1 Introduction

The interest in using fractional differential equations in modeling ecological economy and diffusive processes has wide literature. Especially when one wants to model long-range ecological economy phenomena and diffusive interaction, fractional differential operator has higher accuracy than integer order differential model in depicting the co-evolution process of economic, social and ecological subsystems and the transport of solute in highly heterogeneous porous media. Recently this interest has also been activated by recent progress in the mathematical theory and psycho-socio-economical dynamics, see [1, 2]. On the other hand, since fractional order derivative, which exhibits a long time memory behavior, is nonlocal, thus except for diffusive processes in porous medium flow and ecological economy phenomena, the differential equation with fractional derivative can also describe many other physical phenomena in natural sciences and engineering, such as earthquake, traffic flow, measurement of viscoelastic material properties, polymer rheology and various material processes [3–37].

In this paper, we study the existence of maximum and minimum solutions for the following nonlocal fractional differential system:

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\beta_{1}} (\varphi _{p_{1}} (-\mathscr {D}_{t}^{\alpha_{1}}x_{1} ) )(t)=f_{1}(x_{1}(t), x_{2}(t)), \\ -\mathscr {D}_{t}^{\beta_{2}} (\varphi_{p_{2}} (-\mathscr {D}_{t}^{\alpha_{2}}x_{2} ) )(t)=f_{2}(x_{1}(t), x_{2}(t)) \\ x_{1}(0)=0,\qquad \mathscr {D}_{t}^{\alpha _{1}}x_{1}(0)=\mathscr {D}_{t}^{\alpha _{1}}x_{1}(1)=0,\qquad x_{1}(1)=\int_{0}^{1}x_{1}(t)\,dA_{1}(t), \\ x_{2}(0)=0,\qquad \mathscr {D}_{t}^{\alpha _{2}}x_{2}(0)=\mathscr {D}_{t}^{\alpha _{2}}x_{2}(1)=0, \qquad x_{2}(1)=\int_{0}^{1}x_{2}(t)\,dA_{2}(t), \end{cases} $$
(1.1)

where \(\mathscr {D}_{t}^{\alpha_{i}}\), \(\mathscr {D}_{t}^{\beta_{i}}\) are the standard Riemann-Liouville derivatives satisfying \(1<\alpha _{i}, \beta_{i}\le 2\), \(\int^{1}_{0}x(s)\,dA_{i}(s)\) denotes a Riemann-Stieltjes integral and \(A_{i}\) is a function of bounded variation, \(\varphi_{p_{i}}\) is the p-Laplacian operator defined by \(\varphi_{p_{i}}(s)= |s|^{p_{i}-2}s\), \(p_{i}>1\), where \(i=1,2\). Obviously, \(\varphi_{p_{i}}(s)\) is invertible and its inverse operator is \(\varphi_{q_{i}}(s)\), where \(q_{i}=\frac{p_{i}}{p_{i}-1}\), \(i=1,2\), are conjugate indices of \(p_{i}\).

Recently, some interesting results about the existence of positive solutions for nonlinear fractional equation with p-Laplacian operator have been reported [21–26, 38]. In [23], Chen and Liu investigated the existence of solutions for the anti-periodic fractional order p-Laplacian boundary value problem with the following form:

$$ \textstyle\begin{cases} \mathscr {D}_{t}^{\beta} (\varphi_{p} (\mathscr {D}_{t}^{\alpha}x ) )(t)=f(t, x(t)), \quad t\in (0,1), \\ x(0)=-x(1),\qquad \mathscr {D}_{t}^{\alpha }x(0)=-\mathscr {D}_{t}^{\alpha }x(1), \end{cases} $$
(1.2)

where \(0 < \alpha, \beta\le1\), \(1 < \alpha+ \beta\le2\), \(\mathscr {D}_{t}^{\beta}\) is a Caputo fractional derivative, and \(f : [0, 1]\times R^{2} \to R\) is continuous. Under certain nonlinear growth conditions of the nonlinearity, a new existence result is obtained by using Schaefer’s fixed point theorem. By means of upper and lower solutions method, Wang et al. [26] studied the existence of positive solutions for the following nonlocal fractional p-Laplacian equation:

$$ \textstyle\begin{cases} \mathscr {D}_{t}^{\beta} (\varphi_{p} (\mathscr {D}_{t}^{\alpha}x ) )(t)+f(t, x(t))=0, \quad t\in (0,1), \\ x(0)=0,\qquad x(1)= a x(\xi),\qquad \mathscr {D}_{t}^{\alpha }x(0)=0,\qquad \mathscr {D}_{t}^{\alpha }x(1)=b \mathscr {D}_{t}^{\alpha }x(\eta), \end{cases} $$
(1.3)

where \(1<\alpha, \beta\le2\), \(0 \le a,b\le1\), \(0<\xi,\eta<1\). More recently, Zhang et al. [38] considered the uniqueness of positive solution for the following fractional order differential equation:

$$ \textstyle\begin{cases} \mathscr {D}_{t}^{\beta} (\varphi_{p} (-\mathscr {D}_{t}^{\alpha}x ) )(t)=-f(x(t), \mathscr {D}_{t}^{\gamma}x(t)),\quad t\in(0,1), \\ \mathscr {D}_{t}^{\alpha }x(0)=\mathscr {D}_{t}^{\alpha +1}x(0)=\mathscr {D}_{t}^{\alpha }x(1)=0, \\ \mathscr {D}_{t}^{\gamma}x(0)=0,\qquad \mathscr {D}_{t}^{\gamma}x(1)= \int^{1}_{0}\mathscr {D}_{t}^{\gamma}x(s)\,dA(s), \end{cases} $$
(1.4)

where \(\mathscr {D}_{t}^{\alpha }\), \(\mathscr {D}_{t}^{\beta }\), \(\mathscr {D}_{t}^{\gamma}\) are the standard Riemann-Liouville derivatives, \(\int^{1}_{0}x(s)\,dA(s)\) denotes a Riemann-Stieltjes integral and \(0<\gamma\le1<\alpha \le2<\beta<3\), \(\alpha-\gamma>1\), A is a function of bounded variation and dA can be a signed measure. Under the case where the nonlinearity \(f(u,v)\) may be singular at both \(u=0\) and \(v=0\), the uniqueness of positive solution for equation (1.4) was established via the fixed point theorem of the mixed monotone operator.

However, to the best of our knowledge, there are relatively few results on a fractional order differential system involving the p-Laplacian operator and nonlocal Riemann-Stieltjes integral boundary conditions, and no work has been done concerning the maximal and minimal solutions of system (1.1). Thus, motivated by the above work, in this paper, we consider the maximum and minimum solutions for a fractional order p-Laplacian system subject to a nonlocal Riemann-Stieltjes integral boundary condition. Difference from the above mentioned work is that in this paper we introduce new type growth condition of nonlinearity which covers a large number of nonlinear functions; at the same time, the existence, estimation of the lower and upper bounds and the convergent iterative scheme of minimal and maximal solutions for system (1.1) are also established.

