Multiple Positive Solutions for a Coupled System of Fractional Order BVP with p-Laplacian Operators and Parameters

1. Introduction

The study of nonlinear fractional systems is currently a topic of significant interest, with researchers dedicating substantial efforts to understanding these systems and their applications in various fields. These endeavors are primarily driven by the desire to utilize the findings regarding the existence of positive solutions in practical contexts. This expansion of differential calculus into the realm of complex real-world dynamics has led to the development and evaluation of novel theories based on empirical data [1].

A wide range of materials and processes that exhibit hereditary and memory characteristics find accurate descriptions through the nonlocal nature of fractional calculus (FC) [2,3]. The implications of these findings span across diverse scientific disciplines, including biomathematics [4], viscoelasticity [5], non-Newtonian fluid mechanics [6], and the characterization of anomalous diffusion [7].

The literature in the field of fractional differential equations (FDEs) predominantly centers on established concepts like Riemann-Liouville or Caputo derivatives, with some emerging alternatives such as conformable fractional derivatives [8]. However, it is worth noting that the body of work on FDEs of the conformable type is still evolving. The concept of conformable fractional derivatives was first introduced in 2014 and has gained attention for its computational advantages in solving differential equations. This approach has led to more efficient numerical solutions compared to Riemann-Liouville or Caputo fractional derivatives. Recently, a new formulation of the conformable fractional derivative has been proposed and extensively discussed in the literature [8,9,10]. Researchers have successfully applied conformable fractional derivatives to a wide range of domains, resulting in the establishment of various replicable methodologies [11].

Beyond the realm of mathematical theory, applications of these findings extend to diverse industries, including telecommunication equipment, synthetic chemicals, automobiles, and pharmaceuticals, where boundary value problems (BVPs) play a significant role. In these industrial processes, positive solutions are often highly desirable for practical reasons, as highlighted in recent research [12,13].

In this study, our primary objective is to investigate the following system of fractional differential equations that incorporate $(r_1,r_2,r_3)$-Laplacian operators. We aim to provide a comprehensive analysis of these equations, considering their potential implications and applications:

$$\left\{\begin{array}{c}-{\mathcal{D}}_{h^{+}}^{{\mathit{p}}_1}\left({\varphi }_{{\mathit{r}}_1}\left({\mathcal{D}}_{h^{+}}^{{\mathit{q}}_1}\varpi \left(\xi \right)\right)\right)={\mathit{f}}_1(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right)),\xi \in (h,k),\hfill \\ -{\mathcal{D}}_{h^{+}}^{{\mathit{p}}_2}\left({\varphi }_{{\mathit{r}}_2}\left({\mathcal{D}}_{h^{+}}^{{\mathit{q}}_2}\vartheta \left(\xi \right)\right)\right)={\mathit{f}}_2(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right)),\xi \in (h,k),\hfill \\ -{\mathcal{D}}_{h^{+}}^{{\mathit{p}}_3}\left({\varphi }_{{\mathit{r}}_3}\left({\mathcal{D}}_{h^{+}}^{{\mathit{q}}_3}\omega \left(\xi \right)\right)\right)={\mathit{f}}_3(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right)),\xi \in (h,k),\hfill \end{array}\right.$$

(1)

where h and k are real numbers with $h<k$. The operators $\mathcal{D}{h^{+}}^{qi},\mathcal{D}{h^{+}}^{pi},\mathcal{D}{h^{+}}^{\alpha i}$ correspond to standard Riemann-Liouville fractional order derivatives. Additionally, $q_i\in (1,2]$, $p_i$, ${\alpha }_i\in (0,1]$, and ${\varphi }_{r_i}\left(\zeta \right)={\left|\zeta \right|}^{r_i-1}\zeta$, with $r_i>1$, and ${\varphi }_{r_i}^{-1}={\varphi }_{{\phi }_i}$, where $\frac{1}{{\phi }_i}+\frac{1}{r_i}=1$ for $i=1,2,3$.

The boundary conditions for this system are given as:

$$\left\{\begin{array}{cc}\hfill & \varpi \left(h\right)=0,{\varphi }_{{\mathit{r}}_1}\left({\mathcal{D}}_{h^{+}}^{{\mathit{q}}_1}\varpi \left(h\right)\right)=0,{\mu }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left(k\right)={\psi }_1+{\lambda }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left({\eta }_1\right),\hfill \\ \hfill & \vartheta \left(h\right)=0,{\varphi }_{{\mathit{r}}_2}\left({\mathcal{D}}_{h^{+}}^{q_2}\vartheta \left(h\right)\right)=0,{\mu }_2{\mathcal{D}}_{h^{+}}^{{\alpha }_2}\vartheta \left(k\right)={\psi }_2+{\lambda }_2{\mathcal{D}}_{h^{+}}^{{\alpha }_2}\vartheta \left({\eta }_2\right),\hfill \\ \hfill & \omega \left(h\right)=0,{\varphi }_{{\mathit{r}}_3}\left({\mathcal{D}}_{h^{+}}^{q_3}\omega \left(h\right)\right)=0,{\mu }_3{\mathcal{D}}_{h^{+}}^{{\alpha }_3}\omega \left(k\right)={\psi }_3+{\lambda }_3{\mathcal{D}}_{h^{+}}^{{\alpha }_3}\omega \left({\eta }_3\right),\hfill \end{array}\right.$$

(2)

Here, ${\mu }_i$, ${\lambda }_i$ are positive constants, and ${\eta }_i$ are real numbers within the interval $(h,k)$. It is essential that the conditions ${\mu }_i{(k-h)}^{q_i-{\alpha }_i-1}>{\lambda }_i{({\eta }_i-h)}^{q_i-{\alpha }_i-1}$ hold for all $i=1,2,3$.

To ensure the existence of positive solutions to the system (1)-(2), we make the following assumptions:

(B1)

The functions $f_1$, $f_2$, and $f_3$ are continuous on the specified domains.

(B2)

The parameters ${\alpha }_i$, $q_i$, ${\mu }_i$, ${\lambda }_i$, and ${\eta }_i$ satisfy certain inequalities, ensuring the conditions required for the existence of solutions.

(B3)

We introduce positive constants ${\mathsf{\Phi }}_1$, ${\mathsf{\Phi }}_2$, ${\mathsf{\Phi }}_3$, ${\mathsf{\Theta }}_1$, ${\mathsf{\Theta }}_2$, and ${\mathsf{\Theta }}_3$ with the constraint that $\frac{1}{{\mathsf{\Phi }}_1}+\frac{1}{{\mathsf{\Phi }}_2}+\frac{1}{{\mathsf{\Phi }}_3}+\frac{1}{{\mathsf{\Theta }}_1}+\frac{1}{{\mathsf{\Theta }}_2}+\frac{1}{{\mathsf{\Theta }}_3}\le 1$.

The study of fractional differential equations is a rapidly expanding field with numerous applications in various domains. Our paper provides essential conditions for the functions $f_1$, $f_2$, and $f_3$, as well as intervals for the parameters $({\psi }_1,{\psi }_2,{\psi }_3)$, guaranteeing the existence of at least one and three positive solutions for the specified boundary value problem (1)-(2). A positive solution is defined as a triplet of functions $\left(\varpi \right(\xi ),\vartheta (\xi ),\omega (\xi \left)\right)$ in the space ${(\mathcal{C}[h,k],[0,\infty ))}^3$ that satisfies (1)-(2) with non-negative values for all $\xi \in [h,k]$, and where $\left(\varpi \right(\xi ),\vartheta (\xi ),\omega (\xi \left)\right)$ is not equal to $(0,0,0)$.

For further insights into the applications of fractional calculus in various fields and related literature on positive solutions with different boundary conditions, we recommend reading the referenced books [14,15,16] and exploring the cited papers [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35].

The subsequent sections of this paper are organized as follows: Section 2 introduces foundational concepts and key lemmas essential for underpinning our central results. Moving on to Section 3, we employ various methodological approaches, including cone expansion and compression of functional type, along with the utilization of the Leggett-Williams fixed point theorem, to expound upon our primary findings. Finally, in Section 4, we provide practical insights by presenting two illustrative examples that effectively demonstrate the application and relevance of our core discoveries.

2. Preliminaries and lemmas

We refrain from including definitions and valuable lemmas pertaining to fractional calculus theory in this section, as they can be readily referenced in contemporary literature [2,3].

Definition 1. 

For a function f given on the interval $[h,k]$, the ${\alpha }^{th}$ Riemann-Liouville fractional order derivative of f is defined by

$$\left(D_{h^{+}}^{\alpha }f\right)\left(\xi \right)=\frac{1}{\mathsf{\Gamma }(n-\alpha )}{\left(\frac{d}{d\xi }\right)}^{\alpha }{\int }_h^{\xi }{(\xi -s)}^{n-\alpha -1}f\left(s\right)ds,$$

here $n=\left[\alpha \right]+1$ and $\left[\alpha \right]$ denotes the integral part of α.

Definition 2. 

The functional (arbitrary) order integral of the function $f\in L^1\left([h,k],R_{+}\right)$ of order $\alpha \in R_{+}$ is defined by

$$\left(I_{h^{+}}^{\alpha }f\right)\left(\xi \right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_h^{\xi }{(\xi -s)}^{\alpha -1}f\left(s\right)ds,$$

where Γ is the Gamma function.

Lemma 1. 

