where $\mu >0$ is a parameter and ${\int}_{0}^{1}u(s)\phantom{\rule{0.2em}{0ex}}dA(s)$ is the Stieltjes integral. The function $w\in C((0,1),(0,+\mathrm{\infty}))$ and w may be singular at $t=0$ and/or $t=1$, $f\in C([0,+\mathrm{\infty}),(0,+\mathrm{\infty}))$ and ${f}_{\mathrm{\infty}}=+\mathrm{\infty}$. Some a priori estimates and the existence, multiplicity and nonexistence of positive solutions are obtained. Our proofs are based on the method of global continuous theorem, the lower-upper solutions methods and fixed point index theory. Furthermore, we also discuss the interval of parameter μ such that the problem has a positive solution.

Keywords

singularityglobal continuous theoremsolution of boundednessfixed point indexpositive solution

1 Introduction

We are concerned with the second order nonlocal boundary value problem:

where $\mu >0$ is a parameter and ${\int}_{0}^{1}u(s)\phantom{\rule{0.2em}{0ex}}dA(s)$ is a Stieltjes integral. The function $w\in C((0,1),(0,+\mathrm{\infty}))$ and w may be singular at $t=0$ and/or $t=1$, $f\in C([0,+\mathrm{\infty}),(0,+\mathrm{\infty}))$ and ${f}_{\mathrm{\infty}}={lim}_{u\to \mathrm{\infty}}\frac{f(u)}{u}=+\mathrm{\infty}$.

Integral boundary conditions and multi-point boundary conditions for differential equations come from many areas of applied mathematics and physics [1–7]. Recently, singular boundary value problems have been extensively considered in a lot of literature [1, 2, 5, 8], since they model many physical phenomena including gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical or biological problems. In all these problems, positive solutions are very meaningful.

In [1, 2], Webb and Infante considered the existence of positive solutions of nonlinear boundary value problem:

involving a Stieltjes integral with a signed measure, that is, A has bounded variation. They dealt with many boundary conditions given in the literature in a unified way by utilizing the fixed point index theory in cones.

Recently, many researchers were interested in the global structure of positive solutions for the nonlinear boundary value problem (see, e.g., [3, 6, 7]). In 2009, Ma and An [3] considered the problem (1.1). Assume that

(A0) $A:[0,1]\to \mathbb{R}$ is nondecreasing and $A(t)$ is not a constant on $(0,1)$, $0\le \kappa <1$ with $\kappa :={\int}_{0}^{1}t\phantom{\rule{0.2em}{0ex}}dA(t)$, and ${\int}_{0}^{1}G(t,s)\phantom{\rule{0.2em}{0ex}}dA(t)\ge 0$ for $s\in [0,1]$ (for the definition of $G(t,s)$, see (2.1) below).

(A1) $w:(0,1)\to [0,+\mathrm{\infty})$ is continuous and $w(t)\ne 0$ on any subinterval of $[0,1]$, and $w\mathrm{\Phi}\in {L}^{1}[0,1]\cap C(0,1)$, where $\mathrm{\Phi}(s)=s(1-s)$, $s\in [0,1]$.

(A2) $f\in C([0,+\mathrm{\infty}),[0,+\mathrm{\infty}))$ and $f(s)>0$ for $s>0$.

(A3) ${f}_{0}={f}_{\mathrm{\infty}}=0$, where ${f}_{0}={lim}_{u\to {0}^{+}}\frac{f(u)}{u}$ and ${f}_{\mathrm{\infty}}={lim}_{u\to \mathrm{\infty}}\frac{f(u)}{u}$.

Assume that (A 0)-(A 3) hold. Then there exists a component$\mathfrak{T}$in ∑ which joins$(\mathrm{\infty},\theta )$to$(\mathrm{\infty},\mathrm{\infty})$, and

for some${\rho}^{\ast}>0$. Moreover, there exists${\mu}^{\ast}\ge {\rho}^{\ast}>0$such that (1.1) has at least two positive solutions for$\mu \in ({\mu}^{\ast},\mathrm{\infty})$. Here$\mathfrak{T}$joins$(\mathrm{\infty},\theta )$to$(\mathrm{\infty},\mathrm{\infty})$such that

Here, ∑ is the closure of the set of positive solutions of (1.1) on $[0,\mathrm{\infty})\times X$, $X=\{u\in {C}^{1}[0,1]:u(0)=0,u(1)={\int}_{0}^{1}u(s)\phantom{\rule{0.2em}{0ex}}ds\}$, and the component of a set M is a maximal connected subset of M.

