A General Framework for the Multiplicity of Positive Solutions to Higher-Order Caputo and Hadamard Fractional Functional Differential Coupled Laplacian Systems

Abstract

This paper applies a general framework to explore the existence of multiple positive solutions for the fractional integral boundary value problem of high-order Caputo and Hadamard fractional coupled Laplacian systems with delayed or advanced arguments. We first focus on a generalized fractional homomorphic coupled boundary value problem with Hilfer fractional derivatives. Then we present the Green’s function corresponding to this Hilfer fractional system and its important properties. On this basis, by constructing a positive cone and applying a generalized cone fixed point theorem, we have established some novel criteria to ensure that the generalized fractional system has at least three positive solutions. As applications, we also obtain the multiplicity of the positive solutions of the Caputo and Hadamard fractional-order coupled Laplacian systems under two special Hilfer derivatives, respectively. Finally, we provide several examples to inspect the applicability of the main results.

1. Introduction

In this paper, we mainly consider the following higher-order Caputo fractional Laplacian system (1) and Hadamard fractional Laplacian system (2).

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[{\varphi }_{p1}\left({}^C\mathcal{D}_{0^{+}}^{\alpha }u\left(t\right)\right)\right]}^{\prime }=h_1\left(t\right)f_1(t,u\left({\tau }_1\left(t\right)\right),v\left({\tau }_2\left(t\right)\right)),0<t<1,\hfill \\ {\left[{\varphi }_{p_2}\left({}^C\mathcal{D}_{0^{+}}^{\beta }v\left(t\right)\right)\right]}^{\prime }=h_2\left(t\right)f_2(t,u\left({\tau }_1\left(t\right)\right),v\left({\tau }_2\left(t\right)\right)),0<t<1,\hfill \\ u\left(0\right)=0, {}^C\mathcal{D}_{0^{+}}^{{\alpha }_i}u\left(0\right)=0,u\left(1\right)={\mathcal{J}}_{0^{+}}^{\mu }u\left(1\right),\hfill \\ v\left(0\right)=0, {}^C\mathcal{D}_{0^{+}}^{{\beta }_j}v\left(0\right)=0,v\left(1\right)={\mathcal{J}}_{0^{+}}^{\nu }v\left(1\right),\hfill \end{array}\right.\end{array}$$

(1)

where $\alpha$, $\beta$, ${\alpha }_i$, ${\beta }_j$, $\mu$ and $\nu$ are some real constants satisfying $n-1<\alpha <n$, $m-1<\beta <m$, $i<{\alpha }_i<i+1$, $j<{\beta }_j<j+1$, $i=1,2,\dots ,n-2$, $j=1,2,\dots ,m-2$. ${}^C\mathcal{D}_{0^{+}}^{\ast }$ and ${\mathcal{J}}_{0^{+}}^{\ast }$ stand for *-order Caputo fractional derivatives and integrals, respectively. Laplacian operators and its inverses are ${\varphi }_{p_i}\left(w\right)={\left|w\right|}^{p_i-2}w(p_i>1)$ and ${\varphi }_{p_i}^{-1}={\varphi }_{q_i}$ with ${\displaystyle \frac{1}{p_i}}+{\displaystyle \frac{1}{q_i}}=1$. $h_1,h_2\in C([0,1],{\mathbb{R}}^{+})$, $f_1,f_2\in C([0,1]\times {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+},{\mathbb{R}}^{+})$, ${\tau }_1,{\tau }_2\in C([0,1],[0,1])$. ${\tau }_1\left(t\right),{\tau }_2\left(t\right)>t$ and ${\tau }_1\left(t\right),{\tau }_2\left(t\right)<t$, respectively, indicate advanced and delayed. $\mathbb{R}=(-\infty ,+\infty )$, ${\mathbb{R}}^{+}=(0,+\infty )$ and ${\mathbb{R}}_0^{+}=[0,+\infty )$.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[{\varphi }_{p_1}\left({}^H\mathcal{D}_{1^{+}}^{\alpha }u\left(t\right)\right)\right]}^{\prime }=h_1\left(t\right)f_1(t,u\left({\tau }_1\left(t\right)\right),v\left({\tau }_2\left(t\right)\right)),1<t<e,\hfill \\ {\left[{\varphi }_{p_2}\left({}^H\mathcal{D}_{1^{+}}^{\beta }v\left(t\right)\right)\right]}^{\prime }=h_2\left(t\right)f_2(t,u\left({\tau }_1\left(t\right)\right),v\left({\tau }_2\left(t\right)\right)),1<t<e,\hfill \\ u\left(1\right)=0,{}^H\mathcal{D}_{1^{+}}^{{\alpha }_i}u\left(1\right)=0,u\left(e\right)={}^H\mathcal{J}_{1^{+}}^{\mu }u\left(e\right),\hfill \\ v\left(1\right)=0,{}^H\mathcal{D}_{1^{+}}^{{\beta }_i}v\left(1\right)=0,v\left(e\right)={}^H\mathcal{J}_{1^{+}}^{\nu }v\left(e\right),\hfill \end{array}\right.\end{array}$$

(2)

where ${}^H\mathcal{D}_{1^{+}}^{\ast }$ and ${}^H\mathcal{J}_{1^{+}}^{\ast }$ stand for *-order Hadamard fractional derivatives and integrals, respectively. $h_1,h_2\in C([1,e],{\mathbb{R}}^{+})$, $f_1,f_2\in C([1,e]\times {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+},{\mathbb{R}}^{+})$, ${\tau }_1,{\tau }_2\in C([1,e],[1,e])$. The other data meet the same conditions as (1).

Our focus is to unify system (1) and system (2) within a general framework to study the existence of multiple positive solutions. To this end, we only need to study the multiplicity of positive solutions for a generalized fractional system with the $\psi$-Hilfer fractional derivative as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[{\mathcal{H}}_1\left({\mathcal{D}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)\right)\right]}^{\prime }=h_1\left(t\right)f_1(t,u\left({\tau }_1\left(t\right)\right),v\left({\tau }_2\left(t\right)\right)),a<t<b,\hfill \\ {\left[{\mathcal{H}}_2\left({\mathcal{D}}_{a^{+}}^{\beta ,\psi }v\left(t\right)\right)\right]}^{\prime }=h_2\left(t\right)f_2(t,u\left({\tau }_1\left(t\right)\right),v\left({\tau }_2\left(t\right)\right)),a<t<b,\hfill \\ u\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\alpha }_i,\psi }u\left(a\right)=0,u\left(b\right)={\mathcal{J}}_{a^{+}}^{\mu ,\psi }u\left(b\right),\hfill \\ v\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\beta }_j,\psi }v\left(a\right)=0,v\left(b\right)={\mathcal{J}}_{a^{+}}^{\nu ,\psi }v\left(b\right),\hfill \end{array}\right.\end{array}$$

(3)

where $0\le a<b$, ${\mathcal{D}}_{a^{+}}^{\ast ,\psi }$ and ${\mathcal{J}}_{a^{+}}^{\ast ,\psi }$ stand for *-order $\psi$-Hilfer fractional derivatives and integrals, respectively. ${\mathcal{H}}_1,{\mathcal{H}}_2:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ represent two positive increasing homomorphisms. $h_1,h_2\in C([a,b],{\mathbb{R}}^{+})$, $f_1,f_2\in C([a,b]\times {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+},{\mathbb{R}}^{+})$, ${\tau }_1,{\tau }_2\in C([a,b],[a,b])$. The other data meet the same conditions as (1).

Obviously, when $\psi \left(t\right)=t$, $a=0, b=1$ and ${\mathcal{H}}_i\left(w\right)={\varphi }_i\left(w\right)$, system (3) becomes system (1). When $\psi \left(t\right)=logt$, $a=1, b=e$ and ${\mathcal{H}}_i\left(w\right)={\varphi }_i\left(w\right)$, system (3) also becomes system (2). Additionally, an increasing positive homomorphism $\mathcal{H}:\mathbb{R}\to \mathbb{R}$ must satisfy the following conditions: $\forall x,y\in \mathbb{R}$, $x\le y\Rightarrow \mathcal{H}\left(x\right)\le \mathcal{H}\left(y\right)$; $\mathcal{H}$ and its inverse ${\mathcal{H}}^{-1}$ are both continuous; $\forall x,y\in {\mathbb{R}}^{+}$, $\mathcal{H}\left(xy\right)=\mathcal{H}\left(x\right)\mathcal{H}\left(y\right)$. It is easy to see that the Laplacian operator ${\varphi }_{p_i}\left(w\right)={\left|w\right|}^{p_i-2}w(p_i>1)$ is an increasing and positive homomorphism.

It is well known that fractional differential equations are very useful mathematical models in fields such as physics, chemistry, biology, thermal systems, rheological materials and signal processing. Our system incorporates many bioecological models such as Lotka–Volterra and Ayala–Gilpin models. For instance, the Caputo fractional Lotka–Volterra predation model and Ayala–Gilpin competition model are, respectively, represented as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C\mathcal{D}_{0^{+}}^{\alpha }u\left(t\right)=r_1\left(t\right)u\left(t\right)\left[1-{\displaystyle \frac{u\left(t\right)}{K_1}}-a_{12}\left(t\right){\displaystyle \frac{v\left(t\right)}{K_2}}\right],t>0,\hfill \\ {}^C\mathcal{D}_{0^{+}}^{\beta }v\left(t\right)=r_2\left(t\right)v\left(t\right)\left[1-{\displaystyle \frac{v\left(t\right)}{K_2}}+a_{21}\left(t\right){\displaystyle \frac{u\left(t\right)}{K_1}}\right],t>0,\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^C\mathcal{D}_{0^{+}}^{\alpha }x\left(t\right)=r_1\left(t\right)x\left(t\right)\left[1-{\left({\displaystyle \frac{x\left(t\right)}{K_1}}\right)}^{{\theta }_1}-b_{12}\left(t\right){\left({\displaystyle \frac{y\left(t\right)}{K_2}}\right)}^{{\theta }_2}\right],t>0,\hfill \\ {}^C\mathcal{D}_{0^{+}}^{\beta }y\left(t\right)=r_2\left(t\right)y\left(t\right)\left[1-{\left({\displaystyle \frac{y\left(t\right)}{K_2}}\right)}^{{\theta }_2}-b_{21}\left(t\right){\left({\displaystyle \frac{x\left(t\right)}{K_1}}\right)}^{{\theta }_1}\right],t>0,\hfill \end{array}\right.\end{array}$$

where $u\left(t\right)$ and $v\left(t\right)$, respectively, represent the population density of the prey and the predator. $x\left(t\right)$ and $y\left(t\right)$ express the population densities of the two competing species. $r_1\left(t\right), r_2\left(t\right)>0$ are their intrinsic growth rates. $K_1$ and $K_2$ stand for the maximum capacity of environment to species. $a_{12}\left(t\right)>0$ indicates the predation rate. $a_{21}\left(t\right)>0$ represents the conversion rate of predators after predation. $b_{12}\left(t\right), b_{21}\left(t\right)>0$ are the competition rates between species x and y. ${\theta }_1, {\theta }_2>0$ describe the nonlinear disturbances within the species.

Recently, some excellent papers have been published in establishing fractional differential equation models to solve practical problems such as the transient response of parallel circuit [1] and the mathematical modeling of epidemics [2,3,4,5,6].Therefore, many scholars have extensively and deeply studied the properties of fractional differential equations such as existence [7,8,9,10,11], uniqueness [12,13,14,15,16] and stability [17,18,19,20,21]. In the development of fractional calculus, new fractional derivatives and integrals have been constantly proposed. For example, Hilfer [22] put forward a novel fractional derivative named Hilfer fractional derivative in 2002. He replaced the function t in the Riemann–Liouville definition and the function $logt$ in the Hadamard definition with a general function $\psi \left(t\right)$. Subsequently, Sousa and Oliveira [23,24] conducted a detailed study on Hilfer fractional derivatives and integrals. They provided the definitions of $\psi$-Riemann–Liouville and $\psi$-Caputo derivatives and studied their properties such as the identity, limitation, uniform convergence and the relationship with their corresponding $\psi$-fractional integral. In the past decade, theoretical and applied studies on Hilfer fractional differential equations have been highly sought after by researchers. There have been some papers dealing with the existence and uniqueness [25], controllability [26,27] and stability [28,29] of Hilfer fractional differential equations. In addition, many practical problems such as fluid mechanics, population dynamics, and heat conduction involve integral boundary value problems for fractional differential equations. Therefore, the study of the integral boundary value problem of fractional differential equations has practical significance and has achieved many excellent results (see [25,30,31,32]).

Multiplicity is one of the important properties of fractional differential equations. In particular, for fractional differential equation models with practical application backgrounds, it is important and of practical significance to explore their multiple positive solutions. Lately, Zhang et al. [33] considered the following fractional Laplacian equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{\gamma }\left[{\varphi }_p\left({}^C\mathcal{D}_{0^{+}}^{\delta }v\left(t\right)\right)\right]+g(t,v\left(t\right)))=0,0<t<1,\hfill \\ {\varphi }_p\left({}^C\mathcal{D}_{0^{+}}^{\delta }v\left(0\right)\right)={\left[{\varphi }_p\left({}^C\mathcal{D}_{0^{+}}^{\delta }v\left(0\right)\right)\right]}^{\prime }={\varphi }_p\left({}^C\mathcal{D}_{0^{+}}^{\delta }v\left(1\right)\right)=0,\hfill \\ v^{\prime \prime }\left(0\right)=v^{\prime }\left(0\right)=0,l_{1}v\left(0\right)+l_2{v}^{\prime }\left(0\right)={\int }_0^{1}h\left(s\right)v\left(s\right)ds,\hfill \end{array}\right.\end{array}$$

where $2<\delta , \gamma \le 3$, $5<\delta +\gamma \le 6$ and $l_1, l_2\in (0,+\infty )$. ${\mathcal{D}}_{0^{+}}^{\gamma }$ and ${}^C\mathcal{D}_{0+}^{\delta }$ stand for the Riemann–Liouville and Caputo fractional derivatives, respectively. $h, g\ge 0$ are continuous. Based on the Avery–Henderson fixed point theorem, they discussed the multiple and nonexistence of positive solutions. In 2025, Nyamoradi and Ahmad [34] applied the generalized Leggett-Williams fixed point theorem and found that a generalized Riemann–Liouville fractional equation has at least three positive solutions. In [35], the authors concerned with the three-point boundary value problem of $(r_1,r_2,r_3)$-Laplacian fractional differential equations obtained the existence of multiple positive solutions. There have some latest articles dealing with the multiplicity of fractional differential systems. These papers on multiple solutions of fractional differential equations mainly consider the Caputo fractional derivative [36,37,38,39], the Riemann–Liouville fractional derivative [40,41,42,43] and the Hadamard fractional derivative [44]. These papers mainly use the classic Avery Peterson and Leggett-Williams fixed point theorems to obtain multiple solutions of fractional differential equations. To the best of our knowledge, studies about the multiplicity of solutions for $\psi$-fractional differential equations are still very rare.

