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Symmetric positive solutions for fourth-order n-dimensional m-Laplace systems

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Abstract

This paper investigates the existence, multiplicity, and nonexistence of symmetric positive solutions for the fourth-order n-dimensional m-Laplace system

$$\left \{ \textstyle\begin{array}{l} \phi_{m}(\mathbf{x}{''}(t))){''}=\Psi(t)\mathbf{f}(t,\mathbf{x}(t)), \quad 0< t< 1,\\ \mathbf{x}(0)=\mathbf{x}(1)=\int_{0}^{1}\mathbf{g}(s)\mathbf{x}(s)\, ds,\\ \phi_{m}(\mathbf{x}{''}(0))=\phi_{m}(\mathbf{x}{''}(1))=\int _{0}^{1}\mathbf{h}(s)\phi_{m}(\mathbf{x}{''}(s))\,ds. \end{array}\displaystyle \right . $$

The vector-valued function x is defined by \(\mathbf {x}=[x_{1},x_{2},\dots,x_{n}]^{\top}\), \(\Psi(t)=\operatorname{diag}[\psi_{1}(t), \ldots, \psi _{i}(t), \ldots, \psi_{n}(t)]\), where \(\psi_{i}\in L^{p}[0,1]\) for some \(p\geq1\). Our methods employ the fixed point theorem in a cone and the inequality technique. Finally, an example illustrates our main results.

Introduction

Consider the fourth-order n-dimensional m-Laplace system

$$ \bigl(\phi_{m}\bigl(\mathbf{x} {''}(t) \bigr)\bigr){''}=\Psi(t)\mathbf{f}\bigl(t,\mathbf{x}(t) \bigr),\quad 0< t< 1, $$
(1.1)

subject to the following boundary conditions:

$$ \left \{ \textstyle\begin{array}{l} \mathbf{x}(0)=\mathbf{x}(1)=\int_{0}^{1}\mathbf{g}(s)\mathbf{x}(s)\, ds,\\ \phi_{m}(\mathbf{x}{''}(0))=\phi_{m}(\mathbf{x}{''}(1))=\int _{0}^{1}\mathbf{h}(s)\phi_{m}(\mathbf{x}{''}(s))\,ds, \end{array}\displaystyle \right . $$
(1.2)

where

$$\begin{aligned}& \mathbf{x}(t)=\bigl(x_{1}(t),x_{2}(t),\ldots,x_{n}(t) \bigr)^{T}, \\& \Psi(t)=\operatorname{diag}\bigl[\psi_{1}(t),\psi_{2}(t),\ldots, \psi_{n}(t)\bigr], \\& \mathbf{f}(t,\mathbf{x})=\bigl(f_{1}(t,\mathbf{x}), \ldots,f_{i}(t,\mathbf {x}),\ldots,f_{n}(t,\mathbf{x}) \bigr)^{T}, \\& \phi_{m}\bigl(\mathbf{x} {''}(t)\bigr)= \bigl(\phi_{m}\bigl(x_{1}{''}(t) \bigr),\phi _{m}\bigl(x_{2}{''}(t) \bigr),\ldots,\phi_{m}\bigl(x_{n}{''}(t) \bigr)\bigr)^{T}, \\& \mathbf{g}(s)=\operatorname{diag}\bigl[g_{1}(s),g_{2}(s), \ldots,g_{n}(s)\bigr], \\& \mathbf{h}(s)=\operatorname{diag}\bigl[h_{1}(s),h_{2}(s), \ldots,h_{n}(s)\bigr]. \end{aligned}$$

Here, we understand that \(f_{i}(t,\mathbf{x})\) means that \(f_{i}(t,x_{1},x_{2},\ldots,x_{n})\), \(i=1,2,\ldots,n\).

Therefore, system (1.1) means that

$$ \left ( \textstyle\begin{array}{c} (\phi_{m}(x_{1}{''}(t))){''} \\ (\phi_{m}(x_{2}{''}(t))){''} \\ \vdots\\ (\phi_{m}(x_{n}{''}(t))){''} \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} \psi_{1}(t) & 0 & \cdots& 0 \\ 0 & \psi_{2}(t) & \cdots& 0 \\ \vdots& \vdots& \ddots& \vdots\\ 0 & 0 & \cdots& \psi_{n}(t) \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{c} f_{1}(t,\mathbf{x}) \\ f_{2}(t,\mathbf{x}) \\ \vdots\\ f_{n}(t,\mathbf{x}) \end{array}\displaystyle \right ). $$
(1.3)

Similarly, (1.2) means that

$$ \left \{ \textstyle\begin{array}{l} \left ( {\scriptsize\begin{matrix}{} x_{1}(0) \cr x_{2}(0) \cr \vdots\cr x_{n}(0) \end{matrix}} \right )=\left ( {\scriptsize\begin{matrix}{} x_{1}(1) \cr x_{2}(1) \cr \vdots\cr x_{n}(1) \end{matrix}} \right )= \int_{0}^{1}\left ( {\scriptsize\begin{matrix}{} g_{1}(s) & 0 & \cdots& 0 \cr 0 & g_{2}(s) & \cdots& 0 \cr \vdots& \vdots& \ddots& \vdots\cr 0 & 0 & \cdots& g_{n}(s) \end{matrix}} \right )\left ( {\scriptsize\begin{matrix}{} x_{1}(s) \cr x_{2}(s) \cr \vdots\cr x_{n}(s) \end{matrix}} \right )\,ds, \\ \left ( {\scriptsize\begin{matrix}{} \phi_{m}(x_{1}{''}(0)) \cr \phi_{m}(x_{2}{''}(0)) \cr \vdots\cr \phi_{m}(x_{n}{''}(0)) \end{matrix}} \right )=\left ( {\scriptsize\begin{matrix}{} \phi_{m}(x_{1}{''}(1)) \cr \phi_{m}(x_{2}{''}(1)) \cr \vdots\cr \phi_{m}(x_{n}{''}(1)) \end{matrix}} \right )= \int_{0}^{1}\left ( {\scriptsize\begin{matrix}{} h_{1}(s) & 0 & \cdots& 0 \cr 0 & h_{2}(s) & \cdots& 0 \cr \vdots& \vdots& \ddots& \vdots\cr 0 & 0 & \cdots& h_{n}(s) \end{matrix}} \right )\left ( {\scriptsize\begin{matrix}{} \phi_{m}(x_{1}{''}(s)) \cr \phi_{m}(x_{2}{''}(s)) \cr \vdots\cr \phi_{m}(x_{n}{''}(s)) \end{matrix}} \right )\,ds. \end{array}\displaystyle \right . $$
(1.4)

And then it follows respectively from (1.3) and (1.4) that

$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} (\phi_{m}(x_{1}{''}(t))){''}=\psi_{1}(t)f_{1}(t,x_{1}(t),\ldots ,x_{n}(t)), \quad 0< t< 1,\\ (\phi_{m}(x_{2}{''}(t))){''}=\psi_{2}(t)f_{2}(t,x_{1}(t),\ldots ,x_{n}(t)), \quad 0< t< 1,\\ \vdots\\ (\phi_{m}(x_{n}{''}(t))){''}=\psi_{n}(t)f_{n}(t,x_{1}(t),\ldots ,x_{n}(t)), \quad0< t< 1, \end{array}\displaystyle \right . \end{aligned}$$
(1.5)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} x_{1}(0)=x_{1}(1)=\int_{0}^{1}g_{1}(s)x_{1}(s)\,ds,\\ \phi_{m}(x_{1}{''}(0))=\phi_{m}(x_{1}{''}(1))=\int_{0}^{1}h_{1}(s)\phi _{m}(x_{1}{''}(s))\,ds,\\ x_{2}(0)=x_{2}(1)=\int_{0}^{1}g_{2}(s)x_{2}(s)\,ds,\\ \phi_{m}(x_{2}{''}(0))=\phi_{m}(x_{2}{''}(1))=\int_{0}^{1}h_{2}(s)\phi _{m}(x_{2}{''}(s))\,ds,\\ \vdots\\ x_{n}(0)=x_{n}(1)=\int_{0}^{1}g_{n}(s)x_{n}(s)\,ds,\\ \phi_{m}(x_{n}{''}(0))=\phi_{m}(x_{n}{''}(1))=\int_{0}^{1}h_{n}(s)\phi _{m}(x_{n}{''}(s))\,ds. \end{array}\displaystyle \right . \end{aligned}$$
(1.6)

From above, we know that system (1.1)–(1.2) is equivalent to system (1.3)–(1.4), and system (1.3)–(1.4) is equivalent to system (1.5)–(1.6); thus, system (1.1)–(1.2) is equivalent to (1.5)–(1.6).

A vector-valued function x is called a solution of (1.1)–(1.2) if \(\mathbf{x}\in C^{2}([0,1],\mathcal{R}^{n})\) with \(\phi_{m}(\mathbf{x}{''})\in C^{2}((0,1),\mathcal{R}^{n})\), and satisfies (1.1) and (1.2). If, for each \(i=1,2,\ldots ,n\), \(x_{i}(t)\ge0\) for all \(t\in(0,1)\) and there is at least one nontrivial component of \(\bf{x}\), then we say that \(\mathbf {x}(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}\) is positive on J.