Assume that ${\mathcal{D}}_{h^{+}}^{\sigma }\in L^1[h,k]$ with a fractional derivative of order $\sigma >0$ then

$${\mathcal{I}}_{h^{+}}^{\sigma }{\mathcal{D}}_{h^{+}}^{\sigma }u\left(t\right)=u\left(\xi \right)+c_1{(\xi -h)}^{\sigma -1}+c_2{(\xi -h)}^{\sigma -2}+\cdots \cdots +c_n{(\xi -h)}^{\sigma -n}$$

for some $c_i\in R,i=1,2,3,\cdots ,n$ where n is the smallest integer greater than or equal to $\sigma .$

Definition 3. 

Let φ be a cone in the real Banach space $\mathcal{S}$. A map $\sigma :\phi \to [0,\infty )$ is said to be nonnegative continuous concave functional on φ if σ is continuous and $\sigma (\lambda u+(1-\lambda \left)v\right)\ge \lambda \sigma \left(u\right)+(1-\lambda )\sigma \left(v\right)$ for all $u,v\in \phi$ and $\lambda \in [0,1].$

Definition 4. 

Let φ be a cone in the real Banach space $\mathcal{S}$. A map $\rho :\phi \to [0,\infty )$ is said to be nonnegative continuous convex functional on φ if ρ is continuous and $\rho (\lambda u+(1-\lambda \left)v\right)\le \lambda \rho \left(u\right)+(1-\lambda )\rho \left(v\right)$ for all $u,v\in \phi$ and $\lambda \in [0,1].$

$\mathbf{Rule}{\mathbf{S}}_{\mathbf{1}}$: Let $\kappa$ be a cone in a Banach space $\mathcal{D}$ and x be a bounded open subset of $\mathcal{D}$ and $0\in x$. Then a continuous functional $\sigma :\kappa \to [0,\infty )$ is said to satisfy Rule $S_1$ if one of the following conditions holds:

(i)

$\sigma$ is convex, $\sigma \left(0\right)=0,\sigma \left(t\right)\ne 0$ if $t\ne 0$ and ${inf}_{t\in \kappa \cap \partial x}\sigma \left(t\right)>0$,

(ii)

$\sigma$ is sublinear, $\sigma \left(0\right)=0,\sigma \left(t\right)\ne 0$ if $t\ne 0$ and ${inf}_{t\in \kappa \cap \partial x}\sigma \left(t\right)>0$,

(iii)

$\sigma$ is concave and unbounded.

$\mathbf{Rule}{\mathbf{S}}_{\mathbf{2}}$: Let $\kappa$ be a cone in a Banach space $\mathcal{D}$ and x be a bounded open subset of $\mathcal{D}$ and $0\in x$. Then a continuous functional $\rho :\kappa \to [0,\infty )$ is said to satisfy Rule ${\mathcal{S}}_2$ if one of the following conditions holds:

(i)

$\rho$ is convex, $\rho \left(0\right)=0,\rho \left(t\right)\ne 0$ if $t\ne 0$,

(ii)

$\rho$ is sublinear, $\rho \left(0\right)=0,\rho \left(t\right)\ne 0$ if $t\ne 0$,

(iii)

$\rho (t+s)\ge \rho \left(t\right)+\rho \left(s\right)$ for all $t,s\in \kappa ,\rho \left(0\right)=0,\rho \left(t\right)\ne 0$ if $t\ne 0.$

Theorem 1. 

[17] Consider two bounded open subsets, ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$, within a Banach space denoted as $\mathcal{D}$. It is assumed that 0 belongs to ${\mathsf{\Omega }}_1$, and ${\mathsf{\Omega }}_1$ is a subset of ${\mathsf{\Omega }}_2$. Furthermore, let κ represent a cone within the same Banach space $\mathcal{D}$. We introduce an operator $\mathcal{L}$, which maps from $\kappa \cap ({\overline{\mathsf{\Omega }}}_2\setminus {\mathsf{\Omega }}_1)$ to κ and is characterized as completely continuous. Alongside this, two non-negative continuous functionals, σ and ρ, are defined on κ. The main result is contingent upon one of the following two conditions being satisfied:

(a) σ adheres to Rule $\mathcal{S}1$ with $\sigma \left(\mathcal{L}t\right)\ge \sigma \left(t\right)$ for all t belonging to $\kappa \cap \partial \mathsf{\Omega }1$, and ρ adheres to Rule ${\mathcal{S}}_2$ with $\rho \left(\mathcal{L}t\right)\le \rho \left(t\right)$ for all t in $\kappa \cap \partial \mathsf{\Omega }2$.

(b) Conversely, ρ follows Rule ${\mathcal{S}}_2$ with $\rho \left(\mathcal{L}t\right)\le \rho \left(t\right)$ for all t in $\kappa \cap \partial \mathsf{\Omega }1$, and σ conforms to Rule ${\mathcal{S}}_1$ with $\sigma \left(\mathcal{L}t\right)\ge \sigma \left(t\right)$ for all t in $\kappa \cap \partial \mathsf{\Omega }2$.

In either case, the conclusion is that the operator $\mathcal{L}$ possesses at least one fixed point within the set $\kappa \cap ({\overline{\mathsf{\Omega }}}_2\setminus {\mathsf{\Omega }}_1).$

Theorem 2.(Leggett-Williams [18]) Let $p,q,r$ and s be positive real numbers, let κ be a cone in a real Banach space $\mathcal{D}$, ${\kappa }_s=\{t\in \kappa :\parallel t\parallel <s\}$, ψ be a nonnegative continuous concave functional on κ such that $\psi \left(t\right)\le \parallel t\parallel ,\forall t\in \overline{{\kappa }_s}$ and $\kappa (\psi ,q,r)=\{t\in \kappa ;q\le \psi (t),\parallel t\parallel \le r\}$. Suppose $\mathcal{L}:\overline{{\kappa }_s}\to \overline{{\kappa }_s}$ be a completely continuous operator and there exist constants $0<p<q<r\le s$ such that

(i)

$\{t\in \kappa (\psi ,q,r)\mid \psi (t)>q\}\ne \varnothing$ and $\psi \left(\mathcal{L}t\right)>q$ for $t\in \kappa (\psi ,q,r)$,

(ii)

$\parallel \mathcal{L}t\parallel <p$ for $\parallel t\parallel \le p$,

(iii)

$\psi \left(\mathcal{L}t\right)>q$ for $t\in \kappa (\psi ,q,s)$ with $\parallel \mathcal{L}t\parallel >r.$

Then $\mathcal{L}$ has at least three fixed points $t_1,t_2$ and $t_3$ in $\overline{{\kappa }_s}$ satisfying $\parallel t_1\parallel <p,q<\psi \left(t_2\right),p<\parallel t_3\parallel$ and $\psi \left(t_3\right)<q.$

In what follows, we calculate the Green’s function associate with (1)-(2). Consider the homogeneous boundary value problem:

$$-{\mathcal{D}}_{h^{+}}^{{\mathit{q}}_1}\varpi \left(\xi \right)=0,\xi \in (h,k),$$

(3)

$$\varpi \left(h\right)=0;{\mu }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left(k\right)={\psi }_1+{\lambda }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left({\eta }_1\right).$$

(4)

Lemma 2. 

Let ${\Delta }_1\ne 0.$ If $x\left(\xi \right)\in \mathcal{C}[h,k]$ and $1<{\mathit{q}}_1\le 2$, then the boundary value problem:

$${\mathcal{D}}_{h^{+}}^{{\mathit{q}}_1}\varpi \left(\xi \right)+x\left(\xi \right)=0,h<\xi <k,$$

(5)

satisfying the boundary condition (4), has a unique solution

$$\varpi \left(\xi \right)={\int }_h^k{\mathcal{H}}_1(\xi ,\zeta )x\left(\zeta \right)d\zeta +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1},\xi \in [h,k],$$

where ${\mathcal{H}}_1(\xi ,\zeta )$ is the Green’s function for the BVP (5)-(4) and is given by

$${\mathcal{H}}_1(\xi ,\zeta )=h_1(\xi ,\zeta )+\frac{{\lambda }_1{(\xi -h)}^{q_1-1}}{{\mathcal{N}}_1}{h}_2({\eta }_1,\zeta ).$$

Here ${\Delta }_1=\mathsf{\Gamma }\left(q_1\right){\mathcal{N}}_1\ne 0;{\mathcal{N}}_1={\mu }_1{(k-h)}^{q_1-{\alpha }_1-1}-{\lambda }_1{({\eta }_1-h)}^{q_1-{\alpha }_1-1}$ and

$$\begin{array}{cc}\hfill h_1(\xi ,\zeta )& =\frac{1}{\mathsf{\Gamma }\left(q_1\right)}\left\{\begin{array}{c}\frac{{(\xi -h)}^{q_1-1}{(k-\zeta )}^{q_1-{\alpha }_1-1}}{{(k-h)}^{q_1-{\alpha }_1-1}}-{(\xi -\zeta )}^{q_1-1},h\le \zeta \le \xi \le k,\hfill \\ \frac{{(\xi -h)}^{q_1-1}{(k-\zeta )}^{q_1-{\alpha }_1-1}}{{(k-h)}^{q_1-{\alpha }_1-1}},h\le \xi \le \zeta \le k,\hfill \end{array}\right.\hfill \\ \hfill h_2(\xi ,\zeta )& =\frac{1}{\mathsf{\Gamma }\left(q_1\right)}\left\{\begin{array}{c}\frac{{(\xi -h)}^{q_1-{\alpha }_1-1}{(k-\zeta )}^{q_1-{\alpha }_1-1}}{{(k-h)}^{q_1-{\alpha }_1-1}}-{(\xi -\zeta )}^{q_1-{\alpha }_1-1},h\le \zeta \le \xi \le k,\hfill \\ \frac{{(\xi -h)}^{q_1-{\alpha }_1-1}{(k-\zeta )}^{q_1-{\alpha }_1-1}}{{(k-h)}^{q_1-{\alpha }_1-1}},h\le \xi \le \zeta \le k.\hfill \end{array}\right.\hfill \end{array}$$

(6)

Proof. 

