Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives

Abstract

In this paper we study the existence of unique positive solutions for the following coupled system:

$$\begin{aligned} \textstyle\begin{cases} D_{0^{+}}^{\alpha }x(\tau )+f_{1}(\tau ,x(\tau ),D_{0^{+}}^{\eta }x( \tau ))+g_{1}(\tau ,y(\tau ))=0, \\ D_{0^{+}}^{\beta }y(\tau )+f_{2}(\tau ,y(\tau ),D_{0^{+}}^{\gamma }y( \tau ))+g_{2}(\tau ,x(\tau ))=0, \\ \tau \in (0,1),\qquad n-1< \alpha ,\beta < n; \\ x^{(i)}(0)=y^{(i)}(0)=0,\quad i=0,1,2,\ldots ,n-2; \\ [D_{0^{+}}^{\xi }y(\tau ) ]_{\tau =1}=k_{1}(y(1)),\qquad [D_{0^{+}}^{\zeta }x(\tau ) ]_{\tau =1}=k_{2}(x(1)), \end{cases}\displaystyle \end{aligned}$$

where the integer number $n>3$ and $1\leq \gamma \leq \xi \leq n-2$, $1\leq \eta \leq \zeta \leq n-2$, $f_{1},f_{2}:[0,1]\times \mathbb{R^{+}}\times \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, $g_{1},g_{2}:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ and $k_{1},k_{2}:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ are continuous functions, $D_{0^{+}}^{\alpha }$ and $D_{0^{+}}^{\beta }$ stand for the Riemann–Liouville derivatives. An illustrative example is given to show the effectiveness of theoretical results.

1 Introduction

A lot of fractional differential equations and coupled systems have been studied widely; see [1–19, 24] and the references therein. As is well known, coupled systems with boundary conditions appear in the investigations of many problems such as mathematical biology (see [9, 30]), natural sciences and engineering; for example, we can see beam deformation and steady-state heat flow (see [25, 26]) and heat equations (see [18, 24]). So the subject of coupled systems is gaining much attention and importance. There are a large number of articles dealing with the existence or multiplicity of solutions or positive solutions for some nonlinear coupled systems with boundary conditions; for details, see [7, 8, 10, 11, 20, 21, 27, 29, 32, 33, 35–41].

In [42] Zhang and Tian considered a unique positive solution for the following problem:

$$\begin{aligned} \textstyle\begin{cases} D_{0^{+}}^{\alpha }w(\tau )+f(\tau ,w(\tau ),D_{0^{+}}^{\gamma }w( \tau ))+g(\tau ,w(\tau ))=0,\quad \tau \in (0,1), n-1< \alpha < n; \\ w^{(i)}(0)=0, \quad i=0,1,2,\ldots ,n-2; \\ [D_{0^{+}}^{\beta }w(\tau ) ]_{\tau =1}=k(w(1)), \end{cases}\displaystyle \end{aligned}$$

(1)

where $n>3$, $1\leq \gamma \leq \beta \leq n-2$, $f:[0,1]\times \mathbb{R^{+}} \times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$, $g:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ and $k:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ are continuous functions, $D_{0^{+}}^{\alpha }$ is the Riemann–Liouville fractional derivative and $w^{(i)}$ represents the ith (ordinary) derivative of w.

Continuing their work, we establish the existence of solutions for the following coupled system:

$$\begin{aligned} \textstyle\begin{cases} D_{0^{+}}^{\alpha }x(\tau )+f_{1}(\tau ,x(\tau ), D_{0^{+}}^{\eta }x( \tau ))+g_{1}(\tau ,y(\tau ))=0, \\ D_{0^{+}}^{\beta }y(\tau )+f_{2}(\tau ,y(\tau ),D_{0^{+}}^{\gamma }y( \tau ))+g_{2}(\tau ,x(\tau ))=0, \quad \tau \in (0,1), n-1< \alpha ,\beta < n; \\ x^{(i)}(0)=y^{(i)}(0)=0, \quad i=0,1,2,\ldots ,n-2; \\ [D_{0^{+}}^{\xi }y(\tau ) ]_{\tau =1}=k_{1}(y(1)),\qquad [D_{0^{+}}^{\zeta }x(\tau ) ]_{\tau =1}=k_{2}(x(1)), \end{cases}\displaystyle \end{aligned}$$

(2)

where the integer number $n>3$ and $1\leq \gamma \leq \xi \leq n-2$, $1\leq \eta \leq \zeta \leq n-2$, $f_{1},f_{2}:[0,1]\times \mathbb{R^{+}}\times \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, $g_{1},g_{2}:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ and $k_{1},k_{2}:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ are continuous functions, $D_{0^{+}}^{\alpha }$ and $D_{0^{+}}^{\beta }$ stand for the Riemann–Liouville derivatives.

2 Preliminaries

Suppose $(E,\| \cdot \| )$ is a Banach space which is partially ordered by a cone $P\subseteq E$. We denote the zero element of E by θ. A cone P is called normal if there exists a constant $N>0$ such that $\theta \leq x\leq y$ implies $\| x \| \leq N \| y \|$.

Definition 2.1

([22, 23])

$A:P\times P\rightarrow P$ is said to be a mixed monotone operator if $A(x,y)$ is increasing in x and decreasing in y, i.e., for $x_{i},y_{i} \in P$ ($i=1,2$), $x_{1}\leq x_{2}$, $y_{1}\geq y_{2}$ imply $A(x_{1},y_{1})\leq A(x_{2},y_{2})$. The element $x\in P$ is called a fixed point of A if $A(x,x)=x$.

An element $u^{*}\in D$ is called a fixed point of A if it satisfies $A(u^{*},u^{*})=u^{*}$. Let $h>\theta$, write $P_{h}=\{u\in E|\exists \lambda ,\mu >0 : \lambda h\leq u\leq \mu h\}$.

Let Φ be a class of functions $\varphi :(0,1)\rightarrow (0,1)$ with $\varphi (\tau )>\tau$ for $\tau \in (0,1)$.

Theorem 2.2

([34])

LetPbe a normal cone inE, $\alpha \in (0,1)$$A:P\rightarrow P$is an increasing sub-homogeneous, $B:P\rightarrow P$is a decreasing operator, $C:P\times P\rightarrow P$is a mixed monotone operator and that satisfy the following conditions:

$$\begin{aligned} B\biggl(\frac{1}{\tau }u\biggr)\geq \tau Bu,\qquad C\biggl(\tau u,\frac{1}{\tau }v\biggr)\geq \tau ^{\alpha }C(u,v),\quad u,v\in P. \end{aligned}$$

(3)

Assume that

$(i)$:

∃ $h_{0}\in P_{h}$such that$Ah_{0}\in P_{h}$, $Bh_{0}\in P_{h}$, $C(h_{0},h_{0})\in P_{h}$;

$(\mathit{ii})$:

∃ $\delta _{0}>0$with$C(u,v)\geq \delta _{0}(Au+Bv)$for$u,v\in P$.

Then

$(1)$ :

$A:P_{h}\rightarrow P_{h}$, $B:P_{h}\rightarrow P_{h}$and$C:P_{h}\times P_{h}\rightarrow P_{h}$;

$(2)$ :

∃ $x_{0},y_{0}\in P_{h}$and$r\in (0,1)$with

$$ rx_{0}\leq x_{0}< y_{0}, x_{0}\leq Ax_{0}+By_{0}+C(x_{0},y_{0})\leq Ay_{0}+Bx_{0}+C(y_{0},x_{0}) \leq y_{0}; $$

$(3)$ :

the equation$Au+Bu+C(u,u)=u$has a unique solution$u^{*}$in$P_{h}$;

$(4)$ :

for$x_{0},y_{0}\in P_{h}$, we can construct

$$\begin{aligned} &u_{n}=Ax_{n-1}+By_{n-1}+C(x_{n-1},y_{n-1}), \\ &v_{n}=Ay_{n-1}+Bx_{n-1}+C(y_{n-1},x_{n-1}),\quad n=1,2, \ldots \end{aligned}$$

and$u_{n}\rightarrow u^{*}$and$v_{n}\rightarrow v^{*}$.

Definition 2.3

([28, 31])

The Riemann–Liouville fractional derivative for a continuous function f is defined by

$$ D^{\alpha }f(\tau )=\frac{1}{\varGamma (n-\alpha )}\biggl(\frac{d}{d\tau } \biggr)^{n} \int _{0}^{\tau }\frac{f(\rho )}{(t-\rho )^{\alpha -n+1}}\,d\rho \quad \bigl(n=[\alpha ]+1\bigr), $$

where the right-hand side is point-wise defined on $(0,\infty )$.

