## 1 Introduction

The fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. Also, the quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. It defines q-calculus, where q stands for quantum. Despite the old history of the theories, the investigation of their properties has remained untouched until recent time. After its introduction by Jackson in 1910 [1], many researchers have extended this field (see, for example, [28]). It is important that we increase our abilities by investigating complicate fractional differential equations and applications (see [924]). One of the methods in this way is working on fractional differential inclusions which are an appropriate extension for fractional differential equations [2530]. Finally, it is known that fractional difference equations need time scales as a discrete system [31, 32].

In 2013, Ahmad et al. studied the fractional inclusion problem $${}^{c}\mathcal{D}^{\beta }k(t) \in T(t, k(t))$$ with the integral boundary conditions $$k^{j} (0) - c_{i} k^{j} (\delta ) = a_{i} \int _{0}^{1} f_{j}(r , k(r) ) \,\mathrm{d}r$$ for $$j = 0,1,2$$, where T is a multifunction [26]. In 2014, Ghorbanian et al. investigated the existence of solution for the fractional differential inclusion problems

$${}^{c}\mathcal{D}^{\sigma _{1}} [z](t) \in T_{1} \bigl(t, z(t), z' (t), z''(t) \bigr)$$

and

$${}^{c}\mathcal{D}^{\sigma _{2}} [z](t) \in T_{2} \bigl(t, z(t), \qquad {}^{c} \mathcal{D}^{\beta _{1} } [z](t), \ldots , {}^{c}\mathcal{D}^{\beta _{n}} [z](t)\bigr),$$

with integral boundary value conditions

$$\textstyle\begin{cases} z(0) + z (\eta ) + z(1) = \int _{0}^{1} f_{0}(r, z(r)) \,\mathrm{d}r, \\ {}^{c}\mathcal{D}^{\zeta }[z](0) + {}^{c}\mathcal{D}^{\zeta }[z](\eta ) + {}^{c}\mathcal{D}^{\zeta }[z](1 ) = \int _{0}^{1} f_{1} (r, z(r)) \,\mathrm{d}r, \\ {}^{c}\mathcal{D}^{\beta }[z](0) + {}^{c}\mathcal{D}^{\beta }[z](\eta ) + {}^{c}\mathcal{D}^{\beta }[z](1) = \int _{0}^{1} f_{2} (r, z(r)) \,\mathrm{d}r, \\ z(0) + a z(1) = \sum_{i=1}^{n} \mathcal{I}^{\beta _{i}} [z](\eta ), \\ Z'(0) + b z'(1) = \sum_{i=1}^{n} {}^{c}\mathcal{D}^{ \beta _{i}} [z](\eta ), \end{cases}$$

where $$t \in J$$, $$2<\sigma _{1} \leq 3$$, $$1<\sigma _{2}\leq 2$$, $$0<\eta , \zeta , \beta _{i} < 1$$, $$1< \beta <2$$, $$\sigma _{2} -\beta _{i}\geq 1$$, for $$1\leq i \leq n$$,

$$a > \sum_{i=1}^{n} \frac{ \eta ^{\beta _{i}+1}}{ \varGamma (\beta _{i}+2)}, \qquad b > \sum_{i=1}^{n} \frac{ \eta ^{ 1 - \beta _{i}}}{ \varGamma ( 2 - \beta _{i})},$$

$$n \in \mathbb{N}$$, $$T_{1} : J \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \to P_{\mathrm{cp}}( \mathbb{R})$$, $$T_{2} : J\times \mathbb{R}^{n+1}\to P_{\mathrm{cp}}(\mathbb{R})$$ are multifunctions, $$f_{i} : J \times \mathbb{R} \to \mathbb{R}$$ are continuous functions for $$i=0,1,2$$, and $$P_{\mathrm{cp}}(\mathbb{R})$$ is the set of all compact subsets of $$\mathbb{R}$$ [28]. In 2015, Agarwal et al. reviewed the fractional derivative inclusions $${}^{c}\mathcal{D}^{\beta }[x](t) \in F_{1}(t,x(t))$$ and $${}^{c}\mathcal{D}^{\beta }[x] (t) \in F_{2}(t, x(t), {}^{c}\mathcal{D}^{ \zeta } x(t))$$ with the boundary value conditions $$x(0) = a \int _{0}^{\nu }x(s)ds$$, $$x(1) = b \int _{0}^{\eta }x(s) \,\mathrm{d}s$$ and $$x(1) + x' (1) = \int _{0}^{\eta } x(s) \,\mathrm{d}s$$, $$x(0) = 0$$, respectively, where $$t\in J$$, $$\zeta , \eta , \nu \in (0,1)$$, $$\beta \in (1,2]$$ with $$\beta - \zeta >1$$, $$a, b \in \mathbb{R}$$, $${}^{c}\mathcal{D}^{\beta }$$ is the Caputo differentiation and $$F_{1} : J\times \mathbb{R}\times \mathbb{R}\to 2^{\mathbb{R}}$$, $$F_{2} : J \times \mathbb{R} \times \mathbb{R} \to 2^{\mathbb{R}}$$ are compact-valued multifunctions [25]. In 2019, Samei et al. studied the existence of solutions for the hybrid inclusion

$$\textstyle\begin{cases} {}^{C}_{H}D^{\alpha } [ \frac{x(t)- f (t, x(t), I^{\beta _{1}} h_{1} (t, x(t)), I^{\beta _{2}} h_{2}(t, x(t)), \ldots , I^{\beta _{n}} h_{n}( t, x(t)) )}{ g (t, x(t), I^{\gamma _{1}} [x](t), I^{\gamma _{2}} [x] (t), \ldots , I^{\gamma _{m}} [x](t) )} ] \in K (t, x(t)), \\ x (1) = \mu (x),\quad x(e) = \eta (x), \end{cases}$$

where $${}^{C}_{H} \mathcal{D}^{\alpha }$$ and $${}^{H}\mathcal{I}^{\alpha }$$ denote the Caputo–Hadamard fractional derivative and Hadamard integral of order α, respectively, $$t \in J = [1, e]$$, $$n, m \in \mathbb{N}$$, $$1 < \alpha \leq 2$$, $$\beta _{i} >0$$ for $$i=1, 2, \ldots , n$$, $$\gamma _{i} > 0$$ for $$i=1,2, \ldots , m$$, the functions $$f : J\times \mathbb{R}^{n+1} \to \mathbb{R}$$, $$g : J \times \mathbb{R}^{m+1} \to \mathbb{R} -\{0\}$$, $$h_{i}: J \times \mathbb{R} \to \mathbb{R}$$ for $$i= 1,2, \ldots , n$$, functions μ, η map $$C(J, \mathbb{R})$$ into $$\mathbb{R}$$ and the multifunction $$K: J \times \mathbb{R} \to P(\mathbb{R} )$$ satisfies certain conditions [30]. Also, Ntouyas et al. studied the boundary value problem of first-order fractional differential equations given by

$${}^{c}D_{0^{+}}^{\beta _{1}} [f_{1}] (x) = w_{1} \bigl(x, f_{1}(x), f_{2}(x)\bigr), \qquad {}^{c}D_{0^{+}}^{\beta _{2}} [f_{2}] (x) = w_{2} \bigl(x, f_{1}(x), f_{2}(x) \bigr),\quad \bigl(t \in [0,1]\bigr)$$

with Riemann–Liouville integral boundary conditions of different order $$f_{1}(0) = c_{1} I^{\alpha _{1}} [f_{1}] (a_{1})$$ and $$f_{2}(0) = c_{2} I^{\alpha _{2}} [f_{2}](a_{2})$$ for $$0 < a_{1}, a_{2} <1$$, $$\beta _{i}\in (0, 1]$$, $$\alpha _{i}, c_{i} \in \mathbb{R}$$ where $$i=1,2$$ [33]. On the other hand, Samei and Ntouyas investigated a multi-term nonlinear fractional q-integro-differential equation

$${}^{c}D_{q}^{\alpha } [x](t) = w \bigl( t, x(t), (\varphi _{1} x) (t), ( \varphi _{2} x) (t), \qquad {}^{c}D_{q}^{\beta _{1}} [x](t), {}^{c}D_{q}^{ \beta _{2}} [x](t), \ldots , {}^{c}D_{q}^{ \beta _{n}} [x](t) \bigr)$$

under some boundary conditions [7]. In 2019, Samei et al. discussed the fractional hybrid q-differential inclusions

$${}^{c}\mathcal{D}_{q}^{\alpha } \biggl( \frac{k}{ f ( t, k, \mathcal{I}_{q}^{\alpha _{1}} [k], \ldots , \mathcal{I}_{q}^{\alpha _{n}} [k] )} \biggr) \in F \bigl( t, k, \mathcal{I}_{q}^{ \beta _{1}} [k] , \ldots , \mathcal{I}_{q}^{ \beta _{k}} [k] \bigr)$$

with the boundary conditions $$k(0) = k_{0}$$ and $$k(1) = k_{1}$$, where $$1 < \alpha \leq 2$$, $$q \in (0,1)$$, $$k_{0}, k_{1} \in \mathbb{R}$$, $$\alpha _{i} >0$$ for $$i=1, 2, \ldots , n$$, $$\beta _{j} > 0$$ for $$j=1, 2, \ldots , m$$, $$n, m\in \mathbb{N}$$, $${}^{c}\mathcal{ D}_{q}^{\alpha }$$ denotes Caputo type q-derivative of order α, $$\mathcal{I}_{q}^{\beta }$$ denotes Riemann–Liouville type q-integral of order β, $$f : J \times \mathbb{R}^{n} \to (0,\infty )$$ is continuous, and $$F : J \times \mathbb{R}^{m} \to P(\mathbb{R})$$ is a multifunction [34].

