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Positive solutions to second-order singular nonlocal problems: existence and sharp conditions

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Abstract

In this paper we consider sharp conditions on ω and f for the existence of \(C^{1}[0,1]\) positive solutions to a second-order singular nonlocal problem \(u''(t)+\omega (t)f(t,u(t))=0\), \(u(0)=u(1)=\int _{0} ^{1}g(t)u(t)\,dt\); it turns out that this case is more difficult to handle than two point boundary value problems and needs some new ingredients in the arguments. On the technical level, we adopt the topological degree method.

Introduction

We consider sharp conditions for the second-order singular differential equation with integral boundary conditions

$$ \textstyle\begin{cases} u''(t)+\omega (t)f(t,u(t))=0, \quad t\in J, \\ u(0)=u(1)=\int _{0}^{1}g(t)u(t)\,dt, \end{cases} $$
(1.1)

where \(J=(0,1)\), ω is \(L^{p}\)-integrable on \([0,1]\) for some \(1\leq p\leq +\infty \), f may be singular at \(t=0\) and/or 1.

In addition, ω and f satisfy the following conditions:

\((H_{1})\) :

\(\omega \in L^{p}[0,1]\) and there exists \(\zeta > 0\) such that \(\omega (t)\geq \zeta \) a.e. on J;

\((H_{2})\) :

\(f(t,u):J\times [0,+\infty )\rightarrow [0,+\infty )\) is continuous;

\((H_{3})\) :

\(g\in L^{1}[0,1]\) is positive with \(\mu \in [0,1)\), where

$$ \mu = \int _{0}^{1}g(t)\,dt. $$

The theory of boundary value problems with positive solutions originates from various real life problems, such as plasma physics, gas dynamics, and chemical reaction. The study of boundary value problems with positive solutions has attracted recently the attention of different researchers, and it is a topic of current interest, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references therein. Problems with integral boundary conditions come naturally from thermal conduction problems [29] and hydrodynamic problems [30]. In recent years there has been a lot of investigation of boundary value problems with integral boundary conditions (see for instance [31,32,33,34,35,36,37,38,39,40,41,42,43]). In particular, Boucherif [44] used the fixed point theorem in cones to consider the following problem:

$$ \textstyle\begin{cases} u^{\prime \prime }(t)=f(t,u(t)), \quad 0< t < 1, \\ u(0)-cu^{\prime }(0)=\int _{0}^{1}g_{0}(t)u(t)\,dt, \\ u(1)-du^{\prime }(1)=\int _{0}^{1}g_{1}(t)u(t)\,dt. \end{cases} $$
(1.2)

The author obtained several excellent results on the existence of positive solutions to problem (1.2).

Recently, Feng [45] studied the following boundary value problem:

$$ \textstyle\begin{cases} (g(t)x^{\prime }(t))^{\prime }+w(t)f(t,x(t))=0, \quad 0< t < 1, \\ ax(0)-b\lim_{t\rightarrow 0^{+}}g(t)x^{\prime }(t)=\int _{0}^{1}h(s)x(s)\,ds, \\ ax(1)+b\lim_{t\rightarrow 1^{-}}g(t)x^{\prime }(t)=\int _{0}^{1}h(s)x(s)\,ds. \end{cases} $$
(1.3)

The author got the existence results of symmetric positive solutions to problem (1.3) by applying the theory of fixed point index in cones. For other related results on problem (1.1), we refer the reader to [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61] and the references cited therein.

At the same time, we notice that a type of problem on sharp conditions has received much attention, for instance, see [62,63,64,65,66,67,68,69] and the references cited therein. Specially, by the compressing fixed point theorem, Yang [65] gave the sharp conditions for the existence of positive solutions for the following second-order differential equation:

$$ \textstyle\begin{cases} u''(t)+f(u(t))=0, \quad 0< t< 1, \\ \alpha u(0)-\beta u'(0)=0, \\ \gamma u(1)+\delta u'(1)=0, \end{cases} $$

where α, β, γ, \(\delta \geq 0\), \(\rho =\alpha \beta +\alpha \delta +\gamma \beta >0\), f is singular at \(t=0\) or \(t=1\).

In [66], Pouso considered the following initial value problem:

$$ \textstyle\begin{cases} u''(t)=f(u(t)), \\ u(0)=u_{0},\qquad u'(0)=u_{1}. \end{cases} $$

The author obtained sharp conditions for local and global uniqueness and for the existence of periodic solutions for the above problem which is based on a detailed analysis of time maps. The other recent results concerning sharp condition problems can be found in [70,71,72,73,74,75].

However, as we know, in literature there are no articles on sharp conditions for the analogous second order singular differential equations with integral boundary conditions. This shows that the study in the case of \(\omega \in L^{p}[0,1]\) and \(g\not \equiv 0\) is still open for problem (1.1). The purpose of this paper is to establish sharp conditions over ω and f for the existence of positive solutions of (1.1). More precisely, we will investigate and give sharp conditions on the functions \(\omega (t)\) and \(f(t,u)\) which satisfy

\((H_{4})\) :

\(f(t,1)>0\), \(t\in J\), and there exist constants \(\lambda _{1} \geq \lambda _{2}>1\) and \(0<\lambda _{3}\leq \lambda _{4}<1\) such that, for all \(t\in J\), \(u\in [0,+\infty )\),

$$\begin{aligned}& l^{\lambda _{1}}f(t,u)\leq f(t,lu)\leq l^{\lambda _{2}}f(t,u), \quad \forall l \in J'=[0,1]; \end{aligned}$$
(1.4)
$$\begin{aligned}& l^{\lambda _{4}}f(t,u)\leq f(t,lu)\leq l^{\lambda _{3}}f(t,u),\quad \forall l\in J'. \end{aligned}$$
(1.5)
\((H_{5})\) :
$$ 0< \int _{0}^{1}H(s,s)f(s,1)\,ds< +\infty , $$

where \(H(s,s)\) is defined in (2.2).

