Value recovery from spent lithium-ion batteries: A review on technologies, environmental impacts, economics, and supply chain

Majid Alipanah; Apurba Kumar Saha; Ehsan Vahidi; Hongyue Jin; Majid Alipanah; Apurba Kumar Saha; Ehsan Vahidi; Hongyue Jin

doi:10.3934/ctr.2021008

Clean Technologies and Recycling

2021, Volume 1, Issue 2: 152-184. doi: 10.3934/ctr.2021008

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Review Special Issues

Value recovery from spent lithium-ion batteries: A review on technologies, environmental impacts, economics, and supply chain

1.
Department of System and Industrial Engineering, University of Arizona, 1127 E. James E. Rogers Way, Tucson, Arizona 85721, United States
2.
Department of Mining and Metallurgical Engineering, Mackay School of Earth Sciences and Engineering, University of Nevada, 1664 N. Virginia Street, Reno, Nevada 89557, United States
^# These authors contributed equally

Received: 11 August 2021 Accepted: 30 November 2021 Published: 03 December 2021

The demand for lithium-ion batteries (LIBs) has surged in recent years, owing to their excellent electrochemical performance and increasing adoption in electric vehicles and renewable energy storage. As a result, the expectation is that the primary supply of LIB materials (e.g., lithium, cobalt, and nickel) will be insufficient to satisfy the demand in the next five years, creating a significant supply risk. Value recovery from spent LIBs could effectively increase the critical materials supply, which will become increasingly important as the number of spent LIBs grows. This paper reviews recent studies on developing novel technologies for value recovery from spent LIBs. The existing literature focused on hydrometallurgical-, pyrometallurgical-, and direct recycling, and their advantages and disadvantages are evaluated in this paper. Techno-economic analysis and life cycle assessment have quantified the economic and environmental benefits of LIB reuse over recycling, highlighting the research gap in LIB reuse technologies. The study also revealed challenges associated with changing battery chemistry toward less valuable metals in LIB manufacturing (e.g., replacing cobalt with nickel). More specifically, direct recycling may be impractical due to rapid technology change, and the economic and environmental incentives for recycling spent LIBs will decrease. As LIB collection constitutes a major cost, optimizing the reverse logistics supply chain is essential for maximizing the economic and environmental benefits of LIB recovery. Policies that promote LIB recovery are reviewed with a focus on Europe and the United States. Policy gaps are identified and a plan for sustainable LIB life cycle management is proposed.

Keywords:

Citation: Majid Alipanah, Apurba Kumar Saha, Ehsan Vahidi, Hongyue Jin. Value recovery from spent lithium-ion batteries: A review on technologies, environmental impacts, economics, and supply chain[J]. Clean Technologies and Recycling, 2021, 1(2): 152-184. doi: 10.3934/ctr.2021008

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Abstract

1. Introduction

Over the past decades, the study of nonlinear problems has been the interest of many researchers ^{[5,10,11,14,19,24,25,26]}. Also, study of fractional calculus has recently gained great momentum, and has emerged as a significant research area ^{[5,7,15,20,21,30]}. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes; see, for instance, the contribution ^{[2,3,4,6,16,22,23,28,29]} and references therein.

The authors in ^[8] focused on the study of nonlinear jerk problems due to its various physical applications, as form

$\begin{equation*} \frac{{\mathrm{d}}^{3} {\mathrm{y}}}{ {\mathrm{d}}\mathfrak{t}^{3}} = \mathfrak{T} \left( {\mathrm{y}}, \frac{ {\mathrm{dy}}}{ {\mathrm{d}}\mathfrak{t}}, \frac{{\mathrm{d}}^{2} {\mathrm{y}}}{ {\mathrm{d}} \mathfrak{t}^{2}} \right). \end{equation*}$

In 2020, the authors investigated the existence and uniqueness of solutions for the following nonlocal generalized fractional Sturm-Liouville and Langevin equations:

$\begin{equation*} \left\{ \begin{array}{ll} {^{c}\mathcal{D}}_{\iota_1^{+}}^{\alpha} \left( \left[ p(\mathfrak{t}) {^{c}\mathcal{D}}_{\iota_1^{+}}^{\gamma} + q(t) \right] {\mathrm{y}}(\mathfrak{t}) \right) = \mathfrak{T} (\mathfrak{t}, {\mathrm{y}}(\mathfrak{t})), & \mathfrak{t} \in \lbrack 0, T], \alpha , \gamma \in (0, 1],\\ {\mathrm{y}}(0)+ \varkappa_1({\mathrm{y}}) = {\mathrm{y}}_1\in \mathbb{R}, &\\ {^{c}\mathcal{D}}_{\iota_1^{+}}^{\gamma} {\mathrm{y}}(T) + \varkappa_2({\mathrm{y}}) = {\mathrm{y}}_2\in \mathbb{R},& \end{array}\right. \end{equation*}$

where ${^{c}\mathcal{D}}_{\iota_1^{+}}^{ \alpha}$ , ${^{c}\mathcal{D}}_{\iota_1^{+}}^{ \gamma}$ are the Caputo fractional derivatives, $p, q \in C([0, T])$ with $|p|\geq K > 0$ , $\varkappa_1, \varkappa_2 : C(J) \to \mathbb{R}$ are continuous functions and $\mathfrak{T} \in C([0, T] \times \mathbb{R})$ ^[27]. The second derivative of the accelaration (fourth derivative of position) is a physical quantity called a snap or jounce, which can be modeled as

$\begin{equation} \left\{\begin{array}{l} \frac{{\mathrm{dy}}_{1}}{{\mathrm{d}}\mathfrak{t}} = {\mathrm{y}}_{2}(\mathfrak{t}), \quad \frac{{\mathrm{dv}}_{2}}{{\mathrm{d}}\mathfrak{t}} = {\mathrm{y}}_{3}(\mathfrak{t}), \quad \frac{{\mathrm{dy}}_{3}}{{\mathrm{d}}\mathfrak{t}} = {\mathrm{y}}_{4}(\mathfrak{t}), \\[0.2cm] \frac{{\mathrm{dy}}_{4}}{ {\mathrm{d}}\mathfrak{t}} = \mathfrak{T}({\mathrm{y}}_{1}, {\mathrm{y}}_{2}, {\mathrm{y}}_{3}, {\mathrm{y}}_{4}). \end{array}\right. \end{equation}$

(1.1)

It is obvious that the model (1.1) can be reduced to the following equation:

$\begin{equation} \frac{{\mathrm{d}}^{4} {\mathrm{y}}_{1}}{ {\mathrm{d}}\mathfrak{t}^{4}} = \mathfrak{T} \left( {\mathrm{y}}_{1}, \frac{ {\mathrm{dy}}_{1}}{ {\mathrm{d}}\mathfrak{t}}, \frac{{\mathrm{d}}^{2} {\mathrm{y}}_{1}}{ {\mathrm{d}} \mathfrak{t}^{2}}, \frac{{\mathrm{d}}^{3} {\mathrm{y}}_{1}}{ {\mathrm{d}} \mathfrak{t}^{3}}\right). \end{equation}$

(1.2)

Scientifically, jerk and snap are the third and fourth derivatives of our position with regard to time, respectively. The Eq (1.1) contains a $4^{{\mathrm{th}}}$ -order derivative of the variable ${\mathrm{y}}_{1}$ , and it describes a $4^{{\mathrm{th}}}$ -order dynamical vibration model.

The corresponding fractional model is achieved by using the fractional derivative (of order less than or equal $1$ ) instead of the standard deivative $\frac{{\mathrm{d}}}{{\mathrm{d}}\mathfrak{t}}$ . Many types of fractional derivatives can be used here, such as Riemann-Liouville, Caputo, Hadamard, etc. We prefer to use the generalized fractional derivative ( $\mathbb{GFD}$ ), with respect to differentiable increasing function $\mathbb{G}$ . In 2020, Liu {et al.}, developed two iterative algorithms to determine the periods, and then the periodic solutions of nonlinear jerk equations for two possible cases with initial values unknown and initial values given ^[13]. The authors in a recent article ^[23] considered the $\mathbb{G}$ -fractional snap model ( $\mathbb{GF}$ SM) with constant, initial conditions

$\begin{equation} \left\{\begin{array}{ll} {^{c}\mathcal{D}}_{\iota_1^{+}}^{\alpha; \mathbb{G}}{\mathrm{y}}(\mathfrak{t}) = {\mathrm{y}}_1(\mathfrak{t}),& {\mathrm{y}}(\iota_1) = {\mathrm{v}}_{0},\\[0.4cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{\beta; \mathbb{G}} {\mathrm{y}}_1(\mathfrak{t}) = {\mathrm{y}}_2(\mathfrak{t}), & {\mathrm{y}}_1(\iota_1) = {\mathrm{v}}_{1},\\[0.4cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{\gamma ; \mathbb{G} } {\mathrm{y}}_2( \mathfrak{t}) = {\mathrm{y}}_3 (\mathfrak{t}), & {\mathrm{y}}_2(\iota_1) = {\mathrm{v}}_{2},\\[0.4cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{\delta ;\mathbb{G}} {\mathrm{y}}_3( \mathfrak{t}) = \mathcal{T}(\mathfrak{t}, {\mathrm{y}},{\mathrm{y}}_1, {\mathrm{y}}_2, {\mathrm{y}}_3), & {\mathrm{y}}_3(\iota_1) = {\mathrm{v}}_{3}, \end{array}\right. \end{equation}$

(1.3)

where the $\mathbb{G}$ -Caputo derivatives are illustrated by the symbol

${}^{c}\mathcal{D}_{\iota_1^{+}}^{ \eta ;\mathbb{G}}, \qquad \eta\in \Big\{ \alpha, \beta, \gamma, \delta \Big\}, \, 0 < \eta < 1,$

here and the increasing function $\mathbb{G}\in C^{1}([\iota_1, \iota_2])$ is such that $\mathbb{G}^{\prime }(\mathfrak{t})\neq 0$ , for each $\mathfrak{t}\in \lbrack \iota_1, \iota_2]$ and continuous function $\mathcal{T}$ belongs to $C(\lbrack \iota_1, \iota_2] \times \mathbb{R}^{4})$ and ${\mathrm{y}}_{0}, {\mathrm{y}}_{1}, {\mathrm{y}}_{2}, {\mathrm{y}}_{3} \in \mathbb{R}$ . Abbas {et al.} studied the following coupled system of fractional differential equations:

$\begin{equation*} \left\{ \begin{array}{ll} {^{RL}\mathcal{D}}_{\iota_1^{+}}^{\alpha_1; \varrho} {\mathrm{y}}_1(\mathfrak{t}) = \mathfrak{T}_1(\mathfrak{t}, {\mathrm{y}}_1(\mathfrak{t}), {\mathrm{y}}_2(\mathfrak{t})), &\\ {^{RL}\mathcal{D}}_{\iota_1^{+}}^{\alpha_2; \varrho} {\mathrm{y}}_2(\mathfrak{t}) = \mathfrak{T}_2(\mathfrak{t}, {\mathrm{y}}_1(\mathfrak{t}), {\mathrm{y}}_2(\mathfrak{t})), \end{array}\right. \end{equation*}$

for $\mathfrak{t} \in [ \iota_1, \iota_2]$ equipped with the generalized fractional integral boundary conditions

$\begin{equation*} \left\{ \begin{array}{ll} {\mathrm{y}}_1(\tau_1) = 0, & {\mathrm{y}}_1(\iota_2) = {\mathcal{I}}_{\iota_1^{+}}^{\zeta_1; \varrho} {\mathrm{y}}_1({\mathtt{η}}_1), \\ {\mathrm{y}}_2(\tau_2) = 0, & {\mathrm{y}}_2(\iota_2) = {\mathcal{I}}_{\iota_1^{+}}^{\zeta_2; \varrho} {\mathrm{y}}_2({\mathtt{η}}_2), \end{array}\right. \end{equation*}$

where $\varrho \in (0, 1]$ , ${^{RL}\mathcal{D}}_{\iota_1^{+}}^{\alpha_i; \varrho}$ denotes the generalized proportional fractional derivatives of Riemann-Liouville type of order $1 < \alpha_i\leq 2$ , ${\mathcal{I}}_{\iota_1^{+}}^{\zeta_i; \varrho}$ that denotes the generalized proportional fractional integrals of order $0 < \zeta_i < 1$ and $\tau_i, {\mathtt{η}}_i \in (\iota_1, \iota_2)$ and $\mathfrak{T}_i \in C([\iota_1, \iota_2] \times \mathbb{R}^2)$ ^[1].

We center our consideration on the problem of the existence and uniqueness along with the Hyers-Ulam stability $({\mathrm{U}}$ - ${\mathrm{H}}$ - $\mathbb{S})$ of solutions for fractional nonlinear couple snap system $(\mathbb{C}{\mathrm{S}}\mathbb{S})$ in the $\mathbb{G}$ -Caputo sense $(\mathbb{G}{\mathrm{C}})$ with initial conditions

$\begin{equation} \left\{ \begin{array}{l} {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1}; \mathbb{G}} {\mathrm{v}}_{1}( \mathfrak{t}) = {\mathrm{u}}_{1}( \mathfrak{t}), \quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}} {\mathrm{v}}_{2}( \mathfrak{t}) = {\mathrm{u}}_{2}(\mathfrak{t}), \\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t}) = {\mathrm{w}}_{1}(\mathfrak{t}),\quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}{\mathrm{u}}_{2}(\mathfrak{t}) = {\mathrm{w}}_{2}(\mathfrak{t}), \\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}{\mathrm{w}}_{1}(\mathfrak{t}) = {\mathrm{x}}_{1}(\mathfrak{t}), \quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{2};\mathbb{G}} {\mathrm{w}}_{2}( \mathfrak{t}) = {\mathrm{x}}_{2}(\mathfrak{t}), \\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{s_{1};\mathbb{G}}{\mathrm{x}}_{1}(\mathfrak{t} ) = h_{1}(\mathfrak{t}, {\mathrm{v}}_{1}, {\mathrm{v}}_{2},{\mathrm{u}}_{1},{\mathrm{u}}_{2}, {\mathrm{w}}_{1}, {\mathrm{w}}_{2} {\mathrm{,x}}_{1}, {\mathrm{x}}_{2}), \\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{s_{2};\mathbb{G}}{\mathrm{x}}_{2}(\mathfrak{t} ) = h_{2}(\mathfrak{t}, {\mathrm{v}}_{1}, {\mathrm{v}}_{2}, {\mathrm{u}}_{1}, {\mathrm{u}}_{2}, {\mathrm{w}}_{1}, {\mathrm{w}}_{2}, {\mathrm{x}}_{1}, {\mathrm{x}}_{2}), \end{array}\right. \end{equation}$

(1.4)

subject to the following integral boundary conditions

$\begin{align} {\mathrm{v}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} g_{10}(s)\, {\mathrm d}s, & {\mathrm{v}}_{2}(\iota_1) & = \int_{\iota_1}^{\iota_2} g_{20}(s) \, {\mathrm d}s,\\ {\mathrm{u}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} g_{11}(s)\, {\mathrm d}s, & {\mathrm{u}}_{2}(\iota_1) & = \int_{\iota_1}^{\iota_2} g_{21}(s) \, {\mathrm d}s,\\ {\mathrm{w}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} g_{12}(s) \, {\mathrm d}s,& {\mathrm{w}}_{2}(\iota_1) & = \int_{\iota_1}^{ \iota_2} g_{22}(s) \, {\mathrm d}s, \\ {\mathrm{x}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} g_{13}(s)\, {\mathrm d}s,& {\mathrm{x}}_{2}(\iota_1) & = \int_{\iota_1}^{ \iota_2} g_{23}(s) \, {\mathrm d}s, \end{align}$

(1.5)

where the $\mathbb{G}{\mathrm{C}}$ derivatives are illustrated by symbol

$^{c}\mathcal{D}_{\iota_1^{+}}^{\eta ;\mathbb{G}}, \qquad \eta \in \Big\{ q_{k}, p_{k}, r_{k}, s_{k}\Big\},\, 0 < q_{k}, p_{k}, r_{k}, s_{k}\leq 1,$

here the function $\mathbb{G}\in C^{1}(\Sigma)$ is increasing with $\mathbb{G}^{\prime }(\mathfrak{t}) \neq 0$ , for all $\mathfrak{t}\in \Sigma = \lbrack \iota_1, \iota_2]$ and the functions $h_{k}\in C(\Sigma\times \mathbb{R}^{8}), (k = 1, 2)$ and ${\mathrm{g}}_{kj}\in C(\Sigma, \mathbb{R}), (j = 0, 1, 2, 3;\, k = 1, 2)$ are continuous functions. It is obvious that the $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.4) and (1.5) can be rewritten as

$\begin{equation} \left\{\begin{array}{l} {^{c}\mathcal{D}}_{\iota_1^{+}}^{s_{k};\mathbb{G}} \left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{k}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G} } \left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}{\mathrm{v}}_{k}(\mathfrak{ t})\right) \right) \right) = h_{k}(\mathfrak{t}) \\ {\mathrm{v}}_{k}(\iota_1) = \int_{\iota_1}^{\iota_2} g_{k0}(s)\, {\mathrm d}s, \quad \quad {\left. {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k}; \mathbb{G}}{\mathrm{v}}_{k}(\mathfrak{t} ) \right\vert _{\mathfrak{t} = \iota_1} = \int_{\iota_1}^{\iota_2} g_{k1}(s)\, {\mathrm d}s}, \\[0.5cm] {\left. {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}} {\mathrm{v}}_{k}(\mathfrak{t})\right) \right\vert_{ \mathfrak{t} = \iota_1}} = \int_{\iota_1}^{\iota_2} g_{k2}(s)\, {\mathrm d}s, \\ { \left. {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{k};\mathbb{G}} \left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}} \left( {^{c}\mathcal{D}}_{ \iota_1^{+}}^{q_{k};\mathbb{G}} {\mathrm{v}}_{k}(\mathfrak{t})\right) \right) \right\vert_{ \mathfrak{t} = \iota_1}} = \int_{\iota_1}^{\iota_2} g_{k3}(s)\, {\mathrm d}s, \qquad k = 1,2, \end{array}\right. \end{equation}$

(1.6)

where

$\begin{align*} h_{{\mathrm{v}}_1, {\mathrm{v}}_2,k}(\mathfrak{t} ) & = h_{k}\left( \mathfrak{t}, {\mathrm{v}}_{1}(\mathfrak{t} ),{\mathrm{v}}_{2}(\mathfrak{t} ),{^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} ), {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t}),\right.\\ &\quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )\right) ,{^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}} \left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G} }{\mathrm{v}}_{2}(\mathfrak{t} )\right), \\ &\quad \left. {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G} } {\mathrm{v}}_{1}(\mathfrak{t} )\right) \right), {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{2}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c} \mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )\right) \right)\right)\,. \end{align*}$

The main novelty of this work is that we establish our results with the help of the technique of fixed point theorems for a fractional nonlinear $\mathbb{C}{\mathrm{S}}\mathbb{S}$ furnished with generalized operators, which leads to some general theoretical findings involving the following special cases: $\mathbb{G}$ as $\mathbb{G}_1(\iota) = 2^{\iota}$ , $\mathbb{G}_2(\iota) = \iota$ (Caputo derivative), $\mathbb{G}_3(\iota) = \ln \iota$ (Caputo-Hadamard derivative), $\mathbb{G}_4(\iota) = \sqrt{\iota}$ (Katugampola derivative).

This paper is organized as follows: In Section 2, we present some necessary definitions and lemmas that are needed in the subsequent sections. In Section 3, we adopt some fixed point theorems to prove the existence and uniqueness of solutions for problem (1.4). The stability results are extensively discussed in Section 3.2. An illustrative example is presented in Section 4.