Now by mixing the main ideas of the works, we investigate the existence of solutions for a system of fractional q-differential inclusions via sum of two multi-term functions:

$$\textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{\sigma _{1}} [k_{1}](t) \in \mathcal{T}_{1i} (t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}] (t), \ldots , \mathcal{I}_{q}^{\zeta _{m}} [k_{m}(t)] ) \\ \hphantom{{}^{c}\mathcal{D}_{q}^{\sigma _{1}} [k_{1}](t) \in}{} + \mathcal{T}_{21} ( t, k_{1}(t), \ldots , k_{m}(t), {}^{c} \mathcal{D}_{q}^{\zeta _{1}} [k_{1} ](t), \ldots , {}^{c}\mathcal{D}_{q}^{ \zeta _{m}} [k_{m}] (t) ), \\ {}^{c}\mathcal{D}_{q}^{ \sigma _{2}} [k_{2}](t) \in \mathcal{T}_{12} ( t, k_{1} (t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t) ) \\ \hphantom{{}^{c}\mathcal{D}_{q}^{ \sigma _{2}} [k_{2}](t) \in}{} + \mathcal{T}_{22} ( t, k_{1}(t), \ldots , k_{m}(t), {}^{c} \mathcal{D}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , {}^{c}\mathcal{D}_{q}^{ \zeta _{m}} [k_{m}]( t ) ), \\ \vdots \\ {}^{c}\mathcal{D}_{q}^{\sigma _{m}} [k_{m}](t) \in \mathcal{T}_{1m} ( t, [k_{1}](t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}] (t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t) ) \\ \hphantom{{}^{c}\mathcal{D}_{q}^{\sigma _{m}} [k_{m}](t) \in}{} + \mathcal{T}_{2m} ( t, k_{1}(t), \ldots , k_{m}(t), {}^{c} \mathcal{D}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , {}^{c}\mathcal{D}_{q}^{ \zeta _{m}} [k_{m}](t) ), \end{cases}$$
(1)

with the boundary conditions

\begin{aligned} &k_{i}(0) + \mathcal{I}_{q}^{\zeta _{i}} [k_{i}](0) + {}^{c} \mathcal{D}_{q}^{\zeta _{i}} [k_{i}](0) = -k_{i} (1), \\ &k_{i}(1) + \mathcal{I}_{q}^{\zeta _{i}} [k_{i}](1) + {}^{c} \mathcal{D }_{q}^{\zeta _{i}} [k_{i}](1)= - k_{i}(0), \end{aligned}
(2)

where $$1 < \sigma _{i} \leq 2$$, $$0< \zeta _{i} < 1$$, $$t \in \overline{J} = [0,1]$$, and $$\mathcal{T}_{1i}$$, $$\mathcal{T}_{2i}: \overline{J} \times \mathbb{R}^{2m}\to 2^{ \mathbb{R}}$$ are some set-valued maps for $$i = 1, \ldots , m$$.

## 2 Essential preliminaries

In this work, we apply the time scales calculus notation of the book [32]. In fact, we consider the fractional q-calculus on the specific 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 $$( x - y)_{q}^{(0)}=1$$ and $$(x-y)_{q}^{(n)} = \prod_{k=0}^{n-1} (x - yq^{k})$$ for $$n\geq 1$$, where x and y are real numbers [31]. Also, for $$\alpha \in \mathbb{R}$$ and $$a \neq 0$$, we have $$(x - y)_{q}^{ (\alpha ) } = x^{\alpha }\prod_{k=0}^{\infty }\frac{x - y q^{k}}{x - yq^{ \alpha + k}}$$. If $$y=0$$, then it is clear that $$x^{ (\alpha ) } = x^{\alpha }$$ [31] (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 by

$$I_{q} f(x) = \int _{0}^{x} f(s)\,\mathrm{d}_{q} s = x(1- q) \sum_{k=0}^{ \infty } q^{k} f\bigl(x q^{k}\bigr)$$

for $$0 \leq x \leq b$$, provided the series is absolutely convergent [2, 3]. If x in $$[0, T]$$, then

\begin{aligned} \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], \end{aligned}

whenever the series exists. 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 $$n \geq 1$$ 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

