Infinitely many singularities and denumerably many positive solutions for a second-order impulsive Neumann boundary value problem

Minmin Wang; Meiqiang Feng

doi:10.1186/s13661-017-0784-y

Impact Factor 0.819

Research

Open Access

Infinitely many singularities and denumerably many positive solutions for a second-order impulsive Neumann boundary value problem

Minmin Wang¹ and
Meiqiang Feng¹Email author

Boundary Value Problems20172017:50

https://doi.org/10.1186/s13661-017-0784-y

Received: 23 November 2016

Accepted: 4 April 2017

Published: 12 April 2017

Abstract

Using a fixed point theorem of cone expansion and compression of norm type and a new method to deal with the impulsive term, we prove that the second-order singular impulsive Neumann boundary value problem has denumerably many positive solutions. Noticing that $M>0$, our main results improve many previous results.

Keywords

denumerably many positive solutions infinitely many singularities Neumann impulsive boundary conditions cone expansion and compression

1 Introduction

We are concerned with the existence of denumerably many positive solutions of the second-order singular impulsive Neumann boundary value problem

$$ \textstyle\begin{cases} -x''(t)+ Mx(t)=\omega(t)f(t,x(t)),\quad t\in J,\\ -\Delta x'|_{t=t_{k}}=I_{k}(x(t_{k})), \quad k=1,2,\ldots,m,\\ x'(0)=x'(1)=0, \end{cases} $$

(1.1)

where M is a positive constant, $J=[0,1]$, $t_{k} \in\mathrm{R}$, $k =1,\ldots,m$, $m \in\mathrm{N,}$ satisfy $0=t_{0}< t_{1}<t_{2}<\cdots <t_{m}<t_{m+1}=1$, $-\Delta x'|_{t=t_{k}}$ denotes the jump of $x'(t)$ at $t=t_{k}$, that is, $-\Delta x'|_{t=t_{k}}=x'((t_{k})^{+})-x'((t_{k})^{-})$, here $x'((t_{k})^{+})$ and $x'((t_{k})^{-})$, respectively, represent the right-hand limit and left-hand limit of $x'(t)$ at $t=t_{k}$.

In addition, ω, f and $I_{k}$ satisfy the following conditions:

$(H_{1})$ :: $\omega(t)\in L^{p}[0,1]$ for some $p\in[1,+\infty)$, and there exists $N>0$ such that $\omega(t)\geq N$ a.e. on J;

$(H_{2})$ :: $f\in C(J\times\mathrm{R^{+}}, \mathrm{R^{+}})$, $I_{k}\in C( \mathrm{R^{+}}, \mathrm{R^{+}})$, where $\mathrm {R^{+}}=[0,+\infty)$;

$(H_{3})$ :: there exists a sequence $\{t_{i}'\}_{i=1}^{\infty} $ such that $t_{1}'<\delta$, where $\delta=\min\{t_{1},\frac{1}{2}\} $, $t_{i}' \downarrow t^{*}\geq0 $ and $\lim_{t\rightarrow t_{i}'} \omega(t) =+\infty$ for all $i=1, 2,\dots$.

For the case $M=0$ and $I_{k}=0$ ($k=1,2,\ldots,m$), problem (1.1) reduces to the problem studied by Kaufmann and Kosmatov in [1]. By using Krasnosel’skiĭ’s fixed point theorem and Hölder’s inequality, the authors showed the existence of countably many positive solutions. The other related results can be found in [2–13]. However, there are almost no papers considering second-order impulsive Neumann boundary value problem with infinitely many singularities. To identify a few, we refer the reader to [14–27] and the references therein.

The main reason is that $M\neq0$ in problem (1.1), which shows that the solution of problem (1.1) has no concave properties. On the other hand, under the case $M\neq0$ and $\omega(t)$ with infinitely many singularities, the properties of the corresponding Green’s function for problem (1.1) are more complicated.

Our plan of the paper is as follows: in Section 2, we collect some well-known results to be used in the subsequent sections. In particular, we also present some new properties of Green’s function under the case $M\neq0$ and $\omega(t)$ with infinitely many singularities. In Section 3, we obtain some new sufficient conditions for the existence of denumerably many positive solutions for problem (1.1). In Section 4, we give an example of a family functions $\omega(t)$ such that $(H_{3})$ holds.

2 Preliminaries

In this installment, we list some definitions and lemmas which are needed throughout this paper.

Let $J'=J\setminus \{t_{1},t_{2},\ldots,t_{m} \}$ and $E=C[0,1]$. We define $\mathit {PC}^{1}[0,1]$ in E by

$$ \mathit {PC}^{1}[0,1]= \bigl\{ x\in E:x'(t)\in C(t_{k}, t_{k+1}), \exists x' \bigl(t_{k}^{-} \bigr), x' \bigl(t_{k}^{+} \bigr), k=1,2, \ldots,m \bigr\} . $$

(2.1)

Then $\mathit {PC}^{1}[0,1]$ is a real Banach space with norm

$$ \Vert x \Vert _{\mathit {PC}^{1}}=\max \bigl\{ \Vert x \Vert _{\infty}, \bigl\Vert x' \bigr\Vert _{\infty} \bigr\} , $$

(2.2)

where $\Vert x \Vert _{\infty}=\sup_{t\in J} \vert x(t) \vert $, $\Vert x' \Vert _{\infty}=\sup_{t\in J} \vert x'(t) \vert $.

