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Positive solutions to secondorder singular nonlocal problems: existence and sharp conditions
Boundary Value Problems volume 2019, Article number: 173 (2019)
Abstract
In this paper we consider sharp conditions on ω and f for the existence of \(C^{1}[0,1]\) positive solutions to a secondorder 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 secondorder singular differential equation with integral boundary conditions
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:
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:
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 secondorder differential equation:
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:
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

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

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

(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 \(vu\in K\).
Lemma 2.1
Assume that \((H_{1})\)–\((H_{3})\) hold and \(\mu \neq 1\). Then, for any \(y\in E\), the boundary value problem
has a unique solution u given by
where
Proof
First, suppose that u is a solution of (2.1). It is easy to see by integration of (2.1) that
This shows
Integrating again, we obtain
Letting \(t=1\) in (2.6), we get
Combining the boundary condition \(u(0)=u(1)\), we find
Substituting the boundary condition \(u(0)=\int _{0}^{1}g(t)u(t)\,dt\) and (2.8) into (2.6), we get
where \(G(t,s)\) is defined by (2.4). Multiplying the above equation by \(g(t)\) and integrating it again, we obtain
Then we have
and
Therefore, we have
Let
Then
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
where
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
Case 2. If \(\theta \leq t\leq s< 1\), then
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
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.
Therefore, the proof of (2.15) is complete.
Due to (2.14), we find
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
To establish the existence of positive solutions to problem (1.1), we consider the cone K defined by
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
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
Let \(e\in L^{1}[a,b]\) and \(h\in L^{\infty }[a,b]\). Then \(eh\in L^{1}[a,b]\) and
Lemma 2.4
Assume that \((H_{1})\)–\((H_{5})\) hold. Define \(T:K\rightarrow E\) by
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
Consequently, for all \(t\in J'\), we get
It is obvious that the operator
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

(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

(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
Case 2. If \(\frac{1}{2}< t_{0}<1\), then let
Due to the concavity of u and since \(h(t_{0})<1\), we have
Next we show that \(h(t)>e(t)\) on \(J'\) holds.
It is easy to see by calculating that
On the one hand, when \(0< t_{0}\leq \frac{1}{2}\), we have
It is obvious that
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
It can be easily seen that
similarly we can obtain that \(h(t)>e(t)\) on \(J'\).
At the same time, by Lemma 2.2, we have
where \(b=\frac{u(t_{0})}{\gamma }\).
In order to better understand the above two cases, we draw Figs. 1 and 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 2.1–2.6. 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
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
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
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
So,
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
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
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
Accordingly, for all \(u\in B_{r}\), when \( t_{1}t_{2} < r\), we have
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
Hence,
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
that is, \(f(t,u(t))\geq Nu(t)\), \(t\in J_{0}\), \(u\geq R\), where \(N>0\) satisfies
So, for \(u\in K\) with \(\u\=R\), we get
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
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
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
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
Then problem (1.1) admits at least two \(C^{1}[0,1]\) positive solutions if and only if
Proof
We first prove that the operator
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
Consequently,
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
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
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
Moreover, because of \(u\in K\), \(\u\=1\) and \(u(t)\leq \u\=1\leq 1\), we can obtain
Accordingly,
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 3.1–3.2 is directly inspired by Theorem 1.1 of [63], but there are no papers analyzing sharp conditions of positive solution for secondorder 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 secondorder 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 3.1–3.2, one can prove sharp conditions of positive solution for the following problems:
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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Ma, S., Zhang, X. Positive solutions to secondorder singular nonlocal problems: existence and sharp conditions. Bound Value Probl 2019, 173 (2019) doi:10.1186/s1366101912897
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Keywords
 Sharp conditions
 Singular boundary value problems with integral boundary conditions
 Hölder’s inequality
 Fixed point theorems
 Positive solutions