Assume that $\varpi \in C^{\left[q_1\right]+1}[h,k]$ is a solution of fractional order boundary value problem (5)-(4) and is uniquely expressed by

$$\varpi \left(\xi \right)=-{\int }_h^{\xi }\frac{{(\xi -\zeta )}^{q_1-1}}{\mathsf{\Gamma }\left(q_1\right)}x\left(\zeta \right)d\zeta +c_1{(\xi -h)}^{q_1-1}+c_2{(\xi -h)}^{q_1-2}.$$

In view of conditions (4), one can get $c_2=0$ and

$$c_1=\frac{1}{{\Delta }_1}\left[{\mu }_1{\int }_h^k{(k-h)}^{q_1-{\alpha }_1-1}x\left(\zeta \right)d\zeta -{\lambda }_1{\int }_h^{{\eta }_1}{({\eta }_1-\zeta )}^{q_1-{\alpha }_1-1}x\left(\zeta \right)d\zeta \right]+\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1)}{{\Delta }_1}.$$

Hence, we have

$$\begin{array}{cc}\hfill \varpi \left(\xi \right)& =\frac{-1}{\mathsf{\Gamma }\left({\alpha }_1\right)}{\int }_h^{\xi }{(\xi -\zeta )}^{q_1-1}x\left(\zeta \right)d\zeta +\frac{{(\xi -h)}^{q_1-1}}{{\Delta }_1}{\int }_h^k{(k-\zeta )}^{q_1-{\mu }_1-1}x\left(\zeta \right)d\zeta \hfill \\ & -\frac{{\lambda }_1{(\xi -h)}^{q_1-1}}{{\Delta }_1}{\int }_h^{{\eta }_1}{({\eta }_1-\zeta )}^{q_1-{\alpha }_1-1}x\left(\zeta \right)d\zeta +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1}\hfill \\ & =\frac{-1}{\mathsf{\Gamma }\left(q_1\right)}{\int }_h^{\xi }{(\xi -\zeta )}^{q_1-1}x\left(\zeta \right)d\zeta +\frac{{(\xi -h)}^{q_1-1}}{\mathsf{\Gamma }\left(q_1\right)}{\int }_h^k\frac{{(k-\zeta )}^{q_1-{\alpha }_1-1}}{{(k-h)}^{q_1-{\alpha }_1-1}}x\left(\zeta \right)d\zeta \hfill \\ & +\frac{{\lambda }_1{(\xi -h)}^{q_1-1}}{N}{\int }_h^k\frac{{({\eta }_1-h)}^{q_1-{\alpha }_1-1}{(k-\zeta )}^{q-1-{\alpha }_1-1}}{\mathsf{\Gamma }\left(q_1\right){(k-h)}^{q_1-{\alpha }_1-1}}x\left(\zeta \right)d\zeta \hfill \\ & -\frac{{\lambda }_1{(\xi -h)}^{q_1-1}}{N}{\int }_h^{{\eta }_1}\frac{{({\eta }_1-\zeta )}^{q_1-{\alpha }_1-1}}{\mathsf{\Gamma }\left(q_1\right)}x\left(\zeta \right)d\zeta +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1}\hfill \\ & ={\int }_h^k{\mathcal{H}}_1(\xi ,\zeta )x\left(\zeta \right)d\zeta +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1}.\hfill \end{array}$$

□

Lemma 3. 

Let $1<{\mathit{q}}_1\le 2,0<{\mathit{p}}_1\le 1.$ Then the FBVP

$$\left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{h^{+}}^{{\mathit{p}}_1}\left({\varphi }_{{\mathit{r}}_1}\left({\mathcal{D}}_{h^{+}}^{{\mathit{q}}_1}\varpi \left(\xi \right)\right)\right)+{\mathit{f}}_1(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right))=0,h<\xi <k,\hfill \\ \hfill & \varpi \left(h\right)=0,{\mathcal{D}}_{h^{+}}^{q_1}\varpi \left(h\right)=0,{\mu }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left(k\right)={\psi }_1+{\lambda }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left({\eta }_1\right),\hfill \end{array}\right.$$

(7)

has a unique solution,

$$\begin{array}{cc}\hfill \varpi \left(\xi \right)& ={\int }_h^k{\mathcal{H}}_1(\xi ,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{{\mathit{p}}_1-1}}{\mathsf{\Gamma }\left({\mathit{p}}_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }({\mathit{q}}_1-{\alpha }_1){(\xi -h)}^{{\mathit{q}}_1-1}}{{\Delta }_1},\xi \in [h,k].\hfill \end{array}$$

Proof. 

It follows from Lemma 1 and $0<{\mathit{p}}_1\le 1$ that

$${\varphi }_{r_1}\left({\mathcal{D}}_{h^{+}}^{q_1}\varpi \left(\xi \right)\right)=-{\int }_h^{\xi }\frac{{(\xi -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{f}_1(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right))d\tau +c_1{(\xi -h)}^{{\mathit{p}}_1-1}.$$

By ${\mathcal{D}}_{h^{+}}^{q_1}\varpi \left(h\right)=0,$ we have $c_1=0$. So,

$${\mathcal{D}}_{h^{+}}^{q_1}\varpi \left(\xi \right)+{\varphi }_{{\phi }_1}\left({\int }_h^t\frac{{(\xi -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{f}_1(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right))d\tau \right)=0.$$

Thus, the BVP (7) is equal to the following problem:

$${\mathcal{D}}_{h^{+}}^{q_1}\varpi \left(\xi \right)+{\varphi }_{{\phi }_1}\left({\int }_h^{\xi }\frac{{(\xi -\tau )}^{p_1-1}}{\mathsf{\Gamma }(p_1)}{f}_1(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right))d\tau \right)=0;h<\xi <k;$$

$$\varpi \left(h\right)=0;{\mu }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left(k\right)={\psi }_1+{\lambda }_1{\mathcal{D}}_{h^{+}}^{{\alpha }_1}\varpi \left({\eta }_1\right).$$

By Lemma 2, that boundary value problem (7) has a unique solution

$$\begin{array}{cc}\hfill \varpi \left(\xi \right)& ={\int }_h^k{\mathcal{H}}_1(\xi ,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{{\mathit{p}}_1-1}}{\mathsf{\Gamma }\left({\mathit{p}}_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }({\mathit{q}}_1-{\alpha }_1){(\xi -h)}^{{\mathit{q}}_1-1}}{{\Delta }_1},\xi \in [h,k].\hfill \end{array}$$

□

Lemma 4. 

[25] Suppose that condition $\left(B2\right)$ hold, then Green’s function ${\mathcal{H}}_1$ have the following properties:

$\left(i\right)$${\mathcal{H}}_1(\xi ,\zeta )\ge 0$ for all $(\xi ,\zeta )\in (h,k)\times (h,k),$

$\left(ii\right)$${\mathcal{H}}_1(\xi ,\zeta )\le {\mathcal{H}}_1(k,\zeta ),$ for all $(\xi ,\zeta )\in [h,k]\times [h,k],$

$\left(iii\right)$${\mathcal{H}}_1(\xi ,\zeta )\ge {\left(\frac{1}{4}\right)}^{q_1-1}{\mathcal{H}}_1(k,\zeta ),$ for all $(\xi ,\zeta )\in \mathcal{I}\times (h,k),$ where $\mathcal{I}=\left[\frac{3h+k}{4},\frac{h+3k}{4}\right].$

Remark 1. 

In a similar manner, the results of the Green’s function ${\mathcal{H}}_2(\xi ,\zeta )$ and ${\mathcal{H}}_3(\xi ,\zeta )$ for the homogeneous BVP corresponding to the fractional differential equation are obtained. Consider the following condition:

$${\mathcal{H}}_i(\xi ,\zeta )\ge \aleph {\mathcal{H}}_i(k,\zeta )\mathrm{for}\mathrm{all}(\xi ,\zeta )\in \mathcal{I}\times (h,k);i=1,2,3,$$

where $\mathcal{I}=\left[\frac{3h+k}{4},\frac{h+3k}{4}\right]$ and $\aleph =min\left\{{\left(\frac{1}{4}\right)}^{q_1-1},{\left(\frac{1}{4}\right)}^{q_2-1},{\left(\frac{1}{4}\right)}^{q_3-1}\right\}.$

We consider the Banach space $\mathcal{X}=\mathcal{C}[h,k]$ with the supremum norm $\parallel ·\parallel$ and the Banach space $\mathcal{Y}=\mathcal{X}\times \mathcal{X}\times \mathcal{X}$ with the norm $\parallel (\varpi ,\vartheta ,\omega )\parallel =\parallel \varpi \parallel +\parallel \vartheta \parallel +\parallel \omega \parallel .$ We define the cone

$$\begin{array}{cc}\hfill \mathcal{P}& =\left\{\right(\varpi ,\vartheta ,\omega )\in \mathcal{Y}:\varpi (\xi ),\vartheta (\xi ),\omega (\xi )\ge 0,\forall \xi \in [h,k],\mathrm{and}\hfill \\ & \underset{\xi \in \mathcal{I}}{min}[\varpi \left(\xi \right)+\vartheta \left(\xi \right)+\omega \left(\xi \right)]\ge \aleph \Vert (\varpi ,\vartheta ,\omega )\Vert \},\hfill \end{array}$$

where $\mathcal{I}=\left[\frac{3h+k}{4},\frac{h+3k}{4}\right]$ and $\aleph =min\left\{{\left(\frac{1}{4}\right)}^{q_1-1},{\left(\frac{1}{4}\right)}^{q_2-1},{\left(\frac{1}{4}\right)}^{q_3-1}\right\}$.

