorder integral BVPs have already been obtained by Kong [2], Webb and Infante [3, 4], etc. The following BVP:
is a special case of Webb and Infante’s [4], where we can deduce the result. Suppose \(0<\alpha\eta^{2}<1\); BVP (1.3) has at least one positive solution if one of the following conditions holds:

(i)
\(\bar{f}_{0}<\mu\) and \(\underline{f}_{\infty}>\mu\);

(ii)
\(\bar{f}_{0}>\mu\) and \(\underline{f}_{\infty}<\mu\),
where \(\mu=1/r(L)\) and \(r(L)\) is the spectral radius of the associated linear operator. In [4], the authors used fixed point index theory.
As a numerical method, the shooting method is efficient to find the solution of BVPs [5–7]. Kwong and Wong [7] obtained some results for the Robin boundary condition of the form
where \(\theta\in[0,3\pi/4]\) and \(\theta\neq\pi/2\). Kwong and Wong [7] showed that BVP (1.1) with (1.4) has at least one positive solution if \(\bar{f}_{0}< L_{\theta}\) and \(\underline{f}_{\infty}>L_{\theta}\), where \(L_{\theta}\) is a certain but not specified constant related to the associated linear operator.
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.
When \(a(t)\equiv1\), \(m=3\), \(\eta=1/2\), as a special case of [8], the BVP
was studied by Kwong in [6], where the existence condition is
which is obtained by the shooting method.
Following the main idea in [6, 7], we considered the generalized multipoint integral BCs [9]
where \(0<\eta_{1}<\eta_{2}<\cdots<\eta_{n}<1\), \(\alpha_{i}\geq0\) for \(i= 1,\ldots ,n1\), and \(\alpha_{n}>0\) are given constants.
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
The principle of the shooting method is converting the BVP into an IVP (initial value problem) by finding suitable initial slopes \(m>0\) such that the solution of (1.1) comes with the initial value condition
Denote by \(u(t,m)\) the solution of the IVP (1.1) with (2.1) provided it exists, and define
Then solving the boundary value problem is equivalent to finding a \(m^{*}\) such that \(k(m^{*})=1\) or \(\varphi(m^{*})=0\).
For the sake of convenience, we denote
In this paper, we always assume
 (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)
Let \(\varphi_{1}\) and \(\varphi_{2}\) be nontrivial solutions of the equations
respectively, on an interval I; here \(q_{1}\) and \(q_{2}\) are continuous functions such that \(q_{1}(x)\leq q_{2}(x)\) on I. Then between any two consecutive zeros \(x_{1}\) and \(x_{2}\) of \(\varphi_{1}\), there exists at least one zero of \(\varphi_{2}\) unless \(q_{1}(x) \equiv q_{2}(x)\) on \((x_{1}, x_{2})\).
Lemma 2.2
Let \(y(t,m)\), \(z(t,m)\), \(Z(t,m)\) be the positive solution of the initial value problems, respectively,
Suppose \(g(t)\leq G(t)\) be two piecewise continuous functions defined on \([0, 1]\). If
and suppose that \(Z(t)\) does not vanish in \((0,1]\), then for any \(0\leq s\leq\xi\leq1\), it yields
and hence, for any \(0\leq\eta\leq\xi\leq1\), we have
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). □
Remark 2.1
Lemma 2.2 is also the correction for Theorem 1.1 in [9].
Lemma 2.3
Consider the BVP

(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.
In [1] and [9], the proofs are conducted by contradiction to the concavity of solution (also see [4]). In fact, for \(m>0\), we compare the solution \(u(t,m)\) of the IVP given by (1.1) and (2.1) with the solution \(y(t)=mt\) of
If BVP (1.1)(1.2) has a positive solution \(u(t,m)\), then by Lemma 2.2 and the concavity of \(u(t,m)\), we have
that is, \(\alpha\eta^{2}\leq2\).
In the following, we always assume that
 (H_{2}):

