# Application of the shooting method to second-order multi-point integral boundary-value problems

- Huilan Wang
^{1}Email author, - Zigen Ouyang
^{1}and - Liguang Wang
^{1}

**2013**:205

https://doi.org/10.1186/1687-2770-2013-205

© Wang et al.; licensee Springer 2013

**Received: **3 March 2013

**Accepted: **8 August 2013

**Published: **9 September 2013

## Abstract

In this paper, we focus on the following second-order multi-point integral boundary-value problem:

where $0<{\eta}_{1}<{\eta}_{2}<\cdots <{\eta}_{n}<1$, ${\alpha}_{i}\ge 0$ for $i=1,\dots ,n-1$ and ${\alpha}_{n}>0$ are given constants. The proof is based on the shooting method. By constructing a quadratic function and a sine function as the shooting objects and combining the integral mean value theorem with the comparison principle, we consider the existence of positive solutions to the BVP respectively under the case $0<{\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}\le 1$ and the case ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}>1$. The method is concise and some new criteria are established.

**MSC:**34B10, 34B15, 34B18.

## Keywords

## 1 Introduction

For the study of nonlinear second-order multi-point boundary-value problem, many results have been obtained by using all kinds of fixed point theorems related to a completely continuous map defined in a Banach space. We refer the reader to [1–9] and the references therein. Some of the results are so classical that little work can exceed; however, most of these papers are concerned with problems with boundary conditions of restrictions either on the slope of solutions and the solutions themselves, or on the number of boundary points [2, 5–8, 10].

where $0<{\eta}_{1}<{\eta}_{2}<\cdots <{\eta}_{m-2}<1$, ${\alpha}_{i}\ge 0$ for $i=1,\dots ,m-3$, ${\alpha}_{m-2}>0$, $f\in C([0,\mathrm{\infty});[0,\mathrm{\infty}))$, $a\in C([0,1];[0,\mathrm{\infty}))$, and there exists a ${t}_{0}\in [{\eta}_{m-2},1]$ such that $a({t}_{0})>0$.

The author obtained the existence of a positive solution to (1.1)-(1.2) under the case ${f}_{0}=0$ and ${f}_{\mathrm{\infty}}=\mathrm{\infty}$ (super-linear case) or the case ${f}_{0}=\mathrm{\infty}$ and ${f}_{\mathrm{\infty}}=0$ (sub-linear case) when $0<{\sum}_{i=1}^{m-2}{\alpha}_{i}{\eta}_{i}<1$.

where $0<\eta <1$, $\alpha >0$.

Such a boundary condition might be more realistic in the mathematical models of thermal conductivity, groundwater flow, thermoelectric flexibility and plasma physics, because it describes the fluid properties in a certain continuous medium. Under the assumption that $0<\alpha {\eta}^{2}<2$, Tariboon and the author proved that problem (1.1)-(1.3) has at least one positive solution in the super-linear case or in the sub-linear one.

However, the method used in the previous two papers is Krasnoselskii’s fixed point theorem in a cone, which relates to constructing a completely continuous cone map in a Banach space, and the proof is somewhat procedural.

*m*such that

where $u(t,{m}_{1})>0$, $u(t,{m}_{2})>0$ for $t\in (0,1)$, then there must exist a number *m* between ${m}_{1}$ and ${m}_{2}$ such that $u(t,m)$ is the solution of (1.4)-(1.5). By constructing two sine functions as the shooting objects and combining with the comparison principle, the author obtained some better results than those *via* fixed point techniques for the existence of positive solutions to (1.4)-(1.5).

where $0<{\eta}_{1}<{\eta}_{2}<\cdots <{\eta}_{n}<1$, ${\alpha}_{i}\ge 0$ for $i=1,\dots ,n-1$ and ${\alpha}_{n}>0$ are given constants. Following the principle of the shooting method, there are two obstacles we encounter. The first one is that the boundary condition involves integral from 0 to ${\eta}_{i}$ ($i=1,\dots ,n$), so we transform the integral problem into a single-point problem by using the integral mean value theorem. The other difficulty is that we cannot obtain the existence results by constructing two sine functions as in [12] because of the particularity of $\eta =\frac{1}{2}$ in [12]. Therefore, we construct a quadratic function and a sine function as the objective ones.

The purpose of this article lies in two aspects. One is to explore the application of the shooting method in a more complicated multi-point integral boundary value problem, which demonstrates another way in studying BVPs. The other one is to establish new criteria for the existence of positive solutions to (1.1)-(1.7) under the case $0<{\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}\le 1$ and the case ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}>1$.

In this paper, we always assume:

(H_{1}) $f\in C([0,\mathrm{\infty});[0,\mathrm{\infty}))$, $a\in C([0,1];[0,\mathrm{\infty}))$, ${a}^{l}>0$.