Consider the coupled system of integral equations

$$\begin{array}{cc}\hfill \varpi \left(\xi \right)& ={\int }_h^k{\mathcal{H}}_1(\xi ,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1},\xi \in [h,k],\hfill \\ \hfill \vartheta \left(\xi \right)& ={\int }_h^k{\mathcal{H}}_2(\xi ,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}{\mathit{f}}_2\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_2\mathsf{\Gamma }(q_2-{\alpha }_2){(\xi -h)}^{q_2-1}}{{\Delta }_2},\xi \in [h,k],\hfill \\ \hfill \omega \left(\xi \right)& ={\int }_h^k{\mathcal{H}}_3(\xi ,\zeta ){\varphi }_{{\phi }_3}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}{\mathit{f}}_3\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_3\mathsf{\Gamma }(q_3-{\alpha }_3){(\xi -h)}^{q_3-1}}{{\Delta }_3},\xi \in [h,k].\hfill \end{array}$$

By Lemma 2, $(\varpi ,\vartheta ,\omega )\in \mathcal{P}$ is a solution of boundary value problems (1)-(2) if and only if it is a solution of the system of integral equations.

Define the operators ${\mathcal{T}}_1,{\mathcal{T}}_2,{\mathcal{T}}_3:\mathcal{P}\to \mathcal{X}$ by

$$\begin{array}{cc}\hfill {\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\left(\xi \right)& ={\int }_h^k{\mathcal{H}}_1(\xi ,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1},\xi \in [h,k],\hfill \\ \hfill {\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\left(\xi \right)& ={\int }_h^k{\mathcal{H}}_2(\xi ,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}{\mathit{f}}_2\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_2\mathsf{\Gamma }(q_2-{\alpha }_2){(\xi -h)}^{q_2-1}}{{\Delta }_2},\xi \in [h,k],\hfill \\ \hfill {\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\left(\xi \right)& ={\int }_h^k{\mathcal{H}}_3(\xi ,\zeta ){\varphi }_{{\phi }_3}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}{\mathit{f}}_3\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_3\mathsf{\Gamma }(q_3-{\alpha }_3){(\xi -h)}^{q_3-1}}{{\Delta }_3},\xi \in [h,k],\hfill \end{array}$$

and an operator $\mathcal{T}:\mathcal{Y}\to \mathcal{Y}$ as

$$\mathcal{T}(\varpi ,\vartheta ,\omega )=\left({\mathcal{T}}_1(\varpi ,\vartheta ,\omega ),{\mathcal{T}}_2(\varpi ,\vartheta ,\omega ),{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\right),(\varpi ,\vartheta ,\omega )\in \mathcal{Y}.$$

It is clear that the existence of a positive solution to the system (1)-(2) is equivalent to the existence of a fixed points of the operator $\mathcal{T}.$

3. Main results

We denote the following notations for our convenience:

$$\begin{array}{cc}\hfill \mathcal{D}=max\{& {\varphi }_{{\phi }_1}\left(\frac{4^{p_1}\mathsf{\Gamma }(p_1+1)}{{(k-h)}^{p_1}}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_1(k,\zeta )d\zeta ,{\varphi }_{{\phi }_2}\left(\frac{4^{p_2}\mathsf{\Gamma }(p_2+1)}{{(k-h)}^{p_2}}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_2(k,\zeta )d\zeta ,\hfill \\ & {\varphi }_{{\phi }_3}\left(\frac{4^{p_3}\mathsf{\Gamma }(p_3+1)}{{(k-h)}^{p_3}}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_3(k,\zeta )d\zeta \},\hfill \\ \hfill \mathcal{C}=min\{& {\int }_h^k{\mathcal{H}}_1(k,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}d\tau \right)d\zeta ,{\int }_h^k{\mathcal{H}}_2(k,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}d\tau \right)d\zeta ,\hfill \\ & {\int }_h^k{\mathcal{H}}_3(k,\zeta ){\varphi }_{{\phi }_3}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}d\tau \right)d\zeta \}.\hfill \end{array}$$

Let us define two continuous functionals $\alpha$ and $\beta$ on the cone $\mathcal{P}$ by

$$\begin{array}{cc}\hfill \alpha (\varpi ,\vartheta ,\omega )& =\underset{\xi \in \mathcal{I}}{min}\left\{\left|\varpi \right|+\left|\vartheta \right|+\left|\omega \right|\right\}\mathrm{and}\hfill \\ \hfill \beta (\varpi ,\vartheta ,\omega )& =\underset{\xi \in [h,k]}{max}\left\{\left|\varpi \right|+\left|\vartheta \right|+\left|\omega \right|\right\}=\varpi \left(k\right)+\vartheta \left(k\right)+\omega \left(k\right)=\parallel (\varpi ,\vartheta ,\omega )\parallel .\hfill \end{array}$$

It is clear that $\alpha (\varpi ,\vartheta ,\omega )\le \beta (\varpi ,\vartheta ,\omega )$, for all $(\varpi ,\vartheta ,\omega )\in \mathcal{P}.$

Lemma 5. 

$\mathcal{T}:\mathcal{P}\to \mathcal{P}$ is completely continuous.

Proof. 

By using standard arguments, we can easily show that, the operator $\mathcal{T}$ is completely continuous and we need only to prove $\mathcal{T}\left(\mathcal{P}\right)\subset \mathcal{P}$. Let $(\varpi ,\vartheta ,\omega )\in \mathcal{P}$, by Lemma 3, we have

$$\begin{array}{cc}\hfill \parallel {\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\parallel & \le {\int }_h^k{\mathcal{H}}_1(k,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(s-\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(k-h)}^{q_1-1}}{{\Delta }_1},\hfill \\ \hfill \parallel {\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\parallel & \le {\int }_h^k{\mathcal{H}}_2(k,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}{\mathit{f}}_2\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right))\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_2\mathsf{\Gamma }(q_2-{\alpha }_2){(k-h)}^{q_2-1}}{{\Delta }_2},\hfill \\ \hfill \parallel {\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\parallel & \le {\int }_h^k{\mathcal{H}}_3(k,\zeta ){\varphi }_{{\phi }_3}\left({\int }_a^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}{\mathit{f}}_3\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_3\mathsf{\Gamma }(q_3-{\alpha }_3){(k-h)}^{q_3-1}}{{\Delta }_3},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{\xi \in \mathcal{I}}{min}& {\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\left(\xi \right)=\underset{\xi \in \mathcal{I}}{min}[{\int }_h^k{\mathcal{H}}_1(\xi ,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(\xi -h)}^{q_1-1}}{{\Delta }_1}],\hfill \\ & \ge {\left(\frac{1}{4}\right)}^{q_1-1}[{\int }_h^k{\mathcal{H}}_1(k,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}{\mathit{f}}_1\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_1\mathsf{\Gamma }(q_1-{\alpha }_1){(k-h)}^{q_1-1}}{{\Delta }_1}],\hfill \\ & \ge \aleph \parallel {\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\parallel .\hfill \end{array}$$

Similarly, $\underset{\xi \in \mathcal{I}}{min}{\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\left(\xi \right)\ge \aleph \parallel {\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\parallel$ and $\underset{\xi \in \mathcal{I}}{min}{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\left(\xi \right)\ge \aleph \parallel {\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\parallel .$ Therefore

$$\begin{array}{cc}\hfill \underset{\xi \in I}{min}& \left\{{\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\left(\xi \right)\right\}\hfill \\ & \ge \aleph \parallel {\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\parallel +\aleph \parallel {\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\parallel +\aleph \parallel {\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\parallel \hfill \\ & =\aleph \parallel \left({\mathcal{T}}_1(\varpi ,\vartheta ,\omega ),{\mathcal{T}}_2(\varpi ,\vartheta ,\omega ),{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\right)\parallel \hfill \\ & =\aleph \parallel \mathcal{T}(\varpi ,\vartheta ,\omega )\parallel .\hfill \end{array}$$

Hence, we get $\mathcal{T}\left(\mathcal{P}\right)\subset \mathcal{P}$. By using standard arguments involving the Arzela-Ascoli theorem, we can easily show that ${\mathcal{T}}_1,{\mathcal{T}}_2$ and ${\mathcal{T}}_3$ are completely continuous, and then $\mathcal{T}$ is a completely continuous operator from $\mathcal{P}$ to $\mathcal{P}$. □

Theorem 3. 

Assume that conditions $\left(B1\right)-\left(B3\right)$ holds and suppose that there exist positive real numbers $r,\mathcal{R}$ with $r<\eta \mathcal{R}$ and ${\psi }_j<\frac{r{\Delta }_j}{{\mathsf{\Theta }}_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}\le \frac{\mathcal{R}{\Delta }_j}{{\mathsf{\Theta }}_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}$ such that $f_j;j=1,2,3$ satisfying the following conditions:

(C1)

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )\ge {\varphi }_{r_j}\left(\frac{1}{3}\frac{r}{\aleph \mathcal{D}}\right)$, $\xi \in \mathcal{I}$ and $(\varpi ,\vartheta ,\omega )\in [r,\mathcal{R}]$,

(C2)

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )\le {\varphi }_{r_j}\left(\frac{1}{{\mathsf{\Phi }}_j}\frac{\mathcal{R}}{\mathcal{C}}\right)$, $\xi \in [h,k]$ and $(\varpi ,\vartheta ,\omega )\in [0,\mathcal{R}]$.