\(0< \alpha\eta^{2} <2 \).
3 Main results
Lemma 3.1
Assume that (H_{1})(H_{2}) holds. Then there exist a solution \(x=A_{1}\in(0,\pi)\) such that
and a solution \(x=A_{2}\in(0,\pi)\) such that
Proof
It is not difficult to show that
Since the function \(g_{1}(x)\) is continuous on \((0,\pi)\), there must exist a constant \(A_{1}\in(0,\pi)\) such that \(g_{1}(A_{1})=1\).
Similarly,
Thus, there exists a positive constant \(A_{2}\in(0,\pi)\) such that \(g_{2}(A_{2})=1\). □
Theorem 3.1
Assume that (H_{1})(H_{2}) holds. Suppose one of the following conditions holds:
Then problem (1.1)(1.2) has at least one positive solution, where
and \(A_{1}\), \(A_{2}\) is defined in (3.1) and (3.2), respectively.
Proof
(i) Since \(0\leq \bar{f}_{0}< \frac{\underline{A}^{2}}{a^{L}}\), there exists a positive number r such that
Let \(0< m_{1}^{*}< r\), then from the Sturm comparison theorem and the concavity of \(u(t,m_{1}^{*})\), it follows that \(0\leq u(t,m_{1}^{*})\leq m_{1}^{*}t\leq m_{1}^{*}< r\) for \(t\in[0,1]\). Thus
By Lemma 2.3, it gives \(u(t,m_{1}^{*})>0\) for \(t\in(0,1]\).
Let \(Z(t)=(m_{1}^{*}/A_{1})\sin(A_{1} t)\) for \(t\in[0,1]\), then
From Lemma 2.2 and Lemma 3.1, we have
that is, \(\varphi(m_{1}^{*})\leq0\).
On the other hand, the second inequality in (i) implies that there exists a number L large enough such that
and there exists a positive number \(\epsilon< A_{2}(1\eta)/\eta\) small enough that
Next, we will find a positive number \(m_{2}^{*}\) such that \(\varphi (m_{2}^{*})\geq0\).
Claim. There exist a slope \(m_{2}^{*}\) and two positive numbers ρ and σ such that
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 \(t\in I_{m_{0}}\).
Otherwise, provided \(u(t,m)\leq L\) for all \(t\in[0,1]\) as \(m\rightarrow\infty\), then by integrating both sides of (1.1) from 0 to t, we have
Hence, from (3.3) and the continuity of \(f(u)\), we have
where \(L_{f}=\max_{u\in[0,L]}{f(u)}\). If we choose \(m>L+L_{f} a^{L}\), (3.9) will lead to a contradiction.
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 \(t\in I_{m_{0}}\).
Step 2. There exists a monotonically increasing sequence \(\{m_{k}\} \) such that the sequence \({\underline{\delta}_{m_{k}}}\) is decreasing on \(m_{k}\) and \({\overline{\delta}_{m_{k}}}\) is increasing on \(m_{k}\). That is,
and \(u(t,m_{k})\geq L\) for \(t\in I_{m_{k}}\).
First, we prove that
When \(k=1\), we have
in the case
Otherwise, provided
then from (3.8) and (3.11), we have
which contradicts (3.12).
Hence, for a slope \(m_{1}>m_{0}+2a^{L}L_{f}\underline{\delta}_{m_{0}}\), there exists a number \(0<\underline{\delta}_{m_{1}}<\underline{\delta}_{m_{0}}\) such that
See Figure 1.
By mathematical induction, it is not difficult to show that \(\underline{\delta}_{m_{k}}<\underline{\delta}_{m_{k1}}\), \(k=1,2,\ldots\) .
Further, we turn to the right hand of the interval \(I_{m_{k}}\). Since f guarantees that \(u(t, m)\) is uniquely defined, the solutions \(u(t,m_{k1})\) and \(u(t,m_{k})\) have no intersection in the interval \([\underline{\delta}_{m_{k1}},1)\). It follows from
that
Thus we have
When \(k=1\), also see Figure 1.
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 \(\rho=\underline{\delta}_{m_{0}}\), \(\sigma=\overline{\delta}_{m_{0}}=1\).
Subcase 2. \(\eta\nsubseteq[\underline{\delta}_{m_{0}},\overline {\delta}_{m_{0}}]\) or \(u(1,m_{0})< L\). Following the step 1, step 2, and the extension principle of solutions, there exists a positive integer n large enough such that
If we take \(m_{2}^{*}=m_{n}\) and \(\rho=\underline{\delta}_{m_{n}}\), \(\sigma =\overline{\delta}_{m_{n}}\), then
Two of the possible cases of \(I_{m_{0}}\) can be seen in Figure 2.
In the following, we prove that \(k(m_{2}^{*})\geq1\) or \(\varphi(m_{2}^{*})>0\) for the selected \(m_{2}^{*}\) and ρ, σ.
Set \(z(t)= (m_{2}^{*}/\sigma(A_{2}+\epsilon) )\sin (\sigma (A_{2}+\epsilon)t )\), then
where \(\rho\leq\eta<\sigma\leq1\). From (3.7), we have
Further, noting that \(u(1,m_{2}^{*})>L\) (this time \(\sigma=1\)) or \(u(1,m_{2}^{*})\leq u(\sigma,m_{2}^{*})=L\) and the function
is increasing for \(x\in(0,\pi)\), then by Lemma 2.2, Lemma 3.1, and inequality (3.15), we have
which implies \(\varphi(m_{2}^{*})\geq0\).
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
Assume that (H_{1})(H_{2}) hold. Suppose one of the following conditions holds:
Then problem (1.1) with (1.7) has at least one positive solution, where
and \(A_{1}\), \(A_{2}\) is defined by
and
Proof
Similar to (3.5) and (3.17), it follows from (1.7) and (3.18)(3.19) that
and
where \(\eta_{n}<\sigma\leq1\) and (3.15) holds.
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.
In fact, we can extent our results to [8]. The proof is fit, where
and the constant \(A=A_{1}=A_{2}\in(0,\pi)\) is explicitly determined by
In other words, we can substitute the condition
 (i):

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

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

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

\(0\leq\bar{f}_{\infty}< A^{2}< \underline{f}_{0}\),
where A is defined in (4.1).
Next, we apply the result to the special case BVP (1.5), where \(a^{L}=a^{l}=1\), \(m=3\), \(\alpha=\mu\), \(\eta=1/2\). From (4.1), we have
By plugging it into (i′) and (ii′), we have the same result as (1.6).
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).
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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.
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The authors declare that they have no competing interests.
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The work was carried out in collaboration between all authors. HW and ZO are responsible for the majority of the work. HT contributed to the proof of Theorem 3.1.
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Wang, H., Ouyang, Z. & Tang, H. A note on the shooting method and its applications in the Stieltjes integral boundary value problems. Bound Value Probl 2015, 102 (2015). https://doi.org/10.1186/s1366101503598
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DOI: https://doi.org/10.1186/s1366101503598