Motivated by the above-mentioned reason, this manuscript stresses on the existence of multiple positive solutions for system (3). As an application, we also found that there are multiple positive solutions for systems (1) and (2). The main highlights of the manuscript are outlined as follows:

• We have established a unified framework for the higher-order Caputo fractional Laplacian system (1) and the Hadamard fractional Laplacian system (2).
• We studied the multiple positive solutions of Hilfer fractional differential coupled systems and obtained some novel sufficient criteria. This attempt fills the gap in this research field since there are few papers on the multiple positive solutions of Hilfer fractional differential equations.
• Our model takes into account general time-varying delays and Hilfer fractional integral–derivative boundary conditions. Therefore, our system is a class of more generalized fractional functional differential equations.

The remainder of the manuscript is structured as follows: Section 2 introduces some basic knowledge of Hilfer fractional calculus and constructs the Green function. Section 3 mainly investigates and finds that there are at least three positive solutions for system (3). As applications, Section 4 discusses the existence of multiple positive solutions for systems (1) and (2). Section 5 offers some examples to inspect the correctness and availability of our main findings. Section 6 provides a brief conclusion and expectation for future research.

2. Preliminaries

In this portion, we present an important cone fixed point theorem and review some concepts and results of fractional calculus.

Definition 1. 

Given a real Banach space $\mathbb{X}$, a closed convex subset $\varnothing \ne \mathbb{P}\subset \mathbb{X}$ becomes a cone iff $\left(1\right)$ $x\in \mathbb{P},\lambda >0\Rightarrow \lambda x\in \mathbb{P}$; $\left(2\right)$$x\in \mathbb{P},-x\in \mathbb{P}\Rightarrow x=0$.

Given a real Banach space $\mathbb{X}$, a cone $\mathbb{P}\subset \mathbb{X}$ and a continuous functional $\omega :\mathbb{P}\to {\mathbb{R}}_0^{+}$, for $r>0$, we introduce the following collections:

$${\Omega }_{\omega ,r}=\{w\in \mathbb{P}:\omega \left(w\right)<r\},\partial {\Omega }_{\omega ,r}=\{w\in \mathbb{P}:\omega \left(w\right)=r\},{\overline{\Omega }}_{\omega ,r}=\{w\in \mathbb{P}:\omega \left(w\right)\le r\}.$$

The establishment of multiple positive solutions in this paper requires the following important generalized Avery–Henderson cone fixed point theorem.

Lemma 1 

([44]). Given the real Banach space $\mathbb{X}$ and the cone $\mathbb{P}\subset \mathbb{X}$, let $\mathcal{A},\mathcal{C}:\mathbb{P}\to {\mathbb{R}}_0^{+}$ be increasing and continuous, and let $\mathcal{B}:\mathbb{P}\to {\mathbb{R}}_0^{+}$ be continuous. For some constants $M>0$ and $0<r_1<r_2<r_3$ , let $\mathcal{T}:{\overline{\Omega }}_{\mathcal{C},r_3}\to \mathbb{P}$ be completely continuous. Assume that the following hold:

(i)

$\mathcal{A}\left(0\right)<r_1$, $\mathcal{C}\left(w\right)\le \mathcal{B}\left(w\right)\le \mathcal{A}\left(w\right)$ and $\parallel w\parallel \le M\mathcal{C}\left(w\right)$, $\forall w\in {\overline{\Omega }}_{\mathcal{C},r_3}$.

$\left(\mathrm{ii}\right)$

$\mathcal{B}\left(\kappa w\right)\le \kappa \mathcal{B}\left(w\right)$, $\forall 0<\kappa \le 1$, $w\in \partial {\Omega }_{\mathcal{B},r_2}$.

$\left(\mathrm{iii}\right)$

$\mathcal{C}\left(\mathcal{T}w\right)<r_3$, $\forall w\in \partial {\Omega }_{\mathcal{C},r_3};$ $\mathcal{B}\left(\mathcal{T}w\right)>r_2$, $\forall w\in \partial {\Omega }_{\mathcal{B},r_2};$ $\mathcal{A}\left(\mathcal{T}w\right)<r_1$, $\forall w\in \partial {\Omega }_{\mathcal{A},r_1}.$

Then $\mathcal{T}$ has at least three fixed points $w_1,w_2,w_3\in {\Omega }_{\mathcal{C},r_3}$ satisfying

$$0\le \mathcal{A}\left(w_1\right)<r_1<\mathcal{A}\left(w_2\right)<r_2,\mathcal{B}\left(w_2\right)<r_2<\mathcal{B}\left(w_3\right),\mathcal{C}\left(w_3\right)<r_3.$$

Next we review some fundamental concepts and results of fractional calculus.

Definition 2 

([45]). Let $0<a<b\le +\infty$, $\alpha >0$ and $u\in L^1([a,b],\mathbb{R})$. The α-order Riemann–Liouville fractional integral of u is introduced by

$$\begin{array}{c}\hfill {\mathcal{J}}_{a^{+}}^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(t-s)}^{\alpha -1}u\left(s\right)ds.\end{array}$$

Definition 3 

([45]). Let $0<a<b\le +\infty$, $\alpha >0$, $n-1<\alpha \le n(n=[\alpha ]+1)$ and $u\in C^n([a,b],\mathbb{R})$. The α-order Caputo fractional derivative of u is introduced by

$$\begin{array}{c}\hfill {}^C\mathcal{D}_{a^{+}}^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{(t-s)}^{n-\alpha -1}{u}^{\left(n\right)}\left(s\right)ds.\end{array}$$

Definition 4 

([45]). Let $0<a<b\le +\infty$, $\alpha >0$ and $u\in L^1([a,b],\mathbb{R})$. The α-order Hadamard fractional integral of u is introduced by

$$\begin{array}{c}\hfill {}^H\mathcal{J}_{{\mathrm{a}}^{+}}^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{\alpha -1}u\left(s\right){\displaystyle \frac{ds}{s}}.\end{array}$$

Definition 5 

([45]). Let $0<a<b\le +\infty$, $\alpha >0$, $n-1<\alpha \le n(n=[\alpha ]+1)$ and $u\in C^n([a,b],\mathbb{R})$. The α-order Hadamard fractional derivative of u is introduced by

$$\begin{array}{c}\hfill {}^H\mathcal{D}_{{\mathrm{a}}^{+}}^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{\left(log{\displaystyle \frac{t}{s}}\right)}^{n-\alpha -1}{\left(s{\displaystyle \frac{d}{ds}}\right)}^{n}u\left(s\right){\displaystyle \frac{ds}{s}}.\end{array}$$

Definition 6 

([23]). Let $0<a<b\le +\infty$, $\alpha >0$, $u\in L^1([a,b],\mathbb{R})$, and $\psi \in C^1([0,\infty ),\mathbb{R})$ be increasing with ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in (0,\infty )$. The α-order ψ-fractional integral of u is introduced by

$$\begin{array}{c}\hfill {\mathcal{J}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}u\left(s\right)d\psi \left(s\right).\end{array}$$

Definition 7 

([23]). Let $0<a<b\le +\infty$, $\alpha >0$, $n-1<\alpha \le n(n=[\alpha ]+1)$, $u\in C^n([a,b],\mathbb{R})$, and $\psi \in C^n([0,\infty ),\mathbb{R})$ be increasing with ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in (0,\infty )$. The α-order ψ-fractional derivative of u is introduced by

$$\begin{array}{c}\hfill {\mathcal{D}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{n-\alpha -1}{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n}u\left(s\right)d\psi \left(s\right).\end{array}$$

Remark 1. 

In Definitions 6 and 7, if we take $\psi \left(t\right)=t$ and $\psi \left(t\right)=logt$, respectively, then we can obtain Definitions 2–5. Therefore, ψ-fractional calculus is an extension of the Caputo and Hadamard fractional calculus.

Lemma 2 

([23]). Let $\alpha >0$ and $n-1<\alpha <n(n=[\alpha ]+1)$, then the following assertions hold.

$\left(1\right)$

${\mathcal{J}}_{a^{+}}^{\alpha ,\psi }{\mathcal{D}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)=u\left(t\right)+c_0+{\displaystyle \sum _{i=1}^{n-1}}{c}_i{(\psi \left(t\right)-\psi \left(a\right))}^i,$$\forall u\in C^n([a,b],\mathbb{R})$.

$\left(2\right)$

${\mathcal{D}}_{a^{+}}^{\beta ,\psi }{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)={\mathcal{J}}_{a^{+}}^{\alpha -\beta ,\psi }u\left(t\right)$, $\forall u\in C\left(\right[a,b],\mathbb{R})$, $0<\beta \le \alpha$.

$\left(3\right)$

${\mathcal{J}}_{a^{+}}^{\beta ,\psi }{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)={\mathcal{J}}_{a^{+}}^{\alpha +\beta ,\psi }u\left(t\right)$, $\forall u\in C\left(\right[a,b],\mathbb{R})$, $\beta >0$.

Lemma 3. 

Let $\sigma ,\gamma >0$, $\psi \in C\left(\right[a,b],\mathbb{R})$, ${\psi }^{\prime }\left(t\right)\ne 0$ for all $t\in (a,b)$, then

$\left(\mathrm{i}\right)$

${\mathcal{J}}_{a^{+}}^{\sigma ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -1}={\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma +\sigma )}}{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma +\sigma -1}.$

$\left(\mathrm{ii}\right)$

${\mathcal{D}}_{a^{+}}^{\sigma ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -1}={\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -\sigma )}}{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -\sigma -1},\gamma >\sigma .$

Proof. 

(i) By Definition 6 and the Beta function $B(\sigma ,\gamma )={\int }_0^1{\tau }^{\sigma -1}{(1-\tau )}^{\gamma -1}d\tau ={\displaystyle \frac{\Gamma \left(\gamma \right)\Gamma \left(\sigma \right)}{\Gamma (\gamma +\sigma )}}$, we obtain

$$\begin{array}{cc}\hfill & {\mathcal{J}}_{a^{+}}^{\sigma ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -1}={\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\sigma -1}{(\psi \left(s\right)-\psi \left(a\right))}^{\gamma -1}d\psi \left(s\right)\hfill \\ \hfill & \frac{\underline{\zeta =\psi \left(s\right)-\psi \left(a\right)}}{ }{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{\int }_0^{\psi \left(t\right)-\psi \left(a\right)}{(\psi \left(t\right)-\psi \left(a\right)-\zeta )}^{\sigma -1}{\zeta }^{\gamma -1}d\zeta \hfill \\ \hfill & \frac{\underline{\eta ={\displaystyle \frac{\zeta }{\psi \left(t\right)-\psi \left(a\right)}}}}{ }{\displaystyle \frac{1}{\Gamma \left(\sigma \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma +\sigma -1}{\int }_0^1{(1-\eta )}^{\sigma -1}{\eta }^{\gamma -1}d\eta \hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma +\sigma )}}{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma +\sigma -1}.\hfill \end{array}$$

$\left(\mathrm{ii}\right)$ When $\gamma >\sigma$, the assertion $\left(\mathrm{i}\right)$ gives

$${\mathcal{J}}_{a^{+}}^{\sigma ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -\sigma -1}={\displaystyle \frac{\Gamma (\gamma -\sigma )}{\Gamma \left(\gamma \right)}}{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -1}.$$

According to Lemma 2, when operator ${\mathcal{D}}_{a^{+}}^{\alpha ,\psi }$ acts on both ends of the above equation, we obtain

$${(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -\sigma -1}={\mathcal{D}}_{a^{+}}^{\alpha ,\psi }{\mathcal{J}}_{a^{+}}^{\sigma ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -\sigma -1}={\displaystyle \frac{\Gamma (\gamma -\sigma )}{\Gamma \left(\gamma \right)}}{\mathcal{D}}_{a^{+}}^{\alpha ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{\gamma -1},$$

which indicates that the assertion $\left(\mathrm{ii}\right)$ is true. The proof of Lemma 3 is over. □

Next, we look for the Green’s function corresponding to system (3).

Lemma 4. 

For any given $x\in C\left(\right[a,b],\mathbb{R})$, consider the following fractional system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{D}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)+x\left(t\right)=0,a<t<b,\hfill \\ u\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\alpha }_i,\psi }u\left(a\right)=0,u\left(b\right)={\mathcal{J}}_{a^{+}}^{\mu ,\psi }u\left(b\right),\hfill \end{array}\right.\end{array}$$

(4)

where $0\le a<b$, $n-1<\alpha <n(n\ge 2)$, $i<{\alpha }_i<i+1$, $i=1,2,\dots ,n-2$. Let

$${\rho }_1={\displaystyle \frac{\Gamma \left(n\right)}{\Gamma (n+\mu )}}{(\psi \left(b\right)-\psi \left(a\right))}^{\mu },{\Delta }_1=(1-{\rho }_1){(\psi \left(b\right)-\psi \left(a\right))}^{n-1}.$$

If ${\rho }_1\ne 1$, then system (4) is uniquely solved by

$$\begin{array}{c}\hfill u\left(t\right)={\int }_a^{b}G(t,s)x\left(s\right)d\psi \left(s\right),\end{array}$$

where

$$\begin{array}{c}\hfill G(t,s)=\left\{\begin{array}{cc}{G}_1(t,s),\hfill & a\le s\le t\le b,\hfill \\ G_2(t,s),\hfill & a\le t\le s\le b,\hfill \end{array}\right.\end{array}$$

(5)

$$\begin{array}{cc}\hfill G_1(t,s)=& -{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}+{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}\hfill \\ \hfill & -{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha +\mu -1},\hfill \end{array}$$

$$\begin{array}{cc}\hfill G_2(t,s)=& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}-{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha +\mu -1}.\hfill \end{array}$$

Proof. 