Remark 1.1

It is not difficult to see that

  1. (i)

    (1.4) is equivalent to

    $$ l^{\lambda _{2}}f(t,u)\leq f(t,lu)\leq l^{\lambda _{1}}f(t,u), \quad \forall l\geq 1. $$
    (1.6)
  2. (ii)

    (1.5) is equivalent to

    $$ l^{\lambda _{3}}f(t,u)\leq f(t,lu)\leq l^{\lambda _{4}}f(t,u), \quad \forall l\geq 1. $$
    (1.7)

Remark 1.2

If \(f(t,u)\) satisfies \((H_{4})\), then it follows from (1.4) that, for every \(t\in J\), \(f(t,u)\) is nondecreasing with regard to \(u\in [0,+\infty )\), and

$$ \lim_{u\rightarrow +\infty } \min_{t\in [\xi ,\eta ]} \frac{f(t,u)}{u}=+\infty , \quad \forall [\xi ,\eta ]\subset J. $$

Similarly by (1.5), for every \(t\in J\), \(f(t,u)\) is nondecreasing with regard to \(u\in [0,+\infty )\), and

$$ \lim_{u\rightarrow 0}\max_{t\in [\xi ,\eta ]} \frac{f(t,u)}{u}=0, \quad \forall [\xi ,\eta ]\subset J. $$

The rest of the present paper is structured as follows. In the next section, we introduce some notation and preliminary results. In particular, we give some properties of the Green’s function related to problem (1.1). In Sect. 3, by applying Hölder’s inequality and combining the fixed point theorem, we analyze the sharp conditions for the existence of positive solutions for problem (1.1). Finally, in Sect. 4, we present a few of related remarks and comments.

Preliminaries

In this part, we prove a few lemmas and collect some known results for the convenience of later use and reference. The following definitions can be found in Guo and Lakshmikantham [76], or in Papageorgiou, Rădulescu, and Repovs [77].

Definition 2.1

Let E be a real Banach space over R. A nonempty closed set \(K\subset E\) is said to be a cone provided that

  1. (i)

    \(a'u+b'v\in K \) for all \(u, v\in K \) and all \(a'\geq 0\), \(b' \geq 0\), and

  2. (ii)

    \(u,{-u}\in K\) implies \(u=0\).

Every cone \(K\subset E\) induces an ordering in E given by \(u\leq v\) if and only if \(v-u\in K\).

Lemma 2.1

Assume that \((H_{1})\)\((H_{3})\) hold and \(\mu \neq 1\). Then, for any \(y\in E\), the boundary value problem

$$ \textstyle\begin{cases} -u''(t)=y(t), \quad 0< t< 1, \\ u(0)=u(1)=\int _{0}^{1}g(t)u(t)\,dt \end{cases} $$
(2.1)

has a unique solution u given by

$$ u(t)= \int _{0}^{1}{H(t,s)y(s)\,ds}, $$
(2.2)

where

$$\begin{aligned}& H(t,s)=G(t,s)+\frac{1}{1-\mu } \int _{0}^{1}G(\tau ,s)g(\tau )\,d\tau , \end{aligned}$$
(2.3)
$$\begin{aligned}& G(t,s)= \textstyle\begin{cases} s(1-t), \quad 0\leq s\leq t\leq 1, \\ t(1-s), \quad 0\leq t\leq s\leq 1. \end{cases}\displaystyle \end{aligned}$$
(2.4)

Proof

First, suppose that u is a solution of (2.1). It is easy to see by integration of (2.1) that

$$ u'(t)-u'(0)=- \int _{0}^{t}y(s)\,ds. $$
(2.5)

This shows

$$ u'(t)=u'(0)- \int _{0}^{t}y(s)\,ds. $$
(2.6)

Integrating again, we obtain

$$\begin{aligned} u(t) =&u(0)+u'(0)t- \int _{0}^{t} \int _{0}^{\tau }y(s)\,ds\,d\tau \\ =&u(0)+u'(0)t- \int _{0}^{t} \int _{s}^{t}y(s)\,d\tau \,ds \\ =&u(0)+u'(0)t- \int _{0}^{t}y(s) (t-s)\,ds. \end{aligned}$$
(2.7)

Letting \(t=1\) in (2.6), we get

$$ u(1)=u(0)+u^{\prime }(0)- \int _{0}^{1}(1-s)y(s)\,ds. $$

Combining the boundary condition \(u(0)=u(1)\), we find

$$ u'(0)= \int _{0}^{1}{(1-s)y(s)\,ds}. $$
(2.8)

Substituting the boundary condition \(u(0)=\int _{0}^{1}g(t)u(t)\,dt\) and (2.8) into (2.6), we get

$$\begin{aligned} u(t) =& \int _{0}^{1}g(t)u(t)\,dt+ \int _{0}^{1}{(1-s)y(s)\,ds\,t}- \int _{0}^{t}(t-s)y(s)\,ds \\ =& \int _{0}^{1}g(t)u(t)\,dt+ \int _{0}^{t}{s(1-t)y(s)\,ds}+ \int _{t}^{1}t(1-s)y(s)\,ds \\ =& \int _{0}^{1}g(t)u(t)\,dt+ \int _{0}^{1}G(t,s)y(s)\,ds, \end{aligned}$$
(2.9)

where \(G(t,s)\) is defined by (2.4). Multiplying the above equation by \(g(t)\) and integrating it again, we obtain

$$\begin{aligned} \int _{0}^{1}g(t)u(t)\,dt =& \int _{0}^{1}g(t)\biggl[ \int _{0}^{1}g(t)u(t)\,dt+ \int _{0}^{1}G(t,s)y(s)\,ds\biggr]\,dt \\ =& \int _{0}^{1}g(t)u(t)\,dt \int _{0}^{1}g(t)\,dt+ \int _{0}^{1}g(t)G(t,s)\,dt \int _{0}^{1}y(s)\,ds. \end{aligned}$$