Then the system of fractional order boundary value problem (1)-(2) has at least one positive and nondecreasing solution $({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ satisfying $r\le \alpha ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ with $\beta ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\le \mathcal{R}.$

Proof. 

Let ${\mathsf{\Omega }}_1=\{(\varpi ,\vartheta ,\omega );\alpha (\varpi ,\vartheta ,\omega )<r\}$ and ${\mathsf{\Omega }}_2=\{(\varpi ,\vartheta ,\omega );\beta (\varpi ,\vartheta ,\omega )<\mathcal{R}\}.$ It is easy to see that $0\subset {\mathsf{\Omega }}_1,$ set ${\mathsf{\Omega }}_1,{\mathsf{\Omega }}_2$ are bounded open subsets of $\mathcal{E}$. Letting $(\varpi ,\vartheta ,\omega )\in \mathsf{\Omega },$ we have

$$\begin{array}{cc}\hfill r>\alpha (\varpi ,\vartheta ,\omega )& =\underset{\xi \in I}{min}\{\varpi \left(\xi \right)+\vartheta \left(\xi \right)+\omega \left(\xi \right)\}\hfill \\ & \ge \aleph \{\parallel \varpi \parallel +\parallel \vartheta \parallel +\parallel \omega \parallel \}\hfill \\ & =\aleph \beta (\varpi ,\vartheta ,\omega ).\hfill \end{array}$$

Thus $\mathcal{R}>\frac{r}{\aleph }>\beta (\varpi ,\vartheta ,\omega )$, i.e $(\varpi ,\vartheta ,\omega )\in {\mathsf{\Omega }}_2,$ so ${\mathsf{\Omega }}_1\subseteq {\mathsf{\Omega }}_2$.

Claim 1: If $(\varpi ,\vartheta ,\omega )\in \mathcal{P}\cap \partial {\mathsf{\Omega }}_1,$ then $\alpha \left(\mathcal{T}\right(\varpi ,\vartheta ,\omega \left)\right)\ge \alpha (\varpi ,\vartheta ,\omega )=r,$ for $\zeta \in \mathcal{I}$. It follows $\left(C1\right)$ Lemma 4 that

$$\begin{array}{cc}& \alpha \left(\mathcal{T}(\varpi ,\vartheta ,\omega )\right)=\underset{\xi \in \mathcal{I}}{min}\sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(\xi ,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(\xi -h)}^{q_j-1}}{{\Delta }_j},]\hfill \\ & \ge \sum _{j=1}^3\left[{\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\left(\frac{1}{4}\right)}^{q_j-1}{\mathcal{H}}_j(k,\zeta ){\varphi }_{{\phi }_j}\right({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\varphi }_{r_j}\left(\frac{1}{3}\frac{r}{\aleph \mathcal{D}}\right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){\left(\frac{1}{4}\right)}^{q_j-1}{(k-h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \ge \sum _{j=1}^3{\varphi }_{{\phi }_j}\left(\frac{{(k-h)}^{p_j}}{4^{p_j}\mathsf{\Gamma }(p_j+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}\aleph {\mathcal{H}}_j(k,\zeta )\left(\frac{1}{3}\frac{r}{\aleph \mathcal{D}}\right)d\zeta \hfill \\ & \ge \frac{r}{3\mathcal{D}}{\varphi }_{{\phi }_1}\left(\frac{{(k-h)}^{p_1}}{4^{p_1}\mathsf{\Gamma }(p_1+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_1(k,\zeta )d\zeta +\frac{r}{3\mathcal{D}}{\varphi }_{{\phi }_2}\left(\frac{{(k-h)}^{p_2}}{4^{p_2}\mathsf{\Gamma }(p_2+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_2(k,\zeta )d\zeta \hfill \\ & +\frac{r}{3\mathcal{D}}{\varphi }_{{\phi }_3}\left(\frac{{(k-h)}^{p_3}}{4^{p_3}\mathsf{\Gamma }(p_3+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_3(k,\zeta )d\zeta \hfill \\ & =\frac{r}{3}+\frac{r}{3}+\frac{r}{3}=r=\alpha (\varpi ,\vartheta ,\omega ).\hfill \end{array}$$

Claim 2: If $(\varpi ,\vartheta ,\omega )\in \mathcal{P}\cap \partial {\mathsf{\Omega }}_2,$ then $\beta \left(\mathcal{T}\right(\varpi ,\vartheta ,\omega \left)\right)\le \beta (\varpi ,\vartheta ,\omega ).$ Then

$$\left[\varpi \left(\zeta \right)+\vartheta \left(\zeta \right)+\omega \left(\zeta \right)\right]\le \beta (\varpi ,\vartheta ,\omega )=\mathcal{R},$$

for $\zeta \in [h,k]$. It follows from $\left(C2\right)$ and Lemma 4 that

$$\begin{array}{cc}& \beta \left(\mathcal{T}(\varpi ,\vartheta ,\omega )\right)=\underset{\xi \in [h,k]}{max}\sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(\xi ,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(\xi -h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \le \sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(k,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}{{\Delta }_j}]\hfill \end{array}$$

$$\begin{array}{cc}& <\frac{\mathcal{R}}{{\mathsf{\Phi }}_1\mathcal{C}}{\int }_h^k{\mathcal{H}}_1(k,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}d\tau \right)d\zeta +\frac{\mathcal{R}}{{\mathsf{\Phi }}_2\mathcal{C}}{\int }_h^k{\mathcal{H}}_2(\xi ,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}d\tau \right)d\zeta \hfill \\ & +\frac{\mathcal{R}}{{\mathsf{\Phi }}_3\mathcal{C}}{\int }_h^{\zeta }{\mathcal{H}}_3(\xi ,\zeta ){\varphi }_{{\phi }_3}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}d\tau \right)d\zeta +\frac{\mathcal{R}}{{\mathsf{\Theta }}_1}+\frac{\mathcal{R}}{{\mathsf{\Theta }}_2}+\frac{\mathcal{R}}{{\mathsf{\Theta }}_3}\hfill \\ & =\frac{\mathcal{R}}{{\mathsf{\Phi }}_1}+\frac{\mathcal{R}}{{\mathsf{\Phi }}_2}+\frac{\mathcal{R}}{{\mathsf{\Phi }}_3}+\frac{\mathcal{R}}{{\mathsf{\Theta }}_1}+\frac{\mathcal{R}}{{\mathsf{\Theta }}_2}+\frac{\mathcal{R}}{{\mathsf{\Theta }}_3}\hfill \\ & =\mathcal{R}\left[\frac{1}{{\mathsf{\Phi }}_1}+\frac{1}{{\mathsf{\Phi }}_2}+\frac{1}{{\mathsf{\Phi }}_3}+\frac{1}{{\mathsf{\Theta }}_1}+\frac{1}{{\mathsf{\Theta }}_2}+\frac{1}{{\mathsf{\Theta }}_3}\right]\le \mathcal{R}=\beta (\varpi ,\vartheta ,\omega ).\hfill \end{array}$$

Clearly, $\alpha$ satisfies Rule $\left(S_1\right)\left(iii\right)$ and $\beta$ satisfies Rule $\left(S_2\right)\left(i\right)$. Therefore the condition $\left(a\right)$ of Theorem 1 is satisfied and hence $\mathcal{T}$ has at least one fixed point $({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\in \mathcal{P}\cap \left(\overline{{\mathsf{\Omega }}_2}\setminus {\mathsf{\Omega }}_1\right)$,i.e, the system of fractional order boundary value problem $\left(1\right)-\left(2\right)$ has at least one positive and nondecreasing solution $({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ satisfying $r\le \alpha ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ with $\beta ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\le \mathcal{R}$. □

Theorem 4. 

Assume that conditions $\left(B1\right)-\left(B3\right)$ holds and suppose that there exist positive real numbers $r,\mathcal{R}$ with $r<\mathcal{R}$ and ${\psi }_j<\frac{r{\Delta }_j}{{\mathsf{\Theta }}_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}\le \frac{\mathcal{R}{\Delta }_j}{{\mathsf{\Theta }}_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}$ such that ${\mathit{f}}_j;j=1,2,3$ satisfying the following conditions:

(C3) 

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )\le {\varphi }_{r_j}\left(\frac{1}{{\mathsf{\Phi }}_j}\frac{r}{\mathcal{D}}\right)$, $\xi \in [h,k]$ and $(\varpi ,\vartheta ,\omega )\in [0,r]$,

(C4) 

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )\ge {\varphi }_{r_j}\left(\frac{1}{{\mathsf{\Phi }}_j}\frac{\mathcal{R}}{\aleph \mathcal{C}}\right)$, $\xi \in \mathcal{I}$ and $(\varpi ,\vartheta ,\omega )\in [\mathcal{R},\frac{\mathcal{R}}{\aleph }]$.

Then the system of fractional order boundary value problem $\left(1\right)-\left(2\right)$ has at least one positive and nondecreasing solution $({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ satisfying $r\le \beta ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ with $\alpha ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\le \mathcal{R}.$

Proof. 

