A coupled system of $p$-Laplacian implicit fractional differential equations depending on boundary conditions of integral type

1.   Introduction

Derivatives of integer orders are the particular form of fractional order derivatives. When the notation $\frac{d^{{n}}}{dt^{{n}}}$ was presented by Leibniz at the end of ${17}^{th}$ century for ${n}^{th}$ order derivative, L'Hospital asked from him can we take $n = \frac{1}{2}$? Leibniz replied "this is an apparent Paradox, from which one day useful consequences will be drawn", and this was the origin of fractional derivatives Riemann-Liouville derivative was originated after the great contributions of mathematicians Fourier and Laplace, and due to this notion of fractional derivative the fractional calculus was developed. After that, researchers showed interest in fractional calculus ^{[4,5,6,7,8,9,13,17,18,19,20,28]}. Fractional derivatives are universal operators, which cover different physical phenomena and areas of mathematical modeling such as: control theory ^[26], dynamical process ^[24], electro-chemistry ^[16], image and signal processing ^[22], mathematical biology ^[21], etc.

The existence of solutions is the basic subject for the investigation of the fractional differential equations. Numerous mathematicians worked on the existence of solutions for different Boundary Value Problems, Using different fixed point theorems (see ^[23,29,32]). In ^[2] Ahmad et al. studied the existence of coupled fractional differential equations involving a $p$-Laplacian operator by applying fixed point theorems. In ^[11,12], the authors got theoretical results related to fractional differential equations with $p$-Laplacian operator ($p$-LO).

It can be easily observed that obtaining the exact solution of non linear models is a tough and challenging task, to overcome the difficulty, a lot of approximation methods were established. Hyers-Ulam Stability (${HUS}$) can predict the gap between exactness and approximations of the solutions. This notion was initiated (in 1940) by Ulam ^[25] and further extended by Hyers to abstract spaces, after one year. Recent results shows that different models with different boundary conditions are investigated for ${HUS}$ ^[3,15,30,31].

Mohammed et al. ^[14], explored the existence and uniqueness of solution of a model which involves the Caputo-Katugampola fractional derivative:

where $^{c}D^{\beta}$ is a Caputo fractional derivative of order $\beta\in (1, 2)$, $^{c}D^{\gamma, \rho}_{1}, \ \gamma \in (0, 1)$ and $^{c}D^{\alpha, \rho}$, $\alpha \in (1, 2)$, $\rho > 0$ are Caputo-Katugampola fractional derivative, $\phi_{p}, \ p > 1$ is a $p$-LO.

Hira et al. ^[27], investigated the existence, uniqueness and HU stability of solutions to nonlinear coupled FDEs:

where $2 < \alpha_{i}\leq 3, \ 0 < \beta_{1}\leq 1, \ \eta_{i}, \ \gamma_{i} > 0, \ \psi_{i}\in L[0, T]$, and $^{c}D^{\alpha_{i}}_{0^{+}}$ and $^{c}D^{\beta_{i}}_{0^{+}}$ are the Caputo derivatives of order $\alpha_{i}$ and $\beta_{i}, \ i = 1, 2$, respectively. $L_{p}(s) = |s|^{p-2}s$ is a $p$-LO, where $\frac{1}{p}+\frac{1}{q} = 1$, and $L_{q}$ denotes inverse of $p$-Laplacian. $A_{i}:\mathcal{T}\rightarrow \mathfrak{R}$ are closed bounded linear operators for any $\rho\in \mathcal{T} = [0, T]$, and $\Phi, \ \Psi, \ g_{k}:\mathcal{T}\times \mathfrak{R}\times \mathfrak{R}\rightarrow \mathfrak{R}, \ (k = 1, 2)$ are continuous functions $i = 1, 2$. $C_{k}\in \mathfrak{R}^{n\times n}, \ (k = 1, 2)$.

In ^[12], Lu et al. investigated the nonlinear fractional ${BVP}$ with $p$-Laplacian operator

where $\mathcal{Z}\in(2, 3]$, and $\mathcal{B}\in(1, 2]$. ${D}^{\mathcal{Z}}$, ${D}^{\mathcal{B}}$ represents the standard Riemann-Liouville fractional derivatives.

In ^[2], Ahmad et al. investigated the system:

where $^{c}{D}^{\mathcal{Z}}$ and ${D}^{\mathcal{B}}$ respectively denotes the Caputo and Riemann-Liouville ${FD}$ of order $\mathcal{Z}$ and $\mathcal{B}$, ${m}-1 < \mathcal{Z}_{1}, \mathcal{Z}_{2}, \mathcal{B}\leq {m}$, ${m}\in\{4, 5, \dots\}$, and $1 < \delta\leq2$, ${K}_{1}(\cdot)$, ${K}_{2}(\cdot)$ are linear and bounded operators on ${R}$.

Zhang et al. ^[33], studied the existence of

where $\mathcal{Z}$ and $\mathcal{B}$ are the orders of Caputo fractional derivatives $^{c}{{D}^\mathcal{Z}}$ and $^{c}{{D}^\mathcal{B}}$ respectively.