From Lemma 2 and (4), we have

$$\begin{array}{cc}\hfill u\left(t\right)=& -{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }x\left(t\right)+c_0+\sum _{i=1}^{n-1}{c}_i{(\psi \left(t\right)-\psi \left(a\right))}^i\hfill \\ \hfill =& -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}x\left(s\right)d\psi \left(s\right)+c_0+\sum _{i=1}^{n-1}{c}_i{(\psi \left(t\right)-\psi \left(a\right))}^i.\hfill \end{array}$$

(6)

By $u\left(a\right)=0$, (6) gives $c_0=0$. Applying (6), Lemmas 2 and 3, we obtain

$$\begin{array}{cc}\hfill & {\mathcal{D}}_{a^{+}}^{{\alpha }_1,\psi }u\left(t\right)=-{\mathcal{D}}_{a^{+}}^{{\alpha }_1,\psi }{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }x\left(t\right)+\sum _{i=1}^{n-1}{c}_i{\mathcal{D}}_{a^{+}}^{{\alpha }_1,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^i\hfill \\ \hfill =& -{\mathcal{J}}_{a^{+}}^{\alpha -{\alpha }_1,\psi }x\left(t\right)+{\displaystyle \frac{c_1\Gamma \left(2\right)}{\Gamma (2-{\alpha }_1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{1-{\alpha }_1}+\sum _{i=2}^{n-1}{\displaystyle \frac{c_i\Gamma (i+1)}{\Gamma (i-{\alpha }_1+1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{i-{\alpha }_1}\hfill \\ \hfill =& -{\displaystyle \frac{1}{\Gamma (\alpha -{\alpha }_1)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -{\alpha }_1-1}x\left(s\right)d\psi \left(s\right)+{\displaystyle \frac{c_1}{\Gamma (2-{\alpha }_1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{1-{\alpha }_1}\hfill \\ \hfill & +\sum _{i=2}^{n-1}{\displaystyle \frac{c_i·i!}{\Gamma (i-{\alpha }_1+1)}}{(\psi \left(t\right)-\psi \left(a\right))}^{i-{\alpha }_1}.\hfill \end{array}$$

(7)

In view of $1<{\alpha }_1<2$ and ${\mathcal{D}}_{0^{+}}^{{\alpha }_1,\psi }u\left(a\right)=0$, (7) gives $c_1=0$. Similarly, we derive that $c_i=0$, $i=2,3,\dots ,n-2$. Thus, we have

$$\begin{array}{c}\hfill c_0=c_1=c_2=\dots =c_{n-2}=0.\end{array}$$

(8)

From (6) and (8), we get

$$\begin{array}{c}\hfill u\left(t\right)=-{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }x\left(t\right)+c_{n-1}{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}.\end{array}$$

(9)

On the other hand, it follows from (9), Lemmas 2 and 3 that

$$\begin{array}{cc}\hfill {\mathcal{J}}_{a^{+}}^{\mu ,\psi }u\left(t\right)=& -{\mathcal{J}}_{a^{+}}^{\mu ,\psi }{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }x\left(t\right)+c_{n-1}{\mathcal{J}}_{a^{+}}^{\mu ,\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}\hfill \\ \hfill =& -{\mathcal{J}}_{a^{+}}^{\alpha +\mu ,\psi }x\left(t\right)+{\displaystyle \frac{c_{n-1}\Gamma \left(n\right)}{\Gamma (n+\mu )}}{(\psi \left(t\right)-\psi \left(a\right))}^{n+\mu -1}.\hfill \end{array}$$

(10)

Applying (9), (10) and $u\left(b\right)={\mathcal{J}}_{a^{+}}^{\mu ,\psi }u\left(b\right)$, we obtain

$$\begin{array}{cc}\hfill & -{\mathcal{J}}_{a^{+}}^{\alpha ,\psi }x\left(b\right)+c_{n-1}{(\psi \left(b\right)-\psi \left(a\right))}^{n-1}\hfill \\ \hfill =& -{\mathcal{J}}_{a^{+}}^{\alpha +\mu ,\psi }x\left(b\right)+c_{n-1}{\displaystyle \frac{\Gamma \left(n\right)}{\Gamma (n+\mu )}}{(\psi \left(b\right)-\psi \left(a\right))}^{n+\mu -1},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill c_{n-1}={\displaystyle \frac{1}{{\Delta }_1}}\left({\mathcal{J}}_{a^{+}}^{\alpha ,\psi }x\left(b\right)-{\mathcal{J}}_{a^{+}}^{\alpha +\mu ,\psi }x\left(b\right)\right).\end{array}$$

Substituting $c_{n-1}$ into (9), we obtain

$$\begin{array}{cc}\hfill u\left(t\right)=& -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}x\left(s\right)d\psi \left(s\right)+{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}\hfill \\ \hfill & \times {\int }_a^b{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}x\left(s\right)d\psi \left(s\right)-{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma (\alpha +\mu )}}\hfill \\ \hfill & \times {\int }_a^b{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha +\mu -1}x\left(s\right)d\psi \left(s\right)\hfill \\ \hfill =& -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(\psi \left(t\right)-\psi \left(s\right))}^{\alpha -1}x\left(s\right)d\psi \left(s\right)+{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}\hfill \\ \hfill & \times \left({\int }_a^t+{\int }_t^b\right){(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}x\left(s\right)d\psi \left(s\right)-{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma (\alpha +\mu )}}\hfill \\ \hfill & \times \left({\int }_a^t+{\int }_t^b\right){(\psi \left(b\right)-\psi \left(s\right))}^{\alpha +\mu -1}x\left(s\right)d\psi \left(s\right)\hfill \\ \hfill =& {\int }_a^{b}G(t,s)x\left(s\right)d\psi \left(s\right).\hfill \end{array}$$

The proof is over. □

Similar to Lemma 4, we have the following result. Consider another fractional system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathcal{D}}_{a^{+}}^{\beta ,\psi }v\left(t\right)+y\left(t\right)=0,a<t<b,\hfill \\ v\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\beta }_j,\psi }v\left(a\right)=0,v\left(b\right)={\mathcal{J}}_{a^{+}}^{\nu ,\psi }u\left(b\right),\hfill \end{array}\right.\end{array}$$

(11)

where $0\le a<b$, $m-1<\beta <m(m\ge 2)$, $j<{\beta }_j<j+1$, $\nu >0$, $j=1,2,\dots ,m-2$, and any given $y\in C\left(\right[a,b],\mathbb{R})$. Denote

$${\rho }_2={\displaystyle \frac{\Gamma \left(m\right)}{\Gamma (m+\nu )}}{(\psi \left(b\right)-\psi \left(a\right))}^{\nu },{\Delta }_2=(1-{\rho }_2){(\psi \left(b\right)-\psi \left(a\right))}^{m-1}.$$

Lemma 5. 

If ${\rho }_2\ne 1$, then system (11) admits a unique solution as follows

$$\begin{array}{c}\hfill v\left(t\right)={\int }_a^{b}K(t,s)y\left(s\right)d\psi \left(s\right),\end{array}$$

where

$$\begin{array}{c}\hfill K(t,s)=\left\{\begin{array}{cc}{K}_1(t,s),\hfill & a\le s\le t\le b,\hfill \\ K_2(t,s),\hfill & a\le s\le t\le b,\hfill \end{array}\right.\end{array}$$

(12)

$$\begin{array}{cc}\hfill K_1(t,s)=& -{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(s\right))}^{\beta -1}}{\Gamma \left(\beta \right)}}+{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{m-1}}{{\Delta }_2\Gamma \left(\beta \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\beta -1}\hfill \\ \hfill & -{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{m-1}}{{\Delta }_2\Gamma (\beta +\nu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\beta +\nu -1},\hfill \end{array}$$

$$\begin{array}{cc}\hfill K_2(t,s)=& {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{m-1}}{{\Delta }_2\Gamma \left(\beta \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\beta -1}-{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{m-1}}{{\Delta }_2\Gamma (\beta +\nu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\beta +\nu -1}.\hfill \end{array}$$

From Lemmas 4 and 5, one can easily conclude Lemmas 6 and 7, respectively.

Lemma 6. 

If ${\rho }_1\ne 1$ and $x\in C\left(\right[a,b],\mathbb{R})$, the fractional system (13) with homomorphism

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[{\mathcal{H}}_1\left({\mathcal{D}}_{a^{+}}^{\alpha ,\psi }u\left(t\right)\right)\right]}^{\prime }+x\left(t\right)=0,a<t<b,\hfill \\ u\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\alpha }_i,\psi }u\left(a\right)=0,u\left(b\right)={\mathcal{J}}_{a^{+}}^{\mu ,\psi }u\left(b\right),\hfill \end{array}\right.\end{array}$$

(13)

admits a unique solution

$$\begin{array}{c}\hfill u\left(t\right)=-{\int }_a^{b}G(t,s){\mathcal{H}}_1^{-1}\left(-{\int }_a^{s}x\left(z\right)dz\right)d\psi \left(s\right).\end{array}$$

(14)

Lemma 7. 

If ${\rho }_2\ne 1$ and $y\in C\left(\right[a,b],\mathbb{R})$, the fractional system (15) with homomorphism

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[{\mathcal{H}}_2\left({\mathcal{D}}_{a^{+}}^{\beta ,\psi }v\left(t\right)\right)\right]}^{\prime }+y\left(t\right)=0,a<t<b,\hfill \\ v\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\beta }_j,\psi }v\left(a\right)=0,v\left(b\right)={\mathcal{J}}_{a^{+}}^{\nu ,\psi }v\left(b\right),\hfill \end{array}\right.\end{array}$$

(15)

admits a unique solution

$$\begin{array}{c}\hfill v\left(t\right)=-{\int }_a^{b}K(t,s){\mathcal{H}}_2^{-1}\left(-{\int }_a^{s}y\left(z\right)dz\right)d\psi \left(s\right).\end{array}$$

(16)

Now we present the assumptions needed throughout the paper and the basic properties of Green’s functions $G(t,s)$ and $K(t,s)$.

Lemma 8. 

If $0<{\rho }_1,{\rho }_2<1$, then Green’s functions $G(t,s)$ and $K(t,s)$ have the following important properties:

$\left(\mathrm{i}\right)$

$\forall t,s\in [a,b]$, $G(t,s)$, $K(t,s)\ge 0$, ${\displaystyle \frac{\partial G(t,s)}{\partial t}}$, ${\displaystyle \frac{\partial K(t,s)}{\partial t}}\ge 0$.

$\left(\mathrm{ii}\right)$

$\forall t,s\in [a,b]$, $G(t,s)\le {\Lambda }_1$, $K(t,s)\le {\Lambda }_2$, where ${\Lambda }_1={\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}}{(1-{\rho }_1)\Gamma \left(\alpha \right)}}$, ${\Lambda }_2={\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{\beta -1}}{(1-{\rho }_2)\Gamma \left(\beta \right)}}$.

Proof. 

(i) We now discuss the monotonicity of $G_1(t,s)$ and $G_2(t,s)$ with respect to t on $[a,b]$. To this end, let

$$\begin{array}{cc}\hfill g_1(z,s)=& -{\displaystyle \frac{{(z-\psi \left(s\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}+{\displaystyle \frac{{(z-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}\hfill \\ \hfill & -{\displaystyle \frac{{(z-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha +\mu -1},\hfill \end{array}$$

Then, for $i=1,2,\dots ,n$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^i{g}_1(z,s)}{\partial z^i}}=& -(\alpha -1)(\alpha -2)\dots (\alpha -i){\displaystyle \frac{{(z-\psi \left(s\right))}^{\alpha -i-1}}{\Gamma \left(\alpha \right)}}\hfill \\ \hfill & +(n-1)(n-2)\dots (n-i){\displaystyle \frac{{(z-\psi \left(a\right))}^{n-i-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}\hfill \\ \hfill & -(n-1)(n-2)\dots (n-i){\displaystyle \frac{{(z-\psi \left(a\right))}^{n-i-1}}{{\Delta }_1\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha +\mu -1}.\hfill \end{array}$$

(17)

Noticing that $z=\psi \left(t\right)$ is increasing on $[a,b]$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^n{g}_1(z,s)}{\partial z^n}}=-(\alpha -1)(\alpha -2)\dots (\alpha -n){\displaystyle \frac{{(z-\psi \left(s\right))}^{\alpha -n-1}}{\Gamma \left(\alpha \right)}}\ge 0,\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right),\end{array}$$

which implies that ${\displaystyle \frac{{\partial }^{n-1}{g}_1(z,s)}{\partial z^{n-1}}}$ is increasing with respect to z on $\left[\psi \right(a),\psi (b\left)\right]$, satisfying $\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right)$. When $\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right)$, $z=\psi \left(a\right)$ means that $\psi \left(s\right)=\psi \left(a\right)$. Thus, for $\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right)$, we derive from (17) that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{n-1}{g}_1(z,s)}{\partial z^{n-1}}}\ge {\displaystyle \frac{{\partial }^{n-1}{g}_1(z,s)}{\partial z^{n-1}}}{|}_{z=\psi \left(a\right)}={\displaystyle \frac{(n-1)!}{\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -n}>0,\end{array}$$

which implies that ${\displaystyle \frac{{\partial }^{n-2}{g}_1(z,s)}{\partial z^{n-2}}}$ is increasing with respect to z on $\left[\psi \right(a),\psi (b\left)\right]$, satisfying $\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right)$. Therefore, we get

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^{n-2}{g}_1(z,s)}{\partial z^{n-2}}}\ge {\displaystyle \frac{{\partial }^{n-2}{g}_1(z,s)}{\partial z^{n-2}}}{|}_{z=\psi \left(a\right)}=0,\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right),\end{array}$$

which indicates that ${\displaystyle \frac{{\partial }^{n-3}{g}_1(z,s)}{\partial z^{n-3}}}$ is increasing with respect to z on $\left[\psi \right(a),\psi (b\left)\right]$, satisfying $\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right)$. Repeating the above process, we conclude that $g_1(z,s)\ge 0$ and ${\displaystyle \frac{\partial g_1(z,s)}{\partial z}}\ge 0$ for $\psi \left(a\right)\le \psi \left(s\right)\le z\le \psi \left(b\right)$. Noting that $g_1(\psi \left(t\right),s)=G_1(t,s)$, ${\displaystyle \frac{\partial G_1(t,s)}{\partial t}}={\displaystyle \frac{\partial g_1(z,s)}{\partial z}}·{\psi }^{\prime }\left(t\right)$ and ${\psi }^{\prime }\left(t\right)\ge 0$, we know that $G_1(t,s)\ge 0$, ${\displaystyle \frac{\partial G_1(t,s)}{\partial t}}\ge 0$ for $a\le s\le t\le b$.

On the other hand, when $a\le t\le s\le b$, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial G_2(t,s)}{\partial t}}=(n-1){\psi }^{\prime }\left(t\right){\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-2}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}\hfill \\ \hfill & -(n-1){\psi }^{\prime }\left(t\right){\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-2}}{{\Delta }_1\Gamma (\alpha +\mu )}}{(\psi \left(a_k\right)-\psi \left(s\right))}^{\alpha +\mu -1}.\hfill \\ \hfill =& (n-1){\psi }^{\prime }\left(t\right){\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-2}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}·\Theta \left(s\right),\hfill \end{array}$$

(18)

where

$$\Theta \left(s\right)=1-{\displaystyle \frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\mu }.$$

Based on $\psi \left(s\right)$ is increasing on $[a,b]$, we obtain

$$\begin{array}{c}\hfill {\displaystyle \frac{d\Theta \left(s\right)}{ds}}={\displaystyle \frac{\mu \Gamma \left(\alpha \right)}{\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(s\right))}^{\mu -1}{\psi }^{\prime }\left(s\right)\ge 0,s\in [a,b],\end{array}$$

which means that $\Theta \left(s\right)$ is increasing on $\in [a,b]$, and

$$\begin{array}{cc}\hfill \Theta \left(s\right)\ge & \Theta \left(a\right)=1-{\displaystyle \frac{\Gamma \left(\alpha \right)}{\Gamma (\alpha +\mu )}}{(\psi \left(b\right)-\psi \left(a\right))}^{\mu }=1-{\rho }_1>0.\hfill \end{array}$$

(19)

From (18) and (19), we know that ${\displaystyle \frac{\partial G_2(t,s)}{\partial t}}\ge 0$ and $G_2(t,s)\ge G_2(a,s)=0$. Thus, we obtain $G(t,s)\ge 0$ and ${\displaystyle \frac{\partial G(t,s)}{\partial t}}\ge 0$, $\forall t,s\in [a,b]$. In the same manner, one can obtain $K(t,s)\ge 0$ and ${\displaystyle \frac{\partial K(t,s)}{\partial t}}\ge 0$, $\forall t,s\in [a,b]$.

