Skip to content

Advertisement

  • Research
  • Open Access

A class of second-order nonlocal indefinite impulsive differential systems

Boundary Value Problems20182018:163

https://doi.org/10.1186/s13661-018-1082-z

  • Received: 30 April 2018
  • Accepted: 18 October 2018
  • Published:

Abstract

We consider the second-order nonlocal impulsive differential system
$$ \textstyle\begin{cases} -x''=a(t)xy+\omega (t)f(x), \quad 0< t< 1, t\neq t_{k}, \\ -y''=b(t)x, \quad 0< t< 1, t\neq t_{k}, \\ \Delta x|_{t=t_{k}}=I_{k}(x(t_{k})), \quad k=1,2,\ldots,n, \\ \Delta y|_{t=t_{k}}=J_{k}(y(t_{k})), \quad k=1,2,\ldots,n, \\ x(0)=\int_{0}^{1}h(t)x(t)\,dt, \qquad x'(1)=0, \\ y(0)=\int_{0}^{1}g(t)y(t)\,dt, \qquad y'(1)=0, \end{cases} $$
where the weight functions \(a(t)\), \(b(t)\), and \(\omega (t)\) change sign on \([0,1]\), and \(g(t)\not \equiv 0\) and \(h(t)\not \equiv 0\) on \([0,1]\). By constructing a cone \(K_{1}\times K_{2}\), which is the Cartesian product of two cones in space \(PC[0,1]\), and applying the well-known fixed point theorem of cone expansion and compression in \(K_{1}\times K_{2}\), we obtain conditions for the existence and multiplicity of positive solutions of a nonlocal indefinite impulsive differential system. An example is given to illustrate the main results.

Keywords

  • Indefinite weights
  • Nonlocal impulsive differential system
  • Positive solutions
  • Existence and multiplicity
  • Fixed point technology

1 Introduction

It is generally accepted that the theory and applications of differential equations with impulsive effects are an important area of investigation, since it is far richer than the corresponding theory of differential equations without impulsive effects. Various population models, biological system models, ecology models, biotechnology models, pharmacokinetics models, and optimal control models, which are characterized by the fact that per sudden changing of their state, can be expressed by impulsive differential equations. For an introduction of general theory of impulsive differential equations, we refer the reader to the references [1] and [2], whereas the applications of impulsive differential equations can be found in [35]. Some classical methods have been widely used to study impulsive differential equations: the theory of critical point theory and variational methods [68], fixed point theorems in cones [925], and bifurcation theory [26, 27]. In particular, we would like to mention some results of Lin and Jiang [28] and Feng and Xie [29]. Lin and Jiang [28] considered the following Dirichlet boundary value problem with impulse effects:
$$ \textstyle\begin{cases} -u''(t)=f(t,u(t)), \quad t\in J, t\neq t_{k}, \\ \triangle u'|_{t\neq t_{k}}=-I_{k}(u(t_{k})), \quad k=1,2,\ldots,m, \\ u(0)=u(1)=0, \end{cases} $$
(1.1)
and by means of the fixed point index theory in cones the authors obtained some sufficient conditions for the existence of multiple positive solutions for problem (1.1).
Recently, using fixed point theorems in a cone, Feng and Xie [29] studied the existence of positive solutions for the following problem:
$$ \textstyle\begin{cases} -u''(t)=f(t,u(t)), \quad t\in J, t\neq t_{k}, \\ -\triangle u'|_{t\neq t_{k}}=I_{k}(u(t_{k})), \quad k=1,2,\ldots,n, \\ u(0)=\sum_{i=1}^{m=2}a_{i}u(\xi_{i}), \qquad u(1)=\sum_{i=1}^{m=2}b_{i}u(\xi_{i}). \end{cases} $$
(1.2)
In comparison with numerous results on the impulsive differential equations, it is less known about the impulsive differential systems, even for the nonlocal impulsive differential systems.
Moreover, we see that increasing attention has been paid to the study of boundary value problems with integral boundary conditions; for example, see Liu, Sun, Zhang, and Wu [30], Zhang, Feng, and Ge [31], Zhang and Ge [32], Hao et al. [3335], Yan, Zuo, and Hao [36], Zhang et al. [37, 38], Sun, Liu, and Wu [39], Lin and Zhao [40], and Ahmad, Alsaedi, and Alghamdi [41]. This problem contains two-, three-, and multipoint boundary value problems as particular cases; for instance, see Karakostas and Tsamatos [42], Feng and Ge [43], Jiang, Liu, and Wu [44], Lan [45], Zhang et al. [4650], Feng, Du, and Ge [51], Ahmad and Alsaedi [52], Mao and Zhao [53], Liu, Hao, and Wu [54], and the references therein. Specifically, Boucherif [55] exploited the fixed point theorem in cones to study the following problem:
$$ \textstyle\begin{cases} u''(t)=f(t,u(t)), \quad 0< t< 1, \\ u(0)-cu'(0)=\int_{0}^{1}g_{0}(t)u(t)\,dt, \\ u(1)-du'(1)=\int_{0}^{1}g_{1}(t)u(t)\,dt. \end{cases} $$
(1.3)
The author obtained several excellent results on the existence of positive solutions to problem (1.3).
Feng, Ji, and Ge [56] began to study the following boundary value problem with integral boundary conditions in abstract spaces:
$$ \textstyle\begin{cases} u''(t)+f(t,u(t))=\theta, \quad 0< t< 1, \\ u(0)=\int_{0}^{1}g(t)u(t)\,dt, \qquad u(1)=\theta. \end{cases} $$
(1.4)
Applying the fixed point theory in a cone for strict set contraction operators, the authors investigated the existence, nonexistence, and multiplicity of positive solutions for problem (1.4).
Recently, Kong [57] considered the existence and uniqueness of positive solutions for the second-order singular boundary value problem:
$$ \textstyle\begin{cases} u''(t)+\lambda f(u(t))=0, \quad t\in (0,1), \\ u(0)=\int_{0}^{1}u(s)\,dA(s), \qquad u(1)=\int_{0}^{1}u(s)\,dB(s). \end{cases} $$
(1.5)
The author examined the uniqueness of the solution and its dependence on the parameter λ for problem (1.5) by using the mixed monotone operator theory.
Simultaneously, an indefinite problem has attracted the attention of Ma and Han [58], López-Gómez and Tellini [59], Boscaggin and Zanolin [60], Feltrin and Zanolin [61], Boscaggin et al. [62, 63], Sovrano and Zanolin [64], Bravo and Torres [65], Wang and An [66], and Yao [67]. Ma and Han [58] considered the following boundary value problem:
$$ \textstyle\begin{cases} u''+\lambda a(t)f(u)=0, \quad 0< t< 1, \\ u(0)=u(1)=0, \end{cases} $$
(1.6)
where \(a\in C[0,1]\) may change sign, and λ is a parameter. They proved the existence, multiplicity, and stability of positive solutions for problem (1.6) by applying bifurcation techniques.
Aapplying the shooting method, Sovrano and Zanolin [60] presented a multiplicity result for positive solutions for the Neumann problem
$$ \textstyle\begin{cases} u''+ a(t)f(u)=0, \quad 0< t< 1, \\ u(t)>0, \quad t\in [0,T], \\ u'(0)=u'(T)=0, \end{cases} $$
(1.7)
where the weight function \(a\in C[0,1]\) has indefinite sign.
Recently, Wang and An [66] dealt with the existence and multiplicity of positive solutions for the second-order differential system
$$ \textstyle\begin{cases} -u''= a(t)\varphi u+h(t)f(u), \quad 0< t< 1, \\ -\varphi ''= b(t)u, \quad 0< t< 1, \\ u(0)=u(1)=0, \\ \varphi (0)=\varphi (1)=0, \end{cases} $$
(1.8)
where \(a(t)\), \(b(t)\), and \(g(t)\) are allowed to change sign on \([0,1]\). For the latest results of indefinite problems, please refer to Jiao and Zhang [68], Feltrin and Sovrano [69], and Zhang [70].
For all we know, in the literature there are no papers on multiple positive solutions for analogous indefinite impulsive differential systems with nonlocal boundary value conditions. More precisely, the study of \(a(t)\), \(b(t)\), and \(\omega (t)\) changing sign on \([0,1]\) is still open for the second-order nonlocal impulsive differential system
$$ \textstyle\begin{cases} -x''=a(t)xy+\omega (t)f(x), \quad 0< t< 1, t\neq t_{k}, \\ -y''=b(t)x, \quad 0< t< 1, \\ \Delta x|_{t=t_{k}}=I_{k}(x(t_{k})), \quad k=1,2,\ldots,n, \\ \Delta y|_{t=t_{k}}=J_{k}(y(t_{k})), \quad k=1,2,\ldots,n, \\ x(0)=\int_{0}^{1}h(t)x(t)\,dt, \qquad x'(1)=0, \\ y(0)=\int_{0}^{1}g(t)y(t)\,dt, \qquad y'(1)=0, \end{cases} $$
(1.9)
where \(a(t)\), \(\omega (t)\), \(b(t)\) change sign on \([0,1]\), \(t_{k}\) (\(k=1,2,\ldots,n\); where n is a fixed positive integer) are fixed points such that \(0< t_{1}< t_{2}<\cdots <t_{k}<\cdots <t_{n}<1\), \(\Delta x|_{t=t_{k}}\) denotes the jump of \(x(t)\) at \(t=t_{k}\), that is, \(\Delta x|_{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), where \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) represent the right- and left-hand limits of \(x(t)\) at \(t=t_{k}\), respectively; \(\Delta y|_{t=t_{k}}\) has a similar meaning for \(y(t)\). In addition, a, ω, b, f, \(I_{k}\), and \(J_{k}\) (\(k=1,2,\ldots,n\)) satisfy (\(H_{1}\)) \(a, \omega, b: [0,1]\rightarrow (-\infty,+\infty)\) are continuous, and there exists a constant \(\xi \in (0, 1)\) such that
$$ \textstyle\begin{cases} a(t), \omega (t), b(t)\geq 0, & \forall t\in [0,\xi ], \\ a(t)), \omega (t), b(t)\leq 0, & \forall t\in [\xi,1]. \end{cases} $$
Moreover, \(a(t)\), \(\omega (t)\), \(b(t)\) do not vanish identically on any subinterval of \([0,1]\).
(\(H_{2}\)): 

\(f:[0,+\infty)\rightarrow [0,+\infty)\) is continuous.

(\(H_{3}\)): 

\(I_{k}:[0,+\infty)\rightarrow [0,+\infty)\) is continuous.

(\(H_{4}\)): 

\(J_{k}:[0,+\infty)\rightarrow [0,+\infty)\) is continuous.

(\(H_{5}\)): 
\(h,g\in L^{1}[0,1]\) are nonnegative, and \(\nu,\nu_{1} \in [0,1)\), where
$$ \nu = \int_{0}^{1}g(s)\,ds, \qquad \nu_{1}= \int_{0}^{1}h(s)\,ds. $$
(1.10)
Let \(J=[0,1]\) and \(J'=J\backslash \{t_{1},t_{2},\ldots,t_{n}\}\). The basic space used in this paper is \(PC[0,1]= \{u| u:[0,1]\rightarrow R\mbox{ is continuous at }t\neq t_{k}, u(t_{k}^{-})=u(t_{k}), \mbox{and } u(t_{k}^{+})\mbox{ exists}, k=1,2,\ldots, n \}\). Then \(PC[0,1]\) is a real Banach space with the norm
$$ \Vert u \Vert _{PC_{1}}=\max_{t\in J} \bigl\vert u(t) \bigr\vert . $$
For convenience, consider \(PC_{1}[0,1]= \{x: x\mbox{ is continuous at } t\neq t_{k},x(t_{k}^{-})=x(t_{k}), \mbox{and }x(t_{k}^{+})\ \mbox{exists}, k=1,2,\ldots, n \}\), which is a real Banach space with norm
$$ \Vert x \Vert _{PC_{1}}=\max_{t\in J} \bigl\vert x(t) \bigr\vert , $$
and \(PC_{2}[0,1]= \{y: y\mbox{ is continuous at }t\neq t_{k},x(t_{k} ^{-})=x(t_{k}), \mbox{and }y(t_{k}^{+})\mbox{ exists}, k=1,2,\ldots, n \}\), which is a real Banach space with norm
$$ \Vert y \Vert _{PC_{2}}=\max_{t\in J} \bigl\vert y(t) \bigr\vert . $$
Clearly, \(PC_{1}[0,1]\times PC_{2}[0,1]\) is also a real Banach space with norm
$$ \bigl\Vert (x,y) \bigr\Vert =\max \bigl\{ \Vert x \Vert _{PC_{1}}, \Vert y \Vert _{PC_{2}} \bigr\} . $$

By a positive solution of system (1.9) we mean a pair of functions \((x,y)\) with \(x\in C^{2}(J')\cap PC_{1}[0,1]\) and \(y\in C^{2}(J') \cap PC_{2}[0,1]\) such that \((x,y)\) satisfies system (1.9) and \(x, y\geq 0\), \(t\in J'\), \(x,y\not \equiv 0\).

Remark 1.1

The technique to deal with the impulsive term is completely different from that of [627].

Remark 1.2

When we consider nonlocal differential systems with indefinite weights, another difficulty is to prove \(T: K_{1}\times K _{2}\rightarrow K_{1}\times K_{2}\); for details, see Lemma 2.3.

Remark 1.3

It is not difficult to see that Proposition 2.3 of [67] plays key roles in the proofs of main results of [66] and [67]. However, it is invalid for nonlocal problems; for details, see Corollary 4.1.

Remark 1.4

In comparison with other related indefinite problems [5866], the main features of this paper are as follows.
  1. (i)

    \(I_{k}, J_{k}\neq 0\) (\(k=1,2,\ldots,n\)) are introduced.

     
  2. (ii)

    Nonlocal boundary conditions are introduced.

     
  3. (iii)

    \(K_{1}\times K_{2}\) is the Cartesian product of two cones in the space \(PC[0,1]\).