Following ^[33], in this article, we present existence and stability analysis of the model of the form

where $1 < {m}-1 < \mathcal{B}_{1};\mathcal{B}_{2} < {m}$; $1 < n-1 < \mathcal{Z}_{1};\mathcal{Z}_{2} < n$; $\mathcal{Z}_{1}-\mathcal{B}_{1};\mathcal{Z}_{2}-\mathcal{B}_{2} > 1$, $\mathcal{Z}_{1};\mathcal{Z}_{2};\mathcal{B}_{1}$, and $\mathcal{B}_{2}$ be the orders of Caputo fractional derivatives $^{c}{D}^{\mathcal{Z}_{1}}$, $^{c}{D}^{\mathcal{Z}_{2}}$, $^{c}{D}^{\mathcal{B}_{1}}$ and $^{c}{D}^{\mathcal{B}_{2}}$ respectively, and $0 < \gamma_{1}, \gamma_{2}\leq1$. The $p$-Laplacian operator is represented by $\vartheta_{p}$ and is defined as $\vartheta_{p}(\varpi) = |\varpi|^{p-2}\varpi, p > 1, \vartheta_{p}^{-1} = \vartheta_{q}, \frac{1}{p}+\frac{1}{q}$ = $1$. ${S}_{1}, {S}_{2}, {S}_{3}, {S}_{4}\in\mathcal{C}([0, 1], {R}^{+})$, $\digamma_{1}$, and $\digamma_{2}$ are appropriate functions. $\mathtt{H}_{1}, \mathtt{H}_{2}, \mathtt{H}_{3}, \mathtt{H}_{4}\in\mathcal{C}([0, 1], {R}^{+})$ are perturbation functions.

$(\mathfrak{C}_{1})$ $0 < \int_{0}^{1}\mathtt{H}_{1}(\varpi){d\varpi} < \int_{0}^{1}\mathtt{H}_{2}(\varpi){d\varpi} < 2\int_{0}^{1}\mathtt{H}_{1}(\varpi){d\varpi} < +\infty$, $0\leq {S}_{1}(\alpha)\leq {S}_{2}(\alpha)\leq2{S}_{1}(\alpha)$, $0\leq\int_{0}^{1}{S}_{1}(\varpi){d\varpi},$ $\int_{0}^{1}{S}_{2}(\varpi){d\varpi} < 1$, $0\leq {S}_{3}(\alpha)\leq {S}_{4}(\alpha)\leq2{S}_{3}(\alpha)$, $0\leq\int_{0}^{1}{S}_{3}(\varpi){d\varpi},$ $\int_{0}^{1}{S}_{4}(\varpi){d\varpi} < 1$.

The remaining manuscript is as follows: Section 2 contains basic definitions, auxiliary lemmas and related theorems. In Section 3, we present the existence theory for the problem (1.2) and gives some related properties of Green function. In Section 4, we obtain at least three positive solutions of coupled system (1.2) by using the Avery-Peterson fixed point theorem. Section 5 contains ${HU}$ type stability results, and Section 6 provide an example to authenticate the theoretical result.

2.   Preliminaries

Here, we are presenting an important literature concerning the Caputo fractional derivatives and integral, which gives us help throughout this article, for the details, reader should study ^[1,10].

Definition 2.1. Let ${X}\in L^{1}([0, T], {R}^{+})$ be a function. Then the $\mathcal{Z}$ order fractional integral is defined by

on the condition that the integral on right side exists, where $\Gamma$ is the Euler Gamma function, defined as

Definition 2.2. Let ${X}:[0, T]\rightarrow {R}$ be a function. Then the $\mathcal{Z} >$ order fractional derivative is defined as

on the condition that the integral on the right side exists, and $[\mathcal{Z}]$ denote the integer part of a real number $\mathcal{Z}$, where ${n} = 1+[\mathcal{Z}]$.

Definition 2.3. Let ${X}:[0, T]\rightarrow {R}$ be a function. Then the $\mathcal{Z}$ order sequential fractional derivative is defined as:

where $\mathcal{Z} = (\mathcal{Z}_{1}, \mathcal{Z}_{2}, \mathcal{Z}_{3}, \dots, \mathcal{Z}_{{m}})$ is any multi-index, and the operator ${D}^{\mathcal{Z}}$ can either be Riemann-Liouville or Caputo or any other kind of integro-differential operator.

Lemma 2.1. For any $\mathcal{Z} > 0$, the Caputo ${FDE}$ $^{c}{D}^{\mathcal{Z}}{X}(\alpha) = 0$ has a solution of the form

where ${e}_{i}\in{R}, i = \overline{0, {n}-1},$ and ${n} = 1+[\mathcal{Z}]$.

Lemma 2.2. For any $\mathcal{Z} > 0$, we have

where ${e}_{i}\in{R}, i = \overline{0, {n}-1}$, and ${n} = 1+[\mathcal{Z}]$.

Lemma 2.3. ^[33] Let $\digamma_{1}\in\mathcal{C}([0, 1], {R})$, and $1 < {m}-1 < \mathcal{B}_{1} < {m}$. Then

has a solution (unique)

where

Let

From $(\mathfrak{C}_{1})$, we know for $\alpha\in(0, 1)$,

and

Implies

3.   Existence of solutions

For our results we need an assumption and the lemma.

$(\mathfrak{C}_{2})$ $\delta_{{S}_{1}}^{-1} = 1-2M_{{S}_{1}}+M'_{{S}_{1}}\neq0$, $\delta_{{S}_{2}}^{-1} = 1+M_{{S}_{2}}-M'_{{S}_{2}}\neq0$ and $\delta_{{S}_{1}}\delta_{{S}_{2}}(2M_{{S}_{2}}-M'_{{S}_{2}})(M'_{{S}_{1}}-M_{{S}_{1}})\neq1$.