Suppose that $G(t,s)$ is the Green’s function of the boundary value problem

$$-x''(t)+Mu(t)=0,\quad\quad x'(0)=x'(1)=0, $$

then

$$ G(t,s)= \frac{1}{\gamma\sinh\gamma} \textstyle\begin{cases} \cosh\gamma(1-t)\cosh\gamma s,& 0\leq s\leq t\leq1,\\ \cosh\gamma(1-s)\cosh\gamma t,& 0\leq t\leq s\leq1. \end{cases} $$

(2.3)

Lemma 2.1

By the definition of $G(t,s)$ and the properties of $sinhx$ and $coshx$, we have the following results.

(a)

For any $t, s\in J$, there is
$$ A=\frac{1}{\gamma\sinh\gamma}\leq G(t,s)\leq\frac{\cosh\gamma}{\gamma \sinh\gamma}=B. $$

(2.4)
Then it follows from (2.4) that
$$A\leq G(t,s)\leq G(s,s)\leq B. $$
(b)

For any $\tau\in(0,\delta)$,
$$ \frac{D'_{k}}{\gamma\sinh\gamma}\leq G(t,s)\leq\frac{\cosh\gamma(1-\tau )\cosh\gamma\tau'_{k}}{\gamma\sinh\gamma}, \quad \forall t\in \bigl[ \tau,\tau '_{k} \bigr], s\in J, $$

(2.5)
where
$$\tau'_{k}=\max\{1-\tau, 1-t_{k}\},\quad\quad D'_{k}=\max \bigl\{ \cosh\gamma\tau, \cosh\gamma \bigl(1- \tau'_{k} \bigr) \bigr\} , \quad k=1,2,3,\ldots, m. $$
(c)

$$ G_{t}'(t,s)= \frac{1}{\sinh\gamma} \textstyle\begin{cases} -\sinh\gamma(1-t)\cosh\gamma s,& 0\leq s\leq t\leq1,\\ \sinh\gamma(1-s)\cosh\gamma t,& 0\leq t\leq s\leq1, \end{cases} $$

(2.6)
and
$$ \max_{t,s\in J,t\neq s} \bigl\vert G_{t}'(t,s) \bigr\vert \leq \sinh\gamma. $$

(2.7)

Proof

We can get equations (2.4)-(2.7) by the definition of $G(t,s)$, so we omit it here. □

To establish the existence of positive solutions to problem (1.1), for a fixed $\tau\in(0,\delta)$, we construct the cone $K_{\tau}$ in $\mathit {PC}^{1}[0,1]$ by

$$ K_{\tau}= \Bigl\{ x \in \mathit {PC}^{1}[0,1]: x(t) \geq0,t\in J, \min _{t\in[\tau ,\tau'_{k}]}x(t) \geq\sigma_{k} \Vert x \Vert _{\mathit {PC}^{1}} \Bigr\} , $$

(2.8)

where

$$\begin{aligned}& \sigma_{k}=\frac{D'_{k}}{ \rho\gamma\sinh\gamma}, \quad k=1,2,\ldots,m, \end{aligned}$$

(2.9)

$$\begin{aligned}& \rho=\max\{B, \sinh\gamma\}. \end{aligned}$$

(2.10)

It is easy to see $K_{\tau}$ is a closed convex cone of $\mathit {PC}^{1}[0,1]$.

Let $\{\tau_{i}\}_{i=1}^{\infty}$ be such that $t_{i+1}'<\tau _{i}<t_{i}'$, $i=1,2,\dots$. Then for any $i\in\mathrm{N}$, we define the cone $K_{\tau_{i}}$ by

$$ K_{\tau_{i}}= \Bigl\{ x(t)\in \mathit {PC}^{1}[0,1]: x(t)\geq0,t\in J,\min _{t\in[\tau_{i},\tau'_{ik}]}x(t)\geq\sigma_{ik} \Vert x \Vert _{\mathit {PC}^{1}} \Bigr\} , $$

(2.11)

where

$$\begin{aligned}& \tau'_{ik}=\max\{1-\tau_{i},1-t_{k} \}, \quad\quad \sigma_{ik}=\frac{D'_{ik}}{ \rho\gamma\sinh\gamma}, \end{aligned}$$

(2.12)

$$\begin{aligned}& D'_{ik}=\max \bigl\{ \cosh\gamma\tau_{i}, \cosh \gamma \bigl(1-\tau'_{k} \bigr) \bigr\} , \quad i=1,2, \ldots, k=1,2,\ldots,m. \end{aligned}$$

(2.13)

It is easy to see $K_{\tau_{i}}$ is a closed convex cone of $\mathit {PC}^{1}[0,1]$.

Remark 2.1

For any $i=1,2,\ldots$ , $k=1,2,\ldots,m$, it follows from the definition of $\sigma_{k}$ and $\sigma_{ik}$ that $0<\sigma _{k},\sigma_{ik} <1$.

Lemma 2.2

If $(H_{1})$-$(H_{3})$ hold, then problem (1.1) has a unique solution x given by

$$x(t) = \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr). $$

Proof

The proof is similar to that of Lemma 2.4 in [26]. □

Definition 2.1

A function $x(t)$ is said to be a solution of problem (1.1) on J if:

(i)

$x(t)$ is absolutely continuous on each interval $(0,t_{1}]$ and $(t_{k},t_{k+1}]$, $k =1,2,\ldots,n$;
(ii)

for any $k =1,2,\ldots,m$, $x(t_{k}^{ +})$, $x(t_{k}^{-})$ exist;
(iii)

$x(t)$ satisfies (1).