Let ${\mathsf{\Omega }}_3=\{(\varpi ,\vartheta ,\omega );\beta (\varpi ,\vartheta ,\omega )<r\}$ and ${\mathsf{\Omega }}_4=\{(\varpi ,\vartheta ,\omega );\alpha (\varpi ,\vartheta ,\omega )<\mathcal{R}\}.$ We have $0\in {\mathsf{\Omega }}_3,$ set ${\mathsf{\Omega }}_3\subset {\mathsf{\Omega }}_4$ with ${\mathsf{\Omega }}_3$ and ${\mathsf{\Omega }}_4$ are bounded open subsets of $\mathcal{E}$.

Claim 1: If $(\varpi ,\vartheta ,\omega )\in \mathcal{P}\cap \partial {\mathsf{\Omega }}_3,$ then $\beta \left(\mathcal{T}\right(\varpi ,\vartheta ,\omega \left)\right)\le \beta (\varpi ,\vartheta ,\omega )$. To see this, let $(\varpi ,\vartheta ,\omega )\in \mathcal{P}\cap \partial {\mathsf{\Omega }}_3.$ Then $\left[\varpi \right(\zeta )+\vartheta (\zeta )+\omega (\zeta \left)\right]\le \beta (\varpi ,\vartheta ,\omega )=r,$ for $\zeta \in [h,k]$. It follows from $\left(C3\right)$ and Lemma 4 that

$$\begin{array}{cc}& \beta \left(\mathcal{T}(\varpi ,\vartheta ,\omega )\right)=\underset{\xi \in [h,k]}{max}\sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(\xi ,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(\xi -h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \le \sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(k,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}{{\Delta }_j}]\hfill \end{array}$$

$$\begin{array}{cc}& <\frac{r}{{\mathsf{\Phi }}_1\mathcal{D}}{\int }_h^k{\mathcal{H}}_1(k,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}d\tau \right)d\zeta +\frac{r}{{\mathsf{\Phi }}_2\mathcal{D}}{\int }_h^k{\mathcal{H}}_2(k,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}d\tau \right)d\zeta \hfill \\ & +\frac{r}{{\mathsf{\Phi }}_3\mathcal{D}}{\int }_h^k{\mathcal{H}}_3(k,\zeta ){\varphi }_{{\phi }_3}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}d\tau \right)d\zeta +\frac{r}{{\mathsf{\Theta }}_1}+\frac{r}{{\mathsf{\Theta }}_2}+\frac{r}{{\mathsf{\Theta }}_3}\hfill \\ & =\frac{r}{{\mathsf{\Phi }}_1}+\frac{r}{{\mathsf{\Phi }}_2}+\frac{r}{{\mathsf{\Phi }}_3}+\frac{r}{{\mathsf{\Theta }}_1}+\frac{r}{{\mathsf{\Theta }}_2}+\frac{r}{{\mathsf{\Theta }}_3}\hfill \\ & =r\left[\frac{1}{{\mathsf{\Phi }}_1}+\frac{1}{{\mathsf{\Phi }}_2}+\frac{1}{{\mathsf{\Phi }}_3}+\frac{1}{{\mathsf{\Theta }}_1}+\frac{1}{{\mathsf{\Theta }}_2}+\frac{1}{{\mathsf{\Theta }}_3}\right]\le r=\beta (\varpi ,\vartheta ,\omega ).\hfill \end{array}$$

Claim 2: If $(\varpi ,\vartheta ,\omega )\in \mathcal{P}\cap \partial {\mathsf{\Omega }}_4,$ then $\alpha \left(\mathcal{T}\right(\varpi ,\vartheta ,\omega \left)\right)\ge \alpha (\varpi ,\vartheta ,\omega ).$$\mathcal{T}$ see this, let $(\varpi ,\vartheta ,\omega )\in \mathcal{P}\cap \partial {\mathsf{\Omega }}_4$. Then $\frac{\mathcal{R}}{\aleph }=\frac{\alpha (\varpi ,\vartheta ,\omega )}{\eta }\ge \beta (\varpi ,\vartheta ,\omega )\ge [\varpi \left(\zeta \right)+\vartheta \left(\zeta \right)+\omega \left(\zeta \right)]\ge \alpha (\varpi ,\vartheta ,\omega )=\mathcal{R}$ for $\zeta \in \mathcal{I}.$ It follows from $\left(C4\right),$ and Lemma 4, we have

$$\begin{array}{cc}& \alpha \left(\mathcal{T}(\varpi ,\vartheta ,\omega )\right)=\underset{\xi \in \mathcal{I}}{min}\sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(\xi ,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(\xi -h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \ge \sum _{j=1}^3\left[{\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\left(\frac{1}{4}\right)}^{q_j-1}{\mathcal{H}}_j(k,\zeta ){\varphi }_{{\phi }_j}\right({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\varphi }_{r_j}\left(\frac{1}{3}\frac{\mathcal{R}}{\aleph \mathcal{C}}\right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){\left(\frac{1}{4}\right)}^{q_j-1}{(k-h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \ge \sum _{j=1}^3{\varphi }_{{\phi }_j}\left(\frac{{(k-h)}^{p_j}}{4^{p_j}\mathsf{\Gamma }(p_j+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}\aleph {\mathcal{H}}_j(k,\zeta )\left(\frac{1}{3}\frac{\mathcal{R}}{\aleph \mathcal{C}}\right)d\zeta \hfill \\ & \ge \frac{\mathcal{R}}{3\mathcal{C}}{\varphi }_{{\phi }_1}\left(\frac{{(k-h)}^{p_1}}{4^{p_1}\mathsf{\Gamma }(p_1+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_1(k,\zeta )d\zeta +\frac{\mathcal{R}}{3\mathcal{C}}{\varphi }_{{\phi }_2}\left(\frac{{(k-h)}^{p_2}}{4^{p_2}\mathsf{\Gamma }(p_2+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_2(k,\zeta )d\zeta \hfill \\ & +\frac{\mathcal{R}}{3\mathcal{C}}{\varphi }_{{\phi }_3}\left(\frac{{(k-h)}^{p_3}}{4^{p_3}\mathsf{\Gamma }(p_3+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_3(k,\zeta )d\zeta \hfill \\ & =\frac{\mathcal{R}}{3}+\frac{\mathcal{R}}{3}+\frac{\mathcal{R}}{3}=\mathcal{R}=\alpha (\varpi ,\vartheta ,\omega ).\hfill \end{array}$$

Clearly, $\alpha$ satisfies Rule $\left(S1\right)\left(iii\right)$ and $\beta$ satisfies Rule $\left(S2\right)\left(i\right)$. Therefore the condition $\left(a\right)$ of Theorem 1 is satisfied and hence $\mathcal{T}$ has at least one fixed point $({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\in \mathcal{P}\cap \left(\overline{{\mathsf{\Omega }}_4}\setminus {\mathsf{\Omega }}_2\right)$,i.e, the system of fractional order boundary value problem $\left(1\right)-\left(2\right)$ has at least one positive and nondecreasing solution $({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})$ satisfying $r\le \beta \left(({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\right)$ with $\alpha ({\varpi }^{★},{\vartheta }^{★},{\omega }^{★})\le \mathcal{R}$. □

Theorem 5. 

Assume that $\left(B1\right)-\left(B3\right)$ holds. Suppose that there exist $0<k<l<\aleph d$ and $0<{\psi }_j<\frac{k{\Delta }_j}{{\mathsf{\Theta }}_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}\le \frac{d{\Delta }_j}{{\mathsf{\Theta }}_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}$ such that ${\mathit{f}}_j(j=1,2,3)$ satisfies the following conditions:

$\left(C5\right)$

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )<{\varphi }_{r_j}\left(\frac{d}{{\mathsf{\Phi }}_j\mathcal{C}}\right),$ for all $\xi \in [h,k]$, $(\varpi ,\vartheta ,\omega )\in [0,d]$,

$\left(C6\right)$

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )>{\varphi }_{r_j}\left(\frac{l}{3\aleph \mathcal{D}}\right),$ for all $\xi \in \mathcal{I}$, $(\varpi ,\vartheta ,\omega )\in [l,\frac{l}{\aleph }]$,

$\left(C7\right)$

${\mathit{f}}_j(\xi ,\varpi ,\vartheta ,\omega )<{\varphi }_{r_j}\left(\frac{k}{{\mathsf{\Phi }}_j\mathcal{C}}\right),$ for all $\xi \in [h,k]$, $(\varpi ,\vartheta ,\omega )\in [0,k]$.

Then the system (1)-(2) has at least three positive solution $({\varpi }_1,{\vartheta }_1,{\omega }_1),({\varpi }_2,{\vartheta }_2,{\omega }_2)$ and $({\varpi }_3,{\vartheta }_3,{\omega }_3)$ with $\phi ({\varpi }_1,{\vartheta }_1,{\omega }_1)<k,l<\psi ({\varpi }_2,{\vartheta }_2,{\omega }_2)<\phi ({\varpi }_2,{\vartheta }_2,{\omega }_2)<d,k<\phi ({\varpi }_3,{\vartheta }_3,{\omega }_3)<d$ with $\psi ({\varpi }_3,{\vartheta }_3,{\omega }_3)<l.$

Proof. 