2 Preliminaries and lemmas

A number of definitions for the fractional derivative have emerged over the years, and in this paper, we carry out our work base on the sense of Riemann-Liouville fractional derivatives; for details, see [15, 39, 40]. Here we only recall a famous semigroup property for Riemann-Liouville fractional calculus.

Proposition 2.1

see [15, 39, 40]

Let \(\alpha> 0\), and \(f(t)\) is integrable, then

$$I^{\alpha} \mathscr {D}_{t}^{\alpha}f(t)=f(t)+c_{1}t^{\alpha -1}+c_{2}t^{\alpha-2}+ \cdots+c_{n}t^{\alpha-n}, $$

where \(c_{i}\in\mathbb{R}\) (\(i=1,2,\ldots,n\)), n is the smallest integer greater than or equal to α.

Let \(i=1,2\), we firstly focus on the following linear fractional differential equation subject to the nonlocal Riemann-Stieltjes integral boundary condition:

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\alpha _{i}}x_{i}(t)=h_{i}(t),\quad t\in(0,1), \\ x_{i}(0)=0,\qquad x_{i}(1)=\int _{0}^{1}x_{i}(t)\,dA_{i}(t). \end{cases} $$
(2.1)

In order to establish the existence of positive solutions for system (1.1), it is necessary to find Green’s function of BVP (2.1). The following result has been given in [38].

Lemma 2.1

see [38]

Given \(h_{i}\in L^{1}(0, 1)\) and \(1<\alpha _{i}\le2\), then the following boundary value problem

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\alpha_{i}}x_{i}(t)=h_{i}(t) ,\quad 0< t< 1 , \\ x_{i}(0)=x_{i}(1)=0, \end{cases} $$
(2.2)

has the unique solution

$$x_{i}(t)= \int_{0}^{1}G_{\alpha_{i}}(t,s)h_{i}(s) \,ds, $$

where

$$ G_{\alpha_{i}}(t,s)=\frac{1}{\Gamma(\alpha_{i})} \textstyle\begin{cases} {[t(1-s)]}^{\alpha_{i}-1}, &0\leq t\leq s\leq1 , \\ {[t(1-s)]}^{\alpha_{i}-1}-(t-s)^{\alpha_{i}-1},& 0\leq s\leq t\leq 1.\end{cases} $$
(2.3)

By Proposition 2.1, it is easy to get that the unique solution of the following boundary value problem

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\alpha_{i}}x_{i}(t)=0 ,\quad 0< t< 1, \\ x_{i}(0)=0,\qquad x_{i}(1)=1, \end{cases} $$
(2.4)

is \(t^{\alpha_{i}-1}\). Thus, by [16] and [17], we have the following lemma.

Lemma 2.2

see [16]

If \(1<\alpha _{i}\le2\) and \(h_{i}\in L^{1}[0,1]\), then the boundary value problem (2.1) has the unique solution

$$ x_{i}(t)= \int_{0}^{1}H_{i}(t,s)h_{i}(s) \,ds, $$
(2.5)

where

$$ \begin{aligned} &H_{i}(t,s)=\frac{t^{\alpha_{i}-1}}{1-\mathcal {A}_{i}} \mathcal{G}_{A_{i}}(s)+G_{\alpha_{i}}(t,s), \quad \mathcal{A}_{i}= \int _{0}^{1}t^{\alpha_{i}-1}\,dA_{i}(t), \\ &\mathcal{G}_{A_{i}}(s)= \int_{0}^{1}G_{\alpha_{i}}(t, s) \,dA_{i}(t), \end{aligned} $$
(2.6)

and \(G_{\alpha_{i}}(t,s)\) is defined by (2.3).

Now we introduce the following necessary condition to ensure the nonnegativity of Green’s function.

  1. (H0)

    \(A_{i}\) is functions of bounded variation satisfying \(\mathcal{G}_{A_{i}}(s)\ge0\) for \(s\in[0, 1]\) and \(0\le\mathcal{A}_{i}<1\).

Lemma 2.3

see [16]

Assume (H0) holds, then \(G_{\alpha_{i}}(t,s)\) and \(H_{i}(t,s)\) have the following properties:

  1. (1)

    \(G_{\alpha_{i}}(t,s)\) and \(H_{i}(t,s)\) are nonnegative and continuous for \((t,s)\in[0,1]\times[0,1]\).

  2. (2)

    For any \(t,s\in [0,1]\), \(G_{\alpha_{i}}(t,s)\) satisfies

    $$ \frac{t^{\alpha_{i} -1}(1-t)s(1-s)^{\alpha_{i} -1}}{\Gamma(\alpha_{i} )}\leq G_{\alpha_{i}}(t,s) \leq \frac{\alpha_{i}-1}{\Gamma(\alpha_{i} )}s(1-s)^{\alpha_{i} -1}. $$
    (2.7)
  3. (3)

    There exist two constants a, b such that

    $$ a{t^{\alpha_{i}-1}}\mathcal{G}_{A_{i}}(s)\le H_{i}(t,s)\le b t^{\alpha_{i}-1}, \quad s,t\in[0,1]. $$
    (2.8)

For convenience of writing, for \(h_{i}\in L^{1}[0,1]\) and \(p_{i}>1\), \(h_{i}\ge0\), \(i=1,2\), we rewrite the following linear boundary value problems:

$$\textstyle\begin{cases} -\mathscr {D}_{t}^{\beta_{1}} (\varphi_{p_{1}} (-\mathscr {D}_{t}^{\alpha_{1}}x_{1} ) )(t)=h_{1}(t),\quad t\in(0,1), \\ x_{1}(0)=0,\qquad \mathscr {D}_{t}^{\alpha _{1}}x(0)=\mathscr {D}_{t}^{\alpha _{1}}x_{1}(1)=0,\qquad x_{1}(1)=\int_{0}^{1}x_{1}(t)\,dA_{1}(t), \end{cases} $$

and

$$\textstyle\begin{cases} -\mathscr {D}_{t}^{\beta_{2}} (\varphi_{p_{2}} (-\mathscr {D}_{t}^{\alpha_{2}}x_{2} ) )(t)=h_{2}(t),\quad t\in(0,1), \\ x_{2}(0)=0,\qquad \mathscr {D}_{t}^{\alpha _{2}}x(0)=\mathscr {D}_{t}^{\alpha _{2}}x_{2}(1)=0,\qquad x_{2}(1)=\int_{0}^{1}x_{2}(t)\,dA_{2}(t), \end{cases} $$

as

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\beta _{i}} (\varphi_{p_{i}} (-\mathscr {D}_{t}^{\alpha_{i}}x_{i} ) )(t)=h_{i}(t),\quad t\in(0,1), \\ x_{i}(0)=0, \qquad \mathscr {D}_{t}^{\alpha _{i}}x(0)=\mathscr {D}_{t}^{\alpha _{i}}x_{i}(1)=0,\qquad x_{i}(1)=\int_{0}^{1}x_{i}(t)\,dA_{i}(t). \end{cases} $$
(2.9)

Lemma 2.4

The linear boundary value problem (2.9) has a unique positive solution

$$x_{i}(t)= \int_{0}^{1}H_{i}(t,s) \biggl( \int_{0}^{1}G_{\beta_{i}}(s,\tau)h_{i}( \tau )\,d\tau \biggr)^{q_{i}-1}\,ds, $$

where \(q_{i}\) is conjugate indices of \(p_{i}\).

