In this paper, the existence results of positive solutions for three-point Riemann-Stieltjes 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 problemexistencepositive solutionshooting methodcomparison principle
MSC
34B1034B1834C10
1 Introduction
By applying the shooting method, we establish the criteria for the existence of positive solutions to the following Riemann-Stieltjes integral BVPs:
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 \(f_{\infty}=\infty\) (super-linear case) or in the case \(f_{0}=\infty\) and \(f_{\infty}=0\) (sub-linear case) when \(0<\alpha\eta^{2}<2\).
Some meaningful results of nonlinear second-order 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 \(0\leq\sum_{i=1}^{m-2} \alpha_{i} \eta _{i}<1\), 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 super-linear case or the sub-linear case.
When \(a(t)\equiv1\), \(m=3\), \(\eta=1/2\), as a special case of [8], the BVP
where \(0<\eta_{1}<\eta_{2}<\cdots<\eta_{n}<1\), \(\alpha_{i}\geq0\) for \(i= 1,\ldots ,n-1\), 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
$$ u(0)=0,\qquad u'(0)=m. $$
(2.1)
Denote by \(u(t,m)\) the solution of the IVP (1.1) with (2.1) provided it exists, and define
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 non-trivial 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,
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].
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). □
Assume that (H1) 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
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
$$ \frac{f(u)}{u}< \frac{{\underline{A}}^{2}}{a^{L}}\leq\frac{{A_{1}}^{2}}{a^{L}}, \quad 0< u\leq r. $$
(3.3)
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
and there exists a positive number \(\epsilon< A_{2}(1-\eta)/\eta\) small enough that
$$ \frac{f(u)}{u}\geq\frac{(A_{2}+\epsilon)^{2}}{a^{l}},\quad u\geq L. $$
(3.7)
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
$$0< \rho\leq\eta\leq\frac{A_{2}}{A_{2}+\epsilon} \leq\sigma\leq1 \quad \mbox{and}\quad u \bigl(t,m_{2}^{*} \bigr)\geq L \quad\mbox{for } t\in[\rho, \sigma]. $$
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
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,
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
The relationship ofmand\(\pmb{I_{m}}\).\(m_{1}>m_{0}\), \(I_{m_{0}}\subset I_{m_{1}}\).
By mathematical induction, it is not difficult to show that \(\underline{\delta}_{m_{k}}<\underline{\delta}_{m_{k-1}}\), \(k=1,2,\ldots\) .
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 \(u(t,m_{k-1})\) and \(u(t,m_{k})\) have no intersection in the interval \([\underline{\delta}_{m_{k-1}},1)\). It follows from
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
$$ \sigma(A_{2}+\epsilon)\geq A_{2}. $$
(3.15)
Two of the possible cases of \(I_{m_{0}}\) can be seen in Figure 2.
Figure 2
Two of the possible cases of\(\pmb{I_{m_{0}}}\). Subcase 1: \(\eta\in[\underline{\delta}_{m_{0}},\overline{\delta }_{m_{0}}]\) and \(u(1,m_{0})\geq L\); Subcase 2: \(\eta\nsubseteq[\underline{\delta }_{m_{0}},\overline{\delta}_{m_{0}}]\) or \(u(1,m_{0})< L\).
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
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 (H1)-(H2) hold. Suppose one of the following conditions 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 sup-linear case or the sub-linear 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
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
$$A=2\cos^{-1} \biggl(\frac{\mu}{2} \biggr). $$
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 \(\sum_{i=1}^{m-2} \alpha_{i} \eta_{i}=1\), 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
(1)
School of Mathematics and Physics, University of South China
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