Lemma 3.1. Let $(\mathfrak{C}_{2})$ hold, $n-1 < \mathcal{Z}_{1}\leq n$ and $h_{1}:\mathcal{J}\rightarrow {R}$ are is an appropriate function. The ${BVP}$:

has solution(unique) of the form

where

and

Proof. Consider

Using Lemma 2.2, we get

Using boundary conditions of (3.1), we get

and ${e}_{2} = {e}_{3} = \dots = {e}_{n-1} = 0$. Putting ${e}$'s in (3.7) and

where ${G}_{1}(\alpha, \varpi)$ is given in (3.4), $\mathcal{A}_{{S}_{1}} = \int_{0}^{1}{S}_{1}(\varpi){X}(\varpi){d\varpi}$ and $\mathcal{A}_{{S}_{2}} = \int_{0}^{1}{S}_{2}(\varpi){X}(\varpi){d\varpi}$.

In view of (3.8), we get

Integrating (3.9) from $0$ to $1$, we obtain

Implies that

Similarly, we obtain

Implies that

From (3.10) and (3.12), we get

and

Putting (3.12) and (3.13) in (3.8), we get

From (2.3) and (2.4), we can write (3.14) as

□

Lemma 3.2. The coupled ${BVP}$ (1.2) is equivalent to the following system of integral equations

where ${G}_{\cdot}(\alpha, \varpi)$ and ${K}_{\cdot}(\alpha, \varpi)$ are given by (3.3) and (2.2).

Proof. Using Lemmas 2.3 and 3.1, set $\omega(\alpha) = \vartheta_{p}({D}^{\mathcal{Z}_{1}}{X}(\alpha))$ and $h_{1}(\alpha) = \digamma_{1}(\alpha, {D}^{\gamma_{1}}{X}(\alpha), {D}^{\gamma_{2}}{Y}(\alpha))$, and we have

Similarly, we can set $\omega(\alpha) = \vartheta_{p}({D}^{\mathcal{Z}_{2}}{Y}(\alpha))$ and $h_{2}(\alpha) = \digamma_{2}(\alpha, {D}^{\gamma_{1}}{X}(\alpha), {D}^{\gamma_{2}}{Y}(\alpha))$, we obtain

On the other hand, if $({X}, {Y})$ satisfy (3.15), we can easily prove that $({X}, {Y})$ satisfy the pair of boundary value problem (1.2). Hence, the proof is completed. □

Lemma 3.3. We use ${G}_{\mathcal{Z}}(\alpha, \varpi) = ({G}_{\mathcal{Z}_{1}}(\alpha, \varpi), {G}_{\mathcal{Z}_{2}}(\alpha, \varpi))$ and ${K}_{\mathcal{B}}(\alpha, \varpi) = ({K}_{\mathcal{B}_{1}}(\alpha, \varpi), {K}_{\mathcal{B}_{2}}(\alpha, \varpi))$ as the Green's functions of the proposed system (1.2) having the following properties:

$({\bf{I}})$ ${K}_{\mathcal{B}}(\alpha, \varpi)\geq0$ is continuous for all $\alpha, \varpi\in[0, 1]$;

$({\bf{II}})$ ${K}_{\mathcal{B}}(\alpha, \varpi)\leq {K}_{\mathcal{B}}(s, \varpi)$ for all $\alpha, \varpi\in[0, 1]$;

$({\bf{III}})$ $\int_{0}^{1}{K}_{\mathcal{B}_{1}}(\alpha, \varpi){d\varpi} = \frac{1-\alpha^{\mathcal{B}_{1}}}{\Gamma(\mathcal{B}_{1}+1)}\leq\frac{1}{\Gamma(\mathcal{B}_{1}+1)}$ for all $\alpha, \varpi\in[0, 1]$;

$\int_{0}^{1}{K}_{\mathcal{B}_{2}}(\alpha, \varpi){d\varpi} = \frac{1-\alpha^{\mathcal{B}_{2}}}{\Gamma(\mathcal{B}_{2}+1)}\leq\frac{1}{\Gamma(\mathcal{B}_{2}+1)}$ for all $\alpha, \varpi\in[0, 1]$;

$({\bf{IV}})$ ${G}_{\mathcal{Z}}(\alpha, \varpi)\geq0$ is continuous for all $\alpha, \varpi\in[0, 1]$;

$({\bf{V}})$ $\psi(\alpha) = (\psi_{1}(\alpha), \psi_{2}(\alpha))\geq0$ for all $\alpha\in[0, 1]$;

Proof. The proofs of $(I)$ and $(II)$ can be seen in ^[33].

$(III)$ For $\alpha\in[0, 1]$, we have

Also,

$(IV)$ Firstly, from (3.4), one get ${G}_{1}(\alpha, \varpi)\geq0$, $\alpha, \varpi\in[0, 1]$, and from $(\mathfrak{C}_{1})$, $\delta > 0$ and ${G}_{1}(\alpha, \varpi)\geq0$, we have

Thus ${G}_{2}(\alpha, \varpi)$ is a increasing on $\alpha\in[0, 1]$.

Utilizing (3.5), we get

So that ${G}_{\mathcal{Z}_{1}}(\alpha, \varpi)\geq0$. Similarly ${G}_{\mathcal{Z}_{2}}(\alpha, \varpi)\geq0$. Hence, ${G}_{\mathcal{Z}}(\alpha, \varpi)\geq0$.

$(V)$ From $(\mathfrak{C}_{1})$ and (2.5), we know

Thus $\psi_{1}(\alpha)$, is increasing on $\alpha\in[0, 1]$.

From (2.5), we have

Similarly $\psi_{1}(\alpha)\geq0$, and hence $\psi(\alpha)\geq0$. □

Lemma 3.4. For $\kappa\in(0, \frac{1}{2})$, let

where $\rho_{\mathcal{Z}_{1}} = \frac{2M'_{{S}_{1}}-M'_{{S}_{2}}}{1+M_{{S}_{2}}+M'_{{S}_{1}}-M_{{S}_{1}}}$ and $\rho_{\mathcal{Z}_{2}} = \frac{2M'_{{S}_{3}}-M'_{{S}_{4}}}{1+M_{{S}_{4}}+M'_{{S}_{3}}-M_{{S}_{3}}}$.