Then we have

$$ (1-\mu ) \int _{0}^{1}g(t)u(t)\,dt= \int _{0}^{1}g(t)G(t,s)\,dt \int _{0}^{1}y(s)\,ds $$

and

$$ \int _{0}^{1}g(t)u(t)\,dt=\frac{\int _{0}^{1}g(t)G(t,s)\,dt\int _{0}^{1}y(s)\,ds}{1- \mu }. $$
(2.10)

Therefore, we have

$$\begin{aligned} u(t) =&\frac{1}{1-\mu } \int _{0}^{1}g(\tau )\,d\tau \int _{0}^{1}y(s)G( \tau ,s)\,ds+ \int _{0}^{1}G(t,s)y(s)\,ds \\ =& \int _{0}^{1}\biggl[G(t,s)+\frac{1}{1-\mu } \int _{0}^{1}g(\tau )G(\tau ,s)\,d \tau \biggr]y(s)\,ds. \end{aligned}$$
(2.11)

Let

$$ H(t,s)=G(t,s)+\frac{1}{1-\mu } \int _{0}^{1}G(\tau ,s)g(\tau )\,d\tau . $$
(2.12)

Then

$$ u(t)= \int _{0}^{1}H(t,s)y(s)\,ds. $$
(2.13)

The proof of Lemma 2.1 is complete. □

We can show that \(G(t,s)\) and \(H(t,s)\) have the following properties.

Lemma 2.2

Let \(\theta \in (0,\frac{1}{2})\) and define \(J_{0}=[\theta ,1-\theta ]\). If \(\mu \in [0,1)\), then for all \(t \in J_{0}\), \(s\in J'\), we have

$$\begin{aligned}& 0\leq \theta e(s)\leq G(t,s)\leq e(s)\leq \frac{1}{4}, \end{aligned}$$
(2.14)
$$\begin{aligned}& H(t,s)\leq H(s,s)\leq \gamma e(s)\leq \frac{1}{4}\gamma , \end{aligned}$$
(2.15)
$$\begin{aligned}& H(t,s)\geq \theta H(s,s), \qquad H(t,s)\geq \theta \gamma e(s), \end{aligned}$$
(2.16)

where

$$ \gamma =\frac{1}{1-\mu }, \qquad e(s)=s(1-s). $$

Proof

It is clear to see from the definition of \(G(t,s)\) that \(G(t,s)\leq e(s)\leq \frac{1}{4}\). Now, we show that \(\theta e(s) \leq G(t,s)\) also holds.

In fact, for any \(t\in J_{0}\), \(s\in J'\), we have the following.

Case 1. If \(0< s\leq t\leq 1-\theta \), then

$$ \frac{G(t,s)}{G(s,s)}=\frac{s(1-t)}{s(1-s)}=\frac{1-t}{1-s}\geq 1-t \geq \theta . $$

Case 2. If \(\theta \leq t\leq s< 1\), then

$$ \frac{G(t,s)}{G(s,s)}=\frac{t(1-s)}{s(1-s)}=\frac{t}{s}\geq t\geq \theta . $$

Case 3. If \(s\in \{0,1\}\), then it naturally follows from the definition of \(G(t,s)\) and \(G(s,s)\) that \(G(t,s)=G(s,s)=0\).

This shows that

$$ \theta e(s)\leq G(t,s), \quad t\in J_{0},s\in J'. $$

It is not difficult to see that the inequality \(H(t,s)\leq H(s,s)\) holds. Next, we show that \(H(s,s)\leq \gamma e(s)\) also holds.

$$\begin{aligned} H(s,s) \leq& G(s,s)+\frac{1}{1-\mu } \int _{0}^{1}G(s,s)g(\tau )\,d\tau \\ =&G(s,s)\biggl[1+\frac{1}{1-\mu } \int _{0}^{1}g(\tau )\,d\tau \biggr] \\ =&\gamma e(s). \end{aligned}$$

Therefore, the proof of (2.15) is complete.

Due to (2.14), we find

$$\begin{aligned}& H(t,s)\geq \theta G(s,s)+\frac{\theta }{1-\mu } \int _{0}^{1}G(s,s)g( \tau )\,d\tau \geq \theta H(s,s); \\& H(t,s)\geq \theta G(s,s)\biggl[1+\frac{\int _{0}^{1}g(\tau )\,d\tau }{1-\mu }\biggr]= \frac{ \theta }{1-\mu }G(s,s)=\theta \gamma e(s). \end{aligned}$$

So, for all \(t\in J_{0}\), \(s\in J'\), (2.16) is established.

This concludes the proof of Lemma 2.2. □

Definition 2.2

If a function u satisfies (1.1) and \(u(t)>0\), \(t \in J\), then it is said that \(u\in C[0,1]\cap C^{2}(0,1)\) is a positive solution of problem (1.1); If the positive solution \(u\in C^{1}[0,1]\), namely \(u'(0^{+})\) and \(u'(1^{-})\) exist, then u is said to be a \(C^{1}[0,1]\) positive solution of problem (1.1).

Let \(E=C[0,1]\). Then E is a real Banach space with norm \(\|\cdot \|\) defined by

$$ \Vert u \Vert =\sup_{t\in J'} \bigl\vert u(t) \bigr\vert ,\quad u\in E. $$

To establish the existence of positive solutions to problem (1.1), we consider the cone K defined by

$$ K=\bigl\{ u\in E: u(t)\geq \theta \Vert u \Vert ,t\in J'\bigr\} , $$
(2.17)

where θ is a constant as in Lemma 2.2. It is easy to see that K is a convex cone of E.

