An increasing variables singular system of fractional q-differential equations via numerical calculations

Abstract

We investigate the existence of solutions for an increasing variables singular m-dimensional system of fractional q-differential equations on a time scale. In this singular system, the first equation has two variables and the number of variables increases permanently. By using some fixed point results, we study the singular system under some different conditions. Also, we provide two examples involving practical algorithms, numerical tables, and some figures to illustrate our main results.

1 Introduction

The subject of q-difference equations was introduced by Jackson in the first decade of the last century [1]. The fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. It is known that working on quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. In last decades, some researchers studied q-fractional difference equations [2–5]. Later, q-fractional boundary value problems have been considered by many researchers (see, for example, [6–13]). Nowadays many researchers focus on applications of fractional calculus [14–25] or analytical studies [26–36].

In 2013, Baleanu et al. investigated the coupled system of multi-term singular fractional integro-differential boundary value problem

$$ \textstyle\begin{cases} \mathcal{D}^{\sigma _{1}}_{0^{+}} [k](t)+ w_{1} ( t, k(t), l(t), \psi _{11} [k](t), \psi _{21} [l](t), \\ \quad \mathcal{D}^{\alpha _{1}}_{0^{+}} [k](t), \mathcal{D}^{\beta _{11}}_{0^{+}} [l](t), \mathcal{D}^{\beta _{12}}_{0^{+}} [l](t), \ldots , \mathcal{D}^{\beta _{1m}}_{0^{+}} [l](t) )=0, \\ \mathcal{D}^{\sigma _{2}}_{0^{+}} [l](t)+ w_{2} ( t, k(t), l(t), \psi _{12} [k](t), \psi _{22} [l](t), \\ \quad \mathcal{D}^{\alpha _{2}}_{0^{+}} [l](t), \mathcal{D}^{\beta _{21}}_{0^{+}} [k](t), \mathcal{D}^{\beta _{22}}_{0^{+}} [k](t), \ldots , \mathcal{D}^{\beta _{2m}}_{0^{+}} [k](t) )=0, \end{cases} $$

via boundary conditions $k^{(i)} (0) =l^{(i)} (0)= 0$ for $0 \leq i \leq n-2$, $\mathcal{D}^{\delta _{1}}_{0^{+}} [k](1)=0$ for $2 < \delta _{1} < n-1$, $\sigma _{1} - \delta _{1} \geq 1$, and $\mathcal{D}^{\delta _{2}}_{0^{+}} [l](1)=0$ for $2 < \delta _{2} < n-1$, $\sigma _{2} - \delta _{2} \geq 1$, where $n \geq 4$, $n-1 < \sigma _{i}< n$, $0 < \alpha _{i}< 1$, $1< \beta _{ij} < 2$ for $i=1,2$ and $j = 1, 2, \ldots , m$, $\gamma _{ij}$ is positive-valued continuous functions on $[0,1]\times [0,1]$ ($i, j=1,2$), $\psi {ij}[k](t)= \int _{0}^{t} \gamma _{ij} (t, r)k(r) \,\mathrm{d}r$, $w_{1}$, $w_{2}$ satisfy the local Caratheodory condition on $[0,1]\times D(w_{1}, w_{2} \in \operatorname{Car} ([0,1] \times D))$, where $D \subset \mathbb{R}^{m+5}$ and $w_{i}$ may be singular at the value zero of all its variables [37]. In 2016, Taieb et al. reviewed the fractional coupled system of nonlinear differential equations

$$ \textstyle\begin{cases} \mathcal{D}^{\sigma _{1}} [k](t) + \sum_{i=1}^{m} w_{1i} ( t, k(t), l(t), \mathcal{D}^{\beta _{1}} [k](t), \mathcal{D}^{\beta _{2}} [l](t) )=0, \\ \mathcal{D}^{\sigma _{2}} [l](t) + \sum_{i=1}^{m} w_{2i} ( t, k(t), l(t), \mathcal{D}^{\beta _{1}} [k](t), \mathcal{D}^{\beta _{2}} [l](t) )=0, \end{cases} $$

with boundary conditions $k(0)=k_{0}^{*}$, $l(0)=l_{0}^{*}$, $k'(0)= k''(0)=l'(0)=l''(0)=0$, $k'''(0) = \mathcal{J}^{\alpha _{1}}[k](a_{1})$, and $l'''(0) = \mathcal{J}^{\alpha _{2}}[k](a_{2})$, where $t \in [0,1]$, $m \in \mathbb{N}^{*}$, $\alpha _{j} >0$, $\sigma _{j} \in (3, 4)$, $a_{j} \in (0,1)$, $\mathcal{D}^{\sigma _{j}}$, $\mathcal{D}^{\beta _{j}}$ are the Caputo derivatives and $\mathcal{J}^{\alpha _{j}}$ are the Riemann–Liouville fractional integrals [38]. In 2017, El Abidine studied the coupled system of nonlinear fractional equations

$$ \textstyle\begin{cases} \mathcal{D}^{\sigma _{1}} [k](t) = w_{1i} ( t, l(t), \mathcal{D}^{\beta _{1}} [l](t) ), \\ \mathcal{D}^{\sigma _{2}} [l](t) = w_{2i} ( t, k(t), \mathcal{D}^{\beta _{2}} [k](t) ), \end{cases} $$

with boundary conditions $k(0) = k^{(j)}(0)=0$ and $l(0) = l^{(j)}( 0)=0$ for $1 \leq j\leq m-2$ with $m \geq 2$, where $t \in \mathbb{R}^{+} = (0, \infty )$, $m-1 < \sigma _{i} \leq m$, $\beta _{i}\in (0,3)$ for $i=1,2$, $0 < \beta _{1}\leq \sigma _{2}-1$, $0 < \beta _{2} \leq \sigma _{1}-1$, the differential operator is in the Riemann–Liouville sense and $w_{i}$ are Borel measurable functions in ${\mathbb{R}^{+}}^{3}$ satisfying some conditions [39].

By using the main idea of the above works, we investigate the increasing variables m-dimensional singular system of fractional q-differential equations

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{\sigma _{1}} [k_{1}](t) = w_{1} (t, k_{1}(t) ), \\ {}^{c}\mathcal{D}_{q}^{\sigma _{2}} [k_{2}](t) = w_{2} (t, k_{1}(t), k_{2}(t) ), \\ \quad \vdots \\ {}^{c}\mathcal{D}_{q}^{\sigma _{m}} [k_{m}](t)=w_{m} (t, k_{1}(t), k_{2}(t), \ldots , k_{m}(t) ), \end{cases} $$

(1)

with boundary conditions $k_{1} (0) = {}_{1}b_{0}$, $k_{i}^{(j)} (0) = {}_{i}b_{j}$ for $j=0,1, \ldots i-2$ and $2\leq i \leq m$, ${}^{c}\mathcal{D}_{q}^{\zeta _{i-1}} k_{i} (1) = 0$ for $\zeta _{i-1} \in [i-2, i-1]$ and $2\leq i \leq m$, where $t\in J:=(0,1]$, $m \geq 2$, $\sigma _{i} \in (i-1, i)$ for $1\leq i\leq m$, ${}^{c}\mathcal{D}_{q}^{\sigma _{i}}$ denotes the Caputo fractional q-derivative of order $\sigma _{i}$, $w_{i} : J \times \mathbb{R}^{i} \to \mathbb{R}$ are continuous, $w_{i}(t, k_{1}, k_{2},\ldots , k_{i})$ may be singular at $t =0$ of its space variables, $\lim_{t \to 0^{+}} w_{i} (t, k_{1}, k_{2},\ldots , k_{i})= \infty$, and there exists $0 < \alpha _{1},\dots , \alpha _{m} <1$ such that $t^{\alpha _{1}} w_{1},\dots ,t^{\alpha _{m}} w_{m}$ are continuous on $\overline{J} :=[0,1]$.

2 Essential preliminaries

Throughout this article, we apply the time scales calculus notation [40]. In fact, we consider the fractional q-calculus on the time scale $\mathbb{T}_{t_{0}} = \{0 \} \cup \{ t: t=t_{0}q^{n} \}$, where $n\geq 0$, $t_{0} \in \mathbb{R}$, and $q \in (0,1)$. Let $a \in \mathbb{R}$. Define $[a]_{q}=\frac{1-q^{a}}{1-q}$ [1]. The power function $(x-y)_{q}^{(n)}$ with $n \in \mathbb{N}_{0}$ is defined by $(x-y)_{q}^{(n)}= \prod_{k=0}^{n-1} (x - yq^{k})$ for $n \geq 1$ and $(x-y)_{q}^{(0)}=1$, where x and y are real numbers and $\mathbb{N}_{0} := \{ 0\} \cup \mathbb{N}$ [1, 2]. Also, $(x-y)_{q}^{(\alpha )}= x^{\alpha }\prod_{k=0}^{\infty }(x-yq^{k}) / (x - yq^{\alpha + k})$ for $\alpha \in \mathbb{R}$ and $q \neq 0$. If $y=0$, then it is clear that $x^{(\alpha )}= x^{\alpha }$ [6] (see Algorithm 1). The q-gamma function is given by $\varGamma _{q}(z) = (1-q)^{(z-1)} / (1-q)^{z -1}$, where $z \in \mathbb{R} \backslash \{0, -1, -2, \ldots \}$ [1]. Note that $\varGamma _{q} (z+1) = [z]_{q} \varGamma _{q} (z)$. Algorithm 2 shows a pseudo-code description of the technique for estimating q-gamma function of order n. The q-derivative of function f is defined by $(\mathcal{D}_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}$ and $(\mathcal{D}_{q} f)(0) = \lim_{x \to 0} (\mathcal{D}_{q} f)(x)$, which is shown in Algorithm 3 [2, 3]. Furthermore, the higher order q-derivative of a function f is defined by $(D_{q}^{n} f)(x) = D_{q}(D_{q}^{n-1} f)(x)$ for $n \geq 1$, where $(D_{q}^{0} f)(x) = f(x)$ [2, 3]. The q-integral of a function f is defined on $[0,b]$ by $I_{q} f(x) = \int _{0}^{x} f(s) \,\mathrm{d}_{q} s = x(1- q) \sum_{k=0}^{\infty } q^{k} f(x q^{k})$ for $0 \leq x \leq b$, provided the series absolutely converges [2, 3]. If x in $[0, T]$, then

$$ \int _{x}^{T} f(r) \,\mathrm{d}_{q} r = I_{q} f(T) - I_{q} f(x) = (1- q) \sum _{k=0}^{\infty } q^{k} \bigl[ T f\bigl(T q^{k}\bigr) - x f\bigl(x q^{k}\bigr) \bigr], $$

whenever the series exists. In addition, we can interchange the order of double q-integral by $\int _{0}^{t} \int _{0}^{s} h(r) \,\mathrm{d}_{q} r \,\mathrm{d}_{q} s= \int _{0}^{t} \int _{qr}^{t} h(r) \,\mathrm{d}_{q} s \,\mathrm{d}_{q} r$ [41]. Actually, the interchange of order is true since

$$ \begin{aligned} \int _{0}^{t} \int _{qr}^{t} \,\mathrm{d}_{q} s \, \mathrm{d}_{q} r & = \int _{0}^{t} (t- qr)^{(\sigma - 1)} h(r) \, \mathrm{d}_{q}r \\ & = t(1-q) \sum_{i=0}^{\infty } q^{i} h\bigl(q^{i}t\bigr) \bigl( t - q^{i+1}t \bigr) \\ & = t^{2} (1-q)^{2} \sum_{i=0}^{\infty } q^{i} h\bigl(q^{i} t\bigr) \Biggl( \sum _{i=0}^{\infty } q^{i} \Biggr). \end{aligned} $$

In addition the left-hand side can be written as follows:

$$ \begin{aligned}[b] \int _{0}^{t} \int _{0}^{r} h(s) \,\mathrm{d}_{q} s \,\mathrm{d}_{q} r & = t(1-q) \sum_{i=0}^{\infty } q^{i} \int _{0}^{tq^{i}} h(r) \,\mathrm{d}_{q}r \\ & = t^{2} (1-q)^{2} \sum_{i=0}^{\infty } \sum_{j=0}^{\infty } q^{i+2j} h \bigl(q^{i+j} t\bigr). \end{aligned} $$

(2)