Proof. First Step. We need to get

For $\varpi < \alpha$, where $\varpi\in[0, 1)$ and $\alpha\in[0, \kappa]$, we have

which implies that ${G}_{1}(\alpha, \varpi)$ is decreasing function monotonically with respect to $\alpha\in[\varpi, \kappa]$, so that

and

Now if $\varpi\geq \alpha$ and $\alpha\in[0, \kappa]$,

which implies that ${G}_{1}(\alpha, \varpi)$ is a monotone decreasing function with respect to $\alpha\in[0, \varpi]$ and $\alpha\in[0, \kappa]$, so that

and

Thus, (3.16) is satisfied.

Similarly, we get

Step 2. We prove

From Lemma 3.3, we know that ${G}_{2}(\alpha, \varpi)$ is increasing for $\alpha\in[0, 1]$, in such a way that

By $(\mathfrak{C}_{1})$ and (2.7), we have

Obviously, $\rho_{\mathcal{Z}_{1}} > 0$, and

Thus $0 < \rho_{\mathcal{Z}_{1}} < \frac{1}{2}$ and (3.18) is satisfied.

By the same procedure,

Finally, from (3.16) and (3.18), we can easily show that the following results hold:

and

Similarly, from (3.17) and (3.19), we can also easily show that

and

□

Lemma 3.5. If $(\mathfrak{C}_{1})$ is satisfied, then $\psi(\alpha)$ holds the following properties:

$({\bf{I}})$ $\psi_{1}(\alpha)\leq\psi_{1}(1) = \max_{\alpha\in[0, 1]}\psi_{1}(\alpha)$ and $\psi_{2}(\alpha)\leq\psi_{2}(1) = \max_{\alpha\in[0, 1]}\psi_{2}(\alpha);$

$({\bf{II}})$ $\min_{\alpha\in[0, \kappa]}\psi_{1}(\alpha)\geq\rho_{\mathcal{Z}_{1}}\max_{\alpha\in[0, 1]}\psi_{1}(\alpha)$ and $\min_{\alpha\in[0, \kappa]}\psi_{2}(\alpha)\geq\rho_{\mathcal{Z}_{2}}\max_{\alpha\in[0, 1]}\psi_{2}(\alpha)$.

Proof. Using Lemma 3.3 and (2.5), implies $\psi_{1}(\alpha)$ is increasing on $\alpha\in[0, 1]$, and thus

and

Setting $\mathtt{H}_{2}(0) = 2\mathtt{H}_{1}(0)$ and $\mathtt{H}_{1}(1) = 0$, then (3.20) implies

Also,

Thus,

□

4.   Major findings

Suppose that $\mu_{1}$ and $\mu_{2}$ be nonnegative convex and continuous functionals on $\Theta$, $\mu_{3}$ be nonnegative concave and continuous functional on $\Theta$ and $\mu_{4}$ be a nonnegative continuous functional on $\Theta$. For ${M}_{1}, {M}_{2}, {M}_{3}, {M}_{4} > 0$, let us introduce the convex sets:

and a closed set

Lemma 4.1. Let $\Theta$ be a cone in a real Banach space $\chi$. Let $\mu_{1}$ and $\mu_{2}$ be nonnegative continuous convex functionals on $\Theta$, $\mu_{3}$ be nonnegative continuous concave functional on $\Theta$ and $\mu_{4}$ be a nonnegative continuous functionals on $\Theta$ satisfying $\mu_{4}(\lambda({X}, {Y}))\leq\lambda\mu_{4}({X}, {Y})$ for $0\leq\lambda\leq1$, such that for some positive numbers $\mathcal{N}$ and ${M}_{1}$,

for all $({X}, {Y})\in\overline{\Theta(\mu_{1}, {M}_{1})}$. Suppose

is completely continuous and there exist positive numbers ${M}_{2}$, ${M}_{3}$ and ${M}_{4}$ with ${M}_{4} < {M}_{3}$ such that

$(\mathfrak{C}_{3}):$ $\{({X}, {Y})\in\Theta(\mu_{1}, \mu_{2}, \mu_{3}$; ${M}_{3}, {M}_{2}, {M}_{1}): \mu_{3}({X}, {Y}) > {M}_{3}\}\neq\emptyset$ and $\mu_{3}({X}, {Y}) > {M}_{3}$ for $({X}, {Y})\in\Theta(\mu_{1}, \mu_{2}, \mu_{3}$; ${M}_{3}, {M}_{2}, {M}_{1})$;

$(\mathfrak{C}_{4}):$ $\mu_{3}(\mathcal{T}({X}, {Y})) > {M}_{3}$ for $({X}, {Y})\in\Theta(\mu_{1}, \mu_{3};{M}_{3}, {M}_{1})$ with $\mu_{2}(\mathcal{T}({X}, {Y})) > {M}_{4}$;

$(\mathfrak{C}_{5}):$ $(0, 0)\notin\Theta(\mu_{1}, \mu_{4};{M}_{4}, {M}_{1})$ and $\mu_{4}(\mathcal{T}({X}, {Y})) < {M}_{4}$ for $({X}, {Y})\in\Theta(\mu_{1}, \mu_{4};{M}_{4}, {M}_{1})$ with $\mu_{4}({X}, {Y}) = {M}_{4}$.

