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Positive solutions to one-dimensional quasilinear impulsive indefinite boundary value problems

Advances in Difference Equations20182018:421

https://doi.org/10.1186/s13662-018-1881-7

  • Received: 11 July 2018
  • Accepted: 8 November 2018
  • Published:

Abstract

Consider the one-dimensional quasilinear impulsive boundary value problem involving the p-Laplace operator
$$ \textstyle\begin{cases} -(\phi_{p}(u'))'=\lambda \omega (t)f(u), \quad 0< t< 1, \\ -\Delta u|_{t=t_{k}}=\mu I_{k}(u(t_{k})), \quad k=1,2,\ldots,n, \\ \Delta u'|_{t=t_{k}}=0, \quad k=1,2,\ldots,n, \\ u'(0)=0, \qquad u(1)=\int_{0}^{1}g(t)u(t)\,dt, \end{cases} $$
where \(\lambda, \mu >0\) are two positive parameters, \(\phi_{p}(s)\) is the p-Laplace operator, i.e., \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\), \(\omega (t)\) changes sign on \([0,1]\). Several new results are obtained for the above quasilinear indefinite problem.

Keywords

  • Multiplicity of positive solutions
  • Indefinite weight function
  • p-Laplace operator
  • Quasilinear impulsive differential equation

1 Introduction

Impulsive differential equation is regarded as a critical mathematical tool to provide a natural description of observed evolution processes (see [14]). So the consideration of impulsive differential equations has gained prominence and many authors have begun to take a great interest in the subject of impulsive differential equations, for example, see [522] and the references cited therein.

Meanwhile, the p-Laplace operator equation is a typical quasilinear operator equation, which comes naturally from glaciology, nonlinear flow laws, and non-Newtonian mechanics (see [23, 24]). Recently, various existence, multiplicity, and uniqueness results of positive solutions for differential equations with one-dimensional p-Laplace operator have been considered [2533]. Specially, Zhang and Ge [34] investigated the following second order one-dimensional p-Laplace operator equation
$$ \textstyle\begin{cases} -(\phi_{p}(u^{\prime}(t)))^{\prime}= f(t,u(t)), \quad t\neq t_{k}, t\in (0,1), \\ \triangle u|_{t= t_{k}}=I_{k}(u(t_{k})), \quad k=1,2,\ldots,n, \\ u(0)=\sum_{i=1}^{m-2}a_{i}u(\xi_{i}), \quad u^{\prime}(1)=0, \end{cases} $$
(1.1)
where \(\phi_{p}(s)\) is p-Laplace operator, i.e., \(\phi_{p}(s)=|s|^{p-2}s\), \(p>1\), \((\phi_{p})^{-1}=\phi_{q}\), \(\frac{1}{p}+ \frac{1}{q}=1\), \(t_{k}\) (\(k=1,2,\dots,n\), where n is a fixed positive integer) are fixed points with \(0< t_{1}< t_{2}<\cdots <t_{k}<\cdots <t _{n}<1\), \(\xi_{i}\) \((i=1,2,\ldots, m-2)\in (0,1)\) is given \(0<\xi_{1}< \xi_{2}<\cdots <\xi_{m-2}<1\) and \(\xi_{i}\neq t_{k}\), \(i=1,2,\dots,m-2\), \(k=1,2,\dots,n\), \(\Delta u|_{t=t_{k}}\) denotes the jump of \(u(t)\) at \(t=t_{k}\), i.e.,
$$ \Delta u|_{t=t_{k}}=u\bigl(t_{k}^{+}\bigr)-u \bigl(t_{k}^{-}\bigr), $$
where \(u(t_{k}^{+})\) and \(u(t_{k}^{-})\) represent the right-hand limit and left-hand limit of \(u(t)\) at \({t=t_{k}}\), respectively. Applying the classical fixed-point index theorem for compact maps, the authors got several new multiplicity results of positive solutions.

On the other hand, we observe that many authors (see [3549]) have paid more attention to a class of boundary value problems involving integral boundary conditions, which contains two-point, three-point, and general multi-point boundary value problems as exceptional cases, see [5058] and the references cited therein.

However, in literature there are almost no papers on multiple positive solutions for second order impulsive nonlocal indefinite boundary value problems with one-dimensional p-Laplace operator and multiple parameters. More precisely, the study of \(\lambda >0\), \(\mu >0\), \(p \not \equiv 2\), \(I_{k}\neq 0\) (\(k=1,2,\ldots,n\)) and ω changes sign is still open for the second order nonlocal boundary value problem
$$ \textstyle\begin{cases} -(\phi_{p}(u'))'=\lambda \omega (t)f(u), \quad 0< t< 1, \\ -\Delta u|_{t=t_{k}}=\mu I_{k}(u(t_{k})), \quad k=1,2,\ldots,n, \\ \Delta u'|_{t=t_{k}}=0, \quad k=1,2,\ldots,n, \\ u'(0)=0, \quad u(1)=\int_{0}^{1}g(t)u(t)\,dt, \end{cases} $$
(1.2)
where \(\lambda >0\) and \(\mu >0\) are two parameters, \(\omega (t)\) may change sign, \(\phi_{p}(s)\) is a p-Laplace operator, i.e., \(\phi _{p}(s)=|s|^{p-2}s\), \(p>1\), \((\phi_{p})^{-1}=\phi_{q}\), \(\frac{1}{p}+ \frac{1}{q}=1\). \(t_{k}\) (\(k=1,2,\ldots,n\)) (where n is a fixed positive integer) are fixed points with \(0=t_{0}< t_{1}< t_{2}<\cdots <t_{k}< \cdots <t_{n}<t_{n+1}=1\), \(\Delta u|_{t=t_{k}}\) denotes the jump of \(u(t)\) at \(t=t_{k}\), i.e., \(\Delta u|_{t=t_{k}}=u(t_{k}^{+})-u(t_{k} ^{-})\), where \(u(t_{k}^{+})\) and \(u(t_{k}^{-})\) represent the right-hand limit and left-hand limit of \(u(t)\) at \(t=t_{k}\), respectively.
In addition, set \(J=[0,1]\), \(R_{+}=[0,+\infty)\), \(R=(-\infty,+\infty)\), and let ω, f, \(I_{k}\), and g satisfy the following conditions:
(\(H_{1}\)): 
\(\omega:J\rightarrow R\) is continuous, and there exists a constant \(\xi \in (0,1)\) such that
$$ \omega (t)\geq 0,\quad t\in [0,\xi ],\qquad \omega (t)\leq 0,\quad t\in [ \xi,1]. $$
Moreover, \(\omega (t)\) does not vanish identically on any subinterval of J.
(\(H_{2}\)): 
\(f:R_{+}\rightarrow R_{+}\) is continuous, and \(f(u)>0\) for all \(u>0\), there exists \(0< c\leq 1\) such that
$$ f(x)\geq c\psi (x),\quad x\in R_{+}, $$
where \(\psi (x)=\max \{f(y):0\leq y\leq x\}\);
(\(H_{3}\)): 

\(I_{k}\in C(R_{+},R_{+})\), and \(I_{k}(u)>0\) for all \(u>0\).