     
We define \(a^{\pm }(t)\), \(\omega^{\pm }(t)\), and \(b^{\pm }(t)\) as
$$\begin{aligned}& a^{+}(t)=\max \bigl\{ a(t),0\bigr\} , \qquad a^{-}(t)=-\min \bigl\{ a(t),0\bigr\} , \\& \omega^{+}(t)=\max \bigl\{ \omega (t),0\bigr\} ,\qquad \omega^{-}(t)=-\min \bigl\{ \omega (t),0\bigr\} , \\& b^{+}(t)=\max \bigl\{ b(t),0\bigr\} , \qquad b^{-}(t)=-\min \bigl\{ b(t),0\bigr\} , \end{aligned}$$
so that
$$ a(t)=a^{+}(t)-a^{-}(t), \qquad \omega (t)= \omega^{+}(t)-\omega^{-}(t),\qquad b(t)=b^{+}(t)-b^{-}(t), \quad \forall t\in [0,1]. $$

Inspired by the references mentioned, in this paper, we investigate the existence and multiplicity of positive solutions for system (1.9). By constructing a cone \(K_{1}\times K_{2}\), which is the Cartesian product of two cones in the space \(PC[0,1]\), and using the well-known fixed point theorem of cone expansion and compression, we obtain conditions for the existence and multiplicity of positive solutions of system (1.9). We remark that this is probably the first time that the existence and multiplicity of positive solutions of impulsive differential systems with indefinite weight and integral boundary conditions have been studied.

The rest of this paper is organized as follows. In Sect. 2, we give some preliminary results. Section 3 is devoted to state and prove the main results. Finally, an example is given in Sect. 4.

2 Preliminaries

In this section, we give some preliminary results for the convenience of later use and reference. It is clear that system (1.9) is equivalent to the following two boundary value problems:
$$ \textstyle\begin{cases} -y''=b(t)x, \quad 0< t< 1, t\neq t_{k}, \\ \Delta y|_{t=t_{k}}=J_{k}(y(t_{k})), \quad k=1,2,\ldots,n, \\ y(0)=\int_{0}^{1}g(t)y(t)\,dt, \qquad y'(1)=0, \end{cases} $$
(2.1)
and
$$ \textstyle\begin{cases} -x^{\prime\prime }=a(t)xy+\omega (t)f(x), \quad 0< t< 1, t\neq t_{k}, \\ \Delta x|_{t=t_{k}}=I_{k}(x(t_{k})), \quad k=1,2,\ldots,n, \\ x(0)=\int_{0}^{1}h(t)x(t)\,dt, \qquad x'(1)=0. \end{cases} $$
(2.2)

Lemma 2.1

Assume that (\(H_{1}\)), (\(H_{2}\)), (\(H_{4}\)), and (\(H_{5}\)) hold. Then problem (2.1) has a unique solution y, which can be expressed in the form
$$\begin{aligned}& y(t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t_{k})J _{k} \bigl(y(t_{k}) \bigr), \end{aligned}$$
(2.3)
where
$$\begin{aligned}& H(t,s)=G(t,s)+\frac{1}{1-\nu } \int_{0}^{1}G(\tau,s)g(\tau)\,d\tau, \end{aligned}$$
(2.4)
$$\begin{aligned}& H'_{s}(t,s)=G'_{s}(t,s)+ \frac{1}{1-\nu } \int_{0}^{1}G'_{s}(\tau,s)g( \tau)\,d\tau, \end{aligned}$$
(2.5)
$$\begin{aligned}& G(t,s)=\textstyle\begin{cases} t, \quad 0\leq t\leq s\leq 1, \\ s, \quad 0\leq s\leq t\leq 1, \end{cases}\displaystyle \end{aligned}$$
(2.6)
$$\begin{aligned}& G'_{s}(t,s)=\textstyle\begin{cases} 0, \quad 0\leq t\leq s\leq 1, \\ 1, \quad 0\leq s\leq t\leq 1. \end{cases}\displaystyle \end{aligned}$$
(2.7)

Proof

First, suppose that u is a solution of problem (2.1). It is easy to see by integration of problem (2.1) that
$$\begin{aligned}& y'(t)-y'(0)=- \int_{0}^{t}b(s)x(s)\,ds. \end{aligned}$$
(2.8)
Integrating again, we get
$$\begin{aligned}& y(t)=y(0)+y'(0)t- \int_{0}^{t}(t-s)b(s)x(s)\,ds+\sum _{t_{k}< t}J _{k}\bigl(y(t_{k}) \bigr). \end{aligned}$$
(2.9)
Letting \(t=1\) in (2.8), we find
$$\begin{aligned}& y'(0)= \int_{0}^{1}b(s)x(s)\,ds. \end{aligned}$$
(2.10)
Substituting the boundary condition \(y(0)=\int_{0}^{1}g(t)y(t)\,dt\) and (2.10) into (2.9), we obtain
$$\begin{aligned} y(t) =& \int_{0}^{1}g(s)y(s)\,ds + t \int_{0}^{1}b(s)x(s)\,ds- \int_{0}^{t}(t-s)b(s)x(s)\,ds + \sum _{t_{k}< t}J_{k}\bigl(y(t_{k})\bigr) \\ =& \int_{0}^{1}g(s)y(s)\,ds + \int_{0}^{1}G(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}G'_{s}(t,s)J_{k} \bigl(y(t_{k})\bigr), \end{aligned}$$
where
$$\begin{aligned} \int_{0}^{1}g(s)y(s)\,ds =& \int_{0}^{1}g(s) \Biggl[ \int_{0}^{1}g(\tau)y( \tau)\,d\tau + \int_{0}^{1}G(s,\tau)b(\tau)x(\tau)\,d\tau \\ &{} +\sum_{k=1}^{n}G'_{\tau }(s, \tau)J_{k}\bigl(y(t_{k})\bigr) \Biggr]\,ds \\ =& \int_{0}^{1}g(s)\,ds \int_{0}^{1}g(\tau)y(\tau)\,d\tau + \int_{0}^{1}g(s) \Biggl[ \int_{0}^{1}G(s,\tau)b(\tau)x(\tau)\,d\tau \\ &{} +\sum_{k=1}^{n}G'_{\tau }(s, \tau)J_{k}\bigl(y(t_{k})\bigr) \Biggr]\,ds. \end{aligned}$$
Therefore we have
$$ \int_{0}^{1}g(s)y(s)\,ds=\frac{1}{1-\nu } \int_{0}^{1}g(s) \Biggl[ \int_{0} ^{1}G(s,\tau)b(\tau)x(\tau)\,d\tau +\sum _{k=1}^{n}G'_{\tau }(s, \tau)J_{k}\bigl(y(t_{k})\bigr) \Biggr]\,ds $$
and
$$\begin{aligned} y(t) =& \frac{1}{1-\nu } \int_{0}^{1}g(s) \Biggl[ \int_{0}^{1}G(s,\tau)b( \tau)x(\tau)\,d\tau +\sum _{k=1}^{n}G'_{\tau }(s, \tau)J_{k}\bigl(y(t _{k})\bigr) \Biggr]\,ds \\ &{} + \int_{0}^{1}G(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}G'_{s}(t,s)J_{k} \bigl(y(t _{k})\bigr) \\ = &\frac{1}{1-\nu } \int_{0}^{1} \biggl[ \int_{0}^{1}G(\tau,s)g(\tau)\,d\tau \biggr]b(s)y(s) \,ds \\ &{} + \frac{1}{1-\nu } \int_{0}^{1} \Biggl[\sum _{k=1}^{n}G'_{s}(\tau,s)g( \tau)J_{k}\bigl(y(t_{k})\bigr) \Biggr]\,d\tau \\ &{} + \int_{0}^{1}G(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}G'_{s}(t,s)J_{k} \bigl(y(t _{k})\bigr). \end{aligned}$$
Let
$$\begin{aligned}& H(t,s)=G(t,s)+\frac{1}{1-\nu } \int_{0}^{1}G(\tau,s)g(\tau)\,d\tau, \\& H'_{s}(t,s)=G'_{s}(t,s)+ \frac{1}{1-\nu } \int_{0}^{1}G'_{s}(\tau,s)g( \tau)\,d\tau. \end{aligned}$$
Then
$$ y(t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t_{k})J _{k} \bigl(y(t_{k}) \bigr). $$
The proof of sufficiency is complete.
Conversely, let u be a solution of (2.1). Direct differentiation of (2.4) and (2.5) implies, for \(t\neq t_{k}\),
$$ y'(t)= \int_{0}^{1}b(s)x(s)- \int_{0}^{t}b(s)x(s). $$
Evidently,
$$\begin{aligned}& -y''=b(t)x, \\& \Delta y|_{t=t_{k}}=J_{k}\bigl(y(t_{k})\bigr), \quad k=1,2,\ldots,n, \\& y(0)= \int_{0}^{1}g(t)y(t)\,dt, \qquad y'(1)=0. \end{aligned}$$
The lemma is proved. □

Proposition 2.1

Let \(G(t,s)\), \(G'_{s}(t,s)\), \(H(t,s)\), and \(H'_{s}(t,s)\) be given as in Lemma 2.1. If \(\nu \in [0,1)\), then we get
$$\begin{aligned}& G(t,s)>0, \qquad H(t,s)>0,\quad \forall t,s\in (0,1), \end{aligned}$$
(2.11)
$$\begin{aligned}& G(t,t)G(s,s)\leq G(t,s)\leq G(s,s)=s\leq 1,\quad \forall t,s\in J, \end{aligned}$$
(2.12)
$$\begin{aligned}& \rho G(t,t)G(s,s)\leq H(t,s)\leq H(s,s)=\gamma G(s,s)\leq \gamma,\quad \forall t,s\in J, \end{aligned}$$
(2.13)
$$\begin{aligned}& G'_{s}(t,s)\leq 1,\qquad 0\leq H'_{s}(t,s) \leq \gamma,\quad \forall t,s \in J, \end{aligned}$$
(2.14)
where
$$\begin{aligned}& \gamma =\frac{1}{1-\nu }, \qquad \rho =1+\frac{\int_{0}^{1} \tau g(\tau)\,d\tau }{1-\nu }. \end{aligned}$$
(2.15)

Proof

By the definition of \(G(t,s)\) and \(H(t,s)\), relations (2.11) and (2.12) are simple to prove.

Next, we consider (2.13). In fact, from (1.10) and (2.12) we get
$$\begin{aligned} H(t,s) \leq& G(s,s)+\frac{1}{1-\nu } \int_{0}^{1}G(s,s)g(\tau)\,d\tau \\ =&G(s,s) \biggl(1+\frac{1}{1-\nu } \int_{0}^{1}g(\tau)\,d\tau \biggr) \\ =&\frac{1}{1-\nu }G(s,s) \\ \leq& \gamma \end{aligned}$$
and
$$\begin{aligned} H(t,s) \geq& G(t,t)G(s,s)+\frac{1}{1-\nu } \int_{0}^{1}G(s,s)G(\tau, \tau)g(\tau)\,d\tau \\ =&G(s,s) \biggl(G(t,t)+\frac{1}{1-\nu } \int_{0}^{1}G(\tau,\tau)g( \tau)\,d\tau \biggr) \\ \geq& G(s,s) \biggl(G(t,t)+\frac{G(t,t)}{1-\nu } \int_{0}^{1}G(\tau, \tau)g(\tau)\,d\tau \biggr) \\ =&G(s,s)G(t,t) \biggl(1+\frac{G(t,t)}{1-\nu } \int_{0}^{1}\tau g(\tau)\,d\tau \biggr) \\ =&\rho G(t,t)G(s,s). \end{aligned}$$
This shows that (2.13) holds.

Similarly, by the the definition of \(G'_{s}(t,s)\) and \(H'_{s}(t,s)\), we can prove that (2.14) holds. □

Remark 2.1

From (2.5) we can prove that
$$ H'_{s}(t,s)\geq \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau,\quad \forall t,s\in [0, \xi ]. $$

Proof

It follows from (2.5) and (2.7) that
$$\begin{aligned} H'_{s}(t,s) =&\textstyle\begin{cases} \frac{1}{1-\nu }\int_{0}^{1}G'_{s}(\tau,s)g(\tau)\,d\tau, \quad 0\leq t\leq s\leq 1, \\ 1+\frac{1}{1-\nu }\int_{0}^{1}G'_{s}(\tau,s)g(\tau)\,d\tau, \quad 0\leq s\leq t\leq 1 \end{cases}\displaystyle \\ =&\textstyle\begin{cases} \frac{1}{1-\nu } [\int_{0}^{s}G'_{s}(\tau,s)g(\tau)\,d\tau +\int _{s}^{1}G'_{s}(\tau,s)g(\tau)\,d\tau ], \quad 0\leq t\leq s\leq 1, \\ 1+\frac{1}{1-\nu } [\int_{0}^{s}G'_{s}(\tau,s)g(\tau)\,d\tau +\int _{s}^{1}G'_{s}(\tau,s)g(\tau)\,d\tau ], \quad 0\leq s\leq t\leq 1 \end{cases}\displaystyle \\ =&\textstyle\begin{cases} \frac{1}{1-\nu }\int_{s}^{1}g(\tau)\,d\tau, \quad 0\leq t\leq s\leq 1, \\ 1+\frac{1}{1-\nu }\int_{s}^{1}g(\tau)\,d\tau, \quad 0\leq s\leq t\leq 1 \end{cases}\displaystyle \\ \geq& \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau, \quad \forall t,s\in [0, \xi ]. \end{aligned}$$
 □

Lemma 2.2

Assume that (\(H_{1}\))(\(H_{3}\)) and (\(H_{5}\)) hold. Then problem (2.2) has a unique solution x given by
$$\begin{aligned} x(t) =& \int_{0}^{1}H_{1}(t,s)a(s)x(s)y(s)\,ds+ \int_{0}^{1}H_{1}(t,s) \omega (s)f \bigl(x(s) \bigr)\,ds \\ &+\sum_{k=1}^{n}H'_{1s}(t,t_{k})I_{k} \bigl(x(t_{k}) \bigr), \end{aligned}$$
(2.16)
where
$$\begin{aligned}& H_{1}(t,s)=G(t,s)+\frac{1}{1-\nu_{1}} \int_{0}^{1}G(\tau,s)h(\tau)\,d\tau, \end{aligned}$$
(2.17)
$$\begin{aligned}& H'_{1s}(t,s)=G'_{s}(t,s)+ \frac{1}{1-\nu_{1}} \int_{0}^{1}G'_{s}(\tau,s)h( \tau)\,d\tau. \end{aligned}$$
(2.18)

Proof

The proof of Lemma 2.2 is similar to that of Lemma 2.1. □

Proposition 2.2

Let \(H_{1}\) and \(H'_{1s}\) be given as in Lemma 2.1. If \(\nu_{1}\in [0,1)\), then we get
$$\begin{aligned}& \rho_{1} G(t,t)G(s,s)\leq H_{1}(t,s)\leq H_{1}(s,s)=\gamma_{1} G(s,s) \leq \gamma_{1}, \quad \forall t,s\in J, \end{aligned}$$
(2.19)
$$\begin{aligned}& G'_{s}(t,s)\leq 1,\qquad 0\leq H'_{1s}(t,s) \leq \gamma_{1},\quad \forall t,s \in J, \end{aligned}$$
(2.20)
where
$$\begin{aligned}& \gamma_{1}=\frac{1}{1-\nu_{1}}, \qquad \rho_{1}=1+ \frac{\int _{0}^{1}\tau h(\tau)\,d\tau }{1-\nu_{1}}. \end{aligned}$$
(2.21)