Then, $\mathcal{T}$ has at least three fixed points $({X}_{1}, {Y}_{1}), ({X}_{2}, {Y}_{2}), ({X}_{3}, {Y}_{3})\in\overline{\Theta(\mu_{1}, {M}_{1})}$ such that

Let $\chi = \{({X}, {Y})\in\mathcal{C}([0, 1], {R})\times\mathcal{C}([0, 1], {R})\}$ be endowed with the norm $\|({X}, {Y})\| = \max_{\alpha\in[0, 1]}|({X}, {Y})|,$ then $\chi$ is a Banach space.

We define a set $\Theta\subset\chi$ by

For $({X}, {Y}), (\widetilde{{X}}, \widetilde{{Y}})\in\Theta$ and $\nu_{1}, \nu_{2}\geq0$, it is easy to obtain that $\nu_{1}({X}, {Y})(\alpha)+\nu_{2}(\widetilde{{X}}, \widetilde{{Y}})(\alpha)\geq0$, and

Thus, for $({X}, {Y}), (\widetilde{{X}}, \widetilde{{Y}})\in\Theta$ and $\nu_{1}, \nu_{2}\geq0$, $\nu_{1}({X}, {Y})(\alpha)+\nu_{2}(\widetilde{{X}}, \widetilde{{Y}})(\alpha)\in\Theta$. And if $({X}, {Y})\in\Theta$, $({X}, {Y})\neq0$, it can be seen that $-({X}, {Y})\notin\Theta$. Thus, $\Theta$ is a cone in $\chi$.

Let $\mathcal{T}:\Theta\rightarrow \chi$ be an operator, i.e., $\mathcal{T} = (\mathcal{T}_{1}, \mathcal{T}_{2})$, defined by

Lemma 4.2. If $(\mathfrak{C}_{1})$ is true, then $\mathcal{T}:\Theta\rightarrow \Theta$ is a completely continuous operator.

Proof. For $({X}, {Y})\in\Theta$, it is easy to know that $\mathcal{T}$ is a continuous operator, and $\mathcal{T}({X}, {Y})(\alpha)\geq0$. By (3.15), we have

Similarly, for $\min_{\alpha\in[0, \kappa]}\mathcal{T}_{2}({X}, {Y})(\alpha)\geq\rho_{\mathcal{Z}_{2}}\max_{\alpha\in[0, 1]}\mathcal{T}_{2}({X}, {Y})(\alpha)$. So, $\mathcal{T}(\Theta)\subseteq\Theta$.

Set $\Theta_{\tau}\subset\Theta$ be bounded, i.e., $\exists$ a constant $\tau > 0$ in such a way that $\|({X}, {Y})\|\leq\tau$, for every $({X}, {Y})\in\Theta_{\tau}$. Let $M_{0} = \max_{\alpha\in[0, 1], ({X}, {Y})}|\digamma_{1}(\alpha, {D}^{\gamma_{1}}{X}(\alpha), {D}^{\gamma_{2}}{Y}(\alpha))| > 0$ and $N_{0} = \max_{\alpha\in[0, 1], ({X}, {Y})}|\digamma_{2}(\alpha, {D}^{\gamma_{1}}{X}(\alpha), {D}^{\gamma_{2}}{Y}(\alpha))| > 0$. For $({X}, {Y})\in\Theta_{\tau}$, from Lemmas 3.3 and 3.4, we have

and

Similarly,

and

So that $\mathcal{T}({X}, {Y})\leq\max\{M_{01}, N_{01}\}$, which implies $\mathcal{T}(\Theta_{\tau})$ is uniformly bounded. Next, we will show that $\mathcal{T}(\Theta_{\tau})$ is equicontinuous. Since ${G}_{\mathcal{Z}}(\alpha, \varpi)$, $\psi(\alpha)$ are continuous on $[0, 1]\times[0, 1]$, it is uniformly continuous on $[0, 1]\times[0, 1]$. Thus, for any $\epsilon > 0$, there exists a constant $\varsigma_{1} > 0$, such that

for $\alpha_{1}, \alpha_{2}\in[0, 1]$ with $|\alpha_{1}-\alpha_{2}| < \varsigma_{1}$. Therefore,

Similarly

Combining (4.4) and (4.6), we obtain

Implies, $\mathcal{T}(\Theta_{\tau})$ is equicontinuous.

Thus Arzel$\grave{a}$-Ascoli Theorem implies that $\mathcal{T}$ is a completely continuous operator.

□

Define the functionals

then $\mu_{1}$ and $\mu_{2}$ are continuous nonnegative convex functionals, $\mu_{3}$ is a continuous nonnegative concave functional, $\mu_{4}$ is a continuous nonnegative functional, and

Thus, (4.1) in Lemma 4.1 is fulfilled.

Let

Theorem 4.1. Let $(\mathfrak{C}_{1})$ is true, and $\exists$ constants ${M}_{1}, {M}_{3}, {M}_{4}\geq\psi(1)$ with ${M}_{4} < {M}_{3} < \rho{M}_{1}\min{\frac{l_{2}l_{4}}{l_{1}l_{3}}}$ and ${M}_{2} = \frac{{M}_{3}}{\rho}$, in such a way that

$(\mathfrak{C}_{6})$ $\digamma_{1}(\alpha, {X}, {Y})\leq\vartheta_{p}(\frac{{M}_{1}-\psi_{1}(1)}{l_{1}})$ and $\digamma_{2}(\alpha, {X}, {Y})\leq\vartheta_{p}(\frac{{M}_{1}-\psi_{2}(1)}{l_{2}})$, $(\alpha, {X}, {Y})\in[0, 1]\times[0, {M}_{1}]\times[0, {M}_{1}];$