For the case of \(n=1\) and \(\Psi(t)\equiv1\) for \(t\in J\), system (1.1)–(1.2) reduces to the problem studied by Zhang and Liu in [1]. By using the upper and lower solution method and fixed-point theorems, the authors obtained some sufficient conditions for the existence of positive solutions for the above problem. For the case of \(n=1\), \(m=2\), \(g(t)\equiv0\), \(h(t)\equiv0\) for \(t\in J\) and \(\Psi\in C[0,1]\) not \(\Psi\in L^{p}[0,1]\), system (1.1)–(1.2) reduces to the problem studied by Graef et al. in [2]. By using Krasnosel’skii’s fixed-point theorem, the authors obtained some existence and nonexistence results. For other related results on system (1.1)–(1.2), we refer the reader to the references [327]. Moreover, for the latest development direction of the fourth order differential equations, see the references [2831].

At the same time, we notice that a class of boundary value problems with integral boundary conditions has attracted many authors (see [20, 3242]). It is an important and interesting problem, which contains two-point, three-point, and multi-point boundary value problems as special cases; for instance, see [4358] and the references cited therein.

Here we point out that our problem is new in the sense of fourth-order n-dimensional m-Laplace systems with integral boundary conditions introduced here. To the best of our knowledge, the existence of single or multiple positive solutions for fourth-order n-dimensional m-Laplace systems (1.1)–(1.2) has not yet been studied, especially for the case

$$\Psi(t)=\operatorname{diag}\bigl[\psi_{1}(t), \ldots, \psi_{i}(t), \ldots, \psi_{n}(t)\bigr], $$

where \(\psi_{i}\in L^{p}[0,1]\) for some \(p\geq1\). In consequence, our main results of the present work will be a useful contribution to the existing literature on the topic of fourth-order n-dimensional m-Laplace systems with integral boundary conditions. The existence, multiplicity, and nonexistence of symmetric positive solutions for the given problem are new, though they are proved by applying the well-known method based on the fixed point theorem of cone expansion and compression of norm type.

Throughout this paper, we use \(i = 1, 2,\ldots,n\), unless otherwise stated.

Let \(J=[0,1]\), \(\mathcal{R}_{+}=[0,+\infty)\), \(\mathcal {R}_{+}^{n}=\underbrace{\mathcal{R}_{+}\times\mathcal{R}_{+}\times\cdots \times\mathcal{R}_{+}}_{n}\), and

$$\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{\top}\in\mathcal{R}_{+}^{n}. $$

In addition, let the components of Ψ, f, h, and g satisfy the following conditions:

\((H_{1})\) :

\(\psi_{i}(t)\in L^{p}(J)\) for some \(1\le p\le+\infty \), and \(\psi_{i}(t)\) is nonnegative, symmetric on J, and there exists \(N>0\) such that \(\psi_{i}(t)\ge N\) a.e. on J;

\((H_{2})\) :

\(f_{i}:J\times\mathcal{R}_{+}^{n}\rightarrow\mathcal {R}_{+}\) is continuous, and for all \(\mathbf{x}\in\mathcal{R}_{+}^{n}\), \(f_{i}(t,\mathbf{x})\) is symmetric on J;

\((H_{3})\) :

\(g_{i},h_{i}\in L^{1}(J)\) are nonnegative, symmetric on J with

$$ \mu_{i}:= \int_{0}^{1}g_{i}(s)\,ds\in[0,1),\qquad \nu _{i}:= \int_{0}^{1}h_{i}(s)\,ds \in[0,1). $$
(1.7)

The organization of this article is as follows. In Sect. 2, we present some new properties of Green’s function associated with system (1.1)–(1.2), and we list some definitions and lemmas that will be used to prove our main results. Section 3 is devoted to prove the existence, multiplicity, and nonexistence of symmetric positive solutions for system (1.1)–(1.2). Finally, in Sect. 4, an example illustrating our main results is also presented.

Preliminaries

In this part, we give some properties of Green’s function associated with system (1.1)–(1.2), and we present some definitions and lemmas which are needed throughout this paper.

Definition 2.1

(see [59])

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

  1. (i)

    \(au+bv\in P\) for all \(u, v\in P \) and all \(a\geq0\), \(b\geq0\) and

  2. (ii)

    \(u,-u\in P\) implies \(u=0\).

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

Definition 2.2

If \(x(t)=x(1-t)\), \(t\in J\), then x is said to be symmetric in J.

In our discussion, \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})^{T}\) is symmetric on J if and only if \(x_{i}\) is symmetric on J.

Next, we reduce system (1.1)–(1.2) to an integral system. It follows from system (1.5)–(1.6) that system (1.1)–(1.2) can be written as follows:

$$ \left \{ \textstyle\begin{array}{l} (\phi_{m}(x{''}_{i}(t))){''}=\psi_{i}(t)f_{i}(t,x_{1}(t),\ldots ,x_{n}(t)), \quad0< t< 1,\\ x_{i}(0)=x_{i}(1)=\int_{0}^{1}g_{i}(s)x_{i}(s)\,ds,\\ \phi_{m}(x_{i}{''}(0))=\phi_{m}(x_{i}{''}(1))=\int_{0}^{1}h_{i}(s)\phi _{m}(x_{i}{''}(s))\,ds, \end{array}\displaystyle \right . $$
(2.1)

where \(\phi_{m}(s)=|s|^{m-2}s\), \(m>1\), \(\phi_{m^{*}}=\phi_{m}^{-1}\), \(\frac {1}{m}+\frac{1}{m^{*}}=1\).

Firstly, by means of the transformation

$$ \phi_{m}\bigl(x_{i}{''}(t) \bigr)=-y_{i}(t), $$
(2.2)

we can convert system (2.1) into

$$ \left \{ \textstyle\begin{array}{l} y_{i}{''}(t)=-\psi_{i}(t)f_{i}(t,\mathbf{x}(t)), \quad0< t< 1,\\ y_{i}(0)=y_{i}(1)=\int_{0}^{1}h_{i}(s)y_{i}(s)\,ds, \end{array}\displaystyle \right . $$
(2.3)

and

$$ \left \{ \textstyle\begin{array}{l} x_{i}{''}(t)=-\phi_{m^{*}}(y_{i}(t)), \quad 0< t< 1,\\ x_{i}(0)=x_{i}(1)=\int_{0}^{1}g_{i}(s)x_{i}(s)\,ds. \end{array}\displaystyle \right . $$
(2.4)

Lemma 2.1

Assume that \((H_{3})\) holds. Then system (2.4) has a unique solution \(x_{i}(t)\) and \(x_{i}(t)\) can be expressed in the form

$$ x_{i}(t)=- \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \bigl(y_{i}(s)\bigr)\,ds, $$
(2.5)

where

$$\begin{aligned}& H^{i}(t,s)=G(t,s)+\frac{1}{1-\mu_{i}} \int_{0}^{1}G(\tau,s)g_{i}(\tau)\, d\tau, \end{aligned}$$
(2.6)
$$\begin{aligned}& G(t,s)=\left \{ \textstyle\begin{array}{l} t(1-s), \quad0\le t\le s\le1,\\ s(1-t), \quad0\le s\le t\le1. \end{array}\displaystyle \right . \end{aligned}$$
(2.7)

Proof

The proof is similar to that of Lemma 2.1 in [60]. □

By (2.6) and (2.7), we can show that \(H^{i}(t,s)\) and \(G(t,s)\) have the following properties.

Proposition 2.1

Assume that \((H_{3})\) holds. Then we have

$$ \begin{gathered} H^{i}(t,s)>0, \qquad G(t,s)>0,\quad \forall t,s\in(0,1); \\ H^{i}(t,s)\ge0, \qquad G(t,s)\ge0, \quad\forall t,s\in J. \end{gathered} $$
(2.8)

Proposition 2.2

For all \(t,s\in J\), we have

$$\begin{aligned}& e(t)e(s)\le G(t,s)\le G(t,t)=t(1-t)=e(t)\le\overline{e}=\max _{t\in J}e(t)=\frac{1}{4}, \end{aligned}$$
(2.9)
$$\begin{aligned}& G(1-t,1-s)=G(t,s). \end{aligned}$$
(2.10)

Proposition 2.3

Assume that \((H_{3})\) holds. Then, for all \(t,s\in J\), we have

$$ \rho^{i}e(s)\le H^{i}(t,s)\le\gamma^{i}s(1-s)= \gamma^{i}e(s)\le\frac {1}{4}\gamma^{i}, $$
(2.11)

where

$$ \gamma^{i}=\frac{1}{1-\mu_{i}},\qquad \rho^{i}=\frac{\int_{0}^{1}e(\tau )g_{i}(\tau)\,d\tau}{1-\mu_{i}}. $$
(2.12)

Proof

By (2.6) and (2.9), we have

$$ \begin{aligned}[b] H^{i}(t,s)&=G(t,s)+\frac{1}{1-\mu_{i}} \int_{0}^{1}G(s,\tau)g_{i}(\tau )\,d\tau \\ &\ge\frac{1}{1-\mu_{i}} \int_{0}^{1}G(s,\tau)g_{i}(\tau)\,d\tau \\ &\ge\frac{\int_{0}^{1}e(\tau)g_{i}(\tau)\,d\tau}{1-\mu_{i}}s(1-s) \\ &=\rho^{i}e(s), \quad t\in J. \end{aligned} $$
(2.13)

