Investigating minimal requirements for plants on textile substrates in low-cost hydroponic systems

Bennet Brockhagen; Fabian Schoden; Jan Lukas Storck; Timo Grothe; Christian Eßelmann; Robin Böttjer; Anke Rattenholl; Frank Gudermann; Bennet Brockhagen; Fabian Schoden; Jan Lukas Storck; Timo Grothe; Christian Eßelmann; Robin Böttjer; Anke Rattenholl; Frank Gudermann

doi:10.3934/bioeng.2021016

AIMS Bioengineering

2021, Volume 8, Issue 2: 173-191. doi: 10.3934/bioeng.2021016

Previous Article Next Article

Research article Special Issues

Investigating minimal requirements for plants on textile substrates in low-cost hydroponic systems

1.
Institute for Technical Energy Systems (ITES), Bielefeld University of Applied Sciences, Interaktion 1, 33619 Bielefeld, Germany
2.
Faculty of Engineering and Mathematics, Bielefeld University of Applied Sciences, Interaktion 1, 33619 Bielefeld, Germany

Received: 14 February 2021 Accepted: 08 April 2021 Published: 12 May 2021

With a growing world population and the concentration of citizens in big cities new methods of agriculture are required. Vertical farming attracts more attention in mending these growing problems. To enable a widespread use of low-cost hydroponic systems this study investigates minimal requirements for plants (different herbs and vegetables) in such a hydroponic vertical farming system and the suitability of textiles as sustainable substrates. Therefore, this study aims to investigate plant stress levels, germination rates and water usage in a low-cost hydroponic system with no special lightning in principle comparison with indoor cultivation in soil. The results of the pulse-amplitude-modulation (PAM) measurements as measure of photosynthetic performance indicate that the plants were equally stressed in hydroponic and in soil cultivation. In this respect, the photosynthetic quantum yield in both cultivation systems is on average only slightly lower than the values expected under optimal conditions. It was observed that chive and lovage not only had a significantly higher germination rate in the hydroponic system but also accumulated significantly more fresh as well as dry biomass, while spinach, thyme and marjoram showed higher germination rates in soil cultivation. The water consumption in the setup was considerably higher for the hydroponic system compared to indoor soil cultivation.

Keywords:

Citation: Bennet Brockhagen, Fabian Schoden, Jan Lukas Storck, Timo Grothe, Christian Eßelmann, Robin Böttjer, Anke Rattenholl, Frank Gudermann. Investigating minimal requirements for plants on textile substrates in low-cost hydroponic systems[J]. AIMS Bioengineering, 2021, 8(2): 173-191. doi: 10.3934/bioeng.2021016

Related Papers:

[1]	Dongming Nie, Usman Riaz, Sumbel Begum, Akbar Zada . A coupled system of $p$ -Laplacian implicit fractional differential equations depending on boundary conditions of integral type. AIMS Mathematics, 2023, 8(7): 16417-16445. doi: 10.3934/math.2023839
[2]	Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim . Analytical study of $\mathcal{ABC}$ -fractional pantograph implicit differential equation with respect to another function. AIMS Mathematics, 2023, 8(10): 23635-23654. doi: 10.3934/math.20231202
[3]	Ugyen Samdrup Tshering, Ekkarath Thailert, Sotiris K. Ntouyas . Existence and stability results for a coupled system of Hilfer-Hadamard sequential fractional differential equations with multi-point fractional integral boundary conditions. AIMS Mathematics, 2024, 9(9): 25849-25878. doi: 10.3934/math.20241263
[4]	Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut . On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function. AIMS Mathematics, 2022, 7(5): 7817-7846. doi: 10.3934/math.2022438
[5]	Subramanian Muthaiah, Dumitru Baleanu, Nandha Gopal Thangaraj . Existence and Hyers-Ulam type stability results for nonlinear coupled system of Caputo-Hadamard type fractional differential equations. AIMS Mathematics, 2021, 6(1): 168-194. doi: 10.3934/math.2021012
[6]	Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas . Existence and stability results for $\psi$ -Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244
[7]	Najla Alghamdi, Bashir Ahmad, Esraa Abed Alharbi, Wafa Shammakh . Investigation of multi-term delay fractional differential equations with integro-multipoint boundary conditions. AIMS Mathematics, 2024, 9(5): 12964-12981. doi: 10.3934/math.2024632
[8]	Arjumand Seemab, Mujeeb ur Rehman, Jehad Alzabut, Yassine Adjabi, Mohammed S. Abdo . Langevin equation with nonlocal boundary conditions involving a $\psi$ -Caputo fractional operators of different orders. AIMS Mathematics, 2021, 6(7): 6749-6780. doi: 10.3934/math.2021397
[9]	Qun Dai, Shidong Liu . Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method. AIMS Mathematics, 2022, 7(2): 2498-2511. doi: 10.3934/math.2022140
[10]	Zaid Laadjal, Fahd Jarad . Existence, uniqueness and stability of solutions for generalized proportional fractional hybrid integro-differential equations with Dirichlet boundary conditions. AIMS Mathematics, 2023, 8(1): 1172-1194. doi: 10.3934/math.2023059

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 $\mathbb{G}_2 = \iota$ .

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

| Show Table

DownLoad: CSV

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

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

| Show Table

DownLoad: CSV

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;

| Show Table

DownLoad: CSV

Acknowledgments

The project was partly funded by the Federal Ministry for Economic Affairs and Energy in the scope of the ZIM project ZF4036107 and by the HiF fund of Bielefeld University of Applied Sciences. The APC is funded by the Open Access Publication Fund of Bielefeld University of Applied Sciences. We thank Martina Holt and Karl-Josef Dietz for constructive criticism of the manuscript, lending laboratory equipment and for advice on the implementation of the experiments. Furthermore, we thank Anja Storck for her kind support.

Conflict of interest

The authors declare no conflict of interest.

Author contributions

Bennet Brockhagen: conceptualization, methodology, formal analysis, investigation, visualization; Fabian Schoden: conceptualization, methodology, formal analysis, investigation, visualization, writing–first draft; Jan Lukas Storck: conceptualization, methodology, formal analysis, investigation, visualization; Timo Grothe: conceptualization, investigation, visualization; Christian Eßelmann: investigation, validation; Robin Böttjer: methodology; Anke Rattenholl: validation; Frank Gudermann: validation. All authors read and substantially modified the manuscript.

References

[1]	Kalantari F, Mohd Tahir O, Mahmoudi Lahijani A, et al. (2017) A review of vertical farming technology: A guide for implementation of building integrated agriculture in cities. Adv Eng Forum 24: 76-91. doi: 10.4028/www.scientific.net/AEF.24.76
[2]	Despommier D (2013) Farming up the city: The rise of urban vertical farms. Trends Biotechnol 31: 388-389. doi: 10.1016/j.tibtech.2013.03.008
[3]	Al-Chalabi M (2015) Vertical farming: Skyscraper sustainability? Sustain Cities Soc 18: 74-77. doi: 10.1016/j.scs.2015.06.003
[4]	Benke K, Tomkins B (2017) Future food-production systems: Vertical farming and controlled-environment agriculture. Sustain Sci Pract Policy 13: 13-26.
[5]	Udovichenko A, Fleck BA, Weis T, et al. (2021) Framework for design and optimization of a retrofitted light industrial space with a renewable energy-assisted hydroponics facility in a rural northern canadian community. J Build Eng 37: 102160. doi: 10.1016/j.jobe.2021.102160
[6]	Majid M, Khan JN, Shah QM A, et al. (2021) Evaluation of hydroponic systems for the cultivation of Lettuce (Lactuca sativa L., var. Longifolia) and comparison with protected soil-based cultivation. Agric Water Manage 245: 106572. doi: 10.1016/j.agwat.2020.106572
[7]	Cámara-Zapata JM, Brotons-Martínez JM, Simón-Grao S, et al. (2019) Cost–benefit analysis of tomato in soilless culture systems with saline water under greenhouse conditions. J Sci Food Agric 99: 5842-5851. doi: 10.1002/jsfa.9857
[8]	Storck JL, Böttjer R, Vahle D, et al. (2019) Seed germination and seedling growth on knitted fabrics as new substrates for hydroponic systems. Horticulturae 5: 73. doi: 10.3390/horticulturae5040073
[9]	Vinci G, Rapa M (2019) Hydroponic cultivation: life cycle assessment of substrate choice. Brit Food J 121: 1801-1812. doi: 10.1108/BFJ-02-2019-0112
[10]	B Böttjer R, Storck J L, Vahle D, et al. (2019) Influence of textile and environmental parameters on plant growth on vertically mounted knitted fabrics. Tekstilec 62: 200-207. doi: 10.14502/Tekstilec2019.62.200-207
[11]	Ehrmann A (2019) On the possible use of textile fabrics for vertical farming. Tekstilec 62: 34-41. doi: 10.14502/Tekstilec2019.62.34-41
[12]	Haris I, Fasching A, Punzenberger L, et al. CPS/IoT Ecosystem: Indoor vertical farming system (2019) . doi: 10.1109/ISCE.2019.8900974
[13]	Both AJ, Albright LD, Langhans RW, et al. Hydroponic lettuce production influenced by integrated supplemental light levels in a controlled environment agriculture facility: Experimental results (1997) . doi: 10.17660/ActaHortic.1997.418.5
[14]	Kalantari F, Tahir OM, Joni RA, et al. (2018) Opportunities and challenges in sustainability of vertical farming: a review. J Landscape Ecol 11: 35-60. doi: 10.1515/jlecol-2017-0016
[15]	Naik PK, Gaikwad SP, Gupta MJ, et al. (2013) Low cost devices for hydroponics fodder production. Indian Dairyman 65: 68-72.
[16]	Alatorre-Cobos F, Calderón-Vázquez C, Ibarra-Laclette E, et al. (2014) An improved, low-cost, hydroponic system for growing Arabidopsis and other plant species under aseptic conditions. BMC Plant Biol 14: 69. doi: 10.1186/1471-2229-14-69
[17]	Riggio GM, Jones SL, Gibson KE (2019) Risk of human pathogen internalization in leafy vegetables during lab-scale hydroponic cultivation. Horticulturae 5: 25. doi: 10.3390/horticulturae5010025
[18]	Barbosa GL, Almeida Gadelha FD, Kublik N, et al. (2015) Comparison of land, water, and energy requirements of lettuce grown using hydroponic vs. conventional agricultural methods. Int J Environ Res Public Health 12: 6879-6891. doi: 10.3390/ijerph120606879
[19]	Al-Karaki GN, Al-Hashimi M Green fodder production and water use efficiency of some forage crops under hydroponic conditions (2012) . doi: 10.5402/2012/924672
[20]	Bradley P, Marulanda C Simplified hydroponics to reduce global hunger (2001) . doi: 10.17660/ActaHortic.2001.554.31
[21]	Al-Karaki GN, Al-Momani N (2011) Evaluation of some barley cultivars for green fodder production and water use efficiency under hydroponic conditions. Jordan J Agric Sci 7: 448-457.
[22]	Entezari A, Wang RZ, Zhao S, et al. (2019) Sustainable agriculture for water-stressed regions by air-water-energy management. Energy 181: 1121-1128. doi: 10.1016/j.energy.2019.06.045
[23]	Zhu F (2018) Modifications of konjac glucomannan for diverse applications. Food Chem 256: 419-426. doi: 10.1016/j.foodchem.2018.02.151
[24]	Zhang H, Cui S, Lv H, et al. (2019) A crosslinking strategy to make neutral polysaccharide nanofibers robust and biocompatible: With konjac glucomannan as an example. Carbohyd Polym 215: 130-136. doi: 10.1016/j.carbpol.2019.03.075
[25]	Ellis RH, Hong TD, Roberts EH (1985) Handbook of seed technology for genebanks—volume II. Compendium of specific germination information and test recommendation Rome: International Board for Plant Genetic Resources, 221-237.
[26]	Enza Zaden, Thymian Deutscher Winter (Thymus vulgaris, Labiatae), 2021 Available from: https://www.enzazaden.com/de/products-and-services/our-products/German Winter.
[27]	Pharmasaat, Thymus vulgaris, Thymian, 2021 Available from: https://www.pharmasaat.de/onlineshop/Kraeuter-Saatgut/Thymus-vulgaris-Thymian::171.html?language=de.
[28]	Der Bio-Gärtner, Thymian (Thymus vulgaris), 2021 Available from: https://www.bio-gaertner.de/Pflanzen/Thymian.
[29]	ReinSaat, Majoran (Dost), 2021 Available from: https://www.reinsaat.at/shop/DE/kuechen-_und_gewuerzkraeuter/majoran_dost/.
[30]	Pharmasaat, Origanum majorana, Majoran, 2021 Available from: https://www.pharmasaat.de/onlineshop/Kraeuter-Saatgut/Origanum-majorana-Majoran::128.html?language=de#.
[31]	Gartenjournal, Majoran säen – Tipps und Tricks für die Aussaat, 2021 Available from: https://www.gartenjournal.net/majoran-saeen.
[32]	Gartendialog, Lichtkeimer oder Dunkelkeimer: Unterschiede\|71 Pflanzen, 2021 Available from: https://www.gartendialog.de/lichtkeimer-dunkelkeimer/.
[33]	Oh S, Moon KH, Song EY, et al. (2015) Photosynthesis of Chinese cabbage and radish in response to rising leaf temperature during spring. Hortic Environ Biotechnol 56: 159-166. doi: 10.1007/s13580-015-0122-1
[34]	Schreiber U (2004) Pulse-amplitude-modulation (PAM) fluorometry and saturation pulse method: an overview. Chlorophyll a fluorescence 19: 279-319. doi: 10.1007/978-1-4020-3218-9_11
[35]	Lennard W, Ward JA (2019) A comparison of plant growth rates between an NFT hydroponic system and an NFT aquaponic system. Horticulturae 5: 27. doi: 10.3390/horticulturae5020027
[36]	Smart Garden Guide, What is the best pH for hydroponics?, 2021 Available from: https://smartgardenguide.com/best-ph-for-hydroponics/.
[37]	Ashton J, Geary L (2011) The effects of temperature on pH measurement. TSP 1: 1-7.
[38]	Nakaoka S, Yamada A A system for measuring the photosynthetic activity of water plants based on carbon dioxide absorption (2012) . doi: 10.1109/MHS.2012.6492395
[39]	Shipman LL, Cotton TM, Norris JR, et al. (1976) An analysis of the visible absorption spectrum of chlorophyll a monomer, dimer, and oligomers in solution. J Am Chem Soc 98: 8222-8230. doi: 10.1021/ja00441a056
[40]	Zhang X, He D, Niu G, et al. (2018) Effects of environment lighting on the growth, photosynthesis, and quality of hydroponic lettuce in a plant factory. Int J Agric Biol Eng 11: 33-40.
[41]	Steingröver E, Ratering P, Siesling J (1986) Daily changes in uptake, reduction and storage of nitrate in spinach grown at low light intensity. Physiol Plant 66: 550-556. doi: 10.1111/j.1399-3054.1986.tb05965.x
[42]	Stanghellini ME, Stowell LJ, Bates ML (1984) Control of root rot of spinach caused by Pythium Aphanidermatum in a recirculating hydroponic system by ultraviolet irradiation. Plant Dis 68: 1075-1076. doi: 10.1094/PD-69-1075
[43]	Sutton JC, Sopher CR, Owen-Going TN, et al. (2006) Etiology and epidemiology of Pythium root rot in hydroponic crops: current knowledge and perspectives. Summa Phytopathol 32: 307-321. doi: 10.1590/S0100-54052006000400001

This article has been cited by:

1.	Pradip Ramesh Patle, Moosa Gabeleh, Vladimir Rakočević, Mohammad Esmael Samei, New best proximity point (pair) theorems via MNC and application to the existence of optimum solutions for a system of $\psi$ -Hilfer fractional differential equations, 2023, 117, 1578-7303, 10.1007/s13398-023-01451-5
2.	Shahram Rezapour, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez, Mehran Ghaderi, A computational method for investigating a quantum integrodifferential inclusion with simulations and heatmaps, 2023, 8, 2473-6988, 27241, 10.3934/math.20231394
3.	Amjad Ali, Khezer Hayat, Abrar Zahir, Kamal Shah, Thabet Abdeljawad, Qualitative Analysis of Fractional Stochastic Differential Equations with Variable Order Fractional Derivative, 2024, 23, 1575-5460, 10.1007/s12346-024-00982-5
4.	Sabri T. M. Thabet, Imed Kedim, An investigation of a new Lyapunov-type inequality for Katugampola–Hilfer fractional BVP with nonlocal and integral boundary conditions, 2023, 2023, 1029-242X, 10.1186/s13660-023-03070-5
5.	Thabet Abdeljawad, Sabri T. M. Thabet, Imed Kedim, M. Iadh Ayari, Aziz Khan, A higher-order extension of Atangana–Baleanu fractional operators with respect to another function and a Gronwall-type inequality, 2023, 2023, 1687-2770, 10.1186/s13661-023-01736-z
6.	Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez, Efficient results on unbounded solutions of fractional Bagley-Torvik system on the half-line, 2024, 9, 2473-6988, 5071, 10.3934/math.2024246
7.	Anil Chavada, Nimisha Pathak, Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data, 2024, 4, 2767-8946, 119, 10.3934/mmc.2024011
8.	Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim, Analytical study of $\mathcal{ABC}$ -fractional pantograph implicit differential equation with respect to another function, 2023, 8, 2473-6988, 23635, 10.3934/math.20231202
9.	Sabri T. M. Thabet, Sa'ud Al-Sa'di, Imed Kedim, Ava Sh. Rafeeq, Shahram Rezapour, Analysis study on multi-order $\varrho$ -Hilfer fractional pantograph implicit differential equation on unbounded domains, 2023, 8, 2473-6988, 18455, 10.3934/math.2023938
10.	Kottakkaran Sooppy Nisar, Efficient results on Hilfer pantograph model with nonlocal integral condition, 2023, 80, 11100168, 342, 10.1016/j.aej.2023.08.061
11.	R.N. Premakumari, Chandrali Baishya, Mohammad Esmael Samei, Manisha Krishna Naik, A novel optimal control strategy for nutrient–phytoplankton–zooplankton model with viral infection in plankton, 2024, 137, 10075704, 108157, 10.1016/j.cnsns.2024.108157
12.	Sabri T. M. Thabet, Thabet Abdeljawad, Imed Kedim, M. Iadh Ayari, A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law, 2023, 2023, 1687-2770, 10.1186/s13661-023-01790-7
13.	Xinguang Zhang, Peng Chen, Hui Tian, Yonghong Wu, The Iterative Properties for Positive Solutions of a Tempered Fractional Equation, 2023, 7, 2504-3110, 761, 10.3390/fractalfract7100761
14.	Latif Ur Rahman, Muhammad Arshad, Sabri T. M. Thabet, Imed Kedim, Xiaolong Qin, Iterative Construction of Fixed Points for Functional Equations and Fractional Differential Equations, 2023, 2023, 2314-4785, 1, 10.1155/2023/6677650
15.	Ava Sh. Rafeeq, Sabri T.M. Thabet, Mohammed O. Mohammed, Imed Kedim, Miguel Vivas-Cortez, On Caputo-Hadamard fractional pantograph problem of two different orders with Dirichlet boundary conditions, 2024, 86, 11100168, 386, 10.1016/j.aej.2023.11.081
16.	Brahim Tellab, Abdelkader Amara, Mohammed El-Hadi Mezabia, Khaled Zennir, Loay Alkhalifa, Study of a Coupled Ψ–Liouville–Riemann Fractional Differential System Characterized by Mixed Boundary Conditions, 2024, 8, 2504-3110, 510, 10.3390/fractalfract8090510
17.	Deepesh Kumar Patel, Moosa Gabeleh, Mohammad Esmael Samei, On existence of solutions for $\Psi$ -Hilfer type fractional BVP of generalized higher order, 2024, 43, 2238-3603, 10.1007/s40314-024-02681-y
18.	R. Poovarasan, Mohammad Esmael Samei, V. Govindaraj, Study of three-point impulsive boundary value problems governed by $\Psi$ -Caputo fractional derivative, 2024, 70, 1598-5865, 3947, 10.1007/s12190-024-02122-3
19.	Yanfang Li, Donal O’Regan, Jiafa Xu, Nontrivial Solutions for a First-order Impulsive Integral Boundary Value Problem on Time Scales, 2024, 23, 1575-5460, 10.1007/s12346-024-00954-9
20.	Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim, Mohammad Esmael Samei, M. Iadh Ayari, Solvability of a ϱ-Hilfer Fractional Snap Dynamic System on Unbounded Domains, 2023, 7, 2504-3110, 607, 10.3390/fractalfract7080607
21.	M. Lavanya, B. Sundara Vadivoo, Kottakkaran Sooppy Nisar, Controllability Analysis of Neutral Stochastic Differential Equation Using $\psi$ -Hilfer Fractional Derivative with Rosenblatt Process, 2025, 24, 1575-5460, 10.1007/s12346-024-01178-7
22.	Said Zibar, Brahim Tellab, Abdelkader Amara, Homan Emadifar, Atul Kumar, Sabir Widatalla, Existence, uniqueness and stability analysis of a nonlinear coupled system involving mixed ϕ-Riemann-Liouville and ψ-Caputo fractional derivatives, 2025, 2025, 1687-2770, 10.1186/s13661-025-01994-z
23.	R. Poovarasan, Mohammad Esmael Samei, V. Govindaraj, Analysis of existence, uniqueness, and stability for nonlinear fractional boundary value problems with novel integral boundary conditions, 2025, 1598-5865, 10.1007/s12190-025-02378-3
24.	Abderrahman Elgmairi, M’hamed Elomari, Said Melliani, Hybrid Fractional Differential Equations Involving the $\psi$ -Caputo Derivative, 2025, 11, 2349-5103, 10.1007/s40819-025-01856-3
25.	Kalimuthu Ramalakshmi, Mohammad Esmael Samei, Behnam Mohammadaliee, A prominent optimal control strategy for soil-transmitted helminth infections via generalized mathematical model of fractional differential equation, 2025, 3, 30505178, 100023, 10.1016/j.nls.2025.100023

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)