# Global structure of positive solutions for three-point boundary value problems

- Jia-Ping Gu
^{1}, - Liang-Gen Hu
^{1}and - Huai-Nian Zhang
^{2}Email author

**2013**:174

https://doi.org/10.1186/1687-2770-2013-174

© Gu et al.; licensee Springer 2013

**Received: **29 April 2013

**Accepted: **8 July 2013

**Published: **25 July 2013

## Abstract

In this paper, we are concerned with the three-point boundary value problem for second-order differential equations

where $\beta \ge 0$, $0<\eta <1$, $0<\alpha \eta <1$ and $1+\beta -\alpha \eta -\alpha \beta >0$; $w\in C([0,1],(0,+\mathrm{\infty}))$ and $f\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$, ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$ satisfies $f(u)>0$ for $u>0$. The existence of the continuum of a positive solution is established by utilizing the Leray-Schauder global continuation principle. Furthermore, the interval of *α* about the nonexistence of a positive solution is also given.

**MSC:**34B10, 34B18, 34G20.

## Keywords

## 1 Introduction

where $\beta \ge 0$, $0<\eta <1$, $0<\alpha \eta <1$ and $1+\beta -\alpha \eta -\alpha \beta >0$; $w\in C([0,1],(0,+\mathrm{\infty}))$ and $f\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$, ${\mathbb{R}}_{+}=[0,\mathrm{\infty})$ satisfies $f(u)>0$ for $u>0$.

The existence and multiplicity of positive solutions for multi-point boundary value problems have been studied by several authors and many nice results have been obtained; see, for example, [1–6] and the references therein for more information on this problem. The multi-point boundary conditions of ordinary differential equations arose in different areas of applied mathematics and physics. In addition, they are often used to model many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems, infectious diseases as well as concentration in chemical or biological problems. In all these problems, only *positive* solutions are very meaningful.

*et al.*[1] studied the three-point boundary value problem

where $\mu >0$ is a parameter, $\beta \ge 0$, $0<\eta <1$, $0<\alpha \eta <1$ and $1+\beta -\alpha \eta -\alpha \beta >0$. Based on Krein-Rutmann theorems and the fixed point index theory, they not only established the criteria of the existence and multiplicity of a positive solution, but also obtained the parameter *μ* in relation with the nonlinear term *f* and the first eigenvalue of the linear operator.

*On the other hand*, we note that the nice results in [1] only gave the existence and multiplicity of positive solutions, and if the parameter *α* is regarded as a variable, then an interesting problem as to what happens to the global structure of positive solutions of (1.2) was not considered. However, this relationship is very useful for computing the numerical solution of (1.2) as it can be used to guide the numerical work. For example, the global bifurcation of solutions for second-order differential equations has been extensively studied in the literature, see [4, 7, 8].

Motivated by this, in this paper, we consider the three-point boundary value problem for second-order differential equations (1.1) and make use of the Leray-Schauder global continuation theorem in the frame of techniques nicely employed by Ma and Thompson [4] and convex analysis technique. We consider two cases ${f}_{0}=0$, ${f}_{\mathrm{\infty}}=\mathrm{\infty}$ and ${f}_{0}=\mathrm{\infty}$, ${f}_{\mathrm{\infty}}=0$, and establish the existence of continuum of positive solutions, where ${f}_{0}={lim}_{u\to {0}^{+}}\frac{f(u)}{u}$ and ${f}_{\mathrm{\infty}}={lim}_{u\to \mathrm{\infty}}\frac{f(u)}{u}$. Moreover, the interval of parameter *α* about the nonexistence of positive solutions is also given. Our main results extend and improve the corresponding results [1, 3, 4]. In contrast to [[1], Theorem 3.1 and Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter *α* is regarded as a variable.

The rest of this paper is arranged as follows. In Section 2, we give Green’s function and some lemmas. In Section 3, we consider the case ${f}_{0}=0$, ${f}_{\mathrm{\infty}}=\mathrm{\infty}$, and give the existence of the continuum of positive solutions and the interval of parameter *α* about the nonexistence of positive solutions. In Section 4, we study the case ${f}_{0}=\mathrm{\infty}$, ${f}_{\mathrm{\infty}}=0$, and give the existence of global continuum of positive solutions.