Remark 2.2

From (2.18) we can prove that
$$ H'_{1s}(t,s)\geq \frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau, \quad \forall t,s\in [0, \xi ]. $$

Remark 2.3

Let \((x,y)\) be a solution of system (1.9). Then from Lemma 2.1 and Lemma 2.2 we have
$$\begin{aligned}& \begin{aligned}&x(t)= \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x( \tau)\,d\tau \,ds + \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s) \bigr)\,ds \\ &\hphantom{x(t)=}{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H_{1s}'(t,t _{k})I_{k} \bigl(x(t_{k}) \bigr), \\ &y(t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t_{k})J _{k} \bigl(y(t_{k}) \bigr), \end{aligned} \end{aligned}$$
(2.22)
where
$$ H'_{\tau }(s,\tau)=G'_{\tau }(s,\tau)+ \frac{1}{1-\nu } \int_{0}^{1}G'_{ \tau }(\xi, \tau)g(\xi)\,d\xi. $$
To obtain the existence and multiplicity of a positive solution of system (1.9), we make the following hypotheses:
(\(H_{6}\)): 
There exists a constant \(\sigma_{1}\) satisfying \(0<\sigma _{1}<\xi \) such that
$$ \sigma_{1} \int_{\sigma_{1}}^{\xi }H(t,s)b^{+}(s)\,ds\geq \xi \int_{ \xi }^{1}H(t,s)b^{-}(s)\,ds; $$
(\(H_{7}\)): 
There exists a constant \(\sigma_{2}\) satisfying \(0<\sigma _{2}<\xi \) such that
$$ \rho \sigma_{2} \int_{\sigma_{2}}^{\xi }H_{1}(t,s)G(s,s)a^{+}(s) \,ds \geq \gamma \xi \int_{\xi }^{1}H_{1}(t,s)a^{-}(s) \,ds; $$
(\(H_{8}\)): 
There exists a constant μ satisfying \(0<\mu \leq 1\) such that
$$ f(\omega)\geq \mu \varphi (\omega),\quad \omega \in [0,+\infty), $$
where \(\varphi (\omega)=\max \{f(\rho):0\leq \rho \leq \omega \}\);
(\(H_{9}\)): 
There exist constants \(0<\alpha <+\infty \) with \(\alpha \neq 1\) and \(k_{1},k_{2},l_{1},l_{2},m_{1},m_{2}>0\) such that
$$\begin{aligned}& k_{1}x^{\alpha }\leq f(x)\leq k_{2}x^{\alpha }, \qquad l_{1}x^{\alpha } \leq I_{k}(x)\leq l_{2}x^{\alpha }, \\& m_{1}y^{\alpha }\leq J_{k}(y) \leq m_{2}y^{\alpha }, \quad x, y\in [0,+\infty); \end{aligned}$$
(\(H_{10}\)): 
There exists \(0<\sigma_{3}<\xi \) satisfying \(\frac{\sigma _{3}}{2}< t_{1}<\sigma_{3}\) such that
$$ \sigma^{\alpha }_{3}\mu^{2}k_{1} \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)\,ds\geq k_{2}\xi^{\alpha } \int_{\xi }^{1}H_{1}(t,s)\omega ^{-}(s)\,ds. $$

Obviously, \(\varphi:[0,+\infty)\rightarrow [0,+\infty)\) is nondecreasing. Moreover, if f is nondecreasing, then \(f=\varphi \) and \(\mu =1\).

We denote
$$\begin{aligned}& PC_{1}^{+}[0,1]= \biggl\{ x\in PC_{1}[0,1]: \min _{0\leq t\leq 1}x(t)\geq 0\mbox{ and }x(0)= \int_{0}^{1}h(t)x(t)\,dt, x'(1)=0 \biggr\} , \\& K_{1}= \bigl\{ x\in PC_{1}^{+}[0,1]: x \mbox{ is concave on } [0,\xi ],\mbox{ and convex on }[\xi,1] \bigr\} , \\& PC_{2}^{+}[0,1]= \biggl\{ y\in PC_{2}[0,1]: \min _{0\leq t\leq 1}y(t)\geq 0\mbox{ and }y(0)= \int_{0}^{1}g(t)y(t)\,dt, y'(1)=0 \biggr\} , \\& K_{2}= \bigl\{ y\in PC_{2}^{+}[0,1]: y\mbox{ is concave on } [0,\xi ], \mbox{and convex on }[\xi,1] \bigr\} . \end{aligned}$$
If \(x\in K_{1}\), then it is easy to see that \(\|x\|_{PC_{1}}=\max_{0\leq t\leq \xi }|x(t)|\). Similarly, we have \(\|y\|_{PC_{2}}= \max_{0\leq t\leq \xi }|y(t)|\). Also, for a positive number r, we define
$$ \varOmega_{r}= \bigl\{ (x,y)\in K_{1}\times K_{2}, \bigl\Vert (x,y) \bigr\Vert < r \bigr\} , $$
and then we get
$$ \partial \varOmega_{r}= \bigl\{ (x,y)\in K_{1}\times K_{2}, \bigl\Vert (x,y) \bigr\Vert =r \bigr\} . $$
For any \((x,y)\in K_{1}\times K_{2}\), define the mappings \(T_{1}:K _{1}\rightarrow PC_{1}[0,1]\), \(T_{2}:K_{2}\rightarrow PC_{2}[0,1]\), and \(T:K_{1}\times K_{2}\rightarrow PC_{1}[0,1]\times PC_{2}[0,1]\) as follows:
$$\begin{aligned}& \begin{aligned}&(T_{1}x) (t)= \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x( \tau)\,d\tau \,ds \\ &\hphantom{(T_{1}x)(t)=}{}+ \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s) \bigr)\,ds \\ &\hphantom{(T_{1}x)(t)=}{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds \\ &\hphantom{(T_{1}x)(t)=}{}+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k} \bigl(x(t_{k}) \bigr), \\ &(T_{2}y) (t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr), \\ & \bigl(T(x,y) \bigr) (t)= \bigl((T_{1}x) (t),(T_{2}y) (t) \bigr). \end{aligned} \end{aligned}$$
(2.23)

Remark 2.4

It follows from Lemmas 2.12.2 and Remark 2.3 that \((x,y)\) is a solution of system (1.9) if and only if \((x,y)\) is a fixed point of operator T.

Lemma 2.3

Assume that (\(H_{1}\))(\(H_{10}\)) hold. Then \(T(K_{1}\times K_{2})\subset K_{1}\times K_{2}\), and \(T:K_{1}\times K _{2}\rightarrow K_{1}\times K_{2}\) is completely continuous.

Proof

For any \((x,y)\in K_{1}\times K_{2}\), we prove that \(T(x,y)\in K_{1}\times K_{2}\), that is, \(T_{1}x\in K_{1}\) and \(T_{2}y\in K_{2}\). In view of (2.20), we know that
$$ (T_{1}x)^{\prime}(t)= \int_{0}^{1}z(s)\,ds- \int_{0}^{t}z(s)\,ds, $$
where
$$ z(s)=a(s)xy+\omega (s)f \bigl(x(s) \bigr), $$
and then we have \((T_{1}x)^{\prime}(1)=0\). From (2.14) and (2.20) we get
$$\begin{aligned} (T_{1}x) (0) =& \int_{0}^{1} \int_{0}^{1}H_{1}(0,s)H(s,\tau)a(s)b( \tau)x(s)x( \tau)\,d\tau \,ds+ \int_{0}^{1}H_{1}(0,s)\omega (s)f \bigl(x(s)\bigr)\,ds \\ &{} + \int_{0}^{1}H_{1}(0,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H'_{1s}(0,t _{k})I_{k} \bigl(x(t_{k}) \bigr) \\ =&\frac{1}{1-\nu_{1}} \int_{0}^{1}a(s)x(s) \int_{0}^{1}G(\tau,s)h( \tau)\,d\tau \,ds \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \\ &{} +\frac{1}{1-\nu_{1}} \int_{0}^{1}a(s)x(s) \int_{0}^{1}G(\tau,s)h( \tau)\,d\tau \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t _{k}) \bigr) \Biggr)\,ds \\ &{} +\frac{1}{1-\nu_{1}} \int_{0}^{1}\omega (s)f\bigl(x(s)\bigr) \int_{0}^{1}G(\tau,s)h( \tau)\,d\tau \,ds \\ &{} +\frac{1}{1-\nu_{1}}\sum_{k=1}^{n} \int_{0}^{1}G'_{s}(\tau,s)h( \tau)\,d\tau I_{k} \bigl(x(t_{k}) \bigr) \end{aligned}$$
and
$$\begin{aligned} \int_{0}^{1}h(t) (T_{1}x) (t)\,dt =& \int_{0}^{1}h(t) \Biggl[ \int_{0}^{1} \int _{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\ &{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds \\ &{} + \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k} \bigl(x(t_{k}) \bigr) \Biggr]\,dt \\ =& \int_{0}^{1}h(t)\,dt \int_{0}^{1}H_{1}(t,s)a(s)x(s)\,ds \int_{0}^{1}H(s, \tau)b(\tau)x(\tau)\,d\tau \\ &{} + \int_{0}^{1}h(t)\,dt \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1} ^{n}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds \\ &{} + \int_{0}^{1}h(t)\,dt \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds \\ & +\sum_{k=1}^{n} \int_{0}^{1}H'_{1s}(t,t_{k})h(t) \,dtI_{k} \bigl(x(t_{k}) \bigr) \\ =&\frac{1}{1-\nu_{1}} \int_{0}^{1}a(s)x(s) \int_{0}^{1}G(\tau,s)h( \tau)\,d\tau \,ds \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \\ &{} +\frac{1}{1-\nu_{1}} \int_{0}^{1}a(s)x(s) \int_{0}^{1}G(\tau,s)h( \tau)\,d\tau \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t _{k}) \bigr) \Biggr)\,ds \\ &{} +\frac{1}{1-\nu_{1}} \int_{0}^{1}\omega (s)f\bigl(x(s)\bigr) \int_{0}^{1}G(\tau,s)h( \tau)\,d\tau \,ds \\ &{} +\frac{1}{1-\nu_{1}}\sum_{k=1}^{n} \int_{0}^{1}G'_{s}(\tau,s)h( \tau)\,d\tau I_{k} \bigl(x(t_{k}) \bigr) \\ =&(T_{1}x) (0). \end{aligned}$$
Similarly, we have \((T_{2}y)^{\prime}(1)=0\), \((T_{2}y)(0)=\int_{0}^{1}g(t)(T _{2}y)(t)\,dt\).
Define the function \(q: [0,1]\rightarrow [0,1]\) as follows:
  • if \(x(1)=0\), then
    $$ q(t)=\min \biggl\{ \frac{t}{\xi },\frac{1-t}{1-\xi } \biggr\} ,\quad \forall t \in J; $$
  • if \(x(1)>0\), then
    $$ q(t)=\min \biggl\{ \frac{t}{\xi },1 \biggr\} , \quad \forall t\in J. $$
Since \(\sigma_{1}<\xi \), \(\max_{0\leq t\leq 1}q(t)=1\) and \(\min_{\sigma_{i}\leq t\leq \xi }q(t)=\frac{\sigma_{i}}{\xi }\), \(i=1,2,3\).
Let \(x\in K_{1}\). Then x is concave on \([0,\xi ]\) and convex on \([\xi,1]\). Noticing that \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\), we get
$$ x(t)\geq x(\xi)q(t), \quad t\in [0,\xi ]; \qquad x(t)\leq x(\xi)q(t),\quad t\in [\xi,1]. $$
First of all, for any \(x\in K_{1}\), we show that
$$ \int_{0}^{1}H(t,s)b(s)x(s)\,ds\geq \int_{0}^{\sigma_{1}}H(t,s)b^{+}(s)x(s)\,ds, \quad t\in J. $$
Indeed, for \(x\in K_{1}\), we obtain
$$\begin{aligned}& \int_{0}^{1}H(t,s)b(s)x(s)\,ds- \int_{0}^{\sigma_{1}}H(t,s)b^{+}(s)x(s)\,ds \\& \quad = \int_{\sigma_{1}}^{\xi }H(t,s)b^{+}(s)x(s)\,ds- \int_{\xi }^{1}H(t,s)b ^{-}(s)x(s)\,ds \\& \quad \geq \int_{\sigma_{1}}^{\xi }H(t,s)b^{+}(s)q(s)x(\xi)\,ds- \int_{\xi } ^{1}H(t,s)b^{-}(s)q(s)x(\xi)\,ds \\& \quad \geq x(\xi) \biggl[\min_{s\in [\sigma_{1},\xi ]}q(s) \int_{\sigma_{1}}^{\xi }H(t,s)b^{+}(s)\,ds-\max _{s\in [\xi,1]}q(s) \int_{\xi }^{1}H(t,s)b^{-}(s)\,ds \biggr] \\& \quad =x(\xi) \biggl[\frac{\sigma_{1}}{\xi } \int_{\sigma_{1}}^{\xi }H(t,s)b ^{+}(s)\,ds- \int_{\xi }^{1}H(t,s)b^{-}(s)\,ds \biggr]. \end{aligned}$$
Then by (\(H_{6}\)) we have
$$ \int_{0}^{1}H(t,s)b(s)x(s)\,ds\geq \int_{0}^{\sigma_{1}}H(t,s)b^{+}(s)x(s)\,ds. $$
Secondly, for any \(x\in K_{1}\), we prove that
$$\begin{aligned}& \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \quad \geq \int_{0}^{\sigma_{2}} \int_{0}^{1}H_{1}(t,s) H(s,\tau)a ^{+}(s)b(\tau)x(s)x(\tau)\,d\tau \,ds. \end{aligned}$$
For \(t\in J\), since \(\int_{0}^{1}H(t,s)b(s)x(s)\,ds\geq 0\), we have
$$\begin{aligned}& \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \qquad {} - \int_{0}^{\sigma_{2}} \int_{0}^{1}H_{1}(t,s) H(s, \tau)a^{+}(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \quad = \int_{\sigma_{2}}^{\xi } \int_{0}^{1}H_{1}(t,s)H(s, \tau)a^{+}(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \qquad {} - \int_{\xi }^{1} \int_{0}^{1}H_{1}(t,s) H(s, \tau)a^{-}(s)b(\tau)x(s)x( \tau)\,d\tau \,ds \\& \quad \geq \int_{\sigma_{2}}^{\xi }H_{1}(t,s)a^{+}(s)q(s)x( \xi) \int_{0} ^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \,ds \\& \qquad {} - \int_{\xi }^{1}H_{1}(t,s)a^{-}(s)q(s)x( \xi) \int_{0}^{1}H(s,\tau)b( \tau)x(\tau)\,d\tau \,ds \\& \quad \geq \int_{\sigma_{2}}^{\xi }H_{1}(t,s)a^{+}(s) \min_{s\in [\sigma_{2},\xi ]}q(s)x(\xi) \int_{0}^{1}H(s,\tau)b( \tau)x(\tau)\,d\tau \,ds \\& \qquad {} - \int_{\xi }^{1}H_{1}(t,s)a^{-}(s) \max_{s\in [\xi,1]}q(s)x( \xi) \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \,ds \\& \quad \geq x(\xi)\frac{\sigma_{2}}{\xi } \int_{\sigma_{2}}^{\xi }H_{1}(t,s)a ^{+}(s) \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \,ds \\& \qquad {} -x(\xi) \int_{\xi }^{1}H_{1}(t,s)a^{-}(s) \int_{0}^{1}H(s,\tau)b( \tau)x(\tau)\,d\tau \,ds \\& \quad \geq x(\xi)\frac{\sigma_{2}}{\xi } \int_{\sigma_{2}}^{\xi }H_{1}(t,s)a ^{+}(s) \int_{0}^{1}\rho G(s,s)G(\tau,\tau)b(\tau)x(\tau)\,d \tau \,ds \\& \qquad {} -x(\xi) \int_{\xi }^{1}H_{1}(t,s)a^{-}(s) \int_{0}^{1}\gamma G(\tau, \tau)b(\tau)x(\tau)\,d \tau \,ds \\& \quad =x(\xi) \int_{0}^{1}G(\tau,\tau)b(\tau)x(\tau)\,d\tau \biggl[\frac{ \sigma_{2}}{\xi }\rho \int_{\sigma_{2}}^{\xi }H_{1}(t,s)G(s,s)a^{+}(s) \,ds \\& \qquad {}-\gamma \int_{\xi }^{1}H_{1}(t,s)a^{-}(s) \,ds \biggr], \end{aligned}$$
and then it follows from (\(H_{7}\)) that
$$\begin{aligned}& \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \quad \geq \int_{0}^{\sigma_{2}} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a ^{+}(s)b(\tau)x(s)x(\tau)\,d\tau \,ds. \end{aligned}$$
Similarly, for any \(y\in K_{2}\), since \(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})I_{k} (y(t_{k}) )\geq 0\), we get
$$\begin{aligned}& \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds \\& \quad \geq \int_{0}^{\sigma_{2}}H_{1}(t,s)a(s)x(s) \Biggl( \sum_{k=1}^{n} H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds. \end{aligned}$$
Thirdly, for any \(x\in K_{1}\), we prove that
$$ \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds\geq \int_{0}^{\sigma_{3}}H _{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds,\quad t\in J. $$
Since φ is nondecreasing, we also obtain
$$\begin{aligned}& \varphi \bigl(x(t)\bigr)\geq \varphi \bigl(q(t)x(\xi)\bigr),\quad t\in [0,\xi ], \\& \varphi \bigl(x(t)\bigr)\leq \varphi \bigl(q(t)x(\xi)\bigr),\quad t\in [\xi,1]. \end{aligned}$$
Therefore, for any \(x\in K_{1}\), it follows from (\(H_{8}\))–(\(H_{10}\)) that
$$\begin{aligned}& \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds- \int_{0}^{\sigma_{3}}H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \quad = \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds- \int_{ \xi }^{1}H_{1}(t,s) \omega^{-}(s)f\bigl(x(s)\bigr)\,ds \\& \quad \geq \mu \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)\varphi \bigl(x(s)\bigr)\,ds- \int_{\xi }^{1}H_{1}(t,s) \omega^{-}(s)\varphi \bigl(x(s)\bigr)\,ds \\& \quad \geq \mu \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)\varphi \bigl(x( \xi)q(s)\bigr)\,ds- \int_{\xi }^{1}H_{1}(t,s) \omega^{-}(s)\varphi \bigl(x(\xi)q(s)\bigr)\,ds \\& \quad \geq \mu \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(\xi)q(s)\bigr)\,ds-\frac{1}{ \mu } \int_{\xi }^{1}H_{1}(t,s) \omega^{-}(s)f\bigl(x(\xi)q(s)\bigr)\,ds \\& \quad \geq \mu \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)k_{1}x^{ \alpha }(\xi)q^{\alpha }(s) \,ds-\frac{1}{\mu } \int_{\xi }^{1}H_{1}(t,s) \omega^{-}(s) k_{2}x^{\alpha }(\xi)q^{\alpha }(s) \,ds \\& \quad \geq x^{\alpha }(\xi) \biggl[\min_{\sigma_{3}\leq s\leq \xi }q ^{\alpha }(s)\mu k_{1} \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)\,ds- \max_{\xi \leq s\leq 1}q^{\alpha }(s) \frac{1}{\mu }k_{2} \int _{\xi }^{1}H_{1}(t,s) \omega^{-}(s)\,ds \biggr] \\& \quad \geq x^{\alpha }(\xi) \biggl[\frac{\sigma_{3}^{\alpha }}{\xi^{\alpha }} \mu k_{1} \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)\,ds-\frac{1}{ \mu }k_{2} \int_{\xi }^{1}H_{1}(t,s) \omega^{-}(s)\,ds \biggr] \\& \quad \geq 0, \end{aligned}$$
which shows that
$$ \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds\geq \int_{0}^{\sigma_{3}}H _{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds. $$
Thus, for \((x,y)\in K_{1}\times K_{2}\),
$$\begin{aligned}& (T_{1}x) (t)= \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x( \tau)\,d\tau \,ds+ \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(t_{k}) \bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k} \bigl(x(t_{k}) \bigr) \\& \hphantom{(T_{1}x)(t)} \geq \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{0}^{1}H(s, \tau) b(\tau)x(\tau)\,d\tau \,ds \\& \hphantom{(T_{1}x)(t)=}{}+ \int_{0}^{\sigma_{3}}H_{1}(t,s)\omega ^{+}(s)f \bigl(x(t_{k}) \bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1} ^{n}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1} ^{n}H'_{1s}(t,t_{k})I_{k} \bigl(x(t_{k}) \bigr) \\& \hphantom{(T_{1}x)(t)} \geq 0, \\& (T_{2}y) (t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \geq \int_{0}^{\sigma_{1}}H(t,s)b^{+}(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \geq 0. \end{aligned}$$
Moreover, by direct calculation we derive
$$\begin{aligned}& (T_{1}x)''(t)=-a^{+}(t)x(t)y(t)- \omega^{+}(t)f(x)\leq 0,\quad t\in [0, \xi ], \\& (T_{2}x)''(t)=a^{-}(t)x(t)y(t)+ \omega^{-}(t)f(x)\geq 0,\quad t\in [ \xi,1], \\& (T_{1}y)''(t)=-b^{+}(t)x(t)\leq 0, \quad t\in [0,\xi ], \\& (T_{2}y)''(t)=b^{-}(t)x(t)\geq 0, \quad t\in [\xi,1], \end{aligned}$$
which shows that \(T_{1}x\) and \(T_{2}y\) are concave on \([0,\xi ]\) and convex on \([\xi,1]\). It follows that \(T_{1}x\in K_{1}\) and \(T_{2}y\in K_{2}\). Thus \(T(K_{1}\times K_{2})\subset K_{1}\times K _{2}\).