2. Preliminaries

Some primitive notions, definitions and notations, which will be utilized throughout the manuscript, are recalled here. Consider the function $\mathbb{G}$ with assumptions in system (1.4). We start this part by defining $\mathbb{G}$ -Riemann-Liouville fractional ( $\mathbb{GF}$ - ${\mathrm{RL}}$ ) integrals and derivatives ^[17]. For $\eta > 0$ , the $\eta^{{\mathrm{th}} }$ - $\mathbb{GF}$ - ${\mathrm{RL}}$ integral for an integrable function ${\mathrm{v}}:\Sigma \to \mathbb{R}$ w.r.t $\mathbb{G}$ is illustrated as follows

$\begin{align} \mathcal{I}_{\iota_1^{+}}^{\eta ; \mathbb{G}}{\mathrm{v}}(\mathfrak{t}) = \frac{1}{\Gamma (\eta )} \int_{\iota_1}^{\mathfrak{t}}( \mathbb{G}(\mathfrak{t}) - \mathbb{G} (\sigma))^{\eta -1}\mathbb{G}^\prime (\sigma) {\mathrm{v}}(\sigma) \,{\mathrm{d}}\sigma, \end{align}$

(2.1)

where $\Gamma (\eta) = \int_{0}^{+\infty } e^{-\mathfrak{t}} \mathfrak{t}^{ \eta -1}\, {\mathrm{d}}\mathfrak{t}, \, \eta > 0$ . Let $n\in \mathbb{N}$ and $\mathbb{G}, {\mathrm{v}}\in C^{n}(\Sigma)$ be such that $\mathbb{G}$ has the same properties mentioned above. The $\eta^{{\mathrm{th}}}$ - $\mathbb{GF}$ - ${\mathrm{RL}}$ derivative of ${\mathrm{v}}$ is defined by

$\begin{align*} \mathcal{D}_{\iota_1^{+}}^{\eta ; \mathbb{G}}{\mathrm{v}}(\mathfrak{t}) & = A^{(n)} \mathcal{I}_{\iota_1^{+}}^{n -\eta ;\mathbb{G}} {\mathrm{v}}( \mathfrak{t}) \\ & = \frac{1}{\Gamma (n-\eta )} A^{(n)} \int_{\iota_1}^{ \mathfrak{t}}(\mathbb{G}(\mathfrak{t})-\mathbb{G}( \sigma))^{ n -\eta -1}\mathbb{G}^\prime(\sigma ) {\mathrm{v}}(\sigma)\, {\mathrm{d}}\sigma, \end{align*}$

in which $n = [\eta ]+1$ , where $A = \frac{1}{\mathbb{G}^{\prime }(\mathfrak{t})} \frac{ {\mathrm{d}}}{ {\mathrm{d}}\mathfrak{t}}$ . The $\eta ^{{\mathrm{th}}}$ - $\mathbb{G}$ -fractional-Caputo derivative of ${\mathrm{v}}$ is defined by $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}} {\mathrm{v}}(\mathfrak{t}) = \mathcal{I}_{\iota_1^{+}}^{n-\eta; \mathbb{G}} A^{(n)}{\mathrm{v}}(\mathfrak{t})$ , in which $n = [\eta ]+1$ , $(\eta \notin \mathbb{N})$ , $n = \eta$ for $\eta \in \mathbb{N}$ ^[17]. In other words,

$\begin{equation} {}^{c}\mathcal{D}_{\iota_1^{+}}^{\eta ;\mathbb{G}}{\mathrm{v}}(\mathfrak{t}) = \left\{ \begin{array}{ll} \int_{\iota_1}^{\mathfrak{t}}\frac{(\mathbb{G}(\mathfrak{t})-\mathbb{G}(\xi ))^{n-\eta -1}}{\Gamma (n-\eta )}\,\mathbb{G}^{\prime (n)}{\mathrm{v}}(\xi )\,{\mathrm{d}}\xi , & \eta \notin \mathbb{N}, \\[0.4cm] A^{n}{\mathrm{v}}(\mathfrak{t}), & \eta = n \in \mathbb{N}. \end{array}\right. \end{equation}$

(2.2)

This extension (2.2) gives the Caputo derivative when $\mathbb{G}(\mathfrak{t}) = \mathfrak{t}$ ^[17]. Also, in the case $\mathbb{G}(\mathfrak{t}) = \ln \mathfrak{t}$ , it yields the Caputo-Hadamard derivative. If ${\mathrm{v}}\in C^{n}(\Sigma)$ , the $\eta^{{\mathrm{th}}}$ - $\mathbb{G}$ -fractional-Caputo derivative of ${\mathrm{v}}$ is specified as ^[18]

$\begin{equation*} {}^{c}\mathcal{D}_{\iota_1^{+}}^{\eta ;\mathbb{G}}{\mathrm{v}}(\mathfrak{t}) = \mathcal{D}_{ \iota_1^{+}}^{\eta;\mathbb{G}} \left( {\mathrm{v}}(\mathfrak{t} ) - \sum\limits_{j = 0}^{n-1} \frac{A^{(j)}{\mathrm{v}}(\iota_1)}{j!}(\mathbb{G}(\mathfrak{t})- \mathbb{G}(\iota_1))^{j}\right). \end{equation*}$

The composition rules for above $\mathbb{G}$ -operators are recalled in this lemma.

Lemma 2.1. ^[18] Let $n-1 < \eta < n$ and ${\mathrm{v}}\in C^{n}(\Sigma)$ . Then the following holds

$\begin{equation*} \mathcal{I}_{\iota_1^{+}}^{\eta ; \mathbb{G}} \, ^{c}\mathcal{D}_{ \iota_1^{+}}^{\eta ; \mathbb{G}} {\mathrm{v}}(\mathfrak{t}) = {\mathrm{v}}(\mathfrak{t})-\sum\limits_{j = 0}^{n-1} \frac{A^{(j)}{\mathrm{v}}(\iota_1)}{j!}\left[ \mathbb{G}( \mathfrak{t}) - \mathbb{G}(\iota_1)\right] ^{j}, \end{equation*}$

for all $\mathfrak{t}\in \Sigma$ . Moreover, if $m\in \mathbb{N}$ and ${\mathrm{v}}\in C^{n+m}(\Sigma)$ , then, the following holds

$\begin{equation} A^{(m)}\left( ^{c}\mathcal{D}_{ \iota_1^{+}}^{\eta ;\mathbb{G}} {\mathrm{v}}\right) (\mathfrak{t}) = \; ^{c}\mathcal{D}_{\iota_1^{+}}^{\eta +m; \mathbb{G}}{\mathrm{v}}(\mathfrak{t})+\sum\limits_{j = 0}^{m-1} \frac{[\mathbb{G}( \mathfrak{t})-\mathbb{G} (\iota_1)]^{j+n-\eta -m}}{\Gamma (j+n-\eta -m+1)} A^{(j+n)} {\mathrm{v}}(\iota_1). \end{equation}$

(2.3)

Observe that from Eq (2.3) if $A^{(j)}{\mathrm{v}}(\iota_1) = 0$ , for $j = n, n+1, \dots, n+m-1$ , we can get the following relation

$A^{(m)}\left(^{c}\mathcal{D}_{\iota_1^{+}}^{\eta ;\mathbb{G}}{\mathrm{v}}\right) (\mathfrak{t} ) = \; ^{c} \mathcal{D}_{\iota_1^{+}}^{ \eta +m; \mathbb{G}}{\mathrm{v}}(\mathfrak{t}), \qquad \mathfrak{t}\in \Sigma.$

Lemma 2.2. ^[12] Let $\eta, \nu > 0,$ and ${\mathrm{v}}\in C(\Sigma)$ . Then for each $\mathfrak{t}\in \Sigma$ and by assuming

$\begin{equation} F_{\iota_1}(\mathfrak{t}) = \mathbb{G}(\mathfrak{t}) - \mathbb{G}(\iota_1), \end{equation}$

(2.4)

we have

(1) $\mathcal{I}_{\iota_1^{+}}^{\eta; \mathbb{G}}\left(\mathcal{I}_{\iota_1^{+}}^{\nu; \mathbb{G}}{\mathrm{v}}\right) (\mathfrak{t}) = \mathcal{I}_{\iota_1^{+}}^{\eta +\nu; \mathbb{G}} {\mathrm{v}}(\mathfrak{t})$ ;

(2) $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}}\left(\mathcal{I}_{\iota_1^{+}}^{\eta; \mathbb{G}}{\mathrm{v}}\right) (\mathfrak{t}) = {\mathrm{v}}(\mathfrak{t})$ ;

(3) $\mathcal{I}_{\iota_1^{+}}^{\eta; \mathbb{G}}(F_{\iota_1}(\mathfrak{t}))^{\nu -1} = \dfrac{\Gamma (\nu)}{\Gamma (\nu +\eta)}(F_{\iota_1}(\mathfrak{t}))^{\nu +\eta -1}$ ;

(4) $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}} (F_{a}(\mathfrak{t}))^{\nu -1} = \dfrac{\Gamma (\nu)}{\Gamma (\nu - \eta)}(F_{ \iota_1}(\mathfrak{t}))^{\nu -\eta -1}$ ;

(5) $^{c}\mathcal{D}_{\iota_1^{+}}^{\eta; \mathbb{G}}(F_{\iota_1}(\mathfrak{t}))^{j} = 0, \; n - 1 \leq \eta \leq n, n\in \mathbb{N}, \, j = 0, 1, \dots, n-1$ .

Theorem 2.3. ^[9] (Banach's fixed point theorem) Consider $\Pi:\mathcal{Y}\to \mathcal{Y}$ to be a contraction operator, such that $\mathcal{Y}$ is a Banach space. Then, there are only one ${\mathrm{y}}^\ast \in \mathcal{Y}$ , such that $\Pi({\mathrm{y}}^\ast) = {\mathrm{y}}^\ast$ .

Lemma 2.4. ^[9] (Krasnoselskii's fixed point theorem) Assume that $\mathcal{B}\subset X$ is a closed convex and nonempty, and $\mathfrak{L}_1$ , $\mathfrak{L}_2 : \mathcal{B} \to X$ nonlinear operators, such that:

(ⅰ) $\mathfrak{L}_1u + \mathfrak{L}_2v \in \mathcal{B}$ whenever $u, v \in \mathcal{B}$ ;

(ⅱ) $\mathfrak{L}_1$ is a contraction mapping;

(ⅲ) $\mathfrak{L}_2$ is compact and continuous.

Then, there exists $w\in \mathcal{B}$ , such that $w = \mathfrak{L}_1w + \mathfrak{L}_2 w.$

Definition 2.5. ^[29] Let $X_1, X_2$ be Banach spaces and $\Lambda_1, \Lambda_2: X_1\times X_2\rightarrow X_1\times X_2$ be two operators. Then, the operational equations system provided by

$\begin{equation} \left\{\begin{array}{ll} {\mathrm{u_1}}(\mathfrak{t}) = \Lambda_1({\mathrm{u_1}},{\mathrm{u_2}})(\mathfrak{t}),\\ {\mathrm{u_2}}(\mathfrak{t}) = \Lambda_2({\mathrm{u_1}},{\mathrm{u_2}})(\mathfrak{t}), \end{array}\right. \end{equation}$

(2.5)

is called ${\mathrm{U}}$ - ${\mathrm{H}}$ - $\mathbb{S}$ , if there exist $\alpha_i > 0, \, (i = 1, \dots, 4)$ , such that, $\forall \, \rho_1, \rho_2 > 0$ , and each solution $({\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}}) \in X_1 \times X_2$ of the identities

$\begin{equation*} \left\{\begin{array}{ll} \left\|{\mathrm{u_1^\ast}} - \Lambda_1({\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}}) \right\| \leq\rho_1,\\ \left\|{\mathrm{u_2^\ast}} - \Lambda_2({\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}}) \right\| \leq\rho_2, \end{array}\right. \end{equation*}$

there exists $({\mathrm{v_1^\ast}}, {\mathrm{v_2^\ast}}) \in X_1\times X_2$ a solution of system (2.5), such that

$\begin{equation*} \left\{ \begin{array}{ll} \left\|{\mathrm{u_1^\ast}}-{\mathrm{v_1^\ast}}\right\| \leq \alpha_1\rho_1+\alpha_2\rho_2,\\ \left\| {\mathrm{u_2^\ast}} - {\mathrm{v_2^\ast}}\right\|\leq\alpha_3\rho_1+\alpha_4 \rho_2. \end{array}\right. \end{equation*}$

Theorem 2.6. ^[29] Let $X_1, X_2$ be Banach spaces and $\Lambda_1, \Lambda_2: X_1\times X_2\to X_1\times X_2$ be two operators that satisfy

$\begin{equation} \left\{\begin{array}{ll} \left\| \Lambda_1 ( {\mathrm{u_1}}, {\mathrm{u_2}}) - \Lambda_1 ( {\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}}) \right\| \leq\alpha_1\left\|{\mathrm{u_1}}-{\mathrm{u_1^\ast}} \right\| + \alpha_2 \left\|{\mathrm{u_2}} - {\mathrm{u_2^\ast}}\right\|,\\ \left\| \Lambda_2( {\mathrm{u_1}}, {\mathrm{u_2}})-\Lambda_2({\mathrm{u_1^\ast}},{\mathrm{u_2^\ast}})\right\| \leq\alpha_3\left\|{\mathrm{u_1}}-{\mathrm{u_1^\ast}}\right\| + \alpha_4\left\|{\mathrm{u_2}}-{\mathrm{u_2^\ast}}\right\|, \end{array}\right. \end{equation}$

(2.6)

for each $({\mathrm{u_1}}, {\mathrm{u_2}}), ({\mathrm{u_1^\ast}}, {\mathrm{u_2^\ast}})\in X_1\times X_2$ and if the matrix

$\begin{equation*} \varXi = \left( \begin{matrix} \alpha_1 & \alpha_2 \\ \alpha_3 & \alpha_4 \end{matrix}\right), \end{equation*}$

it converges to zero. Then, the system (2.6) is ${\mathrm{U}}$ - ${\mathrm{H}}$ - $\mathbb{S}$ .

3. Main results

Here, we analyze the existence properties of solutions, and their uniqueness for the proposed fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) using Krasnoselskii and Banach fixed point theorems. We need after lemma, which indicate the corresponding integral equation.

Lemma 3.1. For given continuous mappings $h, g_{k}\, (k = 0, 1, 2, 3)$ belongs to $C(\Sigma)$ , and the solution of the linear $\mathbb{G}$ -snap problem is

$\begin{equation} \left\{\begin{array}{l} {^{c}\mathcal{D}}_{\iota_1^{+}}^{s; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{r; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p;\mathbb{G}}\left( {\ ^{c} \mathcal{D}}_{\iota_1^{+}}^{q;\mathbb{G}}{\mathrm{v}}(\mathfrak{t})\right) \right) \right) = h(\mathfrak{t}), \\[0.5cm] {\mathrm{v}}(\iota_1) = \int_{\iota_1}^{b}g_{0}(\xi )\,{\mathrm{d}}\xi, \quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{q;\mathbb{G}}{\mathrm{v}}(\iota_1) = \int_{\iota_1}^{b}g_{1}(\xi )\,{\mathrm{d}}\xi, \\[0.5cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{p;\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q; \mathbb{G}}{\mathrm{v}}(\iota_1)\right) = \int_{\iota_1}^{\iota_2} g_{2}(\xi )\,{\mathrm{d}}\xi, \\[0.5cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{r;\mathbb{G}} \left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{ p ; \mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q;\mathbb{G}} {\mathrm{v}}(\iota_1) \right) \right) = \int_{\iota_1}^{\iota_2} g_{3}(\xi )\,{\mathrm{d}}\xi, \end{array}\right. \end{equation}$

(3.1)

where $q, p, r, s\in (0, 1]$ , are formulated by

$\begin{align*} {\mathrm{v}}(\mathfrak{t})& = \int_{\iota_1}^{\iota_2} g_{0}(\xi ) \,{\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q} g_{1} (\xi ) }{\Gamma (q+1)} \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{ (F_{\iota_1}(\mathfrak{t}))^{q + p} g_{2}(\xi )}{\Gamma (q+p+1)}\, {\mathrm{d}} \xi \\[0.16in] & \quad + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q+p+r}g_{3}( \xi ) }{\Gamma (q+p+r+1)}\, {\mathrm{d}}\xi + \int_{\iota_1}^{\mathfrak{t}} \mathbb{G}^{\prime }(\xi ) \frac{(F_{\xi}( \mathfrak{t}))^{ q+p+r+s-1}}{ \Gamma (q+p+r+k)} h(\xi )\, {\mathrm{d}}\xi. \end{align*}$

Define the vector space

$\begin{align*} X_{k} = \left\{{\mathrm{v}}_k\in C(\Sigma, \mathbb{R}) \, : \, {}^{c}\mathcal{D}_{\iota_1^{+}}^{q_{k}; \mathbb{G}}{\mathrm{v_k}}, {}^{c}\mathcal{D}_{\iota_1^{+}}^{p_{k};\mathbb{G}}\left( {}^{c}\mathcal{D}_{\iota_1^{+}}^{q_{k};\mathbb{G}}{\mathrm{v}}_k\right),\right. \left. {}{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{k};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{ p_{k}; \mathbb{G}} \left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}{\mathrm{v}}_k(\mathfrak{t})\right) \right) \in C(\Sigma, \mathbb{R})\right\}. \end{align*}$

Then, $X_{k}, \, k = 1, 2,$ are Banach spaces via the norm

$\begin{align*} \Vert {\mathrm{v}}_k\Vert & = \sup\limits_{\mathfrak{t}\in \Sigma} \, | \, {\mathrm{v}}_k(\mathfrak{t})| + \sup\limits_{\mathfrak{t} \in \Sigma} \left\vert {}^{c}\mathcal{D}_{\iota_1^{+}}^{q_{k}; \mathbb{G}}{\mathrm{v}}_k( \mathfrak{t}) \right\vert +\sup\limits_{\mathfrak{t}\in \Sigma} \left\vert {}^{c}\mathcal{D}_{\iota_1^{+}}^{p_{k}; \mathbb{G}}\left( {}^{c}\mathcal{D}_{\iota_1^{+}}^{q_{k}; \mathbb{G}}{\mathrm{v}}_k(\mathfrak{t})\right) \right\vert \\[0.3cm] & \quad +\sup\limits_{\mathfrak{t}\in \Sigma} \left\vert {}{^{c}\mathcal{D}} _{\iota_1^{+}}^{r_{k}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}{\mathrm{v}}_k(\mathfrak{t} )\right) \right) \right\vert. \end{align*}$

Hence, the product space $X_{1}\times X_{2}$ is a Banach space with the norm

$\left\Vert (v_{1},v_{2})\right\Vert = \max \Big\{\Vert {\mathrm{v}}_{1}\Vert ,\Vert {\mathrm{v}}_{2}\Vert \Big\}.$

3.1. Existence and uniqueness

In view of Lemma 3.1, the solution of the coupled system (1.6) can be given as

$\begin{align*} {\mathrm{v}}_{k}(\mathfrak{t})& = \int_{\iota_1}^{\iota_2} g_{k0}(\xi)\, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}}g_{k1}(\xi ) }{ \Gamma (q_{k}+1)}\, {\mathrm{d}}\xi \\ &\quad + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}} g_{k2}(\xi)}{\Gamma( q_{k}+p_{k}+1)} \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{{\mathrm{v}}_{k3}(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}+r_{k}}g_{k3}(\xi)}{\Gamma (q_{k}+p_{k}+r_{k}+1)}\, {\mathrm{d}}\xi \\[0.16in] & \quad + \int_{\iota_1}^{ \mathfrak{t}}\mathbb{G}^{\prime }(\xi )\frac{(F_{\xi}( \mathfrak{t}))^{q_{k}+p_{k} + r_{k}+s_{k}-1}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k})} h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}(\xi {\mathrm{)d}}\xi . \end{align*}$

Define the functional $\Lambda _{k} : X_{k} \to \mathbb{R}$ , such that

$\begin{align} (\Lambda_{k}{\mathrm{v}}_{k})(\mathfrak{t})& = \int_{\iota_1}^{\iota_2} g_{k0}(\xi )\, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}} g_{k1}(\xi)}{\Gamma (q_{k}+1)} \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}}g_{k2}(\xi)}{\Gamma(q_{k}+p_{k}+1)}\, {\mathrm{d}}\xi \\ &\quad +\int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}+r_{k}}g_{k3}(\xi)}{\Gamma(q_{k}+p_{k}+r_{k}+1)} \, {\mathrm{d}}\xi +\int_{\iota_1}^{\mathfrak{t}} \mathbb{G}^{\prime }(\xi ) \frac{(F_{\xi}(\mathfrak{t}))^{q_{k}+p_{k}+r_{k}+s_{k}-1} h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}(\xi )}{ \Gamma(q_{k}+p_{k}+r_{k}+s_{k})} \, {\mathrm{d}}\xi . \end{align}$

(3.2)

Under some conditions, we show next that the functional $\Lambda :X_{1} \times X_{2} \to \mathbb{R}^{2}$ is a contraction, where $\Lambda$ is given as

$\begin{equation*} \Lambda (v_{1},v_{2}) = \big(\Lambda _{1}(v_{1},v_{2}),\Lambda _{2}(v_{1},v_{2})\big). \end{equation*}$

Theorem 3.2. Let $h_{k}\in C(\Sigma\times\mathbb{R}^8), \, (k = 1, 2)$ be continuous functions. Moreover, assume that

(H1) there exist real constants $\ell_k > 0, \, (k = 1, 2)$ , so that

$\begin{equation} |h_{k}\big(\mathfrak{t},{\mathrm{v}}_1,{\mathrm{v}}_2,\dots,{\mathrm{v}}_8 \big)-h_{k}\big(\mathfrak{t},{\mathrm{v}}_1^\ast, {\mathrm{v}}_2^\ast, \dots,{\mathrm{v}}_8^\ast \big) |\leq\ell_k\sum\limits_{i = 1}^{8}| {\mathrm{v}}_i-{\mathrm{v}}_i^\ast|, \end{equation}$

(3.3)

for any $\mathfrak{t}\in \Sigma$ , ${\mathrm{v}}_i, {\mathrm{v}}_i^\ast\in C([a, b])$ and $i = 1, 2, \dots, 8$ .