(ii) It follows from (5) and (12) that

$$\begin{array}{cc}\hfill G(t,s)\le & {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\alpha -1}\le {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{n-1}}{{\Delta }_1\Gamma \left(\alpha \right)}}{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}\hfill \\ \hfill =& {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{n-1}}{\Gamma \left(\alpha \right)(1-{\rho }_1){(\psi \left(b\right)-\psi \left(a\right))}^{n-1}}}{(\psi \left(b\right)-\psi \left(a\right))}^{\alpha -1}={\Lambda }_1,\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill K(t,s)\le & {\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{m-1}}{{\Delta }_2\Gamma \left(\beta \right)}}{(\psi \left(b\right)-\psi \left(s\right))}^{\beta -1}\le {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{m-1}}{{\Delta }_2\Gamma \left(\beta \right)}}{(\psi \left(b\right)-\psi \left(a\right))}^{\beta -1}\hfill \\ \hfill =& {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{m-1}}{\Gamma \left(\beta \right)(1-{\rho }_2){(\psi \left(b\right)-\psi \left(a\right))}^{m-1}}}{(\psi \left(b\right)-\psi \left(a\right))}^{\beta -1}={\Lambda }_2.\hfill \end{array}$$

(21)

(20) and (21) show that $\left(\mathrm{ii}\right)$ is true. The proof is completed. □

Based on Lemmas 6 and 7, we have the following assertion.

Lemma 9. 

If ${\rho }_1\ne 1$ and ${\rho }_2\ne 1$, then the nonlinear Hilfer fractional differential system (3) is equivalent to the nonlinear Hilfer fractional integral system outlined below

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u\left(t\right)=-{\int }_a^{b}G(t,s){\mathcal{H}}_1^{-1}\left(-{\int }_a^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right),\hfill \\ v\left(t\right)=-{\int }_a^{b}K(t,s){\mathcal{H}}_2^{-1}\left(-{\int }_a^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right),\hfill \end{array}\right.\end{array}$$

(22)

where $G(t,s)$ and $K(t,s)$ are defined by (5) and (12), respectively.

Lemma 10. 

Assume that $0<{\rho }_1,{\rho }_2<1$. If $U\left(t\right)=\left(u\right(t),v(t\left)\right)$ is a solution of system (22), then $u\left(t\right),v\left(t\right),u^{\prime }\left(t\right),v^{\prime }\left(t\right)\ge 0$, for $t\in [a,b]$.

Proof. 

Since ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$ are the positive increasing homomorphism, it follows from (22) that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u\left(t\right)={\int }_a^{b}G(t,s){\mathcal{H}}_1^{-1}\left({\int }_a^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right),\hfill \\ v\left(t\right)={\int }_a^{b}K(t,s){\mathcal{H}}_2^{-1}\left({\int }_a^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right),\hfill \end{array}\right.\end{array}$$

(23)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}^{\prime }\left(t\right)={\int }_a^b{\displaystyle \frac{\partial G(t,s}{\partial t}}){\mathcal{H}}_1^{-1}\left({\int }_a^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right),\hfill \\ v^{\prime }\left(t\right)={\int }_a^b{\displaystyle \frac{\partial K(t,s}{\partial t}}){\mathcal{H}}_2^{-1}\left({\int }_a^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right).\hfill \end{array}\right.\end{array}$$

(24)

From $G(t,s),K(t,s)$, ${\displaystyle \frac{\partial G(t,s)}{\partial t}}$, ${\displaystyle \frac{\partial K(t,s)}{\partial t}}\ge 0$ for $t\in [a,b]$ and $h_1,h_2,f_1,f_2>0$, we derive from (23) and (24) that $u\left(t\right),v\left(t\right),u^{\prime }\left(t\right),v^{\prime }\left(t\right)\ge 0$ for all $t\in [a,b]$. The proof is completed. □

3. Main Results

This section focuses on discussing the existence of multiple positive solutions for system (3). Let $\mathbb{X}=C\left(\right[a,b],\mathbb{R})\times C\left(\right[a,b],\mathbb{R})$, then $\mathbb{X}$ is a Banach space with the vector norm

$$\begin{array}{c}\hfill \parallel U\parallel =\parallel (u,v)\parallel =max\left\{\underset{a\le t\le b}{max}\left|u\left(t\right)\right|,\underset{a\le t\le b}{max}\left|v\left(t\right)\right|\right\},\forall U=(u,v)\in \mathbb{X}.\end{array}$$

Due to Lemma 8, the operator $\mathcal{T}:\mathbb{X}\to \mathbb{X}$ is defined by

$$\begin{array}{c}\hfill \left(\mathcal{T}U\right)\left(t\right)=(\left({\mathcal{T}}_{1}U\right)\left(t\right),\left({\mathcal{T}}_{2}U\right)\left(t\right)),\forall U=(u,v)\in \mathbb{X},\end{array}$$

(25)

where

$$\begin{array}{c}\hfill \left({\mathcal{T}}_{1}U\right)\left(t\right)=-{\int }_a^{b}G(t,s){\mathcal{H}}_1^{-1}\left(-{\int }_a^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right),\end{array}$$

(26)

$$\begin{array}{c}\hfill \left({\mathcal{T}}_{2}U\right)\left(t\right)=-{\int }_a^{b}K(t,s){\mathcal{H}}_2^{-1}\left(-{\int }_a^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)d\psi \left(s\right).\end{array}$$

(27)

Then solving system (3) is equivalent to finding the fixed points of operator $\mathcal{T}$ in $\mathbb{X}$.

Lemma 11. 

When $0<{\rho }_1,{\rho }_2<1$, the operator $\mathcal{T}:\mathbb{X}\to \mathbb{X}$ defined as (25)–(27) is completely continuous.

Proof. 

Since $G(t,s)$, $K(t,s)$, $h_1\left(t\right)$, $h_2\left(t\right)$, ${\tau }_1\left(t\right)$, ${\tau }_2\left(t\right)$, $f_1(t,u,v)$ and $f_2(t,u,v)$ are all continuous, $\mathcal{T}:\mathbb{X}\to \mathbb{X}$ is also continuous. Next, it is necessary to prove that $\mathcal{T}$ maps the bounded set to the bounded set. Indeed, when $\parallel U\parallel =\parallel (u,v)\parallel \le M$, $|h_k\left(t\right)|\le N_1$ and $|f_k(t,u,v)|\le N_2(k=1,2)$, we derive from (26), (27) and Lemma 8 that

$$\begin{array}{c}\hfill |\left({\mathcal{T}}_{1}U\right)\left(t\right)|\le (b-a){\mathcal{H}}_1^{-1}\left(N_1{N}_2\right){\int }_a^{b}G(b,s)d\psi \left(s\right),\end{array}$$

(28)

$$\begin{array}{c}\hfill |\left({\mathcal{T}}_{2}U\right)\left(t\right)|\le (b-a){\mathcal{H}}_2^{-1}\left(N_1{N}_2\right){\int }_a^{b}K(b,s)d\psi \left(s\right).\end{array}$$

(29)

(28) and (29) indicate that $\left(\mathcal{T}U\right)\left(t\right)$ is bounded. Finally, it is proved that $\mathcal{T}$ is equicontinuous. Owing to $G(t,s)$ and $K(t,s)$ being uniformly continuous on $[a,b]\times [a,b]$, when $t_2\to t_1$, we have

$$\begin{array}{c}\hfill |\left({\mathcal{T}}_{1}U\right)\left(t_2\right)-\left({\mathcal{T}}_{1}U\right)\left(t_1\right)|\le (b-a){\mathcal{H}}_1^{-1}\left(N_1{N}_2\right){\int }_a^b|G(t_2,s)-G(t_1,s)|d\psi \left(s\right)\to 0,\end{array}$$

(30)

$$\begin{array}{c}\hfill |\left({\mathcal{T}}_{2}U\right)\left(t_2\right)-\left({\mathcal{T}}_{2}U\right)\left(t_1\right)|\le (b-a){\mathcal{H}}_1^{-1}\left(N_1{N}_2\right){\int }_a^b|K(t_2,s)-K(t_1,s)|d\psi \left(s\right)\to 0.\end{array}$$

(31)

(30) and (31) mean that $\mathcal{T}$ is equicontinuous. Therefore, we know from the Arzela–Ascoli theorem that $\mathcal{T}$ is completely continuous. This completes the proof. □

We define a cone $\mathbb{P}$ as follows:

$$\begin{array}{c}\hfill \mathbb{P}=\left\{U=(u,v)\in \mathbb{X}:u^{\prime }\left(t\right)\ge 0,v^{\prime }\left(t\right)\ge 0,u\left(t\right)>0,v\left(t\right)>0\right\}.\end{array}$$

(32)

Lemma 12. 

Define the operator $\mathcal{T}:\mathbb{P}\to \mathbb{X}$ as (25)–(27). If $\left(H1\right)$ and $\left(H2\right)$ hold, then $\mathcal{T}:\mathbb{P}\to \mathbb{P}$ is well defined.

Proof. 

For any $U=(u,v)\in \mathbb{P}$, we derive from (26), (27) and Lemma 10 that $\left({\mathcal{T}}_{i}U\right)\left(t\right)>0$ and ${\left({\mathcal{T}}_{i}U\right)}^{\prime }\left(t\right)\ge 0$, $i=1,2$. That is, $\left(\mathcal{T}U\right)\left(t\right)\in \mathbb{P}$. The proof is completed. □

For convenience, we will make the following preparations. For a fixed $c\in (a,b)$, $U=(u,v)\in \mathbb{P}$, we choose some functionals $\mathcal{A},\mathcal{B},\mathcal{C}:\mathbb{P}\to {\mathbb{R}}_0^{+}$ as follows:

$$\mathcal{A}\left(U\right)=max\left\{\underset{t\in [a,b]}{max}{\displaystyle \frac{u\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\alpha -1}}},\underset{t\in [a,b]}{max}{\displaystyle \frac{v\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\beta -1}}}\right\},$$

$$\mathcal{B}\left(U\right)=min\left\{\underset{t\in [c,b]}{min}{\displaystyle \frac{u\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\alpha -1}}},\underset{t\in [c,b]}{min}{\displaystyle \frac{v\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\beta -1}}}\right\},$$

$$\mathcal{C}\left(U\right)=max\left\{\underset{t\in [a,c]}{max}{\displaystyle \frac{u\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\alpha -1}}},\underset{t\in [a,c]}{max}{\displaystyle \frac{v\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\beta -1}}}\right\}.$$

It is easy to see that $\mathcal{A}$, $\mathcal{C}$ are increasing and continuous with $\mathcal{A}\left(\right(0,0\left)\right)=0$, $\mathcal{B}\left(\right(0,0\left)\right)=0$, $\mathcal{C}\left(\right(0,0\left)\right)=0$, and $\mathcal{B}\left(\kappa U\right)=\kappa \mathcal{B}\left(U\right)$, $\forall 0<\kappa \le 1,U\in \mathbb{P}$. Denote

$$L_i=(\psi \left(b\right)-\psi \left(a\right)){\mathcal{H}}_i^{-1}\left({\int }_a^b{h}_i\left(s\right)ds\right),i=1,2,$$

$$l_1={\mathcal{H}}_1^{-1}\left({\int }_c^b{h}_1\left(s\right)ds\right){\int }_c^b{G}_2(c,s)d\psi \left(s\right),$$

$$l_2={\mathcal{H}}_2^{-1}\left({\int }_c^b{h}_2\left(s\right)ds\right){\int }_c^b{K}_2(c,s)d\psi \left(s\right),$$

$${\lambda }_1=1+{\left(\psi \left({\displaystyle \frac{a+b}{2}}\right)\right)}^{\alpha -1},{\lambda }_2=1+{\left(\psi \left({\displaystyle \frac{a+b}{2}}\right)\right)}^{\beta -1}.$$

Theorem 1. 

Let $0<{\rho }_1,{\rho }_2<1$. Assume that there exists some constants $0<R_1<R_2<R_3$ such that the following assumptions are further fulfilled:

$\left(\mathrm{H}1\right)$

$f_1(t,u,v)<{\mathcal{H}}_1\left({\displaystyle \frac{R_3}{{\Lambda }_1{L}_1}}\right)$, $f_2(t,u,v)<{\mathcal{H}}_2\left({\displaystyle \frac{R_3}{{\Lambda }_2{L}_2}}\right)$, $\forall (t,u,v)\in [a,b]\times [0,R_3]\times [0,R_3];$

$\left(\mathrm{H}2\right)$

$f_1(t,u,v)>{\mathcal{H}}_1\left({\displaystyle \frac{{\lambda }_1{R}_2}{l_1}}\right)$, $f_2(t,u,v)>{\mathcal{H}}_2\left({\displaystyle \frac{{\lambda }_2{R}_2}{l_2}}\right)$, $\forall (t,u,v)\in [c,b]\times [R_2,R_3]\times [R_2,R_3];$

$\left(\mathrm{H}3\right)$

$f_1(t,u,v)<{\mathcal{H}}_1\left({\displaystyle \frac{R_1}{{\Lambda }_1{L}_1}}\right)$, $f_2(t,u,v)<{\mathcal{H}}_2\left({\displaystyle \frac{R_1}{{\Lambda }_2{L}_2}}\right)$, $\forall (t,u,v)\in [a,b]\times [0,R_1]\times [0,R_1].$

Then the nonlinear Hilfer fractional system (3) admits at least three pairs positive solutions $U_i=(u_i,v_i)\in {\overline{\Omega }}_{\mathcal{C},R_3},i=1,2,3$ such that

$$0<\mathcal{A}\left(U_1\right)<R_1<\mathcal{A}\left(U_2\right),\mathcal{B}\left(U_2\right)<R_2<\mathcal{B}\left(U_3\right),\mathcal{C}\left(U_3\right)<R_3.$$

Proof. 