Finally, by standard methods and the Arzelà–Ascoli theorem we can prove that T is completely continuous. □

Remark 2.5

In [66] and [67], it is not difficult to see that the function \(q(t)\) plays an important role in the proof of completely continuous operator. If \(x(0)=x(1)=0\), then we can define \(q(t)= \min \{\frac{t}{\xi },\frac{1-t}{1-\xi } \}\). However, if \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\), then the definition of \(q(t)\) is invalid. This shows that when \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\), we require a special technique to give a fine definition of \(q(t)\).

In fact, a fine definition of \(q(t)\) is very difficult to give when \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\). This is probably the main reason why there is almost no paper studying the existence of positive solutions for the class of second-order nonlocal differential systems with indefinite weights and even for second-order nonlocal impulsive differential systems with indefinite weights.

Remark 2.6

The idea of the proof of Lemma 2.3 comes from Theorem 3.1 of [67].

The following lemma is very crucial in our argument.

Lemma 2.4

(Theorem 2.3.4 of [71], 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 an 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 following two conditions is satisfied:
  1. (a)

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

     
and
  1. (b)

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

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

3 Main results

In this part, applying Lemma 2.4, we obtain the following three existence theorems.

Theorem 3.1

Assume that (\(H_{1}\))(\(H_{10}\)) hold. If \(\alpha >1\), then system (1.9) admits at least one positive solution.