In addition, noticing that \(G(t,s)\le s(1-s)\), we have

$$ \begin{aligned}[b] H^{i}(t,s)&=G(t,s)+\frac{1}{1-\mu_{i}} \int_{0}^{1}G(s,\tau)g_{i}(\tau )\,d\tau \\ &\le s(1-s)+\frac{1}{1-\mu_{i}} \int_{0}^{1}s(1-s)g_{i}(\tau)\,d\tau \\ &\le s(1-s)\biggl[1+\frac{1}{1-\mu_{i}} \int_{0}^{1}g_{i}(\tau)\,d\tau\biggr] \\ &=s(1-s)\frac{1}{1-\mu_{i}} \\ &=\gamma^{i}e(s), \quad t\in J. \end{aligned} $$
(2.14)

 □

Proposition 2.4

Assume that (\(H_{3}\)) holds. Then, for all \(t,s\in J\), we have

$$ H^{i}(1-t,1-s)=H^{i}(t,s). $$
(2.15)

Proof

The proof is similar to that of Proposition 2.1 of [60]. □

Lemma 2.2

Assume that \((H_{1})\)\((H_{3})\) hold. Then system (2.3) has a unique solution

$$ y_{i}(t)=- \int_{0}^{1}H_{1}^{i}(t,s) \psi_{i}(s)f_{i}\bigl(s,\mathbf{x}(s)\bigr)\,ds, $$
(2.16)

where

$$ H_{1}^{i}(t,s)=G(t,s)+\frac{1}{1-\nu_{i}} \int_{0}^{1}G(v,s)h_{i}(v)dv. $$
(2.17)

Proof

The proof is similar to Lemma 2.1 of [60]. □

Remark 2.1

Assume that (\(H_{3}\)) holds. Then, for all \(t,s\in J\), it follows from (2.17) that

$$H_{1}^{i}(t,s)\geq0,\qquad \rho_{1}^{i}e(s) \le H_{1}^{i}(t,s)\le\gamma _{1}^{i}s(1-s) \le\frac{1}{4}\gamma_{1}^{i},\qquad H_{1}^{i}(1-t,1-s)=H_{1}^{i}(t,s), $$

where

$$\rho_{1}^{i}=\frac{\int_{0}^{1}e(\tau)h_{i}(\tau)\,d\tau}{1-\nu_{i}}, \qquad\gamma_{1}^{i}= \frac{1}{1-\nu_{i}}. $$

Assume that \(x_{i}\) is a solution of system (2.1). Then it follows from Lemma 2.1 that

$$ x_{i}(t)=- \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \bigl(y_{i}(s)\bigr)\,ds, $$
(2.18)

and then, it follows from Lemma 2.2 that

$$ x_{i}(t)= \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,ds. $$
(2.19)

Let \(E=C[0,1]\), \(X=\underbrace{E\times E\times\cdots\times E}_{n}\), and for all \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})^{T}\in X\), the norm in X is defined as

$$\|\mathbf{x}\|=\sum_{i=1}^{n}\sup _{t\in J}|x_{i}|. $$

Then \((X,\|\cdot\|)\) is a real Banach space.

Define a cone K in X by

$$ \begin{aligned}[b] K={}& \Biggl\{ \mathbf{x}=(x_{1},x_{2}, \ldots,x_{n})^{T}\in X:x_{i}\ge0, x_{i}(t)\text{ is symmetric and concave on }J, \\ & \min_{t\in J}\sum_{i=1}^{n}x_{i}(t) \ge\delta\|\bf{x}\| \Biggr\} , \end{aligned} $$
(2.20)

where

$$\delta=\min_{1\le i\le n}\delta_{i},\quad \delta_{i}= \frac{\rho^{i}(\rho _{1}^{i})^{m^{*}-1}}{\gamma^{i}(\gamma_{1}^{i})^{m^{*}-1}}. $$

We also define two sets \(K_{r}\), \(K_{r,R}\) by

$$K_{r}= \{x\in K:\|x\|< r \},\qquad K_{r,R}=\{x\in K:r< \|x\|< R\}, $$

where \(0< r< R\).

To make our research significant, let \(g_{i}(t)\not\equiv0\), \(h_{i}(t)\not\equiv0\) for any \(t\in J\), \(i=1,2,\ldots,n\).

Remark 2.2

By the definition of \(\rho^{i}\), \(\rho _{1}^{i}\), \(\gamma^{i}\), \(\gamma_{1}^{i}\), we have \(0<\delta_{i}<1\), and then \(0<\delta<1\).

Let \(\mathbf{T}:K\rightarrow X\) be a map with components \((T_{1},\ldots ,T_{i}, \ldots,T_{n})\). Here, we understand \(\mathbf{Tx}= (T_{1}\mathbf{x},\ldots,T_{i}\mathbf{x},\ldots,T_{n}\mathbf{x} )^{T}\), where

$$ (T_{i}\mathbf{x}) (t)= \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,ds. $$
(2.21)

From the proof of Lemma 2.1 and Lemma 2.2, we have the following remark.

Remark 2.3

From (2.21), we know that \(\mathbf {x}\in X\) is a solution of system (1.1)–(1.2) if and only if x is a fixed point of the map \(\bf{T}\).

Lemma 2.3

Assume that \((H_{1})\)\((H_{3})\) hold. Then we have \(\mathbf{T}(K)\subset K\), and \(\mathbf{T}:K\rightarrow K\) is completely continuous.

Proof

For all \(\mathbf{x}\in K\), from (2.21), we know that

$$ (T_{i}\mathbf{x}){''}=-\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(t,s)\psi _{i}(s)f_{i}\bigl(s,\mathbf{x}(s)\bigr)\,ds \biggr)\le0, $$
(2.22)

which implies that \(T_{i}\mathbf{x}\) is concave on J.

In addition, it follows from (2.21) that

$$(T_{i}\mathbf{x}) (0)=(T_{i}\mathbf{x}) (1)\ge0. $$

Thus, for all \(t\in J\), we have \((T_{i}\mathbf{x})(t)\ge0\). Noticing that \(\psi_{i}(t)\) is symmetric on \((0,1)\), \(x_{i}(t)\) is symmetric on J, and \(f_{i}(\cdot,\bf{x})\) is symmetric on J, we have

$$\begin{aligned} (T_{i}\mathbf{x}) (1-t)&= \int_{0}^{1}H^{i}(1-t,s) \phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,ds \\ &= \int_{1}^{0}H^{i}(1-t,1-s) \phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(1-s, \tau)\psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,d(1-s) \\ &= \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(1-s, \tau)\psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,ds \\ &= \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}}\\ &\quad\times \biggl( \int _{1}^{0}H_{1}^{i}(1-s,1- \tau)\psi_{i}(1-\tau)f_{i}\bigl(1-\tau,\mathbf {x}(1-\tau) \bigr)\,d(1-\tau) \biggr)\,ds \\ &= \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau ) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &=(T_{i}\mathbf{x}) (t), \end{aligned}$$

which shows that \((T_{i}\mathbf{x})(1-t)=(T_{i}\mathbf{x})(t)\), \(t\in J\). And hence \((T_{i}\mathbf{x})(t)\) is symmetric on J.

In addition, according to (2.14), we know that

$$\begin{aligned} (T_{i}\mathbf{x}) (t)&= \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,ds \\ & \le\gamma^{i}\bigl(\gamma_{1}^{i} \bigr)^{m^{*}-1} \int_{0}^{1}e(s)\phi _{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}( \tau)f_{i}\bigl(\tau,\mathbf {x}(\tau)\bigr)\,d\tau \biggr)\,ds,\quad \forall t\in J. \end{aligned} $$

Then

$$\|\mathbf{Tx}\|=\sum_{i=1}^{n}\sup _{t\in J}(T_{i}\mathbf{x}) (t)\le \sum _{i=1}^{n}\gamma^{i}\bigl( \gamma_{1}^{i}\bigr)^{m^{*}-1} \int_{0}^{1}e(s)\phi _{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}( \tau)f_{i}\bigl(\tau,\mathbf {x}(\tau)\bigr)\,d\tau \biggr)\,ds. $$

Similarly, according to (2.13), we know that

$$\begin{aligned} \min_{t\in J}\sum _{i=1}^{n}(T_{i}\mathbf{x}) (t)&=\min _{t\in J}\sum_{i=1}^{n} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\, d\tau \biggr)\,ds \\ &\ge\sum_{i=1}^{n}\rho^{i}\bigl( \rho_{1}^{i}\bigr)^{m^{*}-1} \int_{0}^{1}e(s)\phi _{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}( \tau)f_{i}\bigl(\tau,\mathbf {x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\delta_{i} \gamma^{i}\bigl(\gamma_{1}^{i} \bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}(\tau )f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\ge\delta\|\mathbf{Tx}\|. \end{aligned} $$

Thus, we have \(T_{i}\mathbf{x}\in K\), then there is \(\mathbf {T}(K)\subset K\).

Next, we show T is completely continuous, and we need to show \(T_{i}\) is completely continuous.

Let \(l>0\) and define

$$\widehat{f}_{l}^{i}=\sup_{t\in J}\bigl\{ f_{i}\bigl(t,\mathbf{x}(t)\bigr):\mathbf{x}\in \mathcal{R}_{+}^{n}, \|\mathbf{x}\|\le l\bigr\} >0. $$

We show that \(T_{i}\) is compact.