We define the cone $\mathbb{P}$ and the operator $\mathcal{T}:\mathbb{P}\to \mathbb{P}$ as (32) and (25)–(27), respectively. According to Lemma 11, we know that $\mathcal{T}:{\overline{\Omega }}_{\mathcal{C},R_3}\subset \mathbb{P}\to \mathbb{P}$ is completely continuous. For any $U\left(t\right)=\left(u\right(t),v(t\left)\right)\in \mathbb{P}$, we derive from $u^{\prime }\left(t\right),v^{\prime }\left(t\right),{\psi }^{\prime }\left(t\right)\ge 0$ that

$$\begin{array}{c}\hfill \underset{t\in [a,c]}{max}{\displaystyle \frac{u\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\alpha -1}}}\le {\displaystyle \frac{u\left(c\right)}{1+{\left(\psi \left(\frac{a+b}{2}\right)\right)}^{\alpha -1}}}\le \underset{t\in [c,b]}{min}{\displaystyle \frac{u\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\alpha -1}}},\end{array}$$

(33)

and

$$\begin{array}{c}\hfill \underset{t\in [a,c]}{max}{\displaystyle \frac{v\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\beta -1}}}\le {\displaystyle \frac{v\left(c\right)}{1+{\left(\psi \left(\frac{a+b}{2}\right)\right)}^{\beta -1}}}\le \underset{t\in [c,b]}{min}{\displaystyle \frac{v\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\beta -1}}}.\end{array}$$

(34)

Inequalities (33) and (34) indicate that $\mathcal{C}\left(U\right)\le \mathcal{B}\left(U\right)$. Obviously, $\mathcal{B}\left(U\right)\le \mathcal{A}\left(U\right)$. So we have

$$\begin{array}{c}\hfill \mathcal{C}\left(U\right)\le \mathcal{B}\left(U\right)\le \mathcal{A}\left(U\right),\forall U=(u,v)\in \mathbb{P}.\end{array}$$

(35)

Notice that $u\left(t\right),v\left(t\right)>0$ and $u^{\prime }\left(t\right),v^{\prime }\left(t\right)\ge 0$ for all $t\in [a,b]$, then ${max}_{t\in [a,b]}u\left(t\right)=u\left(b\right)$, ${max}_{t\in [a,b]}v\left(t\right)=v\left(b\right)$, $0<{\displaystyle \frac{u\left(a\right)}{u\left(b\right)}}\le 1$ and $0<{\displaystyle \frac{v\left(a\right)}{v\left(b\right)}}\le 1$. Therefore, there exists constants $0<M_1,M_2\le 1$ such that $M_1\le {\displaystyle \frac{u\left(a\right)}{u\left(b\right)}}$, $M_2\le {\displaystyle \frac{v\left(a\right)}{v\left(b\right)}}$. Thus, we have

$$\begin{array}{c}\hfill \underset{t\in [a,c]}{max}{\displaystyle \frac{u\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\alpha -1}}}\ge {\displaystyle \frac{u\left(a\right)}{1+{\left(\psi (c+\frac{b-a}{2})\right)}^{\alpha -1}}}\ge {\displaystyle \frac{M_{1}u\left(b\right)}{1+{\left(\psi (c+\frac{b-a}{2})\right)}^{\alpha -1}}},\end{array}$$

(36)

and

$$\begin{array}{c}\hfill \underset{t\in [a,c]}{max}{\displaystyle \frac{v\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\beta -1}}}\ge {\displaystyle \frac{v\left(a\right)}{1+{\left(\psi (c+\frac{b-a}{2})\right)}^{\beta -1}}}\ge {\displaystyle \frac{M_{2}v\left(b\right)}{1+{\left(\psi (c+\frac{b-a}{2})\right)}^{\beta -1}}}.\end{array}$$

(37)

From (36) and (37), we obtain

$$\begin{array}{c}\hfill \mathcal{C}\left(U\right)\ge M\parallel U\parallel ,M=min\left\{{\displaystyle \frac{M_1}{1+{\left(\psi (c+\frac{b-a}{2})\right)}^{\alpha -1}}},{\displaystyle \frac{M_2}{1+{\left(\psi (c+\frac{b-a}{2})\right)}^{\beta -1}}}\right\}.\end{array}$$

(38)

By (35), (38) and $\mathcal{B}\left(\kappa U\right)=\kappa \mathcal{B}\left(U\right)$, $\forall 0<\kappa \le 1,U\in {\Omega }_{\mathcal{B},R_2}\subset \mathbb{P}$, we assert that the conditions $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$ in Lemma 1 are true. Next, we will continue to verify that condition $\left(\mathrm{iii}\right)$ in Lemma 1 is also true. Indeed, for any $U=(u,v)\in \partial {\Omega }_{\mathcal{C},R_3}$, we have

$$\begin{array}{c}\hfill \mathcal{C}\left(\mathcal{T}U\right)=max\left\{\underset{t\in [a,c]}{max}{\displaystyle \frac{\left({\mathcal{T}}_{1}U\right)\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\alpha -1}}},\underset{t\in [a,c]}{max}{\displaystyle \frac{\left({\mathcal{T}}_{2}U\right)\left(t\right)}{1+{\left(\psi (t+\frac{b-a}{2})\right)}^{\beta -1}}}\right\}.\end{array}$$

(39)

In view of (26), (27), Lemma 8 and $\left(\mathrm{H}1\right)$, we get for $t\in [a,b]$

$$\begin{array}{cc}\hfill \left({\mathcal{T}}_{i}U\right)\left(t\right)<& {\int }_a^b{\Lambda }_i{\mathcal{H}}_i^{-1}\left({\mathcal{H}}_i\left({\displaystyle \frac{R_3}{{\Lambda }_i{L}_i}}\right){\int }_a^b{h}_i\left(z\right)dz\right)d\psi \left(s\right)\hfill \\ \hfill =& {\Lambda }_i·{\displaystyle \frac{R_3}{{\Lambda }_i{L}_i}}·{\mathcal{H}}_i^{-1}\left({\int }_a^b{h}_i\left(z\right)dz\right){\int }_a^{b}d\psi \left(s\right)=R_3,i=1,2.\hfill \end{array}$$

(40)

Combining (39) and (40), we obtain

$$\begin{array}{c}\hfill \mathcal{C}\left(\mathcal{T}U\right)<R_3,\forall U=(u,v)\in \partial {\Omega }_{\mathcal{C},R_3}.\end{array}$$

(41)

Similar to (39)–(41), it follows from (26), (27), Lemma 8 and $\left(\mathrm{H}3\right)$ that

$$\begin{array}{c}\hfill \mathcal{A}\left(\mathcal{T}U\right)<R_1,\forall U=(u,v)\in \partial {\Omega }_{\mathcal{A},R_1}.\end{array}$$

(42)

For any $U=(u,v)\in \partial {\Omega }_{\mathcal{B},R_2}$, we have

$$\begin{array}{c}\hfill \mathcal{B}\left(\mathcal{T}U\right)=min\left\{\underset{t\in [c,b]}{min}{\displaystyle \frac{\left({\mathcal{T}}_{1}U\right)\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\alpha -1}}},\underset{t\in [c,b]}{min}{\displaystyle \frac{\left({\mathcal{T}}_{2}U\right)\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\beta -1}}}\right\}.\end{array}$$

(43)

Noting that $G(c,s)=G_2(c,s)$ and $K(c,s)=K_2(c,s)$ when $c\le s\le b$, we apply (26), (27), Lemma 8 and $\left(\mathrm{H}2\right)$ to obtain for $t\in [c,b]$

$$\begin{array}{cc}\hfill \left({\mathcal{T}}_{1}U\right)\left(t\right)>& {\int }_c^{b}G(c,s){\mathcal{H}}_1^{-1}\left({\mathcal{H}}_1\left({\displaystyle \frac{{\lambda }_1{R}_2}{l_1}}\right){\int }_c^b{h}_1\left(z\right)dz\right)d\psi \left(s\right)\hfill \\ \hfill =& {\displaystyle \frac{{\lambda }_1{R}_2}{l_1}}·{\mathcal{H}}_1^{-1}\left({\int }_a^b{h}_1\left(z\right)dz\right){\int }_c^b{G}_2(c,s)d\psi \left(s\right)={\lambda }_1{R}_2,\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \left({\mathcal{T}}_{2}U\right)\left(t\right)>& {\int }_c^{b}K(c,s){\mathcal{H}}_2^{-1}\left({\mathcal{H}}_2\left({\displaystyle \frac{{\lambda }_2{R}_2}{l_2}}\right){\int }_c^b{h}_2\left(z\right)dz\right)d\psi \left(s\right)\hfill \\ \hfill =& {\displaystyle \frac{{\lambda }_2{R}_2}{l_1}}·{\mathcal{H}}_2^{-1}\left({\int }_a^b{h}_2\left(z\right)dz\right){\int }_c^b{K}_2(c,s)d\psi \left(s\right)={\lambda }_2{R}_2.\hfill \end{array}$$

(45)

From (44) and (45), we have for $t\in [c,b]$

$$\begin{array}{c}\hfill {\displaystyle \frac{\left({\mathcal{T}}_{1}U\right)\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\alpha -1}}}\ge {\displaystyle \frac{\left({\mathcal{T}}_{1}U\right)\left(t\right)}{1+{\left(\psi \left(\frac{a+b}{2}\right)\right)}^{\alpha -1}}}={\displaystyle \frac{\left({\mathcal{T}}_{1}U\right)\left(t\right)}{{\lambda }_1}}>R_2,\end{array}$$

(46)

$$\begin{array}{c}\hfill {\displaystyle \frac{\left({\mathcal{T}}_{2}U\right)\left(t\right)}{1+{\left(\psi (t-\frac{b-a}{2})\right)}^{\beta -1}}}\ge {\displaystyle \frac{\left({\mathcal{T}}_{2}U\right)\left(t\right)}{1+{\left(\psi \left(\frac{a+b}{2}\right)\right)}^{\beta -1}}}={\displaystyle \frac{\left({\mathcal{T}}_{2}U\right)\left(t\right)}{{\lambda }_2}}>R_2.\end{array}$$

(47)

Equations (43), (46) and (47) give that

$$\begin{array}{c}\hfill \mathcal{B}\left(\mathcal{T}U\right)>R_2,\forall U=(u,v)\in \partial {\Omega }_{\mathcal{B},R_2}.\end{array}$$

(48)

Due to (41), (42) and (48), we know that the condition $\left(\mathrm{iii}\right)$ in Lemma 1 also holds. Thus, we conclude from Lemma 1 that system (3) has at least three pairs positive solutions $U_i=(u_i,v_i)\in {\overline{\Omega }}_{\mathcal{C},R_3},i=1,2,3$ satisfying

$$0<\mathcal{A}\left(U_1\right)<R_1<\mathcal{A}\left(U_2\right),\mathcal{B}\left(U_2\right)<R_2<\mathcal{B}\left(U_3\right),\mathcal{C}\left(U_3\right)<R_3.$$

The proof is completed. □

4. Applications

As some important applications, this section mainly discusses the existence of multiple positive solutions for the Caputo and Hadamard fractional coupled Laplacian systems (1) and (2), respectively.

4.1. Multiplicity of Positive Solutions for System (1)

In systems (3), when $\psi \left(t\right)=t$, ${\mathcal{H}}_i\left(w\right)={\varphi }_{p_i}\left(w\right)={\left|w\right|}^{p_i-2}w(p_i>1)$, and $a=0,b=1$, we can obtain the system (1). Correspondingly, we have

$${\widehat{\rho }}_1={\displaystyle \frac{\Gamma \left(n\right)}{\Gamma (n+\mu )}},{\widehat{\rho }}_2={\displaystyle \frac{\Gamma \left(m\right)}{\Gamma (m+\nu )}},{\widehat{\Delta }}_1=1-{\widehat{\rho }}_1,{\widehat{\Delta }}_2=1-{\widehat{\rho }}_2,$$

$$\begin{array}{c}\hfill \widehat{G}(t,s)=\left\{\begin{array}{cc}{\widehat{G}}_1(t,s),\hfill & 0\le s\le t\le 1,\hfill \\ {\widehat{G}}_2(t,s),\hfill & 0\le t\le s\le 1,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\widehat{G}}_1(t,s)=-{\displaystyle \frac{{(t-s)}^{\alpha -1}}{\Gamma \left(\alpha \right)}}+{\displaystyle \frac{t^{n-1}}{{\widehat{\Delta }}_1\Gamma \left(\alpha \right)}}{(1-s)}^{\alpha -1}-{\displaystyle \frac{t^{n-1}}{{\widehat{\Delta }}_1\Gamma (\alpha +\mu )}}{(1-s)}^{\alpha +\mu -1},\end{array}$$

$$\begin{array}{c}\hfill {\widehat{G}}_2(t,s)={\displaystyle \frac{t^{n-1}}{{\widehat{\Delta }}_1\Gamma \left(\alpha \right)}}{(1-s)}^{\alpha -1}-{\displaystyle \frac{t^{n-1}}{{\widehat{\Delta }}_1\Gamma (\alpha +\mu )}}{(1-s)}^{\alpha +\mu -1},\end{array}$$

$$\begin{array}{c}\hfill \widehat{K}(t,s)=\left\{\begin{array}{cc}{\widehat{K}}_1(t,s),\hfill & 0\le s\le t\le 1,\hfill \\ {\widehat{K}}_2(t,s),\hfill & 0\le t\le s\le 1,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\widehat{K}}_1(t,s)=-{\displaystyle \frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}}+{\displaystyle \frac{t^{m-1}}{{\widehat{\Delta }}_2\Gamma \left(\beta \right)}}{(1-s)}^{\beta -1}-{\displaystyle \frac{t^{m-1}}{{\widehat{\Delta }}_2\Gamma (\beta +\nu )}}{(1-s)}^{\beta +\nu -1},\end{array}$$

$$\begin{array}{c}\hfill {\widehat{K}}_2(t,s)={\displaystyle \frac{t^{m-1}}{{\widehat{\Delta }}_2\Gamma \left(\beta \right)}}{(1-s)}^{\beta -1}-{\displaystyle \frac{t^{m-1}}{{\widehat{\Delta }}_2\Gamma (\beta +\nu )}}{(1-s)}^{\beta +\nu -1}.\end{array}$$

Obviously, $0<{\widehat{\rho }}_1,{\widehat{\rho }}_2<1$. From Lemma 8, we know that for all $t,s\in [0,1]$

$${\displaystyle \frac{\partial \widehat{G}(t,s)}{\partial t}},{\displaystyle \frac{\partial \widehat{K}(t,s)}{\partial t}}\ge 0,0\le \widehat{G}(t,s)\le {\displaystyle \frac{1}{{\widehat{\Delta }}_1\Gamma \left(\alpha \right)}}={\widehat{\Lambda }}_1,0\le \widehat{K}(t,s)\le {\displaystyle \frac{1}{{\widehat{\Delta }}_1\Gamma \left(\beta \right)}}={\widehat{\Lambda }}_2.$$

According to Lemma 9, we have the following corollary.