The operator $I_{q}^{n}$ is given by $(I_{q}^{0} h)(x) = h(x)$ and $(I_{q}^{n} h)(x) = (I_{q} (I_{q}^{n-1} h)) (x)$ for all $n \geq 1$ and $h \in C([0,T])$ [2, 3]. It has been proved that $(D_{q} (I_{q} h))(x) = h(x)$ and $(I_{q} (D_{q} h))(x) = h(x) - h(0)$ whenever h is continuous at $x =0$ [2, 3]. The fractional Riemann–Liouville type q-integral of the function h on $J=(0,1)$ for $\sigma \geq 0$ is defined by $\mathcal{I}_{q}^{0} [h](t) = h(t)$ and

$$\begin{aligned} \mathcal{I}_{q}^{\sigma }[h](t) & = \frac{1}{\varGamma _{q}(\sigma )} \int _{0}^{t} (t- qr)^{(\sigma - 1)} h(r) \, \mathrm{d}_{q}r \\ & = t^{\sigma }(1-q)^{\sigma }\sum_{k=0}^{ \infty } q^{k} \frac{ \prod_{i=1}^{k - 1} (1-q^{\sigma +i } ) }{ \prod_{i=1}^{k - 1} (1 - q^{i +1} ) } h\bigl(t q^{k}\bigr) \end{aligned}$$

for $t \in J$ [42]. Also, the Caputo fractional q-derivative of a function h is defined by

$$ \begin{aligned} {}^{c}\mathcal{D}_{q}^{\sigma }[h](t) & = \mathcal{I}_{q}^{[ \sigma ]-\sigma }\bigl[ \mathcal{D}_{q}^{[\sigma ]} [h]\bigr] (t) \\ & = \frac{1}{\varGamma _{q} ([\sigma ] -\sigma )} \int _{0}^{t} (t- qr)^{ ( [\sigma ]-\sigma -1 )} \mathcal{D}_{q}^{[ \sigma ]} [h] (r) \,\mathrm{d}_{q}r, \end{aligned} $$

where $t \in J$ and $\sigma >0$ [42]. It has been proved that $\mathcal{I}_{q}^{\beta }(\mathcal{I}_{q}^{\alpha } [h]) (x) = \mathcal{I}_{q}^{\alpha + \beta } [h] (x)$ and $\mathcal{D}_{q}^{\alpha } [\mathcal{I}_{q}^{\alpha } [h]](x)= h(x)$, where $\alpha , \beta \geq 0$ [42]. Algorithm 5 shows MATLAB lines for $\mathcal{I}_{q}^{\alpha }[h](x)$.

Algorithm 1

The proposed method for calculated $(a-b)_{q}^{(\alpha )}$

Algorithm 2

The proposed method for calculated $\varGamma _{q}(x)$

Algorithm 3

The proposed method for calculated $(D_{q} f)(x)$

Let $(\mathcal{E}, \rho )$ be a metric space. Denote by $\mathcal{P}( \mathcal{E})$ and $2^{\mathcal{E}}$ the class of all subsets and the class of all nonempty subsets of $\mathcal{E}$, respectively. Thus, $\mathcal{P}_{cl}( \mathcal{E})$, $\mathcal{P}_{bd}( \mathcal{E})$, $\mathcal{P}_{cv}( \mathcal{E})$, and $\mathcal{P}_{cp}( \mathcal{E})$ denote the class of all closed, bounded, convex, and compact subsets of $\mathcal{E}$, respectively. For each i, consider the space $E_{i} = \{ k_{i}(t) : k_{i}(t)\in \mathcal{A} \}$ endowed with the norm $\|k_{i}\|_{\infty }= \max_{t\in \overline{J}} |k_{i}(t)|$, where $\mathcal{A}= C(\overline{J}, \mathbb{R})$. Also, define the product space $\mathcal{E} = E_{1} \times \cdots \times E_{m}$ endowed with the norm $\|( k_{1}, \ldots , k_{m})\| = \max_{1 \leq i \leq m } \|k_{i}\|_{\infty }$. Then $(\mathcal{E}, \|. \|)$ is a Banach space [43]. Similar to the idea of the works [44, 45], define the set of the selections of $\mathcal{S}$ at k by

$$\begin{aligned} S & = \bigl\{ k=(k_{1}, k_{2}, \ldots , k_{m}) : k_{i} \in \mathcal{A} , i=1,2,\ldots , m \bigr\} \end{aligned}$$

for all $t\in \overline{J}$ and $k=(k_{1}, \dots ,k_{m}) \in \mathcal{E}$. One can check that $S \neq \emptyset$ for all $k\in \mathcal{E}$ whenever $\dim \mathcal{E} < \infty$ [46]. We need next results.

Lemma 1

([47, 48])

The general solution of the q-fractional equation ${}^{c}\mathcal{D}_{q}^{\sigma } [k](t) =0$is given by $k(t) = d_{0} + d_{1} t + d_{2} t^{2} + \cdots + d_{m-1} t^{m-1}$for $\sigma >0$, where $d_{i} \in \mathbb{R}$for $i=0, 1, \ldots , m-1$and $m = [\sigma ] + 1$.

Theorem 2

([43], Schauder’s fixed point)

Assume that $(\mathcal{E}, \rho )$is a complete metric space, S is a closed convex subset of $\mathcal{E}$, and $\mathcal{N}: \mathcal{E} \to \mathcal{E}$is a map such that the set $K=\{ \mathcal{N}(k) : k \in S\}$is relatively compact in $\mathcal{E}$. Then $\mathcal{N}$has at least one fixed point.

3 Main results

Now, we are ready to provide our results about the m-dimensional system of singular fractional q-differential equations. First, we prove next basic result to give the integral representation of problem (1).

Lemma 3

Let $m \geq 2$for $i \in \{ 1,2, \ldots , m\}$, $\sigma _{i}\in (i-1, i)$, $\varrho _{1}, \dots , \varrho _{m}\in \mathcal{A}$, and $t\in J$. Then the m-dimensional system

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{\sigma _{1}} [k_{1}](t) = \varrho _{1} (t), \\ {}^{c}\mathcal{D}_{q}^{\sigma _{2}} [k_{2}](t) = \varrho _{2} (t), \\ \quad \vdots \\ {}^{c}\mathcal{D}_{q}^{\sigma _{m}} [k_{m}](t)=\varrho _{m} (t), \end{cases} $$

(3)

under the conditions

$$ \textstyle\begin{cases} k_{1} (0) = {}_{1}b_{0}, & \\ k_{i}^{(j)} (0) = {}_{i}b_{j}, & j=0,1, \ldots i-2, \\ {}^{c}\mathcal{D}_{q}^{\zeta _{i-1}} k_{i} (1) = 0, & i-2 < \zeta _{i-1} < i-1 (2\leq i \leq m), \end{cases} $$

(4)

has a unique solution $k=(k_{1}, k_{2}, \ldots , k_{m})$, where

$$ k_{i}(t) = \textstyle\begin{cases} \mathcal{I}_{q}^{\sigma _{i}}[\varrho _{i}](t) + {}_{1}b_{0}, & i=1, \\ \mathcal{I}_{q}^{\sigma _{i}}[\varrho _{i}](t) + \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} &\\ \quad {}- \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)!} t^{i-1} \mathcal{I}_{q}^{\sigma _{i}-\zeta _{i-1}}[\varrho _{i}](1), & 2 \leq i \leq m. \end{cases} $$

(5)

Proof

By using Lemma 1, we obtain the fractional q-integral equation

$$ k_{i}(t) = \mathcal{I}_{q}^{\sigma _{i}}[ \varrho _{i}](t) - \sum_{j=0}^{i-1} {}_{i}d_{j} t^{j} $$

(6)

for $1 \leq i \leq m$. Let

$$ D= \begin{pmatrix} {}_{1}d_{0}& 0 & 0 & 0 & \cdots & 0 &0 \\ {}_{2}d_{0}& {}_{2}d_{1} & 0 &0 & \cdots & 0& 0 \\ {}_{3}d_{0}& {}_{3}d_{1} & {}_{3}d_{2} & 0 & \cdots &0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & 0 & 0 \\ {}_{m-1}d_{0}& {}_{m-1}d_{1} & {}_{m-1}d_{2} & {}_{m-1}d_{3} & \cdots &{}_{m-1}d_{m-2} & 0 \\ {}_{m}d_{0}& {}_{m}d_{1} & {}_{m}d_{2} & {}_{m}d_{3} & \cdots & {}_{m}d_{m-2} & {}_{m}d_{m-1} \end{pmatrix} \in M_{m} (\mathbb{R}). $$

By using the assumptions, we find $k_{1} (0) = -{}_{1}d_{0} = {}_{1}b_{0}$, $k_{i}^{(j)}(0) = - j! {}_{i}d_{j}= {}_{i}b_{j}$ for $j=0, 1, 2, \ldots , i-2$ and

$$ {}^{c}\mathcal{D}_{q}^{\zeta _{i-1}} [k_{i}] (1) = \mathcal{I}_{q}^{\sigma _{i} - \zeta _{i-1} } [\varrho _{i}](1) - \frac{\varGamma _{q}(i)}{\varGamma _{q}(i - \zeta _{i-1} ) } {}_{i}d_{i-1}=0 $$

for $i-2 < \zeta _{i-1} < i-1$, where $2 \leq i \leq m$. Thus, ${}_{1}d_{0}= - {}_{1}b_{0}$ and

$$ {}_{i}d_{j} = \textstyle\begin{cases} -\frac{{}_{i}b_{j}}{j!}, & j=0, 1, \ldots , i-2, \\ \frac{\varGamma _{q}(i - \zeta _{i-1}) }{\varGamma _{q}(i)} \mathcal{I}_{q}^{\sigma _{i} - \zeta _{i-1} } [\varrho _{i}](1), & j=i-1, \end{cases} $$

(7)

for $2 \leq i \leq m$. By substituting these constants and (7) in (6), we find (5). □

Now, define the nonlinear operator $\mathcal{N}: S \to S$ by

$$ \mathcal{N}[k_{1}, k_{2}, \ldots , k_{m}](t) = \begin{pmatrix} N_{1}(k_{1})(t) \\ N_{2}(k_{1}, k_{2})(t) \\ N_{3}(k_{1}, k_{2}, k_{3})(t) \\ \vdots \\ N_{m}(k_{1}, k_{2}, \ldots , k_{m})(t) \end{pmatrix}, $$

(8)

where

$$ N_{i}(k_{1}, k_{2}, \ldots , k_{i}) (t) = \textstyle\begin{cases} \mathcal{I}_{q}^{\sigma _{i}}[w_{i}](t, k_{1}(t)) + {}_{1}b_{0}, & i=1, \\ \mathcal{I}_{q}^{\sigma _{i}}[w_{i}](t, k_{1}(t), \ldots , k_{i}(t)) &\\ \quad {}+ [ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} ]- \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)!} t^{i-1} & \\ \quad {}\times \mathcal{I}_{q}^{ \sigma _{i} - \zeta _{i-1}}[ w_{i}] (1, k_{1}(1), \ldots , k_{i}(1)),& 2 \leq i \leq m, \end{cases} $$

for $t \in \overline{J}$.

Lemma 4

Let $m \geq 2$, $\sigma _{1} \in (0, 1)$, $\sigma _{1} > \alpha _{1}$, $\sigma _{i} \in (i-1, i)$for $i=2, \ldots , m$, $\alpha _{i} \in (0,1)$for $i=1,2, \ldots , m$, $f_{i} : J \to \mathbb{R}$be a function with $\lim_{t \to 0^{+}} f_{i}(t)= \infty$, and the maps $t^{\alpha _{i}} f_{i} (t)$be continuous on J̅. Then the maps

$$ k_{i}(t) = \textstyle\begin{cases} \mathcal{I}_{q}^{\sigma _{i}}[f_{i}](t) + {}_{i}b_{0}, & i=1, \\ \mathcal{I}_{q}^{\sigma _{i}}[f_{i}](t) + \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} & \\ \quad {}- \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)!} t^{i-1} \mathcal{I}_{q}^{\sigma _{i}-\zeta _{i-1}}[f_{i}](1)& 2\leq i \leq m, \end{cases} $$

are continuous on J̅.

Proof

By using the definition of the maps $k_{i}(t)$, we have

$$\begin{aligned} k_{i}(t) & = \textstyle\begin{cases} \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} f_{i}(r) \,\mathrm{d}_{q}r + {}_{i}b_{0}, & i=1, \\ \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} f_{i}(r) \,\mathrm{d}_{q}r+ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} & \\ \quad {}- \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} t^{i-1} & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} f_{i}(r) \,\mathrm{d}_{q}r,& 2 \leq i\leq m, \end{cases}\displaystyle \\ & = \textstyle\begin{cases} \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r + {}_{i}b_{0}, & i=1, \\ \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{ - \alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=1}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} - \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} t^{i-1} & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r,& 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$

and by the continuity of the maps $t^{\alpha _{i}}f_{i}(t)$, we get $k_{i}(0) = {}_{i}b_{0}$ for $i=1,2,\ldots , m$. Now, we consider some cases.