## 2 Preliminaries and lemmas

then *P* is a cone.

We assume that

(H0) $\beta \ge 0$, $0<\eta <1$, $0<\alpha \eta <1$ and $\ell =1+\beta -\alpha \eta -\alpha \beta >0$.

**Lemma 2.1** (see [[1], Lemma 2.1])

*Suppose that condition*(H0)

*holds and*$x\in L[0,1]$.

*Then the following linear differential equation*

*has a unique solution*

*where*$G(t,s):[0,1]\times [0,1]\to [0,\mathrm{\infty})$

*is defined by*

For the sake of convenience, we list the following hypotheses:

(H1) $w\in C([0,1],(0,+\mathrm{\infty}))$.

(H2) $f\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ satisfies $f(u)>0$ for $u>0$.

(H3) ${f}_{0}={lim}_{u\to {0}^{+}}\frac{f(u)}{u}=0$, ${f}_{\mathrm{\infty}}={lim}_{u\to \mathrm{\infty}}\frac{f(u)}{u}=\mathrm{\infty}$ (superlinear).

(H4) ${f}_{0}={lim}_{u\to {0}^{+}}\frac{f(u)}{u}=\mathrm{\infty}$, ${f}_{\mathrm{\infty}}={lim}_{u\to \mathrm{\infty}}\frac{f(u)}{u}=0$ (sublinear).

**Lemma 2.2**

*Assume that*(H0)

*holds*.

*Let*$x\in C[0,1]$

*with*$x(t)\ge 0$

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

*and let*

*u*

*be a solution of*

*Then* $u(t)\ge 0$ *for* $t\in [0,1]$. *Moreover*, *if* $x(\varsigma )>0$ *for some* $\varsigma \in [0,1]$, *then* $u(t)>0$ *for all* $t\in (0,1)$.

*Proof* We only show that if $x(\varsigma )>0$ for some $\varsigma \in [0,1]$, then $u(t)>0$ for all $t\in (0,1)$.

If $\beta =0$, then we have from [[3], Lemma 2] that the results hold.

We separate the proof into two cases: Case I: ${t}_{1}=0$ and Case II: ${t}_{2}\in (0,1]$.

*u*is concave down in $[0,1]$, we obtain that ${u}^{\prime}(t)<0$ and $u(t)<0$ for all $t\in [0,1]$. Set

*u*leads to

This contradicts the hypothesis $\ell =1+\beta -\alpha \eta -\alpha \beta >0$.

- (1)If ${u}^{\prime}(0)=0$, then $u(0)=\beta {u}^{\prime}(0)=0$ and the concavity of
*u*imply that ${u}^{\prime}(t)\le 0$, $\mathrm{\forall}t\in [0,1]$. Hence, we get that $u(t)\le 0$, $\mathrm{\forall}t\in [0,1]$ and $u(\eta )<0$ (since $u(\eta )=0$, we have that $u(1)=0$ leads to $u\equiv 0$. This contradicts (2.3)). Again, since*u*is concave, we have$\frac{u(1)-u(0)}{1}<\frac{u(\eta )-u(0)}{\eta}\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\frac{\alpha u(\eta )}{1}<\frac{u(\eta )}{\eta}.$

- (2)
If ${u}^{\prime}(0)<0$, then, adopting the same proof as in Case I, we get a contradiction.

- (3)If ${u}^{\prime}(0)>0$, then it follows that $u(0)=\beta {u}^{\prime}(0)>0$. In light of ${u}^{\prime}(0)>0$ and the concavity of
*u*, we get from (2.3) that$u(1)<0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}u(\eta )<0,$

This is a contradiction.

Consequently, we get from Case I and Case II that the conclusion holds. □

**Remark 2.1** If $u(t)$ is positive, then we know from the proof in Lemma 2.2 that $u(t)$ may only have zero point at $t=0$ and $t=1$.