Corollary 1. 

The nonlinear Caputo fractional system (1) is equivalent to the following nonlinear integral system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u\left(t\right)=-{\int }_0^1\widehat{G}(t,s){\varphi }_{q_1}\left(-{\int }_0^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)ds,\hfill \\ v\left(t\right)=-{\int }_0^1\widehat{K}(t,s){\varphi }_{q_2}\left(-{\int }_0^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)ds.\hfill \end{array}\right.\end{array}$$

Define the Banach space $\widehat{\mathbb{X}}=C([0,1],\mathbb{R})\times C([0,1],\mathbb{R})$ and its norm

$$\begin{array}{c}\hfill \parallel U\parallel =\parallel (u,v)\parallel =max\left\{\underset{0\le t\le 1}{max}\left|u\left(t\right)\right|,\underset{0\le t\le 1}{max}\left|v\left(t\right)\right|\right\},\forall U=(u,v)\in \widehat{\mathbb{X}},\end{array}$$

and the cone $\widehat{\mathbb{P}}\subset \widehat{\mathbb{X}}$ as (32). Based on Corollary 1, the operator $\widehat{\mathcal{T}}:\widehat{\mathbb{P}}\to \widehat{\mathbb{P}}$ is defined as

$$\begin{array}{c}\hfill \left(\widehat{\mathcal{T}}U\right)\left(t\right)=(\left({\widehat{\mathcal{T}}}_{1}U\right)\left(t\right),\left({\widehat{\mathcal{T}}}_{2}U\right)\left(t\right)),\forall U=(u,v)\in \widehat{\mathbb{X}},\end{array}$$

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left({\widehat{\mathcal{T}}}_{1}U\right)\left(t\right)=-{\int }_0^1\widehat{G}(t,s){\varphi }_{q_1}\left(-{\int }_0^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)ds,\hfill \\ \left({\widehat{\mathcal{T}}}_{2}U\right)\left(t\right)=-{\int }_0^1\widehat{K}(t,s){\varphi }_{q_2}\left(-{\int }_0^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right)ds.\hfill \end{array}\right.\end{array}$$

For a fixed $c\in (0,1)$, $U=(u,v)\in \widehat{\mathbb{P}}$, some functionals $\widehat{\mathcal{A}},\widehat{\mathcal{B}},\widehat{\mathcal{C}}:\widehat{\mathbb{P}}\to {\mathbb{R}}_0^{+}$ are similarly defined as

$$\widehat{\mathcal{A}}\left(U\right)=max\left\{\underset{t\in [0,1]}{max}{\displaystyle \frac{u\left(t\right)}{1+{(t-\frac{1}{2})}^{\alpha -1}}},\underset{t\in [0,1]}{max}{\displaystyle \frac{v\left(t\right)}{1+{(t-\frac{1}{2})}^{\beta -1}}}\right\},$$

$$\widehat{\mathcal{B}}\left(U\right)=min\left\{\underset{t\in [c,1]}{min}{\displaystyle \frac{u\left(t\right)}{1+{(t-\frac{1}{2})}^{\alpha -1}}},\underset{t\in [c,1]}{min}{\displaystyle \frac{v\left(t\right)}{1+{(t-\frac{1}{2})}^{\beta -1}}}\right\},$$

$$\widehat{\mathcal{C}}\left(U\right)=max\left\{\underset{t\in [0,c]}{max}{\displaystyle \frac{u\left(t\right)}{1+{(t+\frac{1}{2})}^{\alpha -1}}},\underset{t\in [0,c]}{max}{\displaystyle \frac{v\left(t\right)}{1+{(t+\frac{1}{2})}^{\beta -1}}}\right\}.$$

Some constants are introduced as

$${\widehat{L}}_i={\varphi }_{q_i}\left({\int }_0^1{h}_i\left(s\right)ds\right),i=1,2,$$

$${\widehat{l}}_1={\varphi }_{q_1}\left({\int }_c^1{h}_1\left(s\right)ds\right){\int }_c^1{\widehat{G}}_2(c,s)ds,$$

$${\widehat{l}}_2={\varphi }_{q_2}\left({\int }_c^1{h}_2\left(s\right)ds\right){\int }_c^1{\widehat{K}}_2(c,s)ds,$$

$${\widehat{\lambda }}_1=1+{\left({\displaystyle \frac{1}{2}}\right)}^{\alpha -1},{\widehat{\lambda }}_2=1+{\left({\displaystyle \frac{1}{2}}\right)}^{\beta -1}.$$

Similar to the proof process of Theorem 1, we can obtain the following main conclusions.

Theorem 2. 

The nonlinear Caputo fractional system (1) has at least three pairs positive solutions $U_i=(u_i,v_i)\in {\overline{\Omega }}_{\widehat{\mathcal{C}},{\widehat{R}}_3},i=1,2,3$ satisfying

$$0<\widehat{\mathcal{A}}\left(U_1\right)<{\widehat{R}}_1<\widehat{\mathcal{A}}\left(U_2\right),\widehat{\mathcal{B}}\left(U_2\right)<{\widehat{R}}_2<\widehat{\mathcal{B}}\left(U_3\right),\widehat{\mathcal{C}}\left(U_3\right)<{\widehat{R}}_3,$$

provided that there exists some constants $0<{\widehat{R}}_1<{\widehat{R}}_2<{\widehat{R}}_3$ such that

$\left(\widehat{\mathrm{H}1}\right)$

$f_1(t,u,v)<{\varphi }_{q_1}\left({\displaystyle \frac{{\widehat{R}}_3}{{\widehat{\Lambda }}_1{\widehat{L}}_1}}\right)$, $f_2(t,u,v)<{\varphi }_{q_2}\left({\displaystyle \frac{{\widehat{R}}_3}{{\widehat{\Lambda }}_2{\widehat{L}}_2}}\right)$, $\forall (t,u,v)\in [0,1]\times [0,{\widehat{R}}_3]\times [0,{\widehat{R}}_3];$

$\left(\widehat{\mathrm{H}2}\right)$

$f_1(t,u,v)>{\varphi }_{q_1}\left({\displaystyle \frac{{\widehat{\lambda }}_1{\widehat{R}}_2}{{\widehat{l}}_1}}\right)$, $f_2(t,u,v)>{\varphi }_{q_2}\left({\displaystyle \frac{{\widehat{\lambda }}_2{\widehat{R}}_2}{{\widehat{l}}_2}}\right)$, $\forall (t,u,v)\in [c,1]\times [{\widehat{R}}_2,{\widehat{R}}_3]\times [{\widehat{R}}_2,{\widehat{R}}_3];$

$\left(\widehat{\mathrm{H}3}\right)$

$f_1(t,u,v)<{\varphi }_{q_1}\left({\displaystyle \frac{{\widehat{R}}_1}{{\widehat{\Lambda }}_1{\widehat{L}}_1}}\right)$, $f_2(t,u,v)<{\varphi }_{q_2}\left({\displaystyle \frac{{\widehat{R}}_1}{{\widehat{\Lambda }}_2{\widehat{L}}_2}}\right)$, $\forall (t,u,v)\in [0,1]\times [0,{\widehat{R}}_1]\times [0,{\widehat{R}}_1]$.

4.2. Multiplicity of Positive Solutions for System (2)

When $\psi \left(t\right)=log\left(t\right)$, ${\mathcal{H}}_i\left(w\right)={\varphi }_{p_i}\left(w\right)={\left|w\right|}^{p_i-2}w(p_i>1)$, and $a=1,b=e$, system (3) becomes system (2). We correspondingly arrive at the following:

$${\tilde{\rho }}_1={\displaystyle \frac{\Gamma \left(n\right)}{\Gamma (n+\mu )}},{\tilde{\rho }}_2={\displaystyle \frac{\Gamma \left(m\right)}{\Gamma (m+\nu )}},{\tilde{\Delta }}_1=1-{\tilde{\rho }}_1,{\tilde{\Delta }}_2=1-{\tilde{\rho }}_2,$$

$$\begin{array}{c}\hfill \tilde{G}(t,s)=\left\{\begin{array}{cc}{\tilde{G}}_1(t,s),\hfill & 1\le s\le t\le e,\hfill \\ {\tilde{G}}_2(t,s),\hfill & 1\le t\le s\le e,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tilde{G}}_1(t,s)=-{\displaystyle \frac{{(log\frac{t}{s})}^{\alpha -1}}{\Gamma \left(\alpha \right)}}+{\displaystyle \frac{{(logt)}^{n-1}}{{\tilde{\Delta }}_1\Gamma \left(\alpha \right)}}{(1-logs)}^{\alpha -1}-{\displaystyle \frac{{(logt)}^{n-1}}{{\tilde{\Delta }}_1\Gamma (\alpha +\mu )}}{(1-logs)}^{\alpha +\mu -1},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{G}}_2(t,s)={\displaystyle \frac{{(logt)}^{n-1}}{{\tilde{\Delta }}_1\Gamma \left(\alpha \right)}}{(1-logs)}^{\alpha -1}-{\displaystyle \frac{{(logt)}^{n-1}}{{\tilde{\Delta }}_1\Gamma (\alpha +\mu )}}{(1-logs)}^{\alpha +\mu -1},\end{array}$$

$$\begin{array}{c}\hfill \tilde{K}(t,s)=\left\{\begin{array}{cc}{\tilde{K}}_1(t,s),\hfill & 1\le s\le t\le e,\hfill \\ {\tilde{K}}_2(t,s),\hfill & 1\le t\le s\le e,\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill {\tilde{K}}_1(t,s)=-{\displaystyle \frac{{(log\frac{t}{s})}^{\beta -1}}{\Gamma \left(\beta \right)}}+{\displaystyle \frac{{(logt)}^{m-1}}{{\tilde{\Delta }}_2\Gamma \left(\beta \right)}}{(1-logs)}^{\beta -1}-{\displaystyle \frac{{(logt)}^{m-1}}{{\tilde{\Delta }}_2\Gamma (\beta +\nu )}}{(1-logs)}^{\beta +\nu -1},\end{array}$$

$$\begin{array}{c}\hfill {\tilde{K}}_2(t,s)={\displaystyle \frac{{(logt)}^{m-1}}{{\tilde{\Delta }}_2\Gamma \left(\beta \right)}}{(1-logs)}^{\beta -1}-{\displaystyle \frac{{(logt)}^{m-1}}{{\tilde{\Delta }}_2\Gamma (\beta +\nu )}}{(1-logs)}^{\beta +\nu -1}.\end{array}$$

Clearly, $0<{\tilde{\rho }}_1,{\tilde{\rho }}_2<1$. It follows from Lemma 8 that for all $t,s\in [1,e]$

$${\displaystyle \frac{\partial \tilde{G}(t,s)}{\partial t}},{\displaystyle \frac{\partial \tilde{K}(t,s)}{\partial t}}\ge 0,0\le \tilde{G}(t,s)\le {\displaystyle \frac{1}{{\tilde{\Delta }}_1\Gamma \left(\alpha \right)}}={\tilde{\Lambda }}_1,0\le \tilde{K}(t,s)\le {\displaystyle \frac{1}{{\tilde{\Delta }}_1\Gamma \left(\beta \right)}}={\tilde{\Lambda }}_2.$$

We derive the following corollary from Lemma 9.

Corollary 2. 

The nonlinear Hadamard fractional system (2) is equivalent to the following nonlinear integral system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u\left(t\right)=-{\int }_1^e\tilde{G}(t,s){\varphi }_{q_1}\left(-{\int }_1^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right){\displaystyle \frac{ds}{s}},\hfill \\ v\left(t\right)=-{\int }_1^e\tilde{K}(t,s){\varphi }_{q_2}\left(-{\int }_1^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}\right.\end{array}$$

Define the Banach space $\tilde{\mathbb{X}}=C([1,e],\mathbb{R})\times C([1,e],\mathbb{R})$ with the norm

$$\begin{array}{c}\hfill \parallel U\parallel =\parallel (u,v)\parallel =max\left\{\underset{1\le t\le e}{max}\left|u\left(t\right)\right|,\underset{1\le t\le e}{max}\left|v\left(t\right)\right|\right\},\forall U=(u,v)\in \tilde{\mathbb{X}},\end{array}$$

and the cone $\tilde{\mathbb{P}}\subset \tilde{\mathbb{X}}$ as (32). Based on Corollary 2, the operator $\tilde{\mathcal{T}}:\tilde{\mathbb{P}}\to \tilde{\mathbb{P}}$ is given by

$$\begin{array}{c}\hfill \left(\tilde{\mathcal{T}}U\right)\left(t\right)=(\left({\tilde{\mathcal{T}}}_{1}U\right)\left(t\right),\left({\tilde{\mathcal{T}}}_{2}U\right)\left(t\right)),\forall U=(u,v)\in \tilde{\mathbb{X}},\end{array}$$

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left({\tilde{\mathcal{T}}}_{1}U\right)\left(t\right)=-{\int }_1^e\tilde{G}(t,s){\varphi }_{q_1}\left(-{\int }_1^s{h}_1\left(z\right)f_1(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right){\displaystyle \frac{ds}{s}},\hfill \\ \left({\tilde{\mathcal{T}}}_{2}U\right)\left(t\right)=-{\int }_1^e\tilde{K}(t,s){\varphi }_{q_2}\left(-{\int }_1^s{h}_2\left(z\right)f_2(z,u\left({\tau }_1\left(z\right)\right),v\left({\tau }_2\left(z\right)\right))dz\right){\displaystyle \frac{ds}{s}}.\hfill \end{array}\right.\end{array}$$

For a fixed $c\in (1,e)$, $U=(u,v)\in \tilde{\mathbb{P}}$, some functionals $\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{\mathcal{C}}:\tilde{\mathbb{P}}\to {\mathbb{R}}_0^{+}$ are similarly defined as

$$\tilde{\mathcal{A}}\left(U\right)=max\left\{\underset{t\in [1,e]}{max}{\displaystyle \frac{u\left(t\right)}{1+{(log(t-\frac{e-1}{2}))}^{\alpha -1}}},\underset{t\in [1,e]}{max}{\displaystyle \frac{v\left(t\right)}{1+{(log(t-\frac{e-1}{2}))}^{\beta -1}}}\right\},$$

$$\tilde{\mathcal{B}}\left(U\right)=min\left\{\underset{t\in [c,e]}{min}{\displaystyle \frac{u\left(t\right)}{1+{(log(t-\frac{e-1}{2}))}^{\alpha -1}}},\underset{t\in [c,e]}{min}{\displaystyle \frac{v\left(t\right)}{1+{(log(t-\frac{e-1}{2}))}^{\beta -1}}}\right\},$$

$$\tilde{\mathcal{C}}\left(U\right)=max\left\{\underset{t\in [1,c]}{max}{\displaystyle \frac{u\left(t\right)}{1+{(log(t+\frac{e+1}{2}))}^{\alpha -1}}},\underset{t\in [1,c]}{max}{\displaystyle \frac{v\left(t\right)}{1+{(log(t+\frac{e+1}{2}))}^{\beta -1}}}\right\}.$$

Some constants are given as follows:

$${\tilde{L}}_i={\varphi }_{q_i}\left({\int }_1^e{h}_i\left(s\right)ds\right),i=1,2,$$

$${\tilde{l}}_1={\varphi }_{q_1}\left({\int }_c^e{h}_1\left(s\right)ds\right){\int }_c^e{\tilde{G}}_2(c,s)ds,$$

$${\tilde{l}}_2={\varphi }_{q_2}\left({\int }_c^e{h}_2\left(s\right)ds\right){\int }_c^e{\tilde{K}}_2(c,s)ds,$$

$${\tilde{\lambda }}_1=1+{\left(log{\displaystyle \frac{e+1}{2}}\right)}^{\alpha -1},{\tilde{\lambda }}_2=1+{\left(log{\displaystyle \frac{e+1}{2}}\right)}^{\beta -1}.$$

We can prove the following important results by using the same method as Theorem 1.

