Existence of positive solutions for a class of p-Laplacian fractional differential equations with nonlocal boundary conditions

Abstract

This article is devoted to proving the uniqueness of positive solutions for p-Laplacian equations with Caputo and Riemann-Liouville fractional derivative. The uniqueness result and the dependence of the solution on a parameter are established based on the fixed point point theorem of mixed monotone operators. In the end, a numerical simulation is given to verify the main results.

1 Introduction

In this paper, we are concerned with the following p-Laplacian fractional differential equations with nonlocal boundary value problem (BVP)

$$ \left \{ \textstyle\begin{array}{lrl} -^{C}D_{0^{+}}^{\beta}\varphi _{p}\Big(-D_{0^{+}}^{\alpha}u(t)- \varphi _{q}\big(I_{0^{+}}^{\beta }g(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{ \gamma}u(t))\big)\Big) \\ \quad \quad =f(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{\gamma}u(t)),~~t \in (0,1), \\ \varphi _{p}(D_{0^{+}}^{\alpha}u(0))^{(r)}=\varphi _{p}(D_{0^{+}}^{ \alpha}u(1))=0,~~r=1,2,\cdots ,m-1, \\ u(0)=u'(0)=\cdots =u^{(n-2)}(0)=0, \\ D_{0^{+}}^{\alpha -1}u(1)=\sum \limits _{i=1}^{k}a_{i}I_{0^{+}}^{ \tau _{i}}u(\eta )+\lambda \sum \limits _{j=1}^{l}b_{j}u(\xi _{j}), \end{array}\displaystyle \right . $$

(1.1)

where $1< n-1<\alpha <n$, $1< m-1<\beta <m$, $\alpha -\beta >1$, $\gamma <\alpha -1$, $\lambda >0$, $0<\xi _{1}<\xi _{2}<\cdots <\xi _{l}<1$, $0<\eta <1$, $\tau _{i}>0$, $a_{i},~b_{j}>0(i=1,2,\cdots ,k,~j=1,2,\cdots ,l)$, $\Delta \triangleq \Gamma (\alpha )-\sum _{i=1}^{k}a_{i} \frac{\Gamma (\alpha )}{\Gamma (\tau _{i}+\alpha )}\eta ^{\alpha + \tau _{i}-1}-\lambda \sum _{j=1}^{l}b_{j}\xi _{j}^{\alpha -1}>0$, $\varphi _{p}(s)=\lvert{s}\rvert ^{p-2}s$, $p>1$, $\varphi _{q}=\varphi _{p}^{-1}$, $^{C}D_{0^{+}}^{\beta}$, $D_{0^{+}}^{\alpha}$ are the Caputo and Riemann-Liouville fractional derivatives, respectively, and $I_{0^{+}}^{\rho}$ represents Riemann-Liouville fractional integral of order $\rho >0$, $f,g\in C((0,1)\times{\mathbb{R}^{+}_{0}}^{3},\mathbb{R}^{+})$ in which $\mathbb{R}^{+}=[0,+\infty )$, $\mathbb{R}^{+}_{0}=(0,+\infty )$.

Fractional calculus has received extensive attention because it can represent many natural phenomena and help establish accurate models in control phenomena, granular heat flow, signal processing, blood flow phenomena, etc. For details, we refer to [8, 9, 21, 27]. In recent years, people have studied it more and more widely and deeply [2–7, 11, 14–16, 19, 20, 22, 25, 28–32]. In order to solve fractional differential equations, people have created many methods, such as upper and lower solution method, mixed monotone operators method, Leray-Schauder degree, and some fixed point theorems; for more information, see [1, 2, 5, 10, 13, 22–26, 28, 31] and references therein. The p-Laplacian operator can be used to describe turbulence problems in non-Newtonian fluids and multispace media [18, 30], and it has also received substantial attention [7, 15, 20].

In [20], by means of upper and lower solution method, the authors investigated the existence analysis of solutions for the following fractional differential equation with four-point BVP

$$ \left \{ \textstyle\begin{array}{lrl} D_{0^{+}}^{\beta}(\phi _{p}(^{C}D_{0^{+}}^{\alpha}x(t)))=f(t,x(t),^{C}D_{0^{+}}^{ \alpha}x(t)),~~t\in (0,1), \\ ^{C}D_{0^{+}}^{\alpha}x(0)=x'(0)=0, \\ x(1)=r_{1}x(\eta ),~^{C}D_{0^{+}}^{\alpha}x(1)=r_{2}^{C}D_{0^{+}}^{ \alpha}x(\xi ), \end{array}\displaystyle \right . $$

where $1<\alpha$, $\beta \leq 2,~r_{1}$, $r_{2}\geq 0$, $h\in C([0,1]\times \mathbb{R}^{+}\times (-\infty ,0],~\mathbb{R}^{+})$, and $\phi _{p}(s)$ is the p-Laplacian operator.

In [15], the authors studied the following multipoint BVP for p-Laplacian fractional differential equation

$$ \left \{ \textstyle\begin{array}{lrl} D_{0^{+}}^{\beta}(\varphi _{p}(D_{0^{+}}^{\alpha}x(t)))=f(t,x(t)),~~t \in (0,1), \\ x(0)=0,~D_{0^{+}}^{\gamma}x(1)=\sum \limits _{i=1}^{m-2}\xi _{i}D_{0^{+}}^{ \gamma}x(\eta _{i}), \\ D_{0^{+}}^{\alpha}x(0)=0,~\varphi _{p}(D_{0^{+}}^{\alpha}x(1))=\sum \limits _{i=1}^{m-2}\xi _{i}\varphi _{p}(D_{0^{+}}^{\alpha}x(\eta _{i})), \end{array}\displaystyle \right . $$

where $D_{0^{+}}^{\alpha}$, $D_{0^{+}}^{\beta}$, and $D_{0^{+}}^{\gamma}$ are the standard Riemann-Liouville derivatives with $1<\alpha$, $\beta \leq 2$, $0<\gamma \leq 1$, $0<\xi _{i},\eta _{i}$, $\zeta _{i}<1(i=1,2, \cdots ,m-2)$. $f:[0,1]\times \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}$ is continuous, $f(t,x)$ is singular at $x=0$, and $\varphi _{p}(s)=\lvert{s}\rvert ^{p-2}s$, $p>1$. The existence results followed from the fixed point of mixed monotone operators.

Motivated by the works mentioned above, in this paper, we focus on the existence of positive solutions and the dependence of parameters for the BVP (1.1). Using the fixed point theorem of mixed monotone operators, the uniqueness result and the dependence of the solution on a parameter are established. Compared with the existing literature, this paper presents the following new features. First, the significant difference with the existing result [15, 20] lies in that the nonlinear term not only contains the derivative form but also contains the integral form. We obtain the unique positive solution as the nonlinear terms f and g may be singular on the time variable and the space variables. For any given initial value, our established numerical algorithm will generate a series of iterative sequences converging to the unique positive solution. Second, the space variables of the nonlinearity decompose into two quantities with diametrically opposite monotonicity, which is different from the above paper. As far as we know, the results of the fraction order equation by the fixed point of mixed monotone operators usually require a nonlinear term with respect to the spatial variable monotonicity. Third, the BVP (1.1) with multipoint boundary conditions and integral boundary conditions are included in our study; that is to say, the BVP (1.1) is more generalized. Finally, we introduce a parameter σ into problem (1.1) to form problem (3.7) and establish the continuous dependence of the positive solutions on a parameter σ.

The structure of this paper is as follows. In Sect. 2, we show some necessary definitions and lemmas from fractional calculus theory. In Sect. 3, the existence of positive solutions for the BVP (1.1) is proved. In Sect. 4, an example is used to verify the conclusions.

2 Basic definitions and preliminaries

In this section, we will introduce the related definitions of fractional integral, fractional derivative, and some other related results that will be used to prove our main results.

Definition 2.1

[17] The Riemann-Liouville integral of order $\alpha (\alpha >0)$ of a function $f:(a,\infty )\rightarrow \mathbb{R}$ is defined by

$$ I^{\alpha}_{a^{+}}f(x)=\frac{1}{\Gamma (\alpha )}\int _{a}^{x}(x-t)^{ \alpha -1}f(t)dt, $$

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

Definition 2.2

[17] The Riemann-Liouville derivative of order $\alpha (n-1<\alpha <n)$ of a function $f:(a,\infty )\rightarrow \mathbb{R}$ is defined by

$$ D^{\alpha}_{a^{+}}f(x)=\left (\frac{d}{dx}\right )^{n} I^{n-\alpha}_{a^{+}}f(x), $$

where $n=[\alpha ]+1$ and $[\alpha ]$ denotes the integer part of number α, provided that the right-hand side is pointwise defined on $(a,+\infty )$.