(I):

Let $t_{0}=0$ and $t \in J$. Since $t^{\alpha _{i}}f_{i}(t)$ is continuous, there exist $M_{1}, \ldots , M_{n}>0$ such that $|t^{\alpha _{i}}f_{i} (t)| \leq M_{i}$ for all $t \in \overline{J}$. Thus,

$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(0) \bigr\vert \\ &\quad = \textstyle\begin{cases} \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert , & i=1, \\ \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} - \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} t^{i-1} & \\ \quad {}\times \int _{0}^{1} (t-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert ,& 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & i=1, \\ \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=1}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } t^{j} + \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i}}{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} t^{i-1} & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r.& 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Hence, by using the q-beta function, we get

$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(0) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i} t^{\sigma _{i} - \alpha _{i} }}{ \varGamma _{q}(\sigma _{i})} \int _{0}^{1} (1-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & i=1, \\ \frac{M_{i} t^{\sigma _{i} - \alpha _{i} }}{\varGamma _{q}(\sigma _{i})} \int _{0}^{1} (1-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=1}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } t^{j} & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1})} t^{i-1}, & 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i}B_{q}(\sigma _{i}, 1- \alpha _{i}) t^{\sigma _{i} - \alpha _{i} } }{ \varGamma _{q}(\sigma _{i})}, & i=1, \\ \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) t^{\sigma _{i} - \alpha _{i} }}{\varGamma _{q}(\sigma _{i})} + \sum_{j=1}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } t^{j} & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } t^{i-1}, & 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$

which, by assumption $\sigma _{1} >\alpha _{1}$ and the fact $\sigma _{i}> \alpha _{i}$, tend to zero as $t\to 0$ for $i=1,2,\ldots , m$.

(II):

Let $t_{0} \in (0,1)$ and $t \in (t_{0}, 1]$. Then we have

$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(t_{0}) \bigr\vert \\ &\quad = \textstyle\begin{cases} \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}- \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t_{0}} (t_{0} - qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert ,& i=1, \\ \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}- \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t_{0}} (t_{0}-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} ( t^{j} -t_{0}^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} ( t^{i-1} - t_{0}^{i-1} ) & \\ \quad {}\times \vert \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert ,& 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} [ \int _{0}^{t} (t-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r &\\ \quad {}- \int _{0}^{t_{0}} (t_{0} - qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r ],& i=1, \\ \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} [ \int _{0}^{t} (t-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}- \int _{0}^{t_{0}} (t_{0} - qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r ]& \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } ( t^{j} - t_{0}^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i}}{ (i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} ( t^{i-1}- t_{0}^{i-1} ) & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i} - \zeta _{i-1}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r. & 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Hence,

$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(t_{0}) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{ \varGamma _{q}(\sigma _{i})} ( t^{\sigma _{i} - \alpha _{i} } - t_{0}^{\sigma _{i} - \alpha _{i}} ), & i=1, \\ \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{\varGamma _{q}(\sigma _{i})} ( t^{\sigma _{i} - \alpha _{i} } - t_{0}^{\sigma _{i} - \alpha _{i}} ) & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } ( t^{j} - t_{0}^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } & \\ \quad {}\times ( t^{i-1} - t_{0}^{i-1} ), & 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$

which similar to case I tends to zero as $t\to 0$ for $i=1,2,\ldots , m$.

(III):

Let $t_{0}=1$ and $t \in [0, t_{0})$. By using similar arguments as in the previous case, one can obtain

$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(t_{0}) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{ \varGamma _{q}(\sigma _{i})} ( t_{0}^{\sigma _{i} - \alpha _{i} } - t^{\sigma _{i} - \alpha _{i}} ), & i=1, \\ \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{\varGamma _{q}(\sigma _{i})} ( t_{0}^{\sigma _{i} - \alpha _{i} } - t^{\sigma _{i} - \alpha _{i}} ) & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } ( t_{0}^{j} - t^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1} ) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } & \\ \quad {}\times ( t_{0}^{i-1} - t^{i-1} ), & 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$

which similar to the previous case tends to zero as $t\to 1$ for $i=1,2,\ldots , m$. This completes the proof. □

Lemma 5

Let $m \geq 2$, $\sigma _{1} \in (0, 1)$, $\sigma _{1} > \alpha _{1}$, $\sigma _{i} \in (i-1, i)$for $i=2, \ldots , m$, $\alpha _{i} \in (0,1)$for $i=1,2, \ldots , m$, $w_{i} : J \times \mathbb{R}^{i} \to \mathbb{R}$be a function with $\lim_{t \to 0^{+}} w_{i}(t, \ldots )= \infty$, and $t^{\alpha _{i}} w_{i} (t)$be continuous on $\overline{J} \times \mathbb{R}^{i}$. Then the operator $\mathcal{N}: S \to S$defined by Eq. (8) is completely continuous.

Proof

Let $( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \in S$ with

$$ \bigl\| ( k_{1}, k_{2}, \ldots , k_{m}) - ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \bigr\| < 1, $$

and $\| ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \|=l_{0}$ for all $( k_{1}, k_{2}, \ldots , k_{m}) \in S$. Hence,

$$ \bigl\Vert ( k_{1}, k_{2}, \ldots , k_{m}) \bigr\Vert < 1+ l_{0}:= l. $$

By using the continuity of the map $t^{\alpha _{i}} \varrho _{i} (t, k_{1}, k_{2}, \ldots , k_{m})$, we get the map

$$ t^{\alpha _{i}} \varrho _{i} (t, k_{1}, k_{2}, \ldots , k_{m}) $$

is uniformly continuous on $\overline{J} \times [-l, l]^{i}$. For each $\varepsilon > 0$, choose $\lambda \in (0,1)$ such that

$$ \bigl\vert t^{\alpha _{i}} w_{i} \bigl(t, k_{1}(t), k_{2}(t), \ldots , k_{i}(t)\bigr) - t^{\alpha _{i}} w_{i} \bigl(t, {}_{0}k_{1}(t), {}_{0}k_{2}(t), \ldots , {}_{0}k_{i}(t) \bigr) \bigr\vert < \varepsilon $$

(9)

for all $t \in \overline{J}$ whenever $\| ( k_{1}, k_{2}, \ldots , k_{m}) - ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \|< \lambda$. Thus,

$$ \begin{aligned}[b] \bigl\| \mathcal{N} & [ k_{1}, k_{2}, \ldots , k_{m}] (t) - \mathcal{N} [ {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}] (t) \bigr\| \\ & = \max_{1 \leq i\leq m} \| N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{i}) (t)\|_{\infty}\end{aligned} $$

(10)

and

$$\begin{aligned} &\bigl\| N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{i}) (t)\bigr\| _{\infty} \\ &\quad \leq \textstyle\begin{cases} \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t-qr)^{(\sigma _{i}-1) }r^{-\alpha _{i}} }{\varGamma _{q}( \sigma _{i}) } & \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{i}(r))- r^{\alpha _{i}} w_{i} (r, {}_{0}k_{i}(r)) \vert \,\mathrm{d}_{q}r, & i=1, \\ \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t-qr)^{( \sigma _{i}-1) }r^{-\alpha _{i}} }{\varGamma _{q}(\sigma _{i}) } & \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) & \\ \quad {}- r^{\alpha _{i}} w_{i} (r, {}_{0}k_{1}(r), \ldots , {}_{0}k_{i}(r)) \vert \,\mathrm{d}_{q}r& \\ \quad {}+ \max_{ t \in \overline{J}} \frac{\varGamma _{q}(i - \zeta _{i-1}) }{(i-1)! } t^{i-1}& \\ \quad {}\times \int _{0}^{1} \frac{(1-qr)^{(\sigma _{i}- \zeta _{i-1}-1) } r^{-\alpha _{i}} }{\varGamma _{q}(\sigma _{i}- \zeta _{i-1}) } & \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) & \\ \quad {}- r^{\alpha _{i}} w_{i} (r, {}_{0}k_{1}(r), \ldots , {}_{0}k_{i}(r) ) \vert \,\mathrm{d}_{q}r, & 2\leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Now, by using (9), we obtain

$$\begin{aligned} &\bigl\| N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{i}) (t)\bigr\| _{\infty} \\ &\quad \leq \textstyle\begin{cases} \frac{\varepsilon }{\varGamma _{q}(\sigma _{i}) } \max_{ t \in \overline{J}} \int _{0}^{t} (t-qr)^{(\sigma _{i}-1) }r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & i=1, \\ \frac{\varepsilon }{\varGamma _{q}(\sigma _{i}) } \max_{ t \in \overline{J}} \int _{0}^{t} (t-qr)^{(\sigma _{i}-1) }r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ \frac{\varepsilon \varGamma _{q}(i - \zeta _{i-1})}{(i-1)!\varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } & \\ \quad \max_{ t \in \overline{J} } \int _{0}^{t} (1-qr)^{( \sigma _{i}- \zeta _{i-1}-1) } r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{ \varepsilon B_{q}(\sigma _{i}, 1- \alpha _{i})}{ \varGamma _{q}(\sigma _{i}) } \max_{ t \in \overline{J} } t^{\sigma _{i}-\alpha _{i} }, & i=1, \\ \varepsilon [ \frac{ B_{q}(\sigma _{i}, 1- \alpha _{i})}{ \varGamma _{q}(\sigma _{i}) } \max_{ t \in \overline{J}} t^{\sigma _{i}-\alpha _{i} } & \\ \quad {}+ \frac{\varGamma _{q}(i - \zeta _{i-1}) B_{q}(\sigma _{i}- \zeta _{i-1}, 1- \alpha _{i}) }{ (i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } ], & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \varepsilon \varLambda _{i}, \end{aligned}$$

where $\varLambda _{i}= \frac{ \varGamma _{q}(1- \alpha _{i})}{ \varGamma _{q}(\sigma _{i}+ 1- \alpha _{i}) }$ whenever $i=1$ and

$$\begin{aligned} \varLambda _{i} &= \frac{ \varGamma _{q}(1- \alpha _{i})}{ \varGamma _{q}(\sigma _{i}+1-\alpha _{i}) } + \frac{\varGamma _{q}(i - \zeta _{i-1}) \varGamma _{q}( 1- \alpha _{i}) }{ (i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1} +1-\alpha _{i}) }, \end{aligned}$$

(11)

whenever $2 \leq i \leq m$. Now, by applying last result and (11), we get

$$ \bigl\| N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{i}) (t)\bigr\| _{\infty }\leq \textstyle\begin{cases} \varepsilon \varLambda _{1}, & i=1, \\ \varepsilon \varLambda _{i}, & 2\leq i \leq m. \end{cases} $$

(12)

Also, (10) and (11) imply that

$$\begin{aligned} &\bigl\| \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) - \mathcal{N} [ {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}] (t) \bigr\| \leq \varepsilon \max _{1 \leq i\leq m} \varLambda _{i} \end{aligned}$$

for all $t \in \overline{J}$. Hence, $\| \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) - \mathcal{N} [ {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}] (t) \| \to 0$ as

$$ \bigl\Vert ( k_{1}, k_{2}, \ldots , k_{m})-( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \bigr\Vert \to 0. $$

Thus, the operator $\mathcal{N}$ is continuous. Now consider a bounded subset $K \subset S$. Then there exists a positive constant δ such that $\|( k_{1}, k_{2}, \ldots , k_{m}) \| \leq \delta$ for all $( k_{1}, k_{2}, \ldots , k_{m}) \in K$. Since the maps $t^{\alpha _{i}} w_{i}(t, k_{1}, k_{2}, \ldots , k_{i} )$ are continuous on $\overline{J} \times [-\delta , \delta ]^{i}$ for $i = 1, 2, \ldots , m$, there exist positive constants $L_{i}$ such that

$$ \bigl\vert t^{\alpha _{i}} w_{i}\bigl(t, k_{1}(t), k_{2}(t), \ldots , k_{i}(t) \bigr) \bigr\vert \leq L_{i} $$

(13)

for all $t \in \overline{J}$ and $( k_{1}, k_{2}, \ldots , k_{m} ) \in K$. Consider the norm

$$ \bigl\Vert \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) \bigr\Vert = \max_{1 \leq i \leq m} \bigl\Vert N_{i}( k_{1}, k_{2}, \ldots , k_{i}) (t) \bigr\Vert _{\infty }. $$

(14)