Define an operator $T: K_{\tau} \to \mathit {PC}^{1}[0,1]$ by

$$ (Tx) (t) = \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr). $$

(2.14)

From (2.14), we know that $x(t)\in \mathit {PC}^{1}[0,1]$ is a solution of problem (1.1) if and only if x is a fixed point of the operator T. Also, for a positive number r, define $\Omega_{r}$ by

$$\Omega_{r}= \bigl\{ x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}< r \bigr\} . $$

Note that $\partial\Omega_{r}= \{x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}=r \}$ and $\bar{\Omega}_{r}= \{x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}\leq r \}$.

Definition 2.2

An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.

Lemma 2.3

Assume that $(H_{1})$-$(H_{3})$ hold. Then $T(K_{\tau })\subset K_{\tau} $ and $T: K_{\tau} \to K_{\tau}$ is a completely continuous.

Proof

For $t\in J$, $x\in K_{\tau}$, it follows from ((2.5)) and (2.14) that

$$ \begin{aligned}[b] (Tx) (t) &= \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ & \leq B \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds + \sum_{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr]. \end{aligned} $$

(2.15)

On the other hand, it follows from (2.6), (2.7) and (2.14) that

$$ \begin{aligned}[b] \bigl\vert (Tx)'(t) \bigr\vert &= \Biggl\vert \int _{0}^{1}G_{t}'(t,s) \omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum_{k=1}^{m}G_{t}'(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr\vert \\ &\leq \int_{0}^{1} \bigl\vert G_{t}'(t,s) \bigr\vert \omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m} \bigl\vert G_{t}'(t,t_{k}) \bigr\vert I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ &\leq \sinh\gamma \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr]. \end{aligned} $$

(2.16)

For any $t\in J$, combined with (2.15) and (2.16), we have

$$ \Vert Tx \Vert _{\mathit {PC}^{1}} \leq\rho \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr]. $$

(2.17)

Then, by (2.5), (2.8) and (2.17), we have

$$ \begin{aligned}[b] \min_{t\in[\tau,\tau_{k}]}(Tx) (t) &=\min _{t\in[\tau,\tau_{k}]} \Biggl[ \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ &\geq\frac{D'_{k}}{\gamma\sinh\gamma} \Biggl[ \int_{0}^{1}\omega(s)f\bigl(s,x(s)\bigr)\,ds+\sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr] \\ &\geq\frac{D'_{k}}{\rho\gamma\sinh\gamma} \rho \Biggl[ \int_{0}^{1}\omega(s)f\bigl(s,x(s)\bigr)\,ds+\sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr] \\ &\geq\sigma_{k} \Vert Tx \Vert _{\mathit {PC}^{1}}. \end{aligned} $$

(2.18)

Evidently, $T(K_{\tau})\subset K_{\tau}$.

Next, we prove that the operator $T: K_{\tau}\to K_{\tau}$ is a completely continuous.

It is obvious that T is continuous.

Let $B_{d}=\{x\in \mathit {PC}^{1}[0,1] \mid \Vert x \Vert _{\mathit {PC}^{1}}\le d\}$ be bounded set. Then, for all $x\in B_{d}$, by the definition of $\Vert Tx \Vert _{\infty}$, $\Vert Tx' \Vert _{\infty}$, $\Vert Tx \Vert _{\mathit {PC}^{1}}$, we have

$$\begin{aligned}& \begin{aligned} \Vert Tx \Vert _{\infty}&=\sup _{t\in J} \bigl\vert Tx(t) \bigr\vert \\ &\le B \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ &\le B \bigl( \Vert \omega \Vert _{1}L+mL^{*} \bigr) \\ &=\Gamma_{0}, \end{aligned} \\& \begin{aligned} \bigl\Vert Tx' \bigr\Vert _{\infty}&=\sup_{t\in J} \bigl\vert Tx'(t) \bigr\vert \\ &\leq \sinh\gamma \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ &\le\sinh\gamma \bigl( \Vert \omega \Vert _{1}L+mL^{*} \bigr) \\ &=\Gamma_{1}, \end{aligned} \end{aligned}$$

and

$$\Vert Tx \Vert _{\mathit {PC}^{1}}=\max \bigl\{ \Vert Tx \Vert _{\infty}, \bigl\Vert Tx' \bigr\Vert _{\infty} \bigr\} \le\max \{\Gamma_{0}, \Gamma_{1} \}, $$

where

$$\begin{aligned}& L=\max_{t\in J, x\in K_{\tau}, \Vert x \Vert _{\mathit {PC}^{1}}\le d}f(t,x), \quad\quad L^{*}=\max \{L_{k}, k=1,2, \ldots,m\}, \\& L_{k}=\max_{t\in J, x\in K_{\tau}, \Vert x \Vert _{\mathit {PC}^{1}}\le d}I_{k} \bigl(t_{k},x(t_{k}) \bigr). \end{aligned}$$

Therefore $T(B_{d})$ is uniformly bounded.