(\(H_{4}\)): 
\(g\in L^{1}[0,1]\) is nonnegative and \(\eta \in [0,1)\), where
$$ \eta = \int_{0}^{1}g(s)\,ds. $$
(1.3)
(\(H_{5}\)): 
There exist \(0<\theta_{1}\leq +\infty\), \(\theta_{1}\neq p-1\), \(0<\theta_{2}\leq +\infty\), \(\theta_{2}\neq 1\), and \(k_{1},k_{2},k _{3},k_{4}>0\) such that
$$ k_{1}u^{\theta_{1}}\leq f(u)\leq k_{2}u^{\theta_{1}}, \qquad k_{3}u^{\theta _{2}}\leq I_{k}(u)\leq k_{4}u^{\theta_{2}}. $$
(\(H_{6}\)): 
There exists a number \(0<\sigma <\xi \) such that
$$ c^{2}k_{1}\sigma^{\theta_{1}} \int_{\sigma }^{\xi }\omega^{+}(t)\,dt \geq k_{2}\xi^{\theta_{1}} \int_{\xi }^{1}\omega^{-}(t)\,dt. $$

We define \(\omega^{+}(t)=\max \{\omega (t),0\}\), \(\omega^{-}(t)=- \min \{\omega (t),0\}\). Then \(\omega (t)=\omega^{+}(t)-\omega^{-}(t)\).

It is well accepted that the fixed point theorem in a cone is crucial in showing the existence of positive solutions of various boundary value problems for second order differential equations.

Lemma 1.1

(Theorem 2.3.4 of [59])

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 \(T: P\cap (\bar{\varOmega }_{2}\backslash \varOmega_{1})\rightarrow P\) be completely continuous, where P is a cone in E. Suppose that one of the two conditions
  1. (i)

    \(\|Tx\|\leq \|x\|\), \(\forall x\in P\cap \partial \varOmega_{1}\) and \(\|Tx\|\geq \|x\|\), \(\forall x\in P\cap \partial \varOmega_{2}\),

     
or
  1. (ii)

    \(\|Tx\|\geq \|x\|\), \(\forall x\in P\cap \partial \varOmega_{1}\), and \(\|Tx\|\leq \|x\|\), \(\forall x\in P\cap \partial \varOmega_{2}\),

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

This paper is organized in the following fashion. In Sect. 2, we present some lemmas to be used in the subsequent sections. Section 3 is devoted to proving the multiplicity of positive solutions for problem (1.2), and we give an example to illustrate the main results in the final section.

2 Preliminaries

Let \(J'=J\backslash \{t_{1},t_{2},\ldots,t_{n}\}\). The basic space used in this paper \(PC[0,1]=\{u|u:J\rightarrow R { }\mbox{is continuous at } t\neq t_{k}, \mbox{left continuous at }t=t_{k}, \mbox{and }u(t_{k}^{+})\mbox{ exists}, k=1,2,\ldots,n\}\). Then \(PC[0,1]\) is a real Banach space with the norm \(\|\cdot \|_{PC}\) defined by \(\|u\|_{PC}=\sup_{t\in J}|u(t)|\). By a solution of (1.2), we mean that a function \(u\in PC[0,1]\cap C ^{2}(J')\) which satisfies (1.2).

In these main results, we will make use of the following lemmas.

Lemma 2.1

Assume that (\(H_{1}\))(\(H_{4}\)) hold. Then \(u\in PC[0,1]\cap C^{2}(J')\) is a solution of problem (1.2) if and only if \(u\in PC[0,1]\) is a solution of the following impulsive integral equation:
$$\begin{aligned} u(t) =&\frac{1}{1-\eta }\biggl[ \int_{0}^{1}g(t) \int_{t}^{1}\phi_{q} \biggl( \int _{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds\,dt \\ &{}+\mu \int _{0}^{1}g(t) \biggl( \sum _{t\leq t_{k}}I_{k}\bigl(u(t_{k})\bigr) \biggr)\,dt\biggr] \\ &{}+ \int_{t}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u( \tau) \bigr)\,d\tau \biggr)\,ds+\mu \sum_{t\leq t_{k}}I_{k} \bigl(u(t_{k})\bigr). \end{aligned}$$
(2.1)

Proof

The proof is similar to that of Lemma 3.1 in [38]. □

To establish the existence of multiple positive solutions in \(PC[0,1]\cap C^{2}(J')\) of problem (1.2), we denote
$$ PC^{+}[0,1]= \Bigl\{ u\in PC[0,1]: \min_{t\in J}u(t) \geq 0 \Bigr\} , $$
and a cone K in \(PC[0,1]\) by
$$ K= \bigl\{ u\in PC^{+}[0,1]:u\mbox{ is concave on } [0,\xi ], \mbox{and } u \mbox{ is convex on }[\xi,1] \bigr\} . $$
(2.2)

Let \(R>r>0\), define \(K_{r}=\{u\in K:\|u\|< r\}\), \(K_{R,r}=\{u\in K:r<\|u \|<R\}\). Note that \(\partial K_{r}=\{u\in K:\|u\|=r\}\), \(\overline{K} _{R,r}=\{u\in K:r\leq \|u\|\leq R\}\).

We define a map \(T:K \rightarrow PC[0,1]\) by
$$\begin{aligned} (Tu) (t) =&\frac{1}{1-\eta }\biggl[ \int_{0}^{1}g(t) \int_{t}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds\,dt \\ &{}+ \mu \int_{0}^{1}g(t) \biggl( \sum _{t\leq t_{k}}I_{k}\bigl(u(t_{k})\bigr) \biggr)\,dt\biggr] \\ &{}+ \int_{t}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u( \tau) \bigr)\,d\tau \biggr)\,ds+\mu \sum_{t\leq t_{k}}I_{k} \bigl(u(t_{k})\bigr), \end{aligned}$$
(2.3)
where η is defined in (1.3).