Note that

$$\begin{aligned} & \bigl\Vert N_{i}( k_{1}, k_{2}, \ldots , k_{i}) (t) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } & \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{i}(r)) \vert \,\mathrm{d}_{q}r + \vert {}_{1}b_{0} \vert , & i=1, \\ \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } & \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) \vert \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} \max_{ t \in \overline{J}} t^{j} + \frac{ \varGamma _{q}(i - \zeta _{i-1} ) }{(i-1)! } \max_{ t \in \overline{J}} t^{i-1}& \\ \quad {}\times \int _{0}^{1} \frac{(1-qr)^{(\sigma _{i}- \zeta _{i-1}-1) } r^{-\alpha _{i}} }{\varGamma _{q}(\sigma _{i}- \zeta _{i-1}) } & \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) \vert \,\mathrm{d}_{q}r, &2\leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Now, by using (13), we get

$$\begin{aligned} & \bigl\Vert N_{i}( k_{1}, k_{2}, \ldots , k_{i}) (t) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \frac{L_{i}}{\varGamma _{q}(\sigma _{i})} \max_{ t \in \overline{J}} \int _{0}^{t} (t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} \,\mathrm{d}_{q}r + \vert {}_{i}b_{0} \vert , & i=1, \\ \frac{L_{i}}{\varGamma _{q}(\sigma _{i})} \max_{ t \in \overline{J}} \int _{0}^{t} (t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} + \frac{L_{i}\varGamma _{q}(i - \zeta _{i-1})}{ \varGamma _{q}(\sigma _{i} -\zeta _{i-1}) } & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i} - \zeta _{i-1}-1) } r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{L_{i}\varGamma _{q}(1-\alpha _{i})}{\varGamma _{q}(\sigma _{i}+ 1-\alpha _{i})} \max_{ t \in \overline{J}} t^{\sigma _{i} - \alpha _{i}} + \vert {}_{i}b_{0} \vert , & i=1, \\ L_{i} [ \frac{\varGamma _{q}(1-\alpha _{i})}{\varGamma _{q}(\sigma _{i}+1-\alpha _{i})} \max_{ t \in \overline{J}} t^{\sigma _{i}-\alpha _{i}} & \\ \quad {}+ \frac{\varGamma _{q}(i - \zeta _{i-1} ) \varGamma _{q}(1-\alpha _{i})}{ (i-1)!\varGamma _{q}(\sigma _{i} -\zeta _{i-1} + 1-\alpha _{i}) } ] + \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!}, & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} L_{i} \varLambda _{i} + \vert {}_{i}b_{0} \vert , & i=1, \\ L_{i} \varLambda _{i} + \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!}, & 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

(15)

On the other hand, by using (14) and (15), we get

$$ \bigl\Vert \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) \bigr\Vert \leq \max_{1 \leq i\leq m} \Biggl\{ L_{1}\varLambda _{1} + \vert {}_{i}b_{0} \vert , L_{i} \varLambda _{i} + \sum _{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} \Biggr\} . $$

Thus $\mathcal{N} (K)$ is bounded. Let $( k_{1}, k_{2}, \ldots , k_{m}) \in K$ and $t_{1}, t_{2} \in \overline{J}$ with $t_{1} < t_{2}$. Then we have

$$\begin{aligned} & \bigl\Vert \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t_{2}) - \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t_{1}) \bigr\Vert \\ &\quad = \max_{1 \leq i\leq m} \bigl\| N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t_{2}) - N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t_{1})\bigr\| _{\infty} \end{aligned}$$

(16)

and

$$\begin{aligned} & \bigl\Vert N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t_{2}) - N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t_{1}) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \max_{ t \in \overline{J}} \vert \int _{0}^{t_{2}} \frac{(t_{2}-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}( \sigma _{i}) } r^{\alpha _{i}} w_{i} (r, k_{i}(r)) \,\mathrm{d}_{q}r & \\ \quad {}- \int _{0}^{t_{1}} \frac{(t_{1}-qr)^{(\sigma _{i} - 1 ) } r^{-\alpha _{i}} }{ \varGamma _{q}( \sigma _{i}) } r^{\alpha _{i}} w_{i} (r, k_{i}(r)) \,\mathrm{d}_{q}r \vert ,& i=1, \\ \max_{ t \in \overline{J}} \vert \int _{0}^{t_{2}} \frac{(t_{2}-qr)^{(\sigma _{i}-1) }r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) \,\mathrm{d}_{q}r & \\ \quad {}- \int _{0}^{t_{1}} \frac{ (t_{1} - qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}( \sigma _{i}) } r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) \,\mathrm{d}_{q}r \vert & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} (t_{2}^{j} - t_{1}^{j} ) + \frac{\varGamma _{q}(i - \zeta _{i-1})}{(i-1)!} (t_{2}^{i-1} - t_{1}^{i-1} )& \\ \quad {}\times \int _{0}^{1} \frac{ (1 - qr)^{(\sigma _{i}-\zeta _{i-1} -1) } r^{-\alpha _{i}} }{ \varGamma _{q}( \sigma _{i}- \zeta _{i-1}) } \\ \quad {}\times \vert r^{\alpha _{i}} w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) \,\mathrm{d}_{q}r \vert \,\mathrm{d}_{q}r, & 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Hence,

$$\begin{aligned} & \bigl\Vert N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t_{2}) - N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t_{1}) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \frac{L_{i}\varGamma _{q}(1-\alpha _{i})}{\varGamma _{q}( \sigma _{i}+ 1-\alpha _{i})} ( t_{2}^{\sigma _{i}-\alpha _{i}} - t_{1}^{\sigma _{i}-\alpha _{i}} ), & i=1, \\ \frac{L_{i} \varGamma _{q}(1-\alpha _{i})}{ \varGamma _{q}( \sigma _{i}+ 1-\alpha _{i})} ( t_{2}^{\sigma _{i}-\alpha _{i}} - t_{1}^{\sigma _{i}-\alpha _{i}} ) & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} (t_{2}^{j} - t_{1}^{j} ) & \\ \quad {}+ \frac{L_{i} \varGamma _{q} (i - \zeta _{i-1})\varGamma _{q}(1- \alpha _{i}) }{ (i-1)!\varGamma _{q}( \sigma _{i} - \zeta _{i-1} + 1- \alpha _{i}) } ( t_{2}^{i-1} - t_{1}^{i-1} ), & 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

(17)

Now, by using (16) and (17), we obtain

$$\begin{aligned} & \bigl\Vert \mathcal{N} [k_{1}, k_{2}, \ldots , k_{m}] (t_{2}) - \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t_{1}) \bigr\Vert \\ &\quad = \max_{1 \leq i\leq m} \Biggl\{ \frac{L_{i}\varGamma _{q}(1-\alpha _{1})}{\varGamma _{q}( \sigma _{1}+ 1-\alpha _{1})} \bigl( t_{2}^{\sigma _{1}-\alpha _{1}} - t_{1}^{\sigma _{1}-\alpha _{1}} \bigr), \\ & \qquad \frac{L_{i} \varGamma _{q}(1-\alpha _{i})}{\varGamma _{q}( \sigma _{i}+ 1-\alpha _{i})} \bigl( t_{2}^{\sigma _{i}-\alpha _{i}} - t_{1}^{\sigma _{i}-\alpha _{i}} \bigr) + \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} \bigl(t_{2}^{j} - t_{1}^{j} \bigr) \\ & \qquad {}+ \frac{L_{i} \varGamma _{q} (i - \zeta _{i-1}) \varGamma _{q}(1- \alpha _{i}) }{ (i-1)!\varGamma _{q}( \sigma _{i} - \zeta _{i-1} + 1- \alpha _{i}) } \bigl( t_{2}^{i-1} - t_{1}^{i-1} \bigr) \Biggr\} . \end{aligned}$$

(18)

The right-hand side of (18) is independent of $(k_{1}, k_{2}, \ldots , k_{m})$ and, by assumption $\sigma _{1} >\alpha _{1}$ and the fact $\sigma _{i}> \alpha _{i}$, tends to zero as $t_{1} \to t_{2}$. This implies that $\mathcal{N}(K)$ is equicontinuous. Now, by using the Arzelà–Ascoli theorem, we conclude that $\mathcal{N}$ is completely continuous. □

Theorem 6

The m-dimensional system of singular fractional q-differential equations (1) has a unique solution on J̅ whenever there exist nonnegative constants ${}_{i}\eta _{j}$ ($j=1, 2, \ldots , i$, $i=1, 2, \ldots , m$, $m \geq 2$) satisfying

$$ t^{\alpha _{i}} \bigl\vert w_{i} (t, k_{1}, \ldots , k_{i}) - w_{i} (t, l_{1}, \ldots , l_{i}) \bigr\vert \leq \sum _{j=1}^{i} {}_{i}\eta _{j} \vert k_{j} - l_{j} \vert $$

(19)

for all $t \in \overline{J}$and $(k_{1}, \ldots , k_{i})$, $(l_{1}, \ldots , l_{i}) \in \mathbb{R}^{i}$, and also

$$ \varSigma = \max_{2 \leq i\leq m} \Biggl\{ {}_{1}\eta _{1}\varLambda _{1}, \sum _{j=1}^{i} {}_{i}\eta _{j} \varLambda _{i} \Biggr\} < 1, $$

(20)

where the constants $\varLambda _{i}$are defined by (11).

Proof

We prove that $\mathcal{N}$ is a contractive operator on S. Assume that $(k_{1}, k_{2}, \ldots , k_{m}) \in S$ and $(l_{1},l_{2}, \ldots , l_{m}) \in S$. Then we have

$$\begin{aligned} & \bigl\Vert \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) - \mathcal{N} [ l_{1}, l_{2}, \ldots , l_{m}] (t) \bigr\Vert \\ &\quad = \max_{1 \leq i\leq m} \bigl\Vert N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( l_{1}, l_{2}, \ldots , l_{i}) (t) \bigr\Vert _{\infty} \end{aligned}$$

(21)

for almost all $t \in \overline{J}$. Hence,

$$\begin{aligned} & \bigl\Vert N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( l_{1}, l_{2}, \ldots , l_{i}) (t) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t - qr)^{ ( \sigma _{i} - 1 ) } r^{-\alpha _{i}} }{ \varGamma _{q}( \sigma _{i}) } & \\ \quad {}\times r^{\alpha _{i}} \vert w_{i} (r, k_{i}(r)) - w_{i} (r, l_{i}(r)) \vert \,\mathrm{d}_{q}r, & i=1, \\ \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t -qr )^{(\sigma _{i}-1) }r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } & \\ \quad {}\times r^{\alpha _{i}} \vert w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) & \\ \quad {}- w_{i} (r, l_{1}(r), \ldots , l_{i}(r)) \vert \,\mathrm{d}_{q}r & \\ \quad {}+ \max_{t\in \overline{J}} \frac{\varGamma _{q}(i - \zeta _{i-1})}{(i-1)!} t^{i-1} \int _{0}^{1} \frac{ (1 - qr)^{(\sigma _{i}-\zeta _{i-1}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}( \sigma _{i}- \zeta _{i-1}) } & \\ \quad {}\times r^{\alpha _{i}} \vert w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) & \\ \quad {}- w_{i} (r, l_{1}(r), \ldots , l_{i}(r)) \vert \,\mathrm{d}_{q}r,& 2\leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Now, by using (19), we obtain

$$\begin{aligned} & \bigl\Vert N_{i} ( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i} ( l_{1}, l_{2}, \ldots , l_{i}) (t) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \frac{{}_{i}\eta _{1}}{ \varGamma _{q}( \sigma _{i}) } \Vert k_{i}-l_{i} \Vert _{\infty }\max_{ t \in \overline{J}} \int _{0}^{t} (t - qr)^{( \sigma _{i} - 1) } r^{-\alpha _{i}}, & i=1, \\ ( \sum_{j=1}^{i} {}_{i}\eta _{j} \Vert k_{i} - l_{j} \Vert _{\infty } ) [ \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t -qr )^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } & \\ \quad {}+ \frac{ \varGamma _{q}(i - \zeta _{i-1})}{ (i-1)! \varGamma _{q}( \sigma _{i}- \zeta _{i-1}) } & \\ \quad {}\times \int _{0}^{1} (1 - qr)^{(\sigma _{i}-\zeta _{i-1}-1) } r^{- \alpha _{i}} \,\mathrm{d}_{q}r ], & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{{}_{i}\eta _{1} B_{q}(\sigma _{i}, 1-\alpha _{i}) }{ \varGamma _{q}( \sigma _{i}) } \Vert k_{i}-l_{i} \Vert _{\infty }\max_{ t \in \overline{J}} t^{\sigma _{i}- \alpha _{i}}, & i=1, \\ \sum_{j=1}^{i} {}_{i}\eta _{j} \max_{1 \leq i\leq m} \Vert k_{i} - l_{i} \Vert _{\infty } [ \frac{ B_{q}(\sigma _{i}, 1- \alpha _{i}) }{ \varGamma _{q}(\sigma _{i}) } \max_{ t \in \overline{J}} t^{\sigma _{i}-\alpha _{i}} & \\ \quad {}+ \frac{ \varGamma _{q}(i - \zeta _{i-1}) B_{q}(\sigma _{i}-\zeta _{i-1}, 1- \alpha _{i}) }{ (i-1)! \varGamma _{q}( \sigma _{i}- \zeta _{i-1}) } ], & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{{}_{i}\eta _{1} \varGamma _{q}( 1-\alpha _{i}) }{ \varGamma _{q}( \sigma _{i} + 1-\alpha _{i}) } \Vert k_{i} - l_{i} \Vert _{\infty }, & i=1, \\ \sum_{j=1}^{i} {}_{i}\eta _{j} [ \frac{ \varGamma _{q}(1- \alpha _{i}) }{ \varGamma _{q}(\sigma _{i}+1- \alpha _{i}) } & \\ \quad {}+ \frac{ \varGamma _{q}(i - \zeta _{i-1}) \varGamma _{q}( 1- \alpha _{i}) }{ (i-1)! \varGamma _{q}( \sigma _{i}- \zeta _{i-1}+1- \alpha _{i}) } ] & \\ \quad {}\times \Vert (k_{1}-l_{1}, k_{2}-l_{2}, \ldots , k_{i}-l_{i}) \Vert _{\infty }, & 2\leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