Proof

On one hand, considering the case \(\alpha >1\), by (\(H_{9}\)) we get
$$\begin{aligned}& \lim_{x\rightarrow 0}\frac{f(x)}{x}\leq \lim_{x\rightarrow 0} \frac{k _{2}x^{\alpha }}{x}=0,\qquad \lim_{x\rightarrow 0}\frac{I_{k}(x)}{x} \leq \lim_{x\rightarrow 0}\frac{l_{2}x^{\alpha }}{x}=0, \\& \lim_{y\rightarrow 0}\frac{J_{k}(y)}{y}\leq \lim_{y\rightarrow 0} \frac{m _{2}y^{\alpha }}{y}=0. \end{aligned}$$
Furthermore, there exist \(r',r''>0\) such that
$$\begin{aligned}& f(x)\leq \varepsilon_{1} x, \qquad I_{k}(x)\leq \varepsilon_{2} x, \quad k=1,2, \ldots,n, 0\leq x\leq r', \\& J_{k}(y)\leq \varepsilon_{3} y,\quad k=1,2,\ldots,n, 0\leq y\leq r'', \end{aligned}$$
where \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{3}\) satisfy
$$\begin{aligned}& 4\gamma_{1}\varepsilon_{1} \int_{0}^{\xi }\omega^{+}(s)\,ds< 1,\qquad \frac{4n \varepsilon_{2}}{1-\nu_{1}}< 1, \\& \varepsilon_{3}< \min \biggl\{ \frac{1}{4n\gamma r\int_{0}^{\xi }a^{+}(s)\,ds},\frac{1}{n\gamma } \biggl(1-\gamma \int_{0}^{\xi }b^{+}(s)\,ds \biggr) \biggr\} . \end{aligned}$$
Let
$$\begin{aligned}& \begin{aligned} &A=\gamma_{1} \int_{0}^{\xi } \int_{0}^{\xi }H(s,\tau)a^{+}(s)b^{+}( \tau)\,d\tau \,ds, \\ &A'=\gamma n \int_{0}^{\xi }a^{+}(s)\,ds, \end{aligned} \end{aligned}$$
(3.1)
and choose \(r=\min \{(4A)^{-1},(4\varepsilon_{3}A')^{-1},r',r'' \}\). Then for any \((x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{r}\), we have \(\|(x,y)\|=r\), and by (2.20) we get
$$\begin{aligned}& (T_{1}x) (t)= \int_{0}^{1} \int_{0}^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x( \tau)\,d\tau \,ds+ \int_{0}^{1}H_{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{(T_{1}x)(t)} = \int_{0}^{\xi } \int_{0}^{1}H_{1}(t,s)H(s, \tau)a^{+}(s)b(\tau)x(s)x( \tau)\,d\tau \,ds \\& \hphantom{(T_{1}x)(t)=}{} - \int_{\xi }^{1} \int_{0}^{1}H_{1}(t,s)H(s, \tau)a^{-}(s)b(\tau)x(s)x( \tau)\,d\tau \,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds - \int_{\xi }^{1}H _{1}(t,s) \omega^{-}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{\xi }H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} - \int_{\xi }^{1}H_{1}(t,s)a^{-}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{(T_{1}x)(t)} \leq \int_{0}^{\xi } \int_{0}^{1}H_{1}(t,s)H(s, \tau)a^{+}(s)b(\tau)x(s)x( \tau)\,d\tau \,ds + \int_{0}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{\xi }H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{(T_{1}x)(t)} = \int_{0}^{\xi } \int_{0}^{\xi }H_{1}(t,s)H(s, \tau)a^{+}(s)b^{+}( \tau)x(s)x(\tau)\,d\tau \,ds \\& \hphantom{(T_{1}x)(t)=}{} - \int_{0}^{\xi } \int_{0}^{\xi }H_{1}(t,s)H(s, \tau)a^{+}(s)b^{-}( \tau)x(s)x(\tau)\,d\tau \,ds+ \int_{0}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{\xi }H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{(T_{1}x)(t)} \leq \int_{0}^{\xi } \int_{0}^{\xi }H_{1}(t,s)H(s, \tau)a^{+}(s)b^{+}( \tau)x(s)x(\tau)\,d\tau \,ds+ \int_{0}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{(T_{1}x)(t)=}{} + \int_{0}^{\xi }H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{(T_{1}x)(t)} \leq \gamma_{1} \int_{0}^{\xi } \int_{0}^{\xi }H(s,\tau)a^{+}(s)b^{+}( \tau)x(s)x(\tau)\,d\tau \,ds+\gamma_{1} \int_{0}^{\xi }\omega^{+}(s) \varepsilon_{1} x\,ds \\& \hphantom{(T_{1}x)(t)=}{} +\gamma_{1}\frac{1}{1-\nu }\sum _{k=1}^{n}\varepsilon_{3}y \int _{0}^{\xi }a^{+}(s)x(s)\,ds+ \frac{1}{1-\nu_{1}}\sum_{k=1}^{n} \varepsilon_{2}x \\& \hphantom{(T_{1}x)(t)} \leq A \Vert x \Vert _{PC_{1}}^{2}+ \gamma_{1}\varepsilon_{1} \int_{0}^{\xi } \omega^{+}(s)\,ds \Vert x \Vert _{PC_{1}}+\gamma_{1}\frac{1}{1-\nu }n\varepsilon _{3} \int_{0}^{\xi }a^{+}(s)\,ds \Vert x \Vert _{PC_{1}} \Vert y \Vert _{PC_{2}} \\& \hphantom{(T_{1}x)(t)=}{} +\frac{1}{1-\nu_{1}}n\varepsilon_{2} \Vert x \Vert _{PC_{1}} \\& \hphantom{(T_{1}x)(t)} \leq Ar^{2}+\gamma_{1}\varepsilon_{1}r \int_{0}^{\xi }\omega^{+}(s) \,ds+A' \varepsilon_{3}r^{2}+\frac{1}{1-\nu_{1}}n \varepsilon_{2}r \\& \hphantom{(T_{1}x)(t)} < \frac{1}{4}r+\frac{1}{4}r+\frac{1}{4}r+ \frac{1}{4}r \\& \hphantom{(T_{1}x)(t)} =r, \end{aligned}$$
(3.2)
$$\begin{aligned}& (T_{2}y) (t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \int_{0}^{\xi }H(t,s)b^{+}(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)x(s)\,ds+ \frac{1}{1-\nu }\sum_{k=1}^{n}J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)\,ds \Vert x \Vert _{PC_{1}}+\frac{1}{1-\nu }n \varepsilon_{3} \Vert y \Vert _{PC_{2}} \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)\,ds r+\gamma n \varepsilon_{3}r \\& \hphantom{(T_{2}y)(t)} < r. \end{aligned}$$
(3.3)
Consequently,
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert < \bigl\Vert (x,y) \bigr\Vert ,\quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{r}. \end{aligned}$$
(3.4)
On the other hand, since \(\alpha >1\), it follows from (\(H_{9}\)) that
$$\begin{aligned}& \lim_{x\rightarrow \infty }\frac{f(x)}{x}\geq \lim_{x\rightarrow \infty } \frac{k_{1}x^{\alpha }}{x}=\infty, \\& \lim_{x\rightarrow \infty }\frac{I_{k}(x)}{x} \geq \lim_{x\rightarrow \infty }\frac{l_{1}x^{\alpha }}{x}=\infty, \\& \lim_{y\rightarrow \infty }\frac{J_{k}(y)}{y}\geq \lim_{y\rightarrow \infty } \frac{m_{1}y^{\alpha }}{y}=\infty, \end{aligned}$$
which shows that there exist \(R',R''>0\) such that
$$\begin{aligned}& f(x)\geq \varepsilon_{4} x, \qquad I_{k}(x)\geq \varepsilon_{5} x, \quad k=1,2, \ldots,n, x\geq R', \\& J_{k}(y)\geq \varepsilon_{6} y,\quad k=1,2,\ldots,n, y\geq R'', \end{aligned}$$
where \(\varepsilon_{4}\), \(\varepsilon_{5}\), \(\varepsilon_{6}\) satisfy
$$\begin{aligned}& 3\rho_{1}\frac{\sigma_{2}}{2}\varepsilon_{4}\min _{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G_{1}(s,s) \omega^{+}(s)\,ds>1, \\& 3\frac{1}{1-\nu_{1}}\varepsilon_{5}\min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \int_{ \xi }^{1}h(\tau)\,d\tau >1, \\& \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varepsilon_{6} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}} \delta (t)>1, \end{aligned}$$
where
$$\begin{aligned}& \delta (t)=\min \biggl\{ \frac{t}{\xi },\frac{\xi -t}{\xi } \biggr\} ,\quad t \in [0,\xi ]. \end{aligned}$$
(3.5)
If \((x,y)\in K_{1}\times K_{2}\), then x, y are two nonnegative concave functions on \([0,\xi ]\). So we get
$$\begin{aligned}& \begin{aligned} &x(t)\geq \delta (t) \Vert x \Vert _{PC_{1}}, \quad t\in [0,\xi ], \\ &y(t)\geq \delta (t) \Vert y \Vert _{PC_{2}}, \quad t\in [0,\xi ]. \end{aligned} \end{aligned}$$
(3.6)
It follows that \(\min_{\frac{\sigma_{i}}{2}\leq t\leq \sigma _{i}}x(t)\geq \theta_{i}\|x\|_{PC_{1}}\), \(\min_{\frac{\sigma_{i}}{2}\leq t\leq \sigma_{i}}y(t)\geq \theta _{i}\|y\|_{PC_{2}}\), \(i=1,2,3\), where
$$\begin{aligned}& \theta_{i}=\min_{\frac{\sigma_{i}}{2}\leq t\leq \sigma_{i}} \delta (t)=\min \biggl\{ \frac{\sigma_{i}}{2\xi },1-\frac{\sigma_{i}}{ \xi } \biggr\} >0. \end{aligned}$$
(3.7)
Let
$$\begin{aligned}& B=\rho_{1}\frac{\sigma_{2}}{2}\theta_{1}\theta_{2} \int_{\frac{\sigma_{2}}{2}}^{\sigma_{2}} \int_{\frac{\sigma_{1}}{2}} ^{\sigma_{1}}G_{1}(s,s) H(s, \tau)a^{+}(s)b^{+}(\tau)\,d\tau \,ds>0, \end{aligned}$$
(3.8)
\(R_{1}>\max \{(3B)^{-1},\frac{R^{\prime}}{\theta_{3}},r \}\), \(R_{2}>\max \{\frac{R^{\prime\prime }}{\theta_{3}},r \}\), and \(R=\max \{R_{1},R_{2} \}\). Then for any \((x,y)\in (K_{1} \times K_{2})\cap \partial \varOmega_{R}\), we have
$$\begin{aligned}& R= \bigl\Vert (x,y) \bigr\Vert =\max \bigl\{ \Vert x \Vert _{PC_{1}}, \Vert y \Vert _{PC_{2}} \bigr\} =\max \{R_{1},R_{2} \}, \\& \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}x(t)\geq \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \Vert x \Vert _{PC_{1}}\geq \theta_{3} R_{1}> R^{\prime}, \\& \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}y(t)\geq \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \Vert y \Vert _{PC_{2}}\geq \theta_{3} R_{2}> R^{\prime\prime }, \end{aligned}$$
and
$$\begin{aligned}& \Vert T_{1}x \Vert _{PC_{1}} \\& \quad =\max_{t\in J} \Biggl\{ \int_{0}^{1} \int_{0} ^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds+ \int_{0}^{1}H _{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds \\& \qquad {} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \Biggr\} \\& \quad \geq \max_{t\in J} \Biggl\{ \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \,ds+ \int_{0}^{\sigma_{3}}H _{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \qquad {} + \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1} ^{n}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1} ^{n}H'_{1s}(t,t_{k})I_{k} \bigl(x(t_{k})\bigr) \Biggr\} \\& \quad \geq \min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{2}}{2}}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{\frac{\sigma_{1}}{2}}^{\sigma_{1}}H(s,\tau)b^{+}(\tau)x( \tau) \,d\tau \,ds \\& \qquad {} +\min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{2}}{2}}^{\sigma_{2}}H_{1}(t,s)a(s)x(s) \biggl( \sum_{\frac{\sigma_{3}}{2}< t_{k}< \sigma_{3}}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \biggr)\,ds \\& \qquad {} +\min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds +\min_{\sigma_{3}\leq t\leq \xi }\sum _{\frac{t}{2}< t_{k}< t}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \quad \geq \rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{2}}{2}}^{\sigma _{2}}G_{1}(s,s)a^{+}(s) \delta (s) \Vert x \Vert _{PC_{1}} \int_{\frac{\sigma_{1}}{2}}^{\sigma_{1}}H(s,\tau)b^{+}(\tau)\delta ( \tau) \Vert x \Vert _{PC_{1}}\,d\tau \,ds \\& \qquad {} +\rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G _{1}(s,s) \omega^{+}(s)\varepsilon_{3} x(s)\,ds +\frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \sum _{\frac{\sigma_{3}}{2}< t_{k}< \sigma_{3}}\varepsilon_{4} x(t_{k}) \\& \quad \geq \rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{2}}{2}}^{\sigma _{2}}G_{1}(s,s)a^{+}(s) \delta (s) \Vert x \Vert _{PC_{1}} \int_{\frac{\sigma_{1}}{2}}^{\sigma_{1}}H(s,\tau)b^{+}(\tau)\delta ( \tau) \Vert x \Vert _{PC_{1}}\,d\tau \,ds \\& \qquad {} +\rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G _{1}(s,s) \omega^{+}(s)\varepsilon_{3} x(s)\,ds +\frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \varepsilon_{4} \sum_{\frac{\sigma_{3}}{2}< t_{1}< \sigma_{3}} x(t_{1}) \\& \quad \geq B \Vert x \Vert _{PC_{1}}^{2}+\rho_{1} \frac{\sigma_{2}}{2}\varepsilon_{3} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G_{1}(s,s) \omega^{+}(s)\,ds \Vert x \Vert _{PC_{1}} \\& \qquad {} +\frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \varepsilon_{4} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \Vert x \Vert _{PC_{1}} \\& \quad \geq B \Vert x \Vert _{PC_{1}}^{2}+\rho_{1} \frac{\sigma_{2}}{2}\varepsilon_{3} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G_{1}(s,s) \omega^{+}(s)\,ds \Vert x \Vert _{PC_{1}} \\& \qquad {} +\frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \varepsilon_{4} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \Vert x \Vert _{PC_{1}} \\& \quad >\frac{1}{3} \Vert x \Vert _{PC_{1}}+\frac{1}{3} \Vert x \Vert _{PC_{1}}+\frac{1}{3} \Vert x \Vert _{PC_{1}} \\& \quad = \Vert x \Vert _{PC_{1}}, \end{aligned}$$
(3.9)
$$\begin{aligned}& \Vert T_{2}y \Vert _{PC_{2}}=\max_{t\in J} \Biggl\{ \int_{0}^{1}H(t,s)b(s)x(s)\,ds+ \sum _{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \max_{t\in J} \Biggl\{ \int_{0}^{\sigma_{2}}H(t,s)b^{+}(s)x(s)\,ds+ \sum _{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{0} ^{\sigma_{2}}H(t,s)b^{+}(s)x(s)\,ds+\min _{\sigma_{3}\leq t\leq \xi }\sum_{\frac{t}{2}< t_{k}< t}\sum _{k=1}^{n}H'_{s}(t,t_{k})J _{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varepsilon_{6} \sum_{\frac{\sigma_{3}}{2}< t_{1}< \sigma_{3}} y(t_{1}) \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varepsilon_{6} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}} \delta (t) \Vert y \Vert _{PC_{2}} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} = \Vert y \Vert _{PC_{2}}. \end{aligned}$$
(3.10)
Consequently,
$$ \bigl\Vert T(x,y) \bigr\Vert > \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{R}. $$
(3.11)
Therefore, applying Lemma 2.4 to (3.4) and (3.11), we can show that T has at least one fixed point
$$ (x,y)\in (K_{1}\times K_{2})\cap (\bar{\varOmega }_{R}\setminus \varOmega _{r}). $$
The proof of Theorem 3.1 is completed. □
The following theorem deals with the multiplicity of system (1.9). For convenience, we introduce the following notations:
$$\begin{aligned}& D=3\gamma_{1} \int_{0}^{\xi }\omega^{+}(s)\,ds, \qquad \varLambda =3n\gamma_{1},\qquad \varGamma =\frac{1-\nu }{n} \biggl(1-\gamma \int_{0}^{\xi }b^{+}(s)\,ds \biggr), \\& D^{*}=\rho_{1}\sigma_{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G_{1}(s,s) \omega^{+}(s)\,ds,\qquad \varLambda^{*}=2\gamma_{1} \int_{\xi }^{1}h(\tau)\,d\tau, \\& \varGamma^{*}= \biggl(\frac{1}{1-\nu }n \int_{\xi }^{1}g(\tau)\,d\tau \biggr)^{-1}. \end{aligned}$$

Theorem 3.2

Assume that (\(H_{1}\))(\(H_{10}\)) hold. Suppose that \(0<\alpha <1\) and there exist constants d and r satisfying \(0< d<\min \{(4A)^{-1},(4A'\varGamma)^{-1},r\}\) such that
$$\begin{aligned}& \min_{(x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}}f(x)< D ^{-1}d, \\& \min_{(x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}}I _{k}(x)< \varLambda^{-1} d, \\& \min_{(x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}}J _{k}(y)< \varGamma d, \end{aligned}$$
(3.12)
where A and \(A'\) are defined in (3.1). Then system (1.9) admits at least two positive solutions.

Proof

If \(0<\alpha <1\), then by (\(H_{9}\)) we know that
  1. (i)

    \(\lim_{x\rightarrow 0}\frac{f(x)}{x}\geq \lim_{x\rightarrow 0}\frac{k_{1}x^{\alpha }}{x}=\infty\), \(\lim_{x\rightarrow 0}\frac{I_{k}(x)}{x}\geq \lim_{x\rightarrow 0}\frac{l_{1}x^{\alpha }}{x}=\infty\), \(\lim_{y\rightarrow 0}\frac{J_{k}(y)}{y}\geq \lim_{y\rightarrow 0}\frac{m_{1}y^{\alpha }}{y}=\infty\);

     
  2. (ii)

    \(\lim_{x\rightarrow \infty }\frac{f(x)}{x}\leq \lim_{x\rightarrow \infty }\frac{k_{2}x^{\alpha }}{x}=0\), \(\lim_{x\rightarrow \infty }\frac{I_{k}(x)}{x}\leq \lim_{x\rightarrow \infty }\frac{l_{2}x^{\alpha }}{x}=0\), \(\lim_{y\rightarrow \infty }\frac{J_{k}(y)}{y}\leq \lim_{y\rightarrow \infty }\frac{m_{2}y^{\alpha }}{y}=0\).