Also, define, for a given positive number r, the set \(\varOmega _{r}\) by

$$\begin{aligned}& \varOmega _{r}=\bigl\{ u\in K: \Vert u \Vert < r\bigr\} , \\& \partial \varOmega _{r}=\bigl\{ u\in K: \Vert u \Vert =r\bigr\} . \end{aligned}$$

To get some norm inequalities in our main results, we employ Hölder’s inequality.

Lemma 2.3

(Hölder)

Let \(e\in L^{p}[a,b]\) with \(p>1\), \(h\in L^{q}[a,b]\) with \(q>1\), and \(\frac{1}{p}+\frac{1}{q}=1\). Then \(eh\in L^{1}[a,b]\), and

$$ \Vert eh \Vert _{1}\leq \Vert e \Vert _{p} \Vert h \Vert _{q}. $$

Let \(e\in L^{1}[a,b]\) and \(h\in L^{\infty }[a,b]\). Then \(eh\in L^{1}[a,b]\) and

$$ \Vert eh \Vert _{1}\leq \Vert e \Vert _{1} \Vert h \Vert _{\infty }. $$

Lemma 2.4

Assume that \((H_{1})\)\((H_{5})\) hold. Define \(T:K\rightarrow E\) by

$$ (Tu) (t)= \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds,\quad \forall u\in K. $$
(2.18)

Then \(u\in C[0,1]\) is a \(C[0,1]\cap C^{2}(0,1)\) positive solution of (1.1) if and only if u is a fixed point of T.

Proof

Suppose that \((H_{1})\)\((H_{5})\) hold. For fixed \(u\in E\), \(u(t)\geq 0\), \(t\in J'\), taking a constant \(0< a<1\) such that \(a\|u\|<1\), then it follows from (1.4) and (1.5) that

$$\begin{aligned} f\bigl(t,u(t)\bigr) \leq& \biggl(\frac{1}{a}\biggr)^{\lambda _{1}}f \bigl(t,au(t)\bigr) \leq \biggl(\frac{1}{a}\biggr)^{\lambda _{1}}f\bigl(t,a \Vert u \Vert \bigr) \leq a^{\lambda _{2}-\lambda _{1}} \Vert u \Vert ^{\lambda _{2}}f(t,1). \end{aligned}$$

Consequently, for all \(t\in J'\), we get

$$ 0< \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds\leq a^{\lambda _{2}-\lambda _{1}} \Vert u \Vert ^{\lambda _{2}} \Vert \omega \Vert _{1} \int _{0}^{1}H(s,s)f(s,1)\,ds< + \infty . $$

It is obvious that the operator

$$ (Tu) (t)= \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds,\quad \forall u\in K $$

is defined well. And hence the definition of T and the properties of \(G(t,s)\) and \(H(t,s)\) yield that \(u\in C[0,1]\) is a \(C[0,1]\cap C^{2}(0,1)\) positive solution of (1.1) if and only if u is a positive fixed point of operator T. This finishes the proof of Lemma 2.4. □

Lemma 2.5

((Theorem 2.3.4 of [76]) (Fixed point theorem of cone expansion and compression of norm type))

Let \(\varOmega _{1}\) and \(\varOmega _{2}\) be two bounded open sets in a real Banach space E such that \(0 \in \varOmega _{1}\) and \(\bar{\varOmega }_{1}\subset \varOmega _{2}\). Let the operator \(T: K\cap (\bar{\varOmega }_{2}\backslash \varOmega _{1})\rightarrow K\) be completely continuous, where K is a cone in E. Suppose that one of the two conditions

  1. (i)

    \(\|Tx\|\leq \|x\|\), \(\forall x\in K\cap \partial \varOmega _{1}\) and \(\|Tx\|\geq \|x\|\), \(\forall x\in K\cap \partial \varOmega _{2}\),

and

  1. (ii)

    \(\|Tx\|\geq \|x\|\), \(\forall x\in K\cap \partial \varOmega _{1}\), and \(\|Tx\|\leq \|x\|\), \(\forall x\in K\cap \partial \varOmega _{2}\),

is satisfied. Then T has at least one fixed point in \(K\cap (\bar{ \varOmega }_{2}\backslash \varOmega _{1})\).

Lemma 2.6

If u is a \(C^{1}[0,1]\) positive solution of problem (1.1), then there exists \(b>0\) satisfying \(u(s)\geq bH(s,s)\), \(s \in J'\).

Proof

Let u take its maximum at \(t_{0}\). Then we discuss Lemma 2.6 under the following two cases.

Case 1. If \(0< t_{0}\leq \frac{1}{2}\), then let

$$ h(t)= \textstyle\begin{cases} \frac{1-t_{0}}{t_{0}} t, \quad 0\leq t\leq t_{0}, \\ -t+1, \quad t_{0}< t\leq 1. \end{cases} $$

Case 2. If \(\frac{1}{2}< t_{0}<1\), then let

$$ h(t)= \textstyle\begin{cases} t, \quad 0\leq t\leq t_{0}, \\ -\frac{t_{0}}{1-t_{0}}t+\frac{t_{0}}{1-t_{0}}, \quad t_{0}< t\leq 1. \end{cases} $$

Due to the concavity of u and since \(h(t_{0})<1\), we have

$$ u(t)\geq u(t_{0})h(t). $$

Next we show that \(h(t)>e(t)\) on \(J'\) holds.