Then, the fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) admits a unique solution on $\Sigma$ if $\Phi\ell < 1$ , whenever $\ell = \max\{ \ell_1, \ell_2\}$ , $\Phi = \max\{\Phi_1, \Phi_2\}$ and

$\begin{align} \Phi_k & = \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{ \Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}+s_{k}}}{ \Gamma(p_{k} + r_{k}+s_{k}+1)} \\ &\quad +\frac{ (F_{\iota_1}(\iota_2))^{r_{k}+s_{k}}}{ \Gamma (r_{k}+s_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{s_{k}}}{ \Gamma (s_{k}+1)}, \end{align}$

(3.4)

with $\Phi_k\ell_k < 1$ .

Proof. First of all, we define a closed bounded ball

$\mathbb{B}_\varepsilon = \Big\{ ({\mathrm{v}}_1,{\mathrm{v}}_2) \in X_{1}\times X_{2}:\; \; \Vert ({\mathrm{v}}_1,{\mathrm{v}}_2) \Vert\leq \varepsilon \Big\},$

satisfying

$\begin{equation} \varepsilon\geq\max\left\lbrace \dfrac{\Delta_1+h_1^0 \Phi_1}{(1- \ell_1 \Phi_1)},\dfrac{\Delta_2+h_2^0 \Phi_2}{(1- \ell_2 \Phi_2)}\right\rbrace, \end{equation}$

(3.5)

where

$\begin{align} \Delta_k& = M_{k0} +M_{k1}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}}}{ \Gamma (q_{k}+1)}\right) \\ &\quad+M_{k2}\left(1+\frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma (p_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}}}{\Gamma(q_{k}+p_{k}+1)}\right) \\ &\quad +M_{k3}\left( 1 + \frac{(F_{\iota_1}(\iota_2))^{ r_{k}}}{\Gamma (r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma(q_{k} + p_{k} + r_{k}+1)}\right), \end{align}$

(3.6)

and

$\begin{align} \label{eqMh} M_{kj} & = \sup\limits_{\mathfrak{t}\in \Sigma}\int_{\iota_1}^{\iota_2}| g_{kj}(\xi)| \, {\mathrm{d}}\xi, \qquad (j = 0,1,2,3), \\ h_k^0 & = \sup\limits_{\mathfrak{t}\in \Sigma }| h_k( \mathfrak{t},0,0,0,0,0,0,0,0)|, \qquad (k = 1,2). \end{align}$

Now, define the operator

$\Lambda (v_{1},v_{2}) = (\Lambda _{1}(v_{1},v_{2}),\Lambda _{2}(v_{1},v_{2})),\qquad \forall \, (v_{1}, v_{2}) \in X_{1}\times X_{2},$

(3.7)

where $\Lambda_k$ is given in (3.2). To show that $\Lambda(\mathbb{B}_\varepsilon) \subset \mathbb{B}_\varepsilon$ , by using hypotheses (H1), for $({\mathrm{v}}_1, {\mathrm{v}}_2) \in\mathbb{B}_\varepsilon$ and $\mathfrak{t} \in \Sigma$ , we get

$\begin{align} |\Lambda_{k} ( {\mathrm{v}}_{1}, {\mathrm{v}}_{2}) (\mathfrak{t})|& \leq \int_{\iota_1}^{\iota_2} |g_{k0}(\xi)| \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}}| g_{k1}(\xi)| }{\Gamma (q_{k}+1)} \, {\mathrm{d}}\xi \\ &\quad +\int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}}| g_{k2}(\xi)| }{ \Gamma(q_{k}+p_{k}+1)}\, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}( \mathfrak{t}))^{ q_{k} + p_{k} + r_{k}} |g_{k3}(\xi)| }{ \Gamma ( q_{k}+p_{k}+r_{k}+1)}\, {\mathrm{d}}\xi\\ & \quad + \mathcal{I}_{\iota_1^{+}}^{q_{k}+p_{k} + r_{k} + s_{k} ;\mathbb{G}} \Big( |h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}( \mathfrak{t}) - h_{k}(\mathfrak{t},0,0,0,0,0,0,0,0)| \\ & \quad +|h_{k}(\mathfrak{t},0,0,0,0,0,0,0,0)| \Big) \\ & \leq \int_{a}^{b}|g_{k0}(\xi)| \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}}| g_{k1}(\xi)| }{\Gamma (q_{k}+1)} \, {\mathrm{d}}\xi \\ &\quad + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}} |g_{k2}(\xi)|}{ \Gamma(q_{k} + p_{k}+1)} {\mathrm{d}} +\int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{ q_{k}+p_{k}+r_{k}} |g_{k3}(\xi)| }{\Gamma (q_{k}+p_{k}+r_{k}+1)} \, {\mathrm{d}}\xi\\ & \quad + \mathcal{I}_{\iota_1^{+} }^{q_{k}+p_{k} + r_{k}+s_{k} ;\mathbb{G}} \bigg( \ell_k \Big( |{\mathrm{v}}_{1}(\mathfrak{t} )| + |{\mathrm{v}}_{2}(\mathfrak{t} )| \\ & \quad + \left|{ ^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1}; \mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )\right| + \left| {^{c} \mathcal{D}}_{\iota_1^{+}}^{ q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )\right| \\ &\quad + \left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )\right)\right| +\left| {^{c}\mathcal{D}}_{ \iota_1^{+}}^{p_{2}; \mathbb{G}} \left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2}; \mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )\right) \right| \\ &\quad +\left|{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}\left( {^{c}\mathcal{D}} _{\iota_1^{+}}^{p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{a^{+}}^{q_{1};\mathbb{G} }{\mathrm{v}}_{1}(\mathfrak{t} )\right) \right) \right| \\ & \quad +\left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{2}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+} }^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )\right)\right)\right| \Big)\\ &\quad + |h_{k}(\mathfrak{t},0,0,0,0,0,0,0,0)| \bigg) \\ & \leq M_{k0} + M_{k1}\frac{ (F_{\iota_1}(\iota_2))^{q_{k}}}{\Gamma (q_{k}+1)} + M_{k2} \frac{(F_{\iota_1}(\iota_2))^{ q_{k}+p_{k}}}{\Gamma(q_{k}+p_{k}+1)} \\ & \quad + M_{k3}\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+1)}\\ & \quad +\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)}\left(\ell_k\left(\|{\mathrm{v}}_{1}\|+\|{\mathrm{v}}_{2}\|\right)+h_{k}^0 \right). \end{align}$

(3.8)

Also,

$\begin{align} |{^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}} & \left( \Lambda _{k}( {\mathrm{v}}_{1},{\mathrm{v}}_{2})(\mathfrak{t})\right) | \leq \int_{\iota_1}^{\iota_2} |g_{k1}(\xi)|\, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{p_{k}} |g_{k2}(\xi)|}{\Gamma( p_{k}+1)} \, {\mathrm{d}}\xi \\ & \quad + \int_{\iota_1}^{\iota_2} \frac{ ( F_{\iota_1}(\mathfrak{t}))^{ p_{k}+r_{k}} |g_{k3}(\xi)|}{\Gamma (p_{k}+r_{k}+1)}\,{\mathrm{d}}\xi\\ & \quad + \mathcal{I}_{\iota_1^{+}}^{p_{k}+r_{k}+s_{k} ;\mathbb{G}} \Big( |h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k} ( \mathfrak{t}) - h_{k}( \mathfrak{t},0,0,0,0,0,0,0,0)| \\ & \quad + |h_{k}( \mathfrak{t},0,0,0,0,0,0,0,0)|\Big) \\ & \leq \int_{ \iota_1}^{\iota_2} |g_{k1}(\xi)| \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{ (F_{\iota_1}(\mathfrak{t}))^{p_{k}} |g_{k2}(\xi)|}{\Gamma(p_{k}+1)} {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{ (F_{\iota_1}(\mathfrak{t}))^{p_{k}+r_{k}} |g_{k3}(\xi)| }{\Gamma(p_{k}+r_{k}+1)} \, {\mathrm{d}}\xi \\ & \quad + \mathcal{I}_{\iota_1^{+}}^{p_{k}+r_{k}+s_{k} ;\mathbb{G}} \bigg( \ell_k \Big( |{\mathrm{v}}_{1}(\mathfrak{t} )| + |{\mathrm{v}}_{2}(\mathfrak{t} )| + |{^{c}\mathcal{D}}_{\iota_1^{+}}^{ q_{1}; \mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )| + |{^{c} \mathcal{D}}_{\iota_1^{+}}^{ q_{2}; \mathbb{G}}{\mathrm{v}}_{2}( \mathfrak{t} )| \\ &\quad + \left|{^{c}\mathcal{D}}_{ \iota_1^{+}}^{ p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )\right)\right| +\left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G} } {\mathrm{v}}_{2}(\mathfrak{t} )\right) \right| \\ &\quad + \left|{ ^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}} \left( {^{c}\mathcal{D}} _{\iota_1^{+}}^{ p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{ \iota_1^{+}}^{q_{1};\mathbb{G} }{\mathrm{v}}_{1}( \mathfrak{t} ) \right) \right) \right| \\ & \quad + \left| {^{c}\mathcal{D}}_{ \iota_1^{+}}^{r_{2};\mathbb{G}}\left({^{c}\mathcal{D}}_{ \iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c}\mathcal{D}}_{ \iota_1^{+}}^{q_{2}; \mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )\right)\right)\right| \Big)\\ & \quad + | h_{k}( \mathfrak{t},0,0,0,0,0,0,0,0)|\bigg) \\ & \leq M_{k1} + M_{k2} \frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma(p_{k}+1)} + M_{k3}\frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)}\\ & \quad + \mathcal{I}_{\iota_1^{+}}^{p_{k}+r_{k}+s_{k} ;\mathbb{G}} \left( \ell_k \left(\|{\mathrm{v}}_{1}\|+\|{\mathrm{v}}_{2}\|\right)+h_{k}^0\right) \\ & \leq M_{k1} + M_{k2}\frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma (p_{k}+1)} +M_{k3} \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)}\\ & \quad + \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}+s_{k}}}{\Gamma (p_{k}+r_{k}+s_{k}+1)} \left(\ell_k\left(\|{\mathrm{v}}_{1}\|+\|{\mathrm{v}}_{2}\|\right)+h_{k}^0\right), \end{align}$

(3.9)

$\begin{align} \left|{^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}}\left( { ^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left( \Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})(\mathfrak{t})\right) \right) \right| & \leq M_{k2} +M_{k3}\frac{(F_{\iota_1}(\iota_2))^{r_{k}}}{\Gamma (r_{k}+1)}\\ & \quad + \frac{(F_{\iota_1}(\iota_2))^{r_{k}+s_{k}}}{\Gamma (r_{k}+s_{k}+1)} \left( \ell_k\left(\|{\mathrm{v}}_{1}\|+\|{\mathrm{v}}_{2}\|\right)+h_{k}^0\right), \end{align}$

(3.10)

and

(3.11)

Thus, due to (3.8)–(3.11) and (3.5), we obtain

$\begin{align*} \Vert \Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})\Vert & = \sup\limits_{\mathfrak{t}\in \Sigma } |\Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})(\mathfrak{t})|+\sup\limits_{\mathfrak{t}\in \Sigma} \left\vert {}^{c}\mathcal{D}_{\iota_1^{+}}^{ q_{k}; \mathbb{G}}\left( \Lambda _{k}( {\mathrm{v}}_{1},{\mathrm{v}}_{2})\right) (\mathfrak{t})\right\vert \\ & \quad+\sup\limits_{\mathfrak{t} \in \Sigma}\left\vert {}^{c}\mathcal{D}_{\iota_1^{+}}^{p_{k}; \mathbb{G}}\left( {}^{c}\mathcal{D}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left( \Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})\right)(\mathfrak{t})\right) \right\vert \\ & \quad +\sup\limits_{\mathfrak{t}\in \Sigma}\left\vert {}{^{c}\mathcal{D}} _{\iota_1^{+}}^{r_{k};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left( \Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})\right)(\mathfrak{t}) \right) \right) \right\vert \\ &\leq \left[M_{k0} + M_{k1}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}}}{ \Gamma (q_{k}+1)}\right)\right. \\ &\quad+M_{k2}\left(1+\frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma(p_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}}}{\Gamma(q_{k}+p_{k}+1)}\right) \\ &\quad\left. +M_{k3}\left( 1+\frac{(F_{\iota_1}(\iota_2))^{r_{k}}}{ \Gamma(r_{k}+1)}+\frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+1)}\right) \right] \\ &\quad+\left(\ell_k\|\left({\mathrm{v}}_{1},{\mathrm{v}}_{2}\right)\|+h_{k}^0\right)\left[\frac{ (F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k}+1)} \right.\\ &\quad\left. +\frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}+s_{k}}}{\Gamma (p_{k}+r_{k}+s_{k}+1)} +\frac{(F_{\iota_1}(\iota_2))^{r_{k}+s_{k}}}{\Gamma (r_{k}+s_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{s_{k}}}{\Gamma (s_{k}+1)} \right] \\ &\leq \left[M_{k0} +M_{k1}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}}}{ \Gamma (q_{k}+1)}\right)\right. \\ &\quad + M_{k2}\left( 1 +\frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma (p_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}}}{\Gamma(q_{k}+p_{k}+1)}\right) \\ &\quad\left. + M_{k3}\left( 1+\frac{(F_{\iota_1}(\iota_2))^{r_{k}}}{ \Gamma(r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+1)}\right)\right] \\ &\quad+\left( \ell_k\varepsilon+h_{k}^0\right)\left[\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)}+\frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}+s_{k}}}{\Gamma (p_{k}+r_{k}+s_{k}+1)} \right. \\ &\quad\left. + \frac{(F_{\iota_1}(\iota_2))^{r_{k}+s_{k}}}{\Gamma (r_{k}+s_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{s_{k}}}{ \Gamma (s_{k}+1)} \right] \\ &\leq \Delta_k+\left( \ell_k\varepsilon+h_{k}^0\right)\Phi_k\leq\varepsilon. \end{align*}$

Hence, we deduce that $\Vert \Lambda ({\mathrm{v}}_{1}, {\mathrm{v}}_{2})\Vert\leq\varepsilon,$ for $({\mathrm{v}}_1, {\mathrm{v}}_2) \in\mathbb{B}_\varepsilon$ , so $\Lambda(\mathbb{B}_\varepsilon)\subset\mathbb{B}_\varepsilon.$ Next, we prove that $\Lambda$ is a contraction operator, by using ${\mathrm{(H1)}}$ , for $({\mathrm{v}}_1, {\mathrm{v}}_2), ({\mathrm{u}}_1, {\mathrm{u}}_2) \in\mathbb{B}_\varepsilon$ and $\mathfrak{t}\in\Sigma$ , we have

$\begin{align} |\Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})(\mathfrak{t}) & - \Lambda _{k}({\mathrm{u}}_{1},{\mathrm{u}}_{2})(\mathfrak{t})| \\ & \leq \mathcal{I}_{\iota_1^{+}}^{ q_{k}+p_{k}+r_{k}+s_{k} ;\mathbb{G}} | h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k} ( \mathfrak{t}) - h_{{\mathrm{u}}_1,{\mathrm{u}}_2,k} ( \mathfrak{t})|\\ & \leq \mathcal{I}_{\iota_1^{+}}^{q_{k}+p_{k} + r_{k}+s_{k} ;\mathbb{G}} \bigg( \ell_k \Big( |{\mathrm{v}}_{1}(\mathfrak{t} )-{\mathrm{u}}_{1}(\mathfrak{t} )|+|{\mathrm{v}}_{2}(\mathfrak{t} ) - {\mathrm{u}}_{2}(\mathfrak{t} )| \\ & \quad + \left|{^{c}\mathcal{D}}_{\iota_1^{+} }^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} ) - {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}} {\mathrm{u}}_{1}(\mathfrak{t} ) \right| + \left|{^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )-{^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{u}}_{2}(\mathfrak{t} ) \right| \\ & \quad + \left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )\right) - {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t} )\right)\right| \\ &\quad+\left| {^{c}\mathcal{D}} _{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G} } {\mathrm{v}}_{2}(\mathfrak{t} )\right) - {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G} }{\mathrm{u}}_{2}(\mathfrak{t} )\right)\right|\\ &\quad+\left|{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G} } {\mathrm{v}}_{1}(\mathfrak{t} )\right) \right) \right. \\ & \quad \left. -{^{c} \mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t} )\right) \right)\right|\\ &\quad + \left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{2}; \mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c} \mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t})\right)\right) \right. \\ & \quad \left. -{^{c}\mathcal{D}}_{a^{+}}^{r_{2}; \mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c} \mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{u}}_{2}(\mathfrak{t} )\right)\right)\right| \Big) \bigg) \\ &\leq \mathcal{I}_{\iota_1^{+}}^{q_{k}+p_{k}+r_{k}+s_{k} ;\mathbb{G}} \left(\ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right)\\ &\leq\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)} \left(\ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right), \end{align}$

(3.12)