Lemma 2.2

From (2.1), we know that \(u\in PC[0,1]\) is a solution of problem (1.2) if and only if u is a fixed point of the map T.

Lemma 2.3

Assume that (\(H_{1}\))(\(H_{6}\)) hold. Then we have \(T(K)\subset K\), and \(T:K\rightarrow K\) is completely continuous.

Proof

From (2.3), we know that
$$ (Tu)'(t)=-\phi_{q} \biggl( \int_{0}^{t}\lambda \omega (s)f\bigl(u(s)\bigr)\,ds \biggr). $$
(2.4)
Define \(q(t):J\rightarrow J\) as follows:
$$ q(t)=\min \biggl\{ \frac{t}{\xi },\frac{1-t}{1-\xi } \biggr\} , $$
and \(\min_{\sigma \leq t\leq \xi }q(t)=\frac{\sigma }{\xi }\), \(\max_{\xi \leq t\leq 1}q(t)=1\).
Firstly, for any \(u\in K\), we have
$$ \int_{0}^{1}\omega (s)f\bigl(u(s)\bigr)\,ds\geq \int_{0}^{\sigma }\omega^{+}(s)f\bigl(u(s) \bigr)\,ds. $$
(2.5)
In fact, by (2.2), we know that \(u(t)\geq 0\). Since \(u\in K\), \(u(0) \geq 0\), and \(u(1)\geq 0\), we have
$$\begin{aligned}& \frac{u(t)-u(0)}{t-0}\geq \frac{u(\xi)-u(0)}{\xi -0},\quad t\in [0,\xi ] \quad \Rightarrow \quad u(t)\geq q(t)u(\xi),\quad t\in [0,\xi ], \\& \frac{u(t)-u(1)}{t-1}\geq \frac{u(\xi)-u(1)}{\xi -1},\quad t\in [\xi,1] \quad \Rightarrow \quad u(t)\leq q(t)u(\xi),\quad t\in [\xi,1]. \end{aligned}$$
As we all know, ψ is nondecreasing on J, so we have
$$ \psi \bigl(u(t)\bigr)\geq \psi \bigl(q(t)u(\xi)\bigr),\quad t\in [0,\xi ], \qquad \psi \bigl(u(t)\bigr) \leq \psi \bigl(q(t)u(\xi)\bigr),\quad t\in [\xi,1]. $$
So, it follows from (\(H_{5}\)) and (\(H_{6}\)) that
$$\begin{aligned}& \int_{0}^{1}\omega (s)f\bigl(u(s)\bigr)\,ds- \int_{0}^{\sigma }\omega^{+}(s)f\bigl(u(s) \bigr)\,ds \\& \quad = \int_{\sigma }^{\xi }\omega^{+}(s)f\bigl(u(s) \bigr)\,ds- \int_{\xi }^{1}\omega^{-}(s)f\bigl(u(s) \bigr)\,ds \\& \quad \geq c \int_{\sigma }^{\xi }\omega^{+}(s)\psi \bigl(u(s) \bigr)\,ds- \int_{\xi }^{1} \omega^{-}(s)\psi \bigl(u(s) \bigr)\,ds \\& \quad \geq c \int_{\sigma }^{\xi }\omega^{+}(s)\psi \bigl(q(s)u(\xi)\bigr)\,ds- \int_{ \xi }^{1}\omega^{-}(s)\psi \bigl(q(s)u( \xi)\bigr)\,ds \\& \quad \geq c \int_{\sigma }^{\xi }\omega^{+}(s)f\bigl(q(s)u( \xi)\bigr)\,ds-\frac{1}{c} \int_{\xi }^{1}\omega^{-}(s)f\bigl(q(s)u(\xi) \bigr)\,ds \\& \quad \geq ck_{1}u^{\theta }(\xi)\frac{\sigma^{\theta }}{\xi^{\theta }} \int_{\sigma }^{\xi }\omega^{+}(s)\,ds- \frac{1}{c}k_{2}u^{\theta }( \xi) \int_{\xi }^{1}\omega^{-}(s)\,ds \\& \quad \geq u^{\theta }(\xi) \biggl( ck_{1}\frac{\sigma^{\theta }}{\xi^{\theta }} \int_{\sigma }^{\xi }\omega^{+}(s)\,ds- \frac{1}{c}k_{2} \int_{\xi } ^{1}\omega^{-}(s)\,ds \biggr) \\& \quad \geq 0. \end{aligned}$$
Secondly, if \(t\in [0,\xi ]\), we have
$$ \int_{0}^{t}\omega (s)f\bigl(u(s)\bigr)\,ds= \int_{0}^{t}\omega^{+}(s)f\bigl(u(s) \bigr)\,ds \geq 0. $$
Since \(p, q>1\), we get
$$\begin{aligned} (Tu)''(t) =& \biggl( -\phi_{q} \biggl( \int_{0}^{t}\lambda \omega (s)f\bigl(u(s)\bigr)\,ds \biggr) \biggr) ' \\ =& \biggl( - \biggl( \int_{0}^{t}\lambda \omega^{+}(s)f \bigl(u(s)\bigr)\,ds \biggr) ^{q-1} \biggr) ' \\ =&-(q-1) \biggl( \int_{0}^{t}\lambda \omega^{+}(s)f \bigl(u(s)\bigr)\,ds \biggr) ^{q-2} \lambda \omega^{+}(t)f \bigl(u(t)\bigr) \\ \leq& 0. \end{aligned}$$
If \(t\in [\xi,1]\), then we have
$$\begin{aligned} \int_{0}^{t}\omega (s)f\bigl(u(s)\bigr)\,ds =& \int_{0}^{\xi }\omega^{+}(s)f\bigl(u(s) \bigr)\,ds- \int_{\xi }^{t}\omega^{-}(s)f\bigl(u(s) \bigr)\,ds \\ \geq& \int_{0}^{\xi }\omega^{+}(s)f\bigl(u(s) \bigr)\,ds- \int_{\xi }^{1}\omega^{-}(s)f\bigl(u(s) \bigr)\,ds \\ =& \int_{0}^{1}\omega (s)f\bigl(u(s)\bigr)\,ds \\ \geq& \int_{0}^{\sigma }\omega^{+}(s)f\bigl(u(s) \bigr)\,ds \\ \geq& 0. \end{aligned}$$
And then, for \(t\in [\xi,1]\), it follows from \(p,q>1\) that
$$\begin{aligned} (Tu)''(t) =& \biggl( -\phi_{q} \biggl( \int_{0}^{t}\lambda \omega (s)f\bigl(u(s)\bigr)\,ds \biggr) \biggr) ' \\ =& \biggl( -\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(s)f \bigl(u(s)\bigr)\,ds- \int_{\xi }^{t}\lambda \omega^{-}(s)f \bigl(u(s)\bigr) \biggr) \biggr) ' \\ =& \biggl( - \biggl( \int_{0}^{\xi }\lambda \omega^{+}(s)f \bigl(u(s)\bigr)\,ds- \int_{ \xi }^{t}\lambda \omega^{-}(s)f \bigl(u(s)\bigr) \biggr) ^{q-1} \biggr) ' \\ =&-(q-1) \biggl( \int_{0}^{\xi }\lambda \omega^{+}(s)f \bigl(u(s)\bigr)\,ds- \int_{ \xi }^{t}\lambda \omega^{-}(s)f \bigl(u(s)\bigr) \biggr) ^{q-2} \bigl( -\lambda \omega^{-}(t)f \bigl(u(t)\bigr) \bigr) \\ \geq& 0. \end{aligned}$$