For each \(l>0\), let \(B_{l}=\{\mathbf{x}\in K:\|\mathbf{x}\|\leq l\}\). Then \(B_{l}\) is a bounded closed convex set in K. \(\forall(\mathbf{x}_{m})_{m\in\mathcal{N}}\in K\), it follows from (2.21) that

$$\begin{aligned} |T_{i}\mathbf{x}_{m}|&= \biggl\vert \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf {x}_{m}( \tau)\bigr)\,d\tau \biggr)\,ds \biggr\vert \\ &\le\frac{1}{4}\gamma^{i}\biggl(\frac{1}{4} \gamma_{1}^{i}\biggr)^{m^{*}-1} \biggl\vert \int_{0}^{1}\phi_{m^{*}} \biggl( \int_{0}^{1}\psi_{i}( \tau)f_{i}\bigl(\tau ,\mathbf{x}_{m}(\tau)\bigr)\,d\tau \biggr)\,ds \biggr\vert \\ &\le\frac{1}{4}\gamma^{i}\biggl(\frac{1}{4} \gamma_{1}^{i}\biggr)^{m^{*}-1}\bigl(\widehat {f}_{l}^{i}\bigr)^{m^{*}-1} \biggl\vert \int_{0}^{1}\phi_{m^{*}} \biggl( \int _{0}^{1}\psi_{i}(\tau)\,d\tau \biggr)\,ds \biggr\vert \\ &\le\frac{1}{4}\gamma^{i}\biggl(\frac{1}{4} \gamma_{1}^{i}\biggr)^{m^{*}-1}\bigl(\widehat {f}_{l}^{i}\bigr)^{m^{*}-1} \biggl\vert \int_{0}^{1}\phi_{m^{*}}\bigl(\| \psi_{i}\| _{1} \bigr)\,ds \biggr\vert \\ &=\frac{1}{4}\gamma^{i}\biggl(\frac{1}{4} \gamma_{1}^{i}\biggr)^{m^{*}-1}\bigl(\widehat {f}_{l}^{i}\bigr)^{m^{*}-1}\bigl(\|\psi_{i} \|_{1}\bigr)^{m^{*}-1} \\ &=\biggl(\frac{1}{4}\gamma_{1}^{i} \biggr)^{m^{*}}\bigl(\widehat{f}_{l}^{i} \bigr)^{m^{*}-1}\bigl(\| \psi_{i}\|_{1}\bigr)^{m^{*}-1}. \end{aligned}$$

Therefore, \((T_{i}(B_{l}))\) is uniformly bounded.

Next we show the equicontinuity of \((T_{i}\mathbf{x}_{m})_{m\in \mathcal{N}}\). Due to \(H^{i}(t,s)\) is continuous on \(J\times J\), then \(H^{i}(t,s)\) is uniformly continuous. Thus, for any \(\varepsilon>0\), there exist \(l_{1}>0\), \(t_{1},t_{2}\in J\), if \(|t_{1}-t_{2}|< l_{1}\), we have

$$\begin{aligned} \big|(T_{i}\mathbf{x}_{m}) (t_{2})-(T_{i}\mathbf{x}_{m}) (t_{1})\big|={}& \biggl\vert \int_{0}^{1}H^{i}(t_{2},s) \phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}_{m}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &- \int_{0}^{1}H^{i}(t_{1},s) \phi_{m^{*}} \biggl( \int _{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}_{m}(\tau )\bigr)\,d\tau \biggr)\,ds \biggr\vert \\ ={}& \biggl\vert \int_{0}^{1} \bigl[H^{i}(t_{2},s)-H^{i}(t_{1},s) \bigr]\\ &\times\phi _{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau ,\mathbf{x}_{m}( \tau)\bigr)\,d\tau \biggr)\,ds \biggr\vert \\ \le{}&\biggl(\frac{1}{4}\gamma_{1}^{i} \biggr)^{m^{*}-1}\bigl(\|\psi_{i}\| _{1}\bigr)^{m^{*}-1} \bigl(\widehat{f}_{l}^{i}\bigr)^{m^{*}-1} \int_{0}^{1} \big|H^{i}(t_{1},s)-H^{i}(t_{2},s) \big|\,ds \\ \le{}&\varepsilon, \end{aligned} $$

which shows that \((T_{i}\mathbf{x}_{m})_{m\in\mathcal{N}}\) is equicontinuous on J. Therefore, it follows from the Arzelà–Ascoli theorem that there exist a function \(T^{1}_{i}\in C[0,1]\) and a subsequence of \((T_{i}\mathbf{x}_{m})_{m\in\mathcal{N}}\) converging uniformly to \(T^{1}_{i}\) on J.

We prove the continuity of \(T_{i}\). Let \((\mathbf{x}_{m})_{m\in \mathcal{N}}\) be any sequence converging on K to \(\mathbf{x}\in K\), and let \(L>0\) be such that \(\|\mathbf{x}_{m}\|\le L\) for all \(m\in \mathcal{N}\). Note that \(f_{i}(t,\mathbf{x})\) is continuous on \(J\times K_{L}\). It is not difficult to see that the dominated convergence theorem guarantees that

$$ \lim_{m\rightarrow\infty}(T_{i}\mathbf{x}_{m}) (t)=(T_{i}\mathbf {x}) (t) $$
(2.23)

for each \(t\in J\). Moreover, the compactness of \(T_{i}\) implies that \((T_{i}\mathbf{x}_{m})(t)\) converges uniformly to \((T_{i}\mathbf {x})(t)\) on J. If not, then there exist \(\varepsilon_{0}>0\) and a subsequence \((\mathbf{x}_{m_{j}})_{j\in\mathcal{N}}\) of \((\mathbf {x}_{m})_{m\in\mathcal{N}}\) such that

$$ \sup_{t\in J}\big|(T_{i}\mathbf{x}_{m_{j}}) (t)-(T_{i}\mathbf{x}) (t)\big|\ge \varepsilon_{0}, \quad j\in \mathcal{N}. $$
(2.24)

Now, it follows from the compactness of \(T_{i}\) that there exists a subsequence of \((\mathbf{x}_{m_{j}})_{j\in\mathcal{N}}\) (without loss of generality, assume that the subsequence is \((\mathbf {x}_{m_{j}})_{j\in\mathcal{N}}\)) such that \((T_{i}\mathbf {x}_{m_{j}})_{j\in\mathcal{N}}\) converges uniformly to \(y_{0}\in C[0,1]\). Thus, from (2.24), we easily see that

$$ \sup_{t\in J}\big|y_{0}(t)-(T_{i}\mathbf{x}) (t)\big|\ge\varepsilon_{0},\quad j\in \mathcal{N}. $$
(2.25)

On the other hand, from the pointwise convergence (2.23) we obtain

$$y_{0}(t)=(T_{i}\mathbf{x}) (t),\quad t\in J. $$

This is a contradiction to (2.25). Therefore \(T_{i}\) is continuous.

Therefore \(T_{i}:K\rightarrow K\) is completely continuous. This completes the proof of Lemma 2.3. □

In the following lemma, we employ Hölder’s inequality to obtain some of the norm inequalities in our main results.

Lemma 2.4

(Hölder)

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

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

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

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

Finally, we state the well-known fixed point theorem of cone expansion and compression of norm type.

Lemma 2.5

(see [59])

Let P be a cone in a real Banach space E. Assume \(\Omega_{1}\), \(\Omega_{2}\) are bounded open sets in E with \(0\in\overline{\Omega}_{1}\subset\Omega_{2}\), \(A:P\cap (\overline{\Omega}_{2}\setminus\Omega_{1})\rightarrow P\) is completely continuous such that either

  1. (i)

    \(\|Ax\|\le\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\); \(\|Ax\| \ge\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\) or

  2. (ii)

    \(\|Ax\|\ge\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\); \(\|Ax\| \le\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\).

Then A has at least one fixed point in \(P\cap(\overline{\Omega }_{2}\setminus\Omega_{1})\).

Remark 2.4

To make it clear for the reader what \(\Omega _{1}\), \(\Omega_{2}\), \(\partial\Omega_{1}\), \(\partial\Omega_{2}\), and \(\bar{\Omega}_{2}\setminus\Omega_{1}\) mean, we give typical examples of \(\Omega_{1}\) and \(\Omega_{2}\).

$$\begin{gathered} \Omega_{1}=\bigl\{ x\in C[a,b]:\|x\|< r\bigr\} ,\qquad \Omega_{2}= \bigl\{ x\in C[a,b]:\|x\|< R\bigr\} , \\ \overline{\Omega}_{2}\setminus\Omega_{1}=\bigl\{ x\in C[a,b]:r\leq\|x\|\leq R\bigr\} ,\end{gathered} $$

where \(0< r< R\), \(\|x\|=\max_{t\in[a,b]}|x(t)|\).

Main results

In this part, by using Lemmas 2.12.5, we show the existence, multiplicity, and nonexistence of symmetric positive solutions for system (1.1)–(1.2) under the following three cases for \(\psi_{i}\in L^{p}[0,1]:1< p<\infty\), \(p=1\), and \(p=\infty\).

