1 Introduction

In this paper, we investigate the following system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator:

\begin{aligned} \textstyle\begin{cases} -D^{\alpha_{1}}_{0^{+}}(\varphi_{p_{1}}(D^{\beta_{1}}_{0^{+}}u(t)))={\lambda }f(t,u(t),v(t)),\quad 0< t< 1,\\ -D^{\alpha_{2}}_{0^{+}}(\varphi_{p_{2}}(D^{\beta_{2}}_{0^{+}}v(t)))={\mu }g(t,u(t),v(t)),\quad 0< t< 1, \\ u(0)=u(1)=u'(0)=u'(1)=0, \qquad D^{\beta_{1}}_{0^{+}}u(0)=0,\qquad D^{\beta _{1}}_{0^{+}}u(1)=b_{1}D^{\beta_{1}}_{0^{+}}u(\eta_{1}),\\ v(0)=v(1)=v'(0)=v'(1)=0,\qquad D^{\beta_{2}}_{0^{+}}v(0)=0,\qquad D^{\beta _{2}}_{0^{+}}v(1)=b_{2}D^{\beta_{2}}_{0^{+}}v(\eta_{2}), \end{cases}\displaystyle \end{aligned}
(1.1)

where $$\alpha_{i}\in(1,2],\beta_{i}\in(3,4],D^{\alpha_{i}}_{0^{+}}$$ and $$D^{\beta_{i}}_{0^{+}}$$ are the standard Riemann-Liouville derivatives, $$\varphi_{p_{i}}(s) = \vert s \vert ^{p_{i}-2}s,p_{i}>1,\varphi_{p_{i}}^{-1} = \varphi _{q_{i}},\frac{1}{p_{i}}+\frac{1}{q_{i}}=1,\eta_{i}\in(0,1),b_{i}\in (0,\eta_{i}^{\frac{1-\alpha_{i}}{p_{i}-1}} ),i=1,2.f, g\in C([0,1]\times[0,+\infty)^{2}, [0,+\infty)), \lambda$$ and μ are positive parameters.

Fractional differential equation models are proved to be more adequate than integer order models for some problems in science and engineering. Fractional differential equations play a very important role in various fields due to their deep real world background. For an introduction of fractional calculus and fractional differential equations, we refer the reader to [13] and the references therein.

Turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problem, Leibenson [4] introduced differential equations with p-Laplacian operator

\begin{aligned} \bigl(\varphi_{p}\bigl(u'(t)\bigr)\bigr)'=f \bigl(t,u(t)\bigr). \end{aligned}
(1.2)

The study of differential equation with p-Laplacian operator is of significance theoretically and practically. It is quite natural to study fractional differential equation relative to equation (1.2).

Recently, many scholars have paid more attention to the fractional order differential equation boundary value problems with p-Laplacian operator, see [523]. In [15], Lu et al. investigated a class of boundary value problems for fractional differential equations with p-Laplacian

$$\textstyle\begin{cases} D^{\alpha}_{0^{+}}(\varphi_{p}(D^{\beta}_{0^{+}}u(t)))=f(t,u(t)),\quad 0 < t < 1, \\ u(0)=u'(0)=u'(1)=0, \qquad D^{\beta}_{0^{+}}u(0)= D^{\beta}_{0^{+}}u(1)=0, \end{cases}$$

where $$\alpha\in(1,2]$$, $$\beta\in(2,3], D^{\alpha}_{0^{+}}$$ and $$D^{\beta}_{0^{+}}$$ are the standard Riemann-Liouville derivatives, $$f \in C([0,1]\times[0,+\infty),[0,+\infty))$$. The existence and multiplicity results of positive solutions were obtained by using the Guo-Krasnosel’skii fixed point theorem, the Leggett-Williams fixed point theorem and the upper and lower solutions method. Xu and Dong [17] considered the following three point boundary value problem of fractional differential equation with p-Laplacian operator:

$$\textstyle\begin{cases} -D^{\alpha}_{0^{+}}(\varphi_{p}(D^{\beta}_{0^{+}}u(t)))=f(t,u(t)),\quad 0< t< 1, \\ u(0)=u(1)=u'(0)=u'(1)=0,\qquad D^{\beta}_{0^{+}}u(0)=0,\qquad D^{\beta}_{0^{+}}u(1)=bD^{\beta}_{0^{+}}u(\eta), \end{cases}$$

where $$\alpha\in(1,2]$$, $$\beta\in(3,4]$$, $$D_{0^{+}}^{\alpha}$$ and $$D_{0^{+}}^{\beta}$$ are the standard Riemann-Liouville derivatives, $$\eta\in (0,1)$$, $$b\in (0,\eta^{{\frac{1-\alpha}{p-1}}} )$$, $$f\in C([0,1]\times[0,+\infty), [0,+\infty))$$. The existence and uniqueness of positive solutions were obtained by using the upper and lower solutions method and Schauder’s fixed point theorem, the iterative sequences for the unique solution were also given.

In [18], Zhang et al. considered the eigenvalue problems of fractional differential equations with integral boundary conditions and p-Laplacian operator

\begin{aligned} \textstyle\begin{cases} -D^{\alpha}_{0^{+}}(\varphi_{p}(D^{\beta}_{0^{+}}u(t)))=\lambda f(t,u(t),\quad 0< t< 1, \\ u(0)=0, \qquad D^{\beta}_{0^{+}}u(0)=0, \qquad u(1)=\int^{1}_{0}u(s)\,dA(s),\qquad \end{cases}\displaystyle \end{aligned}
(1.3)

where $$\alpha\in(0,1], \beta\in(1,2], D^{\alpha}_{0^{+}}$$ and $$D^{\beta }_{0^{+}}$$ are the standard Riemann-Liouville derivatives, A is a function of bounded variation, $$\int^{1}_{0}u(s)\,dA(s)$$ is the Riemann-Stieltjes integral, $$f(t,u):(0,1)\times(0,+\infty)\rightarrow [0,+\infty)$$ is a continuous function that may be singular at $$t = 0, 1$$ and $$u =0$$. The existence of positive solutions of problem (1.3) was established by using the upper and lower solutions method and Schauder’s fixed point theorem. Lv [20]  discussed an m-point boundary value problem of fractional differential equation with p-Laplacian operator

\begin{aligned} \textstyle\begin{cases} D^{\alpha}_{0^{+}}(\varphi_{p}(D^{\beta}_{0^{+}}u(t)))+\varphi_{p}(\lambda )f(t,u(t))=0, \quad 0< t< 1, \\ u(0)= 0,\qquad D^{\gamma}_{0^{+}}u(1)=\sum^{m-2}_{i=1}\xi_{i}D^{\gamma }_{0^{+}}u(\eta_{i}), \qquad D^{\beta}_{0^{+}}u(0)=0, \end{cases}\displaystyle \end{aligned}
(1.4)

where $$\alpha, \gamma\in(0,1], \beta\in(1,2], \beta-\alpha-1\geq0, \beta-\gamma-1\geq0, \lambda$$ is a positive parameter, $$0 < \xi_{i}, \eta_{i} < 1$$, $$\sum^{m-2}_{i=1}\xi_{i}\eta^{\beta-\alpha-1}_{i} < 1$$, $$f\in C([0,1]\times[0,+\infty), [0,+\infty))$$. The existence and multiplicity of positive solutions for system (1.4) were established via the monotone iterative method and the fixed point index theory.

The system of fractional differential equations boundary value problems with p-Laplacian operator have also received much attention and have developed very rapidly, see [2432]. In [24], Li et al. studied the following fractional differential system involving the p-Laplacian operator and nonlocal boundary conditions:

$$\textstyle\begin{cases} D^{\alpha_{1}}_{0^{+}}(\varphi_{p_{1}}(D^{\beta _{1}}_{0^{+}}u(t)))=f(t,v(t)),\quad 0< t< 1, \\ D^{\alpha_{2}}_{0^{+}}(\varphi_{p_{2}}(D^{\beta _{2}}_{0^{+}}v(t)))=g(t,u(t)),\quad 0< t< 1, \\ u(0)= D^{\beta_{1}}_{0^{+}}u(0)=0,\qquad D^{\gamma_{1}}_{0^{+}}u(1)=\sum^{m-2}_{j=1}a_{1j}D^{\gamma_{1}}_{0^{+}}u(\eta_{j}),\\ v(0)= D^{\beta_{2}}_{0^{+}}v(0)=0,\qquad D^{\gamma_{2}}_{0^{+}}v(1)=\sum^{m-2}_{j=1}a_{2j}D^{\gamma_{2}}_{0^{+}}v(\eta_{j}), \end{cases}$$

where $$\alpha_{i},\gamma_{i}\in(0,1], \beta_{i}\in(1,2], D^{\alpha _{i}}_{0^{+}}, D^{\beta_{i}}_{0^{+}}$$ and $$D^{\gamma_{i}}_{0^{+}}$$ are the standard Riemann-Liouville derivatives, $$i=1, 2$$. The conditions for the existence of the maximal and minimal solutions to the system were established.