Moreover, by direct calculating, we get \((Tu)(t)\geq 0\) for \(t\in J\), \((Tu)''(t)\leq 0\) for \(t\in [0,\xi ]\), and \((Tu)''(t)\geq 0\) for \(t\in [\xi,1]\). Thus, \(T(K)\subset K\).

Then it finally follows from the Arzelà–Ascoli theorem that the operator T is completely continuous. □

From Lemma 2.3, since \((Tu)'(t)\leq 0\), then T is nonincreasing for \(u\in K\). It is not difficult to see that
$$\begin{aligned} \Vert Tu \Vert _{PC} =&(Tu) (0) \\ =&\frac{1}{1-\eta }\biggl[ \int_{0}^{1}g(0) \int_{0} ^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds\,dt \\ &{} +\mu \int_{0}^{1}g(0) \biggl( \sum _{t\leq t_{k}}I _{k}\bigl(u(t_{k})\bigr) \biggr)\,dt\biggr] \\ &{}+ \int_{0}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u( \tau) \bigr)\,d\tau \biggr)\,ds+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr). \end{aligned}$$
(2.6)

Lemma 2.4

If (\(H_{1}\))(\(H_{4}\)) hold, then for \(u\in K\) we get
$$\begin{aligned}& \Vert Tu \Vert _{PC}\leq \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \frac{1}{1-\eta }\sum _{k=1}^{n}I_{k}\bigl(u(t_{k}) \bigr), \end{aligned}$$
(2.7)
$$\begin{aligned}& \Vert Tu \Vert _{PC}\geq \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi _{q} \biggl( \int_{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}(\tau)f \bigl(u( \tau)\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr). \end{aligned}$$
(2.8)

Proof

By (2.6), for \(u\in K\), we have
$$\begin{aligned} \Vert Tu \Vert _{PC} =&\frac{1}{1-\eta }\biggl[ \int_{0}^{1}g(t) \int_{t}^{1}\phi _{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds\,dt \\ &{}+ \mu \int_{0}^{1}g(t) \biggl( \sum _{t\leq t_{k}}I_{k}\bigl(u(t_{k})\bigr) \biggr)\,dt\biggr] \\ &{} + \int_{0}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u( \tau) \bigr)\,d\tau \biggr)\,ds+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \leq& \frac{1}{1-\eta }\Biggl[ \int_{0}^{1}g(t) \int_{0}^{1}\phi_{q} \biggl( \int _{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds\,dt \\ &{}+\mu \int _{0}^{1}g(t) \Biggl( \sum _{k=1}^{n}I_{k}\bigl(u(t_{k}) \bigr) \Biggr)\,dt\Biggr] \\ &{} + \int_{0}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u( \tau) \bigr)\,d\tau \biggr)\,ds+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ =&\frac{1}{1-\eta } \int_{0}^{1}\phi_{q} \biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds+\mu \frac{1}{1-\eta }\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ =&\frac{1}{1-\eta } \biggl[ \int_{0}^{\xi }\phi_{q} \biggl( \int_{0}^{s} \lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds+ \int_{\xi }^{1} \phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \\ &{}- \int_{\xi }^{s}\lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) \biggr]\,ds+\mu \frac{1}{1-\eta }\sum _{k=1}^{n}I_{k}\bigl(u(t _{k}) \bigr) \\ \leq& \frac{1}{1-\eta }\biggl[ \int_{0}^{\xi }\phi_{q} \biggl( \int_{0}^{ \xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+ \int_{\xi } ^{1}\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds\biggr] \\ &{} +\mu \frac{1}{1-\eta }\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ =&\frac{1}{1-\eta } \int_{0}^{1}\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds+\mu \frac{1}{1-\eta } \sum _{k=1}^{n}I_{k}\bigl(u(t_{k}) \bigr) \\ =&\frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \frac{1}{1-\eta }\sum _{k=1} ^{n}I_{k}\bigl(u(t_{k}) \bigr). \end{aligned}$$
Then (2.7) holds.
From (2.5) and (2.6), we have
$$\begin{aligned} \Vert Tu \Vert _{PC} =&\frac{1}{1-\eta }\biggl[ \int_{0}^{1}g(t) \int_{t}^{1}\phi _{q}\biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr)\,ds\,dt \\ &{}+ \mu \int_{0}^{1}g(t) \biggl( \sum _{t\leq t_{k}}I_{k}\bigl(u(t_{k})\bigr)\biggr)\,dt \biggr] \\ &{}+ \int_{0}^{1}\phi_{q}\biggl( \int_{0}^{s}\lambda \omega (\tau)f\bigl(u( \tau) \bigr)\,d\tau \biggr)\,ds+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ =&\frac{1}{1-\eta }\biggl\{ \int_{0}^{\xi }g(t)\biggl[ \int_{t}^{\xi }\phi _{q}\biggl( \int_{0}^{s}\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+ \int_{\xi }^{1}\phi_{q}\biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u( \tau)\bigr)\,d\tau \\ &{} - \int_{\xi }^{s}\lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds\biggr]\,dt+ \int_{\xi }^{1}g(t) \int_{t}^{1}\phi_{q}\biggl( \int _{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \\ &{} - \int_{\xi }^{s}\lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds\,dt\biggr\} + \int_{0}^{\xi }\phi_{q}\biggl( \int_{0}^{s} \lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+ \int_{\xi }^{1} \phi_{q}\biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \\ &{} - \int_{\xi }^{s}\lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds+ \mu \frac{1}{1-\eta } \int_{0}^{1}g(t)\sum_{t\leq t_{k}}I_{k} \bigl(u(t _{k})\bigr)\,dt \\ &{}+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \geq& \frac{1}{1-\eta } \int_{0}^{\xi }g(t) \int_{\xi }^{1}\phi_{q}\biggl( \int _{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau - \int_{\xi }^{s} \lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds\,dt \\ &{}+ \int_{\xi }^{1}\phi_{q}\biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau)f \bigl(u(\tau)\bigr)\,d\tau - \int_{\xi }^{s}\lambda \omega^{-}(\tau)f \bigl(u( \tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ =&\frac{1-\int_{\xi }^{1}g(t)\,dt}{1-\eta } \int_{\xi }^{1}\phi_{q}\biggl( \int _{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau - \int_{\xi }^{s} \lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \geq& \frac{1-\int_{\xi }^{1}g(t)\,dt}{1-\eta } \int_{\xi }^{1}\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau - \int_{\xi }^{1}\lambda \omega^{-}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+ \mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ =&\frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q}\biggl( \int _{0}^{1}\lambda \omega (\tau)f\bigl(u(\tau) \bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q}\biggl( \int _{0}^{\sigma }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) + \mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q}\biggl( \int _{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr). \end{aligned}$$
Then (2.8) holds. □