**Lemma 2.3**

*Let*(H0)

*hold and let*$u\in {C}^{2}([0,1],{\mathbb{R}}_{+})$

*be a function satisfying*

*and*

*Then there exists*$\sigma \in [0,1]$

*such that*

*Proof* If $\beta =0$, then from [[4], Lemma 3.3] the conclusion holds.

- (1)
If $\alpha \in [0,1]$, then $u(\eta )\ge \alpha u(\eta )=u(1)$. This together with the concavity of

*u*yields ${u}^{\prime}(1)\le 0$. Since ${u}^{\prime}(0)=\frac{u(0)}{\beta}\ge 0$, there exists $\sigma \in [0,1]$ such that ${u}^{\prime}(\sigma )=0$. - (2)If $\alpha \in (1,\frac{1+\beta}{\eta +\beta})$, then $u(1)=\alpha u(\eta )>u(\eta )$. Consequently, there exists ${t}_{3}\in [\eta ,1]$ such that$u({t}_{3})=\underset{t\in [0,1]}{max}u(t)$

*u*in $[0,1]$ imply that

This completes the proof. □

Assume that (H0)-(H2) hold, then it is easy to verify that $T:P\to P$ is well defined and completely continuous. We note that *u* is a positive solution of problem (1.1) if and only if $u=T(\alpha ,u)$ on *P*.

By a *positive* solution of (1.1) we mean a solution of (1.1) which is positive on $(0,1)$.

in $\mathbb{R}\times C[0,1]$.

Using the Leray-Schauder global continuation theorem [[8], Theorem 14.C], Ma and Thompson [[4], Lemma 2.2] obtained the following result.

**Lemma 2.4**

*Let*

*P*

*be a cone in a Banach space*

*X*.

*Let*$U\subset P$

*be a bounded and open subset in*

*X*

*with respect to the topology induced by*$\parallel \cdot \parallel $

*on*

*P*.

*Assume that the operator*$T:[{\alpha}_{1},{\alpha}_{2}]\times P\to P$

*is a continuous*,

*compact map satisfying*:

- (1)
*the equation*$u=T(\alpha ,u)$*has no solution on*$[{\alpha}_{1},{\alpha}_{2}]\times (P\mathrm{\setminus}U)$; - (2)
$i(T({\alpha}_{1},\cdot ),U,P)=k$

*with*$k\ne 0$.

*Then the set*

*has a continuum* ℒ *of solutions in* $[{\alpha}_{1},{\alpha}_{2}]\times P$, *which connects the set* $\{{\alpha}_{1}\}\times U$ *with the set* $\{{\alpha}_{2}\}\times U$.

## 3 The superlinear case

**Lemma 3.1** [[1], Theorem 3.2]

*Let conditions*(H0)-(H3)

*hold*.

*Then there exist two constants*${r}_{\alpha}$

*and*${R}_{\alpha}$

*with*${r}_{\alpha}<{R}_{\alpha}$

*such that problem*(1.1)

*has at least one positive solution*${u}_{\alpha}$

*with*${r}_{\alpha}<\parallel {u}_{\alpha}\parallel <{R}_{\alpha}$.

*Furthermore*,

*where* ${P}_{R}=\{u\in P:\parallel u\parallel <R\}$.

*Proof* Since ${f}_{0}=\mathrm{\infty}$ and ${f}_{\mathrm{\infty}}=0$, we take $\mu =1$ in [[1], Theorem 3.2] and all the conditions in [[1], Theorem 3.2] are satisfied. Therefore, the conclusion holds. □

**Lemma 3.2**

*Assume that*(H0)-(H3)

*hold*.

*Let*${r}_{\alpha}$

*and*${R}_{\alpha}$

*be the constants as in Lemma*3.1.

*Then there exists a positive number*${\rho}_{\alpha}$

*with*$({\rho}_{\alpha},{\rho}_{\alpha}^{-1})\supset [{r}_{\alpha},{R}_{\alpha}]$

*such that*

*where* ${\mathrm{\Psi}}_{\alpha}$ *is defined by* (3.2).