It is easy to see by calculating that

$$ e'(0)=1,\qquad e'(1)=-1. $$

On the one hand, when \(0< t_{0}\leq \frac{1}{2}\), we have

$$ h'(t)= \textstyle\begin{cases} \frac{1-t_{0}}{t_{0}}, \quad 0\leq t\leq t_{0}, \\ -1, \quad t_{0}< t\leq 1. \end{cases} $$

It is obvious that

$$ \frac{1-t_{0}}{t_{0}}>1, $$

so by the concavity of e we have that \(h(t)>e(t)\) on \(J'\).

On the other hand, when \(\frac{1}{2}< t_{0}<1\), we have

$$ h'(t)= \textstyle\begin{cases} 1, \quad 0\leq t\leq t_{0}, \\ -\frac{1-t_{0}}{t_{0}}, \quad t_{0}< t\leq 1. \end{cases} $$

It can be easily seen that

$$ -\frac{1-t_{0}}{t_{0}}< -1, $$

similarly we can obtain that \(h(t)>e(t)\) on \(J'\).

At the same time, by Lemma 2.2, we have

$$ u(t)\geq u(t_{0})h(t)\geq u(t_{0})e(t)\geq \frac{u(t_{0})}{\gamma }H(t,t)= bH(t,t),\quad t\in J, $$

where \(b=\frac{u(t_{0})}{\gamma }\).

In order to better understand the above two cases, we draw Figs. 1 and 2.

Figure 1
figure1

Case 1

Figure 2
figure2

Case 2

This gives the proof of Lemma 2.6. □

Sharp conditions for the existence of positive solutions

In this section, we establish sharp conditions for the existence of positive solutions for problem (1.1) by Lemmas 2.12.6. We analyze the following three cases for \(\omega \in L^{p}[0,1]:p>1\), \(p = 1\), and \(p=\infty \). Case \(p>1\) is treated in Theorem 3.1.

Theorem 3.1

Suppose that \((H_{1})\)\((H_{5})\) hold. Then problem (1.1) admits a \(u\in C^{1}[0,1]\) positive solution if and only if

$$ 0< \int _{0}^{1}\omega (s)f(s,1)\,ds< +\infty . $$

Proof

(1) Necessity.

Let \(u\in C^{1}[0,1]\) be a positive solution of problem (1.1), then \(u'(0)\) and \(u'(1)\) exist and are finite.

On the one hand, we know that \(u(t)\) is a concave function on \(J'\) by \(u^{\prime \prime }\leq 0\). Therefore, by Lemma 2.6, there exists \(b>0\) satisfying \(u(s)\geq bH(s,s)\), \(s\in J'\). Setting \(l=\min \{b,1\}\), then \(u(s)\geq lH(s,s)\), \(s\in J'\). And by (1.4) and Remark 1.2, we have

$$\begin{aligned} \int _{0}^{1}\omega (s)f\bigl(s,H(s,s)\bigr)\,ds \leq& \int _{0}^{1}\omega (s)f\biggl(s, \frac{1}{l}u(s)\biggr)\,ds \\ \leq& \bar{l} \int _{0}^{1}\bigl[-u^{\prime \prime }(s)\bigr]\,ds \\ \leq& \bar{l}\bigl[u^{\prime }(0)-u^{\prime }(1)\bigr] \\ < &+\infty , \end{aligned}$$

where \(\bar{l}=(\frac{1}{l})^{\lambda _{1}}\).

On the other hand, if we assume that \(f(s,u(s))\equiv 0\), which shows that \(\omega (s)f(s,u(s))\equiv 0\) by \((H_{1})\). Then, by Lemma 2.4, it is obvious that \(u = 0\), which contradicts u is a positive solution of problem (1.1).

Hence there exists \(t_{1}\in J\) such that \(f(t_{1},u(t_{1}))>0\).

And then it follows from \((H_{1})\), Remarks 1.2 and (1.6) that

Case 1. If \(\omega (t_{1}) > 0\), then

$$\begin{aligned} 0 < &\omega (t_{1})f\bigl(t_{1},u(t_{1}) \bigr) \\ \leq& \omega (t_{1})f\bigl(t_{1}, \Vert u_{t_{1}} \Vert \bigr) \\ \leq& \omega (t_{1}) \Vert u_{t_{1}} \Vert ^{\lambda }f(t_{1},1), \end{aligned}$$

where \(\lambda \in \{\lambda _{2}, \lambda _{3}\}\).

Case 2. If \(\omega (t_{1}) =0\), then it follows from \((H_{1})\) that there exists a small neighborhood \([a_{1},b_{1}]\subset J\) of \(t_{1}\) such that \(\omega (t)>0\) for \(t\in [ a_{1},b_{1}]\).

Hence it is easy to see by integration of f and ω that

$$ \int _{a_{1}}^{b_{1}}\omega (s)f(s,1)\,ds>0. $$

So,

$$\begin{aligned} \int _{0}^{1}\omega (s)f(s,1)\,ds \geq& \int _{0}^{1}\omega (s)f\biggl(s, \frac{u(s)}{ \Vert u \Vert }\biggr)\,ds \\ \geq& \biggl(\frac{1}{ \Vert u \Vert } \biggr)^{\lambda ^{*}} \int _{0}^{1}\omega (s) f\bigl(s,u(s)\bigr)\,ds \\ \geq & \biggl(\frac{1}{ \Vert u \Vert } \biggr)^{\lambda ^{*}} \int _{a_{1}}^{b_{1}} \omega (s)f\bigl(s,u(s)\bigr)\,ds \\ >&0, \end{aligned}$$

where \(\lambda ^{*} \in \{\lambda _{1}, \lambda _{4}\}\).

(2) Sufficiency.

(i) First, we prove that the operator \(T: K\rightarrow K\) is completely continuous. For all \(u\in K\), \(T(u)\geq 0\) on \(J_{0}\), it follows from (2.18) and Lemma 2.2 that

$$\begin{aligned} (Tu) (t) =& \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds \\ \geq& \theta \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds \\ =&\theta \Vert Ty \Vert ,\quad \forall t\in J'. \end{aligned}$$

So we have that \(Tu\in K\), \(\forall u\in K\). Thus \(T(K)\subset K\).