3 Main results

Based on the lemmas mentioned above, we give the following theorems and their proofs.

Theorem 3.1

Assume that (\(H_{1}\))(\(H_{6}\)) hold. If \(\theta_{1}>p-1\) and \(\theta_{2}>1\), there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that problem (1.2) admits two positive solutions for \(\lambda \in [\lambda_{0},+\infty)\), \(\mu \in [\mu_{0},+\infty)\).

Proof

Denote
$$\begin{aligned}& A_{1}=\frac{1}{\int_{0}^{\xi }\lambda \omega^{+}(\tau)\,d\tau }\phi _{p}\biggl(\frac{1-\eta }{2} \biggr),\qquad A_{2}=\frac{1-\eta }{2\mu n}, \\& B_{1}=\frac{1}{\int_{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}( \tau)\,d\tau }\phi_{p}\biggl(\frac{1-\eta }{2\alpha (1-\xi)(1-\int_{\xi } ^{1}g(t)\,dt)} \biggr), \qquad B_{2}=\frac{1}{2n\mu \alpha }. \end{aligned}$$
On the one hand, since \(\theta_{1}>p-1\) and \(\theta_{2}>1\), by (\(H_{5}\)), we get
$$ \lim_{u\rightarrow 0}\frac{f(u)}{\phi_{p}(u)}\leq \lim_{u\rightarrow 0} \frac{k_{2}u^{\theta_{1}}}{u^{p-1}}=0, \qquad \lim_{u\rightarrow 0}\frac{I _{k}(u)}{u}\leq \lim_{u\rightarrow 0}\frac{k_{4}u^{\theta_{2}}}{u}=0. $$
Hence, there exists \(r>0\) such that
$$ f(u)< A_{1}\phi_{p}(u), \qquad I_{k}(u)< A_{2}u, \quad u\in [0,r]. $$
Then from (2.7), for \(u\in \partial K_{r}\), then \(\|u\|_{PC}=r\) and \(0\leq u(t)\leq \|u\|=r\) for all \(t\in J\). It is clear that \(f(u(t))< A_{1}\phi_{p}(u(t))\) and \(I_{k}(u(t))< A_{2}u(t)\) for all \(t\in J\). Then from (2.7), for \(u\in \partial K_{r}\), we get
$$\begin{aligned} \Vert Tu \Vert _{PC} \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \frac{1}{1-\eta } \sum _{k=1}^{n}I_{k}\bigl(u(t_{k}) \bigr) \\ < &\frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau)A_{1}\phi_{p}\bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \frac{1}{1-\eta } \sum_{k=1}^{n}A_{2}u(t_{k}) \\ \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau)A_{1}\phi_{p}\bigl( \Vert u \Vert _{PC} \bigr)\,d\tau \biggr) +\mu \frac{1}{1-\eta } \sum_{k=1}^{n}A_{2} \Vert u \Vert _{PC} \\ =&\frac{ \Vert u \Vert _{PC}}{2}+\frac{ \Vert u \Vert _{PC}}{2} \\ =& \Vert u \Vert _{PC}. \end{aligned}$$
Consequently,
$$ \Vert Tu \Vert _{PC}< \Vert u \Vert _{PC}, \quad \forall u\in \partial K_{r}. $$
(3.1)
On the other hand, we denote \(\delta (t)=\min \{\frac{t}{\xi },\frac{ \xi -t}{\xi } \}\), \(t\in [0,\xi ]\). If \(u\in K\), then u is a nonnegative function on \([0,\xi ]\). So we get
$$ u(t)\geq \delta (t) \Vert u \Vert _{PC}, \quad t\in [0,\xi ]. $$

It follows that \(u(t)\geq \alpha \|u\|_{PC}\), \(t\in [\frac{\sigma }{2}, \sigma ]\), where \(\alpha =\min_{\frac{\sigma }{2}\leq t\leq \sigma }\delta (t)\).

Since \(\theta_{1}>p-1\) and \(\theta_{2}>1\), by (\(H_{5}\)), we have
$$ \lim_{u\rightarrow +\infty }\frac{f(u)}{\phi_{p}(u)}\geq \lim_{u\rightarrow +\infty } \frac{k_{1}u^{\theta_{1}}}{u^{p-1}}=+ \infty,\qquad \lim_{u\rightarrow +\infty }\frac{I_{k}(u)}{u} \geq \lim_{u\rightarrow +\infty }\frac{k_{3}u^{\theta_{2}}}{u}=+\infty. $$
Furthermore, there exists \(0< r< R'\) such that
$$ f(u)\geq B_{1}\phi_{p}(u),\qquad I_{k}(u)\geq B_{2}u,\quad u\in [R',+\infty), $$

Choose \(R\geq \frac{R'}{\alpha }\). Then, for any \(u\in \partial K_{R}\), we have \(\min_{\frac{\sigma }{2}\leq t\leq \sigma }u(t)\geq \min_{\frac{\sigma }{2}\leq t\leq \sigma }\delta (t)\|u\|_{PC}= \alpha R\geq R'\), and \(f(u(t))\geq B_{1}u^{p-1}(t)\), \(I_{k}(u(t)) \geq B_{2}u(t)\), \(t\in [\frac{\sigma }{2},\sigma ]\).