(22)

If we apply (21) and (22), then we get

$$\begin{aligned} & \bigl\Vert \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) - \mathcal{N} [ l_{1}, l_{2}, \ldots , l_{m}] (t) \bigr\Vert \\ &\quad \leq \max_{2 \leq i\leq m} \Biggl\{ {}_{1}\eta _{1}\varLambda _{1}, \sum_{j=1}^{i} {}_{i}\eta _{j} \varLambda _{i} \Biggr\} \bigl\Vert ( k_{1}-l_{1}, k_{2}-l_{2}, \ldots , k_{i}-l_{i}) (t) \bigr\Vert _{\infty }. \end{aligned}$$

Now, by using (20), we have

$$ \varSigma = \max_{2 \leq i\leq m} \Biggl\{ {}_{1}\eta _{1}\varLambda _{1}, \sum_{j=1}^{i} {}_{i}\eta _{j} \varLambda _{i} \Biggr\} < 1. $$

Hence, $\mathcal{N}$ is a contraction. By using the Banach contraction principle, $\mathcal{N}$ has a unique fixed point which is the unique solution for system (1). □

Now, we consider different conditions on system (1).

Theorem 7

Let $m \geq 2$, $\sigma _{1} \in (0, 1)$, $\sigma _{1} > \alpha _{1}$, $\sigma _{i} \in (i-1, i)$for $i=2, \ldots , m$, $\alpha _{i} \in (0,1)$for $i=1,2, \ldots , m$, $w_{i} : J \times \mathbb{R}^{i} \to \mathbb{R}$be functions with $\lim_{t \to 0^{+}} w_{i}(t, \ldots ) = \infty$, and $t^{\alpha _{i}} w_{i} (t, \ldots )$be continuous maps on $\overline{J} \times \mathbb{R}^{i}$. Then system (1) has a solution on J̅.

Proof

Assume that

$$ L_{i} = \max_{ t \in \overline{J}} t^{\alpha _{i}} \bigl\vert w_{i}\bigl(t, k_{1}(t), \ldots , k_{i}(t)\bigr) \bigr\vert $$

(23)

and define the set $K_{r} \subset S$ by

$$ K_{r}= \bigl\{ (k_{1}, k_{2}, \ldots , k_{m}) \in S: \bigl\Vert (k_{1}, k_{2}, \ldots , k_{m}) \bigr\Vert \leq r \bigr\} , $$

where

$$ r = \max_{2 \leq i\leq m} \Biggl\{ L_{1} \varLambda _{1}+ \vert {}_{1}b_{0} \vert , L_{i} \varLambda _{i} + \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} \Biggr\} . $$

(24)

We show that $\mathcal{N}$ maps $K_{r}$ into $K_{r}$. For $(k_{1}, k_{2}, \ldots , k_{m}) \in K_{r}$ and $t \in \overline{J}$, put

$$ \mathcal{N} [k_{1}, k_{2}, \ldots , k_{m}](t) = \max_{1 \leq i\leq m} \bigl\Vert N_{i}(k_{1}, k_{2}, \ldots , k_{i}) (t) \bigr\Vert _{\infty }. $$

(25)

Thus, we have

$$\begin{aligned} & \bigl\Vert N_{i}( k_{1}, k_{2}, \ldots , k_{i}) (t) - N_{i}( l_{1}, l_{2}, \ldots , l_{i}) (t) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } r^{\alpha _{i}} & \\ \quad {}\times \vert w_{i} (r, k_{i}(r)) - w_{i} (r, l_{i}(r)) \vert \,\mathrm{d}_{q}r, & i=1, \\ \max_{ t \in \overline{J}} \int _{0}^{t} \frac{(t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} }{ \varGamma _{q}(\sigma _{i}) } r^{\alpha _{i}} & \\ \quad {}\times \vert w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) - w_{i} (r, l_{1}(r), \ldots , l_{i}(r)) \vert \,\mathrm{d}_{q}r & \\ \quad {}+ \max_{ t \in \overline{J}} \frac{ \varGamma _{q}(i - \zeta _{i-1} ) }{(i-1)! } t^{i-1} & \\ \quad {}\times \int _{0}^{1} \frac{(1-qr)^{(\sigma _{i}- \zeta _{i-1}-1) } r^{-\alpha _{i}} }{\varGamma _{q}(\sigma _{i}- \zeta _{i-1}) } r^{\alpha _{i}} & \\ \quad {}\times \vert w_{i} (r, k_{1}(r), \ldots , k_{i}(r)) & \\ - w_{i} (r, l_{1}(r), \ldots , l_{i}(r)) \vert \,\mathrm{d}_{q}r, & 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{L_{i}}{\varGamma _{q}(\sigma _{i})} \max_{ t \in \overline{J}} \int _{0}^{t} (t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} r^{\alpha _{i}} \,\mathrm{d}_{q}r + \vert {}_{i}b_{0} \vert , & i=1, \\ \frac{L_{i}}{\varGamma _{q}(\sigma _{i})} \max_{ t \in \overline{J}} \int _{0}^{t} (t-qr)^{(\sigma _{i}-1) } r^{-\alpha _{i}} r^{\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ [ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} ] + \frac{L_{i} \varGamma _{q}(i - \zeta _{i-1} ) }{(i-1)! \varGamma _{q}(\sigma _{i}- \zeta _{i-1}) } & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & 2\leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

Hence,

$$\begin{aligned} & \bigl\Vert N_{i}( k_{1}, k_{2}, \ldots , k_{i}) (t) \bigr\Vert _{\infty} \\ &\quad \leq \textstyle\begin{cases} \frac{L_{1} \varGamma _{q}(1-\alpha _{i})}{\varGamma _{q}(\sigma _{i}+ 1-\alpha _{i})} \max_{ t \in \overline{J}} t^{\sigma _{i} -\alpha _{i}} + \vert {}_{i}b_{0} \vert , & i=1, \\ L_{i} [ \frac{\varGamma _{q}(1-\alpha _{i})}{\varGamma _{q}(\sigma _{i}1-\alpha _{i})} \max_{ t \in \overline{J}} t^{\sigma _{i} - \alpha _{i}} & \\ \quad {}+ \frac{ \varGamma _{q}(i - \zeta _{i-1} ) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}1-\alpha _{i})} ]+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!}, & 2\leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} L_{i} \varLambda _{i} + \vert {}_{i}b_{0} \vert , & i=1, \\ L_{i} \varLambda _{i} + \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!}, & 2\leq i \leq m. \end{cases}\displaystyle \end{aligned}$$

(26)

Now, by using (25) and (26), we conclude that

$$ \bigl\Vert \mathcal{N} [k_{1}, k_{2}, \ldots , k_{m}](t) \bigr\Vert \leq \max_{2 \leq i\leq m} \Biggl\{ L_{1}\varLambda _{1} + \vert {}_{1}b_{0} \vert , L_{i} \lambda _{i} + \sum _{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{j!} \Biggr\} , $$

(27)

and so $\| \mathcal{N} [k_{1}, k_{2}, \ldots , k_{m}](t) \| \leq r$. By using Lemma 4, we get

$$ \mathcal{N} [k_{1}, k_{2}, \ldots , k_{m}](t) \in C(\overline{J}). $$

Moreover, $\mathcal{N}[ k_{1}, k_{2}, \ldots , k_{m}](t) \in K_{r}$ for $(k_{1}, k_{2}, \ldots , k_{m}) \in K_{r}$. Thus $\mathcal{N} (K_{r}) \subset K_{r}$, and so $\mathcal{N}$ maps $K_{r}$ into $K_{r}$. On the other hand, by using Lemma 5, $\mathcal{N}$ is completely continuous. Now, by using Lemma 2, the map $\mathcal{N}$ has a fixed point which is a solution for system (1). □

Now, we provide two examples to illustrate our main results. In this way, we give a computational technique for checking the m-dimensional system (1). We need to present a simplified analysis which is able to execute the values of the q-gamma function. For this purpose, we provide a pseudo-code description of the method for calculation of the q-gamma function of order n in Algorithms 2, 3, 5, and 4.

Algorithm 4

The proposed method for calculated $\int _{a}^{b} f(r)\, d_{q} r$

Algorithm 5

The proposed method for calculated $I_{q}^{\sigma}[x]$

Example 1

Consider the increasing variables singular 5-dimensional system of fractional q-differential equations

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{ \frac{7}{9}} [k_{1}](t) = w_{1}(t, k_{1}), \\ {}^{c}\mathcal{D}_{q}^{ \frac{8}{7}} [k_{2}](t) = w_{2}(t, k_{1}, k_{2}), \\ {}^{c}\mathcal{D}_{q}^{ \frac{11}{4}} [k_{3}](t) = w_{3}(t, k_{1}, k_{2}, k_{3}), \\ {}^{c}\mathcal{D}_{q}^{\frac{16}{5}} [k_{4}](t) =w_{4}(t, k_{1}, k_{2}, k_{3}, k_{4}), \\ {}^{c}\mathcal{D}_{q}^{\frac{31}{7}} [k_{5}](t) =w_{5}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}), \end{cases} $$

(28)

under the boundary value conditions $k_{1}(0) = \frac{7}{9}$, $k_{2}(0)=\frac{3}{5}$,

$$ \textstyle\begin{cases} k_{3}(0) = \frac{1}{2}, \qquad k'_{3}(0)=2\sqrt{3}, \\ k_{4}(0)=\sqrt{5},\qquad k'_{4}(0)=\frac{\sqrt{5}}{3}, \qquad k''_{4}(0)= \frac{15}{7}, \\ k_{5}(0)=\frac{\sqrt{3}}{3},\qquad k'_{5}(0)=1, \qquad k''_{5}(0)=0, \qquad k'''_{5}(0)= \frac{13}{4}, \end{cases} $$

and ${}^{c}\mathcal{D}_{q}^{ \frac{1}{2}} [k_{2}](1) = {}^{c}\mathcal{D}_{q}^{\frac{4}{3}} [k_{3}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{5}{2}} [k_{4}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{11}{3}} [k_{5}](1) =0$, where $t\in (0,1]$. Put

$$\begin{aligned} &w_{1}(t, k_{1}) = \frac{ 3 \cos ^{2} k_{1}(t) }{ 20 \sqrt{t} }, \\ &w_{2} (t, k_{1}, k_{2}) = \frac{ 2 ( \vert k_{1}(t) \vert + \vert k_{2}(t) \vert ) }{ 25\pi \sqrt[3]{t} ( 1 + \vert k_{1}(t) \vert + \vert k_{2}(t) \vert ) }, \\ &w_{3}(t, k_{1}, k_{2}, k_{3}) = \frac{\sin k_{1}(t) - \cos k_{2}(t) + \sin k_{3}(t)}{20\pi \sqrt[5]{t^{3}}}, \\ &w_{4}(t, k_{1}, k_{2}, k_{3}, k_{4}) \\ &\quad= \frac{\cos ^{2} k_{1}(t) + \sin ^{2} k_{2}(t) + \cos ^{2} k_{3}(t)+ \sin ^{2} k_{4} (t)}{25\pi \sqrt[7]{t^{5}} ( 1+ \cos ^{2} k_{1}(t) + \sin ^{2} k_{2}(t) + \cos ^{2} k_{3}(t)+ \sin ^{2} k_{4} (t) )}, \\ &w_{5}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}) \\ &\quad= \frac{ \vert k_{1}(t) \vert + \vert k_{2}(t) \vert + \vert k_{3}(t) \vert + \vert k_{4} (t) \vert - \vert k_{5}(t) \vert }{20\pi \sqrt[6]{t^{5}} (1+ \exp ( \vert k_{1}(t) \vert + \vert k_{2}(t) \vert + \vert k_{3}(t) \vert + \vert k_{4} (t) \vert - \vert k_{5}(t) \vert ) )}, \end{aligned}$$