A natural problem arises: How can we consider the global structure of positive solutions for the case ${f}_{0}={f}_{\mathrm{\infty}}=\mathrm{\infty}$?

In this paper, we first obtain the global structure of positive solutions by the use of global continuous theorem, and some a priori estimates. Applying the analysis technique, we construct the lower and upper solutions. Those combined with the fixed point index theory, the existence, multiplicity and nonexistence of positive solutions to (1.1) in the case ${f}_{0}={f}_{\mathrm{\infty}}=\mathrm{\infty}$ are investigated. Finally, we discuss the interval of parameter μ such that the problem (1.1) has positive solutions. The proof of the method which is based on the construction of some bounds of the solution together with global continuous theorem and fixed point index is of independent interest, and is different from the other papers.

This paper is arranged as follows. We will give some hypotheses and lemmas in Section 2. In Section 3, new criteria of the existence, multiplicity and nonexistence of a positive solution are obtained. Moreover, an example is given to illustrate our result.

2 Preliminaries and lemmas

Let X denote the Banach space $C[0,1]$ with the maximum norm

Throughout this paper, we suppose that the following conditions hold:

(H0) $A:[0,1]\to \mathbb{R}$ is nondecreasing, $dA(t)\ge 0$ on $(0,1)$ and $\kappa :={\int}_{0}^{1}t\phantom{\rule{0.2em}{0ex}}dA(t)$ with $0\le \kappa <1$.

(H1) $w\in C((0,1),(0,+\mathrm{\infty}))$ and w may be singular at $t=0$ and/or 1, satisfying

Remark 2.1 It is easy to see from (H0) that $G(s)\le [A(1)-A(0)]G(s,s)$. Therefore, if we assume that $0<{\int}_{0}^{1}G(s,s)w(s)\phantom{\rule{0.2em}{0ex}}ds<\mathrm{\infty}$, then (2.3) obviously holds.

LetXbe a Banach space and letKbe an order cone inX. Consider the equation

$u=T(\mu ,u),$

(2.4)

where$\mu \in {\mathbb{R}}_{+}=[0,\mathrm{\infty})$and$u\in X$. Suppose that$T:{\mathbb{R}}_{+}\times K\to K$is completely continuous and$T(0,u)=\theta $for all$u\in K$, then${\mathfrak{L}}_{+}(K)$, the component of the solution set of (2.4) containing$(0,0)$is unbounded.

Remark 2.2

(1)

We note that u is a positive solution of the problem (1.1) if and only if $u=T(\mu ,u)$ on K.

(2)

If $T(\mu ,\theta )\ne \theta $ for $\mu \ne 0$ and $T(0,u)=\theta $ for all $u\in K$, then we get from Lemma 2.1 that there exists an unbounded continuum $\mathfrak{L}$ emanating from $(0,\theta )$ in the closure of the set of positive solutions (1.1) in ${\mathbb{R}}_{+}\times K$.

LetXbe a Banach space, Kan order cone inXand$\mathcal{S}$an open bounded set inXwith$\theta \in \mathcal{S}$. Suppose that$T:\overline{\mathcal{S}}\cap K\to K$is a completely continuous operator. If$Ty\ne \lambda y$, for all$y\in \partial \mathcal{S}\cap K$and all$\lambda \ge 1$, then$i(T,\mathcal{S}\cap K,K)=1$.

3 Main results

Lemma 3.1Let (H 0)-(H 3) hold and let$\mathcal{J}=[{\mu}_{0},\mathrm{\infty})$with${\mu}_{0}>0$. Then there exists a constant${Q}_{\mathcal{J}}>0$such that for all$\mu \in \mathcal{J}$and all possible positive solutions${u}_{\mu}$of (1.1), the inequality$\parallel {u}_{\mu}\parallel <{Q}_{\mathcal{J}}$holds.