Proof

Let \(w_{i} =-\mathscr {D}_{t}^{\alpha_{i}}x_{i}\), \(v_{i} = \varphi_{p_{i}}(w_{i})\), by Lemma 2.1, the unique solution of the boundary value problem

$$\textstyle\begin{cases} -\mathscr {D}_{t}^{\beta_{i}}v_{i}(t)=h_{i}(t),\quad t\in (0,1), \\ v_{i}(0)=v_{i}(1)=0, \end{cases} $$

is

$$ v_{i}(t)= \int_{0}^{1}G_{\beta_{i}}(t,s) h_{i}(s)\,ds,\quad t \in[0, 1]. $$
(2.10)

Noting that \(-\mathscr {D}_{t}^{\alpha_{i}}x=w_{i}\), \(w_{i} = \varphi^{-1}_{p_{i}}(v_{i})\) as well as (2.10), we get that the solution of (2.9) satisfies

$$\textstyle\begin{cases} -\mathscr {D}_{t}^{\alpha_{i}}x_{i}(t)=\varphi ^{-1}_{p_{i}} (\int_{0}^{1}G_{\beta_{i}}(t,s) h_{i}(s)\,ds ),\quad t\in (0,1), \\ x_{i}(0)=0,\qquad x_{i}(1)=\int_{0}^{1}x_{i}(t)\,dA_{i}(t), \end{cases} $$

which implies that

$$x_{i}(t)= \int_{0}^{1}H_{i}(t,s) \varphi^{-1}_{p_{i}} \biggl( \int_{0}^{1}G_{\beta _{i}}(s,\tau) h_{i}(\tau)\,d\tau \biggr)\,ds,\quad t\in[0,1], $$

that is,

$$x_{i}(t)= \int_{0}^{1}H_{i}(t,s) \biggl( \int_{0}^{1}G_{\beta_{i}}(s,\tau) h_{i}(\tau )\,d\tau \biggr)^{q_{i}-1}\,ds, \quad t\in[0,1]. $$

 □

To establish the existence of positive solution of system (1.1), the following new growth condition for nonlinearity will be used in the rest of the paper.

  1. (H1)

    \(f_{1},f_{2}: [0,+\infty)\times[0,+\infty)\to(0,+\infty )\) are continuous and nondecreasing in the first variable and second variable, and there exist positive constants \(\epsilon_{1}>\frac{1}{q_{1}-1}\), \(\epsilon_{2}>\frac{1}{q_{2}-1}\) and M such that

    $$ \max \biggl\{ \sup_{\substack{{s,t\ge0}\\{s+t\neq0}}}\frac{f_{1} ( s,t ) }{(s+t)^{\epsilon_{1}}}, \sup _{\substack{{s,t\ge0}\\{s+t\neq0}}}\frac{f_{2} ( s,t ) }{(s+t)^{\epsilon_{2}}} \biggr\} \le M. $$
    (2.11)

Remark 2.1

In this work, we introduce the growth condition (2.11) for the first time which differs from previous work [10, 11, 16–26, 38]. In particular, it includes a large number of nonlinear functions, some basic examples of \(f_{1}\), \(f_{2}\) satisfying (H1) are

  1. (1)

    \(f(s,t)=a_{0}+\sum_{i=1}^{m}a_{i}(s+t)^{\gamma _{i}}\), where \(a_{i},\gamma_{i}>0\), \(i=0,1,2,\ldots,m\).

  2. (2)

    \(f(s,t)= [ a+\sum_{i=1}^{m}a_{i}(s+t)^{\mu_{i}} ] ^{\frac{1}{\mu }}\), where a, μ, \(a_{i},\mu_{i}\) (\(i=1,2,\ldots,m\)) are positive constants.

  3. (3)

    \(f(s,t)={(s+t+1)^{\mu+1}}\ln ( 1+\frac{1}{1+s+t} ) +(s+t)^{\mu}+a\), \(a, \mu>0\).

  4. (4)

    \(f(s,t)=\ln(2+s+t)\).

Proof

(1)-(4) are obvious, we omit the proof. □

Remark 2.2

If \(f_{i}(s,t)=g_{i}(s+t)\), then the following interesting cases are also included by (2.11):

  1. Case 1.

    There exists a constant \(\epsilon_{i}>\frac{1}{q_{i}-1}\) such that \(\frac{g_{i}(x)}{x^{\epsilon_{i}}}\) is increasing on x and

    $$\lim_{x\to+\infty}{\frac{g_{i}(x)}{x^{\epsilon_{i}}}}=M>0. $$
  2. Case 2.

    There exists a constant \(\epsilon_{i}>\frac{1}{q_{i}-1}\) such that \(\frac{g_{i}(x)}{x^{\epsilon_{i}}}\) is nonincreasing on x.

Case 1 and Case 2 indicate that \(g_{i}\) can be superlinear or sublinear or mixed cases of them; moreover, this shows that assumption (2.11) is very easy to be satisfied.

Let \(E=C[0,1]\times C[0,1]\) be the Banach space of all continuous functions with the norm

$$\bigl\Vert (x_{1},x_{2}) \bigr\Vert = \Vert x_{1} \Vert + \Vert x_{2} \Vert ,\qquad \Vert x_{i} \Vert =\max\bigl\{ x_{i}(t): t \in[0,1]\bigr\} . $$

Define a cone P in E

$$\begin{aligned} P =& \bigl\{ (x_{1},x_{2})\in E: \mbox{there exist nonnegative numbers } l_{x_{i}}< L_{x_{i}} \mbox{ such that} \\ &l_{x_{i}}t^{\alpha_{i}-1}\leq x_{i}(t)\leq L_{x_{i}} t^{\alpha_{i}-1}, t\in[0, 1], i=1,2 \bigr\} , \end{aligned}$$

and operators \(T_{1}\), \(T_{2}\), T,

$$\begin{aligned}& T_{1}(x_{1},x_{2}) (t) = \int_{0}^{1}H_{1}(t,s) \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau) f_{1}\bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{1}-1}\,ds, \\& T_{2}(x_{1},x_{2}) (t)= \int_{0}^{1}H_{2}(t,s) \biggl( \int_{0}^{1}G_{\beta_{2}}(s,\tau) f_{2}\bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{2}-1}\,ds, \\& T(x_{1},x_{2}) (t)=:\bigl(T_{1}(x_{1},x_{2}) (t),T_{2}(x_{1},x_{2}) (t)\bigr), \end{aligned}$$

then the fixed point of operator T in E is the solution of system (1.1).