$m=5$, $\sigma _{1} = \frac{7}{9} \in (0,1)$, $\sigma _{2} = \frac{8}{7} \in (1,2)$, $\sigma _{3} = \frac{11}{4} \in (2,3)$, $\sigma _{4}= \frac{16}{5} \in (3, 4)$, $\sigma _{5} = \frac{31}{7} \in (4,5)$, $\zeta _{1} = \frac{1}{2} \in [0,1]$, $\zeta _{2} = \frac{4}{3} \in [1,2]$, $\zeta _{3} = \frac{5}{2} \in [2,3]$, $\zeta _{4} = \frac{11}{3} \in [3,4]$, ${}_{1}b_{0} = \frac{7}{9}$, ${}_{2}b_{0} = \frac{3}{5}$, ${}_{3}b_{0} = \frac{1}{2}$, ${}_{4}b_{0} = \sqrt{5}$, ${}_{5}b_{0} = \frac{ \sqrt{3}}{3}$, ${}_{3}b_{1} = 2\sqrt{3}$, ${}_{4}b_{1} = \frac{ \sqrt{5}}{3}$, ${}_{5}b_{1} = 1$, ${}_{4}b_{2} = \frac{15}{7}$, ${}_{5}b_{2} = 0$, and ${}_{5}b_{3} = \frac{13}{4}$. Now, we check inequalities (19) and (20). For each $t \in \overline{J}$, $(k_{1}, k_{2}, \ldots , k_{5})$, and $(l_{1}, l_{2}, \ldots , l_{5}) \in \mathbb{R}^{5}$, we have

$$\begin{aligned} t^{\alpha _{1}} \bigl\vert w_{1} \bigl(t, k_{1}(t) \bigr) - w_{1}\bigl(t, l_{1}(t)\bigr) \bigr\vert &\leq t^{ \frac{4}{7} } \biggl\vert \frac{ 3 \cos ^{2} k_{1}(t) }{ 20\pi \sqrt{t} } - \frac{ 3 \cos ^{2} l_{1}(t) }{ 20\pi \sqrt{t} } \biggr\vert \\ & \leq \frac{3 t^{ \frac{1}{14} }}{20} \bigl\vert \cos ^{2} k_{1}(t) - \cos ^{2} l_{1}(t) \bigr\vert \\ & \leq \frac{3 t^{ \frac{1}{14} }}{10\pi } \bigl\vert \sin k_{1}(t) - \sin l_{1}(t) \bigr\vert \leq \frac{3 t^{ \frac{1}{14} }}{10\pi } \bigl\vert k_{1}(t) - l_{1}(t) \bigr\vert , \end{aligned}$$

$\alpha _{1} = \frac{4}{7}$, ${}_{1}\eta _{1}=\frac{3}{10\pi }$,

$$\begin{aligned} &t^{\alpha _{2}} \bigl\vert w_{2} \bigl(t, k_{1}(t), k_{2}(t)\bigr) - w_{2}\bigl(t, l_{1}(t), l_{2}(t)\bigr) \bigr\vert \\ &\quad \leq t^{ \frac{2}{5} } \biggl\vert \frac{ 2 ( \vert k_{1}(t) \vert + \vert k_{2}(t) \vert ) }{ 25\pi \sqrt[3]{t} ( 1 + \vert k_{1}(t) \vert + \vert k_{2}(t) \vert ) } - \frac{ 2 ( \vert l_{1}(t) \vert + \vert l_{2}(t) \vert ) }{ 25\pi \sqrt[3]{t} ( 1 + \vert l_{1}(t) \vert + \vert l_{2}(t) \vert ) } \biggr\vert \\ &\quad \leq \frac{2 t^{ \frac{1}{15} }}{25\pi } | \bigl\vert k_{1}(t) \bigr\vert + | k_{2}(t) \bigl\vert - \bigl( \bigl\vert l_{1}(t) \bigr\vert + \bigl\vert l_{2}(t) \bigr\vert \bigr) \bigr\vert \\ &\quad \leq \frac{2 t^{ \frac{1}{15} }}{25\pi } \bigl[ \bigl\vert k_{1}(t) - l_{1}(t) \bigr\vert + \bigl\vert k_{2}(t) - l_{2}(t) \bigr\vert \bigr], \end{aligned}$$

$\alpha _{2} = \frac{2}{5}$, ${}_{2}\eta _{1}={}_{2}\eta _{2} = \frac{2}{25\pi }$,

$$\begin{aligned} &t^{\alpha _{3}} \bigl\vert w_{3} \bigl(t, k_{1}(t), k_{2}(t) , k_{3}(t)\bigr) - w_{3}\bigl(t, l_{1}(t), l_{2}(t) , l_{3}(t) \bigr) \bigr\vert \\ &\quad \leq t^{ \frac{5}{8} } \biggl\vert \frac{\sin k_{1}(t) - \cos k_{2}(t) + \sin k_{3}(t)}{20\pi \sqrt[5]{t^{3}}} - \frac{\sin l_{1}(t) - \cos l_{2}(t) + \sin l_{3}(t)}{20\pi \sqrt[5]{t^{3}}} \biggr\vert \\ &\quad \leq \frac{t^{ \frac{1}{40} }}{20\pi } \bigl\vert \bigl( \sin k_{1}(t) - \cos k_{2}(t) + \sin k_{3}(t)\bigr) - \bigl(\sin l_{1}(t) - \cos l_{2}(t) + \sin l_{3}(t)\bigr) \bigr\vert \\ &\quad \leq \frac{t^{ \frac{1}{40} }}{20\pi } \bigl[ \bigl\vert k_{1}(t) - l_{1}(t) \bigr\vert + \bigl\vert k_{2}(t) - l_{2}(t) \bigr\vert + \bigl\vert k_{3}(t) - l_{3}(t) \bigr\vert \bigr], \end{aligned}$$

$\alpha _{3} = \frac{5}{8}$, ${}_{3}\eta _{1}={}_{3}\eta _{2} = {}_{3}\eta _{3} = \frac{1}{20\pi }$,

$$\begin{aligned} &t^{\alpha _{4}} \bigl\vert w_{4} \bigl(t, k_{1}(t), k_{2}(t) , k_{3}(t) k_{4}(t)\bigr) - w_{4}\bigl(t, l_{1}(t), l_{2}(t) , l_{3}(t), l_{4}(t) \bigr) \bigr\vert \\ &\quad \leq t^{ \frac{7}{9} } \biggl\vert \frac{ \cos ^{2} k_{1}(t) + \sin ^{2} k_{2}(t) + \cos ^{2} k_{3}(t)+ \sin ^{2} k_{4} (t)}{25\pi \sqrt[7]{t^{5}} ( 1+ \cos ^{2} k_{1}(t) + \sin ^{2} k_{2}(t) + \cos ^{2} k_{3}(t)+ \sin ^{2} k_{4} (t) )} \\ & \qquad {}- \frac{\cos ^{2} l_{1}(t) + \sin ^{2} l_{2}(t) + \cos ^{2} l_{3}(t)+ \sin ^{2} l_{4} (t)}{25\pi \sqrt[7]{t^{5}} ( 1+ \cos ^{2} l_{1}(t) + \sin ^{2} l_{2}(t) + \cos ^{2} l_{3}(t)+ \sin ^{2} l_{4} (t) )} \biggr\vert \\ &\quad \leq \frac{t^{ \frac{4}{63} }}{25\pi } \bigl\vert \bigl(\cos ^{2} k_{1}(t) + \sin ^{2} k_{2}(t) + \cos ^{2} k_{3}(t)+ \sin ^{2} k_{4} (t) \bigr) \\ & \qquad {}- \bigl(\cos ^{2} l_{1}(t) + \sin ^{2} l_{2}(t) + \cos ^{2} l_{3}(t)+ \sin ^{2} l_{4} (t) \bigr) \bigr\vert \\ &\quad \leq \frac{t^{ \frac{4}{63} }}{25\pi } \bigl[ \bigl\vert \cos ^{2} k_{1}(t) - \cos ^{2} l_{1}(t) \bigr\vert + \bigl\vert \sin ^{2} k_{2}(t)- \sin ^{2} l_{2}(t) \bigr\vert \\ & \qquad {}+ \bigl\vert \cos ^{2} k_{3}(t) - \cos ^{2} l_{3}(t) \bigr\vert + \bigl\vert \sin ^{2} k_{4} (t) - \sin ^{2} l_{4} (t) \bigr\vert \bigr] \\ &\quad \leq \frac{2t^{ \frac{4}{63} }}{25\pi } \bigl[ \bigl\vert \sin k_{1}(t) - \sin l_{1}(t) \bigr\vert + \bigl\vert \sin k_{2}(t) - \sin l_{2}(t) \bigr\vert \\ & \qquad {}+ \bigl\vert \sin k_{3}(t) - \sin l_{3}(t) \bigr\vert + \bigl\vert \sin k_{4} (t) - \sin l_{4} (t) \bigr\vert \bigr], \\ &\quad \leq \frac{2t^{ \frac{4}{63} }}{25\pi } \bigl[ \bigl\vert k_{1}(t) - l_{1}(t) \bigr\vert + \bigl\vert k_{2}(t) - l_{2}(t) \bigr\vert + \bigl\vert k_{3}(t) - l_{3}(t) \bigr\vert + \bigl\vert k_{4} (t) - l_{4} (t) \bigr\vert \bigr], \end{aligned}$$

$\alpha _{4} = \frac{7}{9}$, ${}_{4}\eta _{1} = {}_{4}\eta _{2} = {}_{4}\eta _{3} = {}_{4}\eta _{4} =\frac{2}{25\pi }$,

$$\begin{aligned} &t^{\alpha _{5}} \bigl\vert w_{5} \bigl(t, k_{1}(t), k_{2}(t) , k_{3}(t) k_{4}(t), k_{5}(t)\bigr) - w_{5}\bigl(t, l_{1}(t), l_{2}(t) , l_{3}(t), l_{4}(t) , l_{5}(t)\bigr) \bigr\vert \\ &\quad \leq t^{ \frac{10}{11} } \biggl\vert \frac{ \vert k_{1}(t) \vert + \vert k_{2}(t) \vert + \vert k_{3}(t) \vert + \vert k_{4} (t) \vert - \vert k_{5}(t) \vert }{20 \sqrt[6]{t^{5}} ( 1 + \exp ( \vert k_{1}(t) \vert + \vert k_{2}(t) \vert + \vert k_{3}(t) \vert + \vert k_{4} (t) \vert - \vert k_{5}(t) \vert ) )} \\ & \qquad {}- \frac{ \vert l_{1}(t) \vert + \vert l_{2}(t) \vert + \vert l_{3}(t) \vert + \vert l_{4} (t) \vert - \vert l_{5}(t) \vert }{20\pi \sqrt[6]{t^{5}} (1+ \exp ( \vert l_{1}(t) \vert + \vert l_{2}(t) \vert + \vert l_{3}(t) \vert + \vert l_{4} (t) \vert - \vert l_{5}(t) \vert ) )} \biggr\vert \\ &\quad \leq \frac{t^{ \frac{5}{66} }}{20\pi } | \bigl\vert k_{1}(t) \bigr\vert + \bigl\vert k_{2}(t) \bigr\vert + \bigl\vert k_{3}(t) \bigr\vert + | k_{4} (t)| -| k_{5}(t) \bigl\vert \\ & \qquad {}- \bigl( \bigl\vert l_{1}(t) \bigr\vert + \bigl\vert l_{2}(t) \bigr\vert + \bigl\vert l_{3}(t) \bigr\vert + \bigl\vert l_{4} (t) \bigr\vert - \bigl\vert l_{5}(t) \bigr\vert \bigr) \bigr\vert \\ &\quad \leq \frac{t^{ \frac{5}{66} }}{20\pi } \bigl[ \bigl\vert k_{1}(t) - l_{1}(t) \bigr\vert + \bigl\vert k_{2}(t) - l_{2}(t) \bigr\vert + \bigl\vert k_{3}(t) - l_{3}(t) \bigr\vert \\ & \qquad {}+ \bigl\vert k_{4} (t) - l_{4} (t) \bigr\vert + \bigl\vert k_{5}(t)- l_{5}(t) \bigr\vert \bigr], \end{aligned}$$

and $\alpha _{5} = \frac{10}{11}$, ${}_{5}\eta _{1} = {}_{5}\eta _{2} = {}_{5}\eta _{3} = {}_{5}\eta _{4} = {}_{5}\eta _{5} =\frac{1}{20\pi }$. On the other hand, by using (11), we obtain