On the other hand, for all $t_{1}, t_{2}\in J_{k}$ with $t_{1}< t_{2}$, we have

$$\bigl\vert (Tx) (t_{1})-(Tx) (t_{2}) \bigr\vert = \biggl\vert \int _{t_{1}}^{t_{2}}(Tx)'(t)\,dt \biggr\vert \le\Gamma_{1} \vert t_{1}-t_{2} \vert \rightarrow0 \quad (t_{1}\rightarrow t_{2}). $$

Noting (2.7), we know that $G'(t,s)$ is a constant and

$$\begin{aligned} \bigl\vert (Tx)'(t_{1})-(Tx)'(t_{2}) \bigr\vert &= \Biggl\vert \int _{0}^{1} \bigl[G_{t}'(t_{1},s)-G_{t}'(t_{2},s) \bigr]\omega(s)f \bigl(s,x(s) \bigr)\,ds \\ &\quad{} +\sum_{k=1}^{n} \bigl[G_{t}'(t_{1},t_{k})-G_{t}'(t_{2},t_{k}) \bigr]I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr\vert \\ & \leq \int_{0}^{1} \bigl\vert G_{t}'(t_{1},s)-G_{t}'(t_{2},s) \bigr\vert \omega(s)f \bigl(s,x(s) \bigr)\,ds \\ &\quad{} +\sum_{k=1}^{n} \bigl\vert G_{t}'(t_{1},t_{k})-G_{t}'(t_{2},t_{k}) \bigr\vert I_{k} \bigl(t_{k},x(t_{k}) \bigr) \rightarrow0 \quad (t_{1}\rightarrow t_{2}), \end{aligned}$$

which shows that $T(B_{d})$ is equicontinuous. The Arzelà-Ascoli theorem implies that T is completely continuous, and the lemma is proved. □

Lemma 2.4

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

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

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

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

Lemma 2.5

See [28]; fixed point theorem of cone expansion and compression of norm type

Let E be a Banach space, P be a cone in E. Assume that $\Omega _{1}$, $\Omega_{2}$ are bounded open subsets in E with $\theta \in\Omega_{1}$ and $\bar{\Omega}_{1}\subset\Omega_{2}$, where θ denotes zero operator. Suppose $A : P \cap(\bar{\Omega}_{2} \setminus \Omega_{1})\rightarrow P $ is completely continuous such that either

(i)

$\Vert Ax \Vert \leq \Vert x \Vert $, $\forall x\in P\cap\partial\Omega_{1}$; $\Vert Ax \Vert \geq \Vert x \Vert $, $\forall x\in P\cap\partial\Omega_{2} $;
(ii)

$\Vert Ax \Vert \leq \Vert x \Vert $, $\forall x\in P\cap\partial\Omega_{2}$; $\Vert Ax \Vert \geq \Vert x \Vert $, $\forall x\in P\cap\partial\Omega_{1} $.

Then A has a fixed point in $P \cap(\bar{\Omega}_{2} \setminus \Omega_{1})$.

3 Main results

In this section, using Lemmas 2.1-2.5, we give our main results in the case $\omega\in L^{P}[0,1]$; $p>1$, $p=1$ and $p=\infty$.

For convenience, we write

$$D=\max \bigl\{ \Vert G \Vert _{q} \Vert \omega \Vert _{p}, \Vert G \Vert _{1} \Vert \omega \Vert _{\infty }, B \Vert \omega \Vert _{1} \bigr\} ,\quad\quad \rho_{0}=\min \biggl\{ 1,\frac{A}{\sinh\gamma} \biggr\} . $$

Firstly, we consider the case $p>1$.

Theorem 3.1

Assume that $(H_{1})$-$(H_{3})$ hold. Let $\{r_{i}\} _{i=1}^{\infty}$ and $\{R_{i}\}_{i=1}^{\infty}$ be such that

$$R_{i+1}< \sigma_{ik}r_{i}< r_{i}< L_{0}r_{i}< R_{i}, \quad i=1,2, \ldots, k=1,2,\ldots, m, $$

where

$$L_{0}=\max \biggl\{ \frac{\gamma\sinh\gamma}{A(N+m)D'_{k}},\frac{2\rho _{0}}{D+mB},2 \biggr\} . $$

For each natural number i, we assume that f and $I_{k}$ satisfy:

($H_{4}$):: For any $t\in J$, $x\in[0,R_{i}]$, $f(t,x)\leq M_{0}R_{i}$, and for any $x\in[0,R_{i}]$, $k\in\{1,2,\dots,m\}$, $I_{k}(x(t_{k}))\leq M_{0}R_{i}$, where
$$0< M_{0}\leq\frac{\rho_{0}}{D+mB}. $$

($H_{5}$):: For any $t\in J$, $x\in[\sigma_{ik}r_{i},r_{i}]$, $f(t,x)\geq L_{0}r_{i}$, and for any $x\in[\sigma_{ik}r_{i},r_{i}]$, $k\in\{1,2,\dots,m\}$, $I_{k}(x)\geq L_{0}r_{i}$.