Proof Suppose on the contrary that there exist a sequence $\{{\mu}_{n}\}\subset \mathcal{J}$ and a sequence $\{{u}_{{\mu}_{n}}\}$ of the positive solutions of (1.1) corresponding to ${\mu}_{n}$ such that

Choose $\eta :=\frac{10{\pi}^{2}}{{\mu}_{0}w}$, where $w={min}_{t\in [\frac{1}{4},\frac{3}{4}]}w(t)$. Then we find from (H3) that there exists a constant $R>0$ such that

Lemma 3.2Assume that the hypotheses (H 0)-(H 3) hold. Then there exists$\rho >0$such that if the problem (1.1) has a positive solution for parameterμ, then$\mu \le \rho $.

Proof Let u be a positive solution of (1.1) corresponding to μ. From the hypotheses (H2) and (H3), it follows that there exists $\varrho >0$ such that $f(u)\ge \varrho u$ for all $u>0$. Consequently, we have

and let ${\phi}_{1}$ be the positive eigenfunction corresponding to ${\lambda}_{1}$ (see [8]). It is easy to see that ${\phi}_{1}^{\prime}(1)\le 0$. Multiplying (3.4) by ${\phi}_{1}$ and integrating by parts, we get that

Thus, it implies that $\mu \nrightarrow 0$. This is contradiction. □

Theorem 3.1Assume that the conditions (H 0)-(H 3) hold and${\int}_{0}^{1}dA(t)<1$. Then there exists a constant${\mu}_{1}>0$such that the problem (1.1) has at least two positive solutions for$0<\mu <{\mu}_{1}$, and at least one positive solution for$\mu ={\mu}_{1}$, and no positive solution for$\mu >{\mu}_{1}$.

Proof Define

$\begin{array}{rl}{\mu}_{1}:=& sup\{{\mu}^{\ast}>0:\text{for all}\mu \in (0,{\mu}^{\ast}),\\ \text{there exist at least two positive solutions of (1.1)}\}.\end{array}$

(3.5)

From Lemma 2.1 and Remark 2.2, we can find that there exists an unbounded continuum $\mathfrak{L}$ emanating from $(0,\theta )$ in the closure of the set of positive solutions in ${\mathbb{R}}_{+}\times K$ and $T(0,u)=\theta $ for all $u\in K$. Meanwhile, Lemma 3.1 and Lemma 3.3 respectively imply that ${u}_{\mu}$ is bounded ($\mu >0$, $\mu \nrightarrow 0$) and unbounded ($\mu >0$, $\mu \to 0$ and $\parallel {u}_{\mu}\parallel \ge 1$). Therefore, we conclude that the set of (3.5) is nonempty. Those combined with Lemma 3.2 follows that ${\mu}_{1}>0$ is well defined and ${\mu}_{1}\in (0,\rho ]$. From the definition of ${\mu}_{1}$, it is easy to see that the problem (1.1) has at least two positive solutions for $\mu \in (0,{\mu}_{1})$. Again, since the continuum is a compact connected set and T is a completely continuous operator, the problem (1.1) has at least one positive solution at $\mu ={\mu}_{1}$.

Next, we only show that the problem (1.1) has no positive solution for any $\mu >{\mu}_{1}$. Suppose on the contrary that there exists some ${\mu}_{2}$ ($>{\mu}_{1}$) such that the problem (1.1) has a positive solution ${u}_{2}$ corresponding to ${\mu}_{2}$. Then we will prove that the problem (1.1) has at least two positive solutions for any $\mu \in [{\mu}_{1},{\mu}_{2})$ which contradicts the definition of (3.5).

For the sake of obtaining the contradiction, we divide the proof into four steps.

Step 1. Constructing a modified boundary value problem.

Choose arbitrarily a constant $\mu \in [{\mu}_{1},{\mu}_{2})$. Since f is uniformly continuous on $[0,\parallel {u}_{2}\parallel +1]$, there exists a constant $\sigma >0$ such that

Define a set $\mathcal{S}:=\{u\in C[0,1]|0<u(t)<\zeta (t),t\in (0,1)\}$. Then the set $\mathcal{S}$ is bounded and open in $C[0,1]$. Now, we construct the modified second-order boundary value problem:

Step 2. We will show that if u is a positive solution of (3.9), then $u\in \mathcal{S}\cap K$.