     
From (i) it follows that there exists a sufficiently small positive constant r such that
$$\begin{aligned}& f(x)\geq \varepsilon_{7} x,\qquad I_{k}(x)\geq \varepsilon_{8} x,\quad k=1,2, \ldots,n, 0\leq x\leq r, \\& J_{k}(y)\geq \varepsilon_{9} y, \quad k=1,2,\ldots,n, 0 \leq y\leq r, \end{aligned}$$
where \(\varepsilon_{7}\), \(\varepsilon_{8}\), \(\varepsilon_{9}\) satisfy
$$\begin{aligned}& \rho_{1}\sigma_{2}\varepsilon_{7}\min _{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G_{1}(s,s) \omega^{+}(s)\,ds>1, \\& 2\varepsilon_{8}\frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}} \delta (t)>1, \\& \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varepsilon_{9} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}} \delta (t)>1, \end{aligned}$$
with \(\delta (t)\) defined in (3.5).
Therefore, for any \((x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega _{r}\), we get from (3.6) that
$$\begin{aligned}& \Vert T_{1}x \Vert _{PC_{1}}=\max_{t\in J} \Biggl\{ \int_{0}^{1} \int_{0} ^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{}+ \int_{0}^{1}H _{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k} \bigl(x(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \max_{t\in J} \Biggl\{ \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{}+ \int_{0}^{\sigma_{3}}H _{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} + \int_{0}^{\sigma_{2}}H_{1}(t,s)a(s)x(s) \Biggl( \sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \Biggr\} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{2}}{2}}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{\frac{\sigma_{1}}{2}}^{\sigma_{1}}H(s,\tau)b^{+}(\tau)x( \tau) \,d\tau \,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} +\min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds +\min_{0\leq t\leq \xi }\sum _{0< t_{k}< \xi }H'_{1s}(t,t_{k})I _{k}\bigl(x(t_{k})\bigr) \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma _{3}}G_{1}(s,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds +\frac{1}{1-\nu_{1}} \int_{ \xi }^{1}h(\tau)\,d\tau \sum _{0< t_{k}< \xi }I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma _{3}}G_{1}(s,s) \omega^{+}(s)\varepsilon_{7} x(s)\,ds +\frac{1}{1-\nu _{1}} \int_{\xi }^{1}h(\tau)\,d\tau \sum _{\frac{\sigma_{3}}{2}< t_{1}< \sigma_{3}} \varepsilon_{8}x(t_{1}) \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \rho_{1}\frac{\sigma_{2}}{2}\varepsilon_{7} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}\delta (t) \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}G_{1}(s,s) \omega^{+}(s)\,ds \Vert x \Vert _{PC_{1}} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} +\varepsilon_{8}\frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}} \delta (t) \Vert x \Vert _{PC_{1}} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} >\frac{1}{2} \Vert x \Vert _{PC_{1}}+ \frac{1}{2} \Vert x \Vert _{PC_{1}} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} = \Vert x \Vert _{PC_{1}}, \end{aligned}$$
(3.13)
$$\begin{aligned}& \Vert T_{2}y \Vert _{PC_{2}}=\max_{t\in J} \Biggl\{ \int_{0}^{1}H(t,s)b(s)x(s)\,ds+ \sum _{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \max_{t\in J} \Biggl\{ \int_{0}^{\sigma_{2}}H(t,s)b^{+}(s)x(s)\,ds+ \sum _{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{0} ^{\sigma_{2}}H(t,s)b^{+}(s)x(s)\,ds+\min _{\sigma_{3}\leq t\leq \xi }\sum_{\frac{t}{2}< t_{k}< t}\sum _{k=1}^{n}H'_{s}(t,t_{k})J _{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varepsilon_{9} \sum_{\frac{\sigma_{3}}{2}< t_{1}< \sigma_{3}} y(t_{1}) \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varepsilon_{9} \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}} \delta (t) \Vert y \Vert _{PC_{2}} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} = \Vert y \Vert _{PC_{2}}. \end{aligned}$$
(3.14)
Consequently,
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert > \bigl\Vert (x,y) \bigr\Vert , \quad (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{r}. \end{aligned}$$
(3.15)
Next, let us turn to (ii), which shows that there exist \(R',R''>r\) such that
$$\begin{aligned}& f(x)\leq \varepsilon_{1} x,\qquad I_{k}(x)\leq \varepsilon_{2} x, \quad x \geq R', \\& J_{k}(y)\leq \varepsilon_{3} y,\quad y\geq R', \end{aligned}$$
where \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{3}\) satisfy
$$\begin{aligned}& 5\gamma_{1}\varepsilon_{1} \int_{0}^{\xi }\omega^{+}(s)\,ds< 1, \qquad \frac{5n \varepsilon_{2}}{1-\nu_{1}}< 1, \\& \varepsilon_{3}< \max \biggl\{ \frac{1}{6A'R},\frac{1}{\gamma n}- \frac{1}{n} \int_{0}^{\xi }b^{+}(s)\,ds \biggr\} . \end{aligned}$$
Let
$$\begin{aligned}& \eta_{1}=\max_{x\in [0,R']} \bigl\{ f(x) \bigr\} , \qquad \eta_{2}= \max_{x\in [0,R']} \bigl\{ I_{k}(x) \bigr\} , \\& \eta_{3}= \max_{y\in [0,R'']} \bigl\{ J_{k}(y) \bigr\} ,\quad k=1,2,\ldots,n. \end{aligned}$$
Then
$$\begin{aligned}& f(x)\leq \varepsilon_{1} x+\eta_{1}, \qquad I_{k}(x)\leq \varepsilon _{2} x+\eta_{2}, \qquad J_{k}(y)\leq \varepsilon_{3} y+\eta_{3}, \quad \forall x,y\geq 0. \end{aligned}$$
(3.16)
Let
$$ M=\gamma_{1}\eta_{1} \int_{0}^{\xi }\omega^{+}(s)\,ds, \qquad M ^{*}=\frac{1}{1-\nu_{1}}n\eta_{2}. $$
Choosing \(\max \{6M,6M^{*},r\}< R<(6A)^{-1}\), for any \((x,y)\in (K_{1} \times K_{2})\cap \partial \varOmega_{R}\), similarly to the proof of (3.2) and (3.3), we get
$$\begin{aligned}& (T_{1}x) (t)\leq \int_{0}^{\xi } \int_{0}^{\xi }H_{1}(t,s)H(s,\tau)a ^{+}(s)b^{+}(\tau)x(s)x(\tau)\,d\tau \,ds+ \int_{0}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{T_{1}x)(t)\leq}{} + \int_{0}^{\xi }H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{T_{1}x)(t)} \leq \gamma_{1} \int_{0}^{\xi } \int_{0}^{\xi }H(s,\tau)a^{+}(s)b^{+}( \tau) \Vert x \Vert ^{2}_{PC_{1}}\,d\tau \,ds + \gamma_{1} \int_{0}^{\xi }\omega^{+}(s) \bigl( \varepsilon_{1} \Vert x \Vert _{PC_{1}}+\eta_{1} \bigr)\,ds \\& \hphantom{T_{1}x)(t)\leq}{} +\gamma_{1}\frac{1}{1-\nu }n \int_{0}^{\xi }a^{+}(s)\,ds \Vert x \Vert _{PC_{1}}\bigl( \varepsilon_{3} \Vert y \Vert _{PC_{2}}+\eta_{3}\bigr)+\frac{1}{1-\nu_{1}}\sum _{k=1}^{n}\bigl(\varepsilon_{2} \Vert x \Vert _{PC_{1}}+\eta_{2}\bigr) \\& \hphantom{T_{1}x)(t)} \leq AR^{2} +\gamma_{1}\varepsilon_{1} \int_{0}^{\xi }\omega^{+}(s)\,dsR+M+ \varepsilon_{3}A'R^{2}+ \biggl( \gamma_{1}\frac{1}{1-\nu }n\eta_{3} \int _{0}^{\xi }a^{+}(s)\,ds \\& \hphantom{T_{1}x)(t)\leq}{} +\frac{1}{1-\nu_{1}}n\varepsilon_{2} \biggr)R+M^{*} \\& \hphantom{T_{1}x)(t)} < \frac{1}{6}R+\frac{1}{6}R+\frac{1}{6}R+ \frac{1}{6}R+\frac{1}{6}R+ \frac{1}{6}R \\& \hphantom{T_{1}x)(t)} =R, \end{aligned}$$
(3.17)
$$\begin{aligned}& (T_{2}y) (t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \int_{0}^{\xi }H(t,s)b^{+}(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)x(s)\,ds+ \frac{1}{1-\nu }\sum_{k=1}^{n}J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)\,ds \Vert x \Vert _{PC_{1}}+\frac{1}{1-\nu }n \varepsilon_{3} \Vert y \Vert _{PC_{2}} \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)\,ds R+\gamma n \varepsilon_{3}R \\& \hphantom{(T_{2}y)(t)} =R, \end{aligned}$$
(3.18)
which shows that
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert < \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{R}. \end{aligned}$$
(3.19)
Finally, since \(0< d<\min \{(4A)^{-1},(4A'\varGamma)^{-1},r\}\), for \((x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}\), it follows from (3.12) that
$$\begin{aligned}& (T_{1}x) (t)\leq \int_{0}^{\xi } \int_{0}^{\xi }H_{1}(t,s)H(s,\tau)a ^{+}(s)b^{+}(\tau)x(s)x(\tau)\,d\tau \,ds+ \int_{0}^{\xi }H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{(T_{1}x)(t)\leq}{} + \int_{0}^{\xi }H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \\& \hphantom{(T_{1}x)(t)} \leq \gamma_{1} \int_{0}^{\xi } \int_{0}^{\xi }H(s,\tau)a^{+}(s)b^{+}( \tau) \Vert x \Vert ^{2}_{PC_{1}}\,d\tau \,ds + \gamma_{1} \int_{0}^{\xi }\omega^{+}(s)f\bigl(x(s) \bigr)\,ds \\& \hphantom{(T_{1}x)(t)\leq}{} +\gamma_{1}\frac{1}{1-\nu }n \int_{0}^{\xi }a^{+}(s)J_{k} \bigl(y(t_{k}) \bigr)\,ds \Vert x \Vert _{PC_{1}}+ \frac{1}{1-\nu_{1}}\sum_{k=1}^{n}I_{k} \bigl(x(t _{k})\bigr) \\& \hphantom{(T_{1}x)(t)} \leq Ad^{2} +\gamma_{1} \int_{0}^{\xi }\omega^{+}(s) \,ds(D)^{-1}d+A' \varGamma d^{2}+ \frac{1}{1-\nu_{1}}n(\varLambda)^{-1} d \\& \hphantom{(T_{1}x)(t)} < \frac{1}{4}d+\frac{1}{4}d+\frac{1}{4}d+ \frac{1}{4} d \\& \hphantom{(T_{1}x)(t)} =d, \end{aligned}$$
(3.20)
$$\begin{aligned}& (T_{2}y) (t)= \int_{0}^{1}H(t,s)b(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \int_{0}^{\xi }H(t,s)b^{+}(s)x(s)\,ds+\sum _{k=1}^{n}H'_{s}(t,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)x(s)\,ds+ \frac{1}{1-\nu }\sum_{k=1}^{n}J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{(T_{2}y)(t)} \leq \gamma \int_{0}^{\xi }b^{+}(s)\,dsd+ \frac{1}{1-\nu }n\varGamma d \\& \hphantom{(T_{2}y)(t)} =d, \end{aligned}$$
(3.21)
which shows that
$$ \bigl\Vert T(x,y) \bigr\Vert < \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}. $$
(3.22)
Therefore, applying Lemma 2.4 to (3.15), (3.19), and (3.22), we can show that T has at least two fixed points
$$ (x_{1},y_{1})\in (K_{1}\times K_{2})\cap (\bar{\varOmega }_{R}\setminus \varOmega_{r}), \qquad (x_{2},y_{2})\in (K_{1}\times K_{2})\cap (\bar{ \varOmega }_{r}\setminus \varOmega_{d}). $$
The proof of Theorem 3.2 is completed. □

Corollary 3.1

Assume that (\(H_{1}\))(\(H_{10}\)) hold. If \(0<\alpha <1\), then system (1.9) admits at least one positive solution.

Proof

It follows from the proof of Theorem 3.2 that Corollary 3.1 holds. □

Corollary 3.2

Assume that (\(H_{1}\))(\(H_{10}\)) hold. Suppose that \(\alpha >1\) and there exist two constants \(d_{1}\) and r satisfying \(0< d_{1}< r=\min \{(4A)^{-1},(4\varepsilon_{3}A')^{-1},r',r'' \}\) such that
$$\begin{aligned}& \min_{(x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}}f(x)>\bigl(D ^{*}\bigr)^{-1}d_{1}, \\& \min_{(x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}}I _{k}(x)>\bigl(\varLambda^{*} \bigr)^{-1} d_{1}, \\& \min_{(x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}}J _{k}(y)>\varGamma^{*} d_{1}, \end{aligned}$$
where A, \(A'\), \(\varepsilon_{3}\), \(r'\) and \(r''\) are defined in Theorem 3.1. Then system (1.9) admits at least two positive solutions.

Proof

Similarly to the proof of Theorem 3.1, we can obtain that (3.4) and (3.11) hold. Then, similarly to the proof (3.22), we get
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert > \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d_{1}}. \end{aligned}$$
(3.23)
This finishes the proof of Corollary 3.2. □

Finally, in the case \(0<\alpha <1\), we consider the existence of three positive solutions for system (1.9).

Theorem 3.3

Assume that (\(H_{1}\))(\(H_{10}\)) hold and there exist four positive numbers η, \(\eta_{1}\), \(\eta_{2}\), and γ such that one of the following conditions is satisfied:
(\(H_{11}\)): 
\(0<\alpha <1\), \(0<\eta =\max \{\eta_{1}, \eta_{2}\}< \min \{r,(4A)^{-1},(4A'\varGamma)^{-1}\}\leq r\leq \max \{6M,6M^{*},r\} <R<(6A)^{-1}<\gamma \), and
$$\begin{aligned}& f(x)< (D)^{-1}\eta, \qquad I_{k}(x)< (\varLambda)^{-1} \eta, \\& J_{k}(y)< \varGamma \eta, \quad \forall x\in [\theta_{3} \eta_{1}, \eta_{1}], y \in [\theta_{3} \eta_{2}, \eta_{2}], \\& f(x)>\bigl(D^{*}\bigr)^{-1}\gamma, \qquad I_{k}(x)>\bigl(\varLambda^{*}\bigr)^{-1} \gamma, \\& J _{k}(y)>\varGamma^{*} \gamma,\quad \forall x, y\in [0, \gamma ], \end{aligned}$$
where r, R, A, M, \(M^{*}\), D, \(D^{*}\), γ, \(\gamma^{*}\), \(\theta_{3}\), Λ, and \(\varLambda^{*}\) are defined in Theorems 3.1 and 3.2, respectively. Then system (1.9) admits at least three positive solutions.