$\begin{align} |{^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left(\Lambda_{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})\right)(\mathfrak{t}) &- {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left(\Lambda _{k} ( {\mathrm{u}}_{1},{\mathrm{u}}_{2})\right)(\mathfrak{t}) |\\ & \leq \mathcal{I}_{\iota_1^{+}}^{p_{k}+r_{k}+s_{k} ;\mathbb{G}} \left| h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}(\mathfrak{t})-h_{{\mathrm{u}}_1,{\mathrm{u}}_2,k}(\mathfrak{t}) \right|\\ & \leq \mathcal{I}_{\iota_1^{+}}^{p_{k}+r_{k}+s_{k} ;\mathbb{G}} \bigg( \ell_k \Big( |{\mathrm{v}}_{1}(\mathfrak{t} ) - {\mathrm{u}}_{1}(\mathfrak{t} )|+|{\mathrm{v}}_{2}(\mathfrak{t} )-{\mathrm{u}}_{2}(\mathfrak{t} )| \\ & \quad + \left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )-{ ^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t} )\right| \\ & \quad + \left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )-{^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{u}}_{2}(\mathfrak{t} ) \right| \\ & \quad + \left|{^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}} _{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} ) \right) - {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left( {^{c}\mathcal{D}} _{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t} )\right)\right|\\ &\quad + \left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2}; \mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G} }{\mathrm{v}}_{2}(\mathfrak{t} )\right) - {^{c}\mathcal{D}} _{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G} }{\mathrm{u}}_{2}(\mathfrak{t} )\right)\right|\\ &\quad+\left|{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{v}}_{1}(\mathfrak{t} )\right) \right) \right. \\ & \quad \left. -{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{1};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{1};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{1};\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t} )\right) \right)\right|\\ &\quad +\left| {^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{2}; \mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c} \mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{v}}_{2}(\mathfrak{t} )\right)\right) \right. \\ & \quad \left. -{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{2}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{2};\mathbb{G}}\left( {^{c} \mathcal{D}}_{\iota_1^{+}}^{q_{2};\mathbb{G}}{\mathrm{u}}_{2}(\mathfrak{t} )\right)\right)\right| \Big)\bigg)\\ &\leq \mathcal{I}_{\iota_1^{+}}^{p_{k}+r_{k}+s_{k} ;\mathbb{G}} \left( \ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right)\\ &\leq \frac{ (F_{\iota_1}(\iota_2))^{p_{k}+r_{k}+s_{k}}}{\Gamma (p_{k}+r_{k}+s_{k}+1)} \left(\ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right), \end{align}$

(3.13)

$\begin{align} \left| { ^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k}; \mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+} }^{q_{k};\mathbb{G}}\left( \Lambda _{k}({\mathrm{v}}_{1}, {\mathrm{v}}_{2})\right)\right) (\mathfrak{t}) \right. & \left. - { ^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}} \left({^{c} \mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left(\Lambda_{k} ({\mathrm{u}}_{1},{\mathrm{u}}_{2})\right)\right) (\mathfrak{t}) \right| \\ &\leq \frac{ (F_{\iota_1}(\iota_2))^{r_{k}+s_{k}}}{\Gamma (r_{k}+s_{k}+1)}\left( \ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right), \end{align}$

(3.14)

and

$\begin{align} \left|{ ^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{k};\mathbb{G}} \right. & \left. \left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left( \Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})\right)\right)\right) (\mathfrak{t}) \right. \left. -{^{c}\mathcal{D}}_{\iota_1^{+}}^{r_{k};\mathbb{G}}\left( {^{c}\mathcal{D}}_{\iota_1^{+}}^{p_{k};\mathbb{G}}\left({^{c}\mathcal{D}}_{\iota_1^{+}}^{q_{k};\mathbb{G}}\left(\Lambda _{k}({\mathrm{u}}_{1}, {\mathrm{u}}_{2})\right)\right)\right) (\mathfrak{t}) \right| \\ &\leq \frac{(F_{\iota_1}(\iota_2))^{s_{k}}}{\Gamma (s_{k}+1)} \left(\ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right). \end{align}$

(3.15)

Therefore, due to (3.12)–(3.15), we get

$\begin{align*} \|\Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2}) & - \Lambda _{k}( {\mathrm{u}}_{1},{\mathrm{u}}_{2})\|\\ & \leq\left[ \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}+s_{k}}}{\Gamma (p_{k}+r_{k}+s_{k}+1)} \right. \\ &\quad\left. + \frac{(F_{\iota_1}(\iota_2))^{r_{k}+s_{k}}}{\Gamma (r_{k}+s_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{s_{k}}}{ \Gamma (s_{k}+1)} \right]\left(\ell_k\left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right)\right)\\ &\leq \Phi_k \ell_k \left(\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|\right). \end{align*}$

Consequently,

$\|\Lambda({\mathrm{v}}_{1},{\mathrm{v}}_{2})-\Lambda({\mathrm{u}}_{1},{\mathrm{u}}_{2})\| \leq \Phi\ell\|({\mathrm{v}}_{1},{\mathrm{u}}_{1})-({\mathrm{v}}_{2},{\mathrm{u}}_{2})\|.$

Since $\Phi\ell < 1$ , therefore, $\Lambda$ is a contraction operator. Thus, by Banach's fixed point Theorem 2.3, the operator $\Lambda$ has a unique fixed point, which is the unique solution of fractional G-snap system (1.6) and the proof is finished.

Next, we are ready to study the existence of solution of fractional $\mathbb{G}$ - $(\mathbb{C}{\mathrm{S}}\mathbb{S})$ (1.6). For this regaed, we define the operators $\Omega, \Pi :X_{1}\times X_{2}\rightarrow \mathbb{R}^2, \Omega = (\Omega_1, \Omega_2), \Pi = (\Pi_1, \Pi_2)$ , such that $\Lambda_k = \Omega_k+\Pi_k$ , where

$\begin{align} (\Omega _{k}{\mathrm{v}}_{k})(\mathfrak{t}) = \int_{\iota_1}^{ \mathfrak{t}}\mathbb{G}^{\prime }(\xi ) \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k})}h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}(\xi ){\mathrm{d}}\xi, \end{align}$

(3.16)

and

$\begin{align} (\Pi _{k}{\mathrm{v}}_{k })(\mathfrak{t})& = \int_{\iota_1}^{\iota_2} g_{k0}(\xi) \, {\mathrm{d}}\xi + \frac{ (F_{\iota_1}( \mathfrak{t}))^{q_{k}}}{ \Gamma(q_{k}+1)} \int_{\iota_1}^{\iota_2} g_{k1}(\xi)\, {\mathrm{d}}\xi \\ & \quad +\int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}} g_{k2}(\xi)}{\Gamma(q_{k}+p_{k}+1)} \, {\mathrm{d}}\xi + \int_{\iota_1}^{\iota_2} \frac{(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}+r_{k}} g_{k3}(\xi)}{\Gamma (q_{k}+p_{k}+r_{k}+1)} \, {\mathrm{d}}\xi. \end{align}$

(3.17)

Theorem 3.3. Let $h_{k}\in C(\Sigma \times\mathbb{R}^8), (k = 1, 2)$ be continuous functions. Moreover, assume that

(H2) there exist real constants $\lambda_k > 0, \, (k = 1, 2)$ , so that

$|h_{k} (\mathfrak{t},{\mathrm{ v}}_1,{\mathrm{v}}_2,\dots,{\mathrm{v}}_8 ) |\leq\lambda_k,\qquad \forall \mathfrak{t}\in \Sigma\, \,{\mathrm{v}}_i \in C(\Sigma), \, (i = 1,2,\dots,8).$

Then, the fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) has at least one solution on $\Sigma$ .

Proof. At the beginning, we define a closed bounded ball

$\mathbb{B}_r = \Big\{ ({\mathrm{v}}_1,{\mathrm{v}}_2) \in X_{1}\times X_{2} \, : \, \Vert ({\mathrm{v}}_1,{\mathrm{v}}_2) \Vert\leq r \Big\},$

which satisfying

$\begin{equation} r\geq\max\left\lbrace r_1, r_2 \right\rbrace, \end{equation}$

(3.18)

where

$\begin{align*} r_k & \geq M_{k0} +M_{k1}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}}}{ \Gamma (q_{k}+1)}\right) \nonumber\\ &\quad+M_{k2}\left(1+\frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma(p_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}}}{\Gamma (q_{k}+p_{k}+1)}\right) \nonumber\\ &\quad +M_{k3}\left( 1+\frac{(F_{\iota_1}(\iota_2))^{r_{k}}}{ \Gamma(r_{k}+1)}+\frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)}+\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+1)} \right)\nonumber\\ &\quad+\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k}+1)} \lambda_k. \end{align*}$

Firstly, we will prove $\Omega{\mathrm{v}}+\Pi{\mathrm{u}}\in\mathbb{B}_r$ . By using ${\mathrm{(H2)}}$ , for ${\mathrm{v}} = ({\mathrm{v}}_1, {\mathrm{v}}_2), {\mathrm{u}} = ({\mathrm{u}}_1, {\mathrm{u}}_2) \in\mathbb{B}_r$ and $\mathfrak{t}\in[a, b]$ , we have

$\begin{align} \|\Omega_k({\mathrm{v_1}},{\mathrm{v_2}})\| \leq\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)} \lambda_k, \end{align}$

(3.19)

and

$\begin{align} \|\Pi_k({\mathrm{u_1}}, {\mathrm{u_2}})\| &\leq M_{k0} +M_{k1}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}}}{ \Gamma (q_{k}+1)}\right) \\ &\quad+M_{k2}\left(1+\frac{(F_{\iota_1}(\iota_2))^{p_{k}}}{\Gamma(p_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}}}{\Gamma(q_{k}+p_{k}+1)}\right) \\ &\quad +M_{k3}\left( 1+\frac{ ( F_{\iota_1}(\iota_2))^{r_{k}}}{\Gamma (r_{k}+1)}+\frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma(p_{k}+r_{k}+1)}+\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+1)}\right). \end{align}$

(3.20)

Hence, from (3.19) and (3.20), we have

$\begin{align*} \|\Omega_k({\mathrm{v_1}},{\mathrm{v_2}}) & +\Pi_k({\mathrm{u_1}},{\mathrm{u_2}}) \|\nonumber\\ &\leq M_{k0} +M_{k1}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}}}{\Gamma (q_{k}+1)}\right) \nonumber\\ &\quad+M_{k2} \left( 1+ \frac{ (F_{\iota_1} (\iota_2))^{p_{k}}}{ \Gamma(p_{k}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}}}{\Gamma(q_{k}+p_{k}+1)}\right) \nonumber\\ &\quad +M_{k3}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{r_{k}}}{\Gamma (r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{k}+r_{k}}}{\Gamma (p_{k}+r_{k}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+1)}\right)\\ &\quad+\frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)} \lambda_k\leq r. \end{align*}$

Then,

$\|\Omega({\mathrm{v_1}},{\mathrm{v_2}})+\Pi({\mathrm{u_1}},{\mathrm{u_2}})\|\leq r,$

this implying that $\Omega{\mathrm{v}}+\Pi{\mathrm{u}}\in\mathbb{B}_r$ .

Secondly, we will prove that the operator $\Pi$ is a contraction mapping. It is clearly that $\Pi_k$ is a contraction with the constant zero. Thus, $\Pi$ is a contraction operator.

Third, we will prove that the operator $\Omega$ is a continuous. Let $({\mathrm{v}}_{n, 1}, {\mathrm{v}}_{n, 2})$ be a sequence of a bounded ball $\mathbb{B}_r$ , such that $({\mathrm{v}}_{n, 1}, {\mathrm{v}}_{n, 2})\rightarrow ({\mathrm{v}}_{1}, {\mathrm{v}}_{2})$ as $n\rightarrow \infty$ in $\mathbb{B}_r$ , we find that

$\begin{align*} \label{15} |(\Omega_k({\mathrm{v_{n,1}}},{\mathrm{v_{n,2}}}))(\mathfrak{t}) &- ( \Omega_k({\mathrm{v_{1}}},{\mathrm{v_{2}}}))(\mathfrak{t})|\nonumber \\ &\leq\int_{\iota_1}^{\mathfrak{t}}\frac{\mathbb{G}^{\prime }(\xi )(F_{\iota_1}(\mathfrak{t}))^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k})} \left| h_{{\mathrm{v}}_{n,1},{\mathrm{v}}_{n,2},k}(\xi ) - h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}(\xi )\right| \, {\mathrm{d}}\xi \nonumber\\ &\leq \frac{(F_{\iota_1}(\iota_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)} \left\| h_{{\mathrm{v}}_{n,1},{\mathrm{v}}_{n,2},k}(. )- h_{{\mathrm{v}}_1,{\mathrm{v}}_2,k}(. )\right\|. \end{align*}$

By continuity of $h_{{\mathrm{v}}_1, {\mathrm{v}}_2, k}$ , we have

$\|\Omega_k({\mathrm{v_{n,1}}},{\mathrm{v_{n,2}}})-\Omega_k({\mathrm{v_{1}}},{\mathrm{v_{2}}})\|\rightarrow 0,$

as $n\rightarrow \infty$ . So, $\Omega$ is a continuous operator.

Fourth, we will prove that the operator $\Omega$ is a compact operator. By using ${\mathrm{(H2)}}$ , for ${\mathrm{v}} = ({\mathrm{v}}_1, {\mathrm{v}}_2), \in\mathbb{B}_r$ and $\mathfrak{t_1}, \mathfrak{t_2}\in \Sigma$ with $\mathfrak{t_1} < \mathfrak{t_2}$ , we have

$\begin{align*} |(\Omega_k({\mathrm{v_{1}}},{\mathrm{v_{2}}}))(\mathfrak{t_2})-&(\Omega_k({\mathrm{v_{1}}},{\mathrm{v_{2}}}))(\mathfrak{t_1})| \\ &\leq\left| \int_{\iota_1}^{\mathfrak{t_2}}\frac{\mathbb{G}^{\prime }(\xi )(F_{\iota_1}(\mathfrak{t}_2))^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k})} h_{{\mathrm{v}}_{1},{\mathrm{v}}_{2},k}(\xi ) {\mathrm{d}}\xi \right.\\ &\quad\left. - \int_{\iota_1}^{\mathfrak{t_1}}\frac{\mathbb{G}^{\prime }(\xi )(F_{\iota_1}(\mathfrak{t}_1))^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma( q_{k}+ p_{k}+r_{k}+s_{k})} h_{{\mathrm{v}}_{1}, {\mathrm{v}}_{2},k}(\xi) \, {\mathrm{d}}\xi \right|\\ &\leq\left| \int_{\iota_1}^{ \mathfrak{t_1}} \frac{\mathbb{G}^{\prime }(\xi )(F_{\iota_1}(\mathfrak{t})_2)^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k})} h_{{\mathrm{v}}_{1},{\mathrm{v}}_{2},k}(\xi ) \, {\mathrm{d}}\xi \right.\\ &\quad\left. - \int_{\iota_1}^{ \mathfrak{t_1}}\frac{\mathbb{G}^{\prime }(\xi )(F_{\iota_1}(\mathfrak{t})_2)^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k})} h_{{\mathrm{v}}_{1},{\mathrm{v}}_{2},k}(\xi )\, {\mathrm{d}}\xi \right|\\ &\quad + \left| \int_{\mathfrak{t_1}}^{\mathfrak{t_2}}\frac{\mathbb{G}^{\prime }(\xi) (F_{\iota_1}(\mathfrak{t}_2))^{q_{k}+p_{k}+r_{k}+s_{k}-1}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k})} h_{{\mathrm{v}}_{1},{\mathrm{v}}_{2},k}(\xi )\, {\mathrm{d}}\xi \right|\\ &\leq\lambda_k\left[\frac{(F_{\mathfrak{t}_1}(\mathfrak{t}_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma(q_{k}+p_{k}+r_{k}+s_{k}+1)}-\frac{(F_{\mathfrak{t}_1}(\mathfrak{t}_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)}\right] \\ &\quad+ \lambda_k\frac{(F_{\mathfrak{t}_1}(\mathfrak{t}_2))^{q_{k}+p_{k}+r_{k}+s_{k}}}{\Gamma (q_{k}+p_{k}+r_{k}+s_{k}+1)}. \end{align*}$

As $\mathfrak{t_1} \to \mathfrak{t_2}$ , we obtain

$|( \Omega_k({\mathrm{v_{1}}}, {\mathrm{v_{2}}}))(\mathfrak{t_2})-(\Omega_k({\mathrm{v_{1}}},{\mathrm{v_{2}}}))(\mathfrak{t_1})|\to 0,$

impling that $\Omega$ is equicontinuous. Furthermore, in view of (3.19), $\Omega$ is uniformly bounded. Hence, due to the Arzelá-Ascoli theorem, we deduce that $\Omega$ is a compact operator. Then, all the conditions of Theorem 2.4 are holding. Thus, fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) has at least one solution $({\mathrm{v}}_1, {\mathrm{v}}_2) \in\mathbb{B}_r$ . The proof is completed.

3.2. Ulam-Hyers stability

In this part, we review the stability criterion in the context of the ${\mathrm{U}}$ - ${\mathrm{H}}$ - $\mathbb{S}$ for solutions of the fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6).

Theorem 3.4. Let ${\mathrm{(H1)}}$ and $\Phi_k\ell_k < 1, (k = 1, 2)$ hold. Then, the fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) is ${\mathrm{U}}$ - ${\mathrm{H}}$ - $\mathbb{S}$ .

Proof. According to Theorem 3.2, we have

$\begin{align*} \|\Lambda _{k}({\mathrm{v}}_{1},{\mathrm{v}}_{2})-\Lambda _{k}({\mathrm{u}}_{1},{\mathrm{u}}_{2})\| \leq \Phi_k\ell_k\|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\|+\Phi_k\ell_k\|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\|, \end{align*}$

which yields that

$\begin{align*} \|\Lambda ({\mathrm{v}}_{1},{\mathrm{v}}_{2})-\Lambda ({\mathrm{u}}_{1},{\mathrm{u}}_{2})\| \leq \varXi \times \left( \begin{matrix} \|{\mathrm{v}}_{1}-{\mathrm{u}}_{1}\| \\ \|{\mathrm{v}}_{2}-{\mathrm{u}}_{2}\| \end{matrix}\right), \end{align*}$

where

$\varXi = \left( \begin{matrix} \Phi_1 \ell_1 & \Phi_1 \ell_1 \\ \Phi_2 \ell_2 & \Phi_2 \ell_2 \end{matrix}\right).$

Since $\Phi_k \ell_k < 1$ and each geometric sequences $(\Phi_k \ell_k)^n\rightarrow 0,$ hence $\varXi^n\rightarrow 0$ as $n\rightarrow \infty.$ Therefore, due to Theorem 2.6, the fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ (1.6) is ${\mathrm{U}}$ - ${\mathrm{H}}-\mathbb{S}$ .

4. Application

We allow here a few illustrations of the fractional $\mathbb{G}$ - $\mathbb{C}{\mathrm{S}}\mathbb{S}$ , based on numerical recreation to analyze their solutions. In these cases, we consider distinctive cases of the function $\mathbb{G}$ to cover the Caputo, Caputo-Hadamard and Katugampola adaptations.