Definition 2.3

[17] The Caputo fractional derivative of order α for an at least n-times differential function $f:(a,\infty )\rightarrow \mathbb{R}$, is defined as

$$ ^{C}D_{a^{+}}^{\alpha}f(x)=\frac{1}{\Gamma (n-\alpha )}\int _{a}^{x}(x-s)^{n- \alpha -1}g^{(n)}(s)ds,~n-1< \alpha < n,~n=[\alpha ]+1, $$

where $[\alpha ]$ denotes the integer part of the real number α.

Remark 2.1

If $x,y: \mathbb{R}^{+}_{0}\to \mathbb{R}$ with order $\alpha >0$, then

$$ D_{0^{+}}^{\alpha}(x(t)+y(t))=D_{0^{+}}^{\alpha}x(t)+D_{0^{+}}^{ \alpha}y(t). $$

Lemma 2.1

[17]

(1) If $x\in L(0,1)$, $\nu >\kappa >0$, then

$$ I_{0^{+}}^{\nu }I_{0^{+}}^{\kappa }x(t)=I_{0^{+}}^{\nu +\kappa}x(t), \quad D_{0^{+}}^{\kappa}I_{0^{+}}^{\nu }x(t)=I_{0^{+}}^{\nu -\kappa}x(t), \quad D_{0^{+}}^{\kappa}I_{0^{+}}^{\kappa }x(t)=x(t). $$

(2) If $\nu >0$, $\kappa >0$, then

$$ D_{0^{+}}^{\nu}t^{\kappa -1}= \frac{\Gamma (\kappa )}{\Gamma (\kappa -\nu )}t^{\kappa -\nu -1}. $$

Lemma 2.2

[32] Let $x\in C(0,1)\bigcap L(0,1)$, and $1< m-1<\beta <m$, then the BVP

$$ \left \{ \textstyle\begin{array}{lrl} -^{C}D_{0^{+}}^{\beta}v(t)=x(t),~~t\in (0,1), \\ v'(0)=v''(0)=\cdots =v^{(m-1)}(0)=v(1)=0 \end{array}\displaystyle \right . $$

has a solution of the following form

$$ v(t)=\int ^{1}_{0}H(t,s)x(s)ds, $$

where

$$ H(t,s)=\frac{1}{\Gamma (\beta )} \left \{ \textstyle\begin{array}{lrl} (1-s)^{\beta -1}-(t-s)^{\beta -1},~&0\leq s\leq t\leq 1, \\ (1-s)^{\beta -1},~& 0\leq t\leq s\leq 1. \end{array}\displaystyle \right . $$

(2.1)

Lemma 2.3

If $y\in C(0,1)\bigcap L(0,1)$, and $1< n-1<\alpha <n$, then the BVP

$$ \left \{ \textstyle\begin{array}{lrl} -D_{0^{+}}^{\alpha}u(t)=y(t),~~t\in (0,1), \\ u(0)=u'(0)=\cdots =u^{(n-2)}(0)=0, \\ D_{0^{+}}^{\alpha -1}u(1)=\sum \limits _{i=1}^{k}a_{i}I_{0^{+}}^{ \tau _{i}}u(\eta )+\lambda \sum \limits _{j=1}^{l}b_{j}u(\xi _{j}) \end{array}\displaystyle \right . $$

(2.2)

has a solution of the following form

$$ u(t)=\int ^{1}_{0}G(t,s)y(s)ds $$

in which

$$ G(t,s)=G_{0}(t,s)+\frac{t^{\alpha -1}}{\Delta}\sum \limits _{i=1}^{k}a_{i}G_{i}( \eta ,s)+\frac{\lambda t^{\alpha -1}}{\Delta}\sum \limits _{j=1}^{l}b_{j}G_{0}( \xi _{j},s), $$

(2.3)

where

$$\begin{aligned} &G_{0}(t,s)=\frac{1}{\Gamma (\alpha )}\left \{ \textstyle\begin{array}{lrl} t^{\alpha -1}-(t-s)^{\alpha -1},&0\leq s\leq t\leq 1, \\ t^{\alpha -1},&0\leq t\leq s\leq 1, \end{array}\displaystyle \right . \\ &G_{i}(t,s)=\frac{1}{\Gamma (\alpha +\tau _{i})}\left \{ \textstyle\begin{array}{lrl} t^{\alpha +\tau _{i}-1}-(t-s)^{\alpha +\tau _{i}-1},&0\leq s\leq t \leq 1, \\ t^{\alpha +\tau _{i}-1},&0\leq t\leq s\leq 1, \end{array}\displaystyle \right . i=1,2,\cdots ,k. \end{aligned}$$

Proof

Taking the integral $I_{0^{+}}^{\alpha}$ of order α to equation (2.2), we get

$$ u(t)=-\frac{1}{\Gamma (\alpha )}\int _{0}^{t}(t-s)^{\alpha -1}y(s)ds+C_{1}t^{ \alpha -1}+C_{2}t^{\alpha -2}+\cdots +C_{n}t^{\alpha -n}. $$

From $u(0)=u'(0)=\cdots =u^{(n-2)}(0)=0$, we have $C_{2}=C_{3}=\cdots =C_{n}=0$. Furthermore, by Lemma 2.1, we obtain

$$ D_{0^{+}}^{\alpha -1}u(t)=-\int _{0}^{t}y(s)ds+C_{1}\Gamma (\alpha ), $$

and

$$ I_{0^{+}}^{\tau _{i}}u(t)=-\frac{1}{\Gamma (\alpha +\tau _{i})}\int _{0}^{t}(t-s)^{ \alpha +\tau _{i}-1}y(s)ds+C_{1} \frac{\Gamma (\alpha )}{\Gamma (\alpha +\tau _{i})}t^{\alpha +\tau _{i}-1}. $$

From the boundary conditions $D_{0^{+}}^{\alpha -1}u(1)=\sum \limits _{i=1}^{k}a_{i}I_{0^{+}}^{ \tau _{i}}u(\eta )+\lambda \sum \limits _{j=1}^{l}b_{j}u(\xi _{j})$, we can get

$$ \begin{aligned}C_{1}=&\frac{1}{\Delta}\int _{0}^{1}y(s)ds-\frac{1}{\Delta} \sum \limits _{i=1}^{k}\frac{a_{i}}{\Gamma (\alpha +\tau _{i})}\int _{0}^{ \eta}(\eta -s)^{\alpha +\tau _{i}-1}y(s)ds \\ &-\frac{\lambda}{\Delta \Gamma (\alpha )}\sum \limits _{j=1}^{l}b_{j} \int _{0}^{\xi _{j}}(\xi _{j}-s)^{\alpha -1}y(s)ds. \end{aligned} $$

Hence, we conclude that the equation has solution

$$ u(t)=\int _{0}^{1}G(t,s)y(s)ds, $$

$G(t,s)$ is defined by (2.3). This completes the proof. □

Lemma 2.4

If $y\in C(0,1)\bigcap L(0,1)$ and $G(t,s)$ defined by (2.3), then

$$ D_{0^{+}}^{\gamma}\left (\int _{0}^{1}G(t,s)y(s)ds\right )=\int _{0}^{1}G_{ \gamma}(t,s)y(s)ds, $$

where

$$ G_{\gamma}(t,s)=G_{0\gamma}(t,s)+ \frac{\Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{i=1}^{k}a_{i}G_{i}(\eta ,s)+ \frac{\lambda \Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{j=1}^{l}b_{j}G_{0}(\xi _{j},s), $$

(2.4)

and

$$ G_{0\gamma}(t,s)=\frac{1}{\Gamma (\alpha -\gamma )}\left \{ \textstyle\begin{array}{lrl} t^{\alpha -\gamma -1}-(t-s)^{\alpha -\gamma -1},& 0\leq s\leq t\leq 1, \\ t^{\alpha -\gamma -1},& 0\leq t\leq s\leq 1. \end{array}\displaystyle \right . $$

Proof

By Lemma 2.1, the proof of this lemma is easy, and we omit it. □

Remark 2.2

If $z\in C(0,1)\bigcap L(0,1)$, then

$$ \begin{aligned}I_{0^{+}}^{\beta }z(s)&=\frac{1}{\Gamma (\beta )}\int _{0}^{s} (s-\omega )^{\beta -1}z(\omega )d\omega \\ &\leq \frac{1}{\Gamma (\beta )}\int _{0}^{1} (1-\omega )^{\beta -1}z( \omega )d\omega =\int _{0}^{1} H(\omega ,\omega )z(\omega )d\omega . \end{aligned} $$