Lemma 2.5

Assume that (H0)-(H1) hold. Then \(T: P \to P\) is a continuous, compact operator.

Proof

For any \((x_{1},x_{2})\in P\), we can find four nonnegative numbers \(L_{x_{i}}>l_{x_{i}}\ge0\) such that

$$ l_{x_{i}} t^{\alpha_{i}-1}\leq x_{i}(t)\leq L_{x_{i}} t^{\alpha_{i}-1},\quad t\in [0, 1], i=1,2. $$
(2.12)

If \((x_{1},x_{2})=(0,0)\), notice that \(f_{i}(0,0)\neq0\), then it follows from (2.8) that \(T(x_{1},x_{2})\in P\). Otherwise, by (H1), we know that \(T_{1}\) and \(T_{2}\) are increasing with respect to \(x_{1}\), \(x_{2}\). Now we divide into two cases to prove the right-hand side of inequality of (2.12) is valid for T.

Case 1. If \(x_{1}(t)+x_{2}(t)\ge1\), by (2.8), we get

$$\begin{aligned} T_{1}(x_{1},x_{2}) (t) \le& b t^{\alpha _{1}-1} \int_{0}^{1} \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau)f_{1} \bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& bt^{\alpha _{1}-1} \int_{0}^{1} \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau) \frac{f_{1}(x_{1}(\tau),x_{2}(\tau ))}{(x_{1}(\tau)+x_{2}(\tau))^{\epsilon_{1}}}\bigl(x_{1}(\tau)+x_{2}(\tau ) \bigr)^{\epsilon_{1}} \,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& b M^{\epsilon_{1}(q_{1}-1)}t^{\alpha _{1}-1} \int_{0}^{1} \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau) \bigl(x_{1}(\tau)+x_{2}(\tau)\bigr)^{\epsilon _{1}} \,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& b M^{\epsilon_{1}(q_{1}-1)}t^{\alpha _{1}-1} \biggl( \int_{0}^{1}\frac{\beta_{1}-1}{\Gamma(\beta_{1})}\tau(1-\tau )^{\beta_{1}-1}\bigl(L_{x_{1}}\tau^{\alpha_{1}-1}+ L_{x_{2}} \tau^{\alpha _{2}-1}\bigr)^{\epsilon_{1}} \,d\tau \biggr)^{q_{1}-1} \\ \le& b M^{\epsilon_{1}(q_{1}-1)} \biggl(\frac{\beta_{1}-1}{\Gamma(\beta_{1})}(L_{x_{1}}+ L_{x_{2}})^{\epsilon_{1}} \biggr)^{q_{1}-1}t^{\alpha _{1}-1} \le L^{*}_{x_{1}}t^{\alpha _{1}-1}. \end{aligned}$$
(2.13)

Case 2. If \(x_{1}(t)+x_{2}(t)<1\), we have

$$\begin{aligned} T_{1}(x_{1},x_{2}) (t) \le& b t^{\alpha _{1}-1} \int_{0}^{1} \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau)f_{1} \bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& bt^{\alpha _{1}-1} \int_{0}^{1} \biggl( \int_{0}^{1}G_{\beta_{1}}(s, \tau)f_{1}(1,1) \,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& b t^{\alpha _{1}-1} \biggl( \int_{0}^{1}\frac{\beta_{1}-1}{\Gamma(\beta_{1})}\tau(1-\tau )^{\beta_{1}-1}f_{1}(1,1) \,d\tau \biggr)^{q_{1}-1} \\ \le& b \biggl(\frac{\beta_{1}-1}{\Gamma(\beta_{1})}f_{1}(1,1) \biggr)^{q_{1}-1}t^{\alpha _{1}-1} \le L^{*}_{x_{1}}t^{\alpha _{1}-1}, \end{aligned}$$
(2.14)

where

$$L^{*}_{x_{1}}=\max \biggl\{ b M^{\epsilon_{1}(q_{1}-1)} \biggl( \frac{\beta_{1}-1}{\Gamma(\beta_{1})}(L_{x_{1}}+ L_{x_{2}})^{\epsilon_{1}} \biggr)^{q_{1}-1}, b \biggl(\frac{\beta_{1}-1}{\Gamma(\beta_{1})}f_{1}(1,1) \biggr)^{q_{1}-1} \biggr\} . $$

On the other hand,

$$\begin{aligned} T_{1}(x_{1},x_{2}) (t) \ge& a t^{\alpha _{1}-1} \int _{0}^{1}\mathcal{G}_{A_{1}}(s) \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau)f_{2} \bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{1}-1}\,ds \\ \ge& a t^{\alpha _{1}-1} \int_{0}^{1}\mathcal{G}_{A_{1}}(s) \biggl( \int_{0}^{1}\frac{s^{\beta_{1}-1}(1-s)\tau(1-\tau)^{\beta _{1}-1}}{\Gamma(\beta_{1})}f_{1}(0,0) \biggr)^{q_{1}-1}\,ds \\ =&a \biggl(\frac{f_{1}(0,0)}{\Gamma(\beta_{1})} \biggr)^{q_{1}-1} \int_{0}^{1}\mathcal{G}_{A_{1}}(s)s^{(\beta_{1}-1)(q_{1}-1)}(1-s)^{q_{1}-1} \,ds \\ &{}\times \biggl( \int_{0}^{1}\tau(1-\tau)^{\beta_{1}-1}\,d\tau \biggr)^{q_{1}-1}t^{\alpha _{1}-1} \\ =& a \biggl(\frac{f_{1}(0,0)}{\Gamma(\beta_{1}+2)} \biggr)^{q_{1}-1} \int_{0}^{1}\mathcal{G}_{A_{1}}(s)s^{(\beta _{1}-1)(q_{1}-1)}(1-s)^{q_{1}-1}dst^{\alpha _{1}-1} \\ =&l_{x_{1}}^{*} t^{\alpha _{1}-1}, \end{aligned}$$
(2.15)

where

$$l_{x_{1}}^{*}= a \biggl(\frac{f_{1}(0,0)}{\Gamma(\beta_{1}+2)} \biggr)^{q_{1}-1} \int_{0}^{1}\mathcal{G}_{A_{1}}(s)s^{(\beta _{1}-1)(q_{1}-1)}(1-s)^{q_{1}-1} \,ds. $$

Thus (2.13)-(2.15) yield that

$$l^{*}_{x_{1}}t^{\alpha _{1}-1}\le T_{1}(x_{1},x_{2}) (t)\le L^{*}_{x_{1}}t^{\alpha _{1}-1}. $$

In the same way, there exist two constants \(L^{*}_{x_{2}}>l^{*}_{x_{2}}\ge0\) such that

$$l^{*}_{x_{2}}t^{\alpha _{2}-1}\le T_{2}(x_{1},x_{2}) (t)\le L^{*}_{x_{2}}t^{\alpha _{2}-1}. $$

Therefore T is well defined and uniformly bounded and \(T(P) \subset P\).

