# The shooting method and integral boundary value problems of third-order differential equation

- 1.2k Downloads
- 1 Citations

## Abstract

In this paper, the existence of at least one positive solution for third-order differential equation boundary value problems with Riemann-Stieltjes integral boundary conditions is discussed. By applying the shooting method and the comparison principle, we obtain some new results which extend the known ones. Meanwhile, an example is worked out to demonstrate the main results.

## Keywords

shooting method positive solution third-order boundary conditions including Stieltjes integrals## 1 Introduction

It is well known that third-order equations arise from many branches of applied mathematics and physics. For example, in the deflection of a curved beam having a constant or varying cross section, a three layer beam, electromagnetic waves or gravity driven flows [1]. There have been extensive studies on third-order differential equation BVPs (boundary value problems), for example [2, 3, 4, 5]. Most of these results are obtained via applying the topological degree theory, the fixed point theorems on cones, the lower and upper solution method, the critical point theory and monotone technique. We refer the reader to [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and the references therein.

Recently, the attention has shifted to BVPs with Stieltjes integral boundary condition since this kind of conditions has been considered a single framework of multipoint and integral type boundary conditions. For more comments on the Riemann-Stieltjes integral boundary condition and its importance, we refer the reader to [4, 5] and other related work such as [6, 7].

*λ*denotes a linear functional on \(C(J)\) given by \(\lambda [x]=\int^{1}_{0}x(t)\,d\Lambda(t)\) involving a Stieltjes integral with a suitable function Λ of bounded variation.

In [8], the author applied the method of lower and upper solutions to generate an iterative technique and discussed the existence of solutions of nonlinear third-order ordinary differential equations with integral boundary conditions. Pang and Xie [9] investigated the existence of concave positive solutions and established corresponding iterative schemes for a third-order differential equation with Riemann-Stieltjes integral boundary conditions using the monotone iterative technique.

It is well known that the classical shooting method could be effectively used to establish the existence and multiplicity results for differential equation BVPs. To some extent, this approach has an advantage over the traditional methods. Readers can see [18, 19, 20, 21, 22, 23, 24] and the references therein for details.

- (H
_{1}) -
\(f \in C([0,\infty)\times[0,\infty); [0,\infty))\), \(f(u, v)\not \equiv0\);

- (H
_{2}) -
\(h\in C([0,1];[0,\infty))\);

- (H
_{3}) -
\(\int^{1}_{0}\,dA(t)> 1\), \(0< \int^{1}_{0}\,dB(t)<1\).

## 2 Preliminaries

### Proof

Assume *y* is a positive solution of (2.2), then \(y(t)> 0\) for \(t\in (0,1)\) and it follows from \(u(t)=Ay(t)\) that \(u(t)\) satisfies (1.1). Assume on the contrary that there is a \(t_{0} \in(0,1)\) such that \(u(t_{0})= \min_{t\in(0,1)}u(t)\leq0\), then \(u'(t_{0})=0\) and \(u''(t_{0}) \geq0\), which yields \(y(t_{0})=u'(t_{0})=0\). This contradicts the assumption that *y* is a positive solution of (2.2). Hence, \(u(t)>0\) for all \(t\in(0,1)\). □

*m*such that equation (2.2) comes with the initial value condition as

Under the assumptions (H_{1})-(H_{3}), denote by \(y(t,m)\) the solution of the IVP (2.3). We assume that *f* is strong continuous enough to guarantee that \(y(t,m)\) is uniquely defined and that it depends continuously on both *t* and *m*. The discussion of this problem can be found in [18]. Therefore the solution of IVP (2.3) exists.

Then solving (2.2) is equivalent to finding a \(m^{*}\) such that \(k(m^{*}) = 1 \) or \(\varphi(m^{*})=0\).

### Lemma 2.2

(Sturm comparison theorem) [25]

*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 2.3

*Let*\(y(t,m)\), \(z(t,m)\), \(Z(t,m)\)

*be the solution of the IVPs*,

*respectively*,

*and suppose that*\(F(t)\), \(g(t)\),

*and*\(G(t)\)

*are continuous functions defined on*\([0, 1]\)

*such that*

*If*\(Z(t,m)\)

*does not vanish in*\((0,1]\),

*then for any*\(0 \leq\xi\leq s \leq1\),

*we have*

*and hence*,

*for any*\(0 \leq s \leq1\),

*we have*

### Proof

The proof for (2.4) can be found in [18]. The continuity of the integrands implies the existence of the Riemann integral. In view of the definition of Stieltjes integral, by using the inequality of the limit, we have (2.5). □

### Lemma 2.4

*Assume that* (H_{1})-(H_{2}) *hold and* \(0 < \int^{1}_{0}\,dA(t)< 1\), *then BVP* (2.2) *has no positive solution*.