As in [34], the nonlinear term $f(t,x)$ admits the form $f(t,x)=\hat{f}(t,x,x)$, for example $f(t,x)=x^{\sigma}+x^{-\tau}$, $0<\sigma <1$, $0<\tau <+\infty$. $\hat{f}(t,x,y)$ is nondecreasing on x and nonincreasing on y for each fixed $t\in [0,1]$. We suppose that f, g admit the form $f(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{\gamma}u(t)) =\hat{f}(t,u(t),I_{0^{+}}^{ \rho}u(t),D_{0^{+}}^{\gamma}u(t),u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{ \gamma}u(t))$,

$g(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{\gamma}u(t)) =\hat{g}(t,u(t),I_{0^{+}}^{ \rho}u(t),D_{0^{+}}^{\gamma}u(t),u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{ \gamma}u(t))$. So, throughout this paper, we always assume that the following assumption holds:

$({H}_{0})$:

For all $t\in (0,1)$, $x_{i}\in \mathbb{R}^{+}_{0}~(i=1,2,3)$, there exist functions $\hat{f},\hat{g}\in C((0,1)\times{\mathbb{R}^{+}_{0}}^{6},\mathbb{R}^{+})$ satisfying $f(t,x_{1},x_{2},x_{3})=\hat{f}(t,x_{1},x_{2},x_{3},x_{1},x_{2},x_{3})$, $g(t,x_{1},x_{2},x_{3})= \hat{g}(t,x_{1},x_{2},x_{3},x_{1},x_{2}, x_{3})$.

Lemma 2.5

Suppose that $(H_{0})$ holds, then the solution of (1.1) is equivalent to the following integral form

$$ \begin{aligned}u(t)=&\int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{F}(u,u)(s) \big)ds +\int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{G}(u,u)(s)\big)ds, \end{aligned} $$

(2.5)

and we have

$$ \begin{aligned}D_{0^{+}}^{\gamma}u(t)=&\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q} \big(\widehat{F}(u,u)(s)\big)ds+\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q} \big(\widehat{G}(u,u)(s)\big)ds,\end{aligned} $$

(2.6)

where $G(t,s)$ and $G_{\gamma}(t,s)$ are given by (2.3) and (2.4), respectively, and $\widehat{F}(u,v)(t)$, $\widehat{F}(u,v)(t)$ will be given in Sect. 3.

Proof

From Lemma 2.2 and Lemma 2.3, let $y(t)=\varphi _{q}(\int _{0}^{1}H(t,s)x(s)ds)+\varphi _{q}( \widehat{G}(u,u)(t))$ and $x(t)=\hat{f}(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{\gamma}u(t),u(t),I_{0^{+}}^{ \rho}u(t),D_{0^{+}}^{\gamma}u(t))$, we have

$$\begin{aligned} u(t)=&\int _{0}^{1}G(t,s)y(s)ds \\ =&\int _{0}^{1}G(t,s)\bigg[\varphi _{q}\Big(\int _{0}^{1}H(s,\omega )x( \omega )d\omega \Big)+\varphi _{q}\big(\widehat{G}(u,u)(s)\big)\bigg]ds \\ =&\int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{F}(u,u)(s)\big)ds + \int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{G}(u,u)(s)\big)ds. \end{aligned}$$

Using Lemma 2.4, we have

$$\begin{aligned} D_{0^{+}}^{\gamma}u(t)=&\int _{0}^{1}G_{\gamma}(t,s)y(s)ds \\ =&\int _{0}^{1}G_{\gamma}(t,s)\bigg[\varphi _{q}\Big(\int _{0}^{1}H(s, \omega )x(\omega )d\omega \Big)+\varphi _{q}\big(\widehat{G}(u,u)(s) \big)\bigg]ds \\ =&\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q}\big(\widehat{F}(u,u)(s) \big)ds +\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q}\big(\widehat{G}(u,u)(s) \big)ds. \end{aligned}$$

 □

Lemma 2.6

The functions $H(t,s)$, $G(t,s)$ given by (2.1) and (2.3), respectively, satisfy the following properties:

(1) $0\leq H(t,s)\leq H(s,s)$, for all $t,s \in [0,1]$;

(2) $\int _{0}^{1}H(t,s)ds\leq \frac{1}{\Gamma (\beta +1)}$, for $t\in [0,1]$;

(3) $G(t,s)\geq 0$, for all $t,s \in [0,1]$;

(4) $t^{\alpha -1}\zeta (s)\leq G(t,s)\leq Mt^{\alpha -1}$, for all $t,s \in [0,1]$;

(5) $\zeta (s)D_{0^{+}}^{\gamma}t^{\alpha -1}\leq G_{\gamma}(t,s)\leq MD_{0^{+}}^{ \gamma}t^{\alpha -1}$, for all $t,s \in [0,1]$, where

$$\begin{aligned} &M=\frac{1}{\Gamma (\alpha )}+\frac{1}{\Delta}\sum \limits _{i=1}^{k} \frac{a_{i}}{\Gamma (\alpha +\tau _{i})}\eta ^{\alpha +\tau _{i}-1}+ \frac{\lambda}{\Delta}\sum \limits _{j=1}^{l} \frac{b_{j}}{\Gamma (\alpha )}\xi _{j}^{\alpha -1}, \\ &\zeta (s)=\frac{1}{\Delta}\sum \limits _{i=1}^{k}a_{i}G_{i}(\eta ,s)+ \frac{\lambda}{\Delta}\sum \limits _{j=1}^{l}b_{j}G_{0}(\xi _{j},s). \end{aligned}$$

Proof

For the proof of (1) and (2) see [32]. Since (3) can be derived from (4), it remains to verify (4) and (5).

On the one hand, we have

$$ G_{0}(t,s)\leq \frac{t^{\alpha -1}}{\Gamma (\alpha )},~G_{i}(\eta ,s) \leq \frac{\eta ^{\alpha +\tau _{i}-1}}{\Gamma (\alpha +\tau _{i})},~G_{0}( \xi _{i},s)\leq \frac{\xi _{j}^{\alpha -1}}{\Gamma (\alpha )}, $$

so

$$\begin{aligned} G(t,s)&\leq \frac{t^{\alpha -1}}{\Gamma (\alpha )}+ \frac{t^{\alpha -1}}{\Delta}\sum \limits ^{k}_{i=1} \frac{a_{i}}{\Gamma (\alpha +\tau _{i})}\eta ^{\alpha +\tau _{i}-1}+ \frac{\lambda t^{\alpha -1}}{\Delta}\sum \limits _{j=1}^{l} \frac{b_{j}}{\Gamma (\alpha )}\xi _{j}^{\alpha -1} =Mt^{\alpha -1}. \end{aligned}$$

On the other hand,

$$\begin{aligned} G(t,s)&\geq t^{\alpha -1}\left [\frac{1}{\Delta}\sum \limits ^{k}_{i=1}a_{i} G_{i}(\eta ,s)+\frac{\lambda}{\Delta}\sum \limits ^{l}_{j=1}b_{j} G_{0}( \xi _{j},s)\right ] = t^{\alpha -1}\zeta (s). \end{aligned}$$

From Lemmas 2.4 and 2.1, we can obtain

$$ \begin{aligned}G_{\gamma}(t,s)\leq &\frac{1}{\Gamma (\alpha -\gamma )}t^{ \alpha -\gamma -1}+ \frac{\Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{i=1}^{k}a_{i}G_{i}(\eta ,s) \\ &+ \frac{\lambda \Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{j=1}^{l}b_{j}G_{0}(\xi _{j},s) \\ \leq &\frac{1}{\Gamma (\alpha -\gamma )}t^{\alpha -\gamma -1}+ \frac{\Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{i=1}^{k}\frac{a_{i}}{\Gamma (\alpha +\tau _{i})}\eta ^{ \alpha +\tau _{i}-1} \\ &+ \frac{\lambda \Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{j=1}^{l}\frac{b_{j}}{\Gamma (\alpha )}\xi _{j}^{ \alpha -1} \\ \leq &\frac{\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}t^{\alpha - \gamma -1}\left (\frac{1}{\Gamma (\alpha )}+\frac{1}{\Delta}\sum \limits _{i=1}^{k}\frac{a_{i}}{\Gamma (\alpha +\tau _{i})}\eta ^{ \alpha +\tau _{i}-1}+\frac{\lambda}{\Delta}\sum \limits _{j=1}^{l} \frac{b_{j}}{\Gamma (\alpha )}\xi _{j}^{\alpha -1}\right ) \\ =&MD_{0^{+}}^{\gamma}t^{\alpha -1}. \end{aligned} $$