Next, it follows from Arzelà–Ascoli theorem that \(T : K \rightarrow K\) is completely continuous.

It is clear that T is continuous.

Let \(B_{r}=\{u\in E| \|u\|\leq r\}\) be a bounded set. Then, for all \(u\in B_{r}\), by the definition of \(\|Tu\|\) and by Lemma 2.2 and Lemma 2.3, we get

$$\begin{aligned} \Vert Tu \Vert =&\max_{t\in J} \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds \\ \leq& \gamma \int _{0}^{1}G(s,s)\omega (s)\,ds\,L \\ \leq& \gamma \Vert G \Vert _{q} \Vert \omega \Vert _{p}L \\ =&\varGamma , \end{aligned}$$

where \(L=\max_{t\in J, u\in B_{r}}f(t,u)\), \(\varGamma =\gamma \|G \|_{q}\|\omega \|_{p}L\).

Therefore, the operator \(T:K\longrightarrow K\) is uniformly bounded.

On the other hand, since \(H(t,s)\) is continuous on \(J'\times J'\), we can see that \(H(t,s)\) is uniformly continuous on \(J'\times J'\). Therefore, for any \(\varepsilon >0\), there exists \(r>0\), when \(| t_{1}-t_{2} | < r\), we get

$$ \bigl\vert H(t_{1},s)-H(t_{2},s) \bigr\vert < \frac{\varepsilon }{ \Vert \omega \Vert _{1}\cdot L}. $$

Accordingly, for all \(u\in B_{r}\), when \(| t_{1}-t_{2}| < r\), we have

$$\begin{aligned} \bigl\vert (Tu) (t_{1})-(Tu) (t_{2}) \bigr\vert =& \biggl\vert \int _{0}^{1}H(t_{1},s)\omega (s)f \bigl(s,u(s)\bigr)\,ds- \int _{0}^{1}H(t_{2},s)\omega (s)f \bigl(s,u(s)\bigr)\,ds \biggr\vert \\ =& \biggl\vert \int _{0}^{1}\bigl[H(t_{1},s)-H(t_{2},s) \bigr]\omega (s)f\bigl(s,u(s)\bigr)\,ds \biggr\vert \\ \leq& \bigl\vert \Vert \omega \Vert _{1}\cdot L \bigr\vert \cdot \frac{ \varepsilon }{ \Vert \omega \Vert _{1}\cdot L} \\ \leq &\varepsilon . \end{aligned}$$

This shows that the set \(\{T(u): u\in B_{r}\}\) is equicontinuous, and it follows from Arzelà–Ascoli theorem that operator T is completely continuous.

(ii) Next, we prove that T has at least one fixed point in K.

For \(u\in K\), \(\|u\|\leq 1\), we get \(u(t)\leq \|u\|\leq 1\), and by (1.4) and Remark 1.2, we obtain

$$ f\bigl(t,u(t)\bigr)\leq u^{\lambda _{2}}(t)f(t,1). $$

Hence,

$$\begin{aligned} \Vert Tu \Vert \leq& \int _{0}^{1}\gamma G(s,s)\omega (s)u^{\lambda _{2}}(s)f(s,1)\,ds \\ \leq& \gamma \Vert u \Vert ^{\lambda _{2}} \Vert G \Vert _{q} \Vert \omega \Vert _{p} \int _{0}^{1}f(s,1)\,ds \\ \leq& A \Vert u \Vert ^{\lambda _{2}}, \end{aligned}$$

where \(A=\gamma \| G\| _{q}\| \omega \| _{p}\int _{0}^{1}f(s,1)\,ds\).

If \((\frac{1}{A})^{\frac{1}{\lambda _{2}-1}}\leq 1\), setting \(r_{1}^{*}=(\frac{1}{A})^{\frac{1}{\lambda _{2}-1}}\), then \(\|Tu\| \leq \|u\|\), \(\forall u\in K\), \(\|u\|=r_{1}^{*}\).

If \((\frac{1}{A})^{\frac{1}{\lambda _{2}-1}}>1\), we have \(A<1\). Letting \(r_{1}^{**} =1\), similarly we have \(\|Tu\|\leq \|u\|\), \(\forall u\in K\), \(\|u\|=r_{1}^{**}\).

Set \(r_{1}=\max \{r_{1}^{*},r_{1}^{**}\}\). Then we obtain \(\|Tu\| \leq \|u\|\), \(\forall u\in K\), \(\|u\|=r_{1}\).

Moreover, by Remark 1.2, there exists \(R>r_{1}\) for \(u\geq R\) such that

$$ \frac{f(t,u(t))}{u(t)}\geq \min_{t\in J_{0}} \frac{f(t,u(t))}{u(t)} \geq N, \quad t\in J_{0}, $$

that is, \(f(t,u(t))\geq Nu(t)\), \(t\in J_{0}\), \(u\geq R\), where \(N>0\) satisfies

$$ N\geq \frac{1}{\zeta \theta \min_{t\in J_{0}}\int _{\theta } ^{1-\theta }H(t,s)\,ds}. $$

So, for \(u\in K\) with \(\|u\|=R\), we get

$$\begin{aligned} \Vert Tu \Vert \geq& \min_{t\in J_{0}}(Tu) (t)\geq \min _{t\in J _{0}} \int _{\theta }^{1-\theta }H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds \\ \geq& \min_{t\in J_{0}} \int _{\theta }^{1-\theta }H(t,s)\zeta Nu(s)\,ds \\ \geq& N\zeta \min_{t\in J_{0}} \int _{\theta }^{1-\theta }H(t,s) \Vert u \Vert \theta \,ds \\ \geq& N\zeta \theta \min_{t\in J_{0}} \int _{\theta }^{1-\theta }H(t,s)\,ds \Vert u \Vert \\ \geq& \Vert u \Vert . \end{aligned}$$

Thus, \(\|Tu\|\geq \|u\|\), \(\forall u\in K\), \(\|u\|=R\).