Example 4.1. Based on the system (1.6), by assuming $\Sigma = [0.05, 0.95]$ ,

$q_{1} = 0.73 \in (0,1],\quad q_{2} = 0.36 \in (0,1],\quad p_{1} = 0.92 \in (0,1], \quad p_{2} = 0.45 \in (0,1],$

$r_{1} = 0.12 \in (0,1], \quad r_{2} = 0.87 \in (0,1], \quad s_{1} = 0.54 \in (0,1], \quad s_{2} = 0.27 \in (0,1],$

we consider a fractional $\mathbb{C}{\mathrm{S}}\mathbb{S}$ as

$\begin{equation} \left\{ \begin{array}{l} {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.73; \mathbb{G}} {\mathrm{v}}_{1}( \mathfrak{t}) = {\mathrm{u}}_{1}( \mathfrak{t}), \quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.36;\mathbb{G}} {\mathrm{v}}_{2}( \mathfrak{t}) = {\mathrm{u}}_{2}(\mathfrak{t}), \\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.92;\mathbb{G}}{\mathrm{u}}_{1}(\mathfrak{t}) = {\mathrm{w}}_{1}(\mathfrak{t}),\quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.45;\mathbb{G}}{\mathrm{u}}_{2}(\mathfrak{t}) = {\mathrm{w}}_{2}(\mathfrak{t}), \\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.12;\mathbb{G}}{\mathrm{w}}_{1}(\mathfrak{t}) = {\mathrm{x}}_{1}(\mathfrak{t}), \quad {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.87;\mathbb{G}} {\mathrm{w}}_{2}( \mathfrak{t}) = {\mathrm{x}}_{2}(\mathfrak{t}),\\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.54;\mathbb{G}}{\mathrm{x}}_{1}(\mathfrak{t} ) = \dfrac{5 \mathfrak{t}}{36(\sqrt{15} + \mathfrak{t}^2)} + \dfrac{\arctan^2({\mathrm{v}}_1)}{36 \left(7+\arctan^2({\mathrm{v}}_1)\right)} \\ \quad \quad + \dfrac{ \exp(|{\mathrm{v}}_{2}|+1)}{36 \left( \sqrt{15} + \exp( |{\mathrm{v}}_{2}|)\right)} + \dfrac{1}{72} \arcsin\dfrac{{\mathrm{u}}_{1}}{\sqrt[3]{36+{\mathrm{u}}_{1}}} \\ \quad \quad+ \dfrac{\mathfrak{t}}{36} \dfrac{\sin | {\mathrm{u}}_{2}|}{15 + \sin|{\mathrm{u}}_{2} |} + \dfrac{\exp({\mathrm{w}}_1)}{36(\sqrt{7}+\exp({\mathrm{w}}_1))} + \dfrac{ {\mathrm{w}}_2^2}{108\left({\mathrm{w}}_2+3\right)^2}\\ \quad \quad+ \dfrac{\left({\mathrm{x}}_1 {\mathrm{x}}_2\right)^2}{54\left({\mathrm{x}}_1 {\mathrm{x}}_2 + 21 \right)^2}\\[0.3cm] {^{c}\mathcal{D}}_{\iota_1^{+}}^{0.27;\mathbb{G}}{\mathrm{x}}_{2}(\mathfrak{t} ) = \dfrac{ \mathfrak{t}^2}{2\sqrt{3} (\mathfrak{t}^2+49)} + \dfrac{ \tan^2({\mathrm{v}}_1)}{1.5 \left( 7 + \tan^2( {\mathrm{v}}_1) \right) } \\ \quad \quad + \dfrac{ \cos^2(|{\mathrm{v}}_{2}|+1)}{36 \left( \sqrt{15} + \cos^2( |{\mathrm{v}}_{2}|)\right)} + \dfrac{1}{25} \arctan \dfrac{3{\mathrm{u}}_{1}}{\sqrt[5]{6+{\mathrm{u}}_{1}}} \\ \quad \quad+ \dfrac{\arctan | {\mathrm{u}}_{2}|}{45 + \arctan|{\mathrm{u}}_{2} |} + \dfrac{ |{\mathrm{w}}_1|+3}{48\left(|{\mathrm{w}}_1|+5\right)^2} + \dfrac{ \exp({\mathrm{w}}_2)}{45(\sqrt{12} + \exp({\mathrm{w}}_2))} \\ \quad \quad+ \dfrac{\left({\mathrm{x}}_1 +{\mathrm{x}}_2\right)^2}{\sqrt{10} \left({\mathrm{x}}_1 +{\mathrm{x}}_2 + 21 \right)^2}, \end{array}\right. \end{equation}$

(4.1)

for $\mathfrak{t} \in \Sigma$ and

$\begin{align*} {\mathrm{v}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} \frac{s}{2}\, {\mathrm d}s = 0.2250, & {\mathrm{v}}_{2}(\iota_1) & = \int_{\iota_1}^{\iota_2} \sqrt{s} \, {\mathrm d}s = 0.6098,\\ {\mathrm{u}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} \frac{s^2}{5}\, {\mathrm d}s = 0.0571, & {\mathrm{u}}_{2}(\iota_1) & = \int_{\iota_1}^{\iota_2} \frac{3}{2} s\, {\mathrm d}s = 0.6750,\notag\\ {\mathrm{w}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} \frac{s}{\sqrt{2}} \, {\mathrm d}s = 0.3181,& {\mathrm{w}}_{2}(\iota_1) & = \int_{\iota_1}^{ \iota_2} \frac{\sqrt{s}}{7} \, {\mathrm d}s = 0.0871, \notag\\ {\mathrm{x}}_{1}(\iota_1) & = \int_{\iota_1}^{\iota_2} \sin (\pi s)\, {\mathrm d}s = 0.6287,& {\mathrm{x}}_{2}(\iota_1) & = \int_{\iota_1}^{ \iota_2} \cos (\pi s)\, {\mathrm d}s = 0. \notag \end{align*}$

Clearly,

$\begin{align*} h_{1} & (\mathfrak{t}, {\mathrm{v}}_{1}, {\mathrm{v}}_{2},{\mathrm{u}}_{1},{\mathrm{u}}_{2}, {\mathrm{w}}_{1}, {\mathrm{w}}_{2} {\mathrm{,x}}_{1}, {\mathrm{x}}_{2}) \\ & = \dfrac{5 \mathfrak{t}}{36(\sqrt{15} + \mathfrak{t}^2)} + \dfrac{\arctan^2({\mathrm{v}}_1)}{36 \left(7+\arctan^2({\mathrm{v}}_1)\right)} + \dfrac{ \exp(|{\mathrm{v}}_{2}|+1)}{36 \left( \sqrt{15} + \exp( |{\mathrm{v}}_{2}|)\right)} \\ & \quad + \dfrac{1}{72} \arcsin \dfrac{{\mathrm{u}}_{1}}{\sqrt[3]{36+{\mathrm{u}}_{1}}}+ \dfrac{\mathfrak{t}}{36} \dfrac{\sin | {\mathrm{u}}_{2}|}{15 + \sin|{\mathrm{u}}_{2} |} + \dfrac{\exp({\mathrm{w}}_1)}{36(\sqrt{7}+\exp({\mathrm{w}}_1))} \\ & \quad + \dfrac{{\mathrm{w}}_1^2}{108\left({\mathrm{w}}_1+3\right)^2}+ \dfrac{\left({\mathrm{x}}_1 {\mathrm{x}}_2\right)^2}{54\left({\mathrm{x}}_1 {\mathrm{x}}_2 + 21 \right)^2}, \end{align*}$

and

$\begin{align*} h_{2} & (\mathfrak{t}, {\mathrm{v}}_{1}, {\mathrm{v}}_{2},{\mathrm{u}}_{1},{\mathrm{u}}_{2}, {\mathrm{w}}_{1}, {\mathrm{w}}_{2} {\mathrm{,x}}_{1}, {\mathrm{x}}_{2}) \\ & = \dfrac{ \mathfrak{t}^2}{ 2\sqrt{3} (\mathfrak{t}^2+49)} + \dfrac{ \tan^2({\mathrm{v}}_1)}{1.5 \left( 7 + \tan^2( {\mathrm{v}}_1) \right) } \dfrac{ \cos^2(|{\mathrm{v}}_{2}|+1)}{36 \left( \sqrt{15} + \cos^2( |{\mathrm{v}}_{2}|)\right)} \\ & \quad + + \dfrac{1}{25} \arctan \dfrac{3{\mathrm{u}}_{1}}{\sqrt[5]{6+{\mathrm{u}}_{1}}} + \dfrac{\arctan | {\mathrm{u}}_{2}|}{45 + \arctan|{\mathrm{u}}_{2} |} + \dfrac{ |{\mathrm{w}}_1|+3}{48\left(|{\mathrm{w}}_1|+5\right)^2} \\ & \quad + \dfrac{\exp({\mathrm{w}}_2)}{45(\sqrt{12} + \exp({\mathrm{w}}_2))} + \dfrac{\left({\mathrm{x}}_1 +{\mathrm{x}}_2\right)^2}{\sqrt{10} \left({\mathrm{x}}_1 +{\mathrm{x}}_2 + 21 \right)^2}. \end{align*}$

Thus, we can rewrite the above system as Eq (1.6). At present, we will have

$\begin{align*} \big\vert h_{1} & \big(\mathfrak{t},{\mathrm{v}}_1,{\mathrm{v}}_2,\dots,{\mathrm{v}}_8 \big) - h_{1} \big(\mathfrak{t},{\mathrm{v}}_1^\ast, {\mathrm{v}}_2^\ast, \dots, {\mathrm{v}}_8^\ast \big) \big\vert\\ & = \bigg\vert \dfrac{5 \mathfrak{t}}{36(\sqrt{15} + \mathfrak{t}^2)} + \dfrac{ \arctan^2({\mathrm{v}}_1)}{36 \left( 7 + \arctan^2({\mathrm{v}}_1)\right)} + \dfrac{ \exp(|{\mathrm{v}}_{2}|+1)}{36 \left( \sqrt{15} + \exp( | {\mathrm{v}}_{2}|)\right)} \\ & \quad + \dfrac{1}{72} \arcsin \dfrac{{\mathrm{v}}_3}{\sqrt[3]{36+{\mathrm{v}}_3}}+ \dfrac{\mathfrak{t}}{36} \dfrac{\sin | {\mathrm{v}}_4|}{15 + \sin|{\mathrm{v}}_4 |} + \dfrac{\exp({\mathrm{v}}_5)}{36(\sqrt{7}+\exp({\mathrm{v}}_5))} \\ & \quad + \dfrac{{\mathrm{v}}_6^2}{ 108\left({\mathrm{v}}_6+3\right)^2}+ \dfrac{ ({\mathrm{v}}_7{\mathrm{v}}_8)^2}{ 54 \left({\mathrm{v}}_7{\mathrm{v}}_8 + 21 \right)^2}\\ &\quad - \bigg( \dfrac{5 \mathfrak{t}}{36(\sqrt{15} + \mathfrak{t}^2)} + \dfrac{\arctan^2({\mathrm{v}}_1^\ast)}{36 \left(7+\arctan^2({\mathrm{v}}_1^\ast)\right)} + \dfrac{ \exp(|{\mathrm{v}}_2^\ast|+1)}{36 \left( \sqrt{15} + \exp( |{\mathrm{v}}_2^\ast|)\right)} \\ & \quad + \dfrac{1}{72} \arcsin \dfrac{ {\mathrm{v}}_3^\ast}{\sqrt[3]{36 + {\mathrm{v}}_3^\ast}}+ \dfrac{ \mathfrak{t}}{36} \dfrac{\sin |{\mathrm{v}}_4^\ast|}{15 + \sin|{\mathrm{v}}_4^\ast |} + \dfrac{\exp({\mathrm{v}}_5^\ast)}{36(\sqrt{7} + \exp({\mathrm{v}}_5^\ast))} \\ & \quad + \dfrac{ ({\mathrm{v}}_6^\ast)^2}{108\left( {\mathrm{v}}_6^\ast+3\right)^2}+ \dfrac{ \left({\mathrm{v}}_7^\ast {\mathrm{v}}_8^\ast\right)^2}{ 54 \left({\mathrm{v}}_7^\ast {\mathrm{v}}_8^\ast + 21 \right)^2}\bigg)\bigg\vert \leq \frac{5}{36} \sum\limits_{i = 1}^{8}| {\mathrm{v}}_i-{\mathrm{v}}_i^\ast|, \end{align*}$

with $\ell_1 = \frac{5}{36}$ and

$\begin{align*} \big\vert h_{2} & \big(\mathfrak{t},{\mathrm{v}}_1,{\mathrm{v}}_2,\dots,{\mathrm{v}}_8 \big) - h_{2} \big(\mathfrak{t},{\mathrm{v}}_1^\ast, {\mathrm{v}}_2^\ast, \dots, {\mathrm{v}}_8^\ast \big) \big\vert\\ & = \bigg\vert \dfrac{ \mathfrak{t}^2}{2 \sqrt{3} (\mathfrak{t}^2+49)} + \dfrac{ \tan^2({\mathrm{v}}_1)}{1.5 \left( 7 + \tan^2( {\mathrm{v}}_1) \right) } + \dfrac{ \cos^2(|{\mathrm{v}}_{2}|+1)}{36 \left( \sqrt{15} + \cos^2(|{\mathrm{v}}_{2}|)\right)} \\ & \quad + \dfrac{1}{25} \arctan \dfrac{3{\mathrm{v}}_3}{\sqrt[5]{6 + {\mathrm{v}}_3}} + \dfrac{\arctan | {\mathrm{v}}_4|}{45 + \arctan|{\mathrm{v}}_4 |} + \dfrac{|{\mathrm{v}}_5|+3}{48\left(|{\mathrm{v}}_5|+5\right)^2} \\ & \quad + \dfrac{ \exp({\mathrm{v}}_6)}{ 45(\sqrt{12} + \exp({\mathrm{v}}_6))} + \dfrac{\left({\mathrm{v}}_7 +{\mathrm{v}}_8\right)^2}{\sqrt{10} \left({\mathrm{v}}_7 + {\mathrm{v}}_8 + 21 \right)^2}\\ &\quad - \bigg(\dfrac{ \mathfrak{t}^2}{2 \sqrt{3} (\mathfrak{t}^2+49)} + \dfrac{ \tan^2({\mathrm{v}}_1^\ast)}{1.5 \left( 7 + \tan^2( {\mathrm{v}}_1^\ast) \right) } + \dfrac{ \cos^2(|{\mathrm{v}}_2^\ast|+1)}{36 \left( \sqrt{15} + \cos^2( |{\mathrm{v}}_2^\ast|)\right)} \\ & \quad + \dfrac{1}{25} \arctan \dfrac{3{\mathrm{v}}_3^\ast}{\sqrt[5]{ 6 +{\mathrm{v}}_3^\ast}} + \dfrac{\arctan | {\mathrm{v}}_4^\ast|}{45 + \arctan| {\mathrm{v}}_4^\ast |} + \dfrac{ |{\mathrm{v}}_5^\ast|+3}{48\left( | {\mathrm{v}}_5^\ast|+5\right)^2} \\ & \quad + \dfrac{\exp({\mathrm{v}}_6^\ast)}{45(\sqrt{12} + \exp({\mathrm{v}}_6^\ast))} + \dfrac{\left( {\mathrm{v}}_7^\ast +{\mathrm{v}}_8^\ast\right)^2}{\sqrt{10} \left({\mathrm{v}}_8^\ast +{\mathrm{v}}_8^\ast + 21 \right)^2}\bigg)\bigg\vert \leq \frac{1}{2\sqrt{3}} \sum\limits_{i = 1}^{8}| {\mathrm{v}}_i-{\mathrm{v}}_i^\ast|, \end{align*}$

with $\ell_2 = \frac{1}{2\sqrt{3}}$ . So $\ell = \frac{1}{2\sqrt{3}}$ . Now, from (3.6), we consider four cases for $\mathbb{G}$ as:

● $\mathbb{G}_1(\iota) = 2^{\iota}$ ,

● $\mathbb{G}_2(\iota) = \iota$ (Caputo derivative),

● $\mathbb{G}_3(\iota) = \ln \iota$ (Caputo-Hadamard derivative),

● $\mathbb{G}_4(\iota) = \sqrt{\iota}$ (Katugampola derivative).

Thus,

$\begin{align} \Phi_1 & = \frac{(F_{ \iota_1}( \iota_2))^{ q_{1}+p_{1}+r_{1}+s_{1}}}{ \Gamma (q_{1}+p_{1}+r_{1}+s_{1}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{1}+r_{1}+s_{1}}}{\Gamma(p_{k} + r_{1}+s_{1}+1)} \\ &\quad +\frac{ (F_{\iota_1}(\iota_2))^{r_{1}+s_{1}}}{ \Gamma (r_{1}+s_{1}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{s_{1}}}{ \Gamma (s_{1}+1)}\simeq \left\{\begin{array}{ll} 0.4091, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 0.4106, & \mathbb{G}_2(\iota) = \iota,\\ 1.7597, & \mathbb{G}_3( \iota) = \ln \iota,\\ 0.3473, & \mathbb{G}_4(\iota) = \sqrt{\iota}, \end{array} \right. \end{align}$

(4.2)

and

$\begin{align} \Phi_2 & = \frac{(F_{ \iota_1}( \iota_2))^{q_{2}+p_{2}+r_{2}+s_{2}}}{ \Gamma (q_{2}+p_{2}+r_{2}+s_{2}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{2}+r_{2}+s_{2}}}{ \Gamma(p_{k} + r_{2}+s_{2}+1)} \\ &\quad +\frac{ (F_{\iota_1}(\iota_2))^{r_{2}+s_{2}}}{ \Gamma (r_{2}+s_{2}+1)}+ \frac{(F_{\iota_1}(\iota_2))^{s_{2}}}{ \Gamma (s_{2}+1)}\simeq \left\{\begin{array}{ll} 0.3370, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 0.3383, & \mathbb{G}_2(\iota) = \iota,\\ 1.4912, & \mathbb{G}_3( \iota) = \ln \iota,\\ 0.2826, & \mathbb{G}_4(\iota) = \sqrt{\iota}. \end{array} \right. \end{align}$

(4.3)

Hence,

$\Phi\simeq \left\{\begin{array}{ll} 2.9461, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 2.9567, & \mathbb{G}_2(\iota) = \iota,\\ 12.9144, & \mathbb{G}_3( \iota) = \ln \iota,\\ 2.5009, & \mathbb{G}_4(\iota) = \sqrt{\iota}, \end{array} \right.$

and we have

$\Phi\ell \simeq \left\{ \begin{array}{ll} 0.4091 < 1, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 0.4106 < 1, & \mathbb{G}_2(\iota) = \iota,\\ 1.7936\nleq 1, & \mathbb{G}_3( \iota) = \ln \iota,\\ 0.3473 < 1, & \mathbb{G}_4(\iota) = \sqrt{\iota}.\end{array} \right.$

On the other hand, by using equations in (3.7), we get

$\begin{equation*} M_{1j} = \sup\limits_{\mathfrak{t}\in \Sigma} \int_{\iota_1}^{\mathfrak{t}} |g_{1j}(\xi)| \, {\mathrm{d}} \xi = 0.6328, \qquad M_{2j} = \sup\limits_{\mathfrak{t}\in \Sigma} \int_{\iota_1}^{\iota_2} |g_{1j}(\xi)| \, {\mathrm{d}}\xi = 0.7632, \end{equation*}$

for $j = 0, 1, 2, 3$ and

$\begin{align*} h_1^0 & = \sup\limits_{\mathfrak{t}\in \Sigma }| h_1( \mathfrak{t},0,0,0,0,0,0,0,0)|\simeq \frac{5}{36} + \frac{1}{36(1+\sqrt{7})},\\ \qquad h_2^0 & = \sup\limits_{\mathfrak{t}\in \Sigma }| h_2( \mathfrak{t},0,0,0,0,0,0,0,0)|\simeq \frac{1}{2\sqrt{3}} + \frac{1}{18(1+\sqrt{15})}, \end{align*}$

for $k = 1, 2$ . By employing Eq (3.6), we obtain

$\begin{align*} \Delta_1& = M_{10} +M_{11}\left(1+ \frac{(F_{\iota_1}(\iota_2))^{q_{1}}}{ \Gamma (q_1+1)}\right) + M_{12} \left(1+\frac{(F_{\iota_1}(\iota_2))^{p_{1}}}{\Gamma (p_{1}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_1 + p_1 }}{ \Gamma(q_1+ p_1+1)}\right) \notag\\ &\quad +M_{13}\left( 1 + \frac{(F_{\iota_1}(\iota_2))^{ r_1}}{\Gamma (r_1+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{1}+r_{1}}}{\Gamma (p_{1}+r_{1}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{1} + p_{1} + r_{1}}}{\Gamma(q_{1} + p_{1} + r_{1}+1)}\right),\\ &\simeq \left\{\begin{array}{ll} 3.4102, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 3.4173, & \mathbb{G}_2(\iota) = \iota,\\ 9.3326, & \mathbb{G}_3( \iota) = \ln \iota,\\ 3.1122, & \mathbb{G}_4(\iota) = \sqrt{\iota}, \end{array} \right.\\ \Delta_2& = M_{20} +M_{21}\left(1+ \frac{ (F_{\iota_1}(\iota_2))^{q_{2}}}{\Gamma (q_{2}+1)}\right) + M_{22} \left(1+\frac{(F_{\iota_1}( \iota_2))^{p_{2}}}{\Gamma (p_{2}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{2}+p_{2}}}{\Gamma(q_{2}+p_{2}+1)}\right) \notag\\ &\quad +M_{23}\left( 1 + \frac{(F_{\iota_1}(\iota_2))^{ r_{2}}}{\Gamma (r_{2}+1)} + \frac{(F_{\iota_1}(\iota_2))^{p_{2}+r_{2}}}{\Gamma (p_{2} + r_{2}+1)} + \frac{(F_{\iota_1}(\iota_2))^{q_{2}+p_{2}+r_{2}}}{\Gamma(q_{2} + p_{2} + r_{2}+1)}\right)\\ &\simeq \left\{\begin{array}{ll} 4.4208, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 4.4285, & \mathbb{G}_2(\iota) = \iota,\\ 9.8667, & \mathbb{G}_3( \iota) = \ln \iota,\\ 4.0927, & \mathbb{G}_4(\iota) = \sqrt{\iota}, \end{array} \right. \end{align*}$

and so we can choose

$\begin{align*} \varepsilon\geq \max\left\lbrace \dfrac{\Delta_1+h_1^0 \Phi_1}{(1- \ell_1 \Phi_1)},\dfrac{\Delta_2+h_2^0 \Phi_2}{(1- \ell_2 \Phi_2)}\right\rbrace &\simeq \left\{\begin{array}{ll} 7.2317, & \mathbb{G}_1(\iota) = 2^{\iota},\\ 7.2597, & \mathbb{G}_2(\iota) = \iota,\\ -14.7276, & \mathbb{G}_3( \iota) = \ln \iota,\\ 6.1479, & \mathbb{G}_4(\iota) = \sqrt{\iota}. \end{array} \right. \end{align*}$

We define the Algorithm 1 for obtaining the values of $\Phi\ell$ , $\Delta_i$ and $\varepsilon$ , which is shown in the MATLAB commands. One can check numerical results of $\Phi\ell$ , $\Delta_i$ and $\varepsilon$ in and for $\iota\in \lbrack 0.05, 0.95]$ , and in . Accordingly, all requirements of Theorem 3.2 hold, and so the fractional nonlinear couple snap system $(\mathbb{C}{\mathrm{S}}\mathbb{S})$ in the $\mathbb{G}$ -Caputo sense $(\mathbb{G}{\mathrm{C}})$ with initial conditions (4.1) has one unique solution on the $\lbrack 0.05, 0.95]$ .