By $G_{0\gamma}(t,s)\geq 0$ and Lemma 2.1, we get

$$\begin{aligned} G_{\gamma}(t,s)&\geq \frac{\Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{i=1}^{k}a_{i}G_{i}(\eta ,s)+ \frac{\lambda \Gamma (\alpha )t^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )\Delta} \sum \limits _{j=1}^{l}b_{j}G_{0}(\xi _{j},s) \\ &\geq D_{0^{+}}^{\gamma}t^{\alpha -1}\left (\frac{1}{\Delta}\sum \limits _{i=1}^{k}a_{i}G_{i}(\eta ,s)+\frac{\lambda}{\Delta}\sum \limits _{j=1}^{l}b_{j}G_{0}(\xi _{j},s)\right ) \\ &\geq \zeta (s)D_{0^{+}}^{\gamma}t^{\alpha -1}. \end{aligned}$$

This completes the proof. □

Let $(E,\Vert{\cdot}\Vert )$ be a real Banach space, θ be the zero element of E, and $P\subset E$ be a cone. P is called normal if there exists a constant $N>0$ such that, for any $x,y\in E$, $\theta \leq x\leq y$ implies $\Vert{x}\Vert \leq N\Vert{y}\Vert$, and the smallest N is called the normality constant of P. If P is the normal cone of a Banach space E, and $e\in P$, $e>\theta$, define $P_{e}=\{u\in P:\text{ there exist constants } c,C>0 \text{ such that } ce \leq u\leq Ce\}$, then $P_{e}$ is a component of P.

Definition 2.4

[12] Let $P\subset E$, $T:P\times P\rightarrow P$ is said to be a mixed monotone operator, if

$$ u_{1}\leqslant u_{2},~v_{1}\geqslant v_{2},~\text{ imply }~T(u_{1},v_{1}) \leqslant T(u_{2},v_{2}),~\forall \ \mu _{i},v_{i}\in P(i=1,2). $$

$u\in P$ is called a fixed point of T if $T(u,u)=u$.

Lemma 2.7

[33] Let P be a normal cone in a real Banach space, and then $P_{e}$ is a component of P, $T_{1},~T_{2}:~P_{e}\times P_{e}\rightarrow P_{e}$ are mixed monotone operators. In addition, suppose that $T_{1}$, $T_{2}$ such that

(1) for any $\varepsilon \in (0,1)$, there exists a constant $\psi (\varepsilon )\in (\varepsilon ,1]$, such that

$$ T_{1}(\varepsilon u,\varepsilon ^{-1}v)\geq \psi (\varepsilon )T_{1}(u,v),~ \forall \ u,v\in P_{e}; $$

(2) for any $\varepsilon \in (0,1)$,

$$ T_{2}(\varepsilon u,\varepsilon ^{-1}v)\geq \varepsilon T_{2}(u,v),~ \forall \ u,v\in P_{e}; $$

(3) there exists $d>0$, such that

$$ T_{1}(u,v)\geq dT_{2}(u,v),~\forall \ u,v\in P_{e}. $$

Then, there has a unique positive solution $u^{*}\in P_{e}$, for any $u_{0},v_{0}\in P_{e}$, we have

$$ \begin{aligned}u_{n+1}&=T_{1}(u_{n},v_{n})+T_{2}(u_{n},v_{n}), \\ v_{n+1}&=T_{1}(v_{n},u_{n})+T_{2}(v_{n},u_{n}),\quad n=1,2,\ldots , \end{aligned} $$

and $u_{n}\rightarrow u^{*}$, $v_{n}\rightarrow u^{*}$, as $n\rightarrow \infty$.

Lemma 2.8

[33] If $T_{1}$, $T_{2}$ are as defined in Lemma 2.7, and $\sigma >0$ is a parameter, then the equation $\sigma T_{1}(u,u)+\sigma T_{2}(u,u)=u$ has an unique solution $u^{*}_{\sigma}\in P_{e}$, which satisfies

(1) if there exists a constant $r\in (0,1)$ such that

$$ \psi (\varepsilon )\geq \frac{\varepsilon ^{r}-\varepsilon}{d}+ \varepsilon ^{r},~\forall \ \varepsilon \in (0,1), $$

then for any $\sigma _{0}>0$, $\Vert u^{*}_{\sigma}-u^{*}_{\sigma _{0}}\Vert \rightarrow 0$, as $\sigma \rightarrow \sigma _{0}$;

(2) if

$$ \psi (\varepsilon )> \frac{\varepsilon ^{\frac{1}{2}}-\varepsilon}{d}+ \varepsilon ^{\frac{1}{2}},~\forall \ \varepsilon \in (0,1), $$

then $0<\sigma _{1}<\sigma _{2}\Rightarrow u^{*}_{\sigma _{1}}<u^{*}_{ \sigma _{2}}$;

(3) if there exists $r\in (0,\frac{1}{2})$ such that

$$ \psi (\varepsilon )\geq \frac{\varepsilon ^{r}-\varepsilon}{d}+ \varepsilon ^{r},~\forall \ \varepsilon \in (0,1), $$

then $\lim \limits _{\sigma \rightarrow 0^{+}}\Vert{u^{*}_{\sigma}}\Vert =0$, and $\lim \limits _{\sigma \rightarrow \infty}\Vert{u^{*}_{\sigma}}\Vert =+ \infty$.

3 Main results

Let $E=\{u\in C[0,1],D_{0^{+}}^{\gamma}u\in C[0,1]\}$, endowed with the norm $\Vert{u}\Vert =\max \{\max \limits _{t\in [0,1]}\lvert{u(t)}\rvert , \max \limits _{t\in [0,1]}\lvert{D_{0^{+}}^{\gamma}u(t)}\rvert \}$, observe that $(E,~\Vert \cdot \Vert )$ is a Banach space. Define a normal cone $P\in E$ by $P=\{u\in E:u(t)\geq 0,D_{0^{+}}^{\gamma}u(t)\geq 0\}$. The space E has a partial order $u_{1}\preceq u_{2}\Leftrightarrow u_{1}(t)\leq u_{2}(t)$, $D_{0^{+}}^{ \gamma}u_{1}(t)\leq D_{0^{+}}^{\gamma}u_{2}(t)$, then we define $P_{e}=\{u\in P:\exists \ \mu \in (0,1),~\mu e\preceq u\preceq \mu ^{-1}e \}$, where $e(t)=t^{\alpha -1}$, $t\in [0,1]$.

In this article, the following assumptions will be used

$({H}_{1})$:

For any fixed $t\in (0,1)$, $\hat{f}(t,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})$, $\hat{g}(t,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})$ are increasing in $(x_{1},x_{2},x_{3})$ and decreasing in $(y_{1},y_{2},y_{3})$, $x_{i},y_{i}\in \mathbb{R}^{+}_{0}(i=1,2,3)$.

$({H}_{2})$:

For any $\varepsilon \in (0,1)$, there exists $\psi (\varepsilon )\in (\varepsilon ,1]$, such that for all $t\in (0,1)$, $x_{i},y_{i}\in \mathbb{R}^{+}_{0}~(i=1,2,3)$

$$\begin{aligned} &\hat{f}(t,\varepsilon x_{1},\varepsilon x_{2},\varepsilon x_{3}, \varepsilon ^{-1}y_{1},\varepsilon ^{-1}y_{2},\varepsilon ^{-1}y_{3}) \geq \psi ^{p-1}(\varepsilon )\hat{f}(t,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}), \\ &\hat{g}(t,\varepsilon x_{1},\varepsilon x_{2},\varepsilon x_{3}, \varepsilon ^{-1}y_{1},\varepsilon ^{-1}y_{2},\varepsilon ^{-1}y_{3}) \geq \varepsilon ^{p-1}\hat{g}(t,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}). \end{aligned}$$

$({H}_{3})$:

There exists $d>0$, such that for all $t\in (0,1)$, $x_{i},y_{i}\in \mathbb{R}^{+}_{0}~(i=1,2,3)$

$$ \hat{f}(t,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})\geq d^{p-1}\hat{g}(t,x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}). $$

$({H}_{4})$:

$0<\int _{0}^{1}\zeta (s)\varphi _{q}\Big(I_{0^{+}}^{\beta}\hat{g}(s,s^{ \alpha +\rho -1},s^{\alpha +\rho -1},s^{\alpha +\rho -1},1,1,1)\Big)ds<+ \infty$, and

$$ \begin{aligned}0< \int _{0}^{1}H(\omega ,\omega )\hat{f}(\omega ,1,1,1, \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha + \rho -1})d\omega < +\infty , \\ 0< \int _{0}^{1}H(\omega ,\omega )\hat{g}(\omega ,1,1,1,\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1})d \omega < +\infty . \end{aligned} $$

We introduce the operators $T_{1},~T_{2}:~P\times P\rightarrow E$ by

$$\begin{aligned}& T_{1}(u,v)(t)=\int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{F}(u,v)(s) \big)ds, \end{aligned}$$

(3.1)

$$\begin{aligned}& T_{2}(u,v)(t)=\int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{G}(u,v)(s) \big)ds, \end{aligned}$$

(3.2)

where

$$\begin{aligned}& \widehat{F}(u,v)(s)=\int _{0}^{1}H(s,\omega )\hat{f}\big(\omega ,u( \omega ),I_{0^{+}}^{\rho}u(\omega ),D_{0^{+}}^{\gamma}u(\omega ),v( \omega ),I_{0^{+}}^{\rho}v(\omega ),D_{0_{+}}^{\gamma}v(\omega )\big)d \omega , \end{aligned}$$

(3.3)

$$\begin{aligned}& \widehat{G}(u,v)(s)=I_{0^{+}}^{\beta }\hat{g}\big(s,u(s),I_{0^{+}}^{ \rho}u(s),D_{0^{+}}^{\gamma}u(s),v(s),I_{0^{+}}^{\rho}v(s),D_{0_{+}}^{ \gamma}v(s)\big), \end{aligned}$$

(3.4)

and u is the solution to equation (1.1) if and only if $u=T_{1}(u,u)+T_{2}(u,u)$. By Lemmas 2.4, 2.5 and Remark 2.1, we know that

$$\begin{aligned}& D_{0^{+}}^{\gamma}T_{1}(u,v)(t)=\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q} \big(\widehat{F}(u,v)(s)\big)ds, \end{aligned}$$

(3.5)

$$\begin{aligned}& D_{0^{+}}^{\gamma}T_{2}(u,v)(t)=\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q} \big(\widehat{G}(u,v)(s)\big)ds. \end{aligned}$$

(3.6)

Lemma 3.1

Suppose that $(H_{0})$, $(H_{1})$, $(H_{2})$, and $(H_{4})$ hold, then the operators $T_{1},~T_{2}:~P_{e}\times P_{e}\rightarrow P$ are well-defined.

Proof

From the defined of $P_{e}$, for any $(u,v)\in P_{e}\times P_{e}$, $t\in [0,1]$, there is a constant $\mu \in (0,1)$ such that

$$\begin{aligned}& \mu e(t)\leq u(t)\leq \mu ^{-1}e(t),~\mu D_{0^{+}}^{\gamma}e(t)\leq D_{0^{+}}^{ \gamma}u(t)\leq \mu ^{-1}D_{0^{+}}^{\gamma}e(t),\\& \mu e(t)\leq v(t)\leq \mu ^{-1}e(t),~\mu D_{0^{+}}^{\gamma}e(t)\leq D_{0^{+}}^{ \gamma}v(t)\leq \mu ^{-1}D_{0^{+}}^{\gamma}e(t). \end{aligned}$$

Set $m=\min \left \{\frac{\Gamma (\alpha )}{\Gamma (\alpha +\rho )}, \frac{\Gamma (\alpha -\gamma )}{\Gamma (\alpha )}\right \}<1$, consequently,

$$\begin{aligned}& mt^{\alpha +\rho -1}\leq I_{0^{+}}^{\rho}e(t)= \frac{\Gamma (\alpha )}{\Gamma (\alpha +\rho )}t^{\alpha +\rho -1} \leq m^{-1},\\& mt^{\alpha -\gamma -1}\leq D_{0^{+}}^{\gamma}e(t)= \frac{\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}t^{\alpha -\gamma -1} \leq m^{-1}. \end{aligned}$$

From assumptions $(H_{2})$ and $(H_{4})$, by Lemma 2.6, for any $t\in (0,1)$, we obtain

$$ \begin{aligned}&T_{1}(u,v)(t)=\int _{0}^{1}G(t,s)\varphi _{q}\big( \widehat{F}(u,v)(s)\big)ds \\ \leq & \int _{0}^{1}Me(t)\varphi _{q}\bigg(\int _{0}^{1}H(\omega , \omega )\hat{f}\big(\omega ,\mu ^{-1},\mu ^{-1}m^{-1},\mu ^{-1}m^{-1}, \mu \omega ^{\alpha -1},\mu m\omega ^{\alpha +\rho -1},\\ &\quad \mu m \omega ^{ \alpha -\gamma -1})\big)d\omega \bigg)ds \\ \leq &\int _{0}^{1}Me(t)\varphi _{q}\bigg(\int _{0}^{1}H(\omega , \omega ) \hat{f}\big(\omega ,\mu ^{-1}m^{-1},\mu ^{-1}m^{-1},\mu ^{-1}m^{-1}, \mu m\omega ^{\alpha +\rho -1},\mu m\omega ^{\alpha +\rho -1},\\ &\quad \mu m \omega ^{\alpha +\rho -1})\big)d\omega \bigg)ds \\ \leq &\frac{Me(t)}{\psi (\mu m)}\varphi _{q}\left (\int _{0}^{1}H( \omega ,\omega )\hat{f}\left (\omega ,1,1,1,\omega ^{\alpha +\rho -1}, \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1}\right )d\omega \right )< +\infty , \end{aligned} $$

and

$$\begin{aligned} &D_{0^{+}}^{\gamma}T_{1}(u,v)(t)=\int _{0}^{1}G_{\gamma}(t,s) \varphi _{q}\big(\widehat{F}(u,v)(s)\big)ds \\ \leq & \int _{0}^{1}MD_{0^{+}}^{\gamma}(e(t))\varphi _{q}\bigg(\int _{0}^{1}H( \omega ,\omega )\hat{f}\big(\omega ,\mu ^{-1},\mu ^{-1}m^{-1},\mu ^{-1}m^{-1}, \mu \omega ^{\alpha -1},\mu m\omega ^{\alpha +\rho -1},\\ &\quad \mu m \omega ^{ \alpha -\gamma -1})\big)d\omega \bigg)ds \\ \leq &\int _{0}^{1}MD_{0^{+}}^{\gamma}(e(t))\varphi _{q}\bigg(\int _{0}^{1}H( \omega ,\omega ) \hat{f}\big(\omega ,\mu ^{-1}m^{-1},\mu ^{-1}m^{-1}, \mu ^{-1}m^{-1},\mu m\omega ^{\alpha +\rho -1},\\ &\quad \mu m\omega ^{\alpha + \rho -1},\mu m\omega ^{\alpha +\rho -1})\big)d\omega \bigg)ds \\ \leq &\frac{MD_{0^{+}}^{\gamma}(e(t))}{\psi (\mu m)}\varphi _{q} \left (\int _{0}^{1}H(\omega ,\omega )\hat{f}\left (\omega ,1,1,1, \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha + \rho -1}\right )d\omega \right )< +\infty . \end{aligned}$$

Moreover, noticing Remark 2.2, similarly, we can get

$$ \begin{aligned}&T_{2}(u,v)(t)\leq \frac{Me(t)}{\mu m}\varphi _{q}\left ( \int _{0}^{1}H(\omega ,\omega )\hat{g}\left (\omega ,1,1,1,\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1} \right )d\omega \right )< +\infty , \\ &D_{0^{+}}^{\gamma}T_{2}(u,v)(t)\leq \frac{MD_{0^{+}}^{\gamma}(e(t))}{\mu m}\varphi _{q}\left (\int _{0}^{1}H( \omega ,\omega )\hat{g}\left (\omega ,1,1,1,\omega ^{\alpha +\rho -1}, \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1}\right )d\omega \right )\\ &\hphantom{D_{0^{+}}^{\gamma}T_{2}(u,v)(t)} < +\infty . \end{aligned} $$

Hence, $T_{1},T_{2}:P_{e}\times P_{e}\rightarrow P$ are well-defined. This completes the proof. □