$(\mathfrak{C}_{7})$ $\digamma_{1}(\alpha, {X}, {Y}) > \vartheta_{p}(\frac{{M}_{3}-\rho_{\mathcal{Z}_{1}}\psi_{1}(1)}{\rho_{\mathcal{Z}_{1}}l_{3}})$ and $\digamma_{2}(\alpha, {X}, {Y}) > \vartheta_{p}(\frac{{M}_{3}-\rho_{\mathcal{Z}_{2}}\psi_{2}(1)}{\rho_{\mathcal{Z}_{2}}l_{4}})$, $(\alpha, {X}, {Y})\in[0, \kappa]\times[{M}_{3}, \frac{{M}_{3}}{\rho_{\mathcal{Z}_{1}}}]\times[0, {M}_{1}];$

$(\mathfrak{C}_{8})$ $\digamma_{1}(\alpha, {X}, {Y}) < \vartheta_{p}(\frac{{M}_{4}-\psi_{1}(1)}{l_{1}})$ and $\digamma_{2}(\alpha, {X}, {Y}) < \vartheta_{p}(\frac{{M}_{4}-\psi_{2}(1)}{l_{2}})$, $(\alpha, {X}, {Y})\in[0, 1]\times[0, {M}_{4}]\times[0, {M}_{1}].$

Then, (1.2) has at least three positive solutions $({X}_{1}, {Y}_{1})$, $({X}_{2}, {Y}_{2})$, $({X}_{3}, {Y}_{3})$ satisfying

Proof. Obviously, the fixed points of $\mathcal{T}$ are equivalent to the solutions of (1.2). For $({X}, {Y})\in\overline{\Theta(\mu_{1}, {M}_{1})}$, we get

thus

and then

By $(\mathfrak{C}_{6})$, we have

Similarly,

So,

Thus, $\mathcal{T}:\overline{\Theta(\mu_{1}, {M}_{1})}\rightarrow \overline{\Theta(\mu_{1}, {M}_{1})}$.

For $({X}, {Y}) = \frac{{M}_{3}}{\rho}$, $({X}, {Y})(\alpha)\in\Theta(\mu_{1}, \mu_{2}, \mu_{3};{M}_{3}, {M}_{2}, {M}_{1})$, $\mu_{3}(\frac{{M}_{3}}{\rho}) > {M}_{3}$, we get

For $({X}, {Y})(\alpha)\in\Theta(\mu_{1}, \mu_{2}, \mu_{3};{M}_{3}, {M}_{2}, {M}_{1})$, we know that ${M}_{3}\leq({X}, {Y})(\alpha)\leq{M}_{2} = \frac{{M}_{3}}{\rho}$ for $\alpha\in[0, \kappa]$.

In view of $(\mathfrak{C}_{7})$,

Similarly,

So, $\mu_{3}(\mathcal{T}({X}, {Y})) > {M}_{3}$ for all $({X}, {Y})(\alpha)\in\Theta(\mu_{1}, \mu_{2}, \mu_{3};{M}_{3}, {M}_{2}, {M}_{1})$. Hence, $(\mathfrak{C}_{3})$ of Lemma 4.1 is fulfilled.

By (4.7), for all $({X}, {Y})\in\Theta(\mu_{1}, \mu_{3};{M}_{3}, {M}_{1})$ with $\mu_{2}(\mathcal{T}({X}, {Y})) > {M}_{2} = \frac{{M}_{3}}{\rho}$, one get

Implies $(\mathfrak{C}_{4})$ of Lemma 4.1 is true.

As $\mu_{4}(0, 0) = 0 < {M}_{4}$, thus $0\notin\Theta(\mu_{1}, \mu_{4};{M}_{4}, {M}_{1})$. If $({X}, {Y})\in\Theta(\mu_{1}, \mu_{4};{M}_{4}, {M}_{1})$ with $\mu_{4}({X}, {Y}) = {M}_{4}$, we deduced $\mu_{1}({X}, {Y})\leq{M}_{1}$. Thus, $\max_{\alpha\in[0, 1]}({X}, {Y})(\alpha) = {M}_{4}$ and $0\leq({X}, {Y})(\alpha)\leq{M}_{1}$.

Using $(\mathfrak{C}_{8})$, we get

Similarly,

So, $\mu_{3}(\mathcal{T}({X}, {Y})) < {M}_{3}$ for all $({X}, {Y})(\alpha)\in\Theta(\mu_{1}, \mu_{4};{M}_{4}, {M}_{1})$. Therefore, the condition $(\mathfrak{C}_{4})$ of Lemma 4.1 hold.

To sum up, all the conditions of Lemma 4.1 are verified, and it was noticed that $({X}_{i}, {Y}_{i})(\alpha)\geq$$\psi(0)$$> 0$. Hence, the coupled system (1.2) has at least three positive solutions $({X}_{1}, {Y}_{1})$, $({X}_{2}, {Y}_{2})$, $({X}_{3}, {Y}_{3})$ satisfying (4.7) and (4.8). □

5.   Stability results

Definition 5.1. The system (1.2) is ${HU}$ stable if there exist $\pi_{\mathcal{Z}_{1}, \mathcal{Z}_{2}} = \max\{\pi_{\mathcal{Z}_{1}}, \pi_{\mathcal{Z}_{2}}\} > 0$ in such a way that, for $\varepsilon = \max\{\varepsilon_{\mathcal{Z}_{1}}, \varepsilon_{\mathcal{Z}_{2}}\} > 0$ and for any $({X}, {Y})\in\chi$ satisfying