Proof

Since \(0<\alpha <1\), from the proof of Theorem 3.2 we know that
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert > \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{r}, \end{aligned}$$
(3.24)
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert < \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{R}. \end{aligned}$$
(3.25)
By the first part of (\(H_{11}\)), for any \((x,y)\in (K_{1}\times K_{2}) \cap \partial \varOmega_{\eta }\), we obtain
$$\begin{aligned}& \bigl\Vert (x,y) \bigr\Vert =\max \bigl\{ \Vert x \Vert _{PC_{1}}, \Vert y \Vert _{PC_{2}} \bigr\} =\eta = \max \{\eta_{1}, \eta_{2} \}, \\& \theta_{3}\eta_{1}\leq \theta_{3} \Vert x \Vert _{PC_{1}}\leq \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}x(t)\leq x(t) \leq \eta_{1}, \\& \theta_{3}\eta_{2}\leq \theta_{3} \Vert y \Vert _{PC_{2}}\leq \min_{\frac{\sigma_{3}}{2}\leq t\leq \sigma_{3}}y(t)\leq x(t) \leq \eta_{2}, \end{aligned}$$
and similarly to the proof of (3.22), we get
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert < \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{\eta }. \end{aligned}$$
(3.26)
Considering the second part of (\(H_{11}\)), for any \((x,y)\in (K_{1} \times K_{2})\cap \partial \varOmega_{\gamma }\), we have
$$\begin{aligned}& \Vert T_{1}x \Vert _{PC_{1}}=\max_{t\in J} \Biggl\{ \int_{0}^{1} \int_{0} ^{1}H_{1}(t,s)H(s,\tau)a(s)b( \tau)x(s)x(\tau)\,d\tau \,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{}+ \int_{0}^{1}H _{1}(t,s)\omega (s)f \bigl(x(s)\bigr)\,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} + \int_{0}^{1}H_{1}(t,s)a(s)x(s) \Biggl(\sum _{k=1}^{n}H'_{\tau }(s,t _{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+ \sum_{k=1}^{n}H'_{1s}(t,t _{k})I_{k}\bigl(x(t_{k})\bigr) \Biggr\} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \max_{t\in J} \Biggl\{ \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{0}^{1}H(s,\tau)b(\tau)x(\tau)\,d\tau \,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{}+ \int_{0}^{\sigma_{3}}H _{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} + \int_{0}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \Biggl(\sum_{k=1} ^{n}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr)\,ds+\sum_{k=1} ^{n}H'_{1s}(t,t_{k})I_{k} \bigl(x(t_{k})\bigr) \Biggr\} \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{2}}{2}}^{\sigma_{2}}H_{1}(t,s)a^{+}(s)x(s) \int_{\frac{\sigma_{1}}{2}}^{\sigma_{1}}H(s,\tau)b^{+}(\tau)x( \tau) \,d\tau \,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} +\min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{2}}{2}}^{\sigma_{2}}H_{1}(t,s)a(s)x(s) \biggl( \sum_{\frac{\sigma_{3}}{2}< t_{k}< \sigma_{3}}H'_{\tau }(s,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \biggr)\,ds \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}=}{} +\min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds +\min_{0\leq t\leq \xi }\sum _{0< t_{k}< \xi }H'_{1s}(t,t_{k})I _{k}\bigl(x(t_{k})\bigr) \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \min_{\frac{\sigma_{2}}{2}\leq t\leq \sigma_{2}} \int_{\frac{\sigma_{3}}{2}}^{\sigma_{3}}H_{1}(t,s) \omega^{+}(s)f\bigl(x(s)\bigr)\,ds +\min_{0\leq t\leq \xi }\sum _{0< t_{k}< \xi }H'_{1s}(t,t_{k})I _{k}\bigl(x(t_{k})\bigr) \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma _{3}}G_{1}(s,s) \omega^{+}(s)f(x)\,ds +\frac{1}{1-\nu_{1}} \int_{\xi } ^{1}h(\tau)\,d\tau \sum _{\frac{\sigma_{3}}{2}< t_{1}< \sigma_{3}}I_{k}(x) \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} \geq \rho_{1}\frac{\sigma_{2}}{2} \int_{\frac{\sigma_{3}}{2}}^{\sigma _{3}}G_{1}(s,s) \omega^{+}(s)\,ds\bigl(D^{*}\bigr)^{-1}\gamma + \frac{1}{1-\nu_{1}} \int_{\xi }^{1}h(\tau)\,d\tau \bigl( \varLambda^{*}\bigr)^{-1}\gamma \\& \hphantom{ \Vert T_{1}x \Vert _{PC_{1}}} >\frac{1}{2}\gamma +\frac{1}{2}\gamma =\gamma, \end{aligned}$$
(3.27)
$$\begin{aligned}& \Vert T_{2}y \Vert _{PC_{2}}=\max_{t\in J} \Biggl\{ \int_{0}^{1}H(t,s)b(s)x(s)\,ds+ \sum _{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \max_{t\in J} \Biggl\{ \int_{0}^{\sigma_{2}}H(t,s)b^{+}(s)x(s)\,ds+ \sum _{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \Biggr\} \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \min_{t\in [0,\xi ]} \int_{0}^{\sigma_{2}}H(t,s)b^{+}(s)x(s)\,ds+ \min _{t\in [0,\xi ]}\sum_{k=1}^{n}H'_{s}(t,t_{k})J_{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq \min_{t\in [0,\xi ]}\sum _{k=1}^{n}H'_{s}(t,t_{k})J _{k} \bigl(y(t_{k}) \bigr) \\& \hphantom{ \Vert T_{2}y \Vert _{PC_{2}}} \geq n\frac{1}{1-\nu } \int_{\xi }^{1}g(\tau)\,d\tau \varGamma^{*} \gamma =\gamma. \end{aligned}$$
(3.28)
This shows that
$$\begin{aligned}& \bigl\Vert T(x,y) \bigr\Vert > \bigl\Vert (x,y) \bigr\Vert , \quad \forall (x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{\gamma }. \end{aligned}$$
(3.29)
Therefore, applying Lemma 2.4 to (3.24), (3.25), (3.26), and (3.29) respectively, we can show that T has at least three fixed points \((x_{i},y_{i})\) (\(i=1,2,3\)) satisfying
$$\begin{aligned}& (x_{1},y_{1})\in (K_{1}\times K_{2})\cap (\bar{\varOmega }_{\gamma } \setminus \varOmega_{R}), \\& (x_{2},y_{2})\in (K_{1}\times K_{2})\cap (\bar{\varOmega }_{R}\setminus \varOmega_{r}), \\& (x_{3},y_{3})\in (K_{1}\times K_{2}) \cap (\bar{\varOmega }_{r}\setminus \varOmega_{\eta }). \end{aligned}$$
This gives the proof of Theorem 3.3. □

4 An example

Example 4.1

Let \(n=1\) and \(t_{1}=\frac{1}{10}\). Consider the following system:
$$\begin{aligned}& \textstyle\begin{cases} -x''=a(t)xy+\omega (t)x^{3}, \quad 0< t< 1, t\neq \frac{1}{10}, \\ -y''=b(t)x, \quad 0< t< 1, t\neq \frac{1}{10}, \\ \Delta x|_{t=\frac{1}{10}}=I_{1}(x(\frac{1}{10})), \\ \Delta y|_{t=\frac{1}{10}}=J_{1}(y(\frac{1}{10})), \\ x(0)=\int_{0}^{1}x(t)\,dt, \qquad x'(1)=0, \\ y(0)=\int_{0}^{1}ty(t)\,dt, \qquad y'(1)=0, \end{cases}\displaystyle \end{aligned}$$
(4.1)
where \(I_{1}(x)=\frac{x^{3}}{2}\), \(J_{1}(y)=\frac{y^{3}}{4}\), \(h(t)\equiv 1\), \(g(t)=t\), and
$$\begin{aligned}& b(t)=\textstyle\begin{cases} 48(\frac{1}{3}-t), \quad t\in [0,\frac{1}{3}], \\ -\frac{1}{8}(t-\frac{1}{3}), \quad t\in [\frac{1}{3},1], \end{cases}\displaystyle \\& a(t)=\textstyle\begin{cases} \frac{1728}{5}(\frac{1}{3}-t), \quad t\in [0,\frac{1}{3}], \\ -\frac{1}{8}(t-\frac{1}{3}), \quad t\in [\frac{1}{3},1], \end{cases}\displaystyle \\& g(t)=\textstyle\begin{cases} 192(\frac{1}{3}-t), \quad t\in [0,\frac{1}{3}], \\ -\frac{1}{8}(t-\frac{1}{3}), \quad t\in [\frac{1}{3},1]. \end{cases}\displaystyle \end{aligned}$$

Conclusion 4.1

System (4.1) admits at least one positive solution.

For convenience, we give a corollary of Proposition 2.3 in [67].

Corollary 4.1

Consider the following system:
$$\begin{aligned}& \textstyle\begin{cases} -x''=k(t)x^{\alpha }, \quad 0< t< 1, \\ x(0)=\int_{0}^{1}h(t)x(t)\,dt, \qquad x'(1)=0, \end{cases}\displaystyle \end{aligned}$$
(4.2)
$$\begin{aligned}& \textstyle\begin{cases} -y''=k(t)y^{\alpha }, \quad 0< t< 1, \\ y(0)=\int_{0}^{1}g(t)y(t)\,dt, \qquad y'(1)=0, \end{cases}\displaystyle \end{aligned}$$
(4.3)
where \(\alpha >0\), and \(k(t)\) satisfies the changing-sign condition
$$ \textstyle\begin{cases} k(t)\geq 0, \quad t\in [0,\xi ], \\ k(t)\leq 0, \quad t\in [\xi,1], \end{cases} $$
and
$$ c_{1}x^{\alpha }\leq f(x)=x^{\alpha }\leq c_{2}x^{\alpha }, \qquad c_{1}y ^{\alpha }\leq f(y)=y^{\alpha }\leq c_{2}y^{\alpha }, \quad c_{1},c_{2}>0. $$
If there exists \(0<\sigma <\xi \) such that
$$\begin{aligned}& c_{1}\frac{\xi -\sigma }{1-\xi }\sigma^{\alpha +1}\mu^{2}k^{+} \biggl( \xi -\frac{\xi -\sigma }{1-\xi }\eta \biggr)\geq c_{2} \xi^{\alpha }k ^{-}(\xi +\eta), \quad \eta \in [0,1-\xi ], \end{aligned}$$
(4.4)
then the following inequalities hold:
$$\begin{aligned}& \sigma^{\alpha }\mu^{2} \int_{\sigma }^{\xi }H(t,s)k^{+}(s)\,ds\geq \frac{c _{2}}{c_{1}}\xi^{\alpha } \int_{\xi }^{1}H(t,s)k^{-}(s) \,ds, \end{aligned}$$
(4.5)
$$\begin{aligned}& \sigma^{\alpha }\mu^{2} \int_{\sigma }^{\xi }H_{1}(t,s)k^{+}(s) \,ds \geq \frac{c_{2}}{c_{1}}\xi^{\alpha } \int_{\xi }^{1}H_{1}(t,s)k^{-}(s) \,ds. \end{aligned}$$
(4.6)

Proof

Similarly to the proof of Proposition 2.3 in [67], we can prove that
$$ G \biggl(t,\xi -\frac{\xi -\sigma }{1-\xi }\eta \biggr)\geq \sigma G(t, \xi +\eta), \quad \eta \in [0,1-\xi ]. $$
Hence it follows from (2.4) that
$$\begin{aligned} H \biggl(t,\xi -\frac{\xi -\sigma }{1-\xi }\eta \biggr) =&G \biggl(t,\xi - \frac{ \xi -\sigma }{1-\xi }\eta \biggr)+\frac{1}{1-\nu } \int_{0}^{1}G \biggl( \tau,\xi - \frac{\xi -\sigma }{1-\xi } \eta \biggr)g(\tau)\,d\tau \\ \geq& \sigma G(t,\xi +\eta)+\frac{\sigma }{1-\nu '} \int_{0}^{1}G( \tau,\xi +\eta)g(\tau)\,d\tau \\ =&\sigma \biggl[G(t,\xi +\eta)+\frac{1}{1-\nu } \int_{0}^{1}G(\tau, \xi +\eta)g(\tau)\,d\tau \biggr] \\ =&\sigma H(t,\xi +\eta), \quad \eta \in [0,1-\xi ]. \end{aligned}$$
Next, letting \(s=\xi -\frac{\xi -\sigma }{1-\xi }\eta \), \(\eta \in [0,1- \xi ]\), we get
$$ \int_{\sigma }^{\xi }H(t,s)k^{+}(s)\,ds= \frac{\xi -\sigma }{1-\xi } \int _{0}^{1-\xi }H \biggl(t,\xi -\frac{\xi -\sigma }{1-\xi }\eta \biggr)k^{+} \biggl(\xi -\frac{\xi -\sigma }{1-\xi }\eta \biggr)\,d\eta; $$
letting \(s=\xi +\eta \), \(\eta \in [0,1-\xi ]\), we have
$$ \int_{\xi }^{1}H(t,s)k^{-}(s)\,ds= \int_{0}^{1-\xi }H(t,\xi +\eta)k^{-}( \xi + \eta)\,d\eta. $$
Now, from assumption (4.4), for all \((t,\eta)\in [0,1]\times [0,1- \xi ]\), we have
$$\begin{aligned}& c_{1}\frac{\xi -\sigma }{1-\xi }\sigma^{\alpha }\mu^{2}H \biggl(t,\xi -\frac{ \xi -\sigma }{1-\xi }\eta \biggr)k^{+} \biggl(\xi - \frac{\xi -\sigma }{1- \xi }\eta \biggr)\geq c_{2}\xi^{\alpha }H(t,\xi + \eta)k^{-}(\xi + \eta). \end{aligned}$$
(4.7)

Finally, by integrating in η both sides of (4.7) from 0 to \(1-\xi \) it follows that inequality (4.5) holds. □

Similarly, we can show that inequality (4.6) holds.

Proof of Example 4.1

From the definitions of \(a(t)\), \(b(t)\), and \(g(t)\) we know that \(\xi =\frac{1}{3}\).