Theorem 3.1

Assume that the hypotheses $(H_{0})$, $(H_{1})$, $(H_{2})$, $(H_{3})$, and $(H_{4})$ hold, then BVP (1.1) has a unique positive solution $u^{*}\in P_{e}$, and there exists a constant $\mu ^{*}\in (0,1)$ such that

$$\begin{aligned} \mu ^{*}t^{\alpha -1}&\leq u^{*}(t)\leq (\mu ^{*})^{-1}t^{\alpha -1}, \\ \frac{\mu ^{*}\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}t^{\alpha - \gamma -1}&\leq D_{0^{+}}^{\gamma}u^{*}(t)\leq \frac{(\mu ^{*})^{-1}\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}t^{ \alpha -\gamma -1}. \end{aligned}$$

Furthermore, for any $u_{0},v_{0}\in P_{e}$, we have sequences

$$\begin{aligned}& u_{n+1}(t)=T_{1}(u_{n},v_{n})(t)+T_{2}(u_{n},v_{n})(t),\quad n=0,1,2, \ldots \\& v_{n+1}(t)=T_{1}(v_{n},u_{n})(t)+T_{2}(v_{n},u_{n})(t),\quad n=0,1,2, \ldots \end{aligned}$$

when $n\rightarrow +\infty$,

$$\begin{aligned}& u_{n}(t)\rightarrow u^{*}(t),~D_{0^{+}}^{\gamma}u_{n}(t)\rightarrow D_{0^{+}}^{ \gamma}u^{*}(t),\\& v_{n}(t)\rightarrow u^{*}(t),~D_{0^{+}}^{\gamma}v_{n}(t)\rightarrow D_{0^{+}}^{ \gamma}u^{*}(t). \end{aligned}$$

Proof

First, we show that for any $u,~v\in P_{e}$, $t\in (0,1)$, $T_{1},~T_{2}:P_{e}\times P_{e}\rightarrow P_{e}$. Take $0< Q<1$ such that

$$ \begin{aligned}Q< \min \bigg\{ &\psi (\mu m)\int _{0}^{1}\zeta (s)\varphi _{q} \left (\int _{0}^{1}H(s,\omega )\hat{f}\left (\omega ,\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1},1,1,1 \right )d\omega \right )ds, \\ &\mu m \int _{0}^{1}\zeta (s)\varphi _{q}\Big(I_{0^{+}}^{\beta} \hat{g}(s,s^{\alpha +\rho -1},s^{\alpha +\rho -1},s^{\alpha +\rho -1},1,1,1) \Big)ds, \\ &\left (\frac{M}{\psi (\mu m)}\varphi _{q}\left (\int _{0}^{1}H( \omega ,\omega )\hat{f}\left (\omega ,1,1,1,\omega ^{\alpha +\rho -1}, \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1}\right )d\omega \right )\right )^{-1}, \\ &\left (\frac{M}{\mu m}\varphi _{q}\left (\int _{0}^{1}H(\omega , \omega )\hat{g}\left (\omega ,1,1,1,\omega ^{\alpha +\rho -1},\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1}\right )d\omega \right ) \right )^{-1}\bigg\} . \end{aligned} $$

From assumption, we know that

$$\begin{aligned} \hat{f}(\omega ,&u(\omega ),I_{0^{+}}^{\rho}u(\omega ),D_{0^{+}}^{ \gamma}u(\omega ),v(\omega ),I_{0^{+}}^{\rho}v(\omega ),D_{0^{+}}^{ \gamma}v(\omega )) \\ &\geq \hat{f}(\omega ,\mu \omega ^{\alpha -1},\mu m\omega ^{\alpha + \rho -1},\mu m\omega ^{\alpha -\gamma -1},\mu ^{-1},\mu ^{-1}m^{-1}, \mu ^{-1}m^{-1}) \\ &\geq \hat{f}(\omega ,\mu m\omega ^{\alpha +\rho -1},\mu m\omega ^{ \alpha +\rho -1},\mu m\omega ^{\alpha +\rho -1},\mu ^{-1} m^{-1},\mu ^{-1} m^{-1},\mu ^{-1} m^{-1}) \\ &\geq \psi ^{p-1}(\mu m)\hat{f}(\omega ,\omega ^{\alpha +\rho -1}, \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1},1,1,1), \end{aligned}$$

which imply

$$\begin{aligned}& \begin{aligned}&T_{1}(u,v)(t)=\int _{0}^{1}G(t,s)\varphi _{q}\big( \widehat{F}(u,v)(s)\big)ds \\ \geq & t^{\alpha -1}\psi (\mu m)\int _{0}^{1}\zeta (s)\varphi _{q} \left (\int _{0}^{1}H(s,\omega )\hat{f}\left (\omega ,\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1},1,1,1 \right )d\omega \right )ds \\ \geq &Qt^{\alpha -1}, \end{aligned}\\& \begin{aligned}&D_{0^{+}}^{\gamma}T_{1}(u,v)(t)=\int _{0}^{1}G_{\gamma} \varphi _{q}\big(\widehat{F}(u,v)(s)\big)ds \\ \geq & D_{0^{+}}^{\gamma}t^{\alpha -1}\psi (\mu m)\int _{0}^{1}\zeta (s) \varphi _{q}\left (\int _{0}^{1}H(s,\omega )\hat{f}\left (\omega , \omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha + \rho -1},1,1,1\right )d\omega \right )ds \\ \geq & QD_{0^{+}}^{\gamma}t^{\alpha -1}. \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \hat{g}(\omega ,u(\omega ),&I_{0^{+}}^{\rho}u(\omega ),D_{0^{+}}^{ \gamma}u(\omega ),v(\omega ),I_{0^{+}}^{\rho}v(\omega ),D_{0^{+}}^{ \gamma}v(\omega )) \\ &\geq (\mu m)^{p-1}\hat{g}(\omega ,\omega ^{\alpha +\rho -1},\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},1,1,1), \end{aligned}$$

we can get

$$ \begin{aligned}&T_{2}(u,v)(t)=\int _{0}^{1}G(t,s)\varphi _{q}\big( \widehat{G}(u,v)(s)\big)ds \\ \geq & t^{\alpha -1}\mu m\int _{0}^{1}\zeta (s)\varphi _{q}\left ( \frac{1}{\Gamma (\beta )}\int _{0}^{s}(s-\omega )^{\beta -1} \hat{g} \left (\omega ,\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1}, \omega ^{\alpha +\rho -1},1,1,1\right )d\omega \right )ds \\ \geq & Qt^{\alpha -1}, \end{aligned} $$

and $D_{0^{+}}^{\gamma}T_{2}(u,v)(t)\geq QD_{0^{+}}^{\gamma}t^{\alpha -1}$. On the other hand, in consideration of Lemma 3.1, we have

$$\begin{aligned}& T_{1}(u,v)(t)\leq Q^{-1}e(t),~D_{0^{+}}^{\gamma}T_{1}(u,v)(t)\leq Q^{-1}D_{0^{+}}^{ \gamma}e(t),\\& T_{2}(u,v)(t)\leq Q^{-1}e(t),~D_{0^{+}}^{\gamma}T_{2}(u,v)(t)\leq Q^{-1}D_{0^{+}}^{ \gamma}e(t). \end{aligned}$$

Consequently, $T_{1},~T_{2}:P_{e}\times P_{e}\rightarrow P_{e}$.

Second, we will show that $T_{1}$, $T_{2}$ are mixed monotone operators. By assumption $(H_{1})$, for any $(u_{1},v_{1}),~(u_{2},v_{2})\in P_{e}\times P_{e},~u_{1}\preceq u_{2},~v_{2} \preceq v_{1}$, we have

$$\begin{aligned} T_{1}(u_{1},v_{1})(t)&=\int _{0}^{1}G(t,s)\varphi _{q}\big( \widehat{F}(u_{1},v_{1})(s)\big)ds \\ &\leq \int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{F}(u_{2},v_{2})(s) \big)ds \\ &=T_{1}(u_{2},v_{2})(t), \end{aligned}$$

and similarly, we can easily prove

$$ D_{0^{+}}^{\gamma}T_{1}(u_{1},v_{1})(t)\leq D_{0^{+}}^{\gamma}T_{1}(u_{2},v_{2})(t), $$

and

$$ T_{2}(u_{1},v_{1})(t)\leq T_{2}(u_{2},v_{2})(t),\quad D_{0^{+}}^{ \gamma}T_{2}(u_{1},v_{1})(t)\leq D_{0^{+}}^{\gamma}T_{2}(u_{2},v_{2})(t), $$

which imply that $T_{1}(u_{1},v_{1})\preceq T_{1}(u_{2},v_{2})$ and $T_{2}(u_{1},v_{1})\preceq T_{2}(u_{2},v_{2})$ for $u_{1}\preceq u_{2}$, $v_{2}\preceq v_{1}$.

Third, we will show that (1) and (2) of Lemma 2.7 hold. From assumption $(H_{2})$, for any $\varepsilon \in (0,1)$, there exists $\psi (\varepsilon )\in (\varepsilon ,1]$, for any $(u,v)\in P_{e}\times P_{e}$ such that

$$\begin{aligned}& \begin{aligned}T_{1}(\varepsilon u,\varepsilon ^{-1}v)(t)=&\int _{0}^{1}G(t,s) \varphi _{q}\big(\widehat{F}(\varepsilon u,\varepsilon ^{-1}v)(s) \big)ds \\ \geq & \psi (\varepsilon )\int _{0}^{1}G(t,s)\varphi _{q}\big( \widehat{F}(u,v)(s)\big)ds \\ =&\psi (\varepsilon )T_{1}(u,v)(t), \end{aligned}\\& \begin{aligned}D_{0^{+}}^{\gamma}T_{1}(\varepsilon u,\varepsilon ^{-1}v)(t)=& \int _{0}^{1}G_{\gamma}(t,s)\varphi _{q}\big(\widehat{F}(\varepsilon u, \varepsilon ^{-1}v)(s)\big)ds \\ \geq &\psi (\varepsilon )\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q} \big(\widehat{F}(u,v)(s)\big)ds \\ =&\psi (\varepsilon )D_{0^{+}}^{\gamma}T_{1}(u,v)(t). \end{aligned} \end{aligned}$$

Similarly, we have

$$ T_{2}(\varepsilon u,\varepsilon ^{-1} v)(t)\geq \varepsilon T_{2}(u,v)(t),~D_{0^{+}}^{ \gamma}T_{2}(\varepsilon u,\varepsilon ^{-1}v)(t)\geq \varepsilon D_{0^{+}}^{ \gamma}T_{2}(u,v)(t). $$

Hence, $T_{1}(\varepsilon u,\varepsilon ^{-1}v)\succeq \psi (\varepsilon )T_{1}(u,v)$, $T_{2}( \varepsilon u,\varepsilon ^{-1}v)\succeq \varepsilon T_{2}(u,v)$.

In addition, from assumption $(H_{3})$, we have

$$\begin{aligned}& \begin{aligned}T_{1}(u,v)(t)&=\int _{0}^{1}G(t,s)\varphi _{q}\big( \widehat{F}(u,v)(s)\big)ds \\ &\geq d\int _{0}^{1}G(t,s)\varphi _{q}\big(\widehat{G}(u,v)(s)\big)ds =d T_{2}(u,v)(t), \end{aligned}\\& \begin{aligned}D_{0^{+}}^{\gamma}T_{1}(u,v)(t)&=\int _{0}^{1}G_{\gamma}(t,s) \varphi _{q}\big(\widehat{F}(u,v)(s)\big)ds \\ &\geq d\int _{0}^{1}G_{\gamma}(t,s)\varphi _{q}\big(\widehat{G}(u,v)(s) \big)ds =d D_{0^{+}}^{\gamma}T_{2}(u,v)(t), \end{aligned} \end{aligned}$$

which imply that $T_{1}(u,v)\succeq dT_{2}(u,v)$.

Therefore, using Lemma 2.7, we conclude that $T_{1}$, $T_{2}$ has a unique fixed point $u^{*}\in P_{e}$ with sequences $u_{n}\rightarrow u^{*}$, $v_{n}\rightarrow u^{*}$, as $n\rightarrow +\infty$. This completes the proof. □

Next, we consider BVP (1.1) with a parameter

$$ \left \{ \textstyle\begin{array}{lrl} -^{C}D_{0^{+}}^{\beta}\Big(\varphi _{p}(-D_{0^{+}}^{\alpha}u(t)- \varphi _{q}\big(I_{0^{+}}^{\beta }g(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{ \gamma}u(t))\big)\Big) \\ \quad \quad =\sigma f(t,u(t),I_{0^{+}}^{\rho}u(t),D_{0^{+}}^{\gamma}u(t)),~~t \in (0,1), \\ \varphi _{p}(D_{0^{+}}^{\alpha}u(0))^{(r)}=\varphi _{p}(D_{0^{+}}^{ \alpha}u(1))=0,~~r=1,2,\cdots ,m-1, \\ u(0)=u'(0)=\cdots =u^{(n-2)}(0)=0, \\ D_{0^{+}}^{\alpha -1}u(1)=\sum \limits _{i=1}^{k}a_{i}I_{0^{+}}^{ \tau _{i}}u(\eta )+\lambda \sum \limits _{j=1}^{l}b_{j}u(\xi _{j}). \end{array}\displaystyle \right . $$

(3.7)

Theorem 3.2

Assume that $(H_{0})$, $(H_{1})$, $(H_{2})$, $(H_{3})$, and $(H_{4})$ hold, then for any $\sigma >0$, BVP (3.7) has a unique positive solution $u^{*}_{\sigma}$ and there exists a constant $\mu ^{*}_{\sigma}\in (0,1)$ such that

$$\begin{aligned} \mu ^{*}_{\sigma}t^{\alpha -1}&\leq u^{*}_{\sigma}(t)\leq (\mu ^{*}_{ \sigma})^{-1}t^{\alpha -1}, \\ \frac{\mu ^{*}_{\sigma}\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}t^{ \alpha -\gamma -1}&\leq D_{0^{+}}^{\gamma}u^{*}_{\sigma}(t)\leq \frac{(\mu ^{*}_{\sigma})^{-1}\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}t^{ \alpha -\gamma -1}. \end{aligned}$$

Furthermore, we have the conclusions:

(1) If there exists a constant $r\in (0,1)$ such that

$$ \psi (\varepsilon )\geq \frac{\varepsilon ^{r}-\varepsilon}{d}+ \varepsilon ^{r},~\forall \varepsilon \in (0,1), $$

then for any $\sigma _{0}>0$, $\Vert u^{*}_{\sigma}-u^{*}_{\sigma _{0}}\Vert \rightarrow 0$, as $\sigma \rightarrow \sigma _{0}$.

(2) If

$$ \psi (\varepsilon )> \frac{\varepsilon ^{\frac{1}{2}}-\varepsilon}{d}+ \varepsilon ^{\frac{1}{2}},~\forall \varepsilon \in (0,1), $$

then $0<\sigma _{1}<\sigma _{2}\Rightarrow u^{*}_{\sigma _{1}}<u^{*}_{ \sigma _{2}}$.

(3) If there exists $r\in (0,\frac{1}{2})$ such that

$$ \psi (\varepsilon )\geq \frac{\varepsilon ^{r}-\varepsilon}{d}+ \varepsilon ^{r},~\forall \varepsilon \in (0,1), $$

then $\lim \limits _{\sigma \rightarrow 0^{+}}\Vert{u^{*}_{\sigma}}\Vert =0$, and $\lim \limits _{\sigma \rightarrow \infty}\Vert{u^{*}_{\sigma}}\Vert =+ \infty$.

Proof

Define two new operators $T_{1}^{\sigma}=\sigma ^{q-1}T_{1}$, $T_{2}^{\sigma}=\sigma ^{q-1}T_{2}$. From Theorem 3.1, we obtain that BVP (3.7) has a unique solution $u^{*}_{\sigma}\in P_{e}$. Obviously, $u^{*}_{\sigma}$ is the unique solution to equation $\sigma ^{q-1}T_{1}(u,u)+\sigma ^{q-1}T_{2}(u,u)=T_{1}^{\sigma}(u,u)+T_{2}^{ \sigma}(u,u)=u$. As Lemma 2.8, $u^{*}_{\sigma}$ satisfies conditions (1)–(3) given in the Theorem 3.2, this completes the proof. □

4 Applications

In this section, we construct an example to demonstrate the application of our main results.