$\exists$ $(\widehat{{X}}, \widehat{{Y}})\in\chi$ satisfying (1.2) such that

Definition 5.2. The coupled implicit ${FDE}s$ (1.2) is $\mathcal{GHU}$ stable if there exist $\Phi\in\mathcal{C}({R}^{+}, {R}^{+})$ with $\Phi(0) = 0$ such that, for any solution $({X}, {Y})\in\chi$ of inequality (5.1), there exists a solution $(\widehat{{X}}, \widehat{{Y}})\in\chi$ of (1.2) fulfilling

Let $\Psi_{{\mathcal{Z}_{1}}, {\mathcal{Z}_{2}}} = \max\{\Psi_{\mathcal{Z}_{1}}, \Psi_{\mathcal{Z}_{2}}\}\in\mathcal{C}([0, 1], {R})$, and $\pi_{\Psi_{\mathcal{Z}_{1}}, \Psi_{\mathcal{Z}_{2}}} = \max\{\pi_{\Psi_{\mathcal{Z}_{1}}}, \pi_{\Psi_{\mathcal{Z}_{2}}}\} > 0$.

Definition 5.3. The coupled system of implicit ${FDE}s$ (1.2) is said to be ${HUR}$ stable with respect to $\Psi_{{\mathcal{Z}_{1}}, {\mathcal{Z}_{2}}}$ if there exists constants $\pi_{\Psi_{\mathcal{Z}_{1}}, \Psi_{\mathcal{Z}_{2}}}$ such that, for some $\varepsilon > 0$ and for any solution $({X}, {Y})\in\chi$ of the inequality

there exists $(\widehat{{X}}, \widehat{{Y}})\in\chi$ satisfying (1.2) in such a way that

Definition 5.4. The system (1.2) is said to be ${HUR}$ stable with respect to $\Psi_{\mathcal{Z}_{1}, \mathcal{Z}_{2}}$ if there exists constants $\pi_{\Psi_{\mathcal{Z}_{1}}, \Psi_{\mathcal{Z}_{2}}}$ such that, for any approximate solution $({X}, {Y})\in\chi$ of inequality (5.2), there exists a solution $(\widehat{{X}}, \widehat{{Y}})\in\chi$ of (1.2) fulfilling

Remark 5.1. We say that $({X}, {Y})\in\chi$ is a solution of the system of inequalities (5.1) if there exist functions $\Lambda_{f}$, $\Lambda_{g}\in\mathcal{C}([0, 1], {R})$ depending upon ${X}, {Y},$ respectively, such that

$({\bf{I}})$

$({\bf{II}})$

Lemma 5.1. Under the assumptions given in Remark 5.1, the solution $({X}, {Y})\in\chi$ of coupled system

satisfies the system of inequalities

Proof. In light of Lemma 3.2, a solution of the coupled system (5.1) is

Using (4.2) in (5.6), we have

Similarly, using (4.3) in (5.6), we get

□

For the next result, we suppose the following condition.

$(\mathfrak{C}_{9})$ Let ${N}_{1}, {N}_{2}, {S}_{1}, {S}_{2}\in\mathcal{C}([0, 1], {R})$, and there exists $L_{1}, L_{2}, L_{3}, L_{4} > 0$ such that

Theorem 5.1. Suppose the condition $(\mathfrak{C}_{9})$ and Lemma 5.1 satisfies, then (1.2) is ${HU}$ stable, if $\upsilon_{1}\upsilon_{4}-\upsilon_{2}\upsilon_{3} = \upsilon_{4}-\upsilon_{2} > 0$, where

Proof. Consider $({X}, {Y})\in\chi$ to be any solution of (5.5), and $(\widehat{{X}}, \widehat{{Y}})\in\chi$ is a solution of the coupled system (1.2), then we take

implies that

Similarly, we have

From (5.7) and (5.8), we get

implies

From system (5.9), we have

which implies that

If $\max\{\vartheta_{q}(\varepsilon_{\mathcal{Z}_{1}}), \vartheta_{q}(\varepsilon_{\mathcal{Z}_{2}})\} = \vartheta_{q}(\varepsilon_{\mathcal{Z}})$ and $\frac{\upsilon_{1}}{\upsilon_{4}-\upsilon_{2}}\frac{N_{01}-\psi_{2}(1)}{\vartheta_{q}(N_{0})}+\frac{\upsilon_{2}}{\upsilon_{4}-\upsilon_{2}}\frac{N_{01}-\psi_{2}(1)}{\vartheta_{q}(N_{0})} +\frac{\upsilon_{3}}{\upsilon_{4}-\upsilon_{2}}\frac{M_{01}-\psi_{1}(1)}{\vartheta_{q}(M_{0})}+\frac{\upsilon_{4}}{\upsilon_{4}-\upsilon_{2}}\frac{M_{01}-\psi_{1}(1)}{\vartheta_{q}(M_{0})} = \upsilon_{\mathcal{Z}_{1}, \mathcal{Z}_{2}}$, then

Hence, system (1.2) is ${HU}$ stable. Also, if

with $\Phi(0) = 0$, then the solution of system (1.2) is ${GHU}$ stable. □

Theorem 5.2. Suppose the conditions $(\mathfrak{C}_{9})$ and Lemma 5.1 holds, then (1.2) is ${GHU}$ stable, if $\upsilon_{1}\upsilon_{4}-\upsilon_{2}\upsilon_{3} = \upsilon_{4}-\upsilon_{2} > 0$, where $\upsilon_{1}$, $\upsilon_{2}$, $\upsilon_{3}$ and $\upsilon_{4}$ is defined in Theorem 5.1.