Table 1. Numerical values of $\Phi\ell, \, \Delta_1, \Delta_1$ and $\varepsilon$ in Example 4.1 $\forall \iota \in \lbrack 0.05, 0.95]$ when $\mathbb{G}_1 = 2^{\iota}$ and $\mathbb{G}_2 = \iota$ .

	$\mathbb{G}_1(\iota)=2^{\iota}$				$\mathbb{G}_2(\iota)=\iota$
$\iota$	$\Phi\ell$	$\Delta_1$	$\Delta_2$	$\varepsilon\geq$	$\Phi\ell$	$\Delta_1$	$\Delta_2$	$\varepsilon\geq$
$0.05$	$0.4092$	$0.0000$	$0.6098$	$1.4836$	$0.4107$	$0.0000$	$0.6098$	$1.4886$
$0.13$	$0.4092$	$0.0916$	$0.8907$	$1.9071$	$0.4107$	$0.0918$	$0.8915$	$1.9142$
$0.21$	$0.4092$	$0.2602$	$1.1697$	$2.3280$	$0.4107$	$0.2608$	$1.1713$	$2.3371$
$0.29$	$0.4092$	$0.4976$	$1.4321$	$2.7238$	$0.4107$	$0.4986$	$1.4344$	$2.7347$
$0.37$	$0.4092$	$0.7921$	$1.6660$	$3.0766$	$0.4107$	$0.7938$	$1.6687$	$3.0888$
$0.45$	$0.4092$	$1.1297$	$1.8627$	$3.3733$	$0.4107$	$1.1321$	$1.8658$	$3.3867$
$0.53$	$0.4092$	$1.4944$	$2.0267$	$3.6206$	$0.4107$	$1.4976$	$2.0300$	$3.6348$
$0.61$	$0.4092$	$1.8697$	$2.2517$	$3.9599$	$0.4107$	$1.8736$	$2.2552$	$3.9752$
$0.69$	$0.4092$	$2.2394$	$2.5593$	$4.5209$	$0.4107$	$2.2441$	$2.5634$	$4.5429$
$0.77$	$0.4092$	$2.5889$	$2.9421$	$5.1124$	$0.4107$	$2.5943$	$2.9469$	$5.1371$
$0.85$	$0.4092$	$2.9058$	$3.3892$	$5.6758$	$0.4107$	$2.9119$	$3.3949$	$5.6977$
$0.93$	$0.4092$	$3.1810$	$3.8873$	$6.4269$	$0.4107$	$3.1877$	$3.8940$	$6.4518$

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Table 2. Numerical values of $\Phi\ell, \, \Delta_1, \Delta_1$ and $\varepsilon$ in Example 4.1 $\forall \iota \in \lbrack 0.05, 0.95]$ when $\mathbb{G}_3 (\iota) = \ln \iota$ and $\mathbb{G}_4(\iota) = \sqrt{\iota}$ .

	$\mathbb{G}_3=\ln \iota$				$\mathbb{G}_4=\sqrt{\iota}$
$\iota$	$\Phi\ell$	$\Delta_1$	$\Delta_2$	$\varepsilon\geq$	$\Phi\ell$	$\Delta_1$	$\Delta_2$	$\varepsilon\geq$
$0.05$	$1.7937$	$0.0000$	$0.6098$	$-2.4433$	$0.3474$	$0.0000$	$0.6098$	$1.2925$
$0.13$	$1.7937$	$0.2549$	$1.5187$	$-2.7789$	$0.3474$	$0.0835$	$0.8562$	$1.6360$
$0.21$	$1.7937$	$0.7237$	$2.3707$	$-3.3959$	$0.3474$	$0.2373$	$1.1034$	$1.9805$
$0.29$	$1.7937$	$1.3828$	$3.1158$	$-4.2635$	$0.3474$	$0.4537$	$1.3383$	$2.3081$
$0.37$	$1.7937$	$2.1996$	$3.7133$	$-5.3387$	$0.3474$	$0.7224$	$1.5507$	$2.6041$
$0.45$	$1.7937$	$3.1339$	$4.1339$	$-6.5685$	$0.3474$	$1.0303$	$1.7330$	$2.8582$
$0.53$	$1.7937$	$4.1410$	$4.3933$	$-7.8940$	$0.3474$	$1.3629$	$1.8890$	$3.0758$
$0.61$	$1.7937$	$5.1740$	$4.8174$	$-9.2538$	$0.3474$	$1.7053$	$2.1001$	$3.3700$
$0.69$	$1.7937$	$6.1876$	$5.4819$	$-10.5880$	$0.3474$	$2.0427$	$2.3850$	$3.7672$
$0.77$	$1.7937$	$7.1402$	$6.3609$	$-11.8418$	$0.3474$	$2.3617$	$2.7371$	$4.2581$
$0.85$	$1.7937$	$7.9969$	$7.4169$	$-12.9695$	$0.3474$	$2.6511$	$3.1472$	$4.8298$
$0.93$	$1.7937$	$8.7318$	$8.6033$	$-13.9368$	$0.3474$	$2.9026$	$3.6035$	$5.4660$

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Figure 1. Graphical representation of $\Delta_1, \Delta_1$ and $\varepsilon$ for $\iota \in [0.05, 0.95]$ in Example 4.1.

DownLoad: Full-Size Img PowerPoint

5. Conclusions

In this paper, we defined a new fractional mathematical model of a BVP consisting of a coupled snap equation with integral boundary conditions in the framework of the generalized sequential $\mathbb{G}$ -operators, and turned to the investigation of the qualitative behaviors of its solutions, including existence, uniqueness, stability and inclusion version. To confirm the existence criterion, we used the Krasnoselskii theorem, and to confirm the uniqueness criterion, we utilized the Banach theorem. Different kinds of stability criteria were studied based on the standard definitions of these notions. In the final step, we designed examples, and, by assuming different cases for the function $\mathbb{G}$ and order $q$ , we obtained numerical results of these two suggested fractional coupled snap systems in some versions, such as Caputo, Caputo-Hadamard and Katugampola.

Conflict of interest

We declare that no competing interests.

Appendix

Table Algorithm A1. MATLAB lines for Example 4.1.