$$\begin{aligned} &\varLambda _{1} = \frac{ \varGamma _{q}(1- \alpha _{1})}{ \varGamma _{q}(\sigma _{1}+ 1- \alpha _{1}) }= \frac{ \varGamma _{q} ( 1- \frac{4}{7} ) }{ \varGamma _{q} ( \frac{7}{9}+ 1- \frac{4}{7} ) }= \frac{ \varGamma _{q} ( \frac{3}{7} ) }{ \varGamma _{q} ( \frac{76}{63} ) }, \\ &\begin{aligned} \varLambda _{2} &= \frac{ \varGamma _{q} ( 1 - \alpha _{2} ) }{ \varGamma _{q} ( \sigma _{2} + 1 -\alpha _{2} ) } + \frac{ \varGamma _{q} ( 2 - \zeta _{1} ) \varGamma _{q} ( 1- \alpha _{2} ) }{ \varGamma _{q} ( \sigma _{2} - \zeta _{1} +1-\alpha _{2} ) } \\ & = \frac{ \varGamma _{q} ( 1 - \frac{6}{7} ) }{ \varGamma _{q} ( \frac{8}{7} + 1 - \frac{6}{7} ) } + \frac{ \varGamma _{q} ( 2 - \frac{1}{2} ) \varGamma _{q} ( 1- \frac{6}{7} ) }{ \varGamma _{q} ( \frac{8}{7} - \frac{1}{2} + 1 - \frac{6}{7} ) } \\ & = \frac{ \varGamma _{q} ( \frac{1}{7} ) }{ \varGamma _{q} ( \frac{9}{7} ) } + \frac{ \varGamma _{q} ( \frac{3}{2} ) \varGamma _{q} ( \frac{1}{7} ) }{ \varGamma _{q} ( \frac{11}{14} ) }, \end{aligned} \\ &\begin{aligned} \varLambda _{3} &= \frac{ \varGamma _{q} ( 1 - \alpha _{3} ) }{ \varGamma _{q} ( \sigma _{3} + 1 -\alpha _{3} ) } + \frac{ \varGamma _{q} ( 3 - \zeta _{2} ) \varGamma _{q} ( 1- \alpha _{3} ) }{ 2! \varGamma _{q} ( \sigma _{3} - \zeta _{2} + 1 - \alpha _{3} )} \\ & = \frac{ \varGamma _{q} ( 1 - \frac{5}{8} ) }{ \varGamma _{q} ( \frac{11}{4} + 1 -\frac{5}{8} ) } + \frac{ \varGamma _{q} ( 3 - \frac{4}{3} ) \varGamma _{q} ( 1- \frac{5}{8} ) }{ 2! \varGamma _{q} ( \frac{11}{4} - \frac{4}{3} + 1 - \frac{5}{8} )} \\ & = \frac{ \varGamma _{q} ( \frac{3}{8} ) }{ \varGamma _{q} ( \frac{25}{8} ) } + \frac{ \varGamma _{q} ( \frac{5}{3} ) \varGamma _{q} ( \frac{3}{8} ) }{ 2! \varGamma _{q} ( \frac{43}{24} )}, \end{aligned} \\ &\begin{aligned} \varLambda _{4} &= \frac{ \varGamma _{q} ( 1 - \alpha _{4} ) }{ \varGamma _{q} ( \sigma _{4} + 1 -\alpha _{4} ) } + \frac{ \varGamma _{q} ( 4 - \zeta _{3} ) \varGamma _{q} ( 1- \alpha _{4} ) }{ 3! \varGamma _{q} ( \sigma _{4} - \zeta _{3} + 1 - \alpha _{4} )} \\ &= \frac{ \varGamma _{q} ( 1 - \frac{7}{9} ) }{ \varGamma _{q} ( \frac{16}{5} + 1 - \frac{7}{9} ) } + \frac{ \varGamma _{q} ( 4 - \frac{5}{2} ) \varGamma _{q} ( 1- \frac{7}{9} ) }{ 3! \varGamma _{q} ( \frac{16}{5} - \frac{5}{2} + 1 - \frac{7}{9} )} \\ & = \frac{ \varGamma _{q} ( \frac{2}{9} ) }{ \varGamma _{q} ( \frac{154}{45} ) } + \frac{ \varGamma _{q} ( \frac{3}{2} ) \varGamma _{q} ( \frac{2}{9} ) }{ 3! \varGamma _{q} ( \frac{83}{90} )}, \end{aligned} \\ &\begin{aligned} \varLambda _{5} &= \frac{ \varGamma _{q} ( 1 - \alpha _{5} ) }{ \varGamma _{q} ( \sigma _{5} + 1 -\alpha _{5} ) } + \frac{ \varGamma _{q} ( 5 - \zeta _{4} ) \varGamma _{q} ( 1- \alpha _{5} ) }{ 4! \varGamma _{q} ( \sigma _{5} - \zeta _{4} + 1 - \alpha _{5} )} \\ & = \frac{ \varGamma _{q} ( 1 - \frac{10}{11} ) }{ \varGamma _{q} ( \frac{31}{7} + 1 -\frac{10}{11} ) } + \frac{ \varGamma _{q} ( 5 - \frac{11}{3} ) \varGamma _{q} ( 1- \frac{10}{11} ) }{ 4! \varGamma _{q} ( \frac{31}{7} - \frac{11}{3} + 1 - \frac{10}{11} )} \\ & = \frac{ \varGamma _{q} ( \frac{1}{11} ) }{ \varGamma _{q} ( \frac{348}{77} ) } + \frac{ \varGamma _{q} ( \frac{4}{3} ) \varGamma _{q} ( \frac{1}{11} ) }{ 4! \varGamma _{q} ( \frac{197}{231} )}. \end{aligned} \end{aligned}$$

Tables 1, 2, and 3 show $\varLambda _{i} \approx 1.4269$, 6.1292, 2.1068, 2.2574, 3.8301, $\varLambda _{i} \approx 1.9041$, 9.5549, 2.2455, 2.2349, 2.4713, $\varLambda _{i} \approx 2.1668$, 11.5144, 2.2172, 2.0036, 1.4726 for $1 \leq i \leq 5$ and for $q= \frac{1}{10}$, $\frac{1}{2}$, $\frac{6}{7}$, respectively. It is clear that $\sum_{j=1}^{2} {}_{2}\eta _{j}= \frac{4}{25\pi }$, $\sum_{j=1}^{3} {}_{3}\eta _{j}= \frac{3}{20\pi }$, $\sum_{j=1}^{4} {}_{2}\eta _{j}= \frac{8}{25\pi }$, and $\sum_{j=1}^{5} {}_{2}\eta _{j}= \frac{1}{4\pi }$. In Tables 4, 5, and 6, we can see that $\varSigma =0.3122$, 0.4866, and 0.5864, indeed

$$ \varSigma = \max_{2 \leq i\leq m} \Biggl\{ {}_{1}\eta _{1}\varLambda _{1}, \sum_{j=1}^{i} {}_{i}\eta _{j} \varLambda _{i} \Biggr\} < 1, $$

for $q=\frac{1}{10}$, $\frac{1}{2}$, and $\frac{6}{7}$, respectively (Fig. 1). Thus, the assumptions and conditions of Theorem 6 hold. Hence the singular 5-dimensional system of fractional q-differential equations (28) has a unique solution on $(0,1]$. Note that Algorithm 6 shows us how we can obtain the parameters of Example 1.

Figure 1

Numerical results of Σ for $q= \frac{1}{10}, \frac{1}{2}, \frac{6}{7}$ in Example 1

Algorithm 6

The proposed method for solving problem (28) in Example 1 for which we use the conditions of Theorem 6

Table 1 Some numerical results of $\varLambda _{i}$ in Example 1 for $q=\frac{1}{10}$

Table 2 Some numerical results of $\varLambda _{i}$ in Example 1 for $q= \frac{1}{2}$

Table 3 Some numerical results of $\varLambda _{i}$ in Example 1 for $q= \frac{6}{7}$

Table 4 Some numerical results of Σ in Example 1 for $q= \frac{1}{10}$

Table 5 Some numerical results of Σ in Example 1 for $q= \frac{1}{2}$

Table 6 Some numerical results of Σ in Example 1 for $q= \frac{6}{7}$

Example 2

Consider the singular system of fractional q-differential equations

$$ \textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{ \frac{9}{10}} [k_{1}](t) = w_{1}(t, k_{1}), \\ {}^{c}\mathcal{D}_{q}^{\frac{9}{5}} [k_{2}](t) = w_{2}(t, k_{1}, k_{2}) , \\ {}^{c}\mathcal{D}_{q}^{\frac{17}{6}} [k_{3}](t) =w_{3}(t, k_{1}, k_{2}, k_{3}), \\ {}^{c}\mathcal{D}_{q}^{\frac{24}{7}} [k_{4}](t) =w_{4}(t, k_{1}, k_{2}, k_{3}, k_{4}), \\ {}^{c}\mathcal{D}_{q}^{\frac{13}{3}} [k_{5}](t) = w_{5}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}) , \end{cases} $$

(29)

with boundary value conditions $k_{1}(0) = \frac{2}{3}$,

$$ \textstyle\begin{cases} k_{2}(0) =-1, \\ k_{3}(0) = 1, \qquad k'_{3}(0)=\frac{2}{3}, \\ k_{4}(0)=\sqrt{7},\qquad k'_{4}(0)=\frac{\sqrt{7}}{3}, \qquad k''_{4}(0)= \frac{\sqrt{5}}{3}, \\ k_{5}(0)=\frac{2}{3},\qquad k'_{5}(0)=\frac{6}{5}, \qquad k''_{5}(0)= \frac{3}{8}, \qquad k'''_{5}(0)=\frac{2\sqrt{2}}{5}, \end{cases} $$

${}^{c}\mathcal{D}_{q}^{ \frac{1}{7}} [k_{2}](1) = {}^{c}\mathcal{D}_{q}^{\frac{8}{5}} [k_{3}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{11}{4}} [k_{4}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{7}{2}} [k_{5}](1) =0$, where $t\in (0,1]$. Put

$$\begin{aligned} &w_{1} (t, k_{1}) = \frac{ \cos k_{1}(t) }{ 8\pi \sqrt{t} \exp (t)}, \\ &w_{2} (t, k_{1}, k_{2}) = \frac{ 2 \cos ( k_{1}(t) + k_{2}(t)) }{ 15\pi \sqrt[3]{t} ( 1 + \sin (k_{1}(t) + k_{2}(t) ) }, \\ &w_{3} (t, k_{1}, k_{2}, k_{3}) = \frac{ 5(1+ \sin k_{1}(t) + \sin k_{2}(t) + \sin k_{3}(t)) }{21\pi \sqrt[4]{t}}, \\ &w_{4} (t, k_{1}, k_{2}, k_{3}, k_{4}) = \frac{3\exp (2t) \cos ^{2} (k_{1}(t) + k_{3}(t))}{8\pi \sqrt[5]{t} ( 1+ \cos ^{2} (k_{2}(t) + k_{4}(t)) ) }, \\ &w_{5} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}) = \frac{ \exp (-t) \sin (k_{1}(t) + k_{2}(t) + k_{3}(t)+ k_{4} (t) ) }{9\pi \sqrt[4]{t^{3}} ( 1 + \sin ( k_{5}(t) ) ) }, \end{aligned}$$

$m=5$, $\sigma _{1} =\frac{9}{10} \in (0,1)$, $\sigma _{2} = \frac{9}{5} \in (1,2)$, $\sigma _{3} = \frac{17}{6} \in (2,3)$, $\sigma _{4}= \frac{24}{7} \in (3, 4)$, $\sigma _{5} = \frac{13}{3} \in (4,5)$, $\zeta _{1}= \frac{2}{11} \in [0,1]$, $\zeta _{2}= \frac{5}{3} \in [1,2]$, $\zeta _{3}= \frac{7}{3} \in [2,3]$, $\zeta _{4}= \frac{13}{4} \in [3,4]$, ${}_{1}b_{0} = \frac{2}{3}$, ${}_{2}b_{0} = -1$, ${}_{3}b_{0} = 1$, ${}_{4}b_{0} = \sqrt{7}$, ${}_{5}b_{0} = \frac{2}{3}$, ${}_{3}b_{1} = \frac{2}{3}$, ${}_{4}b_{1} = \frac{ \sqrt{7}}{3}$, ${}_{5}b_{1} = 1$, ${}_{4}b_{2} = \frac{\sqrt{5}}{3}$, ${}_{5}b_{2} = \frac{3}{8}$, and ${}_{5}b_{3} = \frac{2\sqrt{2}}{5}$. Now, we check (23) and (24). For each $t \in \overline{J}$ and $(k_{1}, k_{2}, \ldots , k_{5}) \in \mathbb{R}^{5}$, we have

$$\begin{aligned} L_{1} & = \max_{ t \in [0,1] } t^{\alpha _{1}} \bigl\vert w_{1}\bigl(t, k_{1}(t)\bigr) \bigr\vert \leq \max _{ t \in [0,1] } t^{\frac{3}{4}} \biggl\vert \frac{ \cos k_{1}(t) }{ 8\pi \sqrt{t} \exp (t)} \biggr\vert \leq \frac{1}{8\pi } \end{aligned}$$

for $\alpha _{1}=\frac{3}{4}$,

$$\begin{aligned} L_{2} & = \max_{ t \in [0,1] } t^{\alpha _{2}} \bigl\vert w_{2}\bigl(t, k_{1}(t), k_{2}(t)\bigr) \bigr\vert \\ & \leq \max_{ t \in [0,1] } t^{\frac{2}{3}} \biggl\vert \frac{ 2 \cos ( k_{1}(t) + k_{2}(t)) }{ 15\pi \sqrt[3]{t} ( 1 + \sin (k_{1}(t) + k_{2}(t) ) } \biggr\vert \leq \frac{2}{15\pi } \end{aligned}$$

for $\alpha _{2}=\frac{2}{3}$,

$$\begin{aligned} L_{3} & = \max_{ t \in [0,1] } t^{\alpha _{3}} \bigl\vert w_{3}\bigl(t, k_{1}(t), k_{2}(t), k_{3}(t)\bigr) \bigr\vert \\ & \leq \max_{ t \in [0,1] } t^{\frac{4}{5}} \biggl\vert \frac{ 5(1+ \sin k_{1}(t) + \sin k_{2}(t) + \sin k_{3}(t)) }{21\pi \sqrt[4]{t}} \biggr\vert \leq \frac{20}{21\pi } \end{aligned}$$

for $\alpha _{3} = \frac{4}{5}$,

$$\begin{aligned} L_{4} & = \max_{ t \in [0,1] } t^{\alpha _{4}} \bigl\vert w_{4}\bigl(t, k_{1}(t), k_{2}(t), k_{3}(t), k_{4}(t)\bigr) \bigr\vert \\ & \leq \max_{ t \in [0,1] } t^{\frac{1}{2}} \biggl\vert \frac{3\exp (2t) \cos ^{2} (k_{1}(t) + k_{3}(t))}{8\pi \sqrt[5]{t} ( 1+ \cos ^{2} (k_{2}(t) + k_{4}(t)) ) } \biggr\vert \leq \frac{3e^{2}}{8 \pi } \end{aligned}$$

for $\alpha _{4}=\frac{1}{2}$,

$$\begin{aligned} L_{5} & = \max_{ t \in [0,1] } t^{\alpha _{5}} \bigl\vert w_{5}\bigl(t, k_{1}(t), k_{2}(t), k_{3}(t), k_{4}(t), k_{5}(t)\bigr) \bigr\vert \\ & \leq \max_{ t \in [0,1] } t^{\frac{8}{9}} \biggl\vert \frac{ \exp (-t) \sin (k_{1}(t) + k_{2}(t) + k_{3}(t)+ k_{4} (t) ) }{9\pi \sqrt[4]{t^{3}} ( 1 + \sin ( k_{5}(t) ) ) } \biggr\vert \leq \frac{1}{9\pi } \end{aligned}$$

for $\alpha _{5} = \frac{8}{9}$. Now, by using (11), we get

$$\begin{aligned} &\varLambda _{1} = \frac{ \varGamma _{q}(1- \alpha _{1})}{ \varGamma _{q}(\sigma _{1} + 1- \alpha _{1}) } = \frac{ \varGamma _{q} ( 1- \frac{3}{4} ) }{ \varGamma _{q} ( \frac{9}{10}+ 1- \frac{3}{4} ) }= \frac{ \varGamma _{q} ( \frac{1}{4} ) }{ \varGamma _{q} ( \frac{23}{20} ) }, \\ &\begin{aligned} \varLambda _{2} &= \frac{ \varGamma _{q} ( 1 - \alpha _{2} ) }{ \varGamma _{q} ( \sigma _{2} + 1 -\alpha _{2} ) } + \frac{ \varGamma _{q} ( 2 - \zeta _{1} ) \varGamma _{q} ( 1- \alpha _{2} ) }{ \varGamma _{q} ( \sigma _{2} - \zeta _{1} +1-\alpha _{2} ) } \\ & = \frac{ \varGamma _{q} ( 1 - \frac{2}{3} ) }{ \varGamma _{q} ( \frac{9}{5} + 1 - \frac{2}{3} ) } + \frac{ \varGamma _{q} ( 2 - \frac{1}{7} ) \varGamma _{q} ( 1- \frac{2}{3} ) }{ \varGamma _{q} ( \frac{9}{5} - \frac{1}{7} + 1 - \frac{2}{3} ) } \\ & = \frac{ \varGamma _{q} ( \frac{1}{3} ) }{ \varGamma _{q} ( \frac{32}{15} ) } + \frac{ \varGamma _{q} ( \frac{13}{7} ) \varGamma _{q} ( \frac{1}{3} ) }{ \varGamma _{q} ( \frac{209}{105} ) }, \end{aligned} \\ &\begin{aligned} \varLambda _{3} &= \frac{ \varGamma _{q} ( 1 - \alpha _{3} ) }{ \varGamma _{q} ( \sigma _{3} + 1 -\alpha _{3} ) } + \frac{ \varGamma _{q} ( 3 - \zeta _{2} ) \varGamma _{q} ( 1- \alpha _{3} ) }{ 2! \varGamma _{q} ( \sigma _{3} - \zeta _{2} + 1 - \alpha _{3} )} \\ & = \frac{ \varGamma _{q} ( 1 - \frac{4}{5} ) }{ \varGamma _{q} ( \frac{17}{6} + 1 -\frac{4}{5} ) } + \frac{ \varGamma _{q} ( 3 - \frac{8}{5} ) \varGamma _{q} ( 1- \frac{4}{5} ) }{ 2! \varGamma _{q} ( \frac{17}{6} - \frac{8}{5} + 1 - \frac{4}{5} )} \\ & = \frac{ \varGamma _{q} ( \frac{1}{5} ) }{ \varGamma _{q} ( \frac{79}{30} ) } + \frac{ \varGamma _{q} ( \frac{7}{5} ) \varGamma _{q} ( \frac{1}{5} ) }{ 2! \varGamma _{q} ( \frac{43}{30} )}, \end{aligned} \\ &\begin{aligned} \varLambda _{4} &= \frac{ \varGamma _{q} ( 1 - \alpha _{4} ) }{ \varGamma _{q} ( \sigma _{4} + 1 -\alpha _{4} ) } + \frac{ \varGamma _{q} ( 4 - \zeta _{3} ) \varGamma _{q} ( 1- \alpha _{4} ) }{ 3! \varGamma _{q} ( \sigma _{4} - \zeta _{3} + 1 - \alpha _{4} )} \\ &= \frac{ \varGamma _{q} ( 1 - \frac{1}{2} ) }{ \varGamma _{q} ( \frac{24}{7} + 1 - \frac{1}{2} ) } + \frac{ \varGamma _{q} ( 4 - \frac{11}{4} ) \varGamma _{q} ( 1- \frac{1}{2} ) }{ 3! \varGamma _{q} ( \frac{24}{7} - \frac{11}{4} + 1 - \frac{1}{2} )} \\ & = \frac{ \varGamma _{q} ( \frac{1}{2} ) }{ \varGamma _{q} ( \frac{55}{14} ) } + \frac{ \varGamma _{q} ( \frac{5}{4} ) \varGamma _{q} ( \frac{1}{2} ) }{ 3! \varGamma _{q} ( \frac{5}{4} )}, \end{aligned} \\ &\begin{aligned} \varLambda _{5} &= \frac{ \varGamma _{q} ( 1 - \alpha _{5} ) }{ \varGamma _{q} ( \sigma _{5} + 1 -\alpha _{5} ) } + \frac{ \varGamma _{q} ( 5 - \zeta _{4} ) \varGamma _{q} ( 1- \alpha _{5} ) }{ 4! \varGamma _{q} ( \sigma _{5} - \zeta _{4} + 1 - \alpha _{5} )} \\ & = \frac{ \varGamma _{q} ( 1 - \frac{8}{9} ) }{ \varGamma _{q} ( \frac{13}{3} + 1 -\frac{8}{9} ) } + \frac{ \varGamma _{q} ( 5 - \frac{7}{2} ) \varGamma _{q} ( 1- \frac{8}{9} ) }{ 4! \varGamma _{q} ( \frac{13}{3} - \frac{7}{2} + 1 - \frac{8}{9} )} \\ & = \frac{ \varGamma _{q} ( \frac{1}{9} ) }{ \varGamma _{q} ( \frac{40}{9} ) } + \frac{ \varGamma _{q} ( \frac{3}{2} ) \varGamma _{q} ( \frac{1}{9} ) }{ 4! \varGamma _{q} ( \frac{17}{18} )}. \end{aligned} \end{aligned}$$

Tables 7, 8, and 9 show $\varLambda _{i} \approx 2.0428$, 3.2300, 3.3499, 1.2683, 3.2252, $\varLambda _{i} \approx 3.812$, 4.3215, 4.2023, 0.8837, 2.1222, $\varLambda _{i} \approx 3.6791$, 4.8820, 4.4534, 0.6683, 1.2984 for $1 \leq i \leq 5$ and $q=\frac{1}{10}$, $\frac{1}{2}$, $\frac{6}{7}$, respectively. Now, by using (24) and Algorithm 7, we conclude next results. According to Tables 10, 11, and 12, consider the set $K_{r} \subset S$ as

$$\begin{aligned} &K_{r} = \bigl\{ (k_{1}, k_{2}, \ldots , k_{m}) \in S: \bigl\Vert (k_{1}, k_{2}, \ldots , k_{m}) \bigr\Vert \leq 5.0190 \bigr\} , \\ &K_{r} = \bigl\{ (k_{1}, k_{2}, \ldots , k_{m}) \in S: \bigl\Vert (k_{1}, k_{2}, \ldots , k_{m}) \bigr\Vert \leq 4.6798 \bigr\} , \\ &K_{r} = \bigl\{ (k_{1}, k_{2}, \ldots , k_{m}) \in S: \bigl\Vert (k_{1}, k_{2}, \ldots , k_{m}) \bigr\Vert \leq 4.4896 \bigr\} , \end{aligned}$$

for $q=\frac{1}{10}$, $\frac{1}{2}$, and $\frac{6}{7}$, respectively. Table 10 shows that $L_{1} \varLambda _{1} + |{}_{1}b_{0}| \approx 0.0812$, $L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!} \approx 1.1371$, 2.6822, 5.0190, 2.0625, 5.0190. Table 11 shows $L_{1} \varLambda _{1} + |{}_{1}b_{0}| \approx 0.1226$, $L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!} \approx 1.1834$, 2.9406, 4.6798, 2.0235, Table 12 shows that $L_{1} \varLambda _{1} + |{}_{1}b_{0}| \approx 0.1463$, $L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!} \approx 1.2072$, 3.0164, 4.4898, 1.9944 for $2 \leq i \leq 5$ and $q=\frac{1}{10}$, $\frac{1}{2}$, $\frac{6}{7}$, respectively. Also, Table 13 shows us $r \approx 5.0190$, 4.6798, 4.4898 for $q=\frac{1}{10}$, $\frac{1}{2}$, $\frac{6}{7}$, respectively (Figs. 3 and 2). Now, by using Theorem 7, the singular system of fractional q-differential equations (29) has a solution.

Figure 2

Numerical results of r for $q= \frac{1}{10}, \frac{1}{2}, \frac{6}{7}$ in Example 2

Figure 3

Numerical results of r for $q= \frac{1}{10}, \frac{1}{2}$, and $\frac{6}{7}$ in Example 2

Algorithm 7

The proposed method for solving problem (29) in Example 2 for which we use the conditions of Theorem 7

Table 7 Some numerical results of $\varLambda _{i}$ in Example 2 for $q=\frac{1}{10}$

Table 8 Some numerical results of $\varLambda _{i}$ in Example 2 for $q= \frac{1}{2}$

Table 9 Some numerical results of $\varLambda _{i}$ in Example 2 for $q= \frac{6}{7}$

Table 10 Numerical results of $(1)=L_{1} \varLambda _{1} + |{}_{1}b_{0}|$ and $(i)=L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!}$ with $2 \leq i \leq 5$ in Example 2 for $q=\frac{1}{10}$

Table 11 Numerical results of $(1)=L_{1} \varLambda _{1}+ |{}_{1}b_{0}|$ and $(i)=L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!}$ with $2 \leq i \leq 5$ in Example 2 for $q=\frac{1}{10}$

Table 12 Numerical results of $(1)=L_{1} \varLambda _{1}+ |{}_{1}b_{0}|$ and $(i)= L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!}$ with $2 \leq i \leq 5$ in Example 2 for $q=\frac{7}{6}$

Table 13 Numerical results of r in Example 2 for $q=\frac{1}{10}, \frac{1}{2}, \frac{7}{6}$