*Proof*First we claim that if ${f}_{0}=0$, then there exists a positive number ${\rho}_{1}$ such that

*n*, such that

for some constant ${M}_{2}>0$ independent of *n*. Utilizing the Ascoli-Arzela theorem, we have that $\{{v}_{n}\}$ is a relatively compact set on $C[0,1]$. Assume, taking a subsequence if necessary, that ${v}_{n}\to \tilde{v}$ in $C[0,1]$. Then $\parallel \tilde{v}\parallel =1$ and $\tilde{v}\ge 0$ in $[0,1]$.

contradicts $\parallel \tilde{v}\parallel =1$. Therefore, the claim (3.3) holds.

*ψ*and integrating by parts, we find

This is a contradiction. Consequently, conclusion (3.8) holds.

Combining (3.3) and (3.8), we let ${\rho}_{\alpha}=min\{{\rho}_{1},{\rho}_{2}^{-1}\}$. Thus, the result holds. □

**Theorem 3.1** *Assume that conditions* (H0)-(H3) *hold*. *Then*
*contains a continuum which joins* $\{0\}\times C[0,1]$ *with* $(\frac{1+\beta}{\eta +\beta},0)$.

*Proof* We divide the proof into four steps.

Step 1. We construct a continuum.

From Lemma 3.2, we know that $T(\alpha ,u)=u$ has no solutions in $[0,\alpha ]\times (P\mathrm{\setminus}{U}_{\alpha})$. Therefore, from Lemma 2.4, there exists a continuum ${\zeta}^{\alpha}\subset {\mathrm{\Psi}}_{\alpha}$ which joins ${\mathrm{\Phi}}_{0}$ with ${\mathrm{\Phi}}_{\alpha}$. Here ${\mathrm{\Psi}}_{\alpha}$ is defined by (3.2), and ${\mathrm{\Phi}}_{0}$ and ${\mathrm{\Phi}}_{\alpha}$ are defined by (3.1).

Since ${\zeta}^{(1+\beta )/2(\eta +\beta )}\in \mathcal{L}$, we know that $\mathcal{L}\ne \mathrm{\varnothing}$.

From Lemma 3.2, $T(\alpha ,u)=u$ has no solution in $[0,\tilde{\alpha}+\u03f5]\times (P\mathrm{\setminus}{U}_{\tilde{\alpha}+\u03f5})$. Again, using Lemma 2.4, we find a continuum ${\zeta}^{\tilde{\alpha}+\u03f5}\subset {\mathrm{\Psi}}_{\tilde{\alpha}+\u03f5}$ which joins ${\mathrm{\Phi}}_{0}$ with ${\mathrm{\Phi}}_{\tilde{\alpha}+\u03f5}$. This contradicts (3.15). Therefore, the conclusion in (3.14) holds.

*ζ*be a continuum satisfying (3.14). We claim that

Adopting the same proof as in the second step in Lemma 3.2, we can find a contradiction. Hence, the result in (3.16) holds.

*ζ*be a continuum satisfying (3.14). Next we show that

- (1)If $\beta =0$, then we obtain from the boundary condition of (3.18) that $\tilde{u}(0)=0$ and $\tilde{u}(1)=\frac{1}{\eta}\tilde{u}(\eta )$. From the strict concavity of $\tilde{u}$, it follows that$\frac{\tilde{u}(1)}{1}<\frac{\tilde{u}(\eta )}{\eta}\phantom{\rule{1em}{0ex}}\u27fa\phantom{\rule{1em}{0ex}}\frac{\frac{1}{\eta}\tilde{u}(\eta )}{1}<\frac{\tilde{u}(\eta )}{\eta}$

- (2)If $\beta >0$, then $\tilde{u}(0)>0$ (since $\tilde{u}(0)=0$, we know that ${\tilde{u}}^{\prime}(0)=0$ and the strict concavity of $\tilde{u}$ imply $u(t)<0$ on $(0,1)$, which is a contradiction). Put$L(t):=\tilde{u}(0)+{\tilde{u}}^{\prime}(0)t.$

a contradiction.

Consequently, the conclusion in (3.17) holds. □

**Remark 3.1** In contrast to [[1], Theorem 3.2], we obtain the global structure and behavior of positive solutions, where the parameter *α* is regarded as a variable.