For convenience’s sake, we introduce the notations:

$$\begin{gathered} f_{i}^{\beta}=\limsup_{\|\mathbf{x}\|\rightarrow\beta}\max _{t\in J}\frac{f_{i}(t,\mathbf{x})}{\phi_{m}(\|\mathbf{x}\|)},\qquad f_{i\beta }=\liminf _{\|\mathbf{x}\|\rightarrow\beta}\min_{t\in J}\frac {f_{i}(t,\mathbf{x})}{\phi_{m}(\|\mathbf{x}\|)}, \\ D_{i}=\frac{1}{6}n\gamma^{i}\bigl( \gamma_{1}^{i}\|e\|_{q}\|\omega_{i}\| _{p}\bigr)^{m^{*}-1}, \qquad D_{1}^{i}= \biggl(\frac{1}{6}\biggr)^{m^{*}}n\gamma^{i}\bigl( \gamma_{1}^{i}N\bigr)^{m^{*}-1},\end{gathered} $$

where β is 0 or ∞.

The first existence theorem deals with the case \(1< p<\infty\).

Theorem 3.1

Assume that \((H_{1})\)\((H_{3})\) hold. Furthermore, assume that one of the following conditions is satisfied.

\((C_{1})\) :

There exist two constants r, R with \(0< r\le\delta R\) such that \(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{r}{D_{i}})\) for \(t\in J\), \(0\le\|\mathbf{x}\|\le r\), and \(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac {R}{\delta_{i}D_{1}^{i}})\) for \(t\in J\), \(\delta R\le\|\mathbf{x}\|\le R\);

\((C_{2})\) :

\(f_{i0}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) and \(f_{i}^{\infty}<\phi_{m}(\frac{1}{D_{i}})\) (particularly, \(f_{i0}=\infty \) and \(f_{i}^{\infty}=0\)).

Then, system (1.1)(1.2) has at least one symmetric positive solution.

Proof

Case (1). Considering the condition \((C_{1})\), for all \(\mathbf{x}\in\partial K_{r}\), we have \(\|\mathbf{x}\|= r\) and \(f_{i}(t,\mathbf{x}(t))\le\phi_{m}(\frac{r}{D_{i}})\), \(i=1,2,\ldots,n\). Thus, for all \(t\in J\), we have

$$ \begin{aligned}[b] \|\mathbf{Tx}\|&=\sum _{i=1}^{n}\sup_{t\in J}(T_{i} \mathbf{x}) (t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}(\tau)\phi _{m}\biggl(\frac{r}{D_{i}}\biggr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\bigr)^{m^{*}-1}\frac {r}{D_{i}} \int_{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}\|e\|_{q}\|\psi _{i}\|_{p}\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\frac{1}{6} \gamma^{i}\bigl(\gamma_{1}^{i}\|e\|_{q} \|\psi _{i}\|_{p}\bigr)^{m^{*}-1}\frac{r}{D_{i}} \\ &=\sum_{i=1}^{n}\frac{r}{n}=\| \mathbf{x}\|. \end{aligned} $$
(3.1)

In addition, for all \(\mathbf{x}\in\partial K_{R}\), we have \(\|\mathbf {x}\|= R\), and then it follows from (2.20) and \((C_{1})\) that

$$\begin{aligned} \|\mathbf{Tx}\|&=\sum _{i=1}^{n}\sup_{t\in J}(T_{i} \mathbf{x}) (t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\ge\sum_{i=1}^{n}\rho^{i}\bigl( \rho_{1}^{i}N\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\phi_{m}\biggl( \frac {R}{\delta_{i}D_{1}^{i}}\biggr)\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\biggl(\frac{1}{6} \biggr)^{m^{*}}\delta_{i}\gamma^{i}\bigl(\gamma _{1}^{i}N\bigr)^{m^{*}-1}\frac{R}{\delta_{i}D_{1}^{i}} \\ &=\sum_{i=1}^{n}\frac{R}{n}=\| \mathbf{x}\|. \end{aligned}$$
(3.2)

Case (2). Considering condition \((C_{2})\), it follows from the definition of \(f_{i0}\) and \(f_{i0}>\phi_{m}(\frac{1}{\delta _{i}D_{1}^{i}})\) that there exists \(r_{1}>0\) such that

$$f_{i}(t,\mathbf{x})\ge\bigl(f_{i0}-\varepsilon_{1}^{i} \bigr)\phi_{m}\bigl(\|\mathbf {x}\|\bigr),\quad \forall t\in J, 0\le\|\mathbf{x}\|\le r_{1}, $$

where \(\varepsilon_{1}^{i}>0\) satisfies \(D_{1}^{i}\delta _{i}(f_{i0}-\varepsilon_{1}^{i})^{m^{*}-1}\ge1\), \(i=1,2,\ldots,n\). Then, for all \(t\in J\), \(\mathbf{x}\in\partial K_{r_{1}}\), we have

$$ \begin{aligned}[b] \|\mathbf{Tx}\|&=\sum _{i=1}^{n}\sup_{t\in J}(T_{i} \mathbf{x}) (t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\ge\sum_{i=1}^{n}\rho^{i}\bigl( \rho_{1}^{i}N\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau) \bigl(f_{i0}- \varepsilon _{1}^{i}\bigr)\phi_{m}\bigl(\|\mathbf{x}\|\bigr) \,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\biggl(\frac{1}{6} \biggr)^{m^{*}}\delta_{i}\gamma^{i}\bigl(\gamma _{1}^{i}N\bigr)^{m^{*}-1}\bigl(f_{i0}- \varepsilon_{1}^{i}\bigr)^{m^{*}-1}\|\mathbf{x}\| \\ &\ge\sum_{i=1}^{n}\frac{\|\mathbf{x}\|}{n}=\| \mathbf{x}\|. \end{aligned} $$
(3.3)

Next, turning to \(f_{i}^{\infty}<\phi_{m}(\frac{1}{D_{i}})\), \(i=1,2,\ldots,n\), and we know that there exists \(\overline{R}_{1}>0\) such that

$$f_{i}(t,\mathbf{x})\le\bigl(f_{i}^{\infty}+ \varepsilon_{2}^{i}\bigr)\phi_{m}\bigl(\| \mathbf{x} \|\bigr),\quad \forall t\in J, \|\mathbf{x}\|\ge\overline{R}_{1}, $$

where \(\varepsilon_{2}^{i}>0\) satisfies \(D_{i}(f_{i}^{\infty }+\varepsilon_{2}^{i})^{m^{*}-1}\le1\), \(i=1,2,\ldots,n\).

Let

$$M^{i}=\max_{0\le\|\mathbf{x}\|\le\overline{R}_{1},t\in J}f_{i}(t,\mathbf{x}),\quad i=1,2,\ldots,n. $$

Thus, for all \(t\in J\), \(\mathbf{x}\in K\), we have \(f_{i}(t,\mathbf {x})\le M^{i}+(f_{i}^{\infty}+\varepsilon_{2}^{i})\phi_{m}(\|\mathbf {x}\|)\), \(i=1,2,\ldots,n\).

Letting

$$\frac{R_{1}}{n}>\max \bigl\{ r_{1},\overline {R}_{1}, \bigl(M^{i}\bigr)^{m^{*}-1}D_{i}\bigl(1-D_{i} \bigl(f_{i}^{\infty}+\varepsilon _{2}^{i} \bigr)^{m^{*}-1}\bigr)^{-1} \bigr\} $$

then, for all \(t\in J\), \(\mathbf{x}\in\partial K_{R_{1}}\), we have

$$ \begin{aligned}[b] \|\mathbf{Tx}\|&=\sum _{i=1}^{n}\sup_{t\in J}(T_{i} \mathbf{x}) (t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\|e\|_{q}\|\psi_{i} \| _{p}\bigr)^{m^{*}-1} \int_{0}^{1}e(s)\phi_{m^{*}} \biggl( \int _{0}^{1}\bigl(M^{i}+ \bigl(f_{i}^{\infty}+\varepsilon_{2}^{i} \bigr)\phi_{m}\bigl(\|\mathbf {x}\|\bigr)\bigr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\bigl(M^{i} \bigr)^{m^{*}-1}\frac{D_{i}}{n}+\bigl(f_{i}^{\infty }+ \varepsilon_{2}^{i}\bigr)^{m^{*}-1}\|\mathbf{x}\| \frac{D_{i}}{n} \\ &< \sum_{i=1}^{n}\frac{R_{1}}{n}- \frac{D_{i}}{n}R_{1}\bigl(f_{i}^{\infty }+ \varepsilon_{2}^{i}\bigr)^{m^{*}-1}+ \bigl(f_{i}^{\infty}+\varepsilon _{2}^{i} \bigr)^{m^{*}-1}\|\mathbf{x}\|\frac{D_{i}}{n} \\ &=\sum_{i=1}^{n}\frac{R_{1}}{n}=\| \mathbf{x}\|. \end{aligned} $$
(3.4)

Applying Lemma 2.5 to (3.1) and (3.2), or (3.3) and (3.4) yields that T has at least one fixed point \(\mathbf{x}^{*}\in\overline{K}_{r,R}\), or \(\mathbf {x}^{*}\in\overline{K}_{r_{1},R_{1}}\). Thus it follows from Remark 2.3 that system (1.1)–(1.2) has at least one symmetric positive solution. This finishes the proof of Theorem 3.1. □

The following theorem deals with the case \(p=\infty\).

Theorem 3.2

Assume that \((H_{1})\)\((H_{3})\), \((C_{1})\) or \((H_{1})\)\((H_{3})\), \((C_{2})\) hold. Then system (1.1)(1.2) has at least one symmetric positive solution.

Proof

Let \(\|e\|_{1}\|\psi_{i}\|_{\infty}\) replace \(\|e\|_{q}\|\psi _{i}\|_{p}\) and repeat the argument above. □

Finally, we consider the case of \(p=1\).

Let

$$D_{i}=n\gamma^{i}\bigl(\gamma_{1}^{i} \bigr)^{m^{*}-1}\bigl(\|\psi_{i}\| _{1}\bigr)^{m^{*}-1} \biggl(\frac{1}{4}\biggr)^{m^{*}-1}. $$

Theorem 3.3

Assume that \((H_{1})\)\((H_{3})\), \((C_{1})\) or \((H_{1})\)\((H_{3})\), \((C_{2})\) hold. Then system (1.1)(1.2) has at least one symmetric positive solution.

Proof

Similar to the proof of Theorem 3.1. For all \(t\in J\), \(\mathbf{x}\in\partial K_{r}\), we have

$$\begin{aligned} \|\mathbf{Tx}\|&=\sum _{i=1}^{n}\sup_{t\in J}(T_{i} \mathbf{x}) (t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}(\tau)\phi _{m}\biggl(\frac{r}{D_{i}}\biggr)\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\gamma^{i}\bigl( \gamma_{1}^{i}\bigr)^{m^{*}-1}\frac{r}{D_{i}}\bigl(\| \psi_{i}\|_{1}\bigr)^{m^{*}-1}\biggl(\frac{1}{4} \biggr)^{m^{*}-1} \\ &=\sum_{i=1}^{n}\frac{r}{n}=\| \mathbf{x}\|. \end{aligned}$$
(3.5)

Next, similar to the proof of Theorem 3.1, we can finish the proof. □

In the following theorems we only consider the case of \(1< p<\infty\).

Theorem 3.4

Assume that \((H_{1})\)\((H_{3})\) hold. Furthermore, assume that one of the following conditions is satisfied.

\((C_{3})\) :

There exists two constants r, R, which satisfy \(0< r\le \delta R\), such that \(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac{r}{\delta _{i}D_{1}^{i}})\) for \(t\in J\), \(0\le\|\mathbf{x}\|\le r\), and \(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{R}{D_{i}})\) for \(t\in J\), \(\delta R\le\|\mathbf{x}\|\le R\);

\((C_{4})\) :

\(f_{i}^{0}<\phi_{m}(\frac{1}{D_{i}})\) and \(f_{i\infty }>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) (particularly, \(f_{i}^{0}=0\) and \(f_{i\infty}=\infty\)),

where \(i=1,2,\ldots,n\). Thus, system (1.1)(1.2) has at least one symmetric positive solution \(\mathbf{x}^{*}\).

Proof

The proof is similar to that of Theorem 3.1, so we omit it here. □

Next, we discuss the multiplicity of system (1.1)–(1.2).

Theorem 3.5

Assume that \((H_{1})\)\((H_{3})\) and the following conditions hold.

\((C_{5})\) :

\(f_{i0}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) and \(f_{i\infty}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\) (particularly, \(f_{i0}=f_{i\infty}=\infty\));

\((C_{6})\) :

there exists a constant \(b>0\) such that \(\max_{t\in J,\|\mathbf{x}\|=b}f_{i}(t,\mathbf{x})<\phi_{m}(\frac{b}{D_{i}})\).

Then system (1.1)(1.2) has at least two symmetric positive solutions \(\mathbf{x}^{*}\), \(\mathbf{x}^{**}\) with

$$ 0< \big\| \mathbf{x}^{**}\big\| < b< \big\| \mathbf{x}^{*}\big\| . $$
(3.6)

Proof

Choose two constants r, R with \(0< r< b< R\). It follows from \((C_{5})\) that

if \(f_{i0}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\), then by means of the proof of (3.3), we obtain that

$$ \|\mathbf{Tx}\|>\|\mathbf{x}\|,\quad \forall\mathbf{x}\in\partial K_{r}; $$
(3.7)

if \(f_{i\infty}>\phi_{m}(\frac{1}{\delta_{i}D_{1}^{i}})\), then by means of the proof of (3.3), we obtain that

$$ \|\mathbf{Tx}\|>\|\mathbf{x}\|,\quad \forall\mathbf{x}\in\partial K_{R}. $$
(3.8)

On the other hand, it follows from \((C_{6})\) that

$$\begin{aligned} \|\mathbf{Tx}\|&=\sum _{i=1}^{n}\sup_{t\in J} \int _{0}^{1}H^{i}(t,s) \phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi _{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}\|e\|_{q}\| \psi_{i}\| _{p}f_{i}\bigl(\tau,\mathbf{x}(\tau) \bigr)\,d\tau \biggr)\,ds \\ &< \sum_{i=1}^{n}\gamma^{i}\bigl( \gamma_{1}^{i}\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}\|e\|_{q}\| \psi_{i}\|_{p}\phi _{m}\biggl(\frac{b}{D_{i}} \biggr)\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\frac{1}{6} \gamma^{i}\bigl(\gamma_{1}^{i}\|e\|_{q} \|\psi _{i}\|_{p}\bigr)^{m^{*}-1}\frac{b}{D_{i}} \\ &=b=\|\mathbf{x}\|,\quad \forall\mathbf{x}\in\partial K_{b}. \end{aligned}$$
(3.9)

Applying Lemma 2.5 to (3.7), (3.8), and (3.9) yields that T has a fixed point \(\mathbf{x}^{**}\in \overline{K}_{r,b}\) and a fixed point \(\mathbf{x}^{*}\in\overline {K}_{b,R}\). Then it follows from Remark 2.3 that system (1.1)–(1.2) has at least two symmetric positive solutions \(\mathbf{x}^{*}\) and \(\mathbf{x}^{**}\). Noticing (3.9), we obtain \(\|\mathbf{x}^{*}\|\neq b\) and \(\|\mathbf{x}^{**}\|\neq b\). Therefore (3.6) holds, and the proof of Theorem 3.5 is complete. □

Similarly, the following Theorem 3.6 can be obtained.

Theorem 3.6

Assume that \((H_{1})\)\((H_{3})\) and the following conditions hold.

\((C_{7})\) :

\(f_{i}^{0}<\phi_{m}(\frac{1}{D_{i}})\) and \(f_{i}^{\infty}<\phi_{m}(\frac{1}{D_{i}})\);

\((C_{8})\) :

There exists a constant \(B>0\) such that \(\min_{t\in J,\|\mathbf{x}\|=B}f_{i}(t,\mathbf{x})>\phi_{m}(\frac{B}{\delta_{i}D_{1}^{i}})\).

Then system (1.1)(1.2) has at least two symmetric positive solutions \(\mathbf{x}^{*}\) and \(\mathbf{x}^{**}\) with

$$0< \big\| \mathbf{x}^{**}\big\| < B< \big\| \mathbf{x}^{*}\big\| . $$

Theorem 3.7

Assume that \((H_{1})\)\((H_{3})\) hold. If there exist 2n positive numbers \(b_{k}\), \(d_{k}\), \(k=1,2,\ldots,n\), with \(b_{1}<\delta d_{1}<d_{1}<b_{2}<\delta d_{2}<d_{2}<\cdots<b_{n}<\delta d_{n}<d_{n}\), such that

\((C_{9})\) :

\(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{b_{k}}{D_{i}})\) for \(t\in J\), \(\delta b_{k}\le\|\mathbf{x}\|\le b_{k}\) and \(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac{d_{k}}{\delta_{i}D_{1}^{i}})\) for \(t\in J\), \(\delta d_{k}\le\|\mathbf{x}\|\le d_{k}\), \(k=1,2,\ldots,n\); or

\((C_{10})\) :

\(f_{i}(t,\mathbf{x})\ge\phi_{m}(\frac{b_{k}}{\delta _{i}D_{1}^{i}})\) for \(t\in J\), \(\delta b_{k}\le\|\mathbf{x}\|\le b_{k}\) and \(f_{i}(t,\mathbf{x})\le\phi_{m}(\frac{d_{k}}{D_{i}})\) for \(t\in J\), \(\delta d_{k}\le\|\mathbf{x}\|\le d_{k}\), \(k=1,2,\ldots,n\).

Then system (1.1)(1.2) has at least n symmetric positive solutions \(\mathbf{x}_{k}\), and \(\mathbf{x}_{k}\) satisfy

$$b_{k}\le\|\mathbf{x}_{k}\|\le d_{k},\quad k=1,2, \ldots,n. $$

Finally, we discuss the existence result corresponding to the case when system (1.1)–(1.2) has no symmetric positive solutions.

Theorem 3.8

Assume that conditions \((H_{1})\)\((H_{3})\) hold and \(f_{i}(t,\mathbf{x})<\phi_{m} (\frac {\|\mathbf{x}\|}{D_{i}} )\), \(\forall t\in J\), \(\|\mathbf{x}\|>0\). Then system (1.1)(1.2) has no positive solution.

Proof

Assume to the contrary that x is a positive solution of system (1.1)–(1.2), then for any \(0< t<1\), we have \(\mathbf{x}\in K\), \(x_{i}(t)>0\), and

$$\begin{aligned} \|\mathbf{x}\|&=\sum_{i=1}^{n} \sup_{t\in J}x_{i}(t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}(\tau )f_{i}\bigl(\tau,\mathbf{x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &< \sum_{i=1}^{n}\gamma^{i}\bigl( \gamma_{1}^{i}\bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\psi_{i}(\tau)\phi _{m}\biggl(\frac{\|\mathbf{x}\|}{D_{i}}\biggr)\,d\tau \biggr)\,ds \\ &\le\sum_{i=1}^{n}\gamma^{i} \bigl(\gamma_{1}^{i}\bigr)^{m^{*}-1}\frac{\|\mathbf {x}\|}{D_{i}} \int_{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}\|e\|_{q}\| \psi_{i}\|_{p}\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\frac{1}{6} \gamma^{i}\bigl(\gamma_{1}^{i}\|e\|_{q} \|\psi _{i}\|_{p}\bigr)^{m^{*}-1}\frac{\|\mathbf{x}\|}{D_{i}} \\ &=\|\mathbf{x}\|. \end{aligned}$$

This is a contradiction, and this completes the proof. □

Similarly, we have the following results.

Theorem 3.9

Assume that \((H_{1})\)\((H_{3})\) hold and \(f_{i}(t,\mathbf{x})>\phi_{m} (\frac{\|\mathbf{x}\|}{\delta _{i}D_{1}^{i}} )\), \(\forall t\in J\), \(\|\mathbf{x}\|>0\), \(i=1,2,\ldots,n\). Then system (1.1)(1.2) has no positive solution.

Proof

Assume that x is a positive solution of system (1.1)–(1.2). Then, for any \(0< t<1\), we have \(\mathbf {x}\in K\), \(x_{i}(t)>0\), and

$$\begin{aligned} \|\mathbf{x}\|&=\sum_{i=1}^{n} \sup_{t\in J}x_{i}(t) \\ &=\sum_{i=1}^{n}\sup_{t\in J} \int_{0}^{1}H^{i}(t,s)\phi_{m^{*}} \biggl( \int_{0}^{1}H_{1}^{i}(s,\tau) \psi_{i}(\tau)f_{i}\bigl(\tau,\mathbf{x}(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\ge\sum_{i=1}^{n}\rho^{i}\bigl( \rho_{1}^{i}N\bigr)^{m^{*}-1} \int_{0}^{1}e(s)\phi _{m^{*}} \biggl( \int_{0}^{1}e(\tau)f_{i}\bigl(\tau, \mathbf{x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &=\sum_{i=1}^{n}\delta_{i} \gamma^{i}\bigl(\gamma_{1}^{i}N \bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)f_{i}\bigl(\tau, \mathbf {x}(\tau)\bigr)\,d\tau \biggr)\,ds \\ &>\sum_{i=1}^{n}\delta_{i} \gamma^{i}\bigl(\gamma_{1}^{i}N \bigr)^{m^{*}-1} \int _{0}^{1}e(s)\phi_{m^{*}} \biggl( \int_{0}^{1}e(\tau)\phi_{m}\biggl( \frac{\| \mathbf{x}\|}{\delta_{i}D_{1}^{i}}\biggr)\,d\tau \biggr)\,ds \\ &\ge\sum_{i=1}^{n}\biggl(\frac{1}{6} \biggr)^{m^{*}}\delta_{i}\gamma^{i}\bigl(\gamma _{1}^{i}N\bigr)^{m^{*}-1}\frac{\|\mathbf{x}\|}{\delta_{i}D_{1}^{i}} \\ &=\|\mathbf{x}\|. \end{aligned} $$

This leads to a contradiction, and this finishes the proof. □

An example

In the following example, we select \(n=2\), \(m=2\), \(p=2 \), and \(N=1\).

Example 4.1

Consider the following system:

$$ \left\{ \textstyle\begin{array}{l} ( \phi_{2}(\mathbf{x}{''}(t))){''}=\Psi(t)\mathbf{f}(t,\mathbf{x}(t)), \quad0< t< 1,\\ \mathbf{x}(0)=\mathbf{x}(1)=\int_{0}^{1}\mathbf{g}(s)\mathbf{x}(s)\, ds,\\ \phi_{2}(\mathbf{x}{''}(0))=\phi_{2}(\mathbf{x}{''}(1))=\int _{0}^{1}\mathbf{h}(s) \phi_{2}(\mathbf{x}{''}(s))\,ds, \end{array}\displaystyle \right . $$
(4.1)

where

$$\begin{gathered} \mathbf{g}(t)=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{1}{2} & 0 \\ 0 & \frac{1}{2} \end{array}\displaystyle \right ), \quad\quad \mathbf{h}(t)=\left ( \textstyle\begin{array}{c@{\quad}c} \frac{1}{2} & 0 \\ 0 & \frac{1}{3} \end{array}\displaystyle \right ), \\ \Psi(t)=\left ( \textstyle\begin{array}{c@{\quad}c} 6 & 0 \\ 0 & 3 \end{array}\displaystyle \right ), \quad\quad \mathbf{f}(t,\mathbf{x})=\left ( \textstyle\begin{array}{c} (1+\sin\pi t)x_{1}^{6} \\ x_{2}^{4} \end{array}\displaystyle \right ).\end{gathered} $$

Then, by calculations, we obtain that \(m^{*}=2\), \(\mu_{1}=\mu_{2}=\nu _{1}=\frac{1}{2}\), \(\nu_{2}=\frac{1}{3}\), \(\gamma^{1}=\gamma^{2}=\gamma _{1}^{1}=2\), \(\gamma_{1}^{2}=\frac{3}{2}\), \(\rho^{1}=\rho^{2}=\rho _{1}^{1}=\frac{1}{6}\), \(\rho_{1}^{2}=\frac{1}{12}\), \(\delta_{1}=\frac {1}{144}\), \(\delta_{2}=\frac{1}{216}\), \(\delta=\frac{1}{216}\), \(D_{1}=\frac {8}{\sqrt{30}}\), \(D_{2}=\frac{3}{\sqrt{30}}\), \(D_{1}^{1}=\frac{2}{9}\), \(D_{1}^{2}=\frac{1}{6}\), and

$$\begin{gathered} H^{1}(t,s)=H_{1}^{1}(t,s)=H^{2}(t,s)=G(t,s)+ \int_{0}^{1}G(s,\tau)\,d\tau , \\ H_{1}^{2}(t,s)=G(t,s)+\frac{1}{2} \int_{0}^{1}G(s,\tau)\,d\tau, \end{gathered} $$

where

$$G(t,s)= \left\{ \textstyle\begin{array}{l} t(1-s),\quad 0\le t\le s\le1,\\ s(1-t), \quad0\le s\le t\le1. \end{array}\displaystyle \right . $$

Clearly, conditions \((H_{1})\)\((H_{3})\) hold. Next, we show that the condition \((C_{1})\) of Theorem 3.1 holds. Choosing \(r=\frac {1}{2}\), \(R=6^{5}\times\sqrt[6]{3}\), we obtain that

$$\begin{gathered} \phi_{2}\biggl(\frac{r}{D_{1}}\biggr)= \frac{1}{2}\times\frac{\sqrt{30}}{8}=\frac {\sqrt{30}}{16}, \qquad \phi_{2}\biggl(\frac{R}{\delta _{1}D_{1}^{1}}\biggr)=144\times\frac{9}{2} \times6^{5}\times\sqrt[6]{3}=3\times 6^{8}\times\sqrt[6]{3}, \\ \phi_{2}\biggl(\frac{r}{D_{2}}\biggr)=\frac{1}{2}\times \frac{\sqrt{30}}{3}=\frac {\sqrt{30}}{6}, \qquad\phi_{2}\biggl( \frac{R}{\delta_{2}D_{1}^{2}}\biggr)=216\times 6\times6^{5}\times\sqrt[6]{3}= \times6^{9}\times\sqrt[6]{3}, \\ f_{1}(t,\mathbf{x})=(1+\sin\pi t)x_{1}^{6}\le2 \times r^{6}=2\times \biggl(\frac{1}{2}\biggr)^{6}= \frac{1}{32},\quad \text{for }t\in J, 0\le\|\mathbf{x}\|\le r, \\ f_{1}(t,\mathbf{x})=(1+\sin\pi t)x_{1}^{6}\ge( \delta_{1}R)^{6}=\biggl(\frac {1}{144} \times6^{5}\times\sqrt[6]{3}\biggr)^{6}=3 \times(54)^{6},\\ \quad \text{for }t\in J, \delta R\le\|\mathbf{x}\|\le R, \\ f_{2}(t,\mathbf{x})=x_{2}^{4}\le r^{4}=\biggl(\frac{1}{2}\biggr)^{4}= \frac{1}{16},\quad \text{for }t\in J, 0\le\|\mathbf{x}\|\le r, \\ f_{1}(t,\mathbf{x})=x_{2}^{4}\ge( \delta_{2}R)^{4}=\biggl(\frac{1}{216}\times 6^{5}\times\sqrt[6]{3}\biggr)^{4}=3^{\frac{2}{3}} \times6^{8},\quad \text{for }t\in J, \delta R\le\|\mathbf{x}\|\le R. \end{gathered} $$

Therefore, \(f_{i}(t,\mathbf{x})\le\phi_{2}(\frac{r}{D_{i}})\), for all \(t\in J\), \(0\le\|\mathbf{x}\|\le r\), and \(f_{i}(t,\mathbf{x})\ge\phi _{2}(\frac{R}{\delta_{i}D_{1}^{i}})\), for all \(t\in J\), \(\delta R\le\| \mathbf{x}\|\le R\), \(i=1,2\).