Then problem (1.1) has denumerably many positive solutions $\{x_{i}(t)\}_{i=1}^{\infty}$ such that

$$r_{i}\leq \Vert x_{i} \Vert _{\mathit {PC}^{1}}\leq R_{i},\quad i=1,2,\dots. $$

Proof

We consider the following open subset sequences $\{\Omega_{1,i}\} _{i=1}^{\infty}$ and $\{\Omega_{2,i}\}_{i=1}^{\infty}$ of $\mathit {PC}^{1}[0,1]$:

$$\begin{aligned}& \{\Omega_{1,i}\}_{i=1}^{\infty}= \bigl\{ x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}< R_{i} \bigr\} ; \\& \{\Omega_{2,i}\}_{i=1}^{\infty}= \bigl\{ x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}< r_{i} \bigr\} . \end{aligned}$$

Let $\{\tau_{i}\}_{i=1}^{\infty}$ be as in the hypothesis and note that $0< t_{i+1}'<\tau_{i}<t_{i}'<\delta$, $i=1,2,\dots$.

For fixed i, we assume that $x\in K_{\tau_{i}}\cap\partial\Omega _{2,i}$, then for any $t\in J$

$$r_{i}= \Vert x \Vert _{\mathit {PC}^{1}}\geq x(t)\geq\min _{t\in [\tau_{i},\tau'_{i}k]}x(t) \geq\sigma_{ik} \Vert x \Vert _{\mathit {PC}^{1}}= \sigma_{ik}r_{i}. $$

Noticing (2.5) and (2.14), for all $x\in K_{\tau_{i}}\cap\partial\Omega _{2,i}$, by $(H_{1})$ and $(H_{5})$, we have

$$\begin{aligned} (Tx) (t)&= \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(x(t_{k}) \bigr) \\ &\geq\min_{t\in[\tau,\tau'_{k}]} \Biggl[ \int_{0}^{1}G(t,s)\omega (s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(x(t_{k}) \bigr) \Biggr] \\ &\geq \frac{D'_{k}}{\gamma \sinh \gamma} \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(x(t_{k}) \bigr) \Biggr] \\ &\geq \frac{D'_{k}}{\gamma \sinh \gamma} \Biggl[N \int_{\tau_{i}}^{\tau'_{ik}}f \bigl(s,x(s) \bigr)\,ds+ \min _{t_{k}\in[\tau_{i},\tau'_{ik}]} \sum_{k=1}^{m}I_{k} \bigl(x(t_{k}) \bigr) \Biggr] \\ &\geq\frac{D'_{k}}{\gamma \sinh \gamma} L_{0}(N+m)r_{i} \\ &\geq r_{i}= \Vert x \Vert _{\mathit {PC}^{1}}, \end{aligned}$$

which shows that

$$ \Vert Tx \Vert _{\mathit {PC}^{1}}\geq \Vert x \Vert _{\mathit {PC}^{1}}, \quad \forall x\in K_{\tau_{i}}\cap\partial\Omega_{2,i}. $$

(3.1)

On the other hand, for all $t\in J$, $x\in P_{i}\cap\partial\Omega _{1,i}$, we have $x(t)\leq \Vert x \Vert _{\mathit {PC}^{1}}=R_{i}$.

Noticing (2.4) and (2.14), for all $t\in J$, $x\in K_{\tau_{i}}\cap \partial\Omega_{1,i}$, by $(H_{4})$, we have

$$ \begin{aligned}[b] (Tx) (t)&= \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(x(t_{k}) \bigr) \\ &\leq M_{0}R_{i} \int_{0}^{1}G(s,s)\omega(s)\,ds+ M_{0}R_{i} \sum_{k=1}^{m}G(t,t_{k}) \\ &\leq M_{0}R_{i} \Vert G \Vert _{q} \Vert \omega \Vert _{p}+ M_{0}R_{i}mB \\ &\leq M_{0}(D+mB)R_{i} \\ &\leq R_{i}= \Vert x \Vert _{\mathit {PC}^{1}}. \end{aligned} $$

(3.2)

Moreover, by (2.6), (2.16) and $(H_{4})$, we have

$$ \begin{aligned}[b] \bigl\vert (Tx)'(t) \bigr\vert & \leq \int_{0}^{1} \bigl\vert G_{t}'(t,s) \bigr\vert \omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m} \bigl\vert G_{t}'(t,t_{k}) \bigr\vert I_{k} \bigl(x(t_{k}) \bigr) \\ & \leq \sinh\gamma \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(x(t_{k}) \bigr) \Biggr] \\ & \leq\frac{\sinh\gamma}{ A} \Biggl[ \int_{0}^{1}G(s,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(s,s)I_{k} \bigl(x(t_{k}) \bigr) \Biggr] \\ & \leq\frac{\sinh\gamma}{ A} \Biggl[ \int_{0}^{1} \Vert G \Vert _{q} \Vert \omega \Vert _{p}f \bigl(s,x(s) \bigr)\,ds+ B\sum _{k=1}^{m}I_{k} \bigl(x(t_{k}) \bigr) \Biggr] \\ & \leq\frac{\sinh\gamma}{ A} \bigl(M_{0}R_{i} \Vert G \Vert _{q} \Vert \omega \Vert _{p}+ BmM_{0}R_{i} \bigr) \\ & \leq M_{0}\frac{\sinh\gamma}{ A}(D+ mB)R_{i} \\ & \leq R_{i}= \Vert x \Vert _{\mathit {PC}^{1}}. \end{aligned} $$

(3.3)

From (3.2) and (3.3), we have

$$ \Vert Tx \Vert _{\mathit {PC}^{1}}\leq \Vert x \Vert _{\mathit {PC}^{1}},\quad \forall x\in K_{\tau_{i}}\cap\partial\Omega_{1,i}. $$

(3.4)

Applying Lemma 2.5 to (3.1) and (3.4) shows that the operator T has a fixed point $x_{i}\in K_{\tau_{i}}\cap(\bar{\Omega}_{2,i}/ \Omega_{1,i})$ such that $r_{i}\leq \Vert x_{i} \Vert \leq R_{i} $. Since $i\in\mathrm{N}$ was arbitrary, the proof is complete. □

The following results deal with the case $p=\infty$.