Theorem 3. 

The nonlinear Hadamard fractional system (2) has at least three pairs positive solutions $U_i=(u_i,v_i)\in {\overline{\Omega }}_{\tilde{\mathcal{C}},{\tilde{R}}_3},i=1,2,3$ satisfying

$$0<\tilde{\mathcal{A}}\left(U_1\right)<{\tilde{R}}_1<\tilde{\mathcal{A}}\left(U_2\right),\tilde{\mathcal{B}}\left(U_2\right)<{\tilde{R}}_2<\tilde{\mathcal{B}}\left(U_3\right),\tilde{\mathcal{C}}\left(U_3\right)<{\tilde{R}}_3,$$

provided that there exists some constants $0<{\tilde{R}}_1<{\tilde{R}}_2<{\tilde{R}}_3$ such that

$\left(\widetilde{\mathrm{H}1}\right)$

$f_1(t,u,v)<{\varphi }_{q_1}\left({\displaystyle \frac{{\tilde{R}}_3}{{\tilde{\Lambda }}_1{\tilde{L}}_1}}\right)$, $f_2(t,u,v)<{\varphi }_{q_2}\left({\displaystyle \frac{{\tilde{R}}_3}{{\tilde{\Lambda }}_2{\tilde{L}}_2}}\right)$, $\forall (t,u,v)\in [1,e]\times [0,{\tilde{R}}_3]\times [0,{\tilde{R}}_3];$

$\left(\widetilde{\mathrm{H}2}\right)$

$f_1(t,u,v)>{\varphi }_{q_1}\left({\displaystyle \frac{{\tilde{\lambda }}_1{\tilde{R}}_2}{{\tilde{l}}_1}}\right)$, $f_2(t,u,v)>{\varphi }_{q_2}\left({\displaystyle \frac{{\tilde{\lambda }}_2{\tilde{R}}_2}{{\tilde{l}}_2}}\right)$, $\forall (t,u,v)\in [c,e]\times [{\tilde{R}}_2,{\tilde{R}}_3]\times [{\tilde{R}}_2,{\tilde{R}}_3];$

$\left(\widetilde{\mathrm{H}3}\right)$

$f_1(t,u,v)<{\varphi }_{q_1}\left({\displaystyle \frac{{\tilde{R}}_1}{{\tilde{\Lambda }}_1{\tilde{L}}_1}}\right)$, $f_2(t,u,v)<{\varphi }_{q_2}\left({\displaystyle \frac{{\tilde{R}}_1}{{\tilde{\Lambda }}_2{\tilde{L}}_2}}\right)$, $\forall (t,u,v)\in [1,e]\times [0,{\tilde{R}}_1]\times [0,{\tilde{R}}_1]$.

5. Illustrative Examples

This section provides three interesting examples to illustrate the correctness and validity of our main Theorems 1, 2 and 3, respectively.

Consider the following boundary value problem of the fractional coupled system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\left[{\mathcal{H}}_1\left({\mathcal{D}}_{0^{+}}^{{\displaystyle \frac{5}{2}},\psi }u\left(t\right)\right)\right]}^{\prime }=h_1\left(t\right)f_1(t,u(sint),v(cost)),a<t<b,\hfill \\ {\left[{\mathcal{H}}_2\left({\mathcal{D}}_{0^{+}}^{{\displaystyle \frac{7}{2}},\psi }v\left(t\right)\right)\right]}^{\prime }=h_2\left(t\right)f_2(t,u(sint),v(cost)),a<t<b,\hfill \\ u\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\displaystyle \frac{3}{2}},\psi }u\left(a\right)=0,u\left(b\right)={\mathcal{J}}_{a^{+}}^{{\displaystyle \frac{1}{3}},\psi }u\left(b\right),\hfill \\ v\left(a\right)=0,{\mathcal{D}}_{a^{+}}^{{\displaystyle \frac{5}{4}},\psi }v\left(a\right)={\mathcal{D}}_{a^{+}}^{{\displaystyle \frac{9}{4}},\psi }v\left(a\right)=0,v\left(b\right)={\mathcal{J}}_{a^{+}}^{{\displaystyle \frac{4}{3}},\psi }v\left(b\right),\hfill \end{array}\right.\end{array}$$

(49)

where $\alpha ={\displaystyle \frac{5}{2}}$, $\beta ={\displaystyle \frac{7}{2}}$, ${\alpha }_1={\displaystyle \frac{3}{2}}$, ${\beta }_1={\displaystyle \frac{5}{4}}$, ${\beta }_2={\displaystyle \frac{9}{4}}$, $\mu ={\displaystyle \frac{1}{3}}$, $\nu ={\displaystyle \frac{4}{3}}$, ${\tau }_1\left(t\right)=sint$, ${\tau }_2\left(t\right)=cost$. Then $n=3$ and $m=4$.

Case 1. Take $\psi \left(t\right)=e^t$, then the system (49) becomes a $e^t$-Hilfer fractional differential coupled system. We select $a=0$, $b=1$, $c={\displaystyle \frac{1}{2}}$, ${\mathcal{H}}_1\left(w\right)=w^3$, ${\mathcal{H}}_2\left(w\right)=w^2$, $h_1\left(t\right)=t^2$, $h_2\left(t\right)=t^3$, and

$$f_1(t,u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{u^2+v^2}{4(1+u^2+v^2)}},\hfill & (t,u,v)\in [0,1]\times [0,10]\times [0,10],\hfill \\ {\displaystyle \frac{u^2+v^2}{4(1+u^2+v^2)}}[1+{10}^9(arctan{\displaystyle \frac{u+v}{20}}-{\displaystyle \frac{\pi }{4}})],\hfill & (t,u,v)\in [0,1]\times [10,20]\times [10,20],\hfill \\ 8.0337\times {10}^7(1+arctan{\displaystyle \frac{u+v}{40}}-{\displaystyle \frac{\pi }{4}}),\hfill & (t,u,v)\in [0,1]\times [20,\infty )\times [20,\infty ),\hfill \end{array}\right.$$

$$f_2(t,u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{u+v}{1+u+v}},\hfill & (t,u,v)\in [0,1]\times [0,10]\times [0,10],\hfill \\ {\displaystyle \frac{u+v}{5(1+u+v)}}[1+{10}^9(arctan{\displaystyle \frac{u+v}{20}}-{\displaystyle \frac{\pi }{4}})],\hfill & (t,u,v)\in [0,1]\times [10,20]\times [10,20],\hfill \\ 6.2781\times {10}^7(1+arctan{\displaystyle \frac{u+v}{40}}-{\displaystyle \frac{\pi }{4}}),\hfill & (t,u,v)\in [0,1]\times [20,\infty )\times [20,\infty ).\hfill \end{array}\right.$$

Obviously, ${\mathcal{H}}_1,{\mathcal{H}}_2:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ are two positive increasing homomorphisms with ${\mathcal{H}}_1^{-1}\left(w\right)=\sqrt[3]{w}$, ${\mathcal{H}}_2^{-1}\left(w\right)=\sqrt{w}$. After calculations, we obtain

$${\rho }_1={\displaystyle \frac{\Gamma \left(3\right)}{\Gamma \left(\frac{10}{3}\right)}}{(e-1)}^{{\displaystyle \frac{1}{3}}}\approx 0.8623<1,{\Delta }_1=(1-{\rho }_1){(e-1)}^2\approx 0.4066,$$

$${\rho }_2={\displaystyle \frac{\Gamma \left(4\right)}{\Gamma \left(\frac{16}{3}\right)}}{(e-1)}^{{\displaystyle \frac{4}{3}}}\approx 0.3077<1,{\Delta }_2=(1-{\rho }_2){(e-1)}^3\approx 3.5122,$$

$${\Lambda }_1={\displaystyle \frac{{(e-1)}^{\frac{3}{2}}}{(1-{\rho }_1)\Gamma \left(\frac{5}{2}\right)}}\approx 12.3047,{\Lambda }_2={\displaystyle \frac{{(e-1)}^{\frac{5}{2}}}{(1-{\rho }_2)\Gamma \left(\frac{7}{2}\right)}}\approx 1.6822,$$

$$L_1={\displaystyle \frac{e-1}{\sqrt[3]{3}}}\approx 1.1914,L_2={\displaystyle \frac{e-1}{2}}\approx 0.8591,$$

$${\lambda }_1=1+e^{{\displaystyle \frac{3}{4}}}\approx 3.1170,{\lambda }_2=1+e^{{\displaystyle \frac{5}{4}}}\approx 4.4903,$$

$$l_1=\sqrt[3]{{\displaystyle \frac{7}{24}}}\left[{\displaystyle \frac{{(\sqrt{e}-1)}^2{(e-1)}^{\frac{5}{2}}}{{\Delta }_1\Gamma \left(\frac{7}{2}\right)}}-{\displaystyle \frac{{(\sqrt{e}-1)}^2{(e-1)}^{\frac{17}{6}}}{{\Delta }_1\Gamma \left(\frac{23}{6}\right)}}\right]\approx 0.1482,$$

$$l_2={\displaystyle \frac{\sqrt{15}}{8}}\left[{\displaystyle \frac{{(\sqrt{e}-1)}^3{(e-1)}^{\frac{7}{2}}}{{\Delta }_2\Gamma \left(\frac{9}{2}\right)}}-{\displaystyle \frac{{(\sqrt{e}-1)}^3{(e-1)}^{\frac{29}{6}}}{{\Delta }_2\Gamma \left(\frac{35}{6}\right)}}\right]\approx 0.0158.$$

Choose $R_1=10$, $R_2=20$ and $R_3=2\times {10}^5$, then we derive

$$\begin{array}{c}\hfill f_1(t,u,v)\le f_1(t,10,10)\approx 0.2488<{\mathcal{H}}_1\left({\displaystyle \frac{R_1}{{\Lambda }_1{L}_1}}\right)\approx 0.3174,u,v\in [0,R_1],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\le f_2(t,10,10)\approx 0.9524<{\mathcal{H}}_2\left({\displaystyle \frac{R_1}{{\Lambda }_2{L}_2}}\right)\approx 47.8803,u,v\in [0,R_1],\end{array}$$

$$\begin{array}{cc}\hfill f_1(t,u,v)\ge & f_1(t,20,20)\approx 8.0337\times {10}^7\hfill \\ \hfill >& {\mathcal{H}}_1\left({\displaystyle \frac{{\lambda }_1{R}_2}{l_1}}\right)\approx 7.4431\times {10}^7,u,v\in [R_2,R_3],\hfill \end{array}$$

$$\begin{array}{cc}\hfill f_2(t,u,v)\ge & f_2(t,20,20)\approx 6.2781\times {10}^7\hfill \\ \hfill >& {\mathcal{H}}_2\left({\displaystyle \frac{{\lambda }_2{R}_2}{l_2}}\right)\approx 3.2307\times {10}^7,u,v\in [R_2,R_3],\hfill \end{array}$$

$$\begin{array}{cc}\hfill f_1(t,u,v)\le & 8.0337\times {10}^7(1+{\displaystyle \frac{\pi }{4}})\approx 1.4343\times {10}^8\hfill \\ \hfill <& {\mathcal{H}}_1\left({\displaystyle \frac{R_3}{{\Lambda }_1{L}_1}}\right)\approx 2.5392\times {10}^{12},u,v\in [0,R_3],\hfill \end{array}$$

$$\begin{array}{cc}\hfill f_2(t,u,v)\le & 6.2781\times {10}^7(1+{\displaystyle \frac{\pi }{4}})\approx 1.1209\times {10}^8\hfill \\ \hfill <& {\mathcal{H}}_2\left({\displaystyle \frac{R_3}{{\Lambda }_2{L}_2}}\right)\approx 1.9152\times {10}^{10},u,v\in [0,R_3].\hfill \end{array}$$

So, conditions $\left(\mathrm{H}1\right)$, $\left(\mathrm{H}2\right)$ and $\left(\mathrm{H}3\right)$ are all true. Therefore, it is known from Theorem 1 that the $e^t$-Hilfer fractional system (49) has at least three pairs of positive solutions in Case 1.