On the other hand, according to the Arzela-Ascoli theorem and the Lebesgue dominated convergence theorem, we know that \(T:P \to P\) is completely continuous. □

3 Main results

Lemma 3.1

Suppose \(\epsilon_{i}(q_{i}-1)>1\), \(i=1,2\), then the equation

$$ \biggl(\frac{\beta_{1}-1}{\Gamma{(\beta_{1}+2)}}2^{\epsilon_{1}}M \biggr)^{q_{1}-1}x^{\epsilon_{1}(q_{1}-1)-1}+ \biggl(\frac{\beta_{2}-1}{\Gamma{(\beta_{2}+2)} }2^{\epsilon_{2}}M \biggr)^{q_{2}-1}x^{\epsilon_{2}(q_{2}-1)-1}=b^{-1} $$
(3.1)

has a unique positive solution \(r^{*}\) in \([0,\infty )\).

Proof

Let

$$\begin{aligned} \varphi(x) =&b^{-1}- \biggl(\frac{\beta _{1}-1}{\Gamma{(\beta_{1}+2)}}2^{\epsilon_{1}}M \biggr) ^{q_{1}-1}x^{\epsilon_{1}(q_{1}-1)-1} \\ &{} - \biggl(\frac{\beta_{2}-1}{\Gamma {(\beta_{2}+2)}}2^{\epsilon_{2}}M \biggr)^{q_{2}-1}x^{\epsilon_{2}(q_{2}-1)-1}, \end{aligned}$$
(3.2)

then \(\varphi(x)\) is continuous in \([0,\infty )\), and

$$\begin{aligned}& \varphi(0)=b^{-1}>0, \end{aligned}$$
(3.3)
$$\begin{aligned}& \varphi \biggl( \biggl(\frac{(\beta _{1}-1)2^{\epsilon_{1}}Mb^{\frac{1}{q_{1}-1}}}{\Gamma{(\beta_{1}+2)}} \biggr)^{-\frac{q_{1}-1}{\epsilon_{1}(q_{1}-1)-1}} \biggr) \\& \quad = - \biggl(\frac{\beta_{2}-1}{\Gamma{(\beta_{2}+2)}}2^{\epsilon_{2}}M \biggr)^{q_{2}-1} \biggl(\frac{(\beta_{1}-1)2^{\epsilon_{1}}Mb^{\frac{1}{q_{1}-1}}}{\Gamma{(\beta_{1}+2)}} \biggr)^{-\frac{(q_{1}-1) (\epsilon_{2}(q_{2}-1)-1)}{\epsilon_{1}(q_{1}-1)-1}} < 0. \end{aligned}$$
(3.4)

On the other hand, we also have

$$\begin{aligned} \varphi'(x) =&-\bigl[\epsilon_{1}(q_{1}-1)-1 \bigr] \biggl(\frac{\beta_{1}-1}{\Gamma{(\beta_{1}+2)}}2^{\epsilon_{1}}M \biggr) ^{q_{1}-1}x^{\epsilon_{1}(q_{1}-1)-2} \\ &{}-\bigl[\epsilon_{2}(q_{2}-1)-1\bigr] \biggl( \frac{\beta_{2}-1}{\Gamma{(\beta _{2}+2)}}2^{\epsilon_{2}}M \biggr)^{q_{2}-1}x^{\epsilon_{2}(q_{2}-1)-2} \\ < &0,\quad x\in(0,\infty). \end{aligned}$$
(3.5)

Thus by (3.2)-(3.5), equation (3.1) has a unique positive solution \(r^{*}\) in \([0,\infty )\). □

Theorem 3.1

Suppose conditions (H0) and (H1) hold, and

$$ \frac{(\beta_{i}-1) f_{i}(0,0)}{\Gamma(\beta_{i}+2)}\le \biggl(\frac{r^{*}}{b} \biggr)^{\frac{1}{q_{i}-1}},\quad i=1,2. $$
(3.6)

Then system (1.1) has the minimal and maximal solutions, \(x^{*}=(x^{*}_{1},x^{*}_{2})\) and \(y^{*}=(y^{*}_{1},y^{*}_{2})\), which are positive; and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2,3,4\), such that

$$ \begin{aligned} & m_{1}t^{\alpha-1}\le x^{*}_{1}(t) \le n_{1}t^{\alpha-1},\qquad m_{2}t^{\alpha-1}\le y^{*}_{1}(t) \le n_{2}t^{\alpha-1}, \quad t\in[0,1], \\ &m_{3}t^{\alpha-1}\le x^{*}_{2}(t) \le n_{3}t^{\alpha-1},\qquad m_{4}t^{\alpha-1}\le y^{*}_{2}(t) \le n_{4}t^{\alpha-1}, \quad t\in [0,1]. \end{aligned} $$
(3.7)

Moreover, for initial values \(u^{(0)}(t)=(0,0)\), \(w^{(0)}(t)=(r^{*},r^{*})\), let \(\{{u^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by

$$ u^{(n)}(t) = T{u^{(n-1)}}(t) = T^{n}u^{(0)}(t),\qquad {w^{(n)}}(t) = T{w^{(n-1)}}(t) = T^{n}w^{(0)}(t), $$
(3.8)

then

$$\lim_{n\to+\infty}{u^{(n)}(t)}=x^{*}(t),\qquad \lim _{n\to+\infty }{w^{(n)}(t)}=y^{*}(t) $$

uniformly for \(t\in[0,1]\).

Proof

Take \(P[0, r^{*}] = \{(x_{1},x_{2}) \in P : 0 \le\|x_{1}\| \le r^{*}, 0 \le\|x_{2}\| \le r^{*}\}\), we firstly prove \(T(P[0, r^{*}]) \subset P[0, r^{*}]\).

In fact, for any \((x_{1},x_{2}) \in P[0, r^{*}]\), if \((x_{1},x_{2})\equiv(0,0)\), it follows from (2.8) that

$$\begin{aligned} \bigl\Vert T_{1}(x_{1},x_{2}) \bigr\Vert =& \max_{t\in[0,1]} \biggl\{ \int_{0}^{1}H_{1}(t,s) \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau) f_{1}\bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{1}-1}\,ds \biggr\} \\ \le& b \int_{0}^{1} \biggl( \int_{0}^{1} \frac{\beta_{1}-1}{\Gamma(\beta_{1})}\tau(1-\tau )^{\beta_{1}-1}f_{1}(0,0)\,d\tau \biggr)^{q_{1}-1}\,ds \\ =&b \biggl(\frac{\beta_{1}-1}{\Gamma{(\beta_{1}+2})}f_{1}(0,0) \biggr)^{q_{1}-1}\le r^{*}, \end{aligned}$$
(3.9)

and

$$\begin{aligned} \begin{aligned}[b] \bigl\Vert T_{2}(x_{1},x_{2}) \bigr\Vert & = \max_{t\in [0,1]} \biggl\{ \int_{0}^{1}H_{2}(t,s) \biggl( \int_{0}^{1}G_{\beta_{2}}(s,\tau) f_{2}\bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{2}-1}\,ds \biggr\} \\ &\le b \int_{0}^{1} \biggl( \int_{0}^{1} \frac{\beta_{2}-1}{\Gamma(\beta_{2})}\tau(1-\tau )^{\beta_{2}-1}f_{2}(0,0)\,d\tau \biggr)^{q_{2}-1}\,ds \\ &=b \biggl(\frac{\beta_{2}-1}{\Gamma{(\beta_{2}+2)}}f_{2}(0,0) \biggr)^{q_{2}-1}\le r^{*}. \end{aligned} \end{aligned}$$
(3.10)