Let u be a positive solution of (3.9), then we claim that

$u\in \mathcal{S}\cap K.$

(3.10)

Suppose this fails, that is, $u\notin \mathcal{S}\cap K$. Clearly, we only show that $u(t)\nless \zeta (t)$, for $t\in (0,1]$. Comparing the boundary conditions (3.8) and (3.9), the only following three cases need to be considered. Case I. There exists ${t}_{1}\in (0,1)$ such that $u({t}_{1})=\zeta ({t}_{1})$ and $0<u(t)<\zeta (t)$, for $t\in ({t}_{1}-{\sigma}_{1},{t}_{1})\cup ({t}_{1},{t}_{1}+{\sigma}_{1})$ and some ${\sigma}_{1}>0$; Case II. There exists ${t}_{2}\in (0,1]$ such that $u({t}_{2})=\zeta ({t}_{2})$ and $u(t)\ge \zeta (t)$ for $t\in ({t}_{2},1]$. Case III. There exists $[{t}_{3},{t}_{4}]\subset (0,1]$ such that $u(t)\ge \zeta (t)$, $t\in [{t}_{3},{t}_{4}]$, $u({t}_{3})=\zeta ({t}_{3})$ and $u({t}_{4})=\zeta ({t}_{4})$. See the three Figures 1, 2 and 3.

Case I. From (3.7), it implies that there exists a constant ${\u03f5}_{1}>0$ such that

Since f is uniformly continuous on $[0,\parallel {u}_{2}\parallel +1]$, there exists a ${\sigma}_{1}^{\prime}>0$ such that if $u,v\in [0,\parallel \zeta \parallel ]$ and $|u-v|<{\sigma}_{1}^{\prime}$, then we get

$|f(u)-f(v)|<{\u03f5}_{1}{W}^{-1},$

(3.12)

where $W:=\mu {max}_{t\in [{t}_{1}-{\sigma}_{1},{t}_{1}+{\sigma}_{1}]}w(t)>0$. From the assumption of Case I, it follows that there exists a subinterval $[r,s]\subset ({t}_{1}-{\sigma}_{1},{t}_{1}+{\sigma}_{1})$ with ${t}_{1}\in (r,s)$ such that

Obviously, we have ${x}^{\prime}({t}_{2})\ge 0$. From (3.13), it implies that ${x}^{\prime}(t)>0$, for any $t\in ({t}_{2},1]$. Hence, the function $x(t)$ is strictly increasing in $[{t}_{2},1]$. From (3.8) and the boundary condition (3.9), we find

Case III. Since $\xi (u(t))=\zeta (t)$, for $t\in ({t}_{3},{t}_{4})$, we have that ${(u(t)-\zeta (t))}^{\u2033}>0$, for $t\in ({t}_{3},{t}_{4})$. Again since $(u-\zeta )({t}_{3})=(u-\zeta )({t}_{4})=0$, we know from the maximum principle that $u(t)<\zeta (t)$, for $t\in ({t}_{3},{t}_{4})$. This contradicts the assumption of Case III.

Therefore, we conclude that the claim (3.10) holds.

Step 3. $i({\tilde{T}}_{\mu},\mathcal{S}\cap K,K)=1$ (the definition of ${\tilde{T}}_{\mu}$ see below).

Using (2.2), we define an operator ${\tilde{T}}_{\mu}:K\to X$ by

Then ${\tilde{T}}_{\mu}:K\to K$ is completely continuous and u is a positive solution of (3.9) if and only if $u={\tilde{T}}_{\mu}(u)$ on K. From the definition of $\xi (u(t))$, it implies that there exists ${R}_{1}>0$ such that $\parallel {\tilde{T}}_{\mu}u\parallel <{R}_{1}$, for all $u\in K$. Consequently, we get from Lemma 2.2 that

$i({\tilde{T}}_{\mu},{B}_{{R}_{1}}\cap K,K)=1,$

where ${B}_{{R}_{1}}=\{u\in X:\parallel u\parallel <{R}_{1}\}$. Applying the conclusion of Step 2 and the excision property of fixed point index, we find that

Step 4. We conclude that the problem (1.1) has at least two positive solutions corresponding to μ.