### Proof

Hence, we need \(\int^{1}_{0}\,dA(s)\geq1\), and we assume \(\int ^{1}_{0}\,dA(s)> 1\) in (H_{3}) in order to satisfy (3.1). □

## 3 Main results

In the following, we assume that \(A(t)\) has continuous derivative function \(\alpha(t)\) and \(\alpha(t)> 1\) for \(t\in[0,1]\) such that \(\int^{1}_{0}\,dA(t)=\int^{1}_{0}\alpha(t)\,dt > 1\).

It is obvious that \(\alpha^{L}\geq\alpha^{l} > 1\).

### Lemma 3.1

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*Then there exist a solution*\(x=A_{1}\in (0,\frac{\pi}{2})\)

*such that*

*and a solution*\(x=A_{2} \in(0,\frac{\pi}{2})\)

*such that*

### Theorem 3.2

*Assume that*(H

_{1})-(H

_{3})

*hold*.

*Suppose one of the following conditions holds*:

- (i)
\(0\leq f^{0} < \frac{\underline{A}^{2}}{h^{L}}\), \(f_{\infty} > \frac{\bar{A}^{2}}{h^{l}}\);

- (ii)
\(0\leq f^{\infty} < \frac{\underline{A}^{2}}{h^{L}}\), \(f_{0} > \frac{\bar{A}^{2}}{h^{l}}\).

*Then problem*(1.1)

*has at least one positive solution*,

*where*

*and*\(A_{1}\), \(A_{2}\)

*are defined in*(3.1)

*and*(3.2),

*respectively*.

### Proof

As we mentioned above, BVP (1.1) having a positive solution is equivalent to BVP (2.2) having a positive solution.

*r*such that

*L*large enough such that

Next, we will find a positive number \(m^{*}_{2}\) such that \(\varphi (m^{*}_{2})\geq0\).

*σ*such that

Since the solution \(y(t,m)\) is concave and \(y'(0,m)=0\), it hits the line \(y=L\) at most one time for the constant *L* defined in (3.7) and \(t\in(0,1]\). We denote the intersecting time by \(\bar{\delta}_{m}\) provided it exists. Henceforth, denote \(I_{m}=(0,\bar{\delta}_{m}]\subseteq(0,1]\). If \(y(1,m)\geq L\), then \(\bar{\delta}_{m}=1\).

The discussion is divided into three steps.

*Step* 1. We claim that there exists a value \(m_{0}\) large enough such that \(0\leq y(t,m_{0})\leq L\) for \(t\in[\bar{\delta}_{m_{0}},1]\) and \(y(t,m_{0})\geq L\) for \(t\in I_{m_{0}}\).

*t*, we have

*A*is defined in (2.1) as a continuous operator that depends on

*y*, for \(f(Ay,y)\) there exists a maximum for \(y \in[0,L]\). Denote \(L_{f}=\max_{y\in[0,L]}f(Ay,y)\). If we choose \(m>L+L_{f}h^{L}\), (3.10) will lead to a contradiction.

Since \(y(t,m)\) is continuous and concave, there exists a number \(m_{0}\) large enough such that \(y(t,m_{0})\geq L\) for \(t\in I_{m_{0}}\).

*Step*2. There exists a monotonically increasing sequence \(\{m_{k}\} \) such that the sequence \(\bar{\delta}_{m_{k}}\) is increasing on \(m_{k}\). That is,

*f*guarantees that \(y(t,m)\) is uniquely defined, the solution \(y(t,m_{k-1})\) and \(y(t,m_{k})\) have no intersection in the interval \([\bar{\delta}_{m_{k-1}},1)\). It follows from

*Step* 3. Seek a value \(m^{*}_{2}\) and a positive number *σ* such that \(0< \frac{A_{2}}{A_{2}+\epsilon}\leq\sigma\leq1\) and \(y(t,m^{*}_{2})\geq L\) for \(t\in(0,\sigma]\).

*n*large enough such that

In the following, we prove that \(k(m^{*}_{2})\geq1\) for the selected \(m^{*}_{2}\) and *σ*.

From (3.6) and (3.14), we can find a \(m^{*}\) between \(m^{*}_{1}\) and \(m^{*}_{2}\) such that \(y(t,m^{*})\) is the solution of (2.2). So that \(u(t,m^{*})=Ay(t,m^{*})\) is the solution of (1.1).

Now, we prove for (ii).

From (3.17) and (3.18), we can find a \(m^{*}\) between \(m^{*}_{3}\) and \(m^{*}_{4}\) such that \(y(t,m^{*})\) is the solution of (2.2). So \(u(t,m^{*})=Ay(t,m^{*})\) is the solution of (1.1). The proof of the theorem is complete. □

### Example

_{1})-(H

_{3}) and the condition (ii) of Theorem 3.2 are satisfied. It implies that (3.19) has at least one positive solution \(u(t)\).