Firstly, if $(\varpi ,\vartheta ,\omega )\in \overline{{\mathcal{P}}_d}$, then we may assert that $\mathcal{T}:\overline{{\mathcal{P}}_d}\to \overline{{\mathcal{P}}_d}$ is a completely continuous operator. To see this, suppose $(\varpi ,\vartheta ,\omega )\in \overline{{\mathcal{P}}_d},$ then $\parallel (\varpi ,\vartheta ,\omega )\parallel \le d$. It follows from Lemma 4 and $\left(C5\right)$, that

$$\begin{array}{cc}\hfill \parallel \mathcal{T}& (\varpi ,\vartheta ,\omega )\parallel =\underset{\xi \in [h,k]}{max}\left\{{\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\left(\xi \right)\right\}\hfill \\ & =\underset{\xi \in [h,k]}{max}\sum _{i=1}^3[{\int }_h^k{\mathcal{H}}_j(\xi ,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(\xi -h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \le \sum _{j=1}^3[{\int }_h^k{\mathcal{H}}_j(k,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(k-h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & <\frac{1}{{\mathsf{\Phi }}_1}\frac{d}{C}{\int }_h^k{\mathcal{H}}_1(k,\zeta ){\varphi }_{{\phi }_1}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_1-1}}{\mathsf{\Gamma }\left(p_1\right)}d\tau \right)d\zeta +\frac{1}{{\mathsf{\Phi }}_2}\frac{d}{\mathcal{C}}{\int }_h^k{\mathcal{H}}_2(k,\zeta ){\varphi }_{{\phi }_2}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_2-1}}{\mathsf{\Gamma }\left(p_2\right)}d\tau \right)d\zeta \hfill \\ & +\frac{1}{{\mathsf{\Phi }}_3}\frac{d}{\mathcal{C}}{\int }_h^k{\mathcal{H}}_3(k,\zeta ){\varphi }_{{\phi }_3}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_3-1}}{\mathsf{\Gamma }\left(p_3\right)}d\tau \right)d\zeta +\frac{d}{{\mathsf{\Theta }}_j}+\frac{d}{{\mathsf{\Theta }}_j}+\frac{d}{{\mathsf{\Theta }}_j}\hfill \\ & =d\left[\frac{1}{{\mathsf{\Phi }}_1}+\frac{1}{{\mathsf{\Phi }}_2}+\frac{1}{{\mathsf{\Phi }}_3}+\frac{1}{{\mathsf{\Theta }}_1}+\frac{1}{{\mathsf{\Theta }}_2}+\frac{1}{{\mathsf{\Theta }}_3}\right]\le d.\hfill \end{array}$$

Therefore, $\mathcal{T}:\overline{{\mathcal{P}}_d}\to \overline{{\mathcal{P}}_d}$. This together with Lemma 5 implies that $\mathcal{T}:\overline{{\mathcal{P}}_d}\to \overline{{\mathcal{P}}_d}$ is a completely continuous operator. In the similarly way, if $(\varpi ,\vartheta ,\omega )\in \overline{{\mathcal{P}}_k},$ then from $\left(C7\right)$ yields $\parallel \mathcal{T}(\varpi ,\vartheta ,\omega )\parallel <k$. This shows that condition $\left(ii\right)$ of Theorem 2 is fulfilled.

Now, we let $\varpi \left(\xi \right)+\vartheta \left(\xi \right)+\omega \left(\xi \right)=\frac{l}{\aleph }$ for $\xi \in [h,k].$ It is easy to verify that $\varpi \left(\xi \right)+\vartheta \left(\xi \right)+\omega \left(\xi \right)=\frac{l}{\aleph }\in \mathcal{P}\left(\psi ,l,\frac{l}{\aleph }\right)$ and $\psi (\varpi ,\vartheta ,\omega )=\frac{l}{\aleph }>l,$ and so $\{(\varpi ,\vartheta ,\omega )\in \mathcal{P}(\psi ,l,\frac{l}{\aleph });\psi (\varpi ,\vartheta ,\omega )>l\}\ne \varnothing$. Thus, for all $(\varpi ,\vartheta ,\omega )\in \mathcal{P}(\psi ,l,\frac{l}{\aleph })$, we have that $l\le \varpi \left(\xi \right)+\vartheta \left(\xi \right)+\omega \left(\xi \right)\le \frac{l}{\aleph }$ for $\xi \in \mathcal{I}$ and $\mathcal{T}(\varpi ,\vartheta ,\omega )\in \mathcal{P}$, from $\left(C6\right)$, we have

$$\begin{array}{cc}& \psi \left(\mathcal{T}(\varpi ,\vartheta ,\omega )\left(\xi \right)\right)=\underset{\xi \in \mathcal{I}}{min}\left\{{\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\left(t\right)+{\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\left(\xi \right)\right\}\hfill \\ & =\underset{\xi \in I}{min}\sum _{i=1}^3[{\int }_h^k{\mathcal{H}}_j(\xi ,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\mathit{f}}_j\left(\tau ,\varpi \left(\tau \right),\vartheta \left(\tau \right),\omega \left(\tau \right)\right)d\tau \right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){(t-h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \ge \aleph \sum _{i=1}^3[{\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\left(\frac{1}{4}\right)}^{q_j-1}{\mathcal{H}}_j(k,\zeta ){\varphi }_{{\phi }_j}\left({\int }_h^{\zeta }\frac{{(\zeta -\tau )}^{p_j-1}}{\mathsf{\Gamma }\left(p_j\right)}{\varphi }_{r_j}\left(\frac{l}{3\aleph \mathcal{D}}\right)\right)d\zeta \hfill \\ & +\frac{{\psi }_j\mathsf{\Gamma }(q_j-{\alpha }_j){\left(\frac{1}{4}\right)}^{q_j-1}{(k-h)}^{q_j-1}}{{\Delta }_j}]\hfill \\ & \ge \sum _{j=1}^3{\varphi }_{{\phi }_j}\left(\frac{{(k-h)}^{p_j}}{4^{p_j}\mathsf{\Gamma }(p_j+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}\aleph {\mathcal{H}}_j(k,\zeta )\left(\frac{l}{3\aleph \mathcal{D}}\right)d\zeta \hfill \\ & \ge \frac{l}{3\mathcal{D}}{\varphi }_{{\phi }_1}\left(\frac{{(k-h)}^{p_1}}{4^{p_1}\mathsf{\Gamma }(p_1+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_1(k,\zeta )d\zeta +\frac{l}{3\mathcal{D}}{\varphi }_{{\phi }_2}\left(\frac{{(k-h)}^{p_2}}{4^{p_2}\mathsf{\Gamma }(p_2+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_2(k,\zeta )d\zeta \hfill \\ & +\frac{l}{3\mathcal{D}}{\varphi }_{{\phi }_3}\left(\frac{{(k-h)}^{p_3}}{4^{p_3}\mathsf{\Gamma }(p_3+1)}\right){\int }_{\frac{3h+k}{4}}^{\frac{h+3k}{4}}{\mathcal{H}}_3(k,\zeta )d\zeta \hfill \\ & =\frac{l}{3}+\frac{l}{3}+\frac{l}{3}=l.\hfill \end{array}$$

Hence the condition $\left(i\right)$ of Theorem 2 is verified. Next, we prove that $\left(iii\right)$ of Theorem 2 is satisfied. By Lemma 5, we have $\underset{\xi \in I}{min}|{\mathcal{T}}_1(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_2(\varpi ,\vartheta ,\omega )\left(\xi \right)+{\mathcal{T}}_3(\varpi ,\vartheta ,\omega )\left(t\right)|>\aleph \parallel \mathcal{T}(\varpi ,\vartheta ,\omega )\parallel >d\mathrm{for}(\varpi ,\vartheta ,\omega )\in \mathcal{P}(\psi ,l,d)\mathrm{with}\parallel \mathcal{T}(\varpi ,\vartheta ,\omega )\parallel >\frac{l}{\aleph }$. To sum up, all the conditions of Theorem 2 are satisfied, then there exist three positive solutions $({\varpi }_1,{\vartheta }_1,{\omega }_1),({\varpi }_2,{\vartheta }_2,{\omega }_2)$ and $({\varpi }_3,{\vartheta }_3,{\omega }_3)$ with $\phi ({\varpi }_1,{\vartheta }_1,{\omega }_1)<k,l<\psi ({\varpi }_2,{\vartheta }_2,{\omega }_2)<\phi ({\varpi }_2,{\vartheta }_2,{\omega }_2)<d,k<\phi ({\varpi }_3,{\vartheta }_3,{\omega }_3)<d,$ with $\psi ({\varpi }_3,{\vartheta }_3,{\omega }_3)<l$. □

4. Examples

Let $h=1,k=2,p_1=0.5,p_2=0.6,p_3=0.7,q_1=1.5,q_2=1.6,q_3=1.7,{\alpha }_1=0.5,{\alpha }_2=0.6,{\alpha }_3=0.7,{\eta }_1=1.5,{\eta }_2=1.6,{\eta }_3=1.7,{\mu }_1=2,{\mu }_2=3,{\mu }_3=4,{\lambda }_1=1,{\lambda }_2=2,{\lambda }_3=3,r_1=2,r_2=2,r_3=2.$

We consider the system of fractional differential equations

$$\left\{\begin{array}{c}-{\mathcal{D}}_{1^{+}}^{0.5}\left({\varphi }_2\left({\mathcal{D}}_{1^{+}}^{1.5}\varpi \left(\xi \right)\right)\right)=f_1(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right)),\xi \in (1,2),\hfill \\ -{\mathcal{D}}_{1^{+}}^{0.6}\left({\varphi }_2\left({\mathcal{D}}_{1^{+}}^{1.6}\vartheta \left(\xi \right)\right)\right)=f_2(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right)),\xi \in (1,2),\hfill \\ -{\mathcal{D}}_{1^{+}}^{0.7}\left({\varphi }_2\left({\mathcal{D}}_{1^{+}}^{1.7}\omega \left(\xi \right)\right)\right)=f_3(\xi ,\varpi \left(\xi \right),\vartheta \left(\xi \right),\omega \left(\xi \right)),\xi \in (1,2),\hfill \end{array}\right.$$

