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].

In [

8], Ma investigated the existence of positive solutions of the nonlinear second-order m-point boundary value problem

${u}^{\u2033}(t)+a(t)f(u(t))=0,\phantom{\rule{1em}{0ex}}0<t<1,$

(1.1)

$u(0)=0,\phantom{\rule{2em}{0ex}}\sum _{i=1}^{m-2}{\alpha}_{i}{u}_{i}({\eta}_{i})=u(1),$

(1.2)

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 $a({t}_{0})>0$.

Set

${f}_{0}=\underset{u\to {0}^{+}}{lim}\frac{f(u)}{u},\phantom{\rule{2em}{0ex}}{f}_{\mathrm{\infty}}=\underset{u\to \mathrm{\infty}}{lim}\frac{f(u)}{u}.$

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 $0<{\sum}_{i=1}^{m-2}{\alpha}_{i}{\eta}_{i}<1$.

Recently, Tariboon [

9] considered three-point boundary-value problem (1.1) with the integral boundary condition

$u(0)=0,\phantom{\rule{2em}{0ex}}u(1)=\alpha {\int}_{0}^{\eta}u(s)\phantom{\rule{0.2em}{0ex}}ds,$

(1.3)

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.

Constructively, Agarwal [

11] explored the solution of multi-point boundary value problems by converting BVPs to equivalent IVPs, which is called shooting method. After that Man Kam Kwong [

4,

12] used the shooting method to consider second-order multi-point boundary value problems. In [

12], Kwong studied the existence of a positive solution to the following three-point boundary value problem:

${u}^{\u2033}(t)+f(u(t))=0,\phantom{\rule{1em}{0ex}}0<t<1,$

(1.4)

$u(0)=0,\phantom{\rule{2em}{0ex}}\mu u\left(\frac{1}{2}\right)=u(1).$

(1.5)

The principle of the shooting method used in [

12] is converting BVP (1.4)-(1.5) into finding suitable initial slopes

$m>0$ such that the solution of equation (

1.4) with the initial value condition

$u(0)=0,\phantom{\rule{2em}{0ex}}{u}^{\prime}(0)=m$

(1.6)

vanishes for the first time after

$t>1$. Denote by

$u(t,m)$ the solution of (1.4)-(1.6) provided it exists. Then solving the boundary value problem is equivalent to finding

*m* such that

$\mu u(\frac{1}{2},m)=u(1,m).$

If we can find two solutions

$u(t,{m}_{1})$ and

$u(t,{m}_{2})$ of (1.4) such that

$u(1,{m}_{1})\ge (\text{or}\le )\mu u(\frac{1}{2},{m}_{1})$

and

$u(1,{m}_{2})\le (\text{or}\ge )\mu u(\frac{1}{2},{m}_{2}),$

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).

In this paper, we try to employ the shooting method to establish the existence results of positive solutions for (1.1) with the more generalized multi-point integral boundary condition

$u(0)=0,\phantom{\rule{2em}{0ex}}\sum _{i=1}^{n}{\alpha}_{i}{\int}_{0}^{{\eta}_{i}}u(s)\phantom{\rule{0.2em}{0ex}}ds=u(1),$

(1.7)

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

For the sake of convenience, we denote

$\begin{array}{c}\underset{0\le t\le 1}{max}\{a(t)\}={a}^{L},\phantom{\rule{2em}{0ex}}\underset{0\le t\le 1}{min}\{a(t)\}={a}^{l},\hfill \\ {\overline{f}}_{x}=\underset{u\to x}{lim}sup\frac{f(u)}{u},\phantom{\rule{2em}{0ex}}{\underline{f}}_{x}=\underset{u\to x}{lim}inf\frac{f(u)}{u},\phantom{\rule{1em}{0ex}}x\in \{0,+\mathrm{\infty}\}.\hfill \end{array}$

Let

$u(t,m)$ be the solution of (1.1)-(1.6) and define

$k(m)=\frac{{\sum}_{i=1}^{n}{\alpha}_{i}{\int}_{0}^{{\eta}_{i}}u(s,m)\phantom{\rule{0.2em}{0ex}}ds}{u(1,m)}.$

(1.8)

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*,

$\begin{array}{c}{u}^{\u2033}(t)+F(u(t))=0,\phantom{\rule{2em}{0ex}}u(0)=0,\phantom{\rule{2em}{0ex}}{u}^{\prime}(0)=m,\hfill \\ {Z}^{{}^{\u2033}}(t)+G(Z(t))=0,\phantom{\rule{2em}{0ex}}Z(0)=0,\phantom{\rule{2em}{0ex}}{Z}^{\prime}(0)=m,\hfill \\ {z}^{\u2033}(t)+g(z(t))=0,\phantom{\rule{2em}{0ex}}z(0)=0,\phantom{\rule{2em}{0ex}}{z}^{\prime}(0)=m,\hfill \end{array}$

*and suppose that* *F*,

*G*,

*g* *are nonnegative continuous functions on a certain interval* *I* *for* $t\in [0,1]$ *and such that* $g(\omega )\le F(\omega )\le G(\omega ),\phantom{\rule{1em}{0ex}}\omega \in I.$

*If* $Z(t)$ *does not vanish in* $[0,1]$,

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

*it yields* $\frac{z(\eta )}{z(1)}\le \frac{u(\eta )}{u(1)}\le \frac{Z(\eta )}{Z(1)}.$

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.