Proof. By applying steps of Theorem 5.1, we can easily show that system (1.2) is ${GHU}$ stable, by using Definition 5.4. □

For the next theorem, we assume that

$(\mathfrak{C}_{10})$ $\exists$ non-decreasing functions $\bar{w}_{\mathcal{Z}}, \bar{w}_{\mathcal{B}}\in\mathcal{C}([0, 1], {R}^{+})$ in such a way that

and

where $\mathcal{L}_{\mathcal{Z}}, \mathcal{L}_{\mathcal{B}} > 0$.

Theorem 5.3. Suppose the conditions $(\mathfrak{C}_{9})$, $(\mathfrak{C}_{10})$ and Lemma 5.1 satisfies, then system (1.2) is ${HUR}$ stable if $\upsilon_{1}\upsilon_{4}-\upsilon_{2}\upsilon_{3} = \upsilon_{4}-\upsilon_{2} > 0$.

Proof. By applying steps of Theorem 5.1, we can easily show that system (1.2) is ${HUR}$ stable, by using Definition 5.3. □

Theorem 5.4. Suppose the conditions $(\mathfrak{C}_{9})$, $(\mathfrak{C}_{10})$ and Lemma 5.1 holds, then system (1.2) is ${GHUR}$ stable if $\upsilon_{1}\upsilon_{4}-\upsilon_{2}\upsilon_{3} = \upsilon_{4}-\upsilon_{2} > 0$.

Proof. By applying steps of Theorem 5.1, we can easily show that system (1.2) is ${GHUR}$ stable, by using Definition 5.4. □

6.   Example

In this section, we demonstrate an example to illustrate the main results.

Example 6.1. Consider the following system of implicit ${FDE}s$:

Choosing ${M}_{1} = 25000$, ${M}_{3} = 65$, ${M}_{4} = 3$ and $\kappa = \frac{1}{3}$. Reminding $\psi_{1}$, $\psi_{2}$ from (2.5), (2.6) and $\rho_{\mathcal{Z}_{1}}$ and $\rho_{\mathcal{Z}_{2}}$ from Lemma 3.4, we get

In (6.1), we see that

which satisfies the following conditions:

$(\mathfrak{C}_{6})$ $\sup \digamma_{1}(\alpha, {X}, {Y})\approx246.534\leq\min\vartheta_{p}(\frac{{M}_{1}-\psi_{1}(1)}{l_{1}})\approx291.732$,

$\sup \digamma_{2}(\alpha, {X}, {Y})\approx241.492\leq\min\vartheta_{p}(\frac{{M}_{1}-\psi_{2}(1)}{l_{2}})\approx284.618$, $(\alpha, {X}, {Y})\in[0, 1]\times[0, 25000]\times[0, 25000];$

$(\mathfrak{C}_{7})$ $\inf \digamma_{1}(\alpha, {X}, {Y})\approx240.631 > \vartheta_{p}(\frac{{M}_{3}-\rho_{\mathcal{Z}_{1}}\psi_{1}(1)}{\rho_{\mathcal{Z}_{1}}l_{3}})\approx201.981$ $(\alpha, {X}, {Y})\in[0, \frac{1}{3}]\times[65, \frac{{M}_{3}}{\rho_{\mathcal{Z}_{1}}}]\times[0, 25000];$

$\inf \digamma_{2}(\alpha, {X}, {Y})\approx271 > \vartheta_{p}(\frac{{M}_{3}-\rho_{\mathcal{Z}_{2}}\psi_{2}(1)}{\rho_{\mathcal{Z}_{2}}l_{4}})\approx201.843$, $(\alpha, {X}, {Y})\in[0, \frac{1}{3}]\times[65, \frac{{M}_{3}}{\rho_{\mathcal{Z}_{2}}}]\times[0, 25000];$

$(\mathfrak{C}_{8})$ $\sup \digamma_{2}(\alpha, {X}, {Y})\approx198.938 < \vartheta_{p}(\frac{{M}_{4}-\psi_{1}(1)}{l_{1}})\approx243.861$,

$\sup \digamma_{2}(\alpha, {X}, {Y})\approx183.861 < \vartheta_{p}(\frac{{M}_{4}-\psi_{2}(1)}{l_{2}})\approx202.991$, $(\alpha, {X}, {Y})\in[0, 1]\times[0, 3]\times[0, 25000].$

Then, all assumptions of Theorem 4.1 are satisfied. Thus, ${BVP}$ (6.1) has at least three positive solutions $({X}_{1}, {Y}_{1})$, $({X}_{2}, {Y}_{2})$, $({X}_{3}, {Y}_{3})$ satisfying

For ${HUS}$, we found $L_{1} = L_{2} = \frac{1}{150}$, $L_{3} = L_{4} = \frac{1}{160}$, $\upsilon_{1}\approx0.02961$, $\upsilon_{2}\approx0.9721$, $\upsilon_{3}\approx0.03291$ and $\upsilon_{4}\approx0.9948$. Hence, by Theorem 5.1, we have $\upsilon_{4}-\upsilon_{2}\approx0.9948-0.0227 > 0$.

7.   Conclusions

In this paper, we considered an implicit coupled ${BVP}$ of $p$-Laplacian ${FDE}s$ (1.2), which involved disturbing functions. The relating fractional order derivative is taken in Caputo sense. By using the Avery-Peterson fixed point theorem for the proposed problem, we found at least three solutions, under sufficient conditions. In addition, we presented four types of Ulam's stability, i.e., Hyers-Ulam stability, generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability and generalized Hyers-Ulam-Rassias stability, for the coupled implicit fractional $p$-Laplacian system, and an example is provided to authenticate the theoretical results.

Acknowledgments

This research is supported by the scientific research project of Anhui Provincial Department of Education (KJ2021A1155; KJ2020A0780).

Conflict of interest

The authors declare no conflict of interest.