Under the assumption, it is not difficult to prove that the initial problem (1.1)-(1.6) has at least one solution defined on $[0,1]\times [0,+\mathrm{\infty})$. In fact, after translating second-order differential equation (1.1) into one-order equations, one can draw the conclusion [13].

Further, we introduce the comparison results derived from [4, 12], which evolved from the Sturm comparison theorem.

**Theorem 1.1**

*Let*$u(t,m)$, $z(t,m)$, $Z(t,m)$

*be the solution of the initial value problems*,

*respectively*,

*and suppose that*

*F*,

*G*,

*g*

*are nonnegative continuous functions on a certain interval*

*I*

*for*$t\in [0,1]$

*and such that*

*If*$Z(t)$

*does not vanish in*$[0,1]$,

*then for*$0<\eta <1$,

*it yields*

The paper is arranged as follows. In the next section, we put forward the basic principle of the shooting method used in this paper, and show that BVP (1.1)-(1.7) has no positive solution when ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}^{2}>2$. In Section 3, the general criteria are established for the existence of positive solutions to (1.1)-(1.7) under the case $0<{\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}^{2}<2$. Moreover, we present the special results in the form of corollaries corresponding to the super-linear case or the sub-linear case. Finally, we come to the conclusion and an example is presented to illustrate our results.

## 2 Preliminaries

**Lemma 2.1**

*If there exist two initial slopes*${m}_{1}>0$

*and*${m}_{2}>0$

*such that*

- (i)
*the solution*$u(t,{m}_{1})$*of*(1.1)-(1.6)*remains positive in*$(0,1)$*and*$k({m}_{1})\le 1$; - (ii)
*the solution*$u(t,{m}_{2})$*of*(1.1)-(1.6)*satisfies*$u(t,{m}_{2})>0$*for*$t\in (0,1)$*and*$k({m}_{2})\ge 1$;*then multi*-*point boundary value problem*(1.1)-(1.7)*has a positive solution with the slope*${u}^{\prime}(0)={m}_{0}$*between*${m}_{1}$*and*${m}_{2}$.

*Proof*Since the solutions of (1.1)-(1.6) depend on the initial value continuously, then from (1.8), it implies that $k(m)$ is continuous on

*m*. In view of the intermediate value theorem of continuous functions, there exists a number ${m}_{0}$ between ${m}_{1}$ and ${m}_{2}$ such that $k({m}_{0})=1$, that is,

Therefore, $u(t,{m}_{0})$ is the solution of (1.1)-(1.7). □

**Lemma 2.2** *Let* ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}^{2}>2$, *then* (1.1)-(1.7) *has no positive solution*.

*Proof* Assume that (1.1)-(1.7) has a positive solution *u*.

*u*implies that $u({\eta}_{i})>0$ ($i=1,2,\dots ,n$) and

which contradicts with the convexity of *u*.

If $u(1)=0$, then ${\sum}_{i=1}^{n}{\alpha}_{i}{\int}_{0}^{{\eta}_{i}}u(s)\phantom{\rule{0.2em}{0ex}}ds=0$, that is, $u(t)\equiv 0$ for $t\in [0,{\eta}_{n}]$. If there exists $\tau \in ({\eta}_{n},1)$ such that $u(\tau )>0$, then $u(0)=u({\eta}_{n})=0$ and $u(\tau )>0$, which contradicts with the convexity of *u*. Therefore $u(t)\equiv 0$ for $t\in [0,1]$.

In the rest of this paper, we always assume:

(H_{2}) $0<{\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}^{2}<2$.

□

## 3 Main results

**Theorem 3.1**

*Assume that*(H

_{1})-(H

_{2})

*holds*.

*Suppose*$0<{\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}\le 1$

*and there exists a constant*$A\in [0,\frac{\pi}{2}]$

*such that*

- (i)
${\overline{f}}_{0}<\frac{{A}^{2}}{{a}^{L}}\le \frac{{A}^{2}}{{a}^{l}}<{\underline{f}}_{\mathrm{\infty}}$;

*or* - (ii)
${\overline{f}}_{\mathrm{\infty}}<\frac{{A}^{2}}{{a}^{L}}\le \frac{{A}^{2}}{{a}^{l}}<{\underline{f}}_{0}$.

*Then problem* (1.1)-(1.7) *has a positive solution*.

*Proof*(i) Since ${\overline{f}}_{0}<\frac{{A}^{2}}{{a}^{L}}$, we can choose a positive number ${m}^{\ast}$ such that

where ${\xi}_{i}\in [{\eta}_{i-1},{\eta}_{i}]$ and $\overline{\xi}\in \{{\xi}_{1},\dots ,{\xi}_{n}\}$ such that $u(\overline{\xi},{m}_{1})={max}_{1\le i\le n}u({\xi}_{i},{m}_{1})$.

*M*large enough such that

*M*, there exist two numbers

*δ*and ${M}^{\ast}$ such that

_{2}) and (3.3), it is not difficult to verify that

By Lemma 2.1 and (3.2)-(3.5), there exists a number ${m}_{0}$ between ${m}_{1}$ and ${m}_{2}$ such that $u(t,{m}_{0})$ is the positive solution of (1.1)-(1.7). The proof for (i) is complete.