## Notes

### Acknowledgements

The work is supported by Chinese Universities Scientific Fund (Project No.2016LX002)

## References

- 1.Gregus, M: Third Order Linear Differential Equations. Mathematics and Its Applications. Reidel, Dordrecht (1987) CrossRefMATHGoogle Scholar
- 2.Zhou, C, Ma, D: Existence and iteration of positive solutions for a generalized right-focal boundary value problem with
*p*-Laplacian operator. J. Math. Anal. Appl.**324**, 409-424 (2006) MathSciNetCrossRefMATHGoogle Scholar - 3.Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc.
**74**, 673-693 (2006) MathSciNetCrossRefMATHGoogle Scholar - 4.Webb, JRL, Infante, G: Nonlocal boundary value problems of arbitrary order. J. Lond. Math. Soc. (2)
**79**, 238-258 (2009) MathSciNetCrossRefMATHGoogle Scholar - 5.Webb, JRL: Positive solutions of some higher order nonlocal boundary value problems. Electron. J. Qual. Theory Differ. Equ.
**2009**, 29 (2009) MathSciNetMATHGoogle Scholar - 6.Graef, JR, Webb, JRL: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Anal.
**71**, 1542-1551 (2009) MathSciNetCrossRefMATHGoogle Scholar - 7.Jankowski, T: Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions. Nonlinear Anal.
**75**, 913-923 (2012) MathSciNetCrossRefMATHGoogle Scholar - 8.Boucherif, A, Bouguima, SM, Benbouziane, Z, Al-Malki, N: Third order problems with nonlocal conditions of integral type. Bound. Value Probl.
**2014**, 137 (2014) MathSciNetCrossRefMATHGoogle Scholar - 9.Pang, H, Xie, W, Cao, L: Successive iteration and positive solutions for a third-order boundary value problem involving integral conditions. Bound. Value Probl.
**2015**, 139 (2015) MathSciNetCrossRefMATHGoogle Scholar - 10.Sun, Y: Positive solutions of singular third-order three-point boundary value problem. J. Math. Anal. Appl.
**306**, 589-603 (2005) MathSciNetCrossRefMATHGoogle Scholar - 11.Lin, X, Du, Z, Liu, W: Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem. Appl. Math. Comput.
**205**, 187-196 (2008) MathSciNetMATHGoogle Scholar - 12.El-Shahed, M: Positive solutions for nonlinear singular third order boundary value problem. Commun. Nonlinear Sci. Numer. Simul.
**14**, 424-429 (2009) MathSciNetCrossRefMATHGoogle Scholar - 13.Graef, JR, Kong, L: Positive solutions for third order semipositone boundary value problems. Appl. Math. Lett.
**22**, 1154-1160 (2009) MathSciNetCrossRefMATHGoogle Scholar - 14.Li, S: Positive solutions of nonlinear singular third-order two-point boundary value problem. J. Math. Anal. Appl.
**323**, 413-425 (2006) MathSciNetCrossRefMATHGoogle Scholar - 15.Liu, Z, Ume, JS, Kang, SM: Positive solutions of a singular nonlinear third order two-point boundary value problems. J. Math. Anal. Appl.
**326**, 589-601 (2007) MathSciNetCrossRefMATHGoogle Scholar - 16.Yao, Q: The existence and multiplicity of positive solutions for a third-order three-point boundary value problem. Acta Math. Appl. Sin.
**19**, 117-122 (2003) MathSciNetCrossRefMATHGoogle Scholar - 17.Yao, Q, Feng, Y: The existence of solution for a third-order two-point boundary value problem. Appl. Math. Lett.
**15**, 227-232 (2002) MathSciNetCrossRefMATHGoogle Scholar - 18.Kwong, MK: The shooting method and multiple solutions of two/multi-point BVPs of second-order ODE. Electron. J. Qual. Theory Differ. Equ.
**2006**, 6 (2006) MathSciNetMATHGoogle Scholar - 19.Hopkins, B, Kosmatov, N: Third-order boundary value problems with sign-changing solutions. Nonlinear Anal.
**67**, 126-137 (2007) MathSciNetCrossRefMATHGoogle Scholar - 20.Iturriage, L, Sanchez, J: Exact number of solutions of stationary reaction-diffusion equations. Appl. Math. Comput.
**216**, 1250-1258 (2010) MathSciNetMATHGoogle Scholar - 21.Kwong, MK, Wong, JSW: The shooting method and nonhomogeneous multipoint BVPs of second-order ODE. Bound. Value Probl.
**2007**, 64012 (2007) MathSciNetCrossRefMATHGoogle Scholar - 22.Agarwal, RP: The numerical solution of multipoint boundary value problems. J. Comput. Appl. Math.
**5**, 17-24 (1979) MathSciNetCrossRefMATHGoogle Scholar - 23.Henderson, J: Uniqueness implies existence for three-point boundary value problems for second order differential equations. Appl. Math. Lett.
**18**, 905-909 (2005) MathSciNetCrossRefMATHGoogle Scholar - 24.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) MathSciNetCrossRefMATHGoogle Scholar - 25.Deng, Z: Sturmian theory of ordinary differential equations of the second order. J. Cent. China Normal Univ. Nat. Sci. No. S1 (1982) Google Scholar

## Copyright information

**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.