Ren et al. [25] considered the following nonlocal fractional differential system:

$$\textstyle\begin{cases} -D^{\alpha_{1}}_{0^{+}}(\varphi_{p_{1}}(-D^{\beta _{1}}_{0^{+}}u(t)))=f(u(t),v(t)),\quad 0< t< 1, \\ -D^{\alpha_{2}}_{0^{+}}(\varphi_{p_{2}}(-D^{\beta _{2}}_{0^{+}}v(t)))=g(u(t),v(t)),\quad 0< t< 1, \\ u(0)= D^{\beta_{1}}_{0^{+}}u(0)= D^{\beta_{1}}_{0^{+}}u(1)=0,\qquad u(1)=\int ^{1}_{0}u(s)\,dA(s),\\ v(0)= D^{\beta_{2}}_{0^{+}}v(0)= D^{\beta_{2}}_{0^{+}}v(1)=0, \qquad v(1)=\int^{1}_{0}v(s)\,dB(s), \end{cases}$$

where $$\alpha_{i}, \beta_{i}\in(1,2], D^{\alpha_{i}}_{0^{+}}$$ and $$D^{\beta _{i}}_{0^{+}}$$ are the standard Riemann-Liouville derivatives. A and B are functions of bounded variations, $$\int^{1}_{0}u(s)\,dA(s)$$ and $$\int^{1}_{0}v(s)\,dB(s)$$ are Riemann-Stieltjes integrals. By introducing a new type of growth conditions and using the monotone iterative technique, some new results about the existence of maximal and minimal solutions were established, and the estimation of the lower and upper bounds of the maximum and minimum solutions was also derived.

By means of the Avery-Henderson fixed point theorem and six functionals fixed point theorem, Rao [26] investigated the existence of multiple positive solutions for a coupled system of p-Laplacian fractional order two point boundary value problems

$$\textstyle\begin{cases} -D^{\alpha_{1}}_{a^{+}}(\varphi_{p}(D^{\beta _{1}}_{a^{+}}u(t)))=f(t,u(t),v(t)),\quad a< t< b, \\ -D^{\alpha_{2}}_{a^{+}}(\varphi_{p}(D^{\beta _{2}}_{a^{+}}v(t)))=g(t,u(t),v(t)), \quad a< t< b, \\ \xi u(a)-\eta u'(a)=0, \qquad \gamma u(b)+\delta u'(b)=0, \qquad D^{\beta _{1}}_{a^{+}}u(a)=0,\\ \xi v(a)-\eta v'(a)=0,\qquad \gamma v(b)+\delta v'(b)=0,\qquad D^{\beta_{2}}_{a^{+}}v(a)=0, \end{cases}$$

where $$\alpha_{i}\in(0,1], \beta_{i}\in(1,2], D^{\alpha_{i}}_{a^{+}}$$ and $$D^{\beta_{i}}_{a^{+}}$$ are the standard Riemann-Liouville derivatives.

He and Song [29] discussed the following fractional order differential system with p-Laplacian operator:

$$\textstyle\begin{cases} D^{\alpha_{1}}_{0^{+}}(\varphi_{p_{1}}(D^{\beta_{1}}_{0^{+}}u(t)))={\lambda }f(t,v(t)),\quad 0< t< 1, \\ D^{\alpha_{2}}_{0^{+}}(\varphi_{p_{2}}(D^{\beta_{2}}_{0^{+}}v(t)))={\mu }g(t,u(t)),\quad 0< t< 1, \\ u(0)=0, u(1)=a_{1}u(\xi_{1}), \qquad D^{\beta_{1}}_{0^{+}}u(0)=0,\qquad D^{\beta _{1}}_{0^{+}}u(1)=b_{1}D^{\beta_{1}}_{0^{+}}u(\eta_{1}),\\ v(0)=0, v(1)=a_{2}v(\xi_{2}),\qquad D^{\beta_{2}}_{0^{+}}v(0)=0,\qquad D^{\beta _{2}}_{0^{+}}v(1)=b_{2}D^{\beta_{2}}_{0^{+}}v(\eta_{2}), \end{cases}$$

where $$\alpha_{i}, \beta_{i}\in(1,2], D^{\alpha_{i}}_{0^{+}}$$ and $$D^{\beta _{i}}_{0^{+}}$$ are the standard Riemann-Liouville derivatives, $$\xi_{i}, \eta_{i}\in(0,1), a_{i}, b_{i}\in[0,1], i=1,2$$. λ and μ are positive parameters. The uniqueness of solution was established by using the Banach contraction mapping principle. Khan et al. [31] considered the existence and uniqueness of solutions to a coupled system of fractional differential equations with p-Laplacian operator. The functions involved in the proposed coupled system were continuous and satisfied certain growth conditions. By using topological degree theory, some conditions were established which ensured the existence and uniqueness of solution to the proposed problem.

Motivated by the papers mentioned above, in this paper, we study the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the functions f and g, various existence results for positive solutions are derived in terms of different values of λ and μ via the Guo-Krasnosel’skii fixed point theorem. Moreover, in this paper it is possible to replace the four point boundary conditions by multi-point boundary conditions or integral boundary conditions with minor modifications.

2 Preliminaries and lemmas

We present here the definitions, some lemmas from the theory of fractional calculus and some auxiliary results that will be used to prove our main theorems.

Definition 2.1

([13])

The Riemann-Liouville fractional integral of order $$\alpha>0$$ of a function $$f: (0,+\infty)\to(-\infty,+\infty)$$ is given by

$$I^{\alpha}_{0^{+}}f(t)=\frac{1}{\Gamma(\alpha)} \int^{t}_{0}(t-s)^{\alpha -1}f(s)\,ds$$

provided the right-hand side is pointwise defined on $$(0,+\infty)$$.

Definition 2.2

([13])

The Riemann-Liouville fractional derivative of order $$\alpha>0$$ of a continuous function $$f: (0,+\infty)\to(-\infty,+\infty)$$ is given by

$$D^{\alpha}_{0^{+}}f(t)=\frac{1}{\Gamma(n-\alpha)} \biggl(\frac{d}{dt} \biggr) ^{n} \int^{t}_{0}(t-s)^{n-\alpha-1}f(s)\,ds,$$

where n is the smallest integer not less than α, provided the right-hand side is pointwise defined on $$(0,+\infty)$$.

Lemma 2.1

([13])

Let $$\alpha>0$$. Then the following equality holds for $$u\in L(0,1)$$ and $$D^{\alpha}_{0^{+}}u(t)\in L(0,1)$$:

$$I^{\alpha}_{0^{+}} D^{\alpha}_{0^{+}}u(t)=u(t)+c_{1}t^{\alpha -1}+c_{2}t^{\alpha-2}+ \cdots+c_{n}t^{\alpha-n},$$

where $$c_{1}, c_{2}, \ldots, c_{n}\in(-\infty, +\infty), n-1<\alpha\leq n$$.