Theorem 3.2

Assume that $(H_{1})$-$(H_{3})$ hold. Let $\{a_{i}\} _{i=1}^{\infty}$ and $\{b_{i}\}_{i=1}^{\infty}$ be such that

$$a_{i+1}< \sigma_{ik}b_{i}< b_{i}< L_{0}b_{i}< a_{i}, \quad i=1,2, \ldots, k=1,2,\ldots,m. $$

For each natural number i, we assume that f and $I_{k}$ satisfy ($H_{4}$) and ($H_{5}$), then problem (1.1) has denumerably many positive solutions $\{x_{i}(t)\}_{i=1}^{\infty}$ such that

$$r_{i}\leq \Vert x_{i} \Vert _{\mathit {PC}^{1}}\leq R_{i}, \quad i=1,2,\ldots. $$

Proof

Let $\Vert G \Vert _{1} \Vert \omega \Vert _{\infty}$ replace $\Vert G \Vert _{q} \Vert \omega \Vert _{p}$ and repeat the previous argument. □

Finally, we consider the case of $p=1$.

Theorem 3.3

Assume that $(H_{1})$-$(H_{3})$ hold. Let $\{a_{i}\} _{i=1}^{\infty}$ and $\{b_{i}\}_{i=1}^{\infty}$ be such that

$$a_{i+1}< \sigma_{ik}b_{i}< b_{i}< L_{0}b_{i}< a_{i}, \quad i=1,2, \ldots, k=1,2,\ldots,m. $$

For each natural number i, we assume that f and $I_{k}$ satisfy ($H_{4}$) and ($H_{5}$), then the problem (1.1) has denumerably many positive solutions $\{x_{i}(t)\}_{i=1}^{\infty}$ such that

$$r_{i}\leq \Vert x_{i} \Vert _{\mathit {PC}^{1}}\leq R_{i}, \quad i=1,2,\dots. $$

Proof

Similar to the proof of (3.2) and (3.3), for all $t\in[\tau _{i},\delta-\tau_{i}]$, $x\in K_{\tau_{i}}\cap\partial\Omega_{1,i}$, then $x(t)\leq \Vert x \Vert _{\mathit {PC}^{1}}=R_{i}$.

Since (2.4) and (2.14), for all $x\in K_{\tau_{i}}\cap\partial\Omega _{1,i}$, by $(H_{4})$, we have

$$ \begin{aligned}[b] (Tx) (t)&= \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ &\leq B \Vert \omega \Vert _{1} \int_{0}^{1}f \bigl(s,x(s) \bigr)\,ds+ B\sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ &\leq M_{0}R_{i}B \Vert \omega \Vert _{1}+ mM_{0}R_{i}B \\ &\leq M_{0}(D+mB)R_{i} \\ &\leq R_{i}= \Vert x \Vert _{\mathit {PC}^{1}}, \end{aligned} $$

(3.5)

and by (2.4), (2.7), (2.16) and $(H_{4})$,

$$\begin{aligned} \bigl\vert (Tx)'(t) \bigr\vert & \leq \int_{0}^{1} \bigl\vert G_{t}'(t,s) \bigr\vert \omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m} \bigl\vert G_{t}'(t,t_{k}) \bigr\vert I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ & \leq \sinh\gamma \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ & \leq\frac{\sinh\gamma}{ A} \Biggl[ \int_{0}^{1}G(s,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(s,s)I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ & \leq\frac{\sinh\gamma}{ A} \Biggl[ \int_{0}^{1}B \Vert \omega \Vert _{1}f \bigl(s,x(s) \bigr)\,ds+ B\sum_{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ & \leq\frac{\sinh\gamma}{ A} \bigl(M_{0}R_{i}B \Vert \omega \Vert _{1}+ BmM_{0}R_{i} \bigr) \\ & \leq M_{0}\frac{\sinh\gamma}{A}B(D+ m)R_{i} \\ & \leq R_{i}= \Vert x \Vert _{\mathit {PC}^{1}}. \end{aligned}$$

(3.6)

From (3.5) and (3.6), we have

$$\Vert Tx \Vert _{\mathit {PC}^{1}}\leq \Vert x \Vert _{\mathit {PC}^{1}},\quad \forall x\in K_{\tau_{i}}\cap\partial\Omega_{1,i}. $$

Similarly to the proof of Theorem 3.1, we can finish the proof of Theorem 3.3. □

4 An example

From Section 3, it is not difficult to see that $(H_{3})$ plays an important role in the proof that problem (1.1) has denumerably many positive solutions. As an example, we consider a family of functions $\omega(t)$ as follows.

Example 4.1

Let $k=m=1$, $t_{1}=\frac{1}{3}$, and

$$t_{n}'=t_{1}-\sum _{i=1}^{n} \frac{1}{(i+1)(i+2)(i+3)(i+4)}, \quad n=1,2,\ldots. $$

It is easy to see that

$$\begin{aligned}& t_{1}'=\frac{39}{120}< \frac{1}{3}, \\& t_{n}'-t_{n+1}'= \frac{1}{(n+2)(n+3)(n+4)(n+5)}, \quad n=1,2,\ldots, \end{aligned}$$

and

$$t^{*}=\lim_{n\rightarrow\infty}t_{n}=t_{1}- \sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)(i+3)(i+4)} = \frac{1}{3}- \frac{1}{72}=\frac{23}{72}>\frac{1}{4}, $$

where $\sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)(i+3)(i+4)}=\frac{1}{72}$.

Let

$$\omega(t)=\sum_{i=1}^{\infty} \omega_{n}(t), \quad t\in J, $$

where

$$\omega_{n}(t)= \textstyle\begin{cases} \frac{1}{2n^{4}(t_{n}'+t_{n+1}')},& t\in[0,\frac {t_{n}'+t_{n+1}'}{2}),\\ \frac{1}{\sqrt{t_{n}'-t}}, & t\in[\frac{t_{n}'+t_{n+1}'}{2},t_{n}'),\\ \frac{1}{\sqrt{t-t_{n}'}}, & t\in[t_{n}',\frac{t_{n}'+t_{n-1}'}{2}],\\ \frac{2}{2n^{4}(2-t_{n}'-t_{n-1}')},& t\in(\frac {t_{n}'+t_{n-1}'}{2},1]. \end{cases} $$

From $\sum_{i=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}$ and $\sum_{i=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{4}}{6}$, we have

$$\begin{aligned} \sum_{i=1}^{\infty} \int_{0}^{1}\omega_{n}(t)\,dt &= \sum _{i=1}^{\infty} \biggl\{ \int_{0}^{(t_{n}'+t_{n+1}')/2}\frac {1}{2n^{4}(t_{n}'+t_{n+1}')}\,dt + \int_{(t_{n-1}'+t_{n}')/2}^{1}\frac {2}{2n^{4}(2-t_{n}'-t_{n-1}')}\,dt \\ &\quad{} + \int_{(t_{n}'+t_{n+1}')/2}^{t_{n}}\frac{1}{\sqrt{t_{n}'-t}}\,dt+ \int_{t_{n}}^{(t_{n-1}'+t_{n}')/2}\frac{1}{\sqrt {t-t_{n}'}}\,dt \biggr\} \\ &=\sum_{i=1}^{\infty}\frac{1}{n^{4}}+\sqrt{2} \sum_{i=1}^{\infty} \Bigl(\sqrt{ \bigl(t_{n}'-t_{n+1}' \bigr)}+\sqrt { \bigl(t_{n-1}'-t_{n}' \bigr)} \Bigr) \\ &=\frac{\pi^{4}}{90}+\sqrt{2}\sum_{i=1}^{\infty} \biggl[ \frac {1}{(n+2)(n+3)(n+4)(n+5)} \biggr]^{\frac{1}{2}} \\ &\quad{} +\sqrt{2}\sum_{i=1}^{\infty} \biggl[ \frac {1}{(n+1)(n+2)(n+3)(n+4)} \biggr]^{\frac{1}{2}} \\ &\leq\frac{\pi^{4}}{90}+\sqrt{2}\sum_{i=1}^{\infty} \frac {1}{n^{2}}+\sqrt{2}\sum_{i=1}^{\infty} \frac{1}{n^{2}} \\ &=\frac{\pi^{4}}{90}+\sqrt{2}\frac{\pi^{2}}{3}. \end{aligned}$$

Thus, it is easy to see

$$\int_{0}^{1}\omega(t)\,dt= \int_{0}^{1}\sum_{i=1}^{\infty} \omega _{n}(t)\,dt=\sum_{i=1}^{\infty} \int_{0}^{1}\omega_{n}(t)\,dt< \infty. $$

Therefore, $\omega(t)\in L^{1}[0,1]$, which satisfies condition $(H_{3})$.

Declarations

Acknowledgements

This work is sponsored by the National Natural Science Foundation of China (11301178) and the Beijing Natural Science Foundation (1163007), the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (71E1610973) and the teaching reform project of Beijing Information Science & Technology University (2015JGYB41). The authors are grateful to the anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)