Otherwise, for any \(t\in(0,1)\), we have

$$ 0< x_{1}(t)+x_{2}(t) \le\max _{t\in[0,1]}x_{1}(t)+\max_{t\in[0,1]}x_{2}(t) \le2r^{*}. $$
(3.11)

So, by (H1), we get

$$\begin{aligned} \bigl\Vert T_{1}(x_{1},x_{2}) \bigr\Vert =& \max_{t\in[0,1]} \biggl\{ \int_{0}^{1}H_{1}(t,s) \biggl( \int_{0}^{1}G_{\beta_{1}}(s,\tau) f_{1}\bigl(x_{1}(\tau),x_{2}(\tau)\bigr)\,d\tau \biggr)^{q_{1}-1}\,ds \biggr\} \\ \le& b \int_{0}^{1} \biggl( \int_{0}^{1} \frac{\beta_{1}-1}{\Gamma(\beta_{1})}\tau(1-\tau )^{\beta_{1}-1}M\bigl[x_{1}(\tau)+x_{2}(\tau) \bigr]^{\epsilon_{1}}\,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& b \int_{0}^{1} \biggl( \int_{0}^{1} \frac{\beta_{1}-1}{\Gamma(\beta _{1})}\tau(1- \tau)^{\beta_{1}-1}M2^{\epsilon_{1}}\bigl(r^{*}\bigr)^{\epsilon_{1}}\,d\tau \biggr)^{q_{1}-1}\,ds \\ \le& b \biggl(\frac{\beta_{1}-1}{\Gamma{(\beta_{1}+2)}}2^{\epsilon _{1}}M \biggr)^{q_{1}-1}{ \bigl(r^{*}\bigr)}^{\epsilon_{1}(q_{1}-1)} \\ \le& b \biggl(\frac{\beta_{1}-1}{\Gamma{(\beta_{1}+2)}}2^{\epsilon_{1}}M \biggr)^{q_{1}-1}{ \bigl(r^{*}\bigr)}^{\epsilon_{1}(q_{1}-1)} + b \biggl(\frac{\beta_{2}-1}{\Gamma{(\beta_{2}+2)}}2^{\epsilon _{2}}M \biggr)^{q_{2}-1}{\bigl(r^{*}\bigr)}^{\epsilon_{1}(q_{2}-1)} \\ =&r^{*}. \end{aligned}$$
(3.12)

Similar to (3.12), we have

$$\begin{aligned} \bigl\Vert T_{2}(x_{1},x_{2}) \bigr\Vert \le& b \biggl(\frac{\beta _{1}-1}{\Gamma{(\beta_{1}+2)}}2^{\epsilon_{1}}M \biggr)^{q_{1}-1}\bigl(r^{*} \bigr)^{\epsilon_{1}(q_{1}-1)} \\ &{}+b \biggl(\frac{\beta _{2}-1}{\Gamma{(\beta_{2}+2)}}2^{\epsilon_{2}}M \biggr)^{q_{2}-1} \bigl(r^{*}\bigr)^{\epsilon_{1}(q_{2}-1)}=r^{*}, \end{aligned}$$
(3.13)

which implies that \(T(P[0, r^{*}]) \subset P[0, r^{*}]\).

Let \(u^{(0)}(t) =(u^{(0)}_{1}(t),u^{(0)}_{2}(t))=(0,0)\) and

$$\begin{aligned} u^{(1)}(t)&=:\bigl(u^{(1)}_{1}(t),u^{(1)}_{2}(t) \bigr)= \bigl(\bigl(T_{1}\bigl(u^{(0)}_{1},u^{(0)}_{2} \bigr)\bigr) (t),\bigl(T_{2}\bigl(u^{(0)}_{1},u^{(0)}_{2} \bigr)\bigr) (t)\bigr) \\ &=\bigl(\bigl(T_{1}(0,0)\bigr) (t),\bigl(T_{2}(0,0)\bigr) (t) \bigr), \quad t \in[0, 1], \end{aligned}$$

it follows from \(u^{(0)}(t)\in P([0,r^{*}])\) that \(u^{(1)}(t) \in T(P[0, r^{*}])\).

Denote

$$u^{(n+1)}(t) = Tu^{(n)}(t) = T^{n+1} u^{(0)}(t), \quad n = 1, 2, \ldots. $$

It follows from \(T(P[0, r^{*}]) \subset P[0, r^{*}]\) that \(u_{n}(t) \in P[0, r^{*}]\) for \(n\ge1\). Noticing that T is compact, we get that \(\{u^{(n)}\}\) is a sequentially compact set.

On the other hand, since \(u^{(1)}(t)\ge0=u^{(0)}(t)\), we have

$$u^{(2)}(t) = \bigl(Tu^{(1)}\bigr) (t) \ge\bigl(Tu^{(0)} \bigr) (t)= u^{(1)}(t),\quad t \in[0, 1]. $$

By induction, we get

$$u^{(n+1)} \ge u^{(n)}, \quad n = 1, 2, \ldots. $$

Consequently, there exists \(x^{*}=(x^{*}_{1},x^{*}_{2}) \in P[0, r]\) such that \(u^{(n)} \to x^{*}\). Letting \(n \to+\infty\), from the continuity of T and \(Tu^{(n)} =u^{(n-1)}\), we obtain \(Tx^{*} = x^{*}\), which implies that \(x^{*}=(x^{*}_{1},x^{*}_{2})\) is a nonnegative solution of the nonlinear integral equation (1.1). Since \(x^{*}\in P\), there exist constants \(0< m_{1}< n_{1}\), \(0< m_{2}< n_{2}\) such that

$$0< m_{1}t^{\alpha_{1}-1}\le x_{1}^{*}(t)\le n_{1} t^{\alpha_{1}-1}, 0< m_{2}t^{\alpha _{2}-1}\le x_{2}^{*}(t)\le n_{2} t^{\alpha_{2}-1},\quad t \in(0, 1), $$

and consequently \(x^{*}\) is a positive solution of system (1.1).