**Theorem 3.2**

*Suppose that*$\beta \ge 0$, $\eta \in (0,1)$

*and*$\alpha >\frac{1+\beta}{\eta +\beta}$.

*Let condition*(H1)

*and*$f\in C(\mathbb{R},{\mathbb{R}}_{+})$

*hold*,

*and let*$u\in {C}^{2}[0,1]$

*be a solution of*

*Then problem* (3.19) *has no positive solutions*.

*Proof* If $f(0)=0$, then we know that $u(t)\equiv 0$ for $t\in [0,1]$ is a trivial solution of (3.19).

*i.e.*,

- (1)
Case I. $\beta =0$.

*u*is concave down and

*u*is a positive solution of equation (3.19), we obtain that

*i.e.*,

- (2)
Case II. $\beta >0$.

If ${u}^{\prime}(0)\le 0$, then $u(0)\le 0$ and ${u}^{\prime}(t)\le 0$ imply that $u(t)\le 0$, $\mathrm{\forall}t\in [0,1]$. This contradicts (3.20).

*u*implies that

contradicts (3.21).

This is a contradiction.

Therefore, we conclude that if $\alpha >\frac{1+\beta}{\eta +\beta}$, then problem (3.19) has no positive solutions. □

## 4 The sublinear case

**Lemma 4.1** [[1], Theorem 3.1]

*Let conditions*(H0)-(H2)

*and*(H4)

*hold*.

*Then there exist two constants*${r}_{\alpha}$

*and*${R}_{\alpha}$

*with*${r}_{\alpha}<{R}_{\alpha}$

*such that problem*(1.1)

*has at least one positive solution*${u}_{\alpha}$

*with*${r}_{\alpha}<\parallel {u}_{\alpha}\parallel <{R}_{\alpha}$.

*Furthermore*,

*where* ${P}_{R}=\{u\in P:\parallel u\parallel <R\}$.

**Lemma 4.2**

*Assume that*(H0)-(H2)

*and*(H4)

*hold*.

*Let*${r}_{\alpha}$

*and*${R}_{\alpha}$

*be the constants as in Lemma*4.1.

*Then there exists a positive number*${\varrho}_{\alpha}$

*with*$({\varrho}_{\alpha},{\varrho}_{\alpha}^{-1})\supset [{r}_{\alpha},{R}_{\alpha}]$

*such that*

*where* ${\mathrm{\Psi}}_{\alpha}$ *is defined by* (3.2).

*Proof*First, we claim that if ${f}_{0}=\mathrm{\infty}$, then there exists a positive number ${\varrho}_{1}$ such that

Adopting the same proof as in Lemma 3.2, we get a contradiction. Hence, conclusion (4.1) holds.

uniformly holds for $t\in [0,1]$. Again, applying the proof method as that in Lemma 3.2, we get a contradiction. Consequently, conclusion (4.2) holds.

If we let ${\varrho}_{\alpha}=min\{{\varrho}_{1},{\varrho}_{2}^{-1}\}$, then combining (4.1) and (4.2), we have that the result holds. □

**Theorem 4.1** *Let* (H0)-(H2) *and* (H4) *hold*. *Then*
*contains a continuum which joins* $\{0\}\times C[0,1]$ *with* $(\frac{1+\beta}{\eta +\beta},\mathrm{\infty})$.

*Proof*Applying the method as in Theorem 3.1, we find from Lemma 2.2, Lemma 4.1 and Lemma 4.2 that there exists a continuum $\xi \in \mathbb{L}$ satisfying

Using the same proof as in Lemma 3.2, we get a contradiction.

Hence, the conclusion holds. □

**Remark 4.1** In contrast to [[1], Theorem 3.1], we obtain the global structure and behavior of positive solutions, where the parameter *α* is regarded as a variable.

## Declarations

### Acknowledgements

The work was supported partly by NSFC (No. 11201248), K.C. Wong Magna Fund of Ningbo University and Ningbo Natural Science Foundation (No. 2012A610031).

## Authors’ Affiliations

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