We transform problem (1.1) to its equivalent integral equations. Denote $$\varphi_{p_{1}}(D^{\beta_{1}}_{0^{+}}u(t)) = z(t)$$, then

$$z(0)=0,\qquad z(1)=b_{1}^{p_{1}-1}z(\eta_{1}).$$

We now consider the following fractional differential equation:

\begin{aligned} \textstyle\begin{cases} -D^{\alpha_{1}}_{0^{+}}z(t)=y(t),\quad 0< t< 1, \\ z(0)=0, \qquad z(1)=b_{1}^{p_{1}-1}z(\eta_{1}). \end{cases}\displaystyle \end{aligned}
(2.1)

Lemma 2.2

([17])

If $$y \in C[0,1]$$, then problem (2.1) has a unique solution

$$z(t)= \int_{0}^{1} H_{1}(t,s)y(s)\,ds,$$

where

\begin{aligned} &H_{1}(t,s)=h_{1}(t,s)+\frac{b_{1}^{p_{1}-1}t^{\alpha_{1}-1}}{1-b_{1}^{p_{1}-1}\eta _{1}^{\alpha_{1}-1}}h_{1}( \eta_{1},s), \\ &h_{1}(t,s)=\frac{1}{\Gamma(\alpha_{1})}\textstyle\begin{cases} [t(1-s)]^{\alpha_{1}-1},& 0\leq t\leq s\leq1, \\ [t(1-s)]^{\alpha_{1}-1}-(t-s)^{\alpha_{1}-1},& 0\leq s\leq t\leq1. \end{cases}\displaystyle \end{aligned}

From the above analysis, the boundary value problem

\begin{aligned} \textstyle\begin{cases} -D^{\alpha_{1}}_{0^{+}}(\varphi_{p_{1}}(D^{\beta_{1}}_{0^{+}}u(t)))=y(t),\quad 0< t< 1, \\ u(0)=u(1)=u'(0)=u'(1)=0,\qquad D^{\beta_{1}}_{0^{+}}u(0)=0,\qquad D^{\beta _{1}}_{0^{+}}u(1)=b_{1}D^{\beta_{1}}_{0^{+}}u(\eta_{1}), \end{cases}\displaystyle \end{aligned}
(2.2)

is equal to

$$\textstyle\begin{cases} D^{\beta_{1}}_{0^{+}}u(t) = \varphi_{q_{1}} (\int_{0}^{1} H_{1}(t,s)y(s)\,ds ), \\ u(0) = u(1) = u'(0) = u'(1) = 0. \end{cases}$$

Lemma 2.3

([17])

If $$y \in C[0,1]$$, then problem (2.2) has a unique solution

$$u(t)= \int_{0}^{1} G_{1}(t,s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)y(\tau )\,d\tau \biggr)\,ds,$$

where

\begin{aligned} G_{1}(t,s)=\frac{1}{\Gamma(\beta_{1})}\textstyle\begin{cases} t^{\beta_{1}-2}(1-s)^{\beta_{1}-2}[(s-t)+(\beta_{1}-2)(1-t)s], \\ \quad 0\leq t\leq s\leq1,\\ t^{\beta_{1}-2}(1-s)^{\beta_{1}-2}[(s-t)+(\beta_{1}-2)(1-t)s]+(t-s)^{\beta _{1}-1},\\ \quad 0\leq s\leq t\leq1. \end{cases}\displaystyle \end{aligned}
(2.3)

Lemma 2.4

([17, 33])

The function $$G_{1}(t,s)$$ defined by (2.3) is continuous on $$[0,1]\times[0,1]$$ and has the following properties:

1. (a)

$$G_{1}(t,s)>0$$ for all $$(t,s)\in(0,1)\times(0,1)$$;

2. (b)

$$(\beta_{1}-2)k_{1}(t)l_{1}(s)\leq\Gamma(\beta_{1})G_{1}(t,s)\leq M_{1}l_{1}(s), (t,s)\in(0,1)\times(0,1)$$;

3. (c)

$$(\beta_{1}-2)k_{1}(t)l_{1}(s)\leq\Gamma(\beta_{1})G_{1}(t,s)\leq M_{1}k_{1}(t), (t,s)\in(0,1)\times(0,1)$$;

where

\begin{aligned} &k_{1}(t)=t^{\beta_{1}-2}(1-t)^{2}, \qquad l_{1}(s)=s^{2}(1-s)^{\beta_{1}-2},\\ &M_{1}=\max\bigl\{ \beta_{1}-1,(\beta_{1}-2)^{2} \bigr\} . \end{aligned}

Similarly, we can obtain the following Lemmas 2.5 and 2.6 for the following boundary value problem:

\begin{aligned} \textstyle\begin{cases} -D^{\alpha_{2}}_{0^{+}}(\varphi_{p_{2}}(D^{\beta_{2}}_{0^{+}}v(t)))=y(t),\quad 0< t< 1, \\ v(0)=v(1)=v'(0)=v'(1)=0,\qquad D^{\beta_{2}}_{0^{+}}v(0)=0,\qquad D^{\beta _{2}}_{0^{+}}v(1)=b_{2}D^{\beta_{2}}_{0^{+}}u(\eta_{2}). \end{cases}\displaystyle \end{aligned}
(2.4)

Lemma 2.5

If $$y \in C[0,1]$$, then problem (2.4) has a unique solution

$$v(t)= \int_{0}^{1} G_{2}(t,s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)y(\tau )\,d\tau \biggr)\,ds,$$

where

\begin{aligned} &G_{2}(t,s)=\frac{1}{\Gamma(\beta_{2})}\textstyle\begin{cases} t^{\beta_{2}-2}(1-s)^{\beta_{2}-2}[(s-t)+(\beta_{2}-2)(1-t)s],\\ \quad 0\leq t\leq s\leq1,\\ t^{\beta_{2}-2}(1-s)^{\beta_{2}-2}[(s-t)+(\beta_{2}-2)(1-t)s]+(t-s)^{\beta _{2}-1}, \\ \quad 0\leq s\leq t\leq1, \end{cases}\displaystyle \\ &H_{2}(t,s)=h_{2}(t,s)+\frac{b_{2}^{p_{2}-1}t^{\alpha_{2}-1}}{1-b_{2}^{p_{2}-1}\eta _{2}^{\alpha_{2}-1}}h_{1}( \eta_{2},s), \\ &h_{2}(t,s)=\frac{1}{\Gamma(\alpha_{2})}\textstyle\begin{cases} [t(1-s)]^{\alpha_{2}-1}, & 0\leq t\leq s\leq1,\\ [t(1-s)]^{\alpha_{2}-1}-(t-s)^{\alpha_{2}-1},& 0\leq s\leq t\leq1. \end{cases}\displaystyle \end{aligned}
(2.5)

Lemma 2.6

The function $$G_{2}(t,s)$$ given by (2.5) has the properties:

1. (a)

$$G_{2}(t,s)>0, (t,s)\in(0,1)\times(0,1)$$;

2. (b)

$$(\beta_{2}-2)k_{2}(t)l_{2}(s)\leq\Gamma(\beta_{2})G_{2}(t,s)\leq M_{2}l_{2}(s), (t,s)\in(0,1)\times(0,1)$$;

3. (c)

$$(\beta_{2}-2)k_{2}(t)l_{2}(s)\leq\Gamma(\beta_{2})G_{2}(t,s)\leq M_{2}k_{2}(t), (t,s)\in(0,1)\times(0,1)$$;

where

$$k_{2}(t)=t^{\beta_{2}-2}(1-t)^{2}, \qquad l_{2}(s)=s^{2}(1-s)^{\beta_{2}-2},\qquad M_{2}=\max\bigl\{ \beta_{2}-1,(\beta_{2}-2)^{2} \bigr\} .$$

Let $$X=C[0,1]$$, then X is a Banach space with the norm $$\Vert u \Vert =\sup_{t\in[0,1]} \vert u(t) \vert$$. Let $$Y = X \times X$$, then Y is a Banach space with the norm $$\Vert (u,v) \Vert _{Y} = \Vert u \Vert + \Vert v \Vert$$. For $$\theta_{1}, \theta_{2}\in(0,1)$$ and $$\theta_{1} < \theta_{2}$$, denote

$$P = \Bigl\{ (u,v)\in Y: u(t)\geq0, v(t)\geq0, \forall t\in[0,1], \min _{t\in[\theta_{1},\theta_{2}]}\bigl(u(t)+v(t)\bigr)\geq\gamma \bigl\Vert (u,v) \bigr\Vert _{Y} \Bigr\} ,$$

where $$\gamma= \min\{\gamma_{1},\gamma_{2}\}$$ and $$\gamma_{i} = \frac{(\beta _{i}-2)}{M_{i}}\min_{t\in[\theta_{1},\theta_{2}]}k_{i}(t), i=1,2$$, then P is a cone of Y. Define operators $$T_{1}, T_{2}: Y\rightarrow X$$ and $$Q: Y\rightarrow Y$$ as follows:

\begin{aligned} &T_{1}(u,v) (t)=\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(t,s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds, \quad t \in[0,1], \\ &T_{2}(u,v) (t)=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(t,s) \varphi_{q_{2}} \biggl( \int _{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds,\quad t\in[0,1], \\ &Q(u,v)=\bigl(T_{1}(u,v),T_{2}(u,v)\bigr),\quad (u,v)\in Y. \end{aligned}

It is well known that if $$(u,v)$$ is a fixed point of the operator Q in P, then $$(u,v)$$ is a positive solution of system (1.1).

Lemma 2.7

$$Q : P \rightarrow P$$ is a completely continuous operator.

Proof

For $$(u,v)\in P$$ and $$t\in[0,1]$$, obviously, $$T_{1}(u,v)(t)\geq0, T_{2}(u,v)(t)\geq0$$. It follows from Lemmas 2.4 and 2.6 that

\begin{aligned} T_{1}(u,v) (t)&=\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(t,s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac {M_{1}l_{1}(s)}{\Gamma(\beta_{1})}\varphi_{q_{1}} \biggl( \int_{0}^{1}H_{1}(s,\tau)f\bigl(\tau ,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds, \end{aligned}

then

$$\bigl\Vert T_{1}(u,v) \bigr\Vert \leq\frac{M_{1}}{\Gamma(\beta_{1})} \varphi_{q_{1}}(\lambda) \int _{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau )\bigr)\,d\tau \biggr)\,ds.$$

Similarly,

$$\bigl\Vert T_{2}(u,v) \bigr\Vert \leq\frac{M_{2}}{\Gamma(\beta_{2})} \varphi_{q_{2}}(\mu) \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau )\bigr)\,d\tau \biggr)\,ds.$$

Therefore

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\leq{}&\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda) \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau )f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &{}+\frac{M_{2}}{\Gamma(\beta_{2})}\varphi _{q_{2}}(\mu) \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau ,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds. \end{aligned}

On the other hand, for any $$t\in[\theta_{1},\theta_{2}]$$, by Lemma 2.4, we have

\begin{aligned} T_{1}(u,v) (t)\geq{}&\varphi_{q_{1}}(\lambda) \int_{0}^{1}\frac {\beta_{1}-2}{\Gamma{(\beta_{1})}}k_{1}(t) l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ ={}&\frac{\beta_{1}-2}{\Gamma(\beta_{1})}k_{1}(t)\varphi _{q_{1}}(\lambda) \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau )f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \geq{}&\frac{\beta_{1}-2}{M_{1}}k_{1}(t) \bigl\Vert T_{1}(u,v) \bigr\Vert , \end{aligned}

then

$$\min_{t\in[\theta_{1},\theta_{2}]}T_{1}(u,v) (t)\geq\gamma_{1} \bigl\Vert T_{1}(u,v) \bigr\Vert .$$

Similarly,

$$\min_{t\in[\theta_{1},\theta_{2}]}T_{2}(u,v) (t)\geq\gamma_{2} \bigl\Vert T_{2}(u,v) \bigr\Vert .$$

Hence

\begin{aligned} &\min_{t\in[\theta_{1},\theta_{2}]}\bigl(T_{1}(u,v) (t)+T_{2}(u,v) (t)\bigr) \\ &\quad\geq\min_{t\in[\theta_{1},\theta_{2}]}T_{1}(u,v) (t)+ \min _{t\in[\theta _{1},\theta_{2}]}T_{2}(u,v) (t) \\ &\quad\geq\gamma_{1} \bigl\Vert T_{1}(u,v) \bigr\Vert + \gamma_{2} \bigl\Vert T_{2}(u,v) \bigr\Vert \\ &\quad\geq\gamma\bigl( \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \bigr)=\gamma \bigl\Vert Q(u,v) \bigr\Vert _{Y}, \end{aligned}

that is, $$Q(P)\subset P$$.

By the Ascoli-Arzela theorem and the continuity of $$f, g, G_{i}$$ and $$H_{i}$$, we deduce that $$T_{1}$$ and $$T_{2}$$ are completely continuous operators, then Q is a completely continuous operator. This completes the proof. □

Lemma 2.8

([34])

Let P be a positive cone in a Banach space E, $$\Omega_{1}$$ and $$\Omega_{2}$$ are bounded open sets in E, $$\theta\in\Omega_{1}$$, $$\overline{\Omega }_{1}\subset\Omega_{2}$$, $$A :P\cap(\overline{\Omega}_{2}\backslash\Omega _{1}) \rightarrow P$$ is a completely continuous operator. If the following conditions are satisfied:

1. (i)

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

2. (ii)

$$\Vert Ax \Vert \geq \Vert x \Vert$$, $$\forall{x}\in{P\cap\partial\Omega_{1}}$$, $$\Vert Ax \Vert \leq \Vert x \Vert$$, $$\forall{x}\in{P\cap\partial\Omega _{2}}$$, then A has at least one fixed point in $$P\cap(\overline{\Omega }_{2}\backslash\Omega_{1})$$.

3 Main results

Denote

\begin{aligned} &f^{s}_{0}=\limsup_{u+v\rightarrow0}\max _{t\in[0,1]}\frac{f(t,u,v)}{\varphi _{p_{1}}(u+v)}, \qquad g^{s}_{0}=\limsup _{u+v\rightarrow0}\max_{t\in[0,1]}\frac{g(t,u,v)}{\varphi _{p_{2}}(u+v)}, \\ &f^{i}_{0}=\liminf_{u+v\rightarrow0}\min _{t\in[\theta_{1},\theta_{2}]}\frac {f(t,u,v)}{\varphi_{p_{1}}(u+v)},\qquad g^{i}_{0}=\liminf _{u+v\rightarrow0}\min_{t\in[\theta_{1},\theta_{2}]}\frac {g(t,u,v)}{\varphi_{p_{2}}(u+v)}, \\ &f^{s}_{\infty}=\limsup_{u+v\rightarrow\infty}\max _{t\in[0,1]}\frac {f(t,u,v)}{\varphi_{p_{1}}(u+v)},\qquad g^{s}_{\infty}=\limsup _{u+v\rightarrow\infty}\max_{t\in[0,1]}\frac {g(t,u,v)}{\varphi_{p_{2}}(u+v)}, \\ &f^{i}_{\infty}=\liminf_{u+v\rightarrow\infty}\min _{t\in[\theta_{1},\theta _{2}]}\frac{f(t,u,v)}{\varphi_{p_{1}}(u+v)},\qquad g^{i}_{\infty}=\liminf _{u+v\rightarrow\infty}\min_{t\in[\theta_{1},\theta _{2}]}\frac{g(t,u,v)}{\varphi_{p_{2}}(u+v)}. \\ &A= \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)\,d\tau \biggr) \,ds,\qquad B= \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)\,d\tau \biggr) \,ds, \\ &C= \int_{\theta_{1}}^{\theta_{2}} l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{1}(s,\tau)\,d\tau \biggr) \,ds,\qquad D= \int_{\theta _{1}}^{\theta_{2}} l_{2}(s) \varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau)\,d\tau \biggr) \,ds. \end{aligned}

For $$f^{s}_{0}, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty)$$, we define the symbols $$L_{1}, L_{2}, L_{3}$$ and $$L_{4}$$ as follows:

\begin{aligned} &L_{1}=\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta_{1}-2)C\gamma _{1}\gamma} \biggr) \frac{1}{f_{\infty}^{i}},\qquad L_{2}=\varphi_{p_{1}} \biggl( \frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr)\frac{1}{f_{0}^{s}}, \\ &L_{3}=\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta_{2}-2)D\gamma _{2}\gamma} \biggr) \frac{1}{g_{\infty}^{i}},\qquad L_{4}=\varphi_{p_{2}} \biggl( \frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac{1}{g_{0}^{s}}. \end{aligned}

Theorem 3.1

(1) If $$f^{s}_{0}, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty), L_{1} < L_{2}, L_{3} < L_{4}$$, then for each $$\lambda\in(L_{1},L_{2})$$ and $$\mu\in (L_{3},L_{4})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(2) If $$f^{s}_{0} = 0, g^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty), L_{3} < L_{4}$$, then for each $$\lambda\in(L_{1},\infty)$$ and $$\mu\in(L_{3}, L_{4})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(3) If $$g^{s}_{0} = 0, f^{s}_{0}, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty), L_{1} < L_{2}$$, then for each $$\lambda\in(L_{1}, L_{2})$$ and $$\mu\in(L_{3}, \infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(4) If $$f^{s}_{0} = g^{s}_{0} = 0, f^{i}_{\infty}, g^{i}_{\infty}\in(0,\infty)$$, then for each $$\lambda\in(L_{1},\infty)$$ and $$\mu\in(L_{3},\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(5) If $$\{f^{s}_{0}, g^{s}_{0} \in(0,\infty), f^{i}_{\infty}= \infty\}$$ or $$\{ f^{s}_{0}, g^{s}_{0}\in(0,\infty), g^{i}_{\infty}= \infty\}$$, then for each $$\lambda\in(0,L_{2})$$ and $$\mu\in(0,L_{4})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(6) If $$\{f^{s}_{0} = 0, g^{s}_{0}\in(0,\infty), g^{i}_{\infty}= \infty\}$$ or $$\{ f^{s}_{0} = 0, g^{s}_{0}\in(0,\infty), f^{i}_{\infty}=\infty\}$$, then for each $$\lambda\in(0,\infty)$$ and $$\mu\in(0,L_{4})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(7) If $$\{f^{s}_{0}\in(0,\infty), g^{s}_{0} = 0, g^{i}_{\infty}= \infty\}$$ or $$\{ f^{s}_{0}\in(0,\infty), g^{s}_{0} = 0, f^{i}_{\infty}= \infty\}$$, then for each $$\lambda\in(0,L_{2})$$ and $$\mu\in(0,\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(8) If $$\{f^{s}_{0} = g^{s}_{0} = 0, g^{i}_{\infty}= \infty\}$$ or $$\{f^{s}_{0} = g^{s}_{0} = 0, f^{i}_{\infty}= \infty\}$$, then for each $$\lambda\in(0,\infty )$$ and $$\mu\in(0,\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

Proof

Because the proofs of the above cases are similar, in what follows we will prove two of them, namely cases (1) and (6).

(1) For any $$\lambda\in(L_{1},L_{2})$$ and $$\mu\in(L_{3},L_{4})$$, there exists $$0<\varepsilon< \min\{f^{i}_{\infty},g^{i}_{\infty}\}$$ such that

\begin{aligned} &\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta_{1}-2)C\gamma_{1}\gamma } \biggr)\frac{1}{f_{\infty}^{i}-\varepsilon} \leq \lambda\leq\varphi _{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr) \frac {1}{f_{0}^{s}+\varepsilon}, \\ &\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta_{2}-2)D\gamma_{2}\gamma } \biggr)\frac{1}{g_{\infty}^{i}-\varepsilon} \leq \mu\leq\varphi _{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr) \frac {1}{g_{0}^{s}+\varepsilon}. \end{aligned}

By the definitions of $$f^{s}_{0}$$ and $$g^{s}_{0}$$, there exists $$R_{1} > 0$$ such that

\begin{aligned} &f(t,u,v) < \bigl(f^{s}_{0}+\varepsilon\bigr) \varphi_{p_{1}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{1}, \\ &g(t,u,v) < \bigl(g^{s}_{0}+\varepsilon\bigr) \varphi_{p_{2}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{1}. \end{aligned}

Denote $$\Omega_{1} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{1}\}$$, for any $$(u,v)\in P\cap\partial\Omega_{1}$$ and $$t\in[0,1]$$, we have $$0\leq u(t)+v(t)\leq \Vert u \Vert + \Vert v \Vert = \Vert (u,v) \Vert _{Y} = R_{1}$$, then

\begin{aligned} T_{1}(u,v) (t)={}&\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(t,s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \leq{}&\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac {M_{1}}{\Gamma(\beta_{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau ) \bigl(f^{s}_{0}+\varepsilon\bigr)\varphi_{p_{1}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ \leq{}&\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda)\varphi_{q_{1}} \bigl(f^{s}_{0}+\varepsilon\bigr) \int_{0}^{1}l_{1}(s)\varphi _{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau) \varphi_{p_{1}}\bigl( \Vert u \Vert + \Vert v \Vert \bigr)\,d\tau \biggr)\,ds \\ ={}&\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}\bigl(\lambda\bigl(f^{s}_{0}+ \varepsilon\bigr)\bigr)A \bigl\Vert (u,v) \bigr\Vert _{Y}\\ \leq{}& \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

so $$\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}$$. In a similar manner, we deduce

\begin{aligned} T_{2}(u,v) (t)&=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(t,s) \varphi_{q_{2}} \biggl( \int _{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{2}}(\mu) \int_{0}^{1} \frac {M_{2}}{\Gamma(\beta_{2})}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau ) \bigl(g^{s}_{0}+\varepsilon\bigr)\varphi_{p_{2}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}\bigl(\mu \bigl(g^{s}_{0}+ \varepsilon\bigr)\bigr) \int_{0}^{1}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau )\,d\tau \biggr) \,ds \bigl\Vert (u,v) \bigr\Vert _{Y} \\ &=\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}\bigl(\mu \bigl(g^{s}_{0}+ \varepsilon\bigr)\bigr)B \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}
(3.1)

then $$\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}$$. Hence

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}= \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap \partial\Omega_{1}. \end{aligned}
(3.2)

On the other hand, by the definitions of $$f^{i}_{\infty}$$ and $$g^{i}_{\infty}$$, there exists $$\overline{R_{2}} > 0$$ such that

\begin{aligned} &f(t,u,v) \geq\bigl(f^{i}_{\infty}-\varepsilon\bigr) \varphi_{p_{1}}(u+v),\quad t\in [\theta_{1},\theta_{2}], u, v\geq0, u + v\geq\overline{R_{2}}, \\ &g(t,u,v) \geq\bigl(g^{i}_{\infty}-\varepsilon\bigr) \varphi_{p_{2}}(u+v), \quad t\in [\theta_{1},\theta_{2}], u, v\geq0, u + v\geq\overline{R_{2}}. \end{aligned}

Denote $$R_{2}=\max\{2R_{1},\frac{\overline{R_{2}}}{\gamma}\}$$ and $$\Omega_{2} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{2}\}$$. For any $$(u,v)\in P\cap\partial\Omega _{2}$$, we have $$\min_{t\in[\theta_{1},\theta_{2}]}(u(t)+v(t))\geq\gamma \Vert (u,v) \Vert _{Y} = \gamma R_{2}\geq\overline{R_{2}}$$, then

\begin{aligned} &T_{1}(u,v) (\theta_{1})\\ &\quad=\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(\theta_{1},s) \varphi _{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{1}}(\lambda) \int_{0}^{1}\frac{\beta _{1}-2}{\Gamma(\beta_{1})}\gamma_{1}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta_{2}}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau) \bigl(f^{i}_{\infty}-\varepsilon \bigr)\varphi_{p_{1}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}\bigl(\lambda \bigl(f^{i}_{\infty}-\varepsilon\bigr)\bigr) \int_{\theta_{1}}^{\theta _{2}}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau ) \varphi_{p_{1}}\bigl(\gamma \bigl\Vert (u,v) \bigr\Vert _{Y} \bigr)\,d\tau \biggr)\,ds \\ &\quad=\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}}\bigl( \lambda\bigl(f^{i}_{\infty}-\varepsilon\bigr)\bigr)C \bigl\Vert (u,v) \bigr\Vert _{Y}\geq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

and $$\Vert T_{1}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}$$. Similarly, we have

\begin{aligned} &T_{2}(u,v) (\theta_{2})\\ &\quad=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(\theta_{2},s) \varphi _{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{2}}(\mu) \int_{0}^{1}\frac{\beta _{2}-2}{\Gamma(\beta_{2})}\gamma_{2}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{0}^{1}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau) \bigl(g^{i}_{\infty}-\varepsilon\bigr)\varphi_{p_{2}} \bigl(u(\tau )+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{\theta_{1}}^{\theta_{2}}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau) \bigl(g^{i}_{\infty}-\varepsilon\bigr)\varphi _{p_{2}} \bigl(\gamma \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr) \,ds \\ &\quad=\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\gamma\varphi_{q_{2}}\bigl(\mu \bigl(g^{i}_{\infty}-\varepsilon\bigr)\bigr)D \bigl\Vert (u,v) \bigr\Vert _{Y}\geq\frac {1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

then $$\Vert T_{2}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}$$. So,

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\geq \bigl\Vert (u,v) \bigr\Vert _{Y}, \quad (u,v)\in P\cap\partial\Omega_{2}. \end{aligned}
(3.3)

Therefore, by (3.2), (3.3) and Lemma 2.8, we conclude that Q has at least one fixed point $$(u,v)\in P\cap(\overline{\Omega_{2}}\backslash \Omega_{1})$$ with $$R_{1}\leq \Vert (u,v) \Vert _{Y}\leq R_{2}$$.

(6) Suppose $$f^{s}_{0} = 0, g^{s}_{0}\in(0,\infty), g^{i}_{\infty}= \infty$$, then for any $$\lambda\in(0,\infty)$$ and $$\mu\in(0,L_{4})$$, there exists $$\varepsilon> 0$$ such that

$$0 < \lambda< \varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr) \frac{1}{\varepsilon},\qquad \varphi_{p_{2}} \biggl(\frac{\Gamma(\beta _{2})}{(\beta_{2}-2)D\gamma_{2}\gamma} \biggr) \varepsilon< \mu< \varphi _{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr) \frac {1}{g_{0}^{s}+\varepsilon}.$$

By the definitions of $$f^{s}_{0}$$ and $$g^{s}_{0}$$, there exists $$R_{3} > 0$$ such that

\begin{aligned} &f(t,u,v) < \varepsilon\varphi_{p_{1}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{3}, \\ &g(t,u,v) < \bigl(g^{s}_{0}+\varepsilon\bigr) \varphi_{p_{2}}(u+v),\quad t\in[0,1], 0\leq u+v\leq R_{3}. \end{aligned}

Denote $$\Omega_{3} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{3}\}$$. For any $$(u,v)\in P\cap\partial\Omega_{3}$$ and $$t\in[0,1]$$, we have

\begin{aligned} T_{1}(u,v) (t)&\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac{M_{1}}{\Gamma(\beta _{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)\varepsilon \varphi _{p_{1}}\bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\leq\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda\varepsilon) \int_{0}^{1}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)\,d\tau \biggr) \,ds \bigl\Vert (u,v) \bigr\Vert _{Y} \\ &=\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}(\lambda\varepsilon)A \bigl\Vert (u,v) \bigr\Vert _{Y}< \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

then $$\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{3}$$.

Similar to (3.1) of (1), we get $$\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial\Omega_{3}$$, then

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap\partial\Omega_{3}. \end{aligned}
(3.4)

On the other hand, by $$g^{i}_{\infty}=\infty$$, there exists $$\overline {R_{4}} > 0$$ such that

$$g(t,u,v) \geq\frac{1}{\varepsilon}\varphi_{p_{2}}(u+v), \quad t\in[\theta _{1},\theta_{2}], u, v\geq0, u + v\geq\overline{R_{4}}.$$

Let $$R_{4}=\max\{2R_{3},\frac{\overline{R_{4}}}{\gamma}\}$$ and $$\Omega_{4} = \{ (u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{4}\}$$. For any $$(u,v)\in P\cap\partial\Omega _{4}$$, we have $$\min_{t\in[\theta_{1},\theta_{2}]}(u(t)+v(t))\geq\gamma \Vert (u,v) \Vert _{Y} = \gamma R_{4}\geq\overline{R_{4}}$$, then

\begin{aligned} T_{2}(u,v) (\theta_{2})&=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(\theta_{2},s) \varphi _{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\varphi_{q_{2}}(\mu) \int_{0}^{1}\frac{\beta _{2}-2}{\Gamma(\beta_{2})}\gamma_{2}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{0}^{1}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta _{1}}^{\theta_{2}} H_{2}(s,\tau) \frac{1}{\varepsilon}\varphi_{p_{2}}\bigl(u(\tau )+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{\theta_{1}}^{\theta_{2}}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau) \frac{1}{\varepsilon}\varphi _{p_{2}}\bigl(\gamma \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\gamma\varphi_{q_{2}} \biggl( \frac{\mu}{\varepsilon} \biggr)D \bigl\Vert (u,v) \bigr\Vert _{Y}> \bigl\Vert (u,v) \bigr\Vert _{Y}. \end{aligned}

Therefore

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\geq \bigl\Vert T_{2}(u,v) \bigr\Vert \geq \bigl\Vert (u,v) \bigr\Vert _{Y}, \quad (u,v)\in P\cap \partial\Omega_{4}. \end{aligned}
(3.5)

By (3.4), (3.5) and Lemma 2.8, we conclude that Q has at least one fixed point $$(u,v)\in P\cap(\overline{\Omega_{4}}\backslash\Omega_{3})$$ with $$R_{3}\leq \Vert (u,v) \Vert _{Y}\leq R_{4}$$. This completes the proof. □

For $$f^{i}_{0}, g^{i}_{0}, f^{s}_{\infty}, g^{s}_{\infty}\in(0,\infty)$$, we define the symbols $$\widetilde{L_{1}}, \widetilde{L_{2}}, \widetilde{L_{3}}, \widetilde{L_{4}}$$ as follows:

\begin{aligned} &\widetilde{L_{1}}=\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta _{1}-2)C\gamma_{1}\gamma} \biggr)\frac{1}{f_{0}^{i}},\qquad \widetilde{L_{2}}=\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr)\frac{1}{f_{\infty}^{s}}, \\ &\widetilde{L_{3}}=\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta _{2}-2)D\gamma_{2}\gamma} \biggr)\frac{1}{g_{0}^{i}}, \qquad \widetilde{L_{4}}=\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac{1}{g_{\infty}^{s}}. \end{aligned}

Theorem 3.2

(1) If $$f^{s}_{\infty}, g^{s}_{\infty}, f^{i}_{0}, g^{i}_{0}\in(0,\infty)$$, and $$\widetilde{L_{1}} < \widetilde{L_{2}}, \widetilde{L_{3}} < \widetilde {L_{4}}$$, then for each $$\lambda\in(\widetilde{L_{1}}, \widetilde{L_{2}})$$ and $$\mu\in(\widetilde{L_{3}}, \widetilde{L_{4}})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(2) If $$f^{s}_{\infty}, f^{i}_{0}, g^{i}_{0}\in(0,\infty), g^{s}_{\infty}= 0$$, and $$\widetilde{L_{1}} <\widetilde{L_{2}}$$, then for each $$\lambda\in(\widetilde {L_{1}},\widetilde{L_{2}})$$ and $$\mu\in(\widetilde{L_{3}},\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(3) If $$g^{s}_{\infty}, f^{i}_{0}, g^{i}_{0}\in(0,\infty), f^{s}_{\infty}= 0, \widetilde{L_{3}} < \widetilde{L_{4}}$$, then for each $$\lambda\in(\widetilde {L_{1}},\infty)$$ and $$\mu\in(\widetilde{L_{3}}, \widetilde{L_{4}})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(4) If $$f^{i}_{0}, g^{i}_{0}\in(0,\infty), f^{s}_{\infty}= g^{s}_{\infty}= 0$$, then for each $$\lambda\in(\widetilde{L_{1}},\infty)$$ and $$\mu\in (\widetilde{L_{3}},\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(5) If $$\{f^{s}_{\infty}, g^{s}_{\infty}\in(0,\infty), f^{i}_{0} = \infty\}$$ or $$\{f^{s}_{\infty}, g^{s}_{\infty}\in(0,\infty), g^{i}_{0} = \infty\}$$, then for each $$\lambda\in(0,\widetilde{L_{2}})$$ and $$\mu\in(0,\widetilde{L_{4}})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(6) If $$\{f^{i}_{0} = \infty, f^{s}_{\infty}\in(0,\infty), g^{s}_{\infty}= 0\}$$ or $$\{f^{s}_{\infty}\in(0,\infty), g^{s}_{\infty}= 0, g^{i}_{0} = \infty\}$$, then for each $$\lambda\in(0,\widetilde{L_{2}})$$ and $$\mu\in(0,\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(7) If $$\{f^{i}_{0}=\infty, g^{s}_{\infty}\in(0,\infty), f^{s}_{\infty}= 0\}$$ or $$\{g^{s}_{\infty}\in(0,\infty), g^{i}_{0} = \infty, {f^{s}_{\infty}= 0 }\}$$, then for each $$\lambda\in(0,\infty)$$ and $$\mu\in (0,\widetilde{L_{4}})$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

(8) If $$\{f^{s}_{\infty}= g^{s}_{\infty}= 0, f^{i}_{0} = \infty\}$$ or $$\{ f^{s}_{\infty}= g^{s}_{\infty}= 0, g^{i}_{0} = \infty\}$$, then for each $$\lambda \in(0,\infty)$$ and $$\mu\in(0,\infty)$$, system (1.1) has at least one positive solution $$(u(t),v(t)), t\in(0,1)$$.