Lemma 2.5 yields that T admits at least one fixed point \(u^{*}\) such that \(r_{1} \leq \|u^{*}\|\leq R\). Since \(u^{*}(t)\geq \|u^{*}\| \theta \geq r_{1}\theta >0\), \(0 < t<1\), we see that \(u^{*}\) is a positive solution of problem (1.1).

Moreover, for any \(u^{*}\in K\), we have \(u^{*}(s)\leq \|u^{*}\|\), \(s \in J'\), and then, for \(\lambda \in \{\lambda _{2},\lambda _{3}\}\), we get

$$\begin{aligned} \int _{0}^{1} \bigl\vert \bigl(u^{*} \bigr){''}(s) \bigr\vert \,ds =& \int _{0}^{1}\omega (s)f\bigl(s,u^{*}(s) \bigr)\,ds \\ \leq& \int _{0}^{1}\omega (s)f\bigl(s, \bigl\Vert u^{*} \bigr\Vert \bigr)\,ds \\ \leq& \bigl\Vert u^{*} \bigr\Vert ^{\lambda } \int _{0}^{1}\omega (s)f(s,1)\,ds \\ < & +\infty , \end{aligned}$$

that is, \(u^{*}\) is absolutely integrable on \(J'\). This shows that \((u^{*})'(0^{+})\) and \((u^{*})'(1^{-})\) exist, then \(u^{*}\in C^{1}[0,1]\). The proof above shows that \(u^{*}\in C^{1}[0,1]\) is a positive solution of (1.1). This completes the proof of Theorem 3.1. □

The following corollary handles the case \(p=\infty \).

Corollary 3.1

Assume that \((H_{1})\)\((H_{5})\) hold. Then problem (1.1) admits a \(u\in C^{1}[0,1]\) positive solution if and only if

$$ 0< \int _{0}^{1}\omega (s)f(s,1)\,ds< +\infty . $$

Proof

Let \(\|G\|_{1}\|\omega \|_{\infty }\) replace \(\|G\|_{q}\|\omega \|_{p}\) and repeat the argument of Theorem 3.1. Then we can complete the proof of Corollary 3.1. □

At last, we analyze the case of \(p=1\).

Corollary 3.2

Assume that \((H_{1})\)\((H_{5})\) hold. Then problem (1.1) has a \(u\in C^{1}[0,1]\) positive solution if and only if

$$ 0< \int _{0}^{1}\omega (s)f(s,1)\,ds< +\infty . $$

Proof

Let \(\frac{1}{4}\|\omega \|_{1}\) replace \(\|G\|_{q}\| \omega \|_{p}\) and repeat the argument of Theorem 3.1. Then we can complete the proof of Corollary 3.2. □

The following theorem only considers the case of \(p>1\).

Theorem 3.2

Assume \(f(t,u) =h_{1}(t,u)+h_{2}(t,u)\), where \(h_{1}(t,u)\) and \(h_{2}(t,u)\) satisfy \((H_{4})\), and the other main hypothesis is also needed

$$ 0< \gamma \Vert G \Vert _{q} \Vert \omega \Vert _{p} \int _{0}^{1}f(s,1)\,ds< 1. $$

Then problem (1.1) admits at least two \(C^{1}[0,1]\) positive solutions if and only if

$$ 0< \int _{0}^{1}\omega (s)f(s,1)\,ds< +\infty . $$

Proof

We first prove that the operator

$$\begin{aligned} (Tu) (t) =& \int _{0}^{1}H(t,s)\omega (s)f\bigl(s,u(s) \bigr)\,ds \\ =& \int _{0}^{1}H(t,s)\omega (s) \bigl[h_{1}\bigl(s,u(s)\bigr)+h_{2}\bigl(s,u(s)\bigr) \bigr]\,ds \end{aligned}$$

admits at least two fixed points in K.

Choosing \(J_{1} =[\xi _{1},\eta _{1}]\subset J\), \(\tau _{1}=\min_{t\in J_{1}} h_{1}(t,1)\) and taking a constant \(l>1\) such that \(l\theta >1\), for \(\|u\|>1\), \(u\in K\), \(t\in J_{1}\), we obtain

$$ lu(t)\geq l\theta \Vert u \Vert > l\theta >1,\qquad h_{1}\bigl(t,u(t)\bigr)\geq l^{\lambda _{2}-\lambda _{1}}u(t)^{\lambda _{2}}h_{1}(t,1). $$

Consequently,

$$\begin{aligned} (Tu) (t) \geq& \int _{\xi _{1}}^{\eta _{1}}H(t,s)\omega (s) \bigl[h_{1}\bigl(s,u(s)\bigr)+h _{2}\bigl(s,u(s)\bigr) \bigr]\,ds \\ \geq& \int _{\xi _{1}}^{\eta _{1}}G(t,s)\omega (s) \bigl[h_{1}\bigl(s,u(s)\bigr)+h_{2}\bigl(s,u(s)\bigr) \bigr]\,ds \\ \geq& \theta ^{\lambda _{2}}\tau _{1} l^{(\lambda _{2}-\lambda _{1})}\zeta \int _{\xi _{1}}^{\eta _{1}}G(\xi _{1},s)\,ds \Vert u \Vert ^{\lambda _{2}} \\ \geq& A \Vert u \Vert ^{\lambda _{2}}, \end{aligned}$$

where \(A=\theta ^{\lambda _{2}}\tau _{1} l^{(\lambda _{2}-\lambda _{1})} \zeta \int _{\xi _{1}}^{\eta _{1}}G(\xi _{1},s)\,ds\).