Then by (2.8), for \(u\in \partial K_{R}\), we have
$$\begin{aligned} \Vert Tu \Vert _{PC} \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi _{q} \biggl( \int_{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}(\tau)f \bigl(u( \tau)\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q} \biggl( \int _{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}( \tau)B_{1}\phi_{p}\bigl(u( \tau)\bigr)\,d\tau \biggr) +\mu \sum _{k=1}^{n}B_{2}u(t_{k}) \\ \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q} \biggl( \int _{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}( \tau)B_{1}\phi_{p}\bigl( \alpha \Vert u \Vert _{PC}\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}B_{2} \alpha \Vert u \Vert _{PC} \\ =&\frac{\alpha (1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta } \Vert u \Vert _{PC} \phi_{q} \biggl( \int_{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}( \tau)B_{1}\,d\tau \biggr) +\mu nB_{2}\alpha \Vert u \Vert _{PC} \\ \geq& \frac{1}{2} \Vert u \Vert _{PC}+\frac{1}{2} \Vert u \Vert _{PC} \\ =& \Vert u \Vert _{PC}. \end{aligned}$$
Consequently,
$$ \Vert Tu \Vert _{PC}\geq \Vert u \Vert _{PC}, \quad \forall u\in \partial K_{R}. $$
(3.2)
In addition, choose a number \(r'\in (0,r)\). Noticing that \(f(u)>0\) for all \(u>0\) and \(I_{k}(u)>0\) for all \(u>0\), we can define
$$\begin{aligned}& f_{r'}=\min \bigl\{ f(u):\alpha r'\leq u\leq r'\bigr\} , \qquad I_{kr'}=\min \bigl\{ I _{k}: \alpha r'\leq u \leq r'\bigr\} , \\& I_{r'}=\min \{I_{kr'}:k=1,2,\ldots,n\}. \end{aligned}$$
Let \(\lambda_{0}=\frac{1}{\int_{\frac{\sigma }{2}}^{\sigma }\omega ^{+}(\tau)f_{r'}\,d\tau }\phi_{p} ( \frac{r'(1-\eta)}{2(1-\int_{ \xi }^{1}g(t)\,dt)(1-\xi)} )\), \(\mu_{0}=\frac{r'}{2nI_{r'}}\). Thus we have
$$\begin{aligned}& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q} \biggl( \int_{\frac{ \sigma }{2}}^{\sigma }\lambda_{0} \omega^{+}(\tau)f_{r'}\,d\tau \biggr) = \frac{1}{2}r', \\& \mu_{0}nI_{r'}=\frac{1}{2}r'. \end{aligned}$$

If \(u\in \partial K_{r'}\), then \(\|u\|_{PC}=r'\) and \(\alpha r'= \min_{\frac{\sigma }{2}\leq t\leq \sigma }\delta (t)\|u\|_{PC} \leq u(t)\leq \|u\|_{PC}=r'\), \(t\in [\frac{\sigma }{2},\sigma ]\). It is clear that \(f(u(t))\geq f_{r'}\) and \(I_{k}(u(t))\geq I_{r'}\), \(t\in [\frac{ \sigma }{2},\sigma ]\).

Then from (2.8), for \(u\in \partial K_{r'}\), we have
$$\begin{aligned} \Vert Tu \Vert _{PC} \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi _{q} \biggl( \int_{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}(\tau)f \bigl(u( \tau)\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q} \biggl( \int _{\frac{\sigma }{2}}^{\sigma }\lambda \omega^{+}( \tau)f_{r'}\,d\tau \biggr) + \mu \sum_{k=1}^{n}I_{r'} \\ \geq& \frac{(1-\int_{\xi }^{1}g(t)\,dt)(1-\xi)}{1-\eta }\phi_{q} \biggl( \int _{\frac{\sigma }{2}}^{\sigma }\lambda_{0} \omega^{+}(\tau)f_{r'}\,d\tau \biggr) +\mu_{0}nI_{r'} \\ =&\frac{1}{2}r'+\frac{1}{2}r' \\ =&r'= \Vert u \Vert _{PC}. \end{aligned}$$
Consequently,
$$ \Vert Tu \Vert _{PC}\geq \Vert u \Vert _{PC}, \quad \forall u\in \partial K_{r'}. $$
(3.3)

Therefore, applying Lemma 1.1 to (3.1), (3.2), and (3.3) yields that T has two fixed points \(u_{1}\in \overline{K}_{R}\setminus \overline{K}_{r}\) and \(u_{2}\in K_{r}\setminus K_{r'}\). Thus, if \(\theta_{1}>p-1\) and \(\theta_{2}>1\), there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that problem (1.2) admits two positive solutions for \(\lambda \in [\lambda_{0},+\infty)\) and \(\mu \in [\mu_{0},+\infty)\). The proof of Theorem 3.1 is completed. □

Theorem 3.2

Assume that (\(H_{1}\))(\(H_{6}\)) hold. If \(0<\theta_{1}<p-1\) and \(0<\theta_{2}<1\), there exist \(\lambda^{0}>0\) and \(\mu^{0}>0\) such that problem (1.2) admits two positive solutions for \(\lambda \in (0,\lambda^{0}]\) and \(\mu \in (0,\mu^{0}]\).