Proof

Because the proofs of the above cases are similar, in what follows we will prove two of them, namely cases (1) and (6).

(1) For any $$\lambda\in(\widetilde{L_{1}},\widetilde{L_{2}})$$ and $$\mu\in (\widetilde{L_{3}},\widetilde{L_{4}})$$, there exists $$0<\varepsilon< \min\{ f^{i}_{0},g^{i}_{0}\}$$ such that

\begin{aligned} &\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2(\beta_{1}-2)C\gamma_{1}\gamma } \biggr)\frac{1}{f_{0}^{i}-\varepsilon} \leq \lambda\leq\varphi _{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{2AM_{1}} \biggr) \frac{1}{f_{\infty }^{s}+\varepsilon}, \\ &\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2(\beta_{2}-2)D\gamma_{2}\gamma } \biggr)\frac{1}{g_{0}^{i}-\varepsilon} \leq \mu\leq\varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac{1}{g_{\infty}^{s}+\varepsilon}. \end{aligned}

By the definitions of $$f^{i}_{0}$$ and $$g^{i}_{0}$$, there exists $$R_{1} > 0$$ such that

\begin{aligned} &f(t,u,v) \geq\bigl(f^{i}_{0}-\varepsilon\bigr) \varphi_{p_{1}}(u+v), \quad t\in[\theta _{1},\theta_{2}], u, v\geq0, u + v\leq R_{1}, \\ &g(t,u,v) \geq\bigl(g^{i}_{0}-\varepsilon\bigr) \varphi_{p_{2}}(u+v),\quad t\in[\theta _{1},\theta_{2}], u, v\geq0, u + v\leq R_{1}. \end{aligned}

Denote $$\Omega_{1} = \{(u,v)\in Y: \Vert (u,v) \Vert < R_{1}\}$$, for any $$(u,v)\in P\cap\partial\Omega_{1}$$, we have

\begin{aligned} &T_{1}(u,v) (\theta_{1})\\ &\quad=\varphi_{q_{1}}(\lambda) \int_{0}^{1} G_{1}(\theta_{1},s) \varphi _{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{1}}(\lambda) \int_{\theta _{1}}^{\theta_{2}}\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma_{1}l_{1}(s) \varphi _{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta_{2}}l_{1}(s)\varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau) \bigl(f^{i}_{0}-\varepsilon\bigr)\varphi _{p_{1}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}}\bigl( \lambda\bigl(f^{i}_{0}-\varepsilon\bigr)\bigr)C \bigl\Vert (u,v) \bigr\Vert _{Y}\geq\frac {1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

then $$\Vert T_{1}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}$$. Similarly, we have

\begin{aligned} &T_{2}(u,v) (\theta_{2})\\ &\quad=\varphi_{q_{2}}(\mu) \int_{0}^{1} G_{2}(\theta_{2},s) \varphi _{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\varphi_{q_{2}}(\mu) \int_{\theta _{1}}^{\theta_{2}}\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma_{2}l_{2}(s) \varphi _{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau)g\bigl(\tau,u( \tau),v(\tau )\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\varphi_{q_{2}}(\mu) \int_{\theta_{1}}^{\theta_{2}}l_{2}(s)\varphi_{q_{2}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{2}(s,\tau) \bigl(g^{i}_{0}-\varepsilon\bigr)\varphi _{p_{2}} \bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\quad\geq\frac{\beta_{2}-2}{\Gamma(\beta_{2})}\gamma _{2}\gamma\varphi_{q_{2}}\bigl(\mu \bigl(g^{i}_{0}-\varepsilon\bigr)\bigr)D \bigl\Vert (u,v) \bigr\Vert _{Y}\geq\frac {1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

then $$\Vert T_{2}(u,v) \Vert \geq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{1}$$. Therefore

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}\geq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap\partial\Omega_{1}. \end{aligned}
(3.6)

On the other hand, we define $$f^{*},g^{*}:[0,1]\times[0,+\infty)\rightarrow [0,+\infty)$$ as follows:

$$f^{*}(t,x)=\max_{0\leq u + v \leq x}f(t,u,v),\qquad g^{*}(t,x)=\max _{0\leq u + v \leq x}g(t,u,v),$$

then

\begin{aligned} &f(t,u,v)\leq f^{*}(t,x),\quad t\in[0,1], u,v\geq0, u + v \leq x, \\ &g(t,u,v)\leq g^{*}(t,x),\quad t\in[0,1], u,v\geq0, u + v \leq x. \end{aligned}

Clearly, $$f^{*}(t,x)$$ and $$g^{*}(t,x)$$ are nondecreasing on x, by the proof of [35], we have

$$\limsup_{x\rightarrow+\infty}\max_{t\in[0,1]}\frac{f^{*}(t,x)}{\varphi _{p_{1}}(x)} \leq f^{s}_{\infty}, \qquad \limsup_{x\rightarrow+\infty}\max _{t\in [0,1]}\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}\leq g^{s}_{\infty}.$$

From the above inequalities, there exists $$\overline{R_{2}} > 0$$ such that

\begin{aligned} &\frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)} \leq\limsup_{x\rightarrow+\infty }\max_{t\in[0,1]} \frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)}+\varepsilon\leq f^{s}_{\infty}+\varepsilon,\quad t \in[0,1], x \geq\overline{R_{2}}, \\ &\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}\leq\limsup_{x\rightarrow+\infty}\max_{t\in[0,1]} \frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}+\varepsilon\leq g^{s}_{\infty}+\varepsilon,\quad t \in[0,1], x \geq\overline{R_{2}}. \end{aligned}

Then $$f^{*}(t,x)\leq(f^{s}_{\infty}+\varepsilon)\varphi_{p_{1}}(x), g^{*}(t,x)\leq (g^{s}_{\infty}+\varepsilon)\varphi_{p_{2}}(x), t\in[0,1], x \geq\overline{R_{2}}$$.

Denote $$R_{2} = \max\{2R_{1},\overline{R_{2}}\}$$, $$\Omega_{2} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y} < {R_{2} }\}$$. For any $$(u,v)\in P\cap\partial \Omega_{2}$$, by the definitions of $$f^{*}$$ and $$g^{*}$$, we have

$$f\bigl(t,u(t),v(t)\bigr)\leq f^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr), \qquad g\bigl(t,u(t),v(t)\bigr)\leq g^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr),\quad t\in[0,1],$$

so

\begin{aligned} T_{1}(u,v) (t)&\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac{M_{1}}{\Gamma(\beta _{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{1}}(\lambda)\frac{M_{1}}{\Gamma (\beta_{1})} \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau ) \bigl(f^{s}_{\infty}+\varepsilon\bigr)\varphi_{p_{1}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}\bigl(\lambda\bigl(f^{s}_{\infty}+ \varepsilon\bigr)\bigr)A \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

and $$\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}$$. Similarly, we have

\begin{aligned} T_{2}(u,v) (t)&\leq\varphi_{q_{2}}(\mu) \int_{0}^{1} \frac{M_{2}}{\Gamma(\beta _{2})}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{2}}(\mu)\frac{M_{2}}{\Gamma(\beta _{2})} \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau) \bigl(g^{s}_{\infty }+\varepsilon\bigr)\varphi_{p_{2}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}\bigl(\mu \bigl(g^{s}_{\infty}+ \varepsilon\bigr)\bigr)B \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

so $$\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{2}$$. Therefore

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}= \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap \partial\Omega_{2}. \end{aligned}
(3.7)

From (3.6), (3.7) and Lemma 2.8, we get that Q has at least one fixed point $$(u,v)\in {P\cap(\overline{\Omega_{2}}\backslash\Omega _{1})}$$ with $$R_{1}\leq \Vert (u,v) \Vert _{Y}\leq R_{2}$$.