Therefore, it follows from Theorem 3.1 that system (4.1) has at least one symmetric positive solution.

Conclusion

In this paper, we obtained several sufficient conditions for the existence, multiplicity, and nonexistence of symmetric positive solutions for the fourth-order n-dimensional m-Laplace system with integral boundary conditions. Our results will be a useful contribution to the existing literature on the topic of fourth-order n-dimensional m-Laplace systems.

References

  1. 1.

    Zhang, X., Liu, L.: Positive solutions of fourth-order four-point boundary value problems with p-Laplacian operator. J. Math. Anal. Appl. 336, 1414–1423 (2007)

  2. 2.

    Graef, J.R., Qian, C., Yang, B.: A three point boundary value problem for nonlinear fourth order differential equations. J. Math. Anal. Appl. 287, 217–233 (2003)

  3. 3.

    Sun, J., Wang, X.: Monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. Math. Probl. Eng. 2011, Article ID 609189 (2011)

  4. 4.

    Yao, Q.: Positive solutions of nonlinear beam equations with time and space singularities. J. Math. Anal. Appl. 374, 681–692 (2011)

  5. 5.

    O’Regan, D.: Solvability of some fourth (and higher) order singular boundary value problems. J. Math. Anal. Appl. 161, 78–116 (1991)

  6. 6.

    Yang, B.: Positive solutions for the beam equation under certain boundary conditions. Electron. J. Differ. Equ. 2005, Article ID 78 (2005)

  7. 7.

    Zhang, K.: Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms. Topol. Methods Nonlinear Anal. 40, 53–70 (2012)

  8. 8.

    Gupta, G.P.: Existence and uniqueness theorems for the bending of an elastic beam equation. Appl. Anal. 26, 289–304 (1988)

  9. 9.

    Agarwal, R.P.: On fourth-order boundary value problems arising in beam analysis. Differ. Integral Equ. 2, 91–110 (1989)

  10. 10.

    Bonanno, G., Bella, B.D.: A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. 343, 1166–1176 (2008)

  11. 11.

    Bai, Z.: Positive solutions of some nonlocal fourth-order boundary value problem. Appl. Math. Comput. 215, 4191–4197 (2010)

  12. 12.

    Hao, X., Xu, N., Liu, L.: Existence and uniqueness of positive solutions for fourth-order m-point boundary value problems with two parameters. Rocky Mt. J. Math. 43, 1161–1180 (2013)

  13. 13.

    Feng, M.: Multiple positive solutions of fourth-order impulsive differential equations with integral boundary conditions and one-dimensional p-Laplacian. Bound. Value Probl. 2011, Article ID 654871 (2011)

  14. 14.

    Anderson, D.R., Avery, R.I.: A fourth-order four-point right focal boundary value problem. Rocky Mt. J. Math. 36, 367–380 (2006)

  15. 15.

    Ma, R., Wang, H.: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. Anal. 59, 225–231 (1995)

  16. 16.

    Fan, W., Hao, X., Liu, L., Wu, Y.: Nontrivial solutions of singular fourth-order Sturm–Liouville boundary value problems with a sign-changing nonlinear term. Appl. Math. Comput. 217, 6700–6708 (2011)

  17. 17.

    Cui, Y., Sun, J.: Existence of multiple positive solutions for fourth-order boundary value problems in Banach spaces. Bound. Value Probl. 2012, Article ID 107 (2012)

  18. 18.

    Ma, H.: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal. 68, 645–651 (2008)

  19. 19.

    Webb, J.R.L., Infante, G., Franco, D.: Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions. Proc. R. Soc. Edinb., Sect. A 138, 427–446 (2008)

  20. 20.

    Zhang, X., Ge, W.: Symmetric positive solutions of boundary value problems with integral boundary conditions. Appl. Math. Comput. 219, 3553–3564 (2012)

  21. 21.

    Zhang, X., Feng, M., Ge, W.: Symmetric positive solutions for p-Laplacian fourth-order differential equations with integral boundary conditions. J. Comput. Appl. Math. 222, 561–573 (2008)

  22. 22.

    Cui, Y., Zou, Y.: Existence and uniqueness theorems for fourth-order singular boundary value problems. Comput. Math. Appl. 58, 1449–1456 (2009)

  23. 23.

    Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems. Nonlinear Anal. 67, 1704–1709 (2007)

  24. 24.

    Zhang, X., Cui, Y.: Positive solutions for fourth-order singular-Laplacian differential equations with integral boundary conditions. Bound. Value Probl. 2010, Article ID 862079 (2010)

  25. 25.

    Afrouzi, G.A., Shokooh, S.: Three solutions for a fourth-order boundary-value problem. Electron. J. Differ. Equ. 2015, Article ID 45 (2015)

  26. 26.

    Sun, M., Xing, Y.: Existence results for a kind of fourth-order impulsive integral boundary value problems. Bound. Value Probl. 2016, Article ID 81 (2016)

  27. 27.

    Zhou, Y., Zhang, X.: New existence theory of positive solutions to fourth order p-Laplacian elasticity problems. Bound. Value Probl. 2015, Article ID 205 (2015)

  28. 28.

    Baraket, S., Radulescu, V.: Combined effects of concave-convex nonlinearities in a fourth-order problem with variable exponent. Adv. Nonlinear Stud. 16, 409–419 (2016)

  29. 29.

    Boureanu, M., Radulescu, V., Repovs, D.: On a \(p(\cdot)\)-biharmonic problem with no-flux boundary condition. Comput. Math. Appl. 72, 2505–2515 (2016)

  30. 30.

    Demarque, R., Miyagaki, O.: Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings. Adv. Nonlinear Anal. 4, 135–151 (2015)

  31. 31.

    Kong, L.: Multiple solutions for fourth order elliptic problems with \(p(x)\)-biharmonic operators. Opusc. Math. 36, 253–264 (2016)

  32. 32.

    Boucherif, A.: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 70, 364–371 (2009)

  33. 33.

    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)

  34. 34.

    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)

  35. 35.

    Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252–263 (2015)

  36. 36.

    Zhang, X., Liu, L., Wu, Y.: Variational structure and multiple solutions for a fractional advection-dispersion equation. Comput. Math. Appl. 68, 1794–1805 (2014)

  37. 37.

    Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26–33 (2014)

  38. 38.

    Kong, L.: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2628–2638 (2010)

  39. 39.

    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)

  40. 40.

    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)

  41. 41.

    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)

  42. 42.

    Jiang, J., Liu, L., Wu, Y.: Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions. Adv. Differ. Equ. 2012, Article ID 124 (2012)

  43. 43.

    Wang, M., Feng, M., Li, P.: Infinitely many positive solutions for Neumann boundary value problems of second-order differential equations with infinitely many singularities and advanced deviating argument. J. Beijing Inf. Sci. Technol. Univ. 32, 6–10 (2017)

  44. 44.

    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)

  45. 45.

    Yan, F., Zuo, M., Hao, X.: Positive solution for a fractional singular boundary value problem with p-Laplacian operator. Bound. Value Probl. 2018, Article ID 51 (2018)

  46. 46.

    Karakostas, G.L., Tsamatos, P.Ch.: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002, Article ID 30 (2002)

  47. 47.

    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)

  48. 48.

    Feng, M., Ge, W.: Positive solutions for a class of m-point singular boundary value problems. Math. Comput. Model. 46, 375–383 (2007)

  49. 49.

    Jiang, J., Liu, L., Wu, Y.: Second-order nonlinear singular Sturm–Liouville problems with integral boundary problems. Appl. Math. Comput. 215, 1573–1582 (2009)

  50. 50.

    Lan, K.Q.: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 63, 690–704 (2001)

  51. 51.

    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)

  52. 52.

    Zhang, X., Liu, L., Wu, Y., Lu, Y.: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680–4691 (2013)

  53. 53.

    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)

  54. 54.

    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)

  55. 55.

    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)

  56. 56.

    Mao, J., Zhao, Z.: The existence and uniqueness of positive solutions for integral boundary value problems. Bull. Malays. Math. Sci. Soc. 34, 153–164 (2011)

  57. 57.

    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)

  58. 58.

    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)

  59. 59.

    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)

  60. 60.

    Feng, M.: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett. 24, 1419–1427 (2011)

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Acknowledgements

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

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This work is sponsored by the National Natural Science Foundation of China (11401031), the Beijing Natural Science Foundation (1163007), the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (KM201611232017), and the teaching reform project of Beijing Information Science & Technology University (2018JGZD41).

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Correspondence to Meiqiang Feng.

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Keywords

  • Symmetric positive solutions
  • n-dimensional system
  • m-Laplace operator
  • Matrix theory
  • Fixed point technique