Definition 2.4

([28, 31])

Let $[a,b]$ be an interval in $\mathbb{R}$ and $\alpha >0$. The Riemann–Liouville fractional order integral of a function $f\in L^{1}([a,b],\mathbb{R})$ is defined by

$$\begin{aligned} I^{\alpha }_{a}f(\tau )=\frac{1}{\varGamma (\alpha )} \int _{a}^{\tau }\frac{f(\rho )}{(\tau -\rho )^{1-\alpha }}\,d\rho , \end{aligned}$$

whenever the integral exists.

Lemma 2.5

([42])

Let$h\in C[0,1]$, then the unique solution of the linear problem

$$\begin{aligned}& D^{\alpha }_{0^{+}}x(\tau )+h(\tau )=0,\quad \tau \in (0,1), n-1< \alpha \leq n; \end{aligned}$$

(4)

$$\begin{aligned}& x^{i}(0)=0, \quad i=0,1,2,3,\ldots,n-2; \end{aligned}$$

(5)

$$\begin{aligned}& \bigl[D^{\beta }_{0^{+}}x(\tau )\bigr]_{\tau =1}=k\bigl(x(1) \bigr),\quad 1\leq \beta \leq n-2; \end{aligned}$$

(6)

is given by

$$ x(\tau )= \int ^{1}_{0}G(\tau ,\rho )h(\rho )\,d\rho + \frac{\varGamma (\alpha -\beta )}{\varGamma (\alpha )}k\bigl(x(1)\bigr)\tau ^{\alpha -1}, $$

where

$$ G(\tau ,\rho )= \textstyle\begin{cases} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\beta -1}-(\tau -\rho )^{\alpha -1}}{\varGamma (\alpha )}, & 0 \leq \rho \leq \tau \leq 1; \\ \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\beta -1}}{\varGamma (\alpha )}, & 0 \leq \tau \leq \rho \leq 1; \end{cases} $$

(7)

is the Green function.

Lemma 2.6

([42])

The Green function (7) has the following properties:

$$\begin{aligned}& 0\leq \tau ^{\alpha -1}(1-\rho )^{\alpha -\beta -1}\bigl[1-(1-\rho )^{ \beta }\bigr]\leq \varGamma (\alpha )G(\tau ,\rho ) \leq \tau ^{\alpha -1}(1- \rho )^{\alpha -\beta -1}, \\& \begin{aligned} 0&\leq \tau ^{\alpha -\gamma -1}(1-\rho )^{\alpha -\beta -1}\bigl[1-(1- \rho )^{\beta -\gamma }\bigr]\leq \varGamma (\alpha -\gamma )D_{0^{+}}^{\gamma }G( \tau ,\rho ) \\ &\leq \tau ^{\alpha -\gamma -1}(1-\rho )^{\alpha -\beta -1}, \quad \tau , \rho \in [0,1]. \end{aligned} \end{aligned}$$

Lemma 2.7

([36])

$K_{h}=P_{h_{1}}\times P_{h_{2}}$, where that$K=P\times P$and$h(\tau )=(h_{1},h_{2})$.

3 Main results

Let $E\times E\subset X\times X$ with $X=C[0,1]$ such that $E=\{x|x,D^{\eta }_{0^{+}}x, D^{\gamma }_{0^{+}}x\in X\}$ endowed with the norm $\| x \| =\max \{\max_{\tau \in [0,1]}{|x(\tau )|}, \max_{\tau \in [0,1]}{D^{\eta }_{0^{+}}|x(\tau )|},\max_{\tau \in [0,1]}{D^{ \gamma }_{0^{+}}|x(\tau )|}\}$. For $(x,y)\in E\times E$, let $\| (x,y) \| =\max \{\| x \| , \| y \| \}$. It is easy to see that $(E\times E,\| (x,y) \| )$ is a Banach space. Define $P=\{x\in E:x,D^{\eta }_{0^{+}}x,D^{\gamma }_{0^{+}}x\geq 0\}$, $K=P\times P$, then K is a normal cone equipped with the following partial order:

$$\begin{aligned} &(x_{1},y_{1})\preceq (x_{2},y_{2}) \quad \Leftrightarrow\quad x_{1}\leq x_{2}, y_{1} \leq y_{2}, \end{aligned}$$

(8)

and

$$\begin{aligned} &D^{\eta }_{0^{+}}x_{1}(\tau )\leq D^{\eta }_{0^{+}}x_{2}( \tau ), \qquad D^{ \gamma }_{0^{+}}x_{1}(\tau )\leq D^{\gamma }_{0^{+}}x_{2}(\tau ), \\ &D^{\eta }_{0^{+}}y_{1}(\tau )\leq D^{\eta }_{0^{+}}y_{2}( \tau ), \qquad D^{ \gamma }_{0^{+}}y_{1}(\tau )\leq D^{\gamma }_{0^{+}}y_{2}(\tau ). \end{aligned}$$

By Lemma 2.5 in [42], the unique positive solution for the problem (1) is given by

$$ \begin{aligned} x(\tau )&= \int _{0}^{1}G(\tau ,\rho )f\bigl(\rho ,x(\rho ),D_{0^{+}}^{ \gamma }x(\rho )\bigr)\,d\rho \\ &\quad {}+ \int _{0}^{1}G(\tau ,\rho )g\bigl(\rho ,x(\rho ) \bigr)\,d \rho +\frac{\varGamma (\alpha -\beta )}{\varGamma (\alpha )}k\bigl(x(1)\bigr)\tau ^{\alpha -1}, \end{aligned} $$

where

$$\begin{aligned} G(\tau ,\rho )= \textstyle\begin{cases} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\beta -1}-(\tau -\rho )^{\alpha -1}}{\varGamma (\alpha )}, & 0\leq \rho \leq \tau \leq 1; \\ \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\beta -1}}{\varGamma (\alpha )},& 0\leq \tau \leq \rho \leq 1, \end{cases}\displaystyle \end{aligned}$$

(9)

is a Green function.

Assume that $f_{1}(\tau ,x,y)$, $f_{2}(\tau ,x,y)$ are continuous, then $(x,y)\in X\times X$ is a solution of the system (2) if and only if $(x,y)$ is a solution of the integral equations

$$\begin{aligned} \textstyle\begin{cases} x(\tau )=\int _{0}^{1}G_{1}(\tau ,\rho )f_{1}(\rho ,x(\rho ),D_{0^{+}}^{ \eta }x(\rho ))\,d\rho \\ \hphantom{x(\tau )=}{}+\int _{0}^{1}G_{1}(\tau ,\rho )g_{1}(\rho ,y(\rho ))\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}(x(1))\tau ^{\alpha -1}, \\ y(\tau )=\int _{0}^{1}G_{2}(\tau ,\rho )f_{2}(\rho ,y(\rho ),D_{0^{+}}^{ \gamma }y(\rho ))\,d\rho \\ \hphantom{y(\tau )=}{}+\int _{0}^{1}G_{2}(\tau ,\rho )g_{2}(\rho ,x(\rho ))\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}(y(1))\tau ^{\beta -1}, \end{cases}\displaystyle \end{aligned}$$

(10)

where

$$\begin{aligned} G_{1}(\tau ,\rho )= \textstyle\begin{cases} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\zeta -1}-(\tau -\rho )^{\alpha -1}}{\varGamma (\alpha )}, & 0\leq \rho \leq \tau \leq 1; \\ \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}, & 0\leq \tau \leq \rho \leq 1, \end{cases}\displaystyle \end{aligned}$$

(11)

and

$$\begin{aligned} G_{2}(\tau ,\rho )= \textstyle\begin{cases} \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}-(\tau -\rho )^{\beta -1}}{\varGamma (\beta )}, & 0\leq \rho \leq \tau \leq 1; \\ \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}, & 0\leq \tau \leq \rho \leq 1, \end{cases}\displaystyle \end{aligned}$$

(12)

are Green functions.

Let us define the operators $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, $C_{2}$ by

$$ \begin{gathered} A_{1}(u) (\tau )= \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,v(\rho )\bigr)\,d \rho ,\qquad A_{2}(v) ( \tau )= \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl( \rho ,u(\rho )\bigr)\,d\rho , \\ B_{1}(u) (\tau )=\frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u(1)\bigr) \tau ^{\alpha -1},\qquad B_{2}(v) (\tau )= \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1} \bigl(v(1)\bigr)\tau ^{\beta -1}, \\ C_{1}(v,u) (\tau )= \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,v(\rho ),D_{0^{+}}^{ \eta }u(\rho ) \bigr)\,d\rho , \\ C_{2}(u,v) (\tau )= \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,u(\rho ),D_{0^{+}}^{ \gamma }v(\rho ) \bigr)\,d\rho , \end{gathered} $$

(13)

for $0\leq \tau \leq 1$.

Theorem 3.1

Assume that

$(H_{1})$ :

$f_{1},f_{2} :[0,1]\times \mathbb{R^{+}}\times \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$, $g_{1},g_{2} :[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$and$k_{1},k_{2} :\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$are continuous, also: $f_{1}(\tau ,1,0)\not \equiv 0$, $f_{2}(\tau ,1,0)\not \equiv 0$;

$(H_{2})$ :

$f_{1}(\tau ,x,y)$and$f_{2}(\tau ,x,y)$are increasing respect to$x\in \mathbb{R^{+}}$, decreasing respect to$y\in \mathbb{R^{+}}$, $g_{1}$, $g_{2}$are increasing respect toyfor fixed$0\leq \tau \leq 1$and$k_{1}$, $k_{2}$are decreasing with$k_{1}(y(1)),k_{2}(x(1))\neq 0$;

$(H_{3})$ :

∃ $\alpha _{1},\alpha _{2}\in (0,1)$such that

$$\begin{aligned} &f_{1}\bigl(\tau ,\lambda x,\lambda ^{-1} y \bigr)\geq \lambda ^{\alpha _{1}}f_{1}( \tau ,x,y), \qquad f_{2}\bigl(\tau ,\lambda x,\lambda ^{-1} y\bigr)\geq \lambda ^{ \alpha _{2}}f_{2}(\tau ,x,y), \end{aligned}$$

(14)

and$g_{1}$, $g_{2}$, $k_{1}$, $k_{2}$satisfy

$$\begin{aligned} &g_{i}(\tau ,\lambda x)\geq \lambda g_{i}( \tau ,x),\qquad k_{i}\bigl(\lambda ^{-1}x\bigr) \geq \lambda k_{i}(x),\quad i=1,2, \end{aligned}$$

(15)

for$\lambda \in (0,1)$, $0\leq \tau \leq 1$, $x\in \mathbb{R^{+}}$;

$(H_{4})$ :

$g_{i}(\tau ,0)\not \equiv 0$and there exist positive constants$\delta _{11}$, $\delta _{12}$, $\delta _{21}$and$\delta _{22}$such that

$$\begin{aligned} &f_{i}(\tau ,x,y)\geq \delta _{i1}g_{i}(\tau ,x),\\ &f_{i}(\tau ,x,y) \geq \delta _{i2}\geq k_{i}(y),\quad (i=1,2), 0\leq \tau \leq 1,x,y\in \mathbb{R^{+}}. \end{aligned}$$

Then

$(1)$ ∃ $(u_{01},u_{02}),(v_{01},v_{02})\in K\subset E\times E$and$r\in (0,1)$such that

$$ r(v_{01},v_{02})\preceq (u_{01},u_{02}) \prec (v_{01},v_{02}), $$

that is,

$$\begin{aligned}& \begin{aligned} &r(v_{01},v_{02})\leq (u_{01},u_{02})< (v_{01},v_{02}), \\ &r\bigl(D^{\gamma }_{0^{+}}v_{01},D^{ \gamma }_{0^{+}}v_{02} \bigr)\leq \bigl(D^{\eta }_{0^{+}}u_{01},D^{\eta }_{0^{+}}u_{02} \bigr)< \bigl(D^{\gamma }_{0^{+}}v_{01},D^{ \gamma }_{0^{+}}v_{02} \bigr), \end{aligned} \\& \begin{aligned} (u_{01},u_{02})&\leq \biggl( \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,u_{01}( \rho ),D_{0^{+}}^{\eta }u_{01}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,v_{01}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{01}(1)\bigr)\tau ^{ \alpha -1}, \\ &\quad \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,u_{02}(\rho ),D_{0^{+}}^{ \eta }u_{02}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,v_{02}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{02}(1)\bigr)\tau ^{ \alpha -1}\biggr), \end{aligned} \\& \begin{aligned} D^{\eta }_{0^{+}}\bigl((u_{01},u_{02}) \bigr)&\leq \biggl( \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,u_{01}(\rho ),D_{0^{+}}^{\eta }u_{01}(\rho )\bigr)\,d \rho \\ &\quad{}+ \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,v_{01}( \rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{01}(1)\bigr) \tau ^{\alpha -1}, \\ &\quad \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,u_{02}( \rho ),D_{0^{+}}^{\eta }u_{02}(\rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,v_{02}( \rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{02}(1)\bigr) \tau ^{\alpha -1}\biggr), \end{aligned} \\& \begin{aligned} (v_{01},v_{02})&\geq \biggl( \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,v_{01}( \rho ),D_{0^{+}}^{\gamma }v_{01}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,u_{01}(\rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{01}(1)\bigr)\tau ^{ \beta -1}, \\ &\quad \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,v_{02}(\rho ),D_{0^{+}}^{ \gamma }v_{02}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,u_{02}(\rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{02}(1)\bigr)\tau ^{ \beta -1}\biggr), \end{aligned} \\& \begin{aligned} D^{\gamma }_{0^{+}}(v_{01},v_{02})& \geq \biggl( \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,v_{01}(\rho ),D_{0^{+}}^{\gamma }v_{01}(\rho )\bigr)\,d \rho \\ &\quad{}+ \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )g_{2}\bigl(\rho ,u_{01}( \rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{01}(1)\bigr) \tau ^{\beta -1}, \\ &\quad \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,v_{02}( \rho ),D_{0^{+}}^{\gamma }v_{02}(\rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )g_{2}\bigl(\rho ,u_{02}( \rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{02}(1)\bigr) \tau ^{\beta -1}\biggr), \end{aligned} \end{aligned}$$

where$G_{1}(\tau ,\rho )$, $G_{2}(\tau ,\rho )$are defined by (11) and (12), respectively.

$(2)$The problem (2) has a unique positive solution$(u^{*},v^{*})$in$K_{h}$, with$h(\tau )=(h_{1}(\tau ), h_{2}(\tau ))=(\tau ^{\alpha -1},\tau ^{ \beta -1})$, $0\leq \tau \leq 1$.

$(3)$For$(x_{01},x_{02}),(y_{01},y_{02})\in P_{h}\times P_{h}$, there are two iterative sequences$\{(x_{n1},x_{n2})\}$, $\{(y_{n1},y_{n2})\}$for approximating$(x^{*},y^{*})$, that is, $(x_{n1},x_{n2})\rightarrow (x^{*},y^{*})$, $(y_{n1},y_{n2})\rightarrow (x^{*},y^{*})$, where

$$\begin{aligned}& \begin{aligned} \bigl(x_{n1}(\tau ),x_{n2}(\tau )\bigr)&= \biggl( \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl( \rho ,x_{(n-1)1}(\rho ),D_{0^{+}}^{\eta }x_{(n-1)1}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,y_{(n-1)1}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(x_{(n-1)1}(1)\bigr) \tau ^{\alpha -1}, \\ &\quad \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,x_{(n-1)2}(\rho ),D_{0^{+}}^{ \eta }x_{(n-1)2}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,y_{(n-1)2}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(x_{(n-1)2}(1)\bigr) \tau ^{\alpha -1} \biggr), \end{aligned} \\& \begin{aligned} \bigl(y_{n1}(\tau ),y_{n2}(\tau )\bigr)&= \biggl( \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl( \rho ,y_{(n-1)1}(\rho ),D_{0^{+}}^{\gamma }y_{(n-1)1}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,x_{(n-1)1}(\rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(y_{(n-1)1}(1)\bigr)\tau ^{ \beta -1}, \\ &\quad \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,y_{(n-1)2}(\rho ),D_{0^{+}}^{ \gamma }y_{(n-1)2}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,x_{(n-1)2}(\rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(y_{(n-1)2}(1)\bigr)\tau ^{ \beta -1}\biggr), \\ &\quad n=1,2,\ldots . \end{aligned} \end{aligned}$$

Proof

By Lemma 2.6 we have

$$\begin{aligned} G_{1}(\tau ,\rho ),G_{2}(\tau ,\rho ),D_{0^{+}}^{\eta }G_{1}(\tau , \rho ),D_{0^{+}}^{\gamma }G_{1}( \tau ,\rho ),D_{0^{+}}^{\eta }G_{2}( \tau ,\rho ),D_{0^{+}}^{\gamma }G_{2}(\tau ,\rho )\geq 0. \end{aligned}$$

(16)

Regarding (16) and $(H_{1})$ in (13) we get $A_{1},A_{2},B_{1},B_{2}:P\rightarrow P$ and $C_{1},C_{2}:P\times P\rightarrow P\times P$.

Obviously $A_{1}$, $A_{2}$ are increasing and sub-homogeneous, Because $g_{1}$, $g_{2}$ are increasing and sub-homogeneous. $B_{1}$, $B_{2}$ are decreasing (due to this fact, $k_{1}$ and $k_{2}$ are decreasing) and satisfy in conditions $B_{i}(\lambda ^{-1}x)\geq \lambda B_{i}(x)$, $i=1,2$, by (15). For any $(u_{1},v_{1}),(u_{2},v_{2})\in K$ with $(u_{1},v_{1})\preceq (u_{2},v_{2})$, considering that $f_{1}(\tau ,x,y)$ and $f_{2}(\tau ,x,y)$ are increasing in x and decreasing in y, we have

$$\begin{aligned} &C_{1}(v_{1},u_{1})\leq C_{1}(v_{2},u_{1}) \quad \text{for fixed }u_{1} \quad \text{and}\quad C_{1}(v_{1},u_{1}) \geq C_{1}(v_{1},u_{2})\quad \text{for fixed } v_{1}, \\ &C_{2}(u_{1},v_{1})\leq C_{2}(u_{2},v_{1}) \quad \text{for fixed }v_{1} \quad \text{and} \quad C_{2}(u_{1},v_{1})\geq C_{2}(u_{1},v_{2}) \quad \text{for fixed } u_{1}, \end{aligned}$$

also

$$\begin{aligned} &C_{1}\bigl(\tau ,\lambda x,\lambda ^{-1} y\bigr)\geq \lambda ^{\alpha _{1}}C_{1}( \tau ,x,y), \qquad C_{2}\bigl( \tau ,\lambda x,\lambda ^{-1} y\bigr)\geq \lambda ^{ \alpha _{2}}C_{2}( \tau ,x,y). \end{aligned}$$

Set $A=(A_{1},A_{2}):K\rightarrow K$, $B=(B_{1},B_{2}):K\rightarrow K$, $C=(C_{1},C_{2}): K\times K\rightarrow K$. Then A, B, C satisfy Eq. (3) of Theorem 2.2, with replacing the cone K for the cone P.

From Lemma 2.7, we get $K_{h}=P_{h_{1}}\times P_{h_{2}}$, where $h(\tau )=(h_{1}(\tau ),h_{2}(\tau ))=(\tau ^{\alpha -1},\tau ^{ \beta -1})$, also by condition $(i)$ of Theorem 2.2, we need prove $A_{1_{h_{1}}},B_{1_{h_{1}}}\in P_{h_{1}}$, $A_{2_{h_{2}}},B_{2_{h_{2}}}\in P_{h_{2}}$ and $C_{1}(h_{1},h_{1})\in P_{h_{1}}$, $C_{2}(h_{2},h_{2})\in P_{h_{2}}$.

Indeed

$$\begin{aligned}& \begin{aligned} A_{1_{h_{1}}}(\tau )&= \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,h_{2}( \rho )\bigr)\,d\rho = \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,\rho ^{ \alpha -1}\bigr)\,d\rho \\ &\geq \tau ^{\alpha -1} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta }]}{\varGamma (\alpha )}g_{1}( \rho ,0) \,d\rho > 0, \end{aligned} \\& \begin{aligned} A_{1_{h_{1}}}(\tau )&= \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,h_{2}( \rho )\bigr)\,d\rho = \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,\rho ^{ \alpha -1}\bigr)\,d\rho \\ &\leq \tau ^{\alpha -1} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}g_{1}( \rho ,1) \,d \rho . \end{aligned} \end{aligned}$$

Let

$$\begin{aligned} &a_{11} := \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta }]}{\varGamma (\alpha )}g_{1}( \rho ,0) \,d\rho > 0. \\ &a_{12}:= \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}g_{1}( \rho ,1) \,d \rho . \end{aligned}$$

Then $a_{12}\geq a_{11}>0$ and thus

$$\begin{aligned} &a_{11}h(\tau )\leq A_{1_{h_{1}}}(\tau )\leq a_{12}h(\tau ),\quad 0\leq \tau \leq 1. \end{aligned}$$

(17)

Also,

$$\begin{aligned}& \begin{aligned} A_{2_{h_{2}}}(\tau )&= \int _{0}^{1}G_{1}(\tau ,\rho )g_{2}\bigl(\rho ,h_{1}( \rho )\bigr)\,d\rho = \int _{0}^{1}G_{1}(\tau ,\rho )g_{2}\bigl(\rho ,\rho ^{ \beta -1}\bigr)\,d\rho \\ &\geq \tau ^{\beta -1} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi }]}{\varGamma (\beta )}g_{2}( \rho ,0) \,d\rho > 0, \end{aligned} \\& \begin{aligned} A_{2_{h_{2}}}(\tau )&= \int _{0}^{1}G_{1}(\tau ,\rho )g_{2}\bigl(\rho ,h_{1}( \rho )\bigr)\,d\rho = \int _{0}^{1}G_{1}(\tau ,\rho )g_{2}\bigl(\rho ,\rho ^{ \beta -1}\bigr)\,d\rho \\ &\leq \tau ^{\beta -1} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}g_{2}( \rho ,1) \,d \rho . \end{aligned} \end{aligned}$$

Let

$$\begin{aligned} &a_{21} := \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi }]}{\varGamma (\beta )}g_{1}( \rho ,0) \,d\rho > 0, \\ &a_{22}:= \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}g_{1}( \rho ,1) \,d \rho . \end{aligned}$$

Then $a_{22}\geq a_{21}>0$ and thus

$$ \begin{gathered} a_{21}h(\tau )\leq A_{1_{h_{1}}}(\tau )\leq a_{22}h(\tau ),\quad 0\leq \tau \leq 1, \\ B_{1}(u) (\tau )=\frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u(1)\bigr) \tau ^{\alpha -1},\qquad B_{2}(v) (\tau )= \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1} \bigl(v(1)\bigr)\tau ^{\beta -1}, \end{gathered} $$

(18)

therefore

$$\begin{aligned} B_{1}(h_{1}) (\tau )&=\frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}(1) \tau ^{\alpha -1},\qquad B_{2}(h_{2}) (\tau )= \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}(1)\tau ^{\beta -1}. \end{aligned}$$

From $k_{2}(u(1))\not \equiv 0$ and $k_{1}(v(1))\not \equiv 0$ we get $B_{1_{h_{1}}}\in P_{h_{1}}$, $B_{2_{h_{2}}}\in P_{h_{2}}$. We have

$$\begin{aligned}& \begin{aligned} C_{1}(h_{1},h_{1}) (\tau )&= \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,h( \rho ),D_{0^{+}}^{\eta }h(\rho ) \bigr)\,d\rho \\ &\leq \int _{0}^{1} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}f_{1} \biggl( \rho ,\rho ^{\alpha -1}, \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}\tau ^{\alpha -\eta -1}\biggr)\,d \rho \\ &\leq \tau ^{\alpha -1} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}f_{1}( \rho ,1,0)\,d \rho , \end{aligned} \\& \begin{aligned} C_{1}(h_{1},h_{1}) (\tau )&= \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,h( \rho ),D_{0^{+}}^{\eta }h(\rho ) \bigr)\,d\rho \\ &\geq \int _{0}^{1} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta }]}{\varGamma (\alpha )}f_{1} \biggl( \rho ,\rho ^{\alpha -1}, \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}(\tau )^{\alpha -\eta -1} \biggr)\,d \rho \\ &\geq \tau ^{\alpha -1} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta }]}{\varGamma (\alpha )}f_{1} \biggl( \rho ,0,\frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}\biggr)\,d\rho , \end{aligned} \\& \begin{aligned} C_{2}(h_{2},h_{2}) (\tau )&= \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,u( \rho ),D_{0^{+}}^{\gamma }v(\rho ) \bigr)\,d\rho \\ &\leq \int _{0}^{1} \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2} \biggl( \rho ,\rho ^{\beta -1}, \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}\tau ^{\beta -\gamma -1}\biggr)\,d \rho \\ &\leq \tau ^{\beta -1} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2}( \rho ,1,0)\,d \rho , \end{aligned} \\& \begin{aligned} C_{2}(h_{2},h_{2}) (\tau )&= \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,u( \rho ),D_{0^{+}}^{\gamma }v(\rho ) \bigr)\,d\rho \\ &\geq \int _{0}^{1} \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi }]}{\varGamma (\beta )}f_{2} \biggl( \rho ,\rho ^{\beta -1}, \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}(\tau )^{\beta - \gamma -1} \biggr)\,d\rho \\ &\geq \tau ^{\beta -1} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2} \biggl(\rho ,0, \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}\biggr)\,d\rho . \end{aligned} \end{aligned}$$

We can calculate that

$$\begin{aligned}& D^{\eta }_{0^{+}}A_{1}(u) (\tau )= \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,v(\rho )\bigr)\,d\rho ,\\& D^{\gamma }_{0^{+}}A_{2}\bigl(v( \tau )\bigr)= \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )g_{2}\bigl( \rho ,u(\rho )\bigr)\,d\rho, \\& D^{\eta }_{0^{+}}B_{1}(u) (\tau )= \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha -\eta )}k_{2}\bigl(u(1)\bigr) \tau ^{\alpha -\eta -1},\qquad D^{\gamma }_{0^{+}}B_{2}(v) (\tau )= \frac{\varGamma (\beta -\xi )}{\varGamma (\beta -\gamma )}k_{1} \bigl(v(1)\bigr)\tau ^{ \beta -\gamma -1}, \\& D^{\eta }_{0^{+}}C_{1}(v,u) (\tau )= \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,v(\rho ),D_{0^{+}}^{\eta }u( \rho )\bigr)\,d\rho , \\& D^{\gamma }_{0^{+}}C_{2}(u,v) (\tau )= \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,u(\rho ),D_{0^{+}}^{\gamma }v( \rho )\bigr)\,d\rho , \end{aligned}$$

also

$$\begin{aligned}& \begin{aligned} D^{\eta }_{0^{+}}A_{1}(h) (\tau )&= \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,\rho ^{\alpha -1}\bigr)\,d\rho \\ &\geq \int _{0}^{1} \frac{\tau ^{\alpha -\eta -1}(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta -\eta }]}{\varGamma (\alpha -\eta )}g_{1} \bigl( \rho ,\rho ^{\alpha -1}\bigr) \,d\rho \\ &\geq \tau ^{\alpha -\eta -1} \frac{\varGamma (\alpha )}{{\varGamma (\alpha -\eta )}} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta -\eta }]}{\varGamma (\alpha )}g_{1}( \rho ,0) \,d\rho , \end{aligned} \\& \begin{aligned} D^{\eta }_{0^{+}}A_{1}(h) (\tau )&= \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,\rho ^{\alpha -1}\bigr)\,d\rho \\ &\leq \int _{0}^{1} \frac{\tau ^{\alpha -\eta -1}(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha -\eta )}g_{1} \bigl( \rho ,\rho ^{\alpha -1}\bigr) \,d\rho \\ &\leq \tau ^{\alpha -\eta -1} \frac{\varGamma (\alpha )}{{\varGamma (\alpha -\eta )}} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}g_{1}( \rho ,1) \,d\rho . \end{aligned} \end{aligned}$$

Set $a^{\prime }_{11}=\int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta -\eta }]}{\varGamma (\alpha )}g_{1}(\rho ,0)\,d\rho$ and $a^{\prime }_{12}=\int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}g_{1}(\rho ,1)\,d\rho$, we have

$$ a^{\prime }_{11}D^{\eta }_{0^{+}}h\leq D^{\eta }_{0^{+}}A_{1}(h)\leq a^{ \prime }_{12}D^{\eta }_{0^{+}}h $$

and by (17) and (18) we have $a^{\prime }_{11}h\leq A_{1}(h)\leq a^{\prime }_{12}h$. So $\min \{a_{11},a^{\prime }_{11}\}h\preceq A_{1}(h)\preceq \max \{a_{12}, a^{ \prime }_{12}\}h$. Hence $A_{1}(h)\in P_{h}$.

Again we have

$$\begin{aligned}& \begin{aligned} D^{\gamma }_{0^{+}}A_{2}(h) (\tau )&= \int _{0}^{1}D^{\gamma }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,\rho ^{\beta -1}\bigr)\,d\rho \\ &\geq \int _{0}^{1} \frac{\tau ^{\beta -\gamma -1}(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi -\gamma }]}{\varGamma (\beta -\gamma )}g_{1} \bigl( \rho ,\rho ^{\beta -1}\bigr) \,d\rho \\ &\geq \tau ^{\beta -\gamma -1} \frac{\varGamma (\beta )}{{\varGamma (\beta -\gamma )}} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi -\gamma }]}{\varGamma (\beta )}g_{1}( \rho ,0) \,d\rho , \end{aligned} \\& \begin{aligned} D^{\gamma }_{0^{+}}A_{2}(h) (\tau )&= \int _{0}^{1}D^{\gamma }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,\rho ^{\beta -1}\bigr)\,d\rho \\ &\leq \int _{0}^{1} \frac{\tau ^{\beta -\gamma -1}(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta -\gamma )}g_{1} \bigl( \rho ,\rho ^{\beta -1}\bigr) \,d\rho \\ &\leq \tau ^{\beta -\gamma -1} \frac{\varGamma (\beta )}{{\varGamma (\beta -\gamma )}} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}g_{1}( \rho ,1) \,d\rho . \end{aligned} \end{aligned}$$

Similarly we set $a^{\prime }_{21}=\int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi -\eta }]}{\varGamma (\beta )}g_{1}(\rho ,0)\,d\rho$ and $a^{\prime }_{22}=\int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}g_{1}(\rho ,1)\,d\rho$, we have

$$ a^{\prime }_{21}D^{\gamma }_{0^{+}}h\leq D^{\gamma }_{0^{+}}A_{2}(h) \leq a^{\prime }_{22}D^{\gamma }_{0^{+}}h $$

and by (17) we have $a^{\prime }_{21}h\leq A_{2}(h)\leq a^{\prime }_{22}h$. So $\min \{a_{21},a^{\prime }_{21}\}h\preceq A_{2}(h)\preceq \max \{a_{21},a^{ \prime }_{22}\}h$, hence $A_{2}(h)\in P_{h}$.

Furthermore,

$$\begin{aligned} &B_{1}(h_{1})=\frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}(1) \tau ^{\alpha -1}=\frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}(1)h_{1}( \tau ), \\ &\begin{aligned} D^{\eta }_{0^{+}}B_{1}(h_{1})&= \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha -\eta )}k_{2}(1)\tau ^{ \alpha -\eta -1}= \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}k_{2}(1) \tau ^{\alpha -\eta -1} \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}\\ &= \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}k_{2}(1)D^{\eta }_{0^{+}}h_{1}( \tau ), \end{aligned} \end{aligned}$$

therefore

$$\begin{aligned} &B_{2}(h_{2})= \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}(1) \tau ^{\beta -1}=\frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}(1)h_{2}( \tau ), \\ &\begin{aligned} D^{\gamma }_{0^{+}}B_{2}(h_{2})&= \frac{\varGamma (\beta -\xi )}{\varGamma (\beta -\gamma )}k_{1}(1)\tau ^{ \beta -\gamma -1} \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}k_{1}(1) \tau ^{\beta -\gamma -1}\frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}\\ &= \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}k_{1}(1)D^{\gamma }_{0^{+}}h_{2}( \tau ), \end{aligned} \end{aligned}$$

from $k_{2}(u(1))\not \equiv 0$ and $k_{1}(v(1))\not \equiv 0$ we get $B_{1_{h_{1}}}\in P_{h_{1}}$, $B_{2_{h_{2}}}\in P_{h_{2}}$.

$$\begin{aligned}& \begin{aligned} D^{\eta }_{0^{+}}C_{1}(h_{1},h_{1}) (\tau )&= \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,\rho ^{\alpha -1},D_{0^{+}}^{\eta } \rho ^{ \alpha -1}\bigr)\,d\rho \\ &\leq \int _{0}^{1} \frac{\tau ^{\alpha -\eta -1}(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha -\eta )}f_{1} \biggl( \rho ,\rho ^{\alpha -1}, \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}\tau ^{\alpha -\eta -1}\biggr)\,d \rho \\ &\leq \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}\tau ^{\alpha - \eta -1} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}f_{1}( \rho ,1,0)\,d \rho , \end{aligned} \\& \begin{aligned} D^{\eta }_{0^{+}}C_{1}(h_{1},h_{1}) (\tau )&= \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,\rho ^{\alpha -1},D_{0^{+}}^{\eta } \rho ^{ \alpha -1}\bigr)\,d\rho \\ &\geq \int _{0}^{1} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta -\eta }]}{\varGamma (\alpha )}\\ &\quad {}\times f_{1} \biggl( \rho ,\rho ^{\alpha -1}, \frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}\tau ^{\alpha -\eta -1}\biggr)\,d \rho \\ &\geq \tau ^{\alpha -\eta -1} \int _{0}^{1} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta -\eta }]}{\varGamma (\alpha )}f_{1} \biggl( \rho ,0,\frac{\varGamma (\alpha )}{\varGamma (\alpha -\eta )}\biggr)\,d\rho , \end{aligned} \\& \begin{aligned} D^{\gamma }_{0^{+}}C_{2}(h_{2},h_{2}) (\tau )&= \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,\rho ^{\beta -1},D_{0^{+}}^{\gamma } \rho ^{ \beta -1}\bigr)\,d\rho \\ &\leq \int _{0}^{1} \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2} \biggl( \rho ,\rho ^{\beta -1}, \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}\tau ^{\beta -\gamma -1}\biggr)\,d \rho \\ &\leq \tau ^{\beta -1} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2}( \rho ,1,0)\,d \rho , \end{aligned} \\& \begin{aligned} D^{\gamma }_{0^{+}}C_{2}(h_{2},h_{2}) (\tau )&= \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,\rho ^{\beta -1},D_{0^{+}}^{\gamma } \rho ^{ \beta -1}\bigr)\,d\rho \\ &\geq \int _{0}^{1} \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi -\gamma }]}{\varGamma (\beta )}\\ &\quad {}\times f_{2}\biggl( \rho ,\rho ^{\beta -1}, \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}\tau ^{\beta -\gamma -1}\biggr)\,d \rho \\ &\geq \tau ^{\beta -1} \int _{0}^{1} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2} \biggl(\rho ,0, \frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )}\biggr)\,d\rho . \end{aligned} \end{aligned}$$

Set

$$ \begin{aligned} &c_{1}= \int ^{1}_{0} \frac{(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\alpha -\eta }]f_{1}(\rho ,0,\frac{\varGamma (\alpha )}{\varGamma (\alpha -\gamma )})}{\varGamma (\alpha )} \,d\rho , \\ &c_{2}= \int ^{1}_{0}\frac{(1-\rho )^{\alpha -\zeta -1}}{\varGamma (\alpha )}f_{1}( \rho ,1,0)\,d\rho , \end{aligned} $$

(19)

and

$$ \begin{aligned} &c_{3}= \int ^{1}_{0} \frac{(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\beta -\gamma }]f_{2}(\rho ,0,\frac{\varGamma (\beta )}{\varGamma (\beta -\gamma )})}{\varGamma (\beta )} \,d\rho , \\ &c_{4}= \int ^{1}_{0} \frac{(1-\rho )^{\beta -\xi -1}}{\varGamma (\beta )}f_{2}( \rho ,1,0) \,d\rho . \end{aligned} $$

(20)

From $(H_{2})$ and $(H_{4})$, it is clear that

$$ c_{2}\geq c_{1}\geq \delta _{1}a_{11}>0, \qquad c_{4}\geq c_{3}\geq \delta _{1} a_{21}>0. $$

Consequently,

$$ c_{1}h\preceq C_{1}(h,h)\preceq c_{2}h, \qquad c_{3}h\preceq C_{2}(h,h) \preceq c_{4}h. $$

Next, we show the proof the condition $(A_{2})$ of Lemma 2.5. By $(H_{4})$,

$$\begin{aligned}& \begin{aligned} C_{1}(y,x)&= \int ^{1}_{0}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,y(\rho ),D^{\eta }_{0^{+}}x(\rho ) \bigr)\,d\rho \\ &\geq \delta _{11} \int ^{1}_{0}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,y(\rho )\bigr) \,d\rho \\ &=\delta _{11}A_{1}(x), \end{aligned} \\& \begin{aligned} D_{0^{+}}^{\eta }C_{1}(y,x)&= \int ^{1}_{0}D_{0^{+}}^{\eta }G_{1}( \tau , \rho )f_{1}\bigl(\rho ,y(\rho ),D^{\eta }_{0^{+}}x( \rho ) \bigr)\,d\rho \\ &\geq \delta _{11} \int ^{1}_{0}D_{0^{+}}^{\eta }G_{1}( \tau ,\rho )g_{1}\bigl( \rho ,y(\rho )\bigr)\,d\rho \\ &=\delta _{11}D_{0^{+}}^{\eta }A_{1}(x). \end{aligned} \end{aligned}$$

Then $C_{1}(y,x)\succeq \delta _{11}A_{1}(x)$.

$$\begin{aligned}& \begin{aligned} C_{2}(x,y)&= \int ^{1}_{0}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,x(\rho ),D^{\gamma }_{0^{+}}y(\rho ) \bigr)\,d\rho \\ &\geq \delta _{21} \int ^{1}_{0}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,x(\rho )\bigr) \,d\rho \\ &=\delta _{21}A_{2}(y), \end{aligned} \\& \begin{aligned} D_{0^{+}}^{\gamma }C_{2}(x,y)&= \int ^{1}_{0}D_{0^{+}}^{\gamma }G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,x(\rho ),D^{\gamma }_{0^{+}}y( \rho ) \bigr)\,d \rho \\ &\geq \delta _{21} \int ^{1}_{0}D_{0^{+}}^{\gamma }G_{2}( \tau ,\rho )g_{2}\bigl( \rho ,x(\rho )\bigr)\,d\rho \\ &=\delta _{21}D_{0^{+}}^{\gamma }A_{2}(y). \end{aligned} \end{aligned}$$

Then $C_{2}(y,x)\succeq \delta _{21}A_{2}(y)$. From $(H_{4})$ and Lemma 2.6, we have

$$\begin{aligned}& \begin{aligned} C_{1}(y,x)&= \int ^{1}_{0}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,y(\rho ),D^{\eta }_{0^{+}}x(\rho ) \bigr)\,d\rho \\ &\geq \int ^{1}_{0} \frac{\tau ^{\alpha -1}(1-\rho )^{\alpha -\zeta -1}[1-(1-\rho )^{\zeta }]}{\varGamma (\alpha )} f_{1} \bigl(\rho ,y(\rho ),D^{\eta }_{0^{+}}x(\rho )\bigr)\,d\rho \\ &\geq \frac{\tau ^{\alpha -1}}{\varGamma (\alpha )} \biggl( \frac{1}{\alpha -\zeta }-\frac{1}{\alpha } \biggr) \delta _{12} \\ &\geq \frac{\tau ^{\alpha -1}}{\varGamma (\alpha )} \biggl( \frac{1}{\alpha -\zeta }-\frac{1}{\alpha -\eta } \biggr)k_{1}\bigl(y(1)\bigr) \\ &=\frac{1}{\varGamma (\alpha -\zeta )} \biggl(\frac{1}{\alpha -\zeta }- \frac{1}{\alpha -\eta } \biggr)B_{2}y, \end{aligned} \\& \begin{aligned} D_{0^{+}}^{\eta }C_{1}(y,x)&= \int ^{1}_{0}D_{0^{+}}^{\eta }G(\tau , \rho )f_{1}\bigl( \rho ,y(\rho ),D^{\eta }_{0^{+}}x(\rho ) \bigr)\,d\rho \\ &\geq \frac{\tau ^{\alpha -\eta -1}}{\varGamma (\alpha -\eta )} \int ^{1}_{0}(1- \rho )^{\alpha -\zeta -1}\bigl(1-(1- \rho )^{\zeta -\eta }\bigr) f_{1}\bigl(\rho ,y( \rho ),D^{\eta }_{0^{+}}x(\rho )\bigr)\,d\rho \\ &\geq \frac{\tau ^{\alpha -\eta -1}}{\varGamma (\alpha -\eta )} \biggl( \frac{1}{\alpha -\zeta }-\frac{1}{\alpha -\eta } \biggr)k_{1}\bigl(y(1)\bigr) \\ &=\frac{1}{\varGamma (\alpha -\zeta )} \biggl(\frac{1}{\alpha -\zeta }- \frac{1}{\alpha -\eta } \biggr)D_{0^{+}}^{\eta }B_{2}y. \end{aligned} \end{aligned}$$

That means $C_{1}(x,y)\succeq \frac{1}{\varGamma (\alpha -\zeta )}( \frac{1}{\alpha -\zeta }-\frac{1}{\alpha -\eta })B_{2}y$. Let

$$ \delta _{1}=\min \biggl\{ \delta _{12},\frac{1}{\varGamma (\alpha -\zeta )} \biggl( \frac{1}{\alpha -\zeta }-\frac{1}{\alpha -\eta }\biggr)\biggr\} , $$

then

$$\begin{aligned}& C_{1}(x,y)\succeq \delta _{1}(A_{1}x+B_{2}y), \\& \begin{aligned} C_{2}(x,y)&= \int ^{1}_{0}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,x(\rho ),D^{\gamma }_{0^{+}}y(\rho ) \bigr)\,d\rho \\ &\geq \int ^{1}_{0} \frac{\tau ^{\beta -1}(1-\rho )^{\beta -\xi -1}[1-(1-\rho )^{\xi }]}{\varGamma (\beta )} f_{2} \bigl(\rho ,x(\rho ),D^{\gamma }_{0^{+}}y(\rho )\bigr)\,d\rho \\ &\geq \frac{\tau ^{\beta -1}}{\varGamma (\beta )} \biggl( \frac{1}{\beta -\xi }-\frac{1}{\beta } \biggr) \delta _{22} \\ &\geq \frac{\tau ^{\beta -1}}{\varGamma (\beta )} \biggl( \frac{1}{\beta -\xi }-\frac{1}{\beta -\gamma } \biggr)k_{2}\bigl(x(1)\bigr) \\ &=\frac{1}{\varGamma (\beta -\xi )} \biggl(\frac{1}{\beta -\xi }- \frac{1}{\beta -\gamma } \biggr)B_{1}x, \end{aligned} \\& \begin{aligned} D_{0^{+}}^{\gamma }C_{2}(x,y)&= \int ^{1}_{0}D_{0^{+}}^{\gamma }G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,x(\rho ),D^{\gamma }_{0^{+}}y( \rho ) \bigr)\,d \rho \\ &\geq \frac{\tau ^{\beta -\gamma -1}}{\varGamma (\beta -\gamma )} \int ^{1}_{0}(1- \rho )^{\beta -\gamma -1}\bigl(1-(1- \rho )^{\xi -\gamma }\bigr) f_{2}\bigl(\rho ,x( \rho ),D^{\gamma }_{0^{+}}y(\rho )\bigr)\,d\rho \\ &\geq \frac{\tau ^{\beta -\gamma -1}}{\varGamma (\beta -\gamma )} \biggl( \frac{1}{\beta -\xi }-\frac{1}{\beta -\gamma } \biggr)k_{1}\bigl(y(1)\bigr) \\ &=\frac{1}{\varGamma (\beta -\xi )} \biggl(\frac{1}{\beta -\xi }- \frac{1}{\beta -\gamma } \biggr)D_{0^{+}}^{\gamma }B_{1}x. \end{aligned} \end{aligned}$$

That means $C_{2}(y,x)\succeq \frac{1}{\varGamma (\beta -\xi )}( \frac{1}{\beta -\xi }-\frac{1}{\beta -\gamma })B_{1}x$. Let

$$ \delta _{2}=\min \biggl\{ \delta _{22},\frac{1}{\varGamma (\beta -\xi )} \biggl( \frac{1}{\beta -\xi }-\frac{1}{\beta -\gamma }\biggr)\biggr\} . $$

Then we have

$$ C_{2}(y,x)\succeq \delta _{2}(A_{2}x+B_{1}y). $$

We see that the conclusion (2) in Lemma 2.5 means that there exist $u_{01},u_{02},v_{01},v_{02}\in P_{h}$ and $r\in (0,1)$ such that

$(1)$ ∃ $(u_{01},v_{01}),(u_{02},v_{02})\in K\subset E\times E$ and $r\in (0,1)$ with

$$ r(v_{01},v_{02})\preceq (u_{01},u_{02}) \prec (v_{01},v_{02}), $$

that is,

$$\begin{aligned}& r(v_{01},v_{02})\leq (u_{01},u_{02})< (v_{01},v_{02}),\\& r\bigl(D^{\gamma }_{0^{+}}v_{01},D^{ \gamma }_{0^{+}}v_{02} \bigr)\leq \bigl(D^{\eta }_{0^{+}}u_{01},D^{\eta }_{0^{+}}u_{02} \bigr)< \bigl(D^{ \gamma }_{0^{+}}v_{01},D^{\gamma }_{0^{+}}v_{02} \bigr), \\& \begin{aligned} (u_{01},u_{02})&\leq \biggl( \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,u_{01}( \rho ),D_{0^{+}}^{\eta }u_{01}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,v_{01}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{01}(1)\bigr)\tau ^{ \alpha -1}, \\ &\quad \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,u_{02}(\rho ),D_{0^{+}}^{ \eta }u_{02}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,v_{02}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{02}(1)\bigr)\tau ^{ \alpha -1}\biggr), \end{aligned} \\& \begin{aligned} D^{\eta }_{0^{+}}\bigl((u_{01},u_{02}) \bigr)&\leq \biggl( \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,u_{01}(\rho ),D_{0^{+}}^{\eta }u_{01}(\rho )\bigr)\,d \rho \\ &\quad{}+ \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,v_{01}( \rho )\bigr)\,d\rho \\ &\quad{}+ \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{01}(1)\bigr) \tau ^{\alpha -1}, \\ &\quad \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )f_{1}\bigl(\rho ,u_{02}( \rho ),D_{0^{+}}^{\eta }u_{02}(\rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}D^{\eta }_{0^{+}}G_{1}( \tau ,\rho )g_{1}\bigl(\rho ,v_{02}( \rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(u_{02}(1)\bigr) \tau ^{\alpha -1}\biggr), \end{aligned} \\& \begin{aligned} (v_{01},v_{02})&\geq \biggl( \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,v_{01}( \rho ),D_{0^{+}}^{\gamma }v_{01}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,u_{01}(\rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{01}(1)\bigr)\tau ^{ \beta -1}, \\ &\quad \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,v_{02}(\rho ),D_{0^{+}}^{ \gamma }v_{02}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,u_{02}(\rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{02}(1)\bigr)\tau ^{ \beta -1}\biggr), \end{aligned} \\& \begin{aligned} D^{\gamma }_{0^{+}}(v_{01},v_{02})& \geq \biggl( \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,v_{01}(\rho ),D_{0^{+}}^{\gamma }v_{01}(\rho )\bigr)\,d \rho \\ &\quad{}+ \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )g_{2}\bigl(\rho ,u_{01}( \rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{01}(1)\bigr) \tau ^{\beta -1}, \\ &\quad \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )f_{2}\bigl(\rho ,v_{02}( \rho ),D_{0^{+}}^{\gamma }v_{02}(\rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}D^{\gamma }_{0^{+}}G_{2}( \tau ,\rho )g_{2}\bigl(\rho ,u_{02}( \rho )\bigr)\,d\rho + \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(v_{02}(1)\bigr) \tau ^{\beta -1}\biggr), \end{aligned} \end{aligned}$$

where $h(\tau )=(h_{1}(\tau ),h_{2}(\tau ))=(\tau ^{\alpha -1},\tau ^{ \beta -1})$, $0\leq \tau \leq 1$, and $G_{1}(\tau ,\rho )$, $G_{2}(\tau ,\rho )$ are defined by (11) and (12), respectively.

$(2)$ The problem (2) has a unique positive solution $(u^{*},v^{*})$ in $K_{h}$;

$(3)$ For $(x_{01},x_{02}),(y_{01},y_{02})\in P_{h}\times P_{h}$, there are two iterative sequences $\{(x_{n1},x_{n2})\}$ and $\{(y_{n1},y_{n2})\}$ for approximating $(x^{*},y^{*})$, that is, $(x_{n1},x_{n2})\rightarrow (x^{*},y^{*})$ and $(y_{n1},y_{n2})\rightarrow (x^{*},y^{*})$, where

$$\begin{aligned}& \begin{aligned} \bigl(x_{n1}(\tau ),x_{n2}(\tau )\bigr)&= \biggl( \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl( \rho ,x_{(n-1)1}(\rho ),D_{0^{+}}^{\eta }x_{(n-1)1}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,y_{(n-1)1}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(x_{(n-1)1}(1)\bigr) \tau ^{\alpha -1}, \\ &\quad \int _{0}^{1}G_{1}(\tau ,\rho )f_{1}\bigl(\rho ,x_{(n-1)2}(\rho ),D_{0^{+}}^{ \eta }x_{(n-1)2}( \rho )\bigr)\,d\rho \\ &\quad{}+ \int _{0}^{1}G_{1}(\tau ,\rho )g_{1}\bigl(\rho ,y_{(n-1)2}(\rho )\bigr)\,d\rho + \frac{\varGamma (\alpha -\zeta )}{\varGamma (\alpha )}k_{2}\bigl(x_{(n-1)2}(1)\bigr) \tau ^{\alpha -1} \biggr), \end{aligned} \\& \begin{aligned} \bigl(y_{n1}(\tau ),y_{n2}(\tau )\bigr)&= \biggl( \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl( \rho ,y_{(n-1)1}(\rho ),D_{0^{+}}^{\gamma }y_{(n-1)1}( \rho )\bigr)\,d\rho , \\ &\quad \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,x_{(n-1)1}(\rho )\bigr)\,d\rho , \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(y_{(n-1)1}(1)\bigr)\tau ^{ \beta -1}, \\ &\quad \int _{0}^{1}G_{2}(\tau ,\rho )f_{2}\bigl(\rho ,y_{(n-1)2}(\rho ),D_{0^{+}}^{ \gamma }y_{(n-1)2}( \rho )\bigr)\,d\rho , \\ &\quad \int _{0}^{1}G_{2}(\tau ,\rho )g_{2}\bigl(\rho ,x_{(n-1)2}(\rho )\bigr)\,d\rho , \frac{\varGamma (\beta -\xi )}{\varGamma (\beta )}k_{1}\bigl(y_{(n-1)2}(1)\bigr)\tau ^{ \beta -1}\biggr),\\ & \quad n=1,2,\ldots . \end{aligned} \end{aligned}$$

 □

3.1 Example

Let us consider

$$ \textstyle\begin{cases} D^{\frac{7}{2}}_{0^{+}}x(\tau )+\tau ^{2}+(y(\tau )){^{\frac{1}{4}}}+(x( \tau )){^{\frac{1}{4}}}+(D^{\frac{3}{2}}_{0^{+}}x(\tau )+1)^{- \frac{1}{2}}+1=0, \quad \tau \in (0,1), \\ D^{\frac{10}{3}}_{0^{+}}y(\tau )+\tau +\tau ^{3}+\frac{y}{1+y}+ \frac{x}{1+x}+\frac{1}{D^{\frac{5}{3}}_{0^{+}}y(\tau )+1}=0, \quad \tau \in (0,1), \\ x(0)=x'(0)=x''(0)=0, \\ y(0)=y'(0)=y''(0)=0, \\ [D^{\frac{8}{5}}_{0^{+}}x(\tau )]_{\tau =1}=(x(1))^{-\frac{1}{3}}+5, \qquad [D^{ \frac{11}{6}}_{0^{+}}y(\tau )]_{\tau =1}=\frac{1}{1+y(1)^{\frac{1}{2}}}. \end{cases} $$

(21)

Let $g_{1}(\tau ,y)=(x(\tau ))^{\frac{1}{4}}+\tau ^{2}$, $f_{1}(\tau ,x,y)=(x(\tau ))^{\frac{1}{4}}+(y(\tau )+1)^{-\frac{1}{2}}+1$ and $k_{1}(y)=\frac{1}{1+y^{\frac{1}{2}}}$, also $g_{2}(\tau ,x)=\tau ^{3}+\frac{x}{1+x}$, $f_{2}(\tau ,x,y)=\tau +\frac{x}{1+x}+\frac{1}{y+1}$ and $k_{2}(x)=x^{-\frac{1}{3}}+5$.

Obviously, $g_{1},g_{2}:[0,1]\times \mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$, $f_{1},f_{2}:[0,1]\times \mathbb{R^{+}}\times \mathbb{R^{+}} \rightarrow \mathbb{R^{+}}$ and $k_{1},k_{2}:\mathbb{R^{+}}\rightarrow \mathbb{R^{+}}$ are continuous. It is easy to check that $g_{1}(\tau ,y)$, $g_{2}(\tau ,x)$ are increasing in y, x, respectively, and $k_{1}(y)$, $k_{2}(x)$ are decreasing in $y,x\in \mathbb{R^{+}}$ (respectively) and $f_{1}(\tau ,x,y)$, $f_{2}(\tau ,x,y)$ are increasing in x and decreasing in y for fixed $\tau \in (0,1)$. In addition, for any $\lambda \in (0,1)$ we get

$$\begin{aligned} &g_{1}(\tau ,\lambda y)=\tau ^{2}+\bigl(\lambda y(\tau ) \bigr){^{\frac{1}{4}}} \geq \lambda ^{\frac{1}{4}} 2+\lambda ^{\frac{1}{4}} \bigl(y(\tau )\bigr){^{\frac{1}{4}}}=\lambda ^{\frac{1}{4}}g_{1}(\tau ,y), \\ &g_{2}(\tau ,\lambda x)=\tau ^{3}+\frac{\lambda x}{1+\eta x}\geq \lambda \tau +\lambda \frac{x}{1+x}=\lambda g_{2}(\tau ,x), \\ &\begin{aligned} f_{1}\bigl(\tau ,\lambda x,\lambda ^{-1}y\bigr)&=\lambda ^{\frac{1}{4}}\bigl(x(\tau )\bigr)^{\frac{1}{4}}+\lambda ^{\frac{1}{2}} \bigl(y(\tau )+1\bigr)^{-\frac{1}{2}}+1 \geq \lambda ^{\frac{1}{2}}\bigl(\bigl(x( \tau )\bigr)^{\frac{1}{4}}+\lambda ^{\frac{1}{2}}\bigl(y( \tau )+1 \bigr)^{-\frac{1}{2}}+1\bigr) \\ &=\lambda ^{\frac{1}{2}}f(\tau ,x,y), \end{aligned} \\ &f_{2}\bigl(\tau ,\lambda x,\lambda ^{-1}y\bigr)=\tau + \frac{\lambda x}{1+\lambda x}+\frac{1}{\lambda ^{-1}y+1}\geq \tau + \frac{\lambda x}{1+x}+ \frac{\lambda }{y+1} \geq \lambda f_{2}(\tau ,x,y), \\ &k_{1}\bigl(\lambda ^{-1} y\bigr)=\frac{1}{1+(\lambda ^{-1} y)^{\frac{1}{2}}}\geq \frac{\lambda ^{\frac{1}{2}}}{1+y^{\frac{1}{2}}} \geq \frac{\lambda }{1+y^{\frac{1}{2}}}=\lambda k_{1}(y). \\ &k_{2}\bigl(\lambda ^{-1} x\bigr)=\bigl(\lambda ^{-1} x\bigr)^{-\frac{1}{3}}+5\geq \lambda ^{\frac{1}{3}}k_{2}(x). \end{aligned}$$

Besides, $g_{1}(\tau ,0)=2\not \equiv 0$, $g_{2}(\tau ,0)=\tau \not \equiv 0$ Moreover, set $\delta _{1}=\delta _{2}=1$,

$$\begin{aligned} &f_{1}(\tau ,x,y)=\bigl(x(\tau )\bigr)^{\frac{1}{4}}+\bigl(y(\tau )+1\bigr)^{-\frac{1}{2}}+1 \geq \bigl(x(\tau )\bigr)^{\frac{1}{4}}+\tau ^{2}=\delta _{1}g(\tau ,x), \\ &f_{2}(\tau ,x,y)=\tau +\frac{x}{1+x}+\frac{1}{y+1}\geq \tau ^{3}+ \frac{x}{1+x}=\delta _{1}g(\tau ,x). \end{aligned}$$

Then by Theorem 3.1 we deduce that (21) has a unique positive solution $(x^{*},y^{*})$ in $(P_{h_{1}},P_{h_{2}})$, where $(h_{1},h_{2})=(\tau ^{\frac{5}{2}},\tau ^{\frac{7}{3}})$.

4 Conclusion

In this manuscript, we extend the existence and uniqueness of positive solutions from a class of fractional differential equations with nonlinear boundary conditions for a new class of coupled system of fractional derivatives.