Case 2. Take $\psi \left(t\right)=t$, $a=0$, $b=1$, ${\mathcal{H}}_1\left(w\right)={\varphi }_{{\displaystyle \frac{3}{2}}}\left(w\right)$ and ${\mathcal{H}}_2\left(w\right)={\varphi }_{{\displaystyle \frac{4}{3}}}\left(w\right)$, then the system (49) is changed into the Caputo fractional differential coupled Laplacian system. Let $c={\displaystyle \frac{\pi }{6}}$, $h_1\left(t\right)={\displaystyle \frac{1+cost}{3}}$, $h_2\left(t\right)={\displaystyle \frac{|sint|}{5}}$ and

$$f_1(t,u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{{(tu+v)}^2}{10}},\hfill & (t,u,v)\in [0,1]\times [0,1]\times [0,1],\hfill \\ 0.4+40[{(u-1)}^2+{(v-1)}^2],\hfill & (t,u,v)\in [0,1]\times [1,2]\times [1,2],\hfill \\ 80.4+{10}^{-9}[{(u-2)}^2+{(v-2)}^2],\hfill & (t,u,v)\in [0,1]\times [2,\infty )\times [2,\infty ),\hfill \end{array}\right.$$

$$f_2(t,u,v)=\left\{\begin{array}{cc}2\sqrt{t}u+3v,\hfill & (t,u,v)\in [0,1]\times [0,1]\times [0,1],\hfill \\ 5+250(u+v-2),\hfill & (t,u,v)\in [0,1]\times [1,2]\times [1,2],\hfill \\ 505+{10}^{-4}(u+v-4),\hfill & (t,u,v)\in [0,1]\times [2,\infty )\times [2,\infty ).\hfill \end{array}\right.$$

It is easy to know that ${\mathcal{H}}_1^{-1}\left(w\right)={\varphi }_3\left(w\right)$, ${\mathcal{H}}_2^{-1}\left(w\right)={\varphi }_4\left(w\right)$. It is computed to obtain

$${\widehat{\rho }}_1={\displaystyle \frac{\Gamma \left(3\right)}{\Gamma \left(\frac{10}{3}\right)}}\approx 0.7199<1,{\widehat{\Delta }}_1=1-{\widehat{\rho }}_1\approx 0.2801,$$

$${\widehat{\rho }}_2={\displaystyle \frac{\Gamma \left(4\right)}{\Gamma \left(\frac{16}{3}\right)}}\approx 0.1495<1,{\widehat{\Delta }}_2=1-{\widehat{\rho }}_2\approx 0.8505,$$

$${\widehat{\Lambda }}_1={\displaystyle \frac{1}{{\widehat{\Delta }}_1\Gamma \left(\frac{5}{2}\right)}}\approx 2.6857,{\widehat{\Lambda }}_2={\displaystyle \frac{1}{{\widehat{\Delta }}_2\Gamma \left(\frac{7}{2}\right)}}\approx 0.3538,$$

$${\widehat{L}}_1={\left({\displaystyle \frac{1+sin1}{3}}\right)}^2\approx 0.3768,{\widehat{L}}_2={\left({\displaystyle \frac{1-cos1}{5}}\right)}^3\approx 7.7715\times {10}^{-4},$$

$${\widehat{\lambda }}_1=1+{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{3}{2}}}\approx 1.3536,{\widehat{\lambda }}_2=1+{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{5}{2}}}\approx 1.1768,$$

$${\widehat{l}}_1={\left({\displaystyle \frac{\frac{1}{2}+sin1-\frac{\pi }{6}}{3}}\right)}^3\left[{\displaystyle \frac{{\left(\frac{\pi }{6}\right)}^2{(1-\frac{\pi }{6})}^{\frac{5}{2}}}{{\widehat{\Delta }}_1\Gamma \left(\frac{7}{2}\right)}}-{\displaystyle \frac{{\left(\frac{\pi }{6}\right)}^2{(1-\frac{\pi }{6})}^{\frac{17}{6}}}{{\widehat{\Delta }}_1\Gamma \left(\frac{23}{6}\right)}}\right]\approx 4.3824\times {10}^{-4},$$

$${\widehat{l}}_2={\left({\displaystyle \frac{\frac{\sqrt{3}}{2}-cos1}{5}}\right)}^4\left[{\displaystyle \frac{{\left(\frac{\pi }{6}\right)}^3{(1-\frac{\pi }{6})}^{\frac{7}{2}}}{{\widehat{\Delta }}_2\Gamma \left(\frac{9}{2}\right)}}-{\displaystyle \frac{{\left(\frac{\pi }{6}\right)}^3{(1-\frac{\pi }{6})}^{\frac{29}{6}}}{{\widehat{\Delta }}_2\Gamma \left(\frac{35}{6}\right)}}\right]\approx 1.8570\times {10}^{-8}.$$

Choose ${\widehat{R}}_1=1$, ${\widehat{R}}_2=2$ and ${\widehat{R}}_3=4\times {10}^4$, then we have

$$\begin{array}{c}\hfill f_1(t,u,v)\le f_1(1,1,1)=0.4<{\varphi }_{{\displaystyle \frac{3}{2}}}\left({\displaystyle \frac{{\widehat{R}}_1}{{\widehat{\Lambda }}_1{\widehat{L}}_1}}\right)\approx 0.9941,u,v\in [0,{\widehat{R}}_1],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\le f_2(1,1,1)=5<{\varphi }_{{\displaystyle \frac{4}{3}}}\left({\displaystyle \frac{{\widehat{R}}_1}{{\widehat{\Lambda }}_2{\widehat{L}}_2}}\right)\approx 15.3784,u,v\in [0,{\widehat{R}}_1],\end{array}$$

$$\begin{array}{c}\hfill f_1(t,u,v)\ge f_1(t,2,2)=80.4>{\varphi }_{{\displaystyle \frac{3}{2}}}\left({\displaystyle \frac{{\widehat{\lambda }}_1{\widehat{R}}_2}{{\widehat{l}}_1}}\right)\approx 78.5967,u,v\in [{\widehat{R}}_2,{\widehat{R}}_3],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\ge f_2(t,2,2)=505>{\varphi }_{{\displaystyle \frac{4}{3}}}\left({\displaystyle \frac{{\widehat{\lambda }}_2{\widehat{R}}_2}{{\widehat{l}}_2}}\right)\approx 502.3120,u,v\in [{\widehat{R}}_2,{\widehat{R}}_3],\end{array}$$

$$\begin{array}{c}\hfill f_1(t,u,v)\le f_1(t,{\widehat{R}}_3,{\widehat{R}}_3)\approx 83.5997<{\varphi }_{{\displaystyle \frac{3}{2}}}\left({\displaystyle \frac{{\widehat{R}}_3}{{\widehat{\Lambda }}_1{\widehat{L}}_1}}\right)\approx 198.8135,u,v\in [0,{\widehat{R}}_3],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\le f_2(t,{\widehat{R}}_3,{\widehat{R}}_3)\approx 512.9996<{\varphi }_{{\displaystyle \frac{4}{3}}}\left({\displaystyle \frac{{\widehat{R}}_3}{{\widehat{\Lambda }}_2{\widehat{L}}_2}}\right)\approx 525.9354,u,v\in [0,{\widehat{R}}_3].\end{array}$$

Thus, conditions $\left(\widehat{\mathrm{H}1}\right)$, $\left(\widehat{\mathrm{H}2}\right)$ and $\left(\widehat{\mathrm{H}3}\right)$ hold. From Theorem 2, we know that the Caputo fractional differential coupled Laplacian system (49) has at least three pairs of positive solutions in Case 2.

Case 3. Take $\psi \left(t\right)=logt$, $a=1$, $b=e$, ${\mathcal{H}}_1\left(w\right)={\varphi }_{{\displaystyle \frac{5}{3}}}\left(w\right)$ and ${\mathcal{H}}_2\left(w\right)={\varphi }_{{\displaystyle \frac{7}{4}}}\left(w\right)$, then the system (49) is converted to the Hadamard fractional differential coupled Laplacian system. Let $c=2$, $h_1\left(t\right)={\displaystyle \frac{1}{1+t}}$, $h_2\left(t\right)={\displaystyle \frac{2}{t^2}}$ and

$$f_1(t,u,v)=\left\{\begin{array}{cc}{\displaystyle \frac{{\left(t^{-1}u\right)}^2+v^2}{16}},\hfill & (t,u,v)\in [1,e]\times [0,4]\times [0,4],\hfill \\ 2+14[{(u-4)}^2+{(v-4)}^2],\hfill & (t,u,v)\in [1,e]\times [4,8]\times [4,8],\hfill \\ 450+{10}^{-7}[{(u-8)}^2+{(v-8)}^2],\hfill & (t,u,v)\in [1,e]\times [8,\infty )\times [8,\infty ),\hfill \end{array}\right.$$

$$f_2(t,u,v)=\left\{\begin{array}{cc}\sqrt{{\displaystyle \frac{u}{t}}}+\sqrt{v},\hfill & (t,u,v)\in [1,e]\times [0,4]\times [0,4],\hfill \\ 4+1600(\sqrt{u-4}+\sqrt{v-4}),\hfill & (t,u,v)\in [1,e]\times [4,8]\times [4,8],\hfill \\ 6404+{\displaystyle \frac{1}{4}}(\sqrt{u-8}+\sqrt{v-8}),\hfill & (t,u,v)\in [1,e]\times [8,\infty )\times [8,\infty ).\hfill \end{array}\right.$$

Apparently, ${\mathcal{H}}_1^{-1}\left(w\right)={\varphi }_{{\displaystyle \frac{5}{2}}}\left(w\right)$, ${\mathcal{H}}_2^{-1}\left(w\right)={\varphi }_{{\displaystyle \frac{7}{3}}}\left(w\right)$. By some computations, we obtain

$${\tilde{\rho }}_1={\displaystyle \frac{\Gamma \left(3\right)}{\Gamma \left(\frac{10}{3}\right)}}\approx 0.7199<1,{\tilde{\Delta }}_1=1-{\widehat{\rho }}_1\approx 0.2801,$$

$${\tilde{\rho }}_2={\displaystyle \frac{\Gamma \left(4\right)}{\Gamma \left(\frac{16}{3}\right)}}\approx 0.1495<1,{\tilde{\Delta }}_2=1-{\widehat{\rho }}_2\approx 0.8505,$$

$${\tilde{\Lambda }}_1={\displaystyle \frac{1}{{\widehat{\Delta }}_1\Gamma \left(\frac{5}{2}\right)}}\approx 2.6857,{\tilde{\Lambda }}_2={\displaystyle \frac{1}{{\widehat{\Delta }}_2\Gamma \left(\frac{7}{2}\right)}}\approx 0.3538,$$

$${\tilde{L}}_1={\left(log{\displaystyle \frac{1+e}{2}}\right)}^{{\displaystyle \frac{3}{2}}}\approx 0.4883,{\tilde{L}}_2={\left(2-{\displaystyle \frac{2}{e}}\right)}^{{\displaystyle \frac{4}{3}}}\approx 1.3670,$$

$${\tilde{\lambda }}_1=1+{\left(log{\displaystyle \frac{e+1}{2}}\right)}^{{\displaystyle \frac{3}{2}}}\approx 1.4883,{\tilde{\lambda }}_2=1+{\left(log{\displaystyle \frac{e+1}{2}}\right)}^{{\displaystyle \frac{5}{2}}}\approx 1.3028,$$

$${\tilde{l}}_1={\left(log{\displaystyle \frac{1+e}{3}}\right)}^{{\displaystyle \frac{3}{2}}}\left[{\displaystyle \frac{{(log2)}^2{(1-log2)}^{\frac{5}{2}}}{{\tilde{\Delta }}_1\Gamma \left(\frac{7}{2}\right)}}-{\displaystyle \frac{{(log2)}^2{(1-log2)}^{\frac{17}{6}}}{{\tilde{\Delta }}_1\Gamma \left(\frac{23}{6}\right)}}\right]\approx 0.0014,$$

$${\tilde{l}}_2={\left(1-{\displaystyle \frac{2}{e}}\right)}^{{\displaystyle \frac{4}{3}}}\left[{\displaystyle \frac{{(log2)}^3{(1-log2)}^{\frac{7}{2}}}{{\tilde{\Delta }}_2\Gamma \left(\frac{9}{2}\right)}}-{\displaystyle \frac{{(log2)}^3{(1-log2)}^{\frac{29}{6}}}{{\tilde{\Delta }}_2\Gamma \left(\frac{35}{6}\right)}}\right]\approx 8.8929\times {10}^{-5}.$$

Choose ${\tilde{R}}_1=4$, ${\tilde{R}}_2=8$ and ${\tilde{R}}_3=6\times {10}^4$, then we get

$$\begin{array}{c}\hfill f_1(t,u,v)\le f_1(1,4,4)=2<{\varphi }_{{\displaystyle \frac{5}{3}}}\left({\displaystyle \frac{{\tilde{R}}_1}{{\tilde{\Lambda }}_1{\tilde{L}}_1}}\right)\approx 2.1032,u,v\in [0,{\tilde{R}}_1],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\le f_2(1,4,4)=4<{\varphi }_{{\displaystyle \frac{7}{4}}}\left({\displaystyle \frac{{\tilde{R}}_1}{{\tilde{\Lambda }}_2{\tilde{L}}_2}}\right)\approx 4.8770,u,v\in [0,{\tilde{R}}_1],\end{array}$$

$$\begin{array}{c}\hfill f_1(t,u,v)\ge f_1(t,8,8)=450>{\varphi }_{{\displaystyle \frac{5}{3}}}\left({\displaystyle \frac{{\tilde{\lambda }}_1{\tilde{R}}_2}{{\tilde{l}}_1}}\right)\approx 416.6470,u,v\in [{\tilde{R}}_2,{\tilde{R}}_3],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\ge f_2(t,8,8)=6.404\times {10}^3>{\varphi }_{{\displaystyle \frac{7}{4}}}\left({\displaystyle \frac{{\tilde{\lambda }}_2{\tilde{R}}_2}{{\tilde{l}}_2}}\right)\approx 6.3997\times {10}^3,u,v\in [{\tilde{R}}_2,{\tilde{R}}_3],\end{array}$$

$$\begin{array}{c}\hfill f_1(t,u,v)\le f_1(t,{\tilde{R}}_3,{\tilde{R}}_3)\approx 1.1698\times {10}^3<{\varphi }_{{\displaystyle \frac{5}{3}}}\left({\displaystyle \frac{{\tilde{R}}_3}{{\tilde{\Lambda }}_1{\tilde{L}}_1}}\right)\approx 1.5205\times {10}^3,u,v\in [0,{\tilde{R}}_3],\end{array}$$

$$\begin{array}{c}\hfill f_2(t,u,v)\le f_2(t,{\tilde{R}}_3,{\tilde{R}}_3)\approx 6.5265\times {10}^3<{\varphi }_{{\displaystyle \frac{7}{4}}}\left({\displaystyle \frac{{\tilde{R}}_3}{{\tilde{\Lambda }}_2{\tilde{L}}_2}}\right)\approx 6.6103\times {10}^3,u,v\in [0,{\tilde{R}}_3].\end{array}$$

In this way, we have verified the truth of conditions $\left(\widetilde{\mathrm{H}1}\right)$, $\left(\widetilde{\mathrm{H}2}\right)$ and $\left(\widetilde{\mathrm{H}3}\right)$. We conclude from Theorem 3 that the Hadamard fractional differential coupled Laplacian system (49) has at least three pairs of positive solutions in Case 3.

6. Conclusions

In the manuscript, we emphasize the existence of multiple positive solutions for the higher-order Caputo fractional Laplacian system (1) and the Hadamard fractional Laplacian system (2). Using the $\psi$-Hilfer fractional derivative is sufficient to study the unified system (3). We construct the Green function corresponding to system (3) and prove some of its important properties. On this basis, the multiplicity of the positive solution of system (3) is studied by constructing a positive cone and applying the cone fixed point theorem (see Theorem 1). When $\psi \left(t\right)=t$ and $\psi \left(t\right)=logt$, we apply Theorem 1 to built the multiplicity of systems (1) and (2) (see Theorems 2 and 3). We share some examples to verify the reliability of our main results. Our sufficient criteria are concise, novel, and not affected by time-varying delays. In addition, we are going to study the multiple solutions of more complicated fractional equations with the $\psi$-Hilfer pseudo-fractional derivative and the $(k,\psi )$-Hilfer fractional derivative in the future.