School of Applied Science, Beijing Information Science & Technology University

References

Kaufmann, ER, Kosmatov, N: A multiplicity result for a boundary value problem with infinitely many singularities. J. Math. Anal. Appl. 269, 444-453 (2002) MathSciNetView ArticleMATHGoogle Scholar
Kosmatov, N: On a singular conjugate boundary value problem with infinitely many solutions. Math. Sci. Res. Hot-Line 4, 9-17 (2000) MathSciNetMATHGoogle Scholar
Liang, SH, Zhang, JH: The existence of countably many positive solutions for nonlinear singular m-point boundary value problems. J. Comput. Appl. Math. 214, 78-89 (2008) MathSciNetView ArticleMATHGoogle Scholar
Liang, SH, Zhang, JH: The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary value problems. Nonlinear Anal. 71, 4588-4597 (2009) MathSciNetView ArticleMATHGoogle Scholar
Liu, B: Positive solutions three-points boundary value problems for one-dimensional p-Laplacian with infinitely many singularities. Appl. Math. Lett. 17, 655-661 (2004) MathSciNetView ArticleMATHGoogle Scholar
Su, H, Wei, ZL, Xu, FY: The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator. J. Math. Anal. Appl. 325, 319-332 (2007) MathSciNetView ArticleMATHGoogle Scholar
Xu, F, Wu, Y, Liu, L, Zhou, Y: Positive solutions of three-point boundary value problems for higher-order p-Laplacian with infinitely many singularities. Discrete Dyn. Nat. Soc. 2006, 69073 (2006) MathSciNetView ArticleMATHGoogle Scholar
Liang, SH, Zhang, JH: The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on time scales. J. Comput. Appl. Math. 223, 291-303 (2009) MathSciNetView ArticleMATHGoogle Scholar
Liang, SH, Zhang, JH: The existence of countably many positive solutions for some nonlinear three-point boundary problems on the half-line. Nonlinear Anal. 70, 3127-3139 (2009) MathSciNetView ArticleMATHGoogle Scholar
Ji, D, Bai, Z, Ge, W: The existence of countably many positive solutions for singular multipoint boundary value problems. Nonlinear Anal. 72, 955-964 (2010) MathSciNetView ArticleMATHGoogle Scholar
Liang, SH, Zhang, JH: Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line. Acta Appl. Math. 111, 27-43 (2010) MathSciNetView ArticleMATHGoogle Scholar
Kosmatov, N: Countably many solutions of a fourth order boundary value problem. Electron. J. Qual. Theory Differ. Equ. 2004, 12 (2004) MathSciNetMATHGoogle Scholar
Davis, JM, Erbe, LH, Henderson, J: Multiplicity of positive solutions for higher order Sturm-Liouville problems. Rocky Mt. J. Math. 31, 169-184 (2001) MathSciNetView ArticleMATHGoogle Scholar
Cabada, A, Sanchez, L: A positive operator approach to the Neumann problem for a second order ordinary differential equation. J. Math. Anal. Appl. 204, 774-785 (1996) MathSciNetView ArticleMATHGoogle Scholar
Cabada, A, Pouso, RL: Existence result for the problem $(\phi(u'))' = f(t, u, u')$ with periodic and Neumann boundary conditions. Nonlinear Anal. 30, 1733-1742 (1997) MathSciNetView ArticleMATHGoogle Scholar
Cabada, A, Habets, P, Lois, S: Monotone method for the Neumann problem with lower and upper solutions in the reverse order. Appl. Math. Comput. 117, 1-14 (2001) MathSciNetMATHGoogle Scholar
Gao, S, Chen, L, Nieto, JJ, Torres, A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037-6045 (2006) View ArticleGoogle Scholar
Chu, J, Sun, Y, Chen, H: Positive solutions of Neumann problems with singularities. J. Math. Anal. Appl. 337, 1267-1272 (2008) MathSciNetView ArticleMATHGoogle Scholar
Dang, H, Oppenheimer, SF: Existence and uniqueness results for some nonlinear boundary value problems. J. Math. Anal. Appl. 198, 35-48 (1996) MathSciNetView ArticleMATHGoogle Scholar
Dong, Y: A Neumann problem at resonance with the nonlinearity restricted in one direction. Nonlinear Anal. 51, 739-747 (2002) MathSciNetView ArticleMATHGoogle Scholar
Erbe, LH, Wang, H: On the existence of positive solutions of ordinary differential equations. Proc. Am. Math. Soc. 120, 743-748 (1994) MathSciNetView ArticleMATHGoogle Scholar
Ma, R: Existence of positive radial solutions for elliptic systems. J. Math. Anal. Appl. 201, 375-386 (1996) MathSciNetView ArticleMATHGoogle Scholar
Yazidi, N: Monotone method for singular Neumann problem. Nonlinear Anal. 49, 589-602 (2002) MathSciNetView ArticleMATHGoogle Scholar
Sun, J, Li, W: Multiple positive solutions to second order Neumann boundary value problems. Appl. Math. Comput. 146, 187-194 (2003) MathSciNetMATHGoogle Scholar
Jiang, D, Liu, H: Existence of positive solutions to second order Neumann boundary value problem. J. Math. Res. Expo. 20, 360-364 (2000) MathSciNetMATHGoogle Scholar
Liu, X, Li, Y: Positive solutions for Neumann boundary value problems of second-order impulsive differential equations in Banach spaces. Abstr. Appl. Anal. 2012, 401923 (2012) MathSciNetMATHGoogle Scholar
Zhang, X: Parameter dependence of positive solutions for second-order singular Neumann boundary value problems with impulsive effects. Abstr. Appl. Anal. 2014, 968792 (2014) MathSciNetGoogle Scholar
Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) MATHGoogle Scholar

Infinitely many singularities and denumerably many positive solutions for a second-order impulsive Neumann boundary value problem

Abstract

Keywords

1 Introduction

2 Preliminaries

Lemma 2.1

Proof

Remark 2.1

Lemma 2.2

Proof

Definition 2.1

Definition 2.2

Lemma 2.3

Proof

Lemma 2.4

Lemma 2.5

3 Main results

Theorem 3.1

Proof

Theorem 3.2

Proof

Theorem 3.3

Proof

4 An example

Example 4.1

Declarations

Acknowledgements

Authors’ Affiliations

References

Copyright