1 clear;
2 format short;
3 syms v e;
4 q_1=0.83; q_2=0.36; p_1=0.92; p_2=0.45;
5 r_1=0.12; r_2=0.87; s_1=0.54; s_2=0.27;
6 iota_1=0.05; iota_2=0.95;
7 G1=2^v; G2=v; G3=log(v); G4=sqrt(v);
8 g_10=v/2; g_20=sqrt(v);
9 g_11=v^2/5; g_21=3*v/2;
10 g_12=v/sqrt(2); g_22=sqrt(v)/7;
11 g_13=sin(vpi); g_23=cos(vpi);
12 mathrmv_1=int(g_10, v, iota_1, iota_2);
13 mathrmv_2=int(g_20, v, iota_1, iota_2);
14 mathrmu_1=int(g_11, v, iota_1, iota_2);
15 mathrmu_2=int(g_21, v, iota_1, iota_2);
16 mathrmw_1=int(g_12, v, iota_1, iota_2);
17 mathrmw_2=int(g_22, v, iota_1, iota_2);
18 mathrmx_1=int(g_13, v, iota_1, iota_2);
19 mathrmx_2=int(g_23, v, iota_1, iota_2);
20 ell_1=5/36; ell_2=1/(5*sqrt(3));
21 ell=max(ell_1, ell_2);
22 h_1_0=5/36+1/(36*(1+sqrt(7)));
23 h_2_0=1/(5sqrt(3))+1/(18(1+sqrt(15)));
24 %G1
25 t=iota_1;
26 column=1;
27 nn=1;
28 while t < =iota_2+0.08
29 MI(nn, column) = nn;
30 MI(nn, column+1) = t;
31 Phi_1=(eval(subs(G1, {v}, {iota_2}))...
32 -eval(subs(G1, {v}, {iota_1})))^(q_1+p_1+r_1+s_1)...
33 /gamma(q_1+p_1+r_1+s_1+1)+(eval(subs(G1, {v}, {iota_2}))...
34 -eval(subs(G1, {v}, {iota_1})))^(p_1+r_1+s_1)...
35 /gamma(p_1+r_1+s_1+1)+(eval(subs(G1, {v}, {iota_2}))...
36 -eval(subs(G1, {v}, {iota_1})))^(r_1+s_1)...
37 /gamma(r_1+s_1+1)+(eval(subs(G1, {v}, {iota_2}))...
38 -eval(subs(G1, {v}, {iota_1})))^(s_1)/gamma(s_1+1);
39 MI(nn, column+2)=Phi_1*ell_1;
40 MI(nn, column+3)=Phi_1*ell_1 < 1;
41 Phi_2=(eval(subs(G1, {v}, {iota_2}))...
42 -eval(subs(G1, {v}, {iota_1})))^(q_2+p_2+r_2+s_2)...
43 /gamma(q_2+p_2+r_2+s_2+1)+(eval(subs(G1, {v}, {iota_2}))...
44 -eval(subs(G1, {v}, {iota_1})))^(p_2+r_2+s_2)...
45 /gamma(p_2+r_2+s_2+1)+(eval(subs(G1, {v}, {iota_2}))...
46 -eval(subs(G1, {v}, {iota_1})))^(r_2+s_2)...
47 /gamma(r_2+s_2+1)+(eval(subs(G1, {v}, {iota_2}))...
48 -eval(subs(G1, {v}, {iota_1})))^(s_2)/gamma(s_2+1);
49 MI(nn, column+4)=Phi_2*ell_2;
50 MI(nn, column+5)=Phi_2*ell_2 < 1;
51 Phi=max(Phi_1, Phi_2);
52 MI(nn, column+6)=Phi;
53 MI(nn, column+7)=Phi*ell;
54 MI(nn, column+8)=Phi*ell < 1;
55 M_10=int(abs(g_10), v, iota_1, t);
56 MI(nn, column+9)=M_10;
57 M_11=int(abs(g_11), v, iota_1, t);
58 MI(nn, column+10)=M_11;
59 M_12=int(abs(g_12), v, iota_1, t);
60 MI(nn, column+11)=M_12;
61 M_13=int(abs(g_13), v, iota_1, t);
62 MI(nn, column+12)=M_13;
63 M_20=int(abs(g_20), v, iota_1, iota_2);
64 MI(nn, column+13)=M_20;
65 M_21=int(abs(g_21), v, iota_1, t);
66 MI(nn, column+14)=M_21;
67 M_22=int(abs(g_22), v, iota_1, t);
68 MI(nn, column+15)=M_22;
69 M_23=int(abs(g_23), v, iota_1, t);
70 MI(nn, column+16)=M_23;
71 M_1j=max(max(max(M_10, M_11), M_12), M_13);
72 MI(nn, column+17)=M_1j;
73 M_2j=max(max(max(M_20, M_21), M_22), M_23);
74 MI(nn, column+18)=M_2j;
75 Delta_1=M_10+M_11*(1+(eval(subs(G1, {v}, {iota_2}))...
76 -eval(subs(G1, {v}, {iota_1})))^(q_1)/gamma(q_1+1))...
77 +M_12*(1+(eval(subs(G1, {v}, {iota_2}))...
78 -eval(subs(G1, {v}, {iota_1})))^(p_1)/gamma(p_1+1)...
79 +(eval(subs(G1, {v}, {iota_2}))...
80 -eval(subs(G1, {v}, {iota_1})))^(q_1+p_1)/gamma(q_1+p_1+1))...
81 +M_13*(1+(eval(subs(G1, {v}, {iota_2}))...
82 -eval(subs(G1, {v}, {iota_1})))^(r_1)/gamma(r_1+1)...
83 +(eval(subs(G1, {v}, {iota_2}))...
84 -eval(subs(G1, {v}, {iota_1})))^(p_1+r_1)/gamma(p_1+r_1+1)...
85 +(eval(subs(G1, {v}, {iota_2}))...
86 -eval(subs(G1, {v}, {iota_1})))^(q_1+p_1+r_1)...
87 /gamma(q_1+p_1+r_1+1));
88 MI(nn, column+19)=Delta_1;
89 Delta_2=M_20+M_21*(1+(eval(subs(G1, {v}, {iota_2}))...
90 -eval(subs(G1, {v}, {iota_1})))^(q_2)/gamma(q_2+1))...
91 +M_22*(1+(eval(subs(G1, {v}, {iota_2}))...
92 -eval(subs(G1, {v}, {iota_1})))^(p_2)/gamma(p_2+1)...
93 +(eval(subs(G1, {v}, {iota_2}))...
94 -eval(subs(G1, {v}, {iota_1})))^(q_2+p_2)/gamma(q_2+p_2+1))...
95 +M_23*(1+(eval(subs(G1, {v}, {iota_2}))...
96 -eval(subs(G1, {v}, {iota_1})))^(r_2)/gamma(r_2+1)...
97 +(eval(subs(G1, {v}, {iota_2}))...
98 -eval(subs(G1, {v}, {iota_1})))^(p_2+r_2)/gamma(p_2+r_2+1)...
99 +(eval(subs(G1, {v}, {iota_2}))...
100 -eval(subs(G1, {v}, {iota_1})))^(q_2+p_2+r_2)...
101 /gamma(q_2+p_2+r_2+1));
102 MI(nn, column+20)=Delta_2;
103 D1=(Delta_1+h_1_0Phi_1)/(1-ell_1Phi_1);
104 MI(nn, column+21)=D1;
105 D2=(Delta_2+h_2_0Phi_1)/(1-ell_2Phi_2);
106 MI(nn, column+22)=D2;
107 MI(nn, column+23)=max(D1, D2);
108 t=t+0.08;
109 nn=nn+1;
110 end;
111 %G2
112 t=iota_1;
113 column=25;
114 nn=1;
115 while t < =iota_2+0.08
116 MI(nn, column) = nn;
117 MI(nn, column+1) = t;
118 Phi_1=(eval(subs(G2, {v}, {iota_2}))...
119 -eval(subs(G2, {v}, {iota_1})))^(q_1+p_1+r_1+s_1)...
120 /gamma(q_1+p_1+r_1+s_1+1)+(eval(subs(G2, {v}, {iota_2}))...
121 -eval(subs(G2, {v}, {iota_1})))^(p_1+r_1+s_1)...
122 /gamma(p_1+r_1+s_1+1)+(eval(subs(G2, {v}, {iota_2}))...
123 -eval(subs(G2, {v}, {iota_1})))^(r_1+s_1)...
124 /gamma(r_1+s_1+1)+(eval(subs(G2, {v}, {iota_2}))...
125 -eval(subs(G2, {v}, {iota_1})))^(s_1)/gamma(s_1+1);
126 MI(nn, column+2)=Phi_1*ell_1;
127 MI(nn, column+3)=Phi_1*ell_1 < 1;
128 Phi_2=(eval(subs(G2, {v}, {iota_2}))...
129 -eval(subs(G2, {v}, {iota_1})))^(q_2+p_2+r_2+s_2)...
130 /gamma(q_2+p_2+r_2+s_2+1)+(eval(subs(G2, {v}, {iota_2}))...
131 -eval(subs(G2, {v}, {iota_1})))^(p_2+r_2+s_2)...
132 /gamma(p_2+r_2+s_2+1)+(eval(subs(G2, {v}, {iota_2}))...
133 -eval(subs(G2, {v}, {iota_1})))^(r_2+s_2)...
134 /gamma(r_2+s_2+1)+(eval(subs(G2, {v}, {iota_2}))...
135 -eval(subs(G2, {v}, {iota_1})))^(s_2)/gamma(s_2+1);
136 MI(nn, column+4)=Phi_2*ell_2;
137 MI(nn, column+5)=Phi_2*ell_2 < 1;
138 Phi=max(Phi_1, Phi_2);
139 MI(nn, column+6)=Phi;
140 MI(nn, column+7)=Phi*ell;
141 MI(nn, column+8)=Phi*ell < 1;
142 M_10=int(abs(g_10), v, iota_1, t);
143 MI(nn, column+9)=M_10;
144 M_11=int(abs(g_11), v, iota_1, t);
145 MI(nn, column+10)=M_11;
146 M_12=int(abs(g_12), v, iota_1, t);
147 MI(nn, column+11)=M_12;
148 M_13=int(abs(g_13), v, iota_1, t);
149 MI(nn, column+12)=M_13;
150 M_20=int(abs(g_20), v, iota_1, iota_2);
151 MI(nn, column+13)=M_20;
152 M_21=int(abs(g_21), v, iota_1, t);
153 MI(nn, column+14)=M_21;
154 M_22=int(abs(g_22), v, iota_1, t);
155 MI(nn, column+15)=M_22;
156 M_23=int(abs(g_23), v, iota_1, t);
157 MI(nn, column+16)=M_23;
158 M_1j=max(max(max(M_10, M_11), M_12), M_13);
159 MI(nn, column+17)=M_1j;
160 M_2j=max(max(max(M_20, M_21), M_22), M_23);
161 MI(nn, column+18)=M_2j;
162 Delta_1=M_10+M_11*(1+(eval(subs(G2, {v}, {iota_2}))...
163 -eval(subs(G2, {v}, {iota_1})))^(q_1)/gamma(q_1+1))...
164 +M_12*(1+(eval(subs(G2, {v}, {iota_2}))...
165 -eval(subs(G2, {v}, {iota_1})))^(p_1)/gamma(p_1+1)...
166 +(eval(subs(G2, {v}, {iota_2}))-eval(subs(G2, {v}, {iota_1})))^(q_1+p_1)...
167 /gamma(q_1+p_1+1))+M_13*(1+(eval(subs(G2, {v}, {iota_2}))...
168 -eval(subs(G2, {v}, {iota_1})))^(r_1)/gamma(r_1+1)...
169 +(eval(subs(G2, {v}, {iota_2}))-eval(subs(G2, {v}, {iota_1})))^(p_1+r_1)...
170 /gamma(p_1+r_1+1)+(eval(subs(G2, {v}, {iota_2}))...
171 -eval(subs(G2, {v}, {iota_1})))^(q_1+p_1+r_1)/gamma(q_1+p_1+r_1+1));
172 MI(nn, column+19)=Delta_1;
173 Delta_2=M_20+M_21*(1+(eval(subs(G2, {v}, {iota_2}))...
174 -eval(subs(G2, {v}, {iota_1})))^(q_2)/gamma(q_2+1))...
175 +M_22*(1+(eval(subs(G2, {v}, {iota_2}))...
176 -eval(subs(G2, {v}, {iota_1})))^(p_2)/gamma(p_2+1)...
177 +(eval(subs(G2, {v}, {iota_2}))...
178 -eval(subs(G2, {v}, {iota_1})))^(q_2+p_2)/gamma(q_2+p_2+1))...
179 +M_23*(1+(eval(subs(G2, {v}, {iota_2}))...
180 -eval(subs(G2, {v}, {iota_1})))^(r_2)/gamma(r_2+1)...
181 +(eval(subs(G2, {v}, {iota_2}))...
182 -eval(subs(G2, {v}, {iota_1})))^(p_2+r_2)/gamma(p_2+r_2+1)...
183 +(eval(subs(G2, {v}, {iota_2}))...
184 -eval(subs(G2, {v}, {iota_1})))^(q_2+p_2+r_2)...
185 /gamma(q_2+p_2+r_2+1));
186 MI(nn, column+20)=Delta_2;
187 D1=(Delta_1+h_1_0Phi_1)/(1-ell_1Phi_1);
188 MI(nn, column+21)=D1;
189 D2=(Delta_2+h_2_0Phi_1)/(1-ell_2Phi_2);
190 MI(nn, column+22)=D2;
191 MI(nn, column+23)=max(D1, D2);
192 t=t+0.08;
193 nn=nn+1;
194 end;
195 %G3
196 t=iota_1;
197 column=49;
198 nn=1;
199 while t < =iota_2+0.08
200 MI(nn, column) = nn;
201 MI(nn, column+1) = t;
202 Phi_1=(eval(subs(G3, {v}, {iota_2}))...
203 -eval(subs(G3, {v}, {iota_1})))^(q_1+p_1+r_1+s_1)...
204 /gamma(q_1+p_1+r_1+s_1+1)+(eval(subs(G3, {v}, {iota_2}))...
205 -eval(subs(G3, {v}, {iota_1})))^(p_1+r_1+s_1)...
206 /gamma(p_1+r_1+s_1+1)+(eval(subs(G3, {v}, {iota_2}))...
207 -eval(subs(G3, {v}, {iota_1})))^(r_1+s_1)...
208 /gamma(r_1+s_1+1)+(eval(subs(G3, {v}, {iota_2}))...
209 -eval(subs(G3, {v}, {iota_1})))^(s_1)/gamma(s_1+1);
210 MI(nn, column+2)=Phi_1*ell_1;
211 MI(nn, column+3)=Phi_1*ell_1 < 1;
212 Phi_2=(eval(subs(G3, {v}, {iota_2}))...
213 -eval(subs(G3, {v}, {iota_1})))^(q_2+p_2+r_2+s_2)...
214 /gamma(q_2+p_2+r_2+s_2+1)+(eval(subs(G3, {v}, {iota_2}))...
215 -eval(subs(G3, {v}, {iota_1})))^(p_2+r_2+s_2)...
216 /gamma(p_2+r_2+s_2+1)+(eval(subs(G3, {v}, {iota_2}))...
217 -eval(subs(G3, {v}, {iota_1})))^(r_2+s_2)...
218 /gamma(r_2+s_2+1)+(eval(subs(G3, {v}, {iota_2}))...
219 -eval(subs(G3, {v}, {iota_1})))^(s_2)/gamma(s_2+1);
220 MI(nn, column+4)=Phi_2*ell_2;
221 MI(nn, column+5)=Phi_2*ell_2 < 1;
222 Phi=max(Phi_1, Phi_2);
223 MI(nn, column+6)=Phi;
224 MI(nn, column+7)=Phi*ell;
225 MI(nn, column+8)=Phi*ell < 1;
226 M_10=int(abs(g_10), v, iota_1, t);
227 MI(nn, column+9)=M_10;
228 M_11=int(abs(g_11), v, iota_1, t);
229 MI(nn, column+10)=M_11;
230 M_12=int(abs(g_12), v, iota_1, t);
231 MI(nn, column+11)=M_12;
232 M_13=int(abs(g_13), v, iota_1, t);
233 MI(nn, column+12)=M_13;
234 M_20=int(abs(g_20), v, iota_1, iota_2);
235 MI(nn, column+13)=M_20;
236 M_21=int(abs(g_21), v, iota_1, t);
237 MI(nn, column+14)=M_21;
238 M_22=int(abs(g_22), v, iota_1, t);
239 MI(nn, column+15)=M_22;
240 M_23=int(abs(g_23), v, iota_1, t);
241 MI(nn, column+16)=M_23;
242 M_1j=max(max(max(M_10, M_11), M_12), M_13);
243 MI(nn, column+17)=M_1j;
244 M_2j=max(max(max(M_20, M_21), M_22), M_23);
245 MI(nn, column+18)=M_2j;
246 Delta_1=M_10+M_11*(1+(eval(subs(G3, {v}, {iota_2}))...
247 -eval(subs(G3, {v}, {iota_1})))^(q_1)/gamma(q_1+1))...
248 +M_12*(1+(eval(subs(G3, {v}, {iota_2}))...
249 -eval(subs(G3, {v}, {iota_1})))^(p_1)/gamma(p_1+1)...
250 +(eval(subs(G3, {v}, {iota_2}))...
251 -eval(subs(G3, {v}, {iota_1})))^(q_1+p_1)/gamma(q_1+p_1+1))...
252 +M_13*(1+(eval(subs(G3, {v}, {iota_2}))...
253 -eval(subs(G3, {v}, {iota_1})))^(r_1)/gamma(r_1+1)...
254 +(eval(subs(G3, {v}, {iota_2}))...
255 -eval(subs(G3, {v}, {iota_1})))^(p_1+r_1)/gamma(p_1+r_1+1)...
256 +(eval(subs(G3, {v}, {iota_2}))...
257 -eval(subs(G3, {v}, {iota_1})))^(q_1+p_1+r_1)...
258 /gamma(q_1+p_1+r_1+1));
259 MI(nn, column+19)=Delta_1;
260 Delta_2=M_20+M_21*(1+(eval(subs(G3, {v}, {iota_2}))...
261 -eval(subs(G3, {v}, {iota_1})))^(q_2)/gamma(q_2+1))...
262 +M_22*(1+(eval(subs(G3, {v}, {iota_2}))...
263 -eval(subs(G3, {v}, {iota_1})))^(p_2)/gamma(p_2+1)...
264 +(eval(subs(G3, {v}, {iota_2}))...
265 -eval(subs(G3, {v}, {iota_1})))^(q_2+p_2)/gamma(q_2+p_2+1))...
266 +M_23*(1+(eval(subs(G3, {v}, {iota_2}))...
267 -eval(subs(G3, {v}, {iota_1})))^(r_2)/gamma(r_2+1)...
268 +(eval(subs(G3, {v}, {iota_2}))...
269 -eval(subs(G3, {v}, {iota_1})))^(p_2+r_2)/gamma(p_2+r_2+1)...
270 +(eval(subs(G3, {v}, {iota_2}))...
271 -eval(subs(G3, {v}, {iota_1})))^(q_2+p_2+r_2)...
272 /gamma(q_2+p_2+r_2+1));
273 MI(nn, column+20)=Delta_2;
274 D1=(Delta_1+h_1_0Phi_1)/(1-ell_1Phi_1);
275 MI(nn, column+21)=D1;
276 D2=(Delta_2+h_2_0Phi_1)/(1-ell_2Phi_2);
277 MI(nn, column+22)=D2;
278 MI(nn, column+23)=max(D1, D2);
279 t=t+0.08;
280 nn=nn+1;
281 end;
282 %G4
283 t=iota_1;
284 column=73;
285 nn=1;
286 while t < =iota_2+0.08
287 MI(nn, column) = nn;
288 MI(nn, column+1) = t;
289 Phi_1=(eval(subs(G4, {v}, {iota_2}))...
290 -eval(subs(G4, {v}, {iota_1})))^(q_1+p_1+r_1+s_1)...
291 /gamma(q_1+p_1+r_1+s_1+1)+(eval(subs(G4, {v}, {iota_2}))...
292 -eval(subs(G4, {v}, {iota_1})))^(p_1+r_1+s_1)...
293 /gamma(p_1+r_1+s_1+1)+(eval(subs(G4, {v}, {iota_2}))...
294 -eval(subs(G4, {v}, {iota_1})))^(r_1+s_1)...
295 /gamma(r_1+s_1+1)+(eval(subs(G4, {v}, {iota_2}))...
296 -eval(subs(G4, {v}, {iota_1})))^(s_1)/gamma(s_1+1);
297 MI(nn, column+2)=Phi_1*ell_1;
298 MI(nn, column+3)=Phi_1*ell_1 < 1;
299 Phi_2=(eval(subs(G4, {v}, {iota_2}))...
300 -eval(subs(G4, {v}, {iota_1})))^(q_2+p_2+r_2+s_2)...
301 /gamma(q_2+p_2+r_2+s_2+1)+(eval(subs(G4, {v}, {iota_2}))...
302 -eval(subs(G4, {v}, {iota_1})))^(p_2+r_2+s_2)...
303 /gamma(p_2+r_2+s_2+1)+(eval(subs(G4, {v}, {iota_2}))...
304 -eval(subs(G4, {v}, {iota_1})))^(r_2+s_2)...
305 /gamma(r_2+s_2+1)+(eval(subs(G4, {v}, {iota_2}))...
306 -eval(subs(G4, {v}, {iota_1})))^(s_2)/gamma(s_2+1);
307 MI(nn, column+4)=Phi_2*ell_2;
308 MI(nn, column+5)=Phi_2*ell_2 < 1;
309 Phi=max(Phi_1, Phi_2);
310 MI(nn, column+6)=Phi;
311 MI(nn, column+7)=Phi*ell;
312 MI(nn, column+8)=Phi*ell < 1;
313 M_10=int(abs(g_10), v, iota_1, t);
314 MI(nn, column+9)=M_10;
315 M_11=int(abs(g_11), v, iota_1, t);
316 MI(nn, column+10)=M_11;
317 M_12=int(abs(g_12), v, iota_1, t);
318 MI(nn, column+11)=M_12;
319 M_13=int(abs(g_13), v, iota_1, t);
320 MI(nn, column+12)=M_13;
321 M_20=int(abs(g_20), v, iota_1, iota_2);
322 MI(nn, column+13)=M_20;
323 M_21=int(abs(g_21), v, iota_1, t);
324 MI(nn, column+14)=M_21;
325 M_22=int(abs(g_22), v, iota_1, t);
326 MI(nn, column+15)=M_22;
327 M_23=int(abs(g_23), v, iota_1, t);
328 MI(nn, column+16)=M_23;
329 M_1j=max(max(max(M_10, M_11), M_12), M_13);
330 MI(nn, column+17)=M_1j;
331 M_2j=max(max(max(M_20, M_21), M_22), M_23);
332 MI(nn, column+18)=M_2j;
333 Delta_1=M_10+M_11*(1+(eval(subs(G4, {v}, {iota_2}))...
334 -eval(subs(G4, {v}, {iota_1})))^(q_1)/gamma(q_1+1))...
335 +M_12*(1+(eval(subs(G4, {v}, {iota_2}))...
336 -eval(subs(G4, {v}, {iota_1})))^(p_1)/gamma(p_1+1)...
337 +(eval(subs(G4, {v}, {iota_2}))...
338 -eval(subs(G4, {v}, {iota_1})))^(q_1+p_1)/gamma(q_1+p_1+1))...
339 +M_13*(1+(eval(subs(G4, {v}, {iota_2}))...
340 -eval(subs(G4, {v}, {iota_1})))^(r_1)/gamma(r_1+1)...
341 +(eval(subs(G4, {v}, {iota_2}))...
342 -eval(subs(G4, {v}, {iota_1})))^(p_1+r_1)/gamma(p_1+r_1+1)...
343 +(eval(subs(G4, {v}, {iota_2}))...
344 -eval(subs(G4, {v}, {iota_1})))^(q_1+p_1+r_1)...
345 /gamma(q_1+p_1+r_1+1));
346 MI(nn, column+19)=Delta_1;
347 Delta_2=M_20+M_21*(1+(eval(subs(G4, {v}, {iota_2}))...
348 -eval(subs(G4, {v}, {iota_1})))^(q_2)/gamma(q_2+1))...
349 +M_22*(1+(eval(subs(G4, {v}, {iota_2}))...
350 -eval(subs(G4, {v}, {iota_1})))^(p_2)/gamma(p_2+1)...
351 +(eval(subs(G4, {v}, {iota_2}))...
352 -eval(subs(G4, {v}, {iota_1})))^(q_2+p_2)/gamma(q_2+p_2+1))...
353 +M_23*(1+(eval(subs(G4, {v}, {iota_2}))...
354 -eval(subs(G4, {v}, {iota_1})))^(r_2)/gamma(r_2+1)...
355 +(eval(subs(G4, {v}, {iota_2}))...
356 -eval(subs(G4, {v}, {iota_1})))^(p_2+r_2)/gamma(p_2+r_2+1)...
357 +(eval(subs(G4, {v}, {iota_2}))...
358 -eval(subs(G4, {v}, {iota_1})))^(q_2+p_2+r_2)...
359 /gamma(q_2+p_2+r_2+1));
360 MI(nn, column+20)=Delta_2;
361 D1=(Delta_1+h_1_0Phi_1)/(1-ell_1Phi_1);
362 MI(nn, column+21)=D1;
363 D2=(Delta_2+h_2_0Phi_1)/(1-ell_2Phi_2);
364 MI(nn, column+22)=D2;
365 MI(nn, column+23)=max(D1, D2);
366 t=t+0.08;
367 nn=nn+1;
368 end;