Since the problem (1.1) is equivalent to the problem (3.9) on $\mathcal{S}\cap K$, we get that the problem (1.1) has a positive solution in $\mathcal{S}\cap K$. Without loss of generality, we may suppose that T has no fixed point on $\partial \mathcal{S}\cap K$ (otherwise the proof is completed). Then $i(T,\mathcal{S}\cap K,K)$ is well defined and (3.15) implies

$i(T,\mathcal{S}\cap K,K)=1.$

(3.16)

On the other hand, from Lemma 3.2, we choose ${\mu}_{\ast}>\rho $ such that the problem (1.1) has no positive solution in K. By apriori estimate in $\mathcal{J}=[\mu ,{\mu}_{\ast}]$, there exists ${R}_{2}\phantom{\rule{0.25em}{0ex}}(>{R}_{1})>0$ such that for all possible positive solutions ${u}_{\lambda}$ of (1.1) with $\lambda \in [\mu ,{\mu}_{\ast}]$, we know that $\parallel {u}_{\lambda}\parallel <{R}_{2}$. Define $\mathfrak{G}:[0,1]\times ({\overline{B}}_{{R}_{2}}\cap K)\to K$ by

Then it is easy to verify that $\mathfrak{G}$ is completely continuous on $[0,1]\times K$ and $\mathfrak{G}(\nu ,u)\ne u$ for all $(\nu ,u)\in [0,1]\times (\partial {B}_{{R}_{2}}\cap K)$. From the property of homotopy invariance, it follows that

Let f attain its maximum at the point ${L}^{\ast}$ of $[0,L]$. If ${\int}_{0}^{1}{G}_{1}(t,s)w(s)\phantom{\rule{0.2em}{0ex}}ds<\mathrm{\infty}$, then adopting the similar method as in [11], Theorem 1], we get that for $0<\mu <{(\frac{f({L}^{\ast})}{L}{\int}_{0}^{1}{G}_{1}(t,s)w(s)\phantom{\rule{0.2em}{0ex}}ds)}^{-1}$, the problem (1.1) has at least two positive solutions ${u}_{1}(t)$ and ${u}_{2}(t)$ such that $0<\parallel {u}_{1}\parallel <L<\parallel {u}_{2}\parallel $ by the use of compression of conical shells in [12], Corollary 20.1]. Consequently, we know that ${\mu}_{1}\ge {(\frac{f({L}^{\ast})}{L}{\int}_{0}^{1}{G}_{1}(t,s)w(s)\phantom{\rule{0.2em}{0ex}}ds)}^{-1}$.

(ii)

If u is a positive solution of the equation (1.1) corresponding to μ, then we have

whereμis a positive parameter, ${\alpha}_{i}\in [0,\mathrm{\infty})$, $i=1,2,\dots ,m-2$, $0<{\eta}_{1}<\cdots <{\eta}_{m-2}<1$and$0\le {\sum}_{i=1}^{m-2}{\alpha}_{i}{\eta}_{i}<1$. Then there exists a constant$\overline{\mu}>0$such that the problem (3.18) has at least two positive solutions for$0<\mu <\overline{\mu}$, and at least one positive solution for$\mu =\overline{\mu}$, and no positive solution for$\mu >\overline{\mu}$.

Proof In the boundary condition of (1.1), if we let

Thus, it implies that (2.3) holds. It is easy to verify that the conditions (H2) and (H3) hold. Therefore, by Theorem 3.1, we obtain that there exists a constant ${\mu}_{1}>0$ such that the problem (3.19) has at least two positive solutions for $0<\mu <{\mu}_{1}$, and at least one positive solution for $\mu ={\mu}_{1}$, and no positive solution for $\mu >{\mu}_{1}$.

Author’s contributions

The author typed, read and approved the final manuscript.

Declarations

Acknowledgement

The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work. The work was supported partly by NSCF of Tianyuan Youth Foundation (No. 11126125), K. C. Wong Magna Fund of Ningbo University, Subject Foundation of Ningbo University (No. xkl11044) and Hulan’s Excellent Doctor Foundation of Ningbo University.

Authors’ Affiliations

(1)

Department of Mathematics, Ningbo University

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