Example 4.1

Consider the equation

$$ \left \{ \textstyle\begin{array}{lrl} -^{C}D_{0^{+}}^{\frac{7}{2}}\varphi _{2}\Big(-D_{0^{+}}^{\frac{5}{2}}u(t)- \varphi _{2}\big(I_{0^{+}}^{\frac{7}{2}}g(t,u(t),I_{0^{+}}^{ \frac{1}{2}}u(t),D_{0^{+}}^{\frac{1}{2}}u(t))\big)\Big) \\ \quad \quad =f(t,u(t),I_{0^{+}}^{\frac{1}{2}}u(t),D_{0^{+}}^{ \frac{1}{2}}u(t)),~~t\in (0,1), \\ (\varphi _{2}(D_{0^{+}}^{\frac{5}{2}}u(0)))'=(\varphi _{2}(D_{0^{+}}^{ \frac{5}{2}}u(0)))''=(\varphi _{2}(D_{0^{+}}^{\frac{5}{2}}u(0)))^{(3)} \\ \quad \quad =\varphi _{2}(D_{0^{+}}^{\frac{5}{2}}u(1))=0,\quad u(0)=u'(0)=0, \\ D_{0^{+}}^{\frac{3}{2}}u(1)=\frac{1}{4}I_{0^{+}}^{\frac{1}{2}}u( \frac{1}{2})+3(\frac{1}{6}u(\frac{1}{3})+\frac{2}{3}u(\frac{2}{3})), \end{array}\displaystyle \right . $$

(4.1)

where $\alpha =\frac{5}{2}$, $\beta =\frac{7}{2}$, $\rho =\frac{1}{2}$, $\gamma = \frac{1}{2}$, $p=q=2$, $k=1$, $l=2$, $\tau _{1}=\frac{1}{2}$, $\eta = \frac{1}{2}$, $a_{1}=\frac{1}{4}$, $\lambda =3$, $b_{1}=\frac{1}{6}$, $b_{2}= \frac{2}{3}$, $\xi _{1}=\frac{1}{3}$, $\xi _{2}=\frac{2}{3}$, and $f(t,u,v,\tau )=4u^{\frac{1}{6}}v^{\frac{1}{4}}\tau ^{-\frac{1}{4}}+6 \tau ^{\frac{1}{8}}u^{-\frac{1}{4}}v^{-\frac{1}{8}}$, $g(t,u,v,\tau )=tu^{\frac{1}{6}}v^{\frac{1}{4}}\tau ^{-\frac{1}{4}}+ \tau ^{\frac{1}{12}}u^{-\frac{1}{4}}v^{-\frac{1}{8}}$. By calculating, we can obtain $\Delta =0.102911>0$.

Putting

$$\begin{aligned}& \hat{f}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3})=4u_{1}^{\frac{1}{6}}u_{2}^{ \frac{1}{4}}v_{3}^{-\frac{1}{4}}+6u_{3}^{\frac{1}{8}}v_{1}^{- \frac{1}{4}}v_{2}^{-\frac{1}{8}},\\& \hat{g}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3})=tu_{1}^{\frac{1}{6}}u_{2}^{ \frac{1}{4}}v_{3}^{-\frac{1}{4}}+u_{3}^{\frac{1}{12}}v_{1}^{- \frac{1}{4}}v_{2}^{-\frac{1}{8}}. \end{aligned}$$

It is easy to check

(1) For all $t\in (0,1)$, $u_{i},v_{i}\in \mathbb{R}^{+}_{0}(i=1,2,3)$, $\hat{f}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3})$, $\hat{g}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3})$ are increasing in $(u_{1},u_{2},u_{3})$ and decreasing in $(v_{1},v_{2},v_{3})$.

(2) Let $\psi (\varepsilon )=\varepsilon ^{\frac{2}{3}}$, for any $\varepsilon \in (0,1)$, $t\in (0,1)$, $u_{i},v_{i}\in \mathbb{R}^{+}_{0}(i=1,2,3)$

$$\begin{aligned} \hat{f}(t,\varepsilon u_{1},\varepsilon u_{2},\varepsilon u_{3}, \varepsilon ^{-1}v_{1},\varepsilon ^{-1}v_{2},\varepsilon ^{-1}v_{3})&= \varepsilon ^{\frac{2}{3}}u_{1}^{\frac{1}{6}}u_{2}^{\frac{1}{4}}v_{3}^{- \frac{1}{4}}+(6\varepsilon ^{\frac{1}{8}}u_{3}^{\frac{1}{2}}+1) \varepsilon ^{\frac{3}{8}}v_{1}^{-\frac{1}{4}}v_{2}^{-\frac{1}{8}} \\ &\geq \varepsilon ^{\frac{2}{3}}\hat{f}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}), \\ \hat{g}(t,\varepsilon u_{1},\varepsilon u_{2},\varepsilon u_{3}, \varepsilon ^{-1}v_{1},\varepsilon ^{-1}v_{2},\varepsilon ^{-1}v_{3})&= \varepsilon ^{\frac{2}{3}}u_{1}^{\frac{1}{6}}u_{2}^{\frac{1}{4}}v_{3}^{- \frac{1}{4}} +\varepsilon ^{\frac{11}{24}}u_{3}^{\frac{1}{12}}v_{1}^{- \frac{1}{4}}v_{2}^{-\frac{1}{8}} \\ &\geq \varepsilon ^{2}\hat{g}(t,u_{1},v_{1},u_{2},u_{3},v_{1},v_{2},v_{3}) \\ &\geq \varepsilon ^{\frac{5}{2}}\hat{g}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}). \end{aligned}$$

(3) Let $d=4$, for all $t\in (0,1)$, $u_{i},v_{i}\in \mathbb{R}^{+}_{0}~(i=1,2,3)$,

$$ \hat{f}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3})\geq 4\hat{g}(t,u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}). $$

(4)

$$\begin{aligned} 0&< \int _{0}^{1}\zeta (s)\varphi _{q}\Big(I_{0^{+}}^{\beta} \hat{g}(s,s^{\alpha +\rho -1},s^{\alpha +\rho -1},s^{\alpha +\rho -1},1,1,1) \Big)ds \\ &=\int _{0}^{1}\zeta (s)\bigg[ \frac{s^{\frac{16}{3}\Gamma (\frac{17}{6})}}{\Gamma (\frac{19}{3})}+ \frac{s^{\frac{11}{3}\Gamma (\frac{7}{6})}}{\Gamma (\frac{14}{3})} \bigg]ds\doteq 0.008958 < +\infty , \\ 0&< \int _{0}^{1}H(\omega ,\omega )\hat{f}(\omega ,1,1,1,\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1})d \omega \\ &=\int _{0}^{1}H(\omega ,\omega )(4\omega ^{-\frac{1}{2}}+6\omega ^{- \frac{3}{4}})d\omega =\frac{4\Gamma (\frac{1}{2})}{7!}+ \frac{6\Gamma (\frac{1}{4})}{\Gamma (\frac{15}{4})}\doteq 4.916618< + \infty , \\ 0&< \int _{0}^{1}H(\omega ,\omega )\hat{g}(\omega ,1,1,1,\omega ^{ \alpha +\rho -1},\omega ^{\alpha +\rho -1},\omega ^{\alpha +\rho -1})d \omega \\ &=\int _{0}^{1}H(\omega ,\omega )(\omega ^{\frac{1}{2}}+\omega ^{- \frac{3}{4}})d\omega =\frac{\Gamma (\frac{3}{2})}{4!}+ \frac{\Gamma (\frac{1}{4})}{\Gamma (\frac{15}{4})}\doteq 0.856135< + \infty . \end{aligned}$$

From Theorem 3.1, we know that equation (4.1) has a unique solution. If we let $u_{0}=t^{\frac{3}{2}}$, $v_{0}=\frac{t^{\frac{3}{2}}}{2}$, we know that $u_{0},~v_{0} \in P_{e}$, where $e(t)=t^{\frac{3}{2}}$. We can obtain the following numerical results and approximate results as shown in Table 1, Fig. 1. and Fig. 2.

Figure 1

Iterative process $u_{n}(t)$ and approximate solution $u^{*}(t)$.

Figure 2

Iterative process $v_{n}(t)$ and approximate solution $u^{*}(t)$.

Table 1 Iterative process $u_{n}(t)$, $v_{n}(t)$ ($n = 0$, 1, 2) and approximate of solutions.

Further, taking $r=\frac{7}{9}$, we know $\psi (\varepsilon )\geq \frac{\varepsilon ^{\frac{7}{9}}-\varepsilon}{4}+\varepsilon ^{ \frac{7}{9}}$. The numerical results shown in Fig. 3 and Table 2 indicate that

$$ \max \limits _{t\in [0,1]}\{\lvert x_{0.999}^{*}(t)-x_{1}^{*}(t) \rvert ,\lvert{D_{0^{+}}^{\frac{1}{2}}x_{0.999}^{*}(t)-D_{0^{+}}^{ \frac{1}{2}}x_{1}^{*}(t)}\rvert \}< 0.01, $$

from which we can interpret that $u_{\sigma}^{*}$ is continuous at $\sigma =1$.

Figure 3

Approximate solution $u_{\sigma}^{*}(t)$.

Table 2 $u_{\sigma}^{*}(t)$ for different values of positive parameter σ ($\sigma =0.95,0.96,0.97,0.98,0.99,0.999,1.0$)