Due to \(\|u\|> 1\), \(\lambda _{2} > 1\), there exists arbitrarily large \(R_{2} >1\) such that

$$ \Vert Tu \Vert \geq \Vert u \Vert ,\quad \forall u\in K, \Vert u \Vert =R_{2}. $$

When \(\|u\|<1\), taking \(J_{2} =[\xi _{2},\eta _{2}]\subset J \) and \(\tau _{2}=\min_{t\in J_{2}} h_{2}(t,1) \), we also get that

$$\begin{aligned} (Tu) (t) \geq & \int _{\xi _{2}}^{\eta _{2}}G(t,s)\omega (s) \bigl[h_{1}\bigl(s,u(s)\bigr)+h _{2}\bigl(s,u(s)\bigr) \bigr]\,ds \\ \geq& \theta ^{\lambda _{4}}\tau _{2} \zeta \int _{\xi _{2}}^{\eta _{2}}G(\xi _{2},s)\,ds \Vert u \Vert ^{\lambda _{4}} \\ \geq& A_{1} \Vert u \Vert ^{\lambda _{4}}, \end{aligned}$$

where \(A_{1}=\theta ^{\lambda _{4}}\tau _{2} \zeta \int _{\xi _{2}}^{\eta _{2}}G(\xi _{2},s)\,ds\).

Similarly, due to \(\|u\|< 1\), \(\lambda _{4} <1\), there exists arbitrarily small \(r_{2}<1\) such that

$$ \Vert Tu \Vert \geq \Vert u \Vert ,\quad \forall u\in K, \Vert u \Vert =r_{2}. $$

Moreover, because of \(u\in K\), \(\|u\|=1\) and \(u(t)\leq \|u\|=1\leq 1\), we can obtain

$$ h_{1}\bigl(t,u(t)\bigr)+h_{2}\bigl(t,u(t)\bigr)\leq u(t)^{\lambda _{2}}h_{1}(t,1)+u(t)^{ \lambda _{3}}h_{2}(t,1) \leq \bigl[h_{1}(s,1)+h_{2}(s,1)\bigr]. $$

Accordingly,

$$\begin{aligned} (Tu) (t) \leq& \int _{0}^{1}\gamma G(s,s)\omega (s) \bigl[h_{1}\bigl(s,u(s)\bigr)+h_{2}\bigl(s,u(s)\bigr) \bigr]\,ds \\ \leq& \gamma \Vert G \Vert _{q} \Vert \omega \Vert _{p} \int _{0}^{1}\bigl[h_{1}(s,1)+h_{2}(s,1) \bigr]\,ds \\ < &1= \Vert u \Vert . \end{aligned}$$
(3.1)

That is, \(\|Tu\|<\|u\|\), \(\forall u\in K\cap \partial \varOmega =\{u\in K : \|u\|=1\}\).

Consequently, Lemma 2.5 yields that the operator T admits at least two fixed points \(u_{1}(t)\) and \(u_{2}(t)\) in K, and \(u_{1}(t) \neq u _{2}(t)\) by (3.1).

On the other hand, the proof of necessity is similar to that of Theorem 3.1, so we omit it here. The proof of Theorem 3.2 is complete. □

Remarks and comments

In this section, we provide some remarks and comments related to problem (1.1).

Remark 4.1

The proof of Theorems 3.13.2 is directly inspired by Theorem 1.1 of [63], but there are no papers analyzing sharp conditions of positive solution for second-order boundary value problems with integral boundary conditions, particularly under the case ω is \(L^{p}\)-integrable.

Remark 4.2

In general, it is difficult to analyze sharp conditions of positive solutions for nonlinear second-order differential equations (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and their references).

Remark 4.3

Similar to the proof of Theorems 3.13.2, one can prove sharp conditions of positive solution for the following problems:

$$\begin{aligned}& \textstyle\begin{cases} u''(t)+\omega (t)f(t,u(t))=0, \\ u(0)=\int _{0}^{1}g(t)u(t)\,dt, \qquad u(1)=0, \end{cases}\displaystyle \end{aligned}$$
(4.1)
$$\begin{aligned}& \textstyle\begin{cases} u''(t)+\omega (t)f(t,u(t))=0, \\ u(0)=0,\qquad u(1)=\int _{0}^{1}g(t)u(t)\,dt, \end{cases}\displaystyle \end{aligned}$$
(4.2)

where \(J=(0,1)\), \(\omega \in L^{p}[0,1]\) for some \(1\leq p\leq +\infty \), \(f\in C(J\times R^{+},R^{+})\), \(R^{+}=[0,+\infty )\) (here, f may be singular at \(t=0\) and/or 1), \(g\in L^{1}[0,1]\).

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Acknowledgements

We wish to express our thanks to Prof. Meiqiang Feng, School of Applied Science, Beijing Information Science & Technology University, Beijing, PR China, for his kind help, careful reading, and making useful comments on the earlier version of this paper. The authors are also grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.

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This work is sponsored by the National Natural Science Foundation of China (11301178) and the Beijing Natural Science Foundation of China (1163007).

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The authors contributed equally in this article. They have all read and approved the final manuscript.

Correspondence to Xuemei Zhang.

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Ma, S., Zhang, X. Positive solutions to second-order singular nonlocal problems: existence and sharp conditions. Bound Value Probl 2019, 173 (2019) doi:10.1186/s13661-019-1289-7

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Keywords

  • Sharp conditions
  • Singular boundary value problems with integral boundary conditions
  • Hölder’s inequality
  • Fixed point theorems
  • Positive solutions