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References

[1]	Grand view research (2020) Lithium-ion battery market Size, Share & Trends analysis report by product (LCO, LFP, NCA, LMO, lithium nickel manganese cobalt), by application, by region, and segmented forecasts, 2020-2027.
[2]	Lithium ion battery Market, Size, Share, COVID impact analysis and forecast to 2027. ResearchAndMarket, 2021.
[3]	Boudette NE, Davenport C (2021) GM announcement Shakes Up U.S, Automakers' transition to electric cars. New York Times. Available from: https://www.nytimes.com/2021/01/29/business/general-motors-electric-cars.html.
[4]	Deng J, Bae C, Denlinger A, et al. (2020) Electric vehicles batteries: requirements and challenges. Joule 4: 511-515. doi: 10.1016/j.joule.2020.01.013
[5]	Yu A, Sumangil M (2021) Top electric vehicle markets dominate lithium-ion battery capacity growth. S & P glob mark Intell. Available from: https://www.spglobal.com/marketintelligence/en/news-insights/blog/top-electric-vehicle-markets-dominate-lithium-ion-battery-capacity-growth.
[6]	Roskill (2020) The resurgence of LFP cathodes: A safe and cost-efficient battery active material to support the EV revolution.
[7]	Yan W, Cao H, Zhang Y, et al. (2020) Rethinking Chinese supply resilience of critical metals in lithium-ion batteries. J Clean Prod 256: 120719. doi: 10.1016/j.jclepro.2020.120719
[8]	Forecasts (2021) Lithium, cobalt, graphite, and nickel. Benchmark Mineral Intelligence.
[9]	Tolomeo R, De Feo G, Adami R, et al. (2020) Application of life cycle assessment to lithium ion batteries in the automotive Sector. Sustainability 12: 4628. doi: 10.3390/su12114628
[10]	Melin HE (2019) The lithium-ion battery end-of-life market - A basesline study. Glob Batter Alliance 1-11.
[11]	Arshad F, Li L, Amin K, et al. (2020) A comprehensive review of the advancement in recycling the anode and electrolyte from spent lithium ion batteries. ACS Sustainable Chem Eng 8: 13527-13554. doi: 10.1021/acssuschemeng.0c04940
[12]	Li L, Qu W, Zhang X, et al. (2015) Succinic acid-based leaching system: A sustainable process for recovery of valuable metals from spent Li-ion batteries. J Power Sources 282: 544-551. doi: 10.1016/j.jpowsour.2015.02.073
[13]	Melin HE (2018) The lithium-ion battery end-of-life market 2018-2025. Circular Energy Storage. United Kingdom 6.
[14]	Campagnol N, Eddy J, Hagenbruch T, et al. (2018) Metal mining constraints on the electric mobility horizon, McKinsey Company.
[15]	Roskill (2021) Cathode and precursor materials: outlook to 2030, 1st ed 2021.
[16]	Merriman D, Liang L, Shang KG, et al. (2021) VW Power Day: A roadmap for e-mobility, battery manufacturing and smart grid rollout.
[17]	Gaines L, Dai Q, Vaughey JT, et al. (2021) Direct recycling R & D at the ReCell center. Recycling 6: 31. doi: 10.3390/recycling6020031
[18]	Ding Y, Cano ZP, Yu A, et al. (2019) Automotive Li-ion batteries: current status and future perspectives. Electrochem Energ Rev 2: 1-28. doi: 10.1007/s41918-018-0022-z
[19]	EFORE. Comparison of lithium-ion batteries 2020. Available from: https://www.efore.com/content/uploads/2020/12/Comparison_of_lithium_batteries_20201209.pdf.
[20]	The White House (2021) Building resilient supply chains, revitalizing american manufacturing, and fostering broad-based growth: 100-day reviews under executive order 14017.
[21]	FCAB (2021) National blueprint for lithium batteries 2021-2030. Available from: https://www.energy.gov/sites/default/files/2021-06/FCAB%20National%20Blueprint%20Lithium%20Batteries%200621_0.pdf.
[22]	NAATBatt (2021) Developing a supply chain for lithium-ion batteries in north America. NAATBatt - webinar.
[23]	Gaines L (2018) Lithium-ion battery recycling processes: research towards a sustainable course. Sustain Mater Technol 17: e00068.
[24]	Huang Y, Han G, Liu J, et al. (2016) A stepwise recovery of metals from hybrid cathodes of spent Li-ion batteries with leaching-flotation-precipitation process. J Power Sources 325: 555-564. doi: 10.1016/j.jpowsour.2016.06.072
[25]	Wang L, Wang X, Yang W (2020) Optimal design of electric vehicle battery recycling network - From the perspective of electric vehicle manufacturers. Appl Energy 275.
[26]	Vieceli N, Nogueira CA, Guimarães C, et al. (2018) Hydrometallurgical recycling of lithium-ion batteries by reductive leaching with sodium metabisulphite. Waste Manag 71: 350-361. doi: 10.1016/j.wasman.2017.09.032
[27]	Zhang X, Xie Y, Cao H, et al. (2014) A novel process for recycling and resynthesizing LiNi1/3Co1/3Mn1/3O2 from the cathode scraps intended for lithium-ion batteries. Waste Manag 34: 1715-1724. doi: 10.1016/j.wasman.2014.05.023
[28]	Rothermel S, Evertz M, Kasnatscheew J, et al. (2016) Graphite recycling from spent lithium-ion batteries. ChemSusChem 9: 3473-3484. doi: 10.1002/cssc.201601062
[29]	Ahmadi L, Young SB, Fowler M, et al. (2017) A cascaded life cycle: reuse of electric vehicle lithium-ion battery packs in energy storage systems. Int J Life Cycle Assess 22: 111-124. doi: 10.1007/s11367-015-0959-7
[30]	Pesaran A (2021) Supporting the Li-ion battery supply chain via PEV battery reuse, developing a supply chain for lithium-ion batteries in north America. NAATBatt - webinar.
[31]	Neubauer J, Smith K, Wood E, et al. (2015) Identifying and Overcoming Critical Barriers to Widespread Second Use of PEV Batteries. Natl Renew Energy Lab 23-62.
[32]	Heymans C, Walker SB, Young SB, et al. (2014) Economic analysis of second use electric vehicle batteries for residential energy storage and load-levelling. Energy Policy 71: 22-30. doi: 10.1016/j.enpol.2014.04.016
[33]	Dai Q, Spangenberger J, Ahmed S, et al. (2019) EverBatt: A closed-loop battery recycling cost and environmental impacts model. Argonne Natl Lab 1-88.
[34]	Alfaro-Algaba M, Ramirez FJ (2020) Techno-economic and environmental disassembly planning of lithium-ion electric vehicle battery packs for remanufacturing. Resour Conserv Recycl 154.
[35]	Gentilini L, Mossali E, Angius A, et al. (2020) A safety oriented decision support tool for the remanufacturing and recycling of post-use H & EVs lithium-ion batteries. Procedia CIRP 90: 73-78. doi: 10.1016/j.procir.2020.01.090
[36]	Rallo H, Benveniste G, Gestoso I, et al. (2020) Economic analysis of the disassembling activities to the reuse of electric vehicles Li-ion batteries. Resour Conserv Recycl 159.
[37]	Wegener K, Andrew S, Raatz A, et al. (2014) Disassembly of electric vehicle batteries using the example of the Audi Q5 hybrid system. Procedia CIRP 23: 155-160. doi: 10.1016/j.procir.2014.10.098
[38]	Herrmann C, Raatz A, Mennenga M, et al. (2012) Assessment of automation potentials for the disassembly of automotive lithium ion battery systems. In: Leveraging Technolnology for a sustainable world, Springer, Berlin, Heidelberg, 149-154.
[39]	McIntyre TJ, Harter JJ, Roberts TA (2019) Development and Operation of a High Throughput Computer Hard Drive Recycling Enterprise, 1-18.
[40]	Xia W, Jiang Y, Chen X, et al. (2021) Application of machine learning algorithms in municipal solid waste management: A mini review. Waste Manag Res 2021: 34269157.
[41]	European Parliament (2021) EU Legislation in Progress New EU regulatory framework for batteries Setting sustainability requirements.
[42]	Zheng R, Wang W, Dai Y, et al. (2017) A closed-loop process for recycling LiNi_xCo_yMn_(1-x−y)O₂ from mixed cathode materials of lithium-ion batteries. Green Energy Environ 2: 42-50. doi: 10.1016/j.gee.2016.11.010
[43]	Liang HJ, Hou BH, Li WH, et al. (2019) Staging Na/K-ion de-/intercalation of graphite retrieved from spent Li-ion batteries: in operando X-ray diffraction studies and an advanced anode material for Na/K-ion batteries. Energy Environ Sci 12: 3575-3584. doi: 10.1039/C9EE02759A
[44]	Sloop S, Crandon L, Allen M, et al. (2020) A direct recycling case study from a lithium-ion battery recall. Sustain Mater Technol 25: e00152.
[45]	Shi Y, Chen G, Chen Z (2018) Effective regeneration of LiCoO₂ from spent lithium-ion batteries: A direct approach towards high-performance active particles. Green Chem 20: 851-862. doi: 10.1039/C7GC02831H
[46]	Ji Y, Edwin E, Kpodzro CTJ, et al. (2021) Direct recycling technologies of cathode in spent lithium-ion batteries. Clean Technol Recycl 1: 124-151. doi: 10.3934/ctr.2021007
[47]	Zhang X, Bian Y, Xu S, et al. (2018) Innovative application of acid leaching to regenerate Li(Ni_1/3Co_1/3Mn_1/3)O₂ cathodes from spent lithium-ion batteries. ACS Sustainable Chem Eng 6: 5959-5968. doi: 10.1021/acssuschemeng.7b04373
[48]	Träger T, Friedrich B, Weyhe R (2015) Recovery concept of value metals from automotive lithium-ion batteries. Chem Ing Tech 87: 1550-1557. doi: 10.1002/cite.201500066
[49]	Zhou M, Li B, Li J, et al. (2021) Pyrometallurgical Technology in the Recycling of a Spent Lithium Ion Battery: Evolution and the Challenge. ACS ES & T Eng 1: 1369-1382
[50]	Velázquez-Martínez O, Valio J, Santasalo-Aarnio A, et al. (2019) A critical review of lithium-ion battery recycling processes from a circular economy perspective. Batteries 5: 5-7. doi: 10.3390/batteries5010005
[51]	Accurec Recycling GmbH (2021) Available from: https://accurec.de/lithium.
[52]	Lv W, Wang Z, Cao H, et al. (2018) A critical review and analysis on the recycling of spent lithium-ion batteries. ACS Sustainable Chem Eng 6: 1504-1521. doi: 10.1021/acssuschemeng.7b03811
[53]	Hu J, Zhang J, Li H, et al. (2017) A promising approach for the recovery of high value-added metals from spent lithium-ion batteries. J Power Sources 351: 192-199. doi: 10.1016/j.jpowsour.2017.03.093
[54]	Fan E, Li L, Zhang X, et al. (2018) Selective recovery of Li and fe from spent lithium-ion batteries by an environmentally friendly mechanochemical approach. ACS Sustainable Chem Eng 6: 11029-11035. doi: 10.1021/acssuschemeng.8b02503
[55]	Yang Y, Song S, Lei S, et al. (2019) A process for combination of recycling lithium and regenerating graphite from spent lithium-ion battery. Waste Manag 85: 529-537. doi: 10.1016/j.wasman.2019.01.008
[56]	Wang B, Lin XY, Tang Y, et al. (2019) Recycling LiCoO₂ with methanesulfonic acid for regeneration of lithium-ion battery electrode materials. J Power Sources 436.
[57]	Yang Y, Xu S, He Y (2017) Lithium recycling and cathode material regeneration from acid leach liquor of spent lithium-ion battery via facile co-extraction and co-precipitation processes. Waste Manag 64: 219-227. doi: 10.1016/j.wasman.2017.03.018
[58]	Nayaka GP, Zhang Y, Dong P, et al. (2019) An environmental friendly attempt to recycle the spent Li-ion battery cathode through organic acid leaching. J Environ Chem Eng 7.
[59]	Gao W, Zhang X, Zheng X, et al. (2017) Lithium carbonate recovery from cathode scrap of spent lithium-ion battery: A closed-loop process. Environ Sci Technol 51: 1662-1669. doi: 10.1021/acs.est.6b03320
[60]	Golmohammadzadeh R, Rashchi F, Vahidi E (2017) Recovery of lithium and cobalt from spent lithium-ion batteries using organic acids: process optimization and kinetic aspects. Waste Manag 64: 244-254. doi: 10.1016/j.wasman.2017.03.037
[61]	Ghassa S, Farzanegan A, Gharabaghi M, et al. (2021) Iron scrap, a sustainable reducing agent for waste lithium ions batteries leaching: an environmentally friendly method to treating waste with waste. Resour Conserv Recycl 166: 105348. doi: 10.1016/j.resconrec.2020.105348
[62]	Wang F, Sun R, Xu J, et al. (2016) Recovery of cobalt from spent lithium ion batteries using sulphuric acid leaching followed by solid-liquid separation and solvent extraction. RSC Adv 6: 85303-85311. doi: 10.1039/C6RA16801A
[63]	Chan KH, Anawati J, Malik M, et al. (2021) Closed-loop recycling of lithium, cobalt, Nickel, and manganese from waste lithium-ion batteries of electric vehicles. ACS Sustainable Chem Eng 9: 4398-4410. doi: 10.1021/acssuschemeng.0c06869
[64]	Zeng X, Li J, Shen B (2015) Novel approach to recover cobalt and lithium from spent lithium-ion battery using oxalic acid. J Hazard Mater 295: 112-118. doi: 10.1016/j.jhazmat.2015.02.064
[65]	Li L, Fan E, Guan Y, et al. (2017) Sustainable recovery of cathode materials from spent lithium-ion batteries using lactic acid leaching system. ACS Sustainable Chem Eng 5: 5224-5233. doi: 10.1021/acssuschemeng.7b00571
[66]	Li L, Bian Y, Zhang X, et al. (2018) Economical recycling process for spent lithium-ion batteries and macro- and micro-scale mechanistic study. J Power Sources 377: 70-79. doi: 10.1016/j.jpowsour.2017.12.006
[67]	Ning P, Meng Q, Dong P, et al. (2020) Recycling of cathode material from spent lithium ion batteries using an ultrasound-assisted DL-malic acid leaching system. Waste Manag 103: 52-60. doi: 10.1016/j.wasman.2019.12.002
[68]	Tanong K, Coudert L, Mercier G, et al. (2016) Recovery of metals from a mixture of various spent batteries by a hydrometallurgical process. J Environ Manage 181: 95-107. doi: 10.1016/j.jenvman.2016.05.084
[69]	Natarajan S, Boricha AB, Bajaj HC (2018) Recovery of value-added products from cathode and anode material of spent lithium-ion batteries. Waste Manag 77: 455-465. doi: 10.1016/j.wasman.2018.04.032
[70]	Esmaeili M, Rastegar SO, Beigzadeh R, et al. (2020) Ultrasound-assisted leaching of spent lithium ion batteries by natural organic acids and H2O2. Chemosphere 254: 126670. doi: 10.1016/j.chemosphere.2020.126670
[71]	Montgomery DC (2017) Design and analysis of experiments, 8th ed., John Wiley & Sons.
[72]	Tapani VM (2012) Experimental optimization and response surfaces. Chemom Pract Appl 91-138.
[73]	Gaines L (2019) Profitable recycling of low-cobalt lithium-ion batteries will depend on new process developments. One Earth 1: 413-415. doi: 10.1016/j.oneear.2019.12.001
[74]	Zheng X, Zhu Z, Lin X, et al. (2018) A mini-review on metal recycling from spent lithium ion batteries. Engineering 4: 361-370. doi: 10.1016/j.eng.2018.05.018
[75]	GEM (2021) Available from: http://en.gem.com.cn/en/AboutTheGroup/index.html.
[76]	Brunp (2021) Available from: https://www.catl.com/en/solution/recycling/.
[77]	Putsche V, Witter E, Santhanagopalan S, et al. (2021) NAATBatt lithium-ion battery supply chain database. National Renewable Energy Laboratory. Version 1.
[78]	Larouche F, Tedjar F, Amouzegar K, et al. (2020) Progress and status of hydrometallurgical and direct recycling of Li-Ion batteries and beyond. Materials (Basel) 13.
[79]	Redwood materials (2021) Available from: https://www.redwoodmaterials.com/.
[80]	Intertek (2019) The future of battery technologies-part V environmental considerations for lithium batteries.
[81]	U.S. geological survey (2021) Mineral commodity summaries. Available from: https://pubs.usgs.gov/periodicals/mcs2021/mcs2021-cobalt.pdf.
[82]	Zeuner B (2018) An Obsolescing Bargain in a Rentier State: multinationals, artisanal miners, and cobalt in the Democratic Republic of Congo. Front Energy Res 6: 1-6. doi: 10.3389/fenrg.2018.00001
[83]	Tsurukawa N, Prakash S, Manhart A (2011) Social impacts of artisanal cobalt mining in Katanga, Democratic Republic of Congo. Ö ko-Inst EV - Inst Appl Ecol Freibg 49: 65.
[84]	Huijbregts MAJ, Steinmann ZJN, Elshout PMF, et al. (2017) ReCiPe2016: a harmonised life cycle impact assessment method at midpoint and endpoint level. Int J Life Cycle Assess 22: 138-147. doi: 10.1007/s11367-016-1246-y
[85]	Richa K, Babbitt CW, Nenadic NG, et al. (2017) Environmental trade-offs across cascading lithium-ion battery life cycles. Int J Life Cycle Assess 22: 66-81. doi: 10.1007/s11367-015-0942-3
[86]	Ciez RE, Whitacre JF (2019) Examining different recycling processes for lithium-ion batteries. Nat Sustain 2: 148-156. doi: 10.1038/s41893-019-0222-5
[87]	Wang S, Yu J (2021) A comparative life cycle assessment on lithium-ion battery: case study on electric vehicle battery in China considering battery evolution. Waste Manag Res 39: 156-164. doi: 10.1177/0734242X20966637
[88]	Richa K, Babbitt CW, Gaustad G (2017) Eco-efficiency analysis of a lithium-ion battery waste hierarchy inspired by circular economy. J Ind Ecol 21: 715-730. doi: 10.1111/jiec.12607
[89]	Jin H, Frost K, Sousa I, et al. (2020) Life cycle assessment of emerging technologies on value recovery from hard disk drives. Resour Conserv Recycl 104781.
[90]	EUR-Lex (2006) Directive 2006/66/EC of the European Parliament and of the Council of 6 September 2006 on batteries and accumulators and waste batteries and accumulators and repealing Directive 91/157/EEC. L 266.
[91]	Wang G, Zhao G, Wu W, et al, (2018) Cascade utilization and recycling of driving LiB, 2nd ed. Beijing: China Electric Power Press.
[92]	CBEA Summary of waste driving battery recycling technology and their profits 2018. Available from: http://www.cbea.com/dianchihuishou/201810/274400.html.
[93]	Green & low carbon development foundation. Research on the power battery recycling mechanism and policy for Shenzhen, 2018.
[94]	Wang X, Gaustad G, Babbitt CW, et al. (2014) Economies of scale for future lithium-ion battery recycling infrastructure. Resour Conserv Recycl 83: 53-62. doi: 10.1016/j.resconrec.2013.11.009
[95]	Electric vehicles revolution, China leads the global boom 2017. Available from: https://www.televisory.com/blogs/-/blogs/electric-vehicles-revolution-china-leads-the-global-boom.
[96]	EU (European Union) (2000) Directive 2000/53/EC of the European Parliament and of the council (2000.9.18), Brussels: European Union.
[97]	GPO (1996) US. Public Law 104-142-Mercury Containing and Rechargeable Battery Management Act.
[98]	CA Code (2006) Rechargeable battery recycling act of 2006.
[99]	New York State rechargeable battery law. New York environmental conservation law. In: Title 18. Rechargeable battery recycling, New York, 2010.
[100]	MN PCA (2015) Product stewardship for rechargeable batteries. St. Paul, MN, USA: Minnesota Pollution Control Agency.
[101]	Call2Recycle (2021) Available from: https://www.call2recycle.org/what-can-i-recycle/.
[102]	Mikolajczak C, Kahn M, White K, et al. (2012) Lithium-ion batteries hazard and use assessment. Springer Science & Business Media.
[103]	European Commission (2020) Regulation of the European Parliament and of the Council concerning batteries and waste batteries, repealing. Directive 2006/66/EC and amending Regulation (EU) No 2019/1020;0353.
[104]	Li L, Dababneh F, Zhao J (2018) Cost-effective supply chain for electric vehicle battery remanufacturing. Appl Energy 226: 277-286. doi: 10.1016/j.apenergy.2018.05.115
[105]	Hoyer C, Kieckhäfer K, Spengler TS (2015) Technology and capacity planning for the recycling of lithium-ion electric vehicle batteries in Germany. J Bus Econ 85: 505-544.
[106]	Tadaros M, Migdalas A, Samuelsson B, et al. (2020) Location of facilities and network design for reverse logistics of lithium-ion batteries in Sweden. Oper Res Int J.
[107]	Hendrickson TP, Kavvada O, Shah N, et al. (2015) Life-cycle implications and supply chain logistics of electric vehicle battery recycling in California. Environ Res Lett 10.
[108]	Zhalechian M, Torabi SA, Mohammadi M (2018) Hub-and-spoke network design under operational and disruption risks. Transp Res E 109: 20-43. doi: 10.1016/j.tre.2017.11.001
[109]	Paul SK, Sarker R, Essam D (2017) A quantitative model for disruption mitigation in a supply chain. Eur J Oper Res 257: 881-895. doi: 10.1016/j.ejor.2016.08.035
[110]	Paul SK, Sarker R, Essam D (2014) Real time disruption management for a two-stage batch production-inventory system with reliability considerations. Eur J Oper Res 237: 113-128. doi: 10.1016/j.ejor.2014.02.005
[111]	Lücker F, Seifert RW, Biçer I (2019) Roles of inventory and reserve capacity in mitigating supply chain disruption risk. Int J Prod Res 57: 1238-1249. doi: 10.1080/00207543.2018.1504173
[112]	Li Q, Zeng B, Savachkin A (2013) Reliable facility location design under disruptions. Comput Oper Res 40: 901-909. doi: 10.1016/j.cor.2012.11.012
[113]	Saha AK, Paul A, Azeem A, et al. (2020) Mitigating partial-disruption risk: A joint facility location and inventory model considering customers' preferences and the role of substitute products and backorder offers. Comput Oper Res 117.
[114]	Li-cycle. Spoke & hub technologies, mechanical & hydrometallurgical process, Li-cycle 2021. Available from: https://li-cycle.com.
[115]	Gao G, Luo X, Lou X, et al. (2019) Efficient sulfuric acid-vitamin C leaching system: Towards enhanced extraction of cobalt from spent lithium-ion batteries. J Mater Cycles Waste Manag 21: 942-949. doi: 10.1007/s10163-019-00850-4
[116]	Cheng Q, Chirdon WM, Lin M, et al. (2019) Characterization, modeling, and optimization of a single-step process for leaching metallic ions from LiNi 1/3 Co 1/3 Mn 1/3 O 2 cathodes for the recycling of spent lithium-ion batteries. Hydrometallurgy 185: 1-11. doi: 10.1016/j.hydromet.2019.01.003
[117]	Xie J, Huang K, Nie Z, et al. (2021) An effective process for the recovery of valuable metals from cathode material of lithium-ion batteries by mechanochemical reduction. Resour Conserv Recycl 168: 105261. doi: 10.1016/j.resconrec.2020.105261
[118]	Chen X, Ma H, Luo C, et al. (2017) Recovery of valuable metals from waste cathode materials of spent lithium-ion batteries using mild phosphoric acid. J Hazard Mater 326: 77-86. doi: 10.1016/j.jhazmat.2016.12.021
[119]	Musariri B, Akdogan G, Dorfling C, et al. (2019) Evaluating organic acids as alternative leaching reagents for metal recovery from lithium ion batteries. Miner Eng 137: 108-117. doi: 10.1016/j.mineng.2019.03.027
[120]	Fan E, Yang J, Huang Y, et al. (2020) Leaching mechanisms of recycling valuable metals from spent lithium-ion batteries by a malonic acid-based leaching system. ACS Appl Energy Mater 3: 8532-8542. doi: 10.1021/acsaem.0c01166
[121]	Roshanfar M, Golmohammadzadeh R, Rashchi F (2019) An environmentally friendly method for recovery of lithium and cobalt from spent lithium-ion batteries using gluconic and lactic acids. J Environ Chem Eng 7.
[122]	Sun LY, Liu BR, Wu T. et al. (2021) Hydrometallurgical recycling of valuable metals from spent lithium-ion batteries by reductive leaching with stannous chloride. Int J Miner Metall Mater 28: 991-1000

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