Step 1. We show that (\(H_{6}\)) holds. For fixed \(c_{1}=c_{2}=1\), \(\sigma_{1}=\frac{1}{6}\), \(\mu =1\), and \(\alpha =1\), (4.4) is equivalent to the inequality
$$ \frac{1}{48}b^{+} \biggl(\frac{1}{3}-\frac{1}{4} \eta \biggr)\geq b^{-} \biggl(\frac{1}{3}+\eta \biggr), \quad \eta \in \biggl[0,\frac{2}{3} \biggr]. $$
Letting \(\frac{1}{3}-\frac{1}{4}\tau =\varrho \), this inequality is equivalent to
$$ \frac{1}{48}b^{+}(\varrho)\geq b^{-} \biggl( \frac{5}{3}-4\varrho \biggr),\quad \varrho \in \biggl[\frac{1}{4}, \frac{1}{3} \biggr]. $$
By the definition of \(b(t)\) the last inequality holds obviously. It is clear that by (4.5) (\(H_{6}\)) is reasonable.
Step 2. We show (\(H_{7}\)) holds. Similarly to Step 1, letting \(c_{1}=1\), \(c_{2}=\frac{36}{5}\), \(\sigma_{2}=\frac{1}{6}\), \(\mu =1\), and \(\alpha =1\), by (4.6) we get
$$\begin{aligned}& \frac{1}{6} \int_{\frac{1}{6}}^{\frac{1}{3}}H_{1}(t,s)a^{+}(s) \,ds\geq \frac{12}{5} \int_{\frac{1}{3}}^{1}H_{1}(t,s)a^{-}(s) \,ds. \end{aligned}$$
(4.8)
It is easy to see by calculating that
$$\begin{aligned}& \nu = \int_{0}^{1}g(s)\,ds= \int_{0}^{1}s\,ds=\frac{1}{2}, \\& \gamma =\frac{1}{1-\nu }=2, \qquad \rho =1+\frac{\int_{0}^{1}\tau g( \tau)\,d\tau }{1-\nu }= \frac{5}{3}. \end{aligned}$$
Furthermore, from inequality (4.8) it follows that
$$\begin{aligned}& \frac{1}{6}\cdot \frac{5}{3} \int_{\frac{1}{6}}^{\frac{1}{3}}H_{1}(t,s) \frac{1}{6}a^{+}(s)\,ds \geq 2\cdot \frac{1}{3} \int_{\frac{1}{3}}^{1}H _{1}(t,s)a^{-}(s) \,ds \\& \quad \Leftrightarrow\quad \frac{1}{6}\cdot \frac{5}{3} \int_{\frac{1}{6}}^{ \frac{1}{3}}H_{1}(t,s)\min _{s\in [\frac{1}{6},\frac{1}{3}]}G(s,s)a ^{+}(s)\,ds \geq 2\cdot \frac{1}{3} \int_{\frac{1}{3}}^{1}H_{1}(t,s)a ^{-}(s) \,ds \\& \quad \Rightarrow\quad \frac{1}{6}\cdot \frac{5}{3} \int_{\frac{1}{6}}^{ \frac{1}{3}}H_{1}(t,s)G(s,s)a^{+}(s) \,ds \geq 2\cdot \frac{1}{3} \int_{\frac{1}{3}}^{1}H_{1}(t,s)a^{-}(s) \,ds. \end{aligned}$$
So, (\(H_{7}\)) holds.

Step 3. Similarly to Step 1, letting \(c_{1}=c_{2}=1\), \(\sigma_{3}=\frac{1}{6}\), \(\mu =1\), and \(\alpha =3\), we get that (\(H_{10}\)) holds.

Hence it follows from Theorem 3.1 that system (4.1) admits at least one positive solution for \(\alpha >1\). □

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) and the Beijing Natural Science Foundation (1163007).

Authors’ contributions

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

Ethics approval and consent to participate

Not applicable.

Competing interests

The authors declare that there is no conflict of interest 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)
Department of Mathematics and Physics, North China Electric Power University, Beijing, People’s Republic of China

References

  1. Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) View ArticleGoogle Scholar
  2. Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi, New York (2006) View ArticleGoogle Scholar
  3. Pasquero, S.: Ideality criterion for unilateral constraints in time-dependent impulsive mechanics. J. Math. Phys. 46, 1–83 (2005) MathSciNetView ArticleGoogle Scholar
  4. Liu, X., Willms, A.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277–299 (1996) View ArticleGoogle Scholar
  5. Guo, Y.: Globally robust stability analysis for stochastic Cohen–Grossberg neural networks with impulse control and time-varying delays. Ukr. Math. J. 69, 1049–1060 (2017) MathSciNetGoogle Scholar
  6. Nieto, J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680–690 (2009) MathSciNetView ArticleGoogle Scholar
  7. Tian, Y., Ge, W.: Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 72, 277–287 (2010) MathSciNetView ArticleGoogle Scholar
  8. Zhou, J., Li, Y.: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 71, 2856–2865 (2009) MathSciNetView ArticleGoogle Scholar
  9. Wang, M., Feng, M.: New Green’s function and two infinite families of positive solutions for a second order impulsive singular parametric equation. Adv. Differ. Equ. 2017, 154 (2017) MathSciNetView ArticleGoogle Scholar
  10. Zhang, H., Liu, L., Wu, Y.: Positive solutions for nth-order nonlinear impulsive singular integro-differential equations on infinite intervals in Banach spaces. Nonlinear Anal. 70, 772–787 (2009) MathSciNetView ArticleGoogle Scholar
  11. Hao, X., Liu, L., Wu, Y.: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 16, 101–111 (2011) MathSciNetView ArticleGoogle Scholar
  12. Jiang, J., Liu, L., Wu, Y.: Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions. Adv. Differ. Equ. 2012, 124 (2012) MathSciNetView ArticleGoogle Scholar
  13. Zhang, X., Feng, M., Ge, W.: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 233, 1915–1926 (2010) MathSciNetView ArticleGoogle Scholar
  14. Li, P., Feng, M., Wang, M.: A class of singular n-dimensional impulsive Neumann systems. Adv. Differ. Equ. 2018, 100 (2018) MathSciNetView ArticleGoogle Scholar
  15. Feng, M., Pang, H.: A class of three-point boundary-value problems for secondorder impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 70, 64–82 (2009) MathSciNetView ArticleGoogle Scholar
  16. Wang, M., Feng, M.: Infinitely many singularities and denumerably many positive solutions for a second-order impulsive Neumann boundary value problem. Bound. Value Probl. 2017, 50 (2017) MathSciNetView ArticleGoogle Scholar
  17. Zhang, X., Ge, W.: Impulsive boundary value problems involving the one-dimensional p-Laplacian. Nonlinear Anal. 70, 1692–1701 (2009) MathSciNetView ArticleGoogle Scholar
  18. Hao, X., Liu, L.: Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces. Math. Methods Appl. Sci. 40, 4832–4841 (2017) MathSciNetMATHGoogle Scholar
  19. Bai, Z., Dong, X., Yin, C.: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016, 63 (2016) MathSciNetView ArticleGoogle Scholar
  20. Zhang, X., Yang, X., Ge, W.: Positive solutions of nth-order impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 71, 5930–5945 (2009) MathSciNetView ArticleGoogle Scholar
  21. Hao, X., Zuo, M., Liu, L.: Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities. Appl. Math. Lett. 82, 24–31 (2018) MathSciNetView ArticleGoogle Scholar
  22. Tian, Y., Bai, Z.: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59, 2601–2609 (2010) MathSciNetView ArticleGoogle Scholar
  23. Zhang, X., Feng, M.: Transformation techniques and fixed point theories to establish the positive solutions of second order impulsive differential equations. J. Comput. Appl. Math. 271, 117–129 (2014) MathSciNetView ArticleGoogle Scholar
  24. Zuo, M., Hao, X., Liu, L., Cui, Y.: Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 2017, 161 (2017) MathSciNetView ArticleGoogle Scholar
  25. Liu, J., Zhao, Z.: Multiple solutions for impulsive problems with non-autonomous perturbations. Appl. Math. Lett. 64, 143–149 (2017) MathSciNetView ArticleGoogle Scholar
  26. Liu, Y., O’Regan, D.: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 1769–1775 (2011) MathSciNetView ArticleGoogle Scholar
  27. Ma, R., Yang, B., Wang, Z.: Positive periodic solutions of first-order delay differential equations with impulses. Appl. Math. Comput. 219, 6074–6083 (2013) MathSciNetMATHGoogle Scholar
  28. Lin, X., Jiang, D.: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501–514 (2006) MathSciNetView ArticleGoogle Scholar
  29. Feng, M., Xie, D.: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math. 223, 438–448 (2009) MathSciNetView ArticleGoogle Scholar
  30. Liu, L., Sun, F., Zhang, X., Wu, Y.: Bifurcation analysis for a singular differential system with two parameters via to degree theory. Nonlinear Anal., Model. Control 22, 31–50 (2017) MathSciNetGoogle Scholar
  31. Zhang, X., Feng, M., Ge, W.: Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 353, 311–319 (2009) MathSciNetView ArticleGoogle Scholar
  32. Zhang, X., Ge, W.: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 3553–3564 (2012) MathSciNetMATHGoogle Scholar
  33. Hao, X., Sun, H., Liu, L.: Existence results for fractional integral boundary value problem involving fractional derivatives on an infinite interval. Math. Methods Appl. Sci., 1–13 (2018) Google Scholar
  34. Hao, X., Wang, H.: Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions. Open Math. 16, 581–596 (2018) MathSciNetView ArticleGoogle Scholar
  35. Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, 182 (2017) MathSciNetView ArticleGoogle Scholar
  36. Yan, F., Zuo, M., Hao, X.: Positive solution for a fractional singular boundary value problem with p-Laplacian operator. Bound. Value Probl. 2018, 51 (2018) MathSciNetView ArticleGoogle Scholar
  37. Zhang, X., Mao, C., Liu, L., Wu, Y.: Exact iterative solution for an abstract fractional dynamic system model for bioprocess. Qual. Theory Dyn. Syst. 16, 205–222 (2017) MathSciNetView ArticleGoogle Scholar
  38. Zhang, X., Liu, L., Wiwatanapataphee, B., Wu, Y.: The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. Appl. Math. Comput. 235, 412–422 (2014) MathSciNetMATHGoogle Scholar
  39. Sun, F., Liu, L., Wu, Y.: Infinitely many sign-changing solutions for a class of biharmonic equation with p-Laplacian and Neumann boundary condition. Appl. Math. Lett. 73, 128–135 (2017) MathSciNetView ArticleGoogle Scholar
  40. Lin, X., Zhao, Z.: Iterative technique for a third-order differential equation with three-point nonlinear boundary value conditions. Electron. J. Qual. Theory Differ. Equ. 2016, 12 (2016) MathSciNetView ArticleGoogle Scholar
  41. Ahmad, B., Alsaedi, A., Alghamdi, B.S.: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal., Real World Appl. 9, 1727–1740 (2008) MathSciNetView ArticleGoogle Scholar
  42. Karakostas, G.L., Tsamatos, P.C.: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002, 30 (2002) MathSciNetMATHGoogle Scholar
  43. Feng, M., Ge, W.: Positive solutions for a class of m-point singular boundary value problems. Math. Comput. Model. 46, 375–383 (2007) MathSciNetView ArticleGoogle Scholar
  44. Jiang, J., Liu, L., Wu, Y.: Second-order nonlinear singular Sturm–Liouville problems with integral boundary problems. Appl. Math. Comput. 215, 1573–1582 (2009) MathSciNetMATHGoogle Scholar
  45. Lan, K.: Multive positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 63, 690–704 (2001) View ArticleGoogle Scholar
  46. Zhang, X., Liu, L., Wu, Y.: The eigenvalue problem for a singular higher fractional differential equation involving fractional derivatives. Appl. Math. Comput. 218, 8526–8536 (2012) MathSciNetMATHGoogle Scholar
  47. Zhang, X., Liu, L., Wu, Y.: Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl. Math. Comput. 219, 1420–1433 (2012) MathSciNetMATHGoogle Scholar
  48. Zhang, X., Liu, L., Wu, Y., Lu, Y.: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680–4691 (2013) MathSciNetMATHGoogle Scholar
  49. Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion. Appl. Math. Lett. 66, 1–8 (2017) MathSciNetView ArticleGoogle Scholar
  50. Zhang, X., Liu, L., Wu, Y.: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263–1274 (2012) MathSciNetView ArticleGoogle Scholar
  51. Feng, M., Du, B., Ge, W.: Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian. Nonlinear Anal. 70, 3119–3126 (2009) MathSciNetView ArticleGoogle Scholar
  52. Ahmad, B., Alsaedi, A.: Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions. Nonlinear Anal., Real World Appl. 10, 358–367 (2009) MathSciNetView ArticleGoogle Scholar
  53. Mao, J., Zhao, Z.: The existence and uniqueness of positive solutions for integral boundary balue problems. Bull. Malays. Math. Sci. Soc. 34, 153–164 (2011) MathSciNetGoogle Scholar
  54. Liu, L., Hao, X., Wu, Y.: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 57, 836–847 (2013) MathSciNetView ArticleGoogle Scholar
  55. Boucherif, A.: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 70, 364–371 (2009) MathSciNetView ArticleGoogle Scholar
  56. Feng, M., Ji, D., Ge, W.: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 222, 351–363 (2008) MathSciNetView ArticleGoogle Scholar
  57. Kong, L.: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2628–2638 (2010) MathSciNetView ArticleGoogle Scholar
  58. Ma, R., Han, X.: Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function. Appl. Math. Comput. 215, 1077–1083 (2009) MathSciNetMATHGoogle Scholar
  59. López-Gómez, J., Tellini, A.: Generating an arbitrarily large number of isolas in a superlinear indefnite problem. Nonlinear Anal. 108, 223–248 (2014) MathSciNetView ArticleGoogle Scholar
  60. Boscaggin, A., Zanolin, F.: Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem. Ann. Mat. Pura Appl. 194, 451–478 (2015) MathSciNetView ArticleGoogle Scholar
  61. Feltrin, G., Zanolin, F.: Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems. Adv. Differ. Equ. 20, 937–982 (2015) MathSciNetMATHGoogle Scholar
  62. Boscaggin, A., Feltrin, G., Zanolin, F.: Fabio Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case. Proc. R. Soc. Edinb., Sect. A 146, 449–474 (2016) View ArticleGoogle Scholar
  63. Boscaggin, A., Zanolin, F.: Positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results and complex dynamics. J. Differ. Equ. 252, 2922–2950 (2012) MathSciNetView ArticleGoogle Scholar
  64. Sovrano, E., Zanolin, F.: Indefinite weight nonlinear problems with Neumann boundary conditions. J. Math. Anal. Appl. 452, 126–147 (2017) MathSciNetView ArticleGoogle Scholar
  65. Bravo, J.L., Torres, P.J.: Periodic solutions of a singular equation with indefinite weight. Adv. Nonlinear Stud. 10, 927–938 (2010) MathSciNetView ArticleGoogle Scholar
  66. Wang, F., An, Y.: On positive solutions for a second order differential system with indefinite weight. Appl. Math. Comput. 259, 753–761 (2015) MathSciNetMATHGoogle Scholar
  67. Yao, Q.: Existence and multiplicity of positive radial solutions for a semilinear elliptic equation with change of sign. Appl. Anal. 80, 65–77 (2001) MathSciNetView ArticleGoogle Scholar
  68. Jiao, L., Zhang, X.: Multi-parameter second-order impulsive indefinite boundary value problems. Adv. Differ. Equ. 2018, 158 (2018) MathSciNetView ArticleGoogle Scholar
  69. Feltrin, G., Sovrano, E.: Three positive solutions to an indefinite Neumann problem: a shooting method. Nonlinear Anal. 166, 87–101 (2018) MathSciNetView ArticleGoogle Scholar
  70. Zhang, Q.: Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part. J. Nonlinear Sci. Appl. 11, 368–374 (2018) MathSciNetView ArticleGoogle Scholar
  71. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) MATHGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement