1 Introduction

The topics of q-calculus and fractional calculus have received a lot of interest in view of their role in describing some real-world problems in numerous fields. It is worth noting that both fractional calculus and q-calculus are generalizations of classical calculus for any order. Note that the field of fractional calculus has a large number of fractional operators, but in our paper, we are interested in studying the equations that contain the fractional derivative of the Riemann–Liouville type. Mathematics modeling is used to convert a range of applied problems into a set of fractional differential and integral equation [1,2,3]. The study of theoretical aspects of q-calculus and fractional calculus has been the focus of many studies, and it is now considered a significant area of research. There are numerous papers on the solvability of nonlinear fractional differential equations. At the same time, the study of coupled systems of nonlinear fractional differential equations is also important due to their numerous applications. In Ahmad and Nieto [4] studied the existence of result for the following coupled system:

$$\begin{aligned}&{\mathcal {D}} ^{\alpha } {{{\mathfrak {U}}({{\L } })}}=\mathfrak {f( {{\L } }},w({{\L } }),{\mathcal {D}}^{\lambda } w({{\L } })),\qquad {\mathcal {D}}^{\beta } w({{\L } })={\mathfrak {g}}({{\L } },{\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\gamma } w({{\L } })),\qquad {{\L } }\in (0,1),\\&{\mathfrak {U}}(0)=0,\quad {\mathfrak {U}}(1)=\varsigma {\mathfrak {U}}(\zeta ),\quad w(0)=0,\quad w(1)= \varsigma w(\zeta ), \end{aligned}$$

where \(\alpha , \lambda , \beta , \zeta , \gamma , and \varsigma \) satisfy certain conditions. In [5], Zhang et al. applied a variety of fixed-point theorems to the following coupled system of nonlinear fractional differential equations to investigate the existence and uniqueness of solutions:

$$\begin{aligned}&{\mathcal {D}}^{\rho _1} {\mathfrak {U}}({{\L } })={\mathfrak {g}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } })\biggl ), \quad {{\L } }\in (0, 1),\\&{\mathcal {D}}^{\rho _2} w({{\L } })={\mathfrak {g}}_2\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } })\biggl ), \quad {{\L } }\in (0, 1),\\&{\mathfrak {U}}(0)={\mathfrak {U}}'(0)=0,w(0)=w'(0)=0,\\&{\mathfrak {U}}(1)=\gamma _1 I^{\beta _1} {\mathfrak {U}}(\mu _1),w(1)=\gamma _2 I^{\beta _2} w(\mu _2), \end{aligned}$$

where \({\mathcal {D}}^{\rho _i}\) and \({\mathcal {D}}^{\alpha _i}\) represent the standard Riemann–Liouville fractional derivative, \(2<\rho _i\le 3,\) \(0<\alpha _i\le 1,\) \(0<\mu _i\le 1,\) \(\gamma _i,\beta _i>0, i=1,2\). For more details, see [6,7,8,9,10,11,12,13,14,15,16,17,18,19].

In contrast, several studies have been written about the existence and uniqueness of solutions for fractional q-integro-differential equations; for further information, see [20,21,22,23,24]. In [25, 26], the authors discussed the numerical and analytical solutions of the Fredholm and Fredholm–Volterra integro-differential equations of the first and second orders, respectively. In addition, they study the numerical solution using a merge of finite difference with Simpson’s and finite difference with trapezoidal methods. In [27], they discussed the existence and uniqueness of a solution for the nonlocal fractional q-integro-differential equation:

$$\begin{aligned} {\mathfrak {U}}''({{\L } })={\mathfrak {g}}\biggl ({{\L } },{\mathfrak {U}}({{\L } }),{^{CF}{\mathcal {D}}^{\beta } {\mathfrak {U}}({{\L } })}, I_q^{\alpha } {\mathfrak {f}}({{\L } }, {\mathfrak {U}}'({{\L } }))\biggl ), \quad {{\L } }\in (0, 1], \end{aligned}$$

under the q-nonlocal condition:

$$\begin{aligned} (1-q) \nu \sum _{l=0}^n q^l {\mathfrak {U}}(q^i \nu )=\rho _0, \quad {\mathfrak {U}}'(0)= \alpha _0,\quad \nu \in (0,1). \end{aligned}$$

where \({^{CF}{\mathcal {D}}^{\beta }}{\mathfrak {U}}({{\L } })\) is the fractional derivative of Caputo–Fabrizio type, \(I_q^{\alpha }\) is the Riemann–Liouville fractional q-integral, and \(\rho _0,\alpha _0\) are constants, \(q,\beta \in (0,1)\). In addition, they solved it numerically by using two methods: the finite-trapezoidal method and the cubic-trapezoidal method.

Inspired by the aforementioned results, we examine in this work the following coupled system of a nonlocal fractional q-integro-differential equation:

$$\begin{aligned}&{\mathcal {D}}^{\beta _1} {\mathfrak {U}}({{\L } })={\mathcal {F}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl ), \quad {{\L } }\in (0, 1),\\&{\mathcal {D}}^{\beta _2} w({{\L } })={\mathcal {F}}_2\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl ), \quad {{\L } }\in (0, 1), \end{aligned}$$
(1)

considering the q-nonlocal conditions:

$$\begin{aligned}&(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)=\rho _1, \quad {\mathfrak {U}}(0)= {\mathfrak {U}}'(0)=0, \quad \chi _1\in (0,1),\\&(1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2} q_2^\kappa w(q_2^\kappa \chi _2)=\rho _2, \quad w(0)=w'(0)=0, \quad \chi _2\in (0,1), \end{aligned}$$
(2)

where \({\mathcal {D}}^{\beta _i}\) and \({\mathcal {D}}^{\alpha _i}\) represent the standard fractional derivative of the Riemann–Liouville type, \(I_{q_i}^{\delta _i}\) represent the Riemann–Liouville fractional q-integrals of the order \(\delta _i>0,\) \(2<\beta _i\le 3,\) \(0<\alpha _i\le 1,\) \(\rho _i\) are constants, and \(q_i,\alpha _i \in (0,1), i=1,2\).

This is how the essay is organized: We list several lemmas and definitions that are indeed in this work in Sect. 2. The existence and uniqueness of the solution to the nonlocal coupled system of a fractional q-integro-differential equation (1)–(2) were examined in Sect. 3. Section 4 contains applications. In Sect. 5, the conclusion is presented.

2 Preliminaries

We now go through some fundamental ideas in q-calculus and fractional calculus, as well as some lemmas that will be applied in this article.

Definition 2.1

[20, 21] Let z be a function that is defined on the interval [0, 1]. The Riemann–Liouville fractional q-integral of order \(\varrho > 0\) can be defined as

$$\begin{aligned} (I_q^\varrho z)({{\L } })=\left\{ \begin{array}{lcl} z({{\L } }), \quad \varrho =0, &{} \\ \frac{1}{\Gamma _q(\varrho )}\int _{0}^{{{\L } }} ({{\L } }-qs)^{(\varrho -1)} z(s) \rm{d}_qs, \end{array} \right. \end{aligned}$$
(3)

where

$$\begin{aligned}&\quad ({{\L } }-qs)^{(0)}=1, \quad ({{\L } }-qs)^{(\phi )}=\prod _{j=0}^{\phi -1}({{\L } }-q^{j+1} s), \phi \in {\mathbb {N}},\quad ({{\L } }-qs)^{(\Omega )}=\prod _{j=0}^{\infty } \frac{({{\L } }-q^{j+1}s)}{({{\L } }-q^{j+\Omega +1} s)},\quad \Omega \in {\mathbb {R}}. \end{aligned}$$

Lemma 2.2

[20] As a result of q-integration by parts, we obtain

$$\begin{aligned} (I_q^\varrho 1) ({{\L } })=\frac{{{\L } }^{(\varrho )}}{\Gamma _q (\varrho +1)}, \qquad \varrho >0. \end{aligned}$$
(4)

Definition 2.3

[28] The Riemann–Liouville fractional derivative of order \(\beta >0\) of a function \({\mathcal {F}}:(0,\infty )\rightarrow {\mathbb {R}} \) can be defined as

$$\begin{aligned} {{\mathcal {D}}^{\beta } {\mathcal {F}}({{\L } })} =\frac{1}{\Gamma (\nu -\beta )} \left( \frac{\rm{d}}{\rm{d}t}\right) ^\nu \int _{0}^{{{\L } }} ({{\L } }-\theta )^{\nu -\beta -1 }{\mathcal {F}}(\theta ) \rm{d}\theta . \end{aligned}$$

where \( \nu = [\beta ]+1, [\beta ]\) represents the integer part of number \(\beta \), provided that the right-hand side is pointwise defined on \((0,\infty )\).

Definition 2.4

[28] The Riemann–Liouville fractional integral of order \(\beta >0\) of a function \({\mathcal {F}}:(0,\infty )\rightarrow {\mathbb {R}} \) is given by

$$\begin{aligned} I^{\beta }{\mathcal {F}}({{\L } })=\frac{1}{\Gamma (\beta )} \int _{0}^{{{\L } }} ({{\L } }-\theta )^{\beta -1 }{\mathcal {F}}(\theta ) \rm{d}\theta . \end{aligned}$$

Lemma 2.5

[28] Suppose that \(\beta >0\), \(\nu -1<\beta <\nu , \nu \in N.\) Then

  1. 1.

    For any \(z \in L^1(c,d), D^{\beta }(I^{\beta }z)=z.\)

  2. 2.

    If \(I^{\nu -\beta }z\in AC^\nu [c,d]\), then

    $$\begin{aligned} I^{\beta } {\mathcal {D}}^{\beta } z({{\L } })=z({{\L } })+k_1 {{\L } }^{\beta -1}+k_2 {{\L } }^{\beta -2}+\cdots +k_\nu {{\L } }^{\beta -\nu }, \end{aligned}$$

where \( k_i \in {\mathbb {R}} (i=1,2,\ldots , \nu ), \nu \) is the lowest integer smaller than or equal to \(\beta .\)

Lemma 2.6

[28]

  1. 1.

    If \( {\mathcal {F}} \in L^1 (c,d), \sigma>\varphi >0,\) then

    $$\begin{aligned} I^\sigma I^\varphi {\mathcal {F}}({{\L } }) = I^{\sigma +\varphi } {\mathcal {F}}({{\L } }),\quad {\mathcal {D}}^\varphi I^\sigma {\mathcal {F}}({{\L } })=I^{\sigma -\varphi } {\mathcal {F}}({{\L } }),\quad {\mathcal {D}}^\sigma I^\sigma {\mathcal {F}}({{\L } })= {\mathcal {F}}({{\L } }). \end{aligned}$$
  2. 2.

    If \(\sigma> \varphi >0,\) then

    $$\begin{aligned} I^{\sigma } {{\L } }^{\varphi }=\frac{\Gamma (\varphi +1)}{\Gamma (\varphi +1+\sigma )} {{\L } }^{\sigma +\varphi },\quad {\mathcal {D}}^{\varphi } {{\L } }^{\sigma }=\frac{\Gamma (\sigma +1)}{\Gamma (\sigma +1-\varphi )} {{\L } }^{\sigma -\varphi }, \quad {\mathcal {D}}^{\sigma } {{\L } }^{\varphi }=0. \end{aligned}$$

Lemma 2.7

[29] Suppose that \(\sigma >0, {\mathfrak {U}}\in L^1([c,d],{\mathbb {R}}).\) Then, we have

$$\begin{aligned} I^{\nu +1} {\mathfrak {U}}({{\L } }) \le \Vert I^{\nu } {\mathfrak {U}}({{\L } }) \Vert _{L^1}, {{\L } }\in [c,d]. \end{aligned}$$

For the convenience, we set

$$\begin{aligned} \Lambda _i=\frac{1}{(1-q_i)\chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa (q_i^\kappa \chi _i)^{(\beta _i-1)}},\quad i=1,2. \end{aligned}$$

Lemma 2.8

Suppose that \({\mathfrak {U}} \in AC^3[0,1]\), and \(2<\beta _1\le 3.\) Therefore, we can get a unique solution to the following nonlocal problem:

$$\begin{aligned} \left\{ \begin{array}{lcl} {\mathcal {D}}^{\beta _1} {\mathfrak {U}}({{\L } })={{\mathcal {F}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl )}, \quad {{\L } }\in (0,1),\\ &{} \\ (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)=\rho _1, \quad {\mathfrak {U}}(0)={\mathfrak {U}}'(0)=0, \quad \chi _1\in (0,1), \end{array} \right. \end{aligned}$$
(5)

as

$$\begin{aligned} {\mathfrak {U}}({{\L } })= {\mathfrak {U}}({{\L } })&= I^{\beta _1} {{\mathcal {F}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl )}\\&+ \Lambda _1\bigg [\rho _1-(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa I^{\beta _1} {{\mathcal {F}}_1\biggl (q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}(q_1^\kappa \chi _1)},\\&{{\mathcal {D}}^{\alpha _2} w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1} {\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2} w(q_1^\kappa \chi _1)\biggl )}\bigg ] {{\L } }^{\beta _1-1}. \end{aligned}$$
(6)

Proof

Considering the lemma (), we can get the solution of (5) as follows:

$$\begin{aligned}&{\mathfrak {U}}({{\L } })=I^{\beta _1}{{\mathcal {F}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl )} +k_1 {{\L } }^{\beta _1-1}+k_2 {{\L } }^{\beta _1-2}+k_3 {{\L } }^{\beta _1-3}. \end{aligned}$$

Using the condition \({\mathfrak {U}}(0)={\mathfrak {U}}'(0)=0\), we get \(k_2=k_3=0\). Therefore,

$$\begin{aligned} {\mathfrak {U}}({{\L } })= I^{\beta _1} {{\mathcal {F}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl )}+k_1 {{\L } }^{\beta _1-1}. \end{aligned}$$
(7)

Using the q-nonlocal condition \((1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)=\rho _1\), we obtain

$$\begin{aligned} (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)&=(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa I^{\beta _1} {{\mathcal {F}}_1\biggl (q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}(q_1^\kappa \chi _1)},\\&{{\mathcal {D}}^{\alpha _2} w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1} {\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2} w(q_1^\kappa \chi _1)\biggl )}+k_1 (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa (q_1^\kappa \chi _1)^{\beta _1-1}. \end{aligned}$$

Therefore,

$$\begin{aligned} k_1=&\Lambda _1\bigg [\rho _1- (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa I^{\beta _1} {{\mathcal {F}}_1\biggl (q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}(q_1^\kappa \chi _1)},\\&{{\mathcal {D}}^{\alpha _2} w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1} {\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2} w(q_1^\kappa \chi _1)\biggl )}\bigg ]. \end{aligned}$$

Substituting \(k_1\) into (7), we obtain

$$\begin{aligned} {\mathfrak {U}}({{\L } })&= I^{\beta _1} {{\mathcal {F}}_1\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl )}\\&\quad + \Lambda _1\bigg [\rho _1-(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa I^{\beta _1} {{\mathcal {F}}_1\biggl (q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}(q_1^\kappa \chi _1)},\\&{{\mathcal {D}}^{\alpha _2} w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1} {\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2} w(q_1^\kappa \chi _1)\biggl )}\bigg ] {{\L } }^{\beta _1-1}. \end{aligned}$$

The proof is finished. \(\square \)

In the same way, the solution of

$$\begin{aligned} \left\{ \begin{array}{lcl} {\mathcal {D}}^{\beta _2} w({{\L } })={{\mathcal {F}}_2\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w{{\L } })\biggl )}, \quad {{\L } }\in (0,1),\\ &{} \\ (1-q_2) \chi _2\sum _{j=0}^{\nu _2} q_2^j w(q_2^j \chi _2)=\rho _2, \quad w(0)=w'(0)=0,\quad \chi _2 \in (0,1), \end{array} \right. \end{aligned}$$

is

$$\begin{aligned} w({{\L } })&= I^{\beta _2} {{\mathcal {F}}_2\biggl ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2} w({{\L } }),I_{q_1}^{\delta _1} {\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2} w({{\L } })\biggl )}\\&\quad + \Lambda _2\bigg [\rho _2-(1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2} q_2^\kappa I^{\beta _2} {{\mathcal {F}}_2\biggl (q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1} {\mathfrak {U}}(q_1^\kappa \chi _1)},\\&{{\mathcal {D}}^{\alpha _2} w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1} {\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2} w(q_1^\kappa \chi _1)\biggl )}\bigg ] {{\L } }^{\beta _2-1}. \end{aligned}$$

Denote that \(L^1([0, 1],{\mathbb {R}} )\) is the Banach space of Lebesgue integrable functions from \([0, 1]\rightarrow {\mathbb {R}} \) with the norm \( \Vert {\mathfrak {U}}\Vert = \int _{0}^{1} |{\mathfrak {U}}({{\L } })| d {{\L } }.\)

3 Main results

The spaces \({\mathcal {X}}\) and \({\mathcal {Y}}\) should now be introduced as follows: \({\mathcal {X}}=\{{\mathfrak {U}} | {\mathfrak {U}}\in {\mathcal {C}}[0,1], {\mathcal {D}}^{\beta _1}{\mathfrak {U}}\in {\mathcal {C}}[0,1] \) and \( I_{q_1}^{\delta _1}{\mathfrak {U}} \in {\mathcal {C}}[0,1]\}\) equipped with the norm \(\Vert {\mathfrak {U}}\Vert =\max _{{{\L } }\in [0,1]}|{\mathfrak {U}}({{\L } })|+\max _{{{\L } }\in [0,1]}|{\mathcal {D}}^{\beta _1} {\mathfrak {U}}({{\L } })|+\max _{{{\L } }\in [0,1]}|I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } })|,\) and also \({\mathcal {Y}}=\{w | w\in {\mathcal {C}}[0,1],{\mathcal {D}}^{\beta _2}w\in {\mathcal {C}}[0,1] \) and \( I_{q_2}^{\delta _2}w \in {\mathcal {C}}[0,1]\}\) equipped with the norm \(\Vert w\Vert =\max _{{{\L } }\in [0,1]}|w({{\L } })|+\max _{{{\L } }\in [0,1]}|{\mathcal {D}}^{\beta _2} w({{\L } })|+\max _{{{\L } }\in [0,1]}|I_{q_2}^{\delta _2}w({{\L } })|.\) Evidently, \(({\mathcal {X}},\Vert .\Vert )\) and \(({\mathcal {Y}},\Vert .\Vert )\) are Banach spaces. Therefore, the product space \(({\mathcal {X}} \times {\mathcal {Y}}, \Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}) \) is also a Banach space equipped with the norm \( \Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}=\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+\Vert w\Vert _{\mathcal {Y}}.\) Now, we define the operator \( G: {\mathcal {X}} \times {\mathcal {Y}}\rightarrow {\mathcal {X}} \times {\mathcal {Y}}\) by

$$\begin{aligned} G({\mathfrak {U}},w)({{\L } })=\bigg (G_1({\mathfrak {U}},w)({{\L } }),G_2({\mathfrak {U}},w)({{\L } })\bigg ), \end{aligned}$$
(8)

where

$$\begin{aligned} G_1({\mathfrak {U}},w)({{\L } })= I^{\beta _1} \vartheta ({{\L } }) + \Lambda _1\bigg [\rho _1-(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa I^{\beta _1} \vartheta (q_1^\kappa \chi _1)\bigg ] {{\L } }^{\beta _1-1}, \end{aligned}$$

and

$$\begin{aligned} G_2({\mathfrak {U}},w)({{\L } })=I^{\beta _2} z({{\L } }) + \Lambda _2\bigg [\rho _2-(1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2} q_2^\kappa I^{\beta _2} z(q_2^\kappa \chi _2)\bigg ] {{\L } }^{\beta _2-1}. \end{aligned}$$

Conveniently, we set

  1. 1.
    $$\begin{aligned} {\mathcal {A}}_{i j}= \bigg (1+|\Lambda _i|(1-q_i) \chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa \bigg ) \Vert I^{\beta _i-1}b_{ij}\Vert _{L^1}, i = 1,2,\quad j = 1, \ldots ,7. \end{aligned}$$
  2. 2.
    $$\begin{aligned} {\mathcal {B}}_{i j}=\Vert I^{\beta _i-\alpha _i-1}b_{ij}\Vert _{L^1}+\frac{|\Lambda _i|\Gamma (\beta _i)}{\Gamma (\beta _i-\alpha _i)} (1-q_i) \chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa \Vert I^{\beta _i-1}b_{ij}\Vert _{L^1}, \end{aligned}$$

                            \(i = 1,2,\quad j = 1, \ldots ,7.\)

  3. 3.
    $$\begin{aligned} C_{ij}=\bigg (1+\frac{1}{\Gamma _{q_i}(\delta _i+1)}\bigg ) {\mathcal {A}}_{i j} +{\mathcal {B}}_{i j}, i = 1,2,\quad j = 1, \ldots ,7. \end{aligned}$$
  4. 4.
    $$\begin{aligned} E_i=\bigg (1+\frac{1}{\Gamma _{q_i}(\delta _i+1)} +\frac{\Gamma (\beta _i)}{\Gamma (\beta _i-\alpha _i)} \bigg )|\rho _i\Lambda _i|, \qquad i = 1,2. \end{aligned}$$
  5. 5.
    $$\begin{aligned} D_i=\max \{C_{i1},C_{i2},C_{i3},C_{i4},C_{i5},C_{i6}\},\qquad i = 1,2. \end{aligned}$$

Theorem 3.1

Suppose that \({\mathcal {F}}_i\) are continuous for almost every \( {{\L } } \in (0,1)\) and measurable in \( {{\L } } \) for any \( w_1, w_2,w_3,w_4,w_5,w_6 \in {\mathbb {R}}\). There exist nonnegative functions \(b_{i j}({{\L } }) \in L^1([0,1],\mathbb {R_+}),i = 1, 2, j=1,2,\ldots ,7,\) such that

  1. (a)
    $$\begin{aligned} |{\mathcal {F}}_i({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)|\le \sum _{j=1}^{6} b_{i j}({{\L } })|w_j|^{\tau _{i j}}+b_{i 7}({{\L } }),\quad \tau _{i j} \in (0,1). \end{aligned}$$

Consequently, there is at least one solution for the nonlocal coupled system (1)-(2).

Proof

To demonstrate that the operator G defined by (8) has a fixed point, we shall employ the Schauder fixed-point theorem.

First, let \( Q_{{\mathfrak {r}}}\subset {\mathcal {X}}\times {\mathcal {Y}} = \lbrace ({\mathfrak {U}},w)({{\L } }) \in {\mathbb {R}}^2: \Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}} \le {\mathfrak {r}} \rbrace \), where

$$\begin{aligned} {\mathfrak {r}} \ge \max \{(16 C_{i 1})^{\frac{1}{1-\tau _{i 1}}},(16 C_{i 2})^{\frac{1}{1-\tau _{i 2}}},(16 C_{i 3})^{\frac{1}{1-\tau _{i 3}}},(16 C_{i 4})^{\frac{1}{1-\tau _{i 4}}},(16 C_{i 5})^{\frac{1}{1-\tau _{i 5}}},(16 C_{i 6})^{\frac{1}{1-\tau _{i 6}}}, 16C_{i 7}, i=1,2\}. \end{aligned}$$

It is obvious that \(Q_{\mathfrak {r}}\) is nonempty, bounded, closed and convex subset of \({\mathcal {C}}[0, 1]\). We demonstrate \(G: Q_{\mathfrak {r}}\rightarrow Q_{\mathfrak {r}}.\) For any \(({\mathfrak {U}},w)({{\L } })\in Q_{\mathfrak {r}},\) we use Lemma () and condition (a), to obtain

$$\begin{aligned} |G_1({\mathfrak {U}},w)({{\L } })|\le&I^{\beta _1} |{\mathcal {F}}_1({{\L } },\mathfrak { U}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}} ({{\L } }),{\mathcal {D}}^{\alpha _2}w({{\L } }),I_{q_1}^{\delta _1}{\mathfrak {U}} ({{\L } }),I_{q_2}^{\delta _2}w({{\L } }))|+ |\rho _1\Lambda _1|\\&+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} |{\mathcal {F}}_1(q_1^\kappa \chi _1,{\mathfrak {U}} (q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}} (q_1^\kappa \chi _1),\\&{\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1}{\mathfrak {U}} (q_1^\kappa \chi _1),I_{q_2}^{\delta _2}w(q_1^\kappa \chi _1))|\\&\le I^{\beta _1} \bigg (\sum _{j=1}^{6} b_{1 j}({{\L } }) R^{\tau _{1 j}}+b_{1 7}({{\L } })\bigg )+|\rho _1 \Lambda _1 |\\&+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} \bigg (\sum _{j=1}^{6} b_{1 j}(q_1^\kappa \chi _1) R^{\tau _{1 j}} +b_{1 7}(q_1^\kappa \chi _1)\bigg ) \\&\le \sum _{j=1}^{6}\bigg [\bigg (1+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa \bigg ) \Vert I^{\beta _1-1}b_{1j}\Vert _{L^1}\bigg ]R^{\tau _{1j }}+|\rho _1\Lambda _1|\\&+\bigg (1+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa \bigg ) \Vert I^{\beta _1-1}b_{17}\Vert _{L^1}\\&=\sum _{j=1}^{6} {\mathcal {A}}_{1j} R^{\tau _{1 j}}+ {\mathcal {A}}_{17}+ |\rho _1\Lambda _1|. \end{aligned}$$

Similarly, by using Lemma (), we get

$$ \begin{aligned} |{\mathcal {D}}^{\alpha _1} G_1({\mathfrak {U}},w)({{\L } })|=&\bigg |I^{\beta _1-\alpha _1}{\mathcal {F}}_1({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2}w({{\L } }),I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2}w({{\L } }))\\&+\frac{\Lambda _1\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} {{\L } }^{\beta _1-\alpha _1-1}\bigg [ \rho _1-(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} {\mathcal {F}}_1(q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),\\&{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1}{\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2}w(q_1^\kappa \chi _1))\bigg ]\bigg |\\&\le I^{\beta _1-\alpha _1}\bigg |{\mathcal {F}}_1({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2}w({{\L } }),I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2}w({{\L } }))\bigg |\\&+\frac{|\rho _1 \Lambda _1| \Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} +\frac{|\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} (1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1}\bigg | {\mathcal {F}}_1(q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),\\&{\mathcal {D}}^{\alpha _1}u(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1}{\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2}w(q_1^\kappa \chi _1))\bigg |\\&\le I^{\beta _1-\alpha _1}\bigg (\sum _{j=1}^{6}\ b_{1j}({{\L } }) R^{\tau _{1 j}} +b_{17}({{\L } })\bigg )+\frac{|\rho _1 \Lambda _1| \Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)}+\frac{|\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} \\&(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1}\bigg (\sum _{j=1}^{6}\ b_{1j}(q_1^\kappa \chi _1) R^{\tau _{1 j}} +b_{17}(q_1^\kappa \chi _1)\bigg )\\&\le \sum _{j=1}^{6}\bigg [ \Vert I^{\beta _1-\alpha _1-1}b_{1j}\Vert _{L_1}+\frac{|\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} (1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa \Vert I^{\beta _1-1}b_{1j}\Vert _{L^1}\bigg ]R^{\tau _{1j}}\\&+\bigg [\Vert I^{\beta _1-\alpha _1-1}b_{17}\Vert _{L^1}+\frac{|\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} (1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa \Vert I^{\beta _1-1}b_{17} \Vert _{L^1}\bigg ]\\&+\frac{|\rho _1 \Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} \\&=\sum _{j=1}^{6} {\mathcal {B}}_{1 j} R^{\tau _{1 j}}+ {\mathcal {B}}_{17}+\frac{|\rho _1\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)}. \end{aligned}$$
(9)

Analogously, we get

$$\begin{aligned} |I_{q_1}^{\delta _1} G_1({\mathfrak {U}},w)({{\L } })|\le&I_{q_1}^{\delta _1} I^{\beta _1} |{\mathcal {F}}_1({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2}w({{\L } }),I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2}w({{\L } }))|+I_{q_1}^{\delta _1} |\rho _1 \Lambda _1 |\\ {}&+I_{q_1}^{\delta _1}|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} |{\mathcal {F}}_1(q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}(q_1^\kappa \chi _1),\\ {}&{\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1}{\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2}w(q_1^\kappa \chi _1))|\\ {}&\le I_{q_1}^{\delta _1}\bigg (\sum _{j=1}^{6} {\mathcal {A}}_{1j} R^{\tau _{1 j}}+ {\mathcal {A}}_{17}+ |\rho _1 \Lambda _1|\bigg ). \end{aligned}$$

Using lemma (), we get

$$\begin{aligned} |I_{q_1}^{\delta _1} G_1({\mathfrak {U}},w)({{\L } })|=\frac{1}{\Gamma _{q_1}(\delta _1+1)}\bigg (\sum _{j=1}^{6} {\mathcal {A}}_{1j} R^{\tau _{1 j}}+ {\mathcal {A}}_{17}+ |\rho _1 \Lambda _1|\bigg ). \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert G_1({\mathfrak {U}},w)\Vert _{\mathcal {X}}\le \sum _{j=1}^{6}C_{1 j} R^{\tau _{1 j}}+C_{1 7}+E_1\le \frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}=\frac{{\mathfrak {r}}}{2}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert G_2({\mathfrak {U}},w) \Vert _{\mathcal {X}}\le \sum _{j=1}^{6}C_{2j} R^{\tau _{2 j}}+C_{2 7}+E_2\le \frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}+\frac{1}{16}{\mathfrak {r}}=\frac{{\mathfrak {r}}}{2}. \end{aligned}$$

Thus,

$$\begin{aligned}&\Vert G({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}= \Vert G_1({\mathfrak {U}},w)\Vert _{\mathcal {X}}+\Vert G_2({\mathfrak {U}},w)\Vert _{\mathcal {Y}}\le {\mathfrak {r}}. \end{aligned}$$

Then, \( G: Q_{{\mathfrak {r}}}\rightarrow Q_{{\mathfrak {r}}}\), and also, the class of functions \( \lbrace G({\mathfrak {U}},w)({{\L } }) \rbrace \) is uniformly bounded in \( Q_{{\mathfrak {r}}}\). Observe that \(G_1({\mathfrak {U}},w)({{\L } }),G_2({\mathfrak {U}},w)({{\L } }),{\mathcal {D}}^{\alpha _1}G_1({\mathfrak {U}},w)({{\L } }),{\mathcal {D}}^{\alpha _2}G_2({\mathfrak {U}},w)({{\L } }),I_{q_1}^{\delta _1}G_1({\mathfrak {U}},w)({{\L } }),I_{q_2}^{\delta _2}G_2({\mathfrak {U}},w)({{\L } })\) are continuous on [0, 1]. Clearly, G is also continuous. Next, we demonstrate that G is equicontinuous. Let

$$\begin{aligned} N_i=\max _{{{\L } }\in [0,1]}\bigg \{|{\mathcal {F}}_i({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2}w({{\L } }),I_{q_1}^{\delta _1} U({{\L } }),I_{q_2}^{\delta _2}w({{\L } }))|\bigg \},\forall ({\mathfrak {U}},w)\in Q_{\mathfrak {r}}, i=1,2. \end{aligned}$$

For \( {{\L } }_1, {{\L } }_2\in [0,1], {{\L } }_1< {{\L } }_2\), we obtain

$$\begin{aligned}{} & {} |G_1({\mathfrak {U}}, w)({{\L } }_2)- G_1({\mathfrak {U}}, w)({{\L } }_1)|\le I^{\beta _1} \bigg | {\mathcal {F}}_1\bigg ({{\L } }_2,{\mathfrak {U}}({{\L } }_2),w({{\L } }_2),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }_2),{\mathcal {D}}^{\alpha _2}w({{\L } }_2),\\ {}{} & {} I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }_2),I_{q_2}^{\delta _2}w({{\L } }_2)\bigg )-{\mathcal {F}}_1\bigg ({{\L } }_1,{\mathfrak {U}}({{\L } }_1),w({{\L } }_1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }_1),{\mathcal {D}}^{\alpha _2}w({{\L } }_1),I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }_1),I_{q_2}^{\delta _2}w({{\L } }_1)\bigg )\bigg |\\ {}{} & {} +\bigg [|\Lambda _1 \rho _1|+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} \bigg |{\mathcal {F}}_1(q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}(q_1^\kappa \chi _1),\\ {}{} & {} {\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1),I_{q_1}^{\delta _1}{\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2}w(q_1^\kappa \chi _1))\bigg |\bigg ]({{\L } }_2^{\beta _1-1}-{{\L } }_1^{\beta _1-1})\\ {}{} & {} \le N_1 \bigg [\int _{0}^{{{\L } }_1}\frac{({{\L } }_2-s)^{\beta _1-1}-({{\L } }_1-s)^{\beta _1-1}}{\Gamma (\beta _1)} ds+\int _{{{\L } }_1}^{{{\L } }_2}\frac{({{\L } }_2-s)^{\beta _1-1}}{\Gamma (\beta _1)}ds\bigg ]\\ {}{} & {} + \bigg [|\Lambda _1 \rho _1|+N_1 |\Lambda _1| (1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa \int _{0}^{q_1 ^\kappa \chi _1}\frac{(q_1^\kappa \chi _1-s)^{\beta _1-1}}{\Gamma (\beta _1)} ds \bigg ]( {{\L } }_2^{\beta _1-1}- {{\L } }_1^{\beta _1-1})\\ {}{} & {} \le \frac{N_1}{\Gamma (\beta _1+1)}({{\L } }_2^{\beta _1}- {{\L } }_1^{\beta _1})+\bigg [ |\rho _1\Lambda _1|+N_1 |\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1 }q_1^\kappa \frac{(q_1^\kappa \chi _1)^{\beta _1}}{\Gamma (\beta _1+1)} \bigg ]\\ {}{} & {} ({{\L } }_2^{\beta _1-1}- {{\L } }_1^{\beta _1-1}). \end{aligned}$$

In contrast, we obtain

$$\begin{aligned}{} & {} |{\mathcal {D}}^{\alpha _1} G_1({\mathfrak {U}}, w)({{\L } }_2)- {\mathcal {D}}^{\alpha _1} G_1({\mathfrak {U}}, w)({{\L } }_1)|\le \bigg |I^{\beta _1-\alpha _1}\bigg [{\mathcal {F}}_1\bigg ({{\L } }_2,{\mathfrak {U}}({{\L } }_2),w({{\L } }_2),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }_2),{\mathcal {D}}^{\alpha _2}w({{\L } }_2),\\ {}{} & {} I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }_2),I_{q_2}^{\delta _2}w({{\L } }_2)\bigg )-{\mathcal {F}}_1\bigg ({{\L } }_1,{\mathfrak {U}}({{\L } }_1),w({{\L } }_1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }_1),{\mathcal {D}}^{\alpha _2}w({{\L } }_1),I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }_1),I_{q_2}^{\delta _2}w({{\L } }_1)\bigg )\bigg ]\bigg |\\ {}{} & {} +\bigg [\frac{|\rho _1 \Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)} +|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} \\ {}{} & {} \bigg |{\mathcal {F}}_1(q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1))\bigg |\bigg ] ({{\L } }_2^{\beta _1-\alpha _1-1}- {{\L } }_1^{\beta _1-\alpha _1-1})\\ {}{} & {} \le N_1\bigg [\int _{0}^{{{\L } }_1}\frac{({{\L } }_2-s)^{\beta _1-\alpha _1-1}-({{\L } }_1-s)^{\beta _1-\alpha _1-1}}{\Gamma (\beta _1-\alpha _1)} ds+\int _{{{\L } }_1}^{{{\L } }_2}\frac{({{\L } }_2-s)^{\beta _1-\alpha _1-1}}{\Gamma (\beta _1-\alpha _1)}ds\bigg ]\\ {}{} & {} + \bigg [ \frac{|\rho _1\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)}+N_1 |\Lambda _1| (1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa \int _{0}^{q_1^\kappa \chi _1}\frac{(q_1^\kappa \chi _1-s)^{\beta _1-1}}{\Gamma (\beta _1)} \bigg ]\\ {}{} & {} ({{\L } }_2^{\beta _1-\alpha _1-1}- {{\L } }_1^{\beta _1-\alpha _1-1})\\ {}{} & {} \le \frac{N_1}{\Gamma (\beta _1-\alpha _1+1)}({{\L } }_2^{\beta _1-\alpha _1}- {{\L } }_1^{\beta _1-\alpha _1})+\bigg [ \frac{|\rho _1\Lambda _1|\Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)}+N_1 |\Lambda _1 |(1-q_1) \chi _1\\ {}{} & {} \sum _{\kappa =0}^{\nu _1}q_1^\kappa \frac{(q_1^\kappa \chi _1)^{\beta _1}}{\Gamma (\beta _1+1)} \bigg ]({{\L } }_2^{\beta _1-\alpha _1-1}- {{\L } }_1^{\beta _1-\alpha _1-1}), \end{aligned}$$

and also,

$$\begin{aligned}&|I_{q_1}^{\delta _1} G_1({\mathfrak {U}}, w)({{\L } }_2)- I_{q_1}^{\delta _1} G_1({\mathfrak {U}}, w)({{\L } }_1)|\le I_{q_1}^{\delta _1} \frac{N_1}{\Gamma (\beta _1+1)}({{\L } }_2^{\beta _1}- {{\L } }_1^{\beta _1})+I_{q_1}^{\delta _1}\bigg [|\rho _1\Lambda _1|\\ {}&+N_1 |\Lambda _1| (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1}q_1^{\kappa }\frac{(q_1^{\kappa } \chi _1)^{\beta _1}}{\Gamma (\beta _1+1)}\bigg ]({{\L } }_2^{\beta _1-1}- {{\L } }_1^{\beta _1-1}) \\ {}&\le \frac{N_1}{\Gamma (\beta _1+1)}({{\L } }_2^{\beta _1}- {{\L } }_1^{\beta _1})\frac{ {{\L } }^{\delta _1}}{\Gamma _{q_1}(\delta _1+1)}+\frac{{{\L } }^{\delta _1}}{\Gamma _{q_1}(\delta _1+1)}\bigg [|\rho _1\Lambda _1|\\ {}&+N_1 |\Lambda _1| (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1}q_1^{\kappa }\frac{(q_1^{\kappa } \chi _1)^{\beta _1}}{\Gamma (\beta _1+1)}\bigg ]({{\L } }_2^{\beta _1-1}- {{\L } }_1^{\beta _1-1})\\ {}&\le \frac{N_1}{\Gamma (\beta _1+1) \Gamma _{q_1}(\delta _1+1)}({{\L } }_2^{\beta _1}- {{\L } }_1^{\beta _1})+\frac{1}{\Gamma _{q_1}(\delta _1+1)}\bigg [|\rho _1\Lambda _1|\\ {}&+N_1 |\Lambda _1| (1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1}q_1^{\kappa }\frac{(q_1^{\kappa } \chi _1)^{\beta _1}}{\Gamma (\beta _1+1)}\bigg ]( {{\L } }_2^{\beta _1-1}- {{\L } }_1^{\beta _1-1}). \end{aligned}$$

Similar to that, we can demonstrate

$$\begin{aligned}&|G_2({\mathfrak {U}}, w)({{\L } }_2)- G_2({\mathfrak {U}}, w)({{\L } }_1)|\le \frac{N_2}{\Gamma (\beta _2+1)}({{\L } }_2^{\beta _2}-{{\L } }_1^{\beta _2})\\ {}&+\bigg [ |\rho _2\Lambda _2|+N_2 |\Lambda _2|(1-q_2) \chi _2 \sum _{\kappa =0}^{\nu _2 }q_2^\kappa \frac{(q_2^\kappa \chi _2)^{\beta _2}}{\Gamma (\beta _2+1)} \bigg ]({{\L } }_2^{\beta _2-1}-{{\L } }_1^{\beta _2-1}), \end{aligned}$$
$$\begin{aligned}{} & {} |{\mathcal {D}}^{\alpha _2} G_2({\mathfrak {U}}, w)({{\L } }_2)- {\mathcal {D}}^{\alpha _2} G_2({\mathfrak {U}}, w)({{\L } }_1)|\le \frac{N_2}{\Gamma (\beta _2-\alpha _2+1)}({{\L } }_2^{\beta _2-\alpha _2}-{{\L } }_1^{\beta _2-\alpha _2})\\ {}{} & {} +\bigg [|\rho _2\Lambda _2| \frac{\Gamma (\beta _2)}{\Gamma (\beta _2-\alpha _2)}+N_2 |\Lambda _2 |(1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2}q_2^\kappa \frac{(q_2^\kappa \chi _2)^{\beta _2}}{\Gamma (\beta _2+1)} \bigg ]({{\L } }_2^{\beta _2-\alpha _2-1}-{{\L } }_1^{\beta _2-\alpha _2-1}), \end{aligned}$$
$$\begin{aligned}&|I_{q_2}^{\delta _2} G_2({\mathfrak {U}}, w)({{\L } }_2)- I_{q_2}^{\delta _2} G_2({\mathfrak {U}}, w)({{\L } }_1)|\le \frac{N_2}{\Gamma (\beta _2+1) \Gamma _{q_2}(\delta _2+1)}({{\L } }_2^{\beta _2}-{{\L } }_1^{\beta _2})+\frac{1}{\Gamma _{q_2}(\delta _2+1)}\bigg [|\rho _2\Lambda _2|\\ {}&+N_2 |\Lambda _2| (1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2}q_2^{\kappa }\frac{(q_2^{\kappa } \chi _2)^{\beta _2}}{\Gamma (\beta _2+1)}\bigg ]({{\L } }_2^{\beta _2-1}-{{\L } }_1^{\beta _2-1}). \end{aligned}$$

Letting \({{\L } }_1\rightarrow {{\L } }_2\), then

$$\begin{aligned}{} & {} |G_1({\mathfrak {U}}, w)({{\L } }_2)- G_1({\mathfrak {U}}, w)({{\L } }_1)|\rightarrow 0, \quad |{\mathcal {D}}^{\alpha _1}G_1({\mathfrak {U}}, w)({{\L } }_2)-{\mathcal {D}}^{\alpha _1} G_1({\mathfrak {U}}, w)({{\L } }_1)|\rightarrow 0,\\{} & {} |I_{q_1}^{\delta _1}G_1({\mathfrak {U}}, w)({{\L } }_2)-I_{q_1}^{\delta _1} G_1({\mathfrak {U}}, w)({{\L } }_1)|\rightarrow 0,\\{} & {} |G_2({\mathfrak {U}}, w)({{\L } }_2)- G_2({\mathfrak {U}}, w)({{\L } }_1)|\rightarrow 0, \quad |{\mathcal {D}}^{\alpha _2}G_2({\mathfrak {U}}, w)({{\L } }_2)-{\mathcal {D}}^{\alpha _2} G_2({\mathfrak {U}}, w)({{\L } }_1)|\rightarrow 0,\\{} & {} |I_{q_2}^{\delta _2}G_2({\mathfrak {U}}, w)({{\L } }_2)- I_{q_2}^{\delta _2}G_2({\mathfrak {U}}, w)({{\L } }_1)|\rightarrow 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert G_1({\mathfrak {U}}, w)({{\L } }_2)- G_1({\mathfrak {U}}, w)({{\L } }_1)\Vert _{\mathcal {X}}\rightarrow 0,\quad \Vert G_2({\mathfrak {U}}, w)({{\L } }_2)- G_2({\mathfrak {U}}, w)({{\L } }_1)\Vert _{\mathcal {Y}}\rightarrow 0. \end{aligned}$$

That is, as \({{\L } }_1 \rightarrow {{\L } }_2\),

$$\begin{aligned} \Vert G({\mathfrak {U}}, w)({{\L } }_2)- G({\mathfrak {U}},w)({{\L } }_1)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}\rightarrow 0. \end{aligned}$$

As a result, we establish that the operator G is equicontinuous. Then, using the Arzela–Ascoli theorem, we conclude that G is a completely continuous operator. So, it follows from the Schauder fixed theorem that (1)–(2) possesses at least one solution \(({\mathfrak {U}}, w) \in Q_{\mathfrak {r}}\). This proof has been established. \(\square \)

Now, we demonstrate the uniqueness of solutions using the Banach contraction principle.

Theorem 3.2

Suppose that \({\mathcal {F}}_i\) are continuous for almost all \( {{\L } } \in (0,1)\) and measurable in t for any \( w_1, w_2,w_3,w_4,w_5,w_6 \in {\mathbb {R}}\). There exist nonnegative functions \(b_{i j}({{\L } }) \in L^1([0,1],\mathbb {R_+}),i = 1, 2, j=1,2,\ldots ,7,\) such that the following requirements are fulfilled:

\((H_1)\):

\(|{\mathcal {F}}_i({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)-{\mathcal {F}}_i({{\L } },z_1,z_2,z_3,z_4,z_5,z_6)|\le \sum _{j=1}^{6}b_{i j}({{\L } })|w_j-z_j|,i=1,2.\)

\((H_1)\):

\(3 D_1+3 D_2<1.\)

Following that, the coupled system (1)–(2) possesses a unique solution.

Proof

Let \(\sup _{{{\L } }\in [0,1]} {\mathcal {F}}_i({{\L } },0,0,0,0,0,0)=\mu _i<\infty ,i=1,2\) and take

$$\begin{aligned} R\ge \frac{\mu '_1+\mu '_2}{1-3D_1-3D_2}, \end{aligned}$$

where \(\mu _i'=|\rho _i\Lambda _i |+\frac{ |\rho _i\Lambda _i| \Gamma (\beta _i)}{\Gamma (\beta _i-\alpha _i)}+\frac{ |\rho _i\Lambda _i|}{\Gamma _{q_i}(\delta _i+1)}\) \(+\bigg [\frac{1}{\Gamma (\beta _i-\alpha _i+1)}+\frac{(1+|\Lambda _i|(1-q_i)\chi _i \sum _{\kappa =0}^{\nu _i} q_i^\kappa )}{\Gamma (\beta _i+1)}(1+\frac{1}{\Gamma _{q_i}(\delta _i+1)})\) \(+\frac{|\Lambda _i|(1-q_i) \chi _i \sum _{\kappa =0}^{\nu _i}q_i^\kappa }{\Gamma (\beta _i-\alpha _i) \Gamma (\beta _i+1)}\bigg ]\mu _i.\) First, we prove that \(G(Q_R)\subset Q_R\), where \( Q_R=\{({\mathfrak {U}},w)|({\mathfrak {U}},w)\in {\mathcal {X}} \times {\mathcal {Y}}:\Vert ({\mathfrak {U}},w)\Vert _{{\mathcal {X}}\times {\mathcal {Y}}}\le R\}\). For any \(({\mathfrak {U}},w)({{\L } }) \in Q_R\), we can obtain

$$\begin{aligned} |G_1({\mathfrak {U}},w)({{\L } })|\le&I^{\beta _1}\bigg [ \bigg |{\mathcal {F}}_1\bigg ({{\L } },{\mathfrak {U}}({{\L } }),w({{\L } }),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}({{\L } }),{\mathcal {D}}^{\alpha _2}w({{\L } }),I_{q_1}^{\delta _1}{\mathfrak {U}}({{\L } }),I_{q_2}^{\delta _2}w({{\L } })\bigg )-{\mathcal {F}}_1\bigg ({{\L } },0,0,0,0,0,0\bigg )\bigg |\\ {}&+|{\mathcal {F}}_1({{\L } },0,0,0,0,0,0)|\bigg ]+ |\rho _1\Lambda _1 |\\ {}&+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} \bigg |{\mathcal {F}}_1\bigg (q_1^\kappa \chi _1,{\mathfrak {U}}(q_1^\kappa \chi _1),w(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _1}{\mathfrak {U}}(q_1^\kappa \chi _1),{\mathcal {D}}^{\alpha _2}w(q_1^\kappa \chi _1),\\ {}&I_{q_1}^{\delta _1}{\mathfrak {U}}(q_1^\kappa \chi _1),I_{q_2}^{\delta _2}w(q_1^\kappa \chi _1)\bigg )-{\mathcal {F}}_1\bigg (q_1^\kappa \chi _1,0,0,0,0,0,0\bigg )\bigg |+|{\mathcal {F}}_1(q_1^\kappa \chi _1,0,0,0,0,0,0)|\bigg ] \end{aligned}$$
$$\begin{aligned}&\le I^{\beta _1} \bigg [(b_{11}({{\L } }) +b_{13}({{\L } })+b_{15}({{\L } }))\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+(b_{12}({{\L } }) +b_{14}({{\L } })+b_{16}({{\L } }))\Vert w\Vert _{\mathcal {Y}}+\mu _1\bigg ]\\ {}&+ |\rho _1\Lambda _1 |+|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa I^{\beta _1} \bigg [(b_{11}({{\L } }) +b_{13}({{\L } })+b_{15}({{\L } }))\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}\\ {}&+(b_{12}({{\L } }) +b_{14}({{\L } })+b_{16}({{\L } }))\Vert w\Vert _{\mathcal {Y}}+\mu _1\bigg ] \\ {}&\le ({\mathcal {A}}_{11}+{\mathcal {A}}_{13}+{\mathcal {A}}_{15})\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+( {\mathcal {A}}_{12}+ {\mathcal {A}}_{14}+ {\mathcal {A}}_{16})\Vert w\Vert _{\mathcal {Y}}+ |\rho _1\Lambda _1 |\\ {}&+\frac{(1+|\Lambda _1|(1-q_1)\chi _1 \sum _{\kappa =0}^{\nu _1} q_1^\kappa )\mu _1}{\Gamma (\beta _1+1)}. \end{aligned}$$

As an alternative, using (9), we may get

$$\begin{aligned} |{\mathcal {D}}^{\alpha _1}G_1({\mathfrak {U}},w)({{\L } })|\le&({\mathcal {B}}_{11}+{\mathcal {B}}_{13}+{\mathcal {B}}_{15})\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+({\mathcal {B}}_{12}+{\mathcal {B}}_{14}+{\mathcal {B}}_{16})\Vert w\Vert _{\mathcal {Y}}\\ {}&+\frac{\mu _1}{\Gamma (\beta _1-\alpha _1+1)}+\frac{|\rho _1\Lambda _1| \Gamma (\beta _1)}{\Gamma (\beta _1-\alpha _1)}+\frac{|\Lambda _1|(1-q_1) \chi _1 \sum _{\kappa =0}^{\nu _1}q_1^\kappa }{\Gamma (\beta _1-\alpha _1) \Gamma (\beta _1+1)}\mu _1. \end{aligned}$$

Also,

$$\begin{aligned} |I_{q_1}^{\delta _1}G_1({\mathfrak {U}},w)({{\L } })|&\le \frac{1}{\Gamma _{q_1}(\delta _1+1)}( {\mathcal {A}}_{11}+ {\mathcal {A}}_{13}+ {\mathcal {A}}_{15})\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+\frac{1}{\Gamma _{q_1}(\delta _1+1)}( {\mathcal {A}}_{12}+ {\mathcal {A}}_{14}+ {\mathcal {A}}_{16})\Vert w\Vert _{\mathcal {Y}}\\ {}&\quad +\frac{|\rho _1 \Lambda _1|}{\Gamma _{q_1}(\delta _1+1)}+\frac{(1+|\Lambda _1|(1-q_1)\chi _1 \sum _{\kappa =0}^{\nu _1} q_1^\kappa )\mu _1}{\Gamma (\beta _1+1)\Gamma _{q_1}(\delta _1+1)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert G_1({\mathfrak {U}},w)({{\L } })\Vert \le&(C_{11}+C_{13}+C_{15})\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+(C_{12}+C_{14}+C_{16})\Vert w\Vert _{\mathcal {Y}}+\mu _1'\\ {}&\le 3 D_1 R+\mu '_1. \end{aligned}$$

Similarly, we can get

$$\begin{aligned} \Vert G_2({\mathfrak {U}},w)({{\L } })\Vert \le&(C_{21}+C_{23}+C_{25})\Vert {\mathfrak {U}}\Vert _{\mathcal {X}}+(C_{22}+C_{24}+C_{26})\Vert w\Vert _{\mathcal {Y}}+\mu _2'\\ {}&\le 3 D_2 R+\mu '_2. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert G({\mathfrak {U}},w)\Vert _{X\times Y}=\Vert G_1({\mathfrak {U}},w)\Vert _{\mathcal {X}}+ \Vert G_2({\mathfrak {U}},w)\Vert _{\mathcal {Y}}\le&3(D_1+D_2)R+\mu '_1+\mu '_2\le R. \end{aligned}$$

Second, for any \(({\mathfrak {U}}_1,w_1)({{\L } })\), \((\mathfrak { U}_2,w_2)({{\L } })\in Q_R,\) we have

$$\begin{aligned} |G_1({\mathfrak {U}}_2,w_2)({{\L } })-G_1(\mathfrak { U}_1,w_1)({{\L } })| \le ( {\mathcal {A}}_{11}+ {\mathcal {A}}_{13}+ {\mathcal {A}}_{15})\Vert {\mathfrak {U}}_2-{\mathfrak {U}}_1\Vert _{\mathcal {X}}+( {\mathcal {A}}_{12}+ {\mathcal {A}}_{14}+ {\mathcal {A}}_{16})\Vert w_2-w_1\Vert _{\mathcal {Y}}, \end{aligned}$$

and

$$\begin{aligned} \Vert {\mathcal {D}}^{\alpha _1}G_1(\mathfrak { U}_2,w_2)({{\L } })-{\mathcal {D}}^{\alpha _1}G_1(\mathfrak { U}_1,w_1)({{\L } })\Vert \le&({\mathcal {B}}_{11}+{\mathcal {B}}_{13}+{\mathcal {B}}_{15})\Vert \mathfrak { U}_2-\mathfrak { U}_1\Vert _{\mathcal {X}}+({\mathcal {B}}_{12}+{\mathcal {B}}_{14}+{\mathcal {B}}_{16})\Vert w_2-w_1\Vert _{\mathcal {Y}}, \end{aligned}$$

also,

$$\begin{aligned} |I_{q_1}^{\delta _1}G_1(\mathfrak { U}_2,w_2)({{\L } })-I_{q_1}^{\delta _1}G_1(\mathfrak { U}_1,w_1)({{\L } })|&\le \frac{1}{\Gamma _{q_1}(\delta _1+1)}( {\mathcal {A}}_{11}+ {\mathcal {A}}_{13}+ {\mathcal {A}}_{15})\Vert \mathfrak { U}_2-\mathfrak { U}_1\Vert _{\mathcal {X}}\\ {}&+\frac{1}{\Gamma _{q_1}(\delta _1+1)}( {\mathcal {A}}_{12}+ {\mathcal {A}}_{14}+ {\mathcal {A}}_{16})\Vert w_2-w_1\Vert _{\mathcal {Y}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert G_1({\mathfrak {U}}_2,w_2)-G_1(\mathfrak { U}_1,w_1)\Vert&\le (C_{11}+C_{13}+C_{15})\Vert {\mathfrak {U}}_2-\mathfrak { U}_1\Vert _{\mathcal {X}}+(C_{12}+C_{14}+C_{16})\Vert w_2-w_1\Vert _{\mathcal {Y}}\\ {}&\le 3D_1\Vert {\mathfrak {U}}_2-{\mathfrak {U}}_1\Vert _{\mathcal {X}}+3 D_1\Vert w_2-w_1\Vert _{\mathcal {Y}}. \end{aligned}$$

Similarly, we can get

$$\begin{aligned} \Vert G_2({\mathfrak {U}}_2,w_2)-G_2({\mathfrak {U}}_1,w_1)\Vert \le 3D_2\Vert {\mathfrak {U}}_2-{\mathfrak {U}}_1\Vert _{\mathcal {X}}+3 D_2\Vert w_2-w_1\Vert _{\mathcal {Y}}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert G({\mathfrak {U}}_2,w_2)-G({\mathfrak {U}}_1,w_1)\Vert _{{\mathcal {X}}\times {\mathcal {Y}} }\le (3 D_1+3 D_2)\bigg [\Vert {\mathfrak {U}}_2-\mathfrak { U}_1\Vert _{\mathcal {X}}+\Vert w_2-w_1\Vert _{\mathcal {Y}}\bigg ]. \end{aligned}$$

Since \(3 D_1 + 3 D_2 < 1\), G is a contraction operator. This implies that G has a unique fixed point, and thus, (1)–(2) has a unique solution. \(\square \)

4 Applications

We now give the two examples below to demonstrate our findings.

Test problem 1: Consider the following coupled system:

$$\begin{aligned}&{\mathcal {D}}^{\frac{9}{4}} {\mathfrak {U}}({{\L } })=b_{11}({{\L } })({{\mathfrak {U}}({{\L } }))}^{\tau _{11}}+b_{12}({{\L } }){(w({{\L } }))}^{\tau _{12}}+b_{13}({{\L } }){({\mathcal {D}}^{\frac{1}{2}}{\mathfrak {U}}({{\L } }))}^{\tau _{13}}+b_{14}({{\L } }){({\mathcal {D}}^{\frac{1}{5}}w({{\L } }))}^{\tau _{14}}\\ {}&\qquad \qquad \quad +b_{15}({{\L } }){(I_{\frac{1}{7}}^{\frac{1}{4}}{\mathfrak {U}}({{\L } }))}^{\tau _{15}}+b_{16}({{\L } }){(I_{\frac{1}{3}}^{\frac{1}{2}}w({{\L } }))}^{\tau _{16}}+b_{17}({{\L } }), {{\L } }\in (0,1),\\ {}&{\mathcal {D}}^{\frac{5}{2}} w({{\L } })=b_{21}({{\L } })({{\mathfrak {U}}({{\L } }))}^{\tau _{21}}+b_{22}({{\L } }){(w({{\L } }))}^{\tau _{22}}+b_{23}({{\L } }){({\mathcal {D}}^{\frac{1}{2}}{\mathfrak {U}}({{\L } }))}^{\tau _{23}}+b_{24}({{\L } }){({\mathcal {D}}^{\frac{1}{5}}w({{\L } }))}^{\tau _{24}}\\ {}&\qquad \qquad \quad +b_{25}({{\L } }){(I_{\frac{1}{7}}^{\frac{1}{4}}{\mathfrak {U}}({{\L } }))}^{\tau _{25}}+b_{26}({{\L } }){(I_{\frac{1}{3}}^{\frac{1}{2}}w({{\L } }))}^{\tau _{26}}+b_{27}({{\L } }), {{\L } }\in (0,1),\\ {}&(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)=\rho _1, \quad {\mathfrak {U}}(0)={\mathfrak {U}}'(0)=0, \quad \chi _1\in (0,1),\\ {}&(1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2} q_2^\kappa w(q_2^\kappa \chi _2)=\rho _2, \quad w(0)=w'(0)=0, \quad \chi _2\in (0,1), \end{aligned}$$

where \( {\beta _1=\frac{9}{4},} \quad {\beta _2=\frac{5}{2},} \quad \alpha _1=\frac{1}{2},\quad \alpha _2=\frac{1}{5},\quad q_1=\frac{1}{7},\quad q_2=\frac{1}{3},\quad \delta _1=\frac{1}{4},\quad \delta _2=\frac{1}{2},\quad \chi _1=0.5, \quad \chi _2=0.4,\quad \nu _1=1,\quad \nu _2= 2, \quad 0<\tau _{i j}<1(j=1,2,\ldots ,6,\quad i=1,2)\) and \(b_{i j}({{\L } }) ( j=1,2,\ldots ,7,\quad i=1,2)\) are nonnegative functions. The test problem 1 must have at least one solution, according to Theorem ().

Test problem 2: Consider the following coupled system:

$$\begin{aligned}&{\mathcal {D}}^{\frac{5}{2}} {\mathfrak {U}}({{\L } })=\frac{{{\L } }^5}{30}{\mathfrak {U}}({{\L } })+\frac{{{\L } }^4}{60}w({{\L } })+\frac{{{\L } }^4}{40}{\mathcal {D}}^{\frac{1}{2}}{\mathfrak {U}}({{\L } })+\frac{{{\L } }^6}{25} {\mathcal {D}}^{\frac{1}{5}}w({{\L } })\\ {}&\qquad \quad \quad \quad +\frac{{{\L } }^7}{45}I_{0.2}^\frac{1}{4}{\mathfrak {U}}({{\L } })+\frac{{{\L } }^5}{50}I_{0.5}^{\frac{1}{3}}w({{\L } })+\frac{1}{50} (1-{{\L } })^5, {{\L } }\in (0,1),\\ {}&{\mathcal {D}}^{\frac{7}{3}} w({{\L } })=\frac{(1-{{\L } })^3}{150}{\mathfrak {U}}({{\L } })+\frac{(1-{{\L } })^5}{100}w({{\L } })+\frac{{{\L } }^7}{100}{\mathcal {D}}^{\frac{1}{2}}{\mathfrak {U}}({{\L } })+\frac{{{\L } }^5}{80} {\mathcal {D}}^{\frac{1}{5}}w({{\L } })\\ {}&\qquad \quad \quad \quad +\frac{{{\L } }^8}{50}I_{0.2}^{\frac{1}{4}}{\mathfrak {U}}({{\L } })+\frac{{{\L } }^8}{100}I_{0.5}^{\frac{1}{3}}w({{\L } })+\frac{1}{100} (1-{{\L } })^8,{{\L } }\in (0,1),\\ {}&(1-q_1) \chi _1\sum _{\kappa =0}^{\nu _1} q_1^\kappa {\mathfrak {U}}(q_1^\kappa \chi _1)=\rho _1, \quad {\mathfrak {U}}(0) ={\mathfrak {U}}'(0)=0, \quad \chi _1\in (0,1),\\ {}&(1-q_2) \chi _2\sum _{\kappa =0}^{\nu _2} q_2^\kappa w(q_2^\kappa \chi _2)=\rho _2, \quad w(0)= w'(0)=0, \quad \chi _2\in (0,1), \end{aligned}$$

where \( {\beta _1=\frac{5}{2},} \quad {\beta _2=\frac{7}{3},} \quad \alpha _1=\frac{1}{2},\quad \alpha _2=\frac{1}{5},\quad q_1=0.2,\quad q_2=0.5,\quad \delta _1=\frac{1}{4},\quad \delta _2=\frac{1}{3},\) \( \chi _1=0.5, \quad \chi _2=0.4,\quad \nu _1=\nu _2=4. \)

Then, we have

$$\begin{aligned} |{\mathcal {F}}_1({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)|=&\frac{{{\L } }^5}{30}|w_1|+\frac{{{\L } }^4}{60}|w_2|+\frac{{{\L } }^4}{40}|w_3|+\frac{{{\L } }^6}{25} |w_4|\\ {}&+\frac{{{\L } }^7}{45}|w_5|+\frac{{{\L } }^5}{50}|w_6|+\frac{1}{50} (1-{{\L } })^5, \end{aligned}$$
$$\begin{aligned} |{\mathcal {F}}_2({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)|\le&\frac{(1-{{\L } })^3}{150}|w_1|+\frac{(1-{{\L } })^5}{100}|w_2|+\frac{{{\L } }^7}{100}|w_3|+\frac{{{\L } }^5}{80} |w_4| \\ {}&+\frac{{{\L } }^8}{50}|w_5|+\frac{{{\L } }^8}{100}|w_6|+\frac{1}{100} (1-{{\L } })^8, \end{aligned}$$

and

$$\begin{aligned}&|{\mathcal {F}}_1({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)-{\mathcal {F}}_1({{\L } },z_1,z_2,z_3,z_4,z_5,z_6)|\le \frac{{{\L } }^5}{30}|w_1-z_1|+\frac{{{\L } }^4}{60}|w_2-z_2|+\frac{{{\L } }^4}{40}|w_3-z_3|\\ {}&+\frac{{{\L } }^6}{25} |w_4-z_4| +\frac{{{\L } }^7}{45}|w_5-z_5|+\frac{{{\L } }^5}{50} |w_6-z_6|+\frac{1}{50}(1-{{\L } })^5,\\ {}&|{\mathcal {F}}_2({{\L } },w_1,w_2,w_3,w_4,w_5,w_6)-{\mathcal {F}}_2({{\L } },z_1,z_2,z_3,z_4,z_5,z_6)|\le \frac{(1-{{\L } })^3}{150}|w_1-z_1|+\frac{(1-{{\L } })^5}{100}|w_2-z_2|\\ {}&+\frac{{{\L } }^7}{100}|w_3-z_3|+\frac{{{\L } }^5}{80} |w_4-z_4| +\frac{{{\L } }^8}{50}|w_5-z_5|+\frac{{{\L } }^8}{100}|w_6-z_6|+\frac{1}{100} (1-{{\L } })^8, \end{aligned}$$

where

\( b_{11}=\frac{{{\L } }^5}{30},\quad b_{12}=\frac{{{\L } }^4}{60},\quad b_{13}=\frac{{{\L } }^4}{40}, \quad b_{14}=\frac{{{\L } }^6}{25}, \quad b_{15}=\frac{{{\L } }^7}{45},\quad b_{16}=\frac{{{\L } }^5}{50},\quad b_{17}=\frac{(1-{{\L } })^5}{50},\quad b_{21}=\frac{(1-{{\L } })^3}{150},\quad b_{22}=\frac{(1-{{\L } })^5}{100},\quad b_{23}=\frac{{{\L } }^7}{100},\quad b_{24}=\frac{{{\L } }^5}{80},\quad b_{25}=\frac{{{\L } }^8}{50},\quad b_{26}=\frac{{{\L } }^8}{50},\quad b_{27}=\frac{(1-{{\L } })^8}{100}.\) By direct calculation, we get

\( {\mathcal {A}}_{11}=0.0095576,\quad {\mathcal {A}}_{12}=0.0062124,\quad {\mathcal {A}}_{13}=0.0093187,\quad {\mathcal {A}}_{14}=0.0091753, \quad {\mathcal {A}}_{15}=0.00419785, \) \( {\mathcal {A}}_{16}=0.0057346,\quad {\mathcal {A}}_{17}=0.0155236,\quad {\mathcal {A}}_{21}=0.0108042,\quad {\mathcal {A}}_{22}=0.0110885,\quad {\mathcal {A}}_{23}=0.00381566,\) \( {\mathcal {A}}_{24}=0.0069399,\quad {\mathcal {A}}_{25}=0.0065411,\quad {\mathcal {A}}_{26}=0.0032706,\quad {\mathcal {A}}_{27}=0.0075244,\quad {\mathcal {B}}_{11}=0.0154193,\) \( {\mathcal {B}}_{12}=0.00974474,\quad {\mathcal {B}}_{13}=0.0146171,\quad {\mathcal {B}}_{14}=0.0151834,\quad B_{15}=0.00711007,\quad {\mathcal {B}}_{16}=0.00925156, \) \({\mathcal {B}}_{17}=0.01935415,\quad {\mathcal {B}}_{21}=0.0118813,\quad {\mathcal {B}}_{22}=0.0121674,\quad {\mathcal {B}}_{23}=0.00452789,\quad {\mathcal {B}}_{24}=0.00814884,\) \( {\mathcal {B}}_{25}=0.00779734,\quad {\mathcal {B}}_{26}=0.0038987,\quad {\mathcal {B}}_{27}=0.008244,\quad C_{11}=0.0195134,\quad C_{12}=0.0126837,\) \( C_{13}=0.0190256,\quad C_{14}=0.0339163,\quad C_{15}=0.0156807,\quad C_{16}=0.0209596,\quad C_{17}=0.0510482,\) \( C_{21}=0.0343762,\)

\( C_{22}=0.0352543,\quad C_{23}=0.0124723,\quad C_{24}=0.022598,\quad C_{25}=0.0214163,\quad \) \(C_{26}=0.0107081,\quad C_{27}=0.0239103.\) Thus, \(D_1=0.051048,\quad D_2=0.03525\). Therefore, \(3D_1+3D_2=0.258907<1.\) The coupled system (1)–(2) must have at least one solution, according to Theorem ().

5 Conclusion

In the present work, we have discussed the existence of solutions for a coupled system of Riemann–Liouville fractional q-integro-differential equations. We have established the conditions under which these solutions exist. Furthermore, we have demonstrated the uniqueness of the solution by utilizing the contraction principle. By employing the powerful tools of fixed-point theorems, specifically the Schauder fixed theorem and the Banach contraction principle, we have provided a rigorous analysis of the coupled system. These theorems have allowed us to establish the existence and uniqueness of solutions, showcasing their effectiveness in addressing complex mathematical problems. Additionally, we have presented two illustrative examples that highlight the practical applications of our results. These examples serve to demonstrate how the findings of our work can be applied to real-world scenarios, further emphasizing the relevance and significance of our research. In conclusion, this work contributes to the understanding of coupled systems of Riemann–Liouville fractional q-integro-differential equations by discussing the existence and uniqueness of solutions. Our findings provide a solid foundation for future research in this area and offer practical insights for solving similar problems in various fields of study.