Proof

On the one hand, since \(0<\theta_{1}<p-1\) and \(0<\theta _{2}<1\), by (\(H_{5}\)), we get
$$ \lim_{u\rightarrow 0}\frac{f(u)}{\phi_{p}(u)}\geq \lim_{u\rightarrow 0} \frac{k_{1}u^{\theta_{1}}}{u^{p-1}}=+\infty,\qquad \lim_{u\rightarrow 0}\frac{I _{k}(u)}{u} \geq \lim_{u\rightarrow 0}\frac{k_{3}u^{\theta_{2}}}{u}=+ \infty. $$
Hence, there exists \(r_{1}>0\) such that
$$ f(u)> B_{1} \phi_{p}(u),\qquad I_{k}(u)> B_{2}u,\quad u\in [0,r_{1}]. $$
Then we have \(\min \{f(u):\alpha r_{1}\leq u\leq r_{1}\}> B_{1}\phi _{p}(u)\) and \(\min \{I_{k}(u):\alpha r_{1}\leq u\leq r_{1}\}> B_{2}u\).
If \(u\in \partial K_{r_{1}}\), then \(\|u\|_{PC}=r_{1}\) and \(\alpha r _{1}=\min_{\frac{\sigma }{2}\leq t\leq \sigma }\delta (t)\|u\| _{PC}\leq u(t)\leq \|u\|_{PC}=r_{1}\), \(t\in [\frac{\sigma }{2},\sigma ]\). It is easy to see that \(f(u(t))> B_{3}\phi_{p}(u(t))\), \(I_{k}(u(t))> B_{4}u(t)\), \(t\in [\frac{\sigma }{2},\sigma ]\). Then from (2.8), for \(u\in \partial K_{r_{1}}\), similar to (3.2), we have
$$ \Vert Tu \Vert _{PC}> \Vert u \Vert _{PC}, \quad \forall u\in \partial K_{r_{1}}. $$
(3.4)
On the other hand, since \(0<\theta_{1}<p-1\) and \(0<\theta_{2}<1\), by (\(H_{5}\)), we have
$$ \lim_{u\rightarrow +\infty }\frac{f(u)}{\phi_{p}(u)}\leq \lim_{u\rightarrow +\infty } \frac{k_{2}u^{\theta }}{u^{p-1}}=0,\qquad \lim_{u\rightarrow +\infty }\frac{I_{k}(u)}{u}\leq \lim_{u\rightarrow +\infty }\frac{k_{4}u^{\theta }}{u}=0. $$
Furthermore, there exists \(0< r_{1}< R'_{1}<+\infty \) such that
$$ f(u)\leq \frac{A_{1}}{2}\phi_{p}(u),\qquad I_{k}(u)\leq \frac{A_{2}}{2}u, \quad u\in \bigl[R'_{1},+\infty\bigr). $$
Let \(M_{1}=\max \{f(u):0\leq u\leq R'_{1}\}\) and \(M_{2}=\max \{I_{k}:0 \leq u\leq R'_{1},k=1,2,\ldots,n\}\). It implies that
$$ f(u)\leq \frac{A_{1}}{2}\phi_{p}(u)+M_{1},\qquad I_{k}(u)\leq \frac{A_{2}}{2}u+M_{2}, \quad u\in [0,+ \infty). $$
Choose \(R_{1}\geq \{R'_{1},\frac{2\phi_{q}(2\int_{0}^{\xi }\lambda \omega^{+}(\tau)M_{1}\,d\tau)}{1-\eta },4\mu nM_{2}\}\). If \(u\in \partial K_{R_{1}}\), then \(\|u\|=R_{1}\) and \(0\leq u(t)\leq R_{1}\), \(t \in J\). It is easy to see that \(f(u(t))\leq \frac{A_{1}}{2}\phi_{p}(u(t))+M _{1}\), \(I_{k}(u(t))\leq \frac{A_{2}}{2}u(t)+M_{2}\), \(t\in J\). Then from (2.7), for \(u\in \partial K_{R_{1}}\), we have
$$\begin{aligned} \Vert Tu \Vert _{PC} \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I _{k}\bigl(u(t_{k})\bigr) \\ \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau) \biggl(\frac{A_{1}}{2}\phi_{p}\bigl(u(\tau)\bigr)+M_{1} \biggr)\,d\tau \biggr) +\mu \sum_{k=1}^{n} \biggl(\frac{A_{2}}{2}u(t_{k})+M_{2}\biggr) \\ \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau) \frac{A_{1}}{2}\phi_{p}\bigl( \Vert u \Vert _{PC} \bigr)\,d\tau + \int_{0}^{\xi }\lambda \omega^{+}( \tau)M_{1}\,d\tau \biggr) \\ &{} +\mu \sum_{k=1}^{n}\frac{A _{2}}{2} \Vert u \Vert _{PC}+\mu nM_{2} \\ \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \frac{1}{2} \phi_{p}\biggl(\frac{ \Vert u \Vert _{PC}(1-\eta)}{2}\biggr)+\frac{1}{2} \phi_{p}\biggl(\frac{R_{1}(1-\eta)}{2}\biggr) \biggr) +\frac{ \Vert u \Vert _{PC}}{4}+ \frac{R_{1}}{4} \\ =&\frac{1}{2}R_{1}+\frac{1}{2}R_{1} \\ =&R_{1}= \Vert u \Vert _{PC}. \end{aligned}$$
Consequently,
$$ \Vert Tu \Vert _{PC}\leq \Vert u \Vert _{PC},\quad \forall u\in \partial K_{R_{1}}. $$
(3.5)
In addition, choosing a number \(r'_{1}\in (0,r_{1})\), we can define
$$\begin{aligned}& f^{r'_{1}}=\max \bigl\{ f(u):0< u\leq r'_{1}\bigr\} , \qquad I_{k}^{r'_{1}}=\max \bigl\{ I _{k}(u):0< u\leq r'_{1}\bigr\} , \\& I^{r'_{1}}=\max \bigl\{ I_{k}^{r'_{1}}:k=1,2,\ldots,n\bigr\} . \end{aligned}$$
Let \(\lambda^{0}=\frac{1}{\int_{0}^{\xi }\omega^{+}(\tau)f^{r'_{1}}\,d\tau }\phi_{p} ( \frac{r'_{1}(1-\eta)}{2} ) \) and \(\mu^{0}=\frac{r'_{1}}{2nI ^{r'_{1}}}\). It is clear that
$$ \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda^{0} \omega^{+}( \tau)f^{r'_{1}}\,d\tau \biggr) \leq \frac{1}{2}r'_{1}, \qquad \mu^{0}nI^{r'_{1}} \leq \frac{1}{2}r'_{1}. $$
If \(u\in \partial K_{r'_{1}}\), then \(\|u\|_{PC}=r'_{1}\) and \(0\leq u(t)\leq \|u\|_{PC}=r'_{1}\), \(t\in J\). It is clear that \(f(u(t))\leq f^{r'_{1}}\), \(I_{k}(u(t))\leq I^{r'_{1}}\), \(t\in J\). Then from (2.7), for \(u\in \partial K_{r'_{1}}\), we have
$$\begin{aligned} \Vert Tu \Vert _{PC} \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}(\tau)f \bigl(u(\tau)\bigr)\,d\tau \biggr) +\mu \sum_{k=1}^{n}I _{k}\bigl(u(t_{k})\bigr) \\ \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda \omega^{+}( \tau)f^{r'_{1}}\,d\tau \biggr) +\mu \sum_{k=1}^{n}I^{r'_{1}} \\ \leq& \frac{1}{1-\eta }\phi_{q} \biggl( \int_{0}^{\xi }\lambda^{0}\omega ^{+}(\tau)f^{r'_{1}}\,d\tau \biggr) +\mu^{0}nI^{r'_{1}} \\ =&\frac{1}{2}r'_{1}+\frac{1}{2}r'_{1} \\ =&r'_{1}= \Vert u \Vert _{PC}. \end{aligned}$$
Consequently,
$$ \Vert Tu \Vert _{PC}\leq \Vert u \Vert _{PC}, \quad \forall u\in \partial K_{r'_{1}}. $$
(3.6)