(8)

$$\left\{\begin{array}{cc}\hfill & \varpi \left(1\right)=0;{\varphi }_2\left({\mathcal{D}}_{1^{+}}^{1.5}\varpi \left(1\right)\right)=0;2{\mathcal{D}}_{1^{+}}^{0.5}\varpi \left(2\right)={\psi }_1+1{\mathcal{D}}_{1^{+}}^{0.5}\varpi \left(1.5\right),\hfill \\ \hfill & \vartheta \left(1\right)=0;{\varphi }_2\left({\mathcal{D}}_{1^{+}}^{1.6}\vartheta \left(1\right)\right)=0;3{\mathcal{D}}_{1^{+}}^{0.6}\vartheta \left(2\right)={\psi }_2+2{\mathcal{D}}_{1^{+}}^{0.6}\vartheta \left(1.6\right),\hfill \\ \hfill & \omega \left(1\right)=0;{\varphi }_2\left({\mathcal{D}}_{1^{+}}^{1.7}\omega \left(1\right)\right)=0;4{\mathcal{D}}_{1^{+}}^{0.7}\omega \left(2\right)={\psi }_3+3{\mathcal{D}}_{1^{+}}^{0.7}\omega \left(1.7\right),\hfill \end{array}\right.$$

(9)

where ${\psi }_1,{\psi }_2,{\psi }_3$ are parameters. We have $\aleph =0.378929;$${\Delta }_1=0.886227>0;{\Delta }_2=0.893515;{\Delta }_3=0.908639$, so assumption $\left(A2\right)$ satisfied. Beside we found $\mathcal{D}=max\{2.0,3.446,5.278\}=5.278;\mathcal{C}=min\{1.858929,2.620029,3.237839\}=1.858929.$

Example 1. 

We consider the functions

$$\begin{array}{cc}\hfill {\mathit{f}}_1(\xi ,\varpi ,\vartheta ,\omega )& =\left\{\begin{array}{c}\frac{1}{18}{e}^{-(\varpi +\vartheta +\omega )}+sin\xi ,0\le \varpi ,\vartheta ,\omega <5,\hfill \\ \frac{1}{2}(e^{-(\varpi +\vartheta +\omega )}+1)+\frac{1}{3}log\xi ,5\le \varpi ,\vartheta ,\omega \le 10,\hfill \end{array}\right.\hfill \\ \hfill {\mathit{f}}_2(\xi ,\varpi ,\vartheta ,\omega )& =\left\{\begin{array}{c}\frac{1}{3}(e^{-(\varpi +\vartheta +\omega )}+1)+\frac{t}{7}sin\xi ,0\le \varpi ,\vartheta ,\omega <5,\hfill \\ \frac{1}{2}(e^{-\xi }+1)+\frac{1}{\xi +1}(e^{-(\varpi +\vartheta +\omega )}log\xi ),5\le \varpi ,\vartheta ,\omega \le 10,\hfill \end{array}\right.\hfill \\ \hfill {\mathit{f}}_3(\xi ,\varpi ,\vartheta ,\omega )& =\left\{\begin{array}{c}\frac{1}{\xi +1}+log\xi {(e^{-(\varpi +\vartheta +\omega )}+2)}^{-1},0\le \varpi ,\vartheta ,\omega <5,\hfill \\ \frac{1}{\xi +1}(log\xi +1)+\frac{\xi e^{-\xi }}{\varpi +\vartheta +\omega }+\frac{10}{119},5\le \varpi ,\vartheta ,\omega \le 10.\hfill \end{array}\right.\hfill \end{array}$$

Choosing $r=1,\mathcal{R}=10,$ with $\frac{1}{{\mathsf{\Phi }}_1}=\frac{1}{{\mathsf{\Phi }}_2}=\frac{1}{{\mathsf{\Phi }}_3}=\frac{1}{{\mathsf{\Theta }}_1}=\frac{1}{{\mathsf{\Theta }}_2}=\frac{1}{{\mathsf{\Theta }}_3}=\frac{1}{6}$ then $r<\aleph \mathcal{R}$ and $f_i(i=1,2,3)$ fulfilling the following conditions:

(C1) 

$f_i(\xi ,\varpi ,\vartheta ,\omega )\ge 0.166668={\varphi }_{r_i}\left(\frac{1}{3}\frac{r}{\aleph \mathcal{D}}\right)$, $\xi \in [2.25,2.75]$ and $(\varpi ,\vartheta ,\omega )\in [1,10]$

(C2) 

$f_i(\xi ,\varpi ,\vartheta ,\omega )\le 0.896574={\varphi }_{r_i}\left(\frac{1}{{\mathsf{\Phi }}_j}\frac{\mathcal{R}}{\mathcal{C}}\right)$, $\xi \in [1,2]$ and $(\varpi ,\vartheta ,\omega )\in [0,10]$.

Thus, all conditions of Theorem 3 are fulfilled. Hence, for ${\psi }_1\le 1.47705$, ${\psi }_2\le 1.489192$, ${\psi }_3\le 1.514398$, the system of (8) and (9) has at least three positive solutions.

Example 2. 

We consider the functions

$$\begin{array}{cc}\hfill {\mathit{f}}_1(\xi ,\varpi ,\vartheta ,\omega )& =\left\{\begin{array}{c}\frac{1}{\xi +25}log(\varpi +\vartheta +\omega +1)+\frac{e^{-\xi }}{12},0\le \varpi ,\vartheta ,\omega \le 1,\hfill \\ \frac{2}{{\xi }^2+1}+\frac{log\xi +1}{e^{-\xi }+5},1<\varpi ,\vartheta ,\omega \le 10,\hfill \\ \frac{1}{5}(\xi +e^{-(\varpi +\vartheta +\omega )})+\frac{1}{9}(\xi +sin\xi ),10<\varpi ,\vartheta ,\omega \le 20,\hfill \end{array}\right.\hfill \\ \hfill {\mathit{f}}_2(\xi ,\varpi ,\vartheta ,\omega )& =\left\{\begin{array}{c}\frac{1}{40+{\xi }^5}{e}^{(\varpi +\vartheta +\omega )}+\frac{log\xi }{10},0\le \varpi ,\vartheta ,\omega \le 1,\hfill \\ \frac{1}{\xi +1}[log\xi +e^{-(\varpi +\vartheta +\omega })]+\frac{e^{-\xi }}{\mathit{u}+\mathit{v}+\mathit{w}}+\frac{9}{24},1<\varpi ,\vartheta ,\omega \le 10,\hfill \\ \frac{log\xi +1}{\varpi +\vartheta +\omega }+\frac{e^{\xi }}{{\xi }^2+1}-sin\xi ,10<\varpi ,\vartheta ,\omega \le 20,\hfill \end{array}\right.\hfill \\ \hfill {\mathit{f}}_3(\xi ,\varpi ,\vartheta ,\omega )& =\left\{\begin{array}{c}\frac{1}{\xi +6}[e^{-(\varpi +\vartheta +\omega )}log\xi ],0\le \varpi ,\vartheta ,\omega \le 1,\hfill \\ \frac{1}{2}[e^{-(\varpi +\vartheta +\omega )}+1]+\frac{2}{\xi +1}[e^{-\xi }sin\xi ],1<\varpi ,\vartheta ,\omega \le 10,\hfill \\ log\xi +\frac{2}{5}(1+\xi )-\frac{2}{\varpi +\vartheta +\omega },10<\varpi ,\vartheta ,\omega \le 20.\hfill \end{array}\right.\hfill \end{array}$$

Choosing $k=4,l=5,d=727.55$, $\frac{1}{{\Im }_1}=\frac{1}{{\Im }_2}=\frac{1}{{\Im }_3}=\frac{1}{{\Re }_1}=\frac{1}{{\Re }_2}=\frac{1}{{\Re }_3}=\frac{1}{6}$ then $0<k<l<\aleph d$ and ${\mathit{f}}_i(i=1,2,3)$ fulfilling the following conditions:

$\left(C5\right)$

${\mathit{f}}_i(\xi ,\varpi ,\vartheta ,\omega )<1.793147={\varphi }_{r_j}\left(\frac{d}{{\mathsf{\Phi }}_j\mathcal{C}}\right),$ for all $\xi \in [1,2]$, $(\varpi ,\vartheta ,\omega )\in [0,20]$,

$\left(C6\right)$

${\mathit{f}}_i(\xi ,\varpi ,\vartheta ,\omega )>0.333335={\varphi }_{r_j}\left(\frac{l}{3\aleph \mathcal{D}}\right),$ for all $\xi \in [2.25,2.75]$, $(\varpi ,\vartheta ,\omega )\in [2,5.278034]$,

$\left(C7\right)$

${\mathit{f}}_i(\xi ,\varpi ,\vartheta ,\omega )<0.089657={\varphi }_{r_j}\left(\frac{k}{{\mathsf{\Phi }}_j\mathcal{C}}\right),$ for all $\xi \in [1,2]$, $(\varpi ,\vartheta ,\omega )\in [0,1]$.

Thus, all conditions of Theorem 5 are fulfilled. Hence, for ${\sigma }_1\le 2.95409$, ${\sigma }_2\le 2.978384$, ${\sigma }_3\le 3.028796,$ the system of (8) and (9) has at least three positive solutions.