Now, we prove for (ii).

*N*large enough such that

*N*, there exist a number

*ϵ*small enough and a number ${m}_{1}$ large enough such that $0<\u03f5<{\eta}_{1}$ and $u(t,{m}_{1})\ge N$ for $t\in [\u03f5,1-\u03f5]$. Therefore

Obviously, $\u03f5\to 0$ as ${m}_{1}\to \mathrm{\infty}$. Thus $u(t,{m}_{1})\ge N$ approximately for $t\in [0,1]$ as ${m}_{1}\to \mathrm{\infty}$.

where ${\xi}_{i}\in [{\eta}_{i-1},{\eta}_{i}]$ and $\overline{\xi}\in \{{\xi}_{1},\dots ,{\xi}_{n}\}$ such that $u(\overline{\xi},{m}_{1})={max}_{1\le i\le n}u({\xi}_{i},{m}_{1})$.

*σ*small enough such that

*σ*and ${m}_{2}$, there exists a positive number

*τ*small enough such that

By Lemma 2.1, the proof for (ii) is complete. □

**Theorem 3.2**

*Assume that*(H

_{1})-(H

_{2})

*holds*.

*Suppose*${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}>1$

*and there exists a constant*$A\in [0,\frac{\pi}{2}]$

*such that*

*Then problem*(1.1)-(1.7)

*has a positive solution under the case*

- (i)
${\overline{f}}_{0}<\frac{{A}^{2}}{{a}^{L}}\le \frac{{A}^{2}}{{a}^{l}}<{\underline{f}}_{\mathrm{\infty}}$;

*or* - (ii)
${\overline{f}}_{\mathrm{\infty}}<\frac{{A}^{2}}{{a}^{L}}\le \frac{{A}^{2}}{{a}^{l}}<{\underline{f}}_{0}$.

*Proof*Note the computation of $k({m}_{1})$ in Theorem 3.1. In (3.2), if we substitute ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}$ with

then $k({m}_{1})\le 1$, and all the steps in the following are the same as in Theorem 3.1. □

Now, let us consider the special super-linear case or the sub-linear case. It is not difficult to verify the following corollaries.

**Corollary 3.1**

*Assume that*$0<{\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}\le 1$

*and*

- (i)
${f}_{0}=0$, ${f}_{\mathrm{\infty}}=\mathrm{\infty}$;

*or* - (ii)
${f}_{0}=\mathrm{\infty}$, ${f}_{\mathrm{\infty}}=0$.

*Then problem* (1.1)-(1.7) *has a positive solution*.

**Corollary 3.2**

*If*${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}>1$

*and there exists a constant*$A\in [0,\frac{\pi}{2}]$

*such that*

*Then*,

*problem*(1.1)-(1.7)

*has a positive solution under the case*

- (i)
${f}_{0}=0$, ${f}_{\mathrm{\infty}}=\mathrm{\infty}$ ;

*or* - (ii)
${f}_{0}=\mathrm{\infty}$, ${f}_{\mathrm{\infty}}=0$.

## 4 Conclusion and examples

The tool which we used for the analysis in this article is the shooting method derived from [4, 12]; however, we considered a more general problem which involves integral boundary-value and multiplicity of boundary-point. The meaningful work that we have done lies in the following three aspects. The first one is that we transform the integral problem into a single-point value one by using the integral mean value theorem. The other one is that we construct a quadratic function and a sine function as the comparison functions because it does not take effect to construct two sine functions as in [12]. Finally, we established the new criteria for the existence of positive solutions to (1.1)-(1.7) under the case ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}\le 1$ and the case ${\sum}_{i=1}^{n}{\alpha}_{i}{\eta}_{i}>1$. Obviously, (1.7) vanishes to (1.3) when $n=1$ and the sup-linear case or the sub-linear case is sufficient for the conditions in Theorem 3.1 and Theorem 3.2, so some of our results are more general or better than those via fixed point techniques. However, in Theorem 3.2, whether the transcendental equation has a solution is somewhat difficult to verify. It can be seen that each method has its pros and cons.

**Example 4.1**Consider the BVP

## Declarations

### Acknowledgements

The authors would like to thank the editors and the anonymous referees for their valuable suggestions on the improvement of this paper. First author was partially supported by the Scientific Research Fund of Hunan Provincial Educational Department (1200361), Project of Science and Technology Bureau of Hengyang, Hunan Province (2012KJ2). Second author was partially supported by the Doctor Foundation of University of South China ( No. 5-XQD-2006-9), the Foundation of Science and Technology Department of Hunan Province (No. 2009RS3019), the Natural Science Foundation of Hunan Province (No. 13JJ3074) and the Subject Lead Foundation of University of South China (No. 2007XQD13).

## Authors’ Affiliations

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