Therefore, applying Lemma 1.1 to (3.4), (3.5), and (3.6) yields that T has two fixed points \(u'_{1}\in \overline{K}_{R_{1}}\setminus \overline{K}_{r_{1}}\) and \(u'_{2}\in K_{r_{1}}\setminus K_{r'_{1}}\). Thus, if \(0<\theta_{1}<p-1\) and \(0<\theta_{2}<1\), there exist \(\lambda^{0}>0\) and \(\mu^{0}>0\) such that problem (1.2) admits two positive solutions for \(\lambda \in (0,\lambda^{0}]\) and \(\mu \in (0, \mu^{0}]\). The proof of Theorem 3.2 is finished. □

Remark 3.1

If \(I_{k}=0\) (\(k=1,2,\ldots,n\)), even for the case \(g(t)\equiv 0\) on J, the results of the present paper are still novel.

Remark 3.2

Comparing with Li, Feng, and Qin [60], the main features of this paper are as follows:
  1. (i)

    \(p> 1\) is considered, not only \(p\equiv 2\).

     
  2. (ii)

    \(I_{k}\neq 0\) (\(k=1,2,\ldots,n\)) is considered.

     
  3. (iii)

    The basic space \(PC[0,1]\) is available, not \(C[0,1]\).

     

4 An example

We give an example to illustrate our main conclusions.

Example 4.1

Let \(p=\frac{3}{2}\), \(n=1\), \(t_{1}=\frac{1}{2}\). Consider the following problem:
$$ \textstyle\begin{cases} -(\phi_{p}(u'))'=\lambda \omega (t)(u+\sin u), \quad 0< t< 1, \\ -\Delta u|_{t=t_{1}}=\mu I_{1}(u(t_{1})), \\ \Delta u'|_{t=t_{1}}=0, \\ u'(0)=0, \quad u(1)=\int_{0}^{1}g(t)u(t)\,dt, \end{cases} $$
(4.1)
where
$$ \omega (t)= \textstyle\begin{cases} 12(\frac{2}{3}-t), & t\in [0,\frac{2}{3}], \\ \frac{2}{3}-t, & t\in [\frac{2}{3},1], \end{cases}\displaystyle \qquad I_{1}(u)=u^{2}, \qquad g(t)=t. $$
From the definition of \(\omega (t)\) and \(g(t)\), we know that \(\xi =\frac{1}{2}\) and \(\eta =\int_{0}^{1}t\,dt=\frac{1}{2}\). From \(p=\frac{3}{2}\), we can get that \(q=3\).

Since f is nondecreasing, then \(c=1\). For fixed \(k_{1}=1\), \(k_{2}=2\), \(\theta_{1}=1\), \(k_{3}=k_{4}=1\), \(\theta_{2}=2\), \(\sigma =\frac{1}{4}\), we can prove that (\(H_{5}\)) holds.

In fact,
$$\begin{aligned} \frac{1}{2} \int_{\frac{1}{2}}^{\frac{2}{3}}12\biggl(\frac{2}{3}-\tau \biggr)\,d\tau =&6 \int_{\frac{1}{2}}^{\frac{2}{3}}\biggl(\frac{2}{3}-\tau\biggr)\,d\tau \\ =&6\biggl(\frac{2}{3}\tau -\frac{\tau^{2}}{2}\biggr)\bigg|_{\frac{1}{2}}^{\frac{2}{3}}\,d\tau \\ =&\frac{1}{12}, \end{aligned}$$
and
$$ 2\times \frac{2}{3} \int_{\frac{2}{3}}^{1}\biggl(\tau -\frac{2}{3}\biggr)\,d\tau = \frac{4}{3}\biggl(\frac{\tau^{2}}{2}-\frac{2}{3}\tau \biggr)\bigg|_{\frac{2}{3}}^{1}= \frac{2}{27}. $$
Obviously, \(\frac{1}{12}>\frac{2}{27}\). Thus
$$ \frac{1}{2} \int_{\frac{1}{2}}^{\frac{2}{3}}12\biggl(\frac{2}{3}-\tau \biggr)\,d\tau \geq \frac{4}{3} \int_{\frac{2}{3}}^{1}\biggl(\tau -\frac{2}{3}\biggr)\,d\tau. $$
This shows that (\(H_{6}\)) holds.

Let \(\lambda_{0}=\frac{12}{7}\sqrt{\frac{3}{13}}(\frac{1}{8}+\sin \frac{1}{8})^{-1}\), \(\mu_{0}=16\). Then it follows from Theorem 3.1 that problem (4.1) admits two positive solutions for \(\lambda \in [ \frac{12}{7}\sqrt{\frac{3}{13}}(\frac{1}{8}+\sin \frac{1}{8})^{-1},+ \infty)\), \(\mu \in [16,\infty)\).

Declarations

Acknowledgements

The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.

Availability of data and materials

Not applicable.

Funding

This work is sponsored by the National Natural Science Foundation of China (11301178), the Beijing Natural Science Foundation (1163007), the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (KM201611232017), the key research and cultivation project of the improvement of scientific research level of BISTU (2018ZDPY18/521823903), and the teaching reform project of BISTU (2018JGYB32).

Authors’ contributions

All authors contributed equally and read and approved the final version of the manuscript.

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that there is no conflict of interests regarding the publication of this manuscript. The authors declare that they have no competing interests.

Consent for publication

Not applicable.

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, Beijing, People’s Republic of China

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