Now let \(w^{(0)}(t) =(w^{(0)}_{1}(t),w^{(0)}_{2}(t))=(r^{*},r^{*})\) and

$$w^{(1)}(t)=:\bigl(w^{(1)}_{1}(t),w^{(1)}_{2}(t) \bigr)= \bigl(\bigl(T_{1}\bigl(w^{(0)}_{1},w^{(0)}_{2} \bigr)\bigr) (t),\bigl(T_{2}\bigl(w^{(0)}_{1},w^{(0)}_{2} \bigr) \bigr)(t)\bigr), \quad t \in[0, 1]. $$

Since \(w^{(0)}(t)=(r^{*},r^{*}) \in P[0, r^{*}]\), and then \(w^{(1)}(t)\in P[0, r^{*}]\). Thus denote

$$w^{(n+1)}(t) = Tw^{(n)}(t) = T^{n+1}w^{(0)}(t), \quad n = 1, 2, \ldots. $$

It follows from \(T(P[0, r^{*}]) \subset P[0, r^{*}]\) that

$$w^{(n)}(t) \in P\bigl[0, r^{*}\bigr], \quad n = 0, 1, 2, \ldots. $$

From Lemma 2.5, T is compact, consequently \(\{w^{(n)}\}\) is a sequentially compact set.

Now, since \(w^{(1)}(t) \in P[0, r^{*}]\), we get

$$0 \le w^{(1)}_{1}(t) \le \bigl\Vert w^{(1)}_{1} \bigr\Vert \le r = w^{(0)}_{1}(t), \qquad 0 \le w^{(1)}_{2}(t) \le \bigl\Vert w^{(1)}_{2} \bigr\Vert \le r^{*} = w^{(0)}_{2}(t). $$

It follows from (H1) that \(w^{(2)}(t) = Tw^{(1)}(t) \le Tw^{(0)}(t)=w^{(1)}(t)\). By induction, we obtain

$$w^{(n+1)} (t)\le w^{(n)}(t),\quad n = 0, 1, 2, \ldots. $$

Consequently, there exists \(y^{*}(t)\in P[0, r^{*}]\) such that \(w^{(n)}(t) \to y^{*}=(y^{*}_{1},y^{*}_{2})\). Letting \(n \to+\infty\), from the continuity of T and \(Tw^{(n)}(t) = w^{(n-1)}(t)\), we have \(Ty^{*} = y^{*}\), which implies that \(y^{*}\) is another nonnegative solution of the boundary value problem (1.1) and \(y^{*}\) also satisfies (3.7) since \(y^{*}\in P\).

In the end, we prove that \(x^{*}\) and \(y^{*}\) are maximum and minimum solutions for system (1.1). Let x̃ be any positive solution of system (1.1), then \(u^{(0)}=0\le\tilde{x}\le r^{*}=w^{(0)}\), and \(u^{(1)}=Tu^{(0)}\le T\tilde{x}=\tilde{x}\le T(w^{(0)})=w^{(1)}\). By induction, we have \(u^{(n)}\le\tilde{x}\le w^{(n)}\), \(n=1,2,3,\ldots\) . Taking limit, we have \(x^{*}\le\tilde{x}\le y^{*}\). This implies that \(x^{*}\) and \(y^{*}\) are the maximal and minimal solutions of system (1.1), respectively. The proof is completed. □

Remark

In particular, if \(\alpha_{i}=\beta_{i}=p_{i}=2\), then the nonlocal fractional system (1.1) will reduce to a fourth order classical beam system of ordinary differential equation,

$$ \textstyle\begin{cases} -x^{(4)}_{1}(t)=f_{1}(x_{1}(t), x_{2}(t)), \\ -x_{2}^{(4)}(t)=f_{2}(x_{1}(t), x_{2}(t)) \\ x_{1}(0)=0,\qquad x_{1}''(0)=x_{1}''(1)=0,\qquad x_{1}(1)=\int_{0}^{1}x_{1}(t)\,dA_{1}(t), \\ x_{2}(0)=0,\qquad x_{2}''(0)=x_{2}''(1)=0, \qquad x_{2}(1)=\int_{0}^{1}x_{2}(t)\,dA_{2}(t), \end{cases} $$
(3.14)

we have the following good result.

Corollary 3.1

Suppose conditions (H0) and (H1) hold, and

$$ f_{i}(0,0)\le\frac{6}{b}r^{*},\quad i=1,2, $$
(3.15)

then system (3.14) has the minimal and maximal solutions, \(x^{*}=(x^{*}_{1},x^{*}_{2})\) and \(y^{*}=(y^{*}_{1},y^{*}_{2})\), which are positive; and there exist some nonnegative numbers \(m_{i}< n_{i}\), \(i=1,2,3,4\), such that

$$ \begin{aligned} &m_{1}t\le x^{*}_{1}(t) \le n_{1}t, m_{2}t\le y^{*}_{1}(t) \le n_{2}t,\quad t\in[0,1], \\ &m_{3}t\le x^{*}_{2}(t) \le n_{3}t, m_{4}t\le y^{*}_{2}(t) \le n_{4}t, \quad t\in[0,1]. \end{aligned} $$
(3.16)

Moreover, for initial values \(u^{(0)}(t)=(0,0)\), \(w^{(0)}(t)=(r^{*},r^{*})\), let \(\{{u^{(n)}}\}\) and \(\{{w^{(n)}}\}\) be the iterative sequences generated by

$$ u^{(n)}(t) = T{u^{(n-1)}}(t) = T^{n}u^{(0)}(t),\qquad {w^{(n)}}(t) = T{w^{(n-1)}}(t) = T^{n}w^{(0)}(t), $$
(3.17)

then

$$\lim_{n\to+\infty}{u^{(n)}(t)}=x^{*}(t),\qquad \lim _{n\to+\infty }{w^{(n)}(t)}=y^{*}(t) $$

uniformly for \(t\in[0,1]\).

In the end, we know that fractional order integral and derivative operators can describe an important characteristics exhibiting long-memory in time in many complex processes and systems. With this advantage, in many eco-economical systems and diffusive processes with long time memory behavior [2, 3, 15, 39], fractional calculus provides an excellent tool to describe the hereditary properties of them. Here we give a specific example arising from the above complex processes.

Example

Consider the following nonlocal boundary value problem of the fractional p-Laplacian equation:

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\frac{4}{3}} (\varphi_{\frac{5}{2}} (-\mathscr {D}_{t}^{\frac{3}{2}}x_{1} ) )(t) \\ \quad =(x_{1}+x_{2})^{2}+{(x_{1}+x_{2})^{2}}(x_{1}+x_{2}+1)\ln ( 1+\frac {1}{1+x_{1}+x_{2}} )+1, \\ -\mathscr {D}_{t}^{\frac{6}{5}} (\varphi_{\frac{3}{2}} (-\mathscr {D}_{t}^{\frac{7}{6}}x_{2} ) )(t) =\ln(2+x_{1}+x_{2}),\quad t\in(0,1), \\ x_{1}(0)=0,\qquad \mathscr {D}_{t}^{\frac{3}{2}}x_{1}(0)=\mathscr {D}_{t}^{\frac{3}{2}}x_{1}(1)=0,\qquad x_{1}(1)=\int^{1}_{0}x_{1}(s)\,dA_{1}(s), \\ x_{2}(0)=0,\qquad \mathscr {D}_{t}^{\frac{7}{6}}x_{2}(0)=\mathscr {D}_{t}^{\frac{7}{6}}x_{2}(1)=0,\qquad x_{2}(1)=\int^{1}_{0}x_{2}(s)\,dA_{2}(s), \end{cases} $$
(3.18)

where

$$A_{1}(t)= \textstyle\begin{cases} 0,& t\in [0,\frac{1}{2} ), \\ 2,& t\in [\frac{1}{2},\frac{3}{4} ), \\ 1,& t\in [\frac{3}{4},1 ], \end{cases}\displaystyle \qquad A_{2}(t)= \textstyle\begin{cases} 0,& t\in [0,\frac{1}{3} ), \\ \frac{3}{2}, &t\in [\frac{1}{3},\frac{2}{3} ), \\ \frac{1}{2}, &t\in [\frac{2}{3},1 ]. \end{cases} $$

Then system (3.18) has the positive minimal and maximal solutions, \(x^{*}=(x^{*}_{1},x^{*}_{2})\) and \(y^{*}=(y^{*}_{1},y^{*}_{2})\); and there exist some nonnegative numbers \(m_{i}\le n_{i}\), \(i=1,2,3,4\), such that

$$ \begin{aligned} &m_{1}t^{\frac{1}{2}}\le x^{*}_{1}(t) \le n_{1}t^{\frac{1}{2}},\qquad m_{2}t^{\frac{1}{2}}\le y^{*}_{1}(t) \le n_{2}t^{\frac{1}{2}}, \quad t\in[0,1], \\ &m_{3}t^{\frac{1}{6}}\le x^{*}_{2}(t) \le n_{3}t^{\frac{1}{6}},\qquad m_{4}t^{\frac{1}{6}}\le y^{*}_{2}(t) \le n_{4}t^{\frac{1}{6}}, \quad t\in[0,1]. \end{aligned} $$
(3.19)

By simple computation, problem (3.18) is equivalent to the following multipoint boundary value problem:

$$ \textstyle\begin{cases} -\mathscr {D}_{t}^{\frac{4}{3}} (\varphi_{\frac{5}{2}} (-\mathscr {D}_{t}^{\frac{3}{2}}x_{1} ) )(t) \\ \quad =(x_{1}+x_{2})^{2} +{(x_{1}+x_{2})^{2}}(x_{1}+x_{2}+1)\ln ( 1+\frac {1}{1+x_{1}+x_{2}} )+1, \\ -\mathscr {D}_{t}^{\frac{6}{5}} (\varphi_{\frac{3}{2}} (-\mathscr {D}_{t}^{\frac{7}{6}}x_{2} ) )(t) =\ln(2+x_{1}+x_{2}), \quad t\in(0,1), \\ x_{1}(0)=0,\qquad \mathscr {D}_{t}^{\frac{3}{2}}x_{1}(0)=\mathscr {D}_{t}^{\frac{3}{2}}x_{1}(1)=0,\qquad x_{1}(1)=2x_{1} (\frac{1}{2} )-x_{1} (\frac{3}{4} ), \\ x_{2}(0)=0,\qquad \mathscr {D}_{t}^{\frac{7}{6}}x_{2}(0)=\mathscr {D}_{t}^{\frac{7}{6}}x_{2}(1)=0, \qquad x_{2}(1)=\frac{3}{2}x (\frac{1}{3} )-x (\frac{2}{3} ). \end{cases} $$

Let

$$\alpha_{1}=\frac{3}{2}, \qquad \alpha_{2}= \frac{7}{6},\qquad \beta_{1}=\frac{4}{3}, \qquad \beta _{2}=\frac{6}{5},\qquad p_{1}=\frac{5}{2}, \qquad p_{2}=\frac{3}{2}, $$

and

$$f_{1}(s,t)={(s+t)^{2}}(s+t+1)\ln \biggl( 1+ \frac{1}{1+s+t} \biggr) +(s+t)^{2}+1, \qquad f_{2}(s,t)= \ln(2+s+t). $$

Firstly, we have

$$\begin{aligned}& \mathcal{A}_{1}= \int_{0}^{1}t^{\alpha_{1}-1}\,dA_{1}(t)=2 \times \biggl(\frac{1}{2} \biggr)^{\frac{1}{2}}- \biggl(\frac{3}{4} \biggr)^{\frac{1}{2}}=0.5482< 1, \\& \mathcal{A}_{2}= \int_{0}^{1}t^{\alpha_{2}-1}\,dA_{2}(t)= \frac{3}{2}\times \biggl(\frac{1}{3} \biggr)^{\frac{1}{6}}- \biggl( \frac{2}{3} \biggr)^{\frac{1}{6}}=0.3053< 1, \end{aligned}$$

and by simple computation, we have \(\mathcal{G}_{A_{i}}(s)\ge0\), \(i=1,2\), and so (H0) holds.

Obviously, \(f_{1},f_{2}: [0,+\infty)\times[0,+\infty)\to[0,+\infty)\) are continuous and nondecreasing in the first variable and second variable and \(f_{i}(0,0)\neq0\), \(i=1,2\). Noticing that

$$\begin{aligned}& \sup_{{s+t\ge1}}\frac{f_{1} ( s,t ) }{(s+t)^{2}}\le\sup_{{s+t\ge1}} \biggl\{ \ln \biggl( 1+\frac{1}{1+s+t} \biggr)^{1+s+t}+2 \biggr\} \le 3, \\& \sup_{{s+t\ge1}}\frac{f_{2} ( s,t ) }{(s+t)}=\sup_{{s+t\ge1}} \frac{\ln(2+s+t) }{(s+t)}\le\ln3\le3, \end{aligned}$$

and then (2.11) is satisfied with \(\epsilon_{1}=2>\frac{1}{q_{1}-1}=\frac{3}{2}\), \(\epsilon_{2}=1>\frac{1}{q_{2}-1}=\frac{1}{2}\), \(M=3\). Thus, by Theorem 3.1, system (3.18) has maximal and minimal solutions which satisfy (3.19).

4 Conclusion

In this work, we have established an existence result on the maximum and minimum solutions for a class of fractional order differential systems involving a p-Laplacian operator and nonlocal boundary conditions. This type of differential systems actually arise from some complex natural processes such as ecological economy phenomena and diffusive interaction, moreover fractional differential operator can more accurately depict the co-evolution process of economic, social and ecological subsystems and the transport of solute in highly heterogeneous porous media. The main contribution is that we introduced some new type growth conditions for nonlinearity and adopted the monotone iterative technique to establish the existence and estimation of the lower and upper bounds of the maximum and minimum solutions. Furthermore, the iterative schemes converging to the exact maximum and minimum solutions, which start from some explicit initial values, are also constructed.