(6) Suppose $$f^{i}_{0} = \infty, f^{s}_{\infty}\in(0,\infty), g^{s}_{\infty}= 0$$, for any $$\lambda\in(0,\widetilde{L_{2}})$$ and $$\mu\in(0,\infty)$$, there exists $$\varepsilon> 0$$ such that

$$\varphi_{p_{1}} \biggl(\frac{\Gamma(\beta_{1})}{(\beta_{1}-2)C\gamma_{1}\gamma } \biggr)\varepsilon< \lambda< \varphi_{p_{1}} \biggl(\frac{\Gamma(\beta _{1})}{2AM_{1}} \biggr)\frac{1}{f_{\infty}^{0}+\varepsilon},\quad 0 < \mu< \varphi_{p_{2}} \biggl(\frac{\Gamma(\beta_{2})}{2BM_{2}} \biggr)\frac {1}{\varepsilon}.$$

By $$f^{i}_{0}=\infty$$, there exists $$R_{3} > 0$$ such that

$$f(t,u,v) \geq\frac{1}{\varepsilon}\varphi_{p_{1}}(u+v), \quad t\in[\theta _{1},\theta_{2}], u,v\geq0, 0 \leq u + v\leq R_{3}.$$

Choose $$\Omega_{3} = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y}< R_{3}\}$$, then for any $$(u,v)\in P\cap\partial\Omega_{3}$$, we have

\begin{aligned} T_{1}(u,v) (\theta_{1})&\geq\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta _{2}}\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma_{1}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau)f\bigl(\tau,u( \tau),v(\tau)\bigr)\,d\tau \biggr)\,ds \\ &\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\varphi_{q_{1}}(\lambda) \int_{\theta_{1}}^{\theta_{2}}l_{1}(s)\varphi _{q_{1}} \biggl( \int_{\theta_{1}}^{\theta_{2}} H_{1}(s,\tau) \frac{1}{\varepsilon }\varphi_{p_{1}}\bigl(u(\tau)+v(\tau)\bigr)\,d\tau \biggr) \,ds \\ &\geq\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}} \biggl( \frac{\lambda}{\varepsilon} \biggr) \int _{\theta_{1}}^{\theta_{2}}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{\theta_{1}}^{\theta _{2}} H_{1}(s,\tau)\,d\tau \biggr) \,ds \bigl\Vert (u,v) \bigr\Vert _{Y} \\ &=\frac{\beta_{1}-2}{\Gamma(\beta_{1})}\gamma _{1}\gamma\varphi_{q_{1}} \biggl( \frac{\lambda}{\varepsilon} \biggr)C \bigl\Vert (u,v) \bigr\Vert _{Y}\geq \bigl\Vert (u,v) \bigr\Vert _{Y}. \end{aligned}

Thus,

\begin{aligned} {\bigl\Vert Q(u,v) \bigr\Vert _{Y} }\geq \bigl\Vert T_{1}(u,v) \bigr\Vert \geq \bigl\Vert (u,v) \bigr\Vert _{Y}, \quad (u,v)\in P\cap\partial\Omega_{3}. \end{aligned}
(3.8)

On the other hand, we define $$f^{*},g^{*}:[0,1]\times[0,+\infty)\rightarrow [0,+\infty)$$ as follows:

$$f^{*}(t,x)=\max_{0\leq u + v \leq x}f(t,u,v), \qquad g^{*}(t,x)=\max _{0\leq u + v \leq x}g(t,u,v).$$

By the proof of [35], we have

$$\limsup_{x\rightarrow+\infty}\max_{t\in[0,1]}\frac{f^{*}(t,x)}{\varphi _{p_{1}}(x)} \leq f^{s}_{\infty},\qquad \lim_{x\rightarrow+\infty}\max _{t\in [0,1]}\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}=0.$$

For above $$\varepsilon>0$$, there exists $$\overline{R_{4}}>0$$ such that, for any $$t\in[0,1], x\geq\overline{R_{4}}$$, we have

\begin{aligned} &\frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)} \leq\limsup_{x\rightarrow+\infty }\max_{t\in[0,1]} \frac{f^{*}(t,x)}{\varphi_{p_{1}}(x)}+\varepsilon\leq f^{s}_{\infty}+\varepsilon, \\ &\frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)} \leq\lim_{x\rightarrow+\infty}\max_{t\in[0,1]} \frac{g^{*}(t,x)}{\varphi_{p_{2}}(x)}+\varepsilon=\varepsilon, \end{aligned}

hence $$f^{*}(t,x)\leq(f^{s}_{\infty}+\varepsilon)\varphi_{p_{1}}(x), g^{*}(t,x) \leq\varepsilon\varphi_{p_{2}}(x)$$.

Let $$R_{4} = \max\{2R_{3},\overline{R_{4}}\}$$ and $${\Omega_{4} } = \{(u,v)\in Y: \Vert (u,v) \Vert _{Y} < R_{4}\}$$. For any $$(u,v)\in P\cap\partial \Omega_{4}$$ and $$t\in[0,1]$$, we have

$$f\bigl(t,u(t),v(t)\bigr)\leq f^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr),\qquad g\bigl(t,u(t),v(t)\bigr)\leq g^{*}\bigl(t, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr),$$

then

\begin{aligned} T_{1}(u,v) (t)&\leq\varphi_{q_{1}}(\lambda) \int_{0}^{1} \frac{M_{1}}{\Gamma(\beta _{1})}l_{1}(s) \varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau)f^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{1}}(\lambda)\frac{M_{1}}{\Gamma (\beta_{1})} \int_{0}^{1} l_{1}(s)\varphi_{q_{1}} \biggl( \int_{0}^{1} H_{1}(s,\tau ) \bigl(f^{s}_{\infty}+\varepsilon\bigr)\varphi_{p_{1}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &=\frac{M_{1}}{\Gamma(\beta_{1})}\varphi _{q_{1}}\bigl(\lambda\bigl(f^{s}_{\infty}+ \varepsilon\bigr)\bigr)A \bigl\Vert (u,v) \bigr\Vert _{Y}\leq \frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

so $$\Vert T_{1}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{4}$$. In a similar manner, we deduce

\begin{aligned} T_{2}(u,v) (t)&\leq\varphi_{q_{2}}(\mu) \int_{0}^{1} \frac{M_{2}}{\Gamma(\beta _{2})}l_{2}(s) \varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)g^{*}\bigl(\tau, \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr)\,d\tau \biggr)\,ds \\ &\leq\varphi_{q_{2}}(\mu)\frac{M_{2}}{\Gamma(\beta _{2})} \int_{0}^{1} l_{2}(s)\varphi_{q_{2}} \biggl( \int_{0}^{1} H_{2}(s,\tau)\varepsilon \varphi_{p_{2}}\bigl( \bigl\Vert (u,v) \bigr\Vert _{Y}\bigr) \,d\tau \biggr)\,ds \\ &=\frac{M_{2}}{\Gamma(\beta_{2})}\varphi_{q_{2}}(\mu \varepsilon)B \bigl\Vert (u,v) \bigr\Vert _{Y}\leq\frac{1}{2} \bigl\Vert (u,v) \bigr\Vert _{Y}, \end{aligned}

so $$\Vert T_{2}(u,v) \Vert \leq\frac{1}{2} \Vert (u,v) \Vert _{Y}, (u,v)\in P\cap\partial \Omega_{4}$$. Therefore

\begin{aligned} \bigl\Vert Q(u,v) \bigr\Vert _{Y}= \bigl\Vert T_{1}(u,v) \bigr\Vert + \bigl\Vert T_{2}(u,v) \bigr\Vert \leq \bigl\Vert (u,v) \bigr\Vert _{Y},\quad (u,v)\in P\cap \partial\Omega_{4}. \end{aligned}
(3.9)

From (3.8), (3.9) and Lemma 2.8, we conclude that Q has at least one fixed point $$(u,v)\in P\cap(\overline{\Omega_{4}}\backslash\Omega_{3})$$ with $$R_{3}\leq \Vert (u,v) \Vert _{Y}\leq R_{4}$$. This completes the proof. □

4 Conclusion

In this paper, we study the existence of positive solutions for a system of nonlinear fractional differential equations nonlocal boundary value problems with parameters and p-Laplacian operator. Under different combinations of superlinearity and sublinearity of the functions f and g, various existence results for positive solutions are derived in terms of different values of λ and μ via the Guo-Krasnosel’skii fixed point theorem.