\begin{aligned} \mathcal{I}_{q}^{\sigma }[h](t)& = \frac{1}{\varGamma _{q}(\sigma )} \int _{0}^{t} (t- qs)^{(\sigma - 1)} h(s) \, \mathrm{d}_{q}s \\ & = 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}
(3)

and $$\mathcal{I}_{q}^{0} [h](t) = h(t)$$ for $$t \in J$$ [5]. Also, the Caputo fractional q-derivative of a function w is defined by

\begin{aligned} {}^{c}\mathcal{D}_{q}^{\sigma }[w] (t) & = \mathcal{I}_{q}^{[ \sigma ]-\sigma } \bigl[ {}^{c} \mathcal{D}_{q}^{[\sigma ]} [w] \bigr] (t) \\ & = \frac{1}{\varGamma _{q} ([\sigma ] - \alpha )} \int _{0}^{t} (t- qs)^{ ( [\sigma ]-\sigma -1 )} {}^{c}\mathcal{D}_{q}^{[ \sigma ]} [w] (s) \, \mathrm{d}_{q}s \\ & = \frac{1}{t^{\sigma }(1-q)^{\sigma }} \sum_{k=0}^{ \infty } q^{k} \frac{ \prod_{i=1}^{k - 1} (1-q^{ i-\sigma } )}{ \prod_{i=1}^{k - 1} (1 - q^{i +1} ) } w\bigl(t q^{k}\bigr), \end{aligned}
(4)

where $$t \in J$$ and $$\sigma >0$$ [5, 35]. 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 $$\alpha , \beta \geq 0$$ [5]. Algorithm 4 shows pseudo-code $$\mathcal{I}_{q}^{\alpha }[h](x)$$.

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, $$\mathcal{P}_{\mathrm{cl}}( \mathcal{E})$$, $$\mathcal{P}_{\mathrm{bd}}( \mathcal{E})$$, $$\mathcal{P}_{\mathrm{cv}}( \mathcal{E})$$, and $$\mathcal{P}_{\mathrm{cp}}( \mathcal{E})$$ denote the class of all closed, bounded, convex, and compact subsets of $$\mathcal{E}$$, respectively. A mapping $$\mathcal{T}: \mathcal{E}\to 2^{\mathcal{E}}$$ is called a multifunction on $$\mathcal{E}$$ and $$e\in \mathcal{E}$$ is called a fixed point of $$\mathcal{T}$$ whenever $$e\in \mathcal{T}(e)$$. A multifunction $$\mathcal{T} : \mathcal{E}\to \mathcal{P}_{\mathrm{cl}}( \mathcal{E})$$ is lower semi-continuous if, for any open set $$\mathcal{O}$$ of $$\mathcal{E}$$, the set

$$\mathcal{T}^{-1} (\mathcal{O}) : = \bigl\{ z\in \mathcal{E} : \mathcal{T} (z) \cap \mathcal{O} \neq \emptyset \bigr\}$$

is open [29]. If the set $$\{ z\in \mathcal{X} : \mathcal{T}(z) \subset \mathcal{O}\}$$ is open for every open set $$\mathcal{O}$$ of $$\mathcal{E}$$, then we say that $$\mathcal{T}$$ is upper semi-continuous [29]. Also, $$\mathcal{T} : \mathcal{E}\to \mathcal{P}_{\mathrm{cp}}( \mathcal{E})$$ is called compact if $$\overline{ \mathcal{T} (\mathcal{B})}$$ is compact for each bounded subset $$\mathcal{B}$$ of $$\mathcal{E}$$ [29]. A multifunction $$\mathcal{T}=[0,1] : \overline{J} \to \mathcal{P}_{\mathrm{cl}}( \mathbb{R})$$ is said to be measurable whenever, for each $$y \in \mathbb{R}$$, the function $$t \mapsto \rho ( y, \mathcal{T}(t)) = \inf \{ | y - z| : z \in \mathcal{T}(t) \}$$ is measurable [36]. The Pompeiu–Hausdorff metric $$P_{\rho }: 2^{\mathcal{E}}\times 2^{\mathcal{E}}\to [0,\infty )$$ is defined by

$$P_{\rho }(S,T) = \max \Bigl\{ \sup _{s\in S} \rho (s,T), \sup_{t\in T} \rho (S,t) \Bigr\} ,$$
(5)

where $$\rho (S,t)= \inf_{s\in S} \rho (s; t)$$ [29]. Then $$(\mathcal{P}_{b,\mathrm{cl}}( \mathcal{E}), P_{\rho })$$ is a metric space and $$(\mathcal{P}_{\mathrm{cl}}( \mathcal{E}), P_{\rho })$$ is a generalized metric space [29]. A multifunction $$\mathcal{T} : \mathcal{E} \to \mathcal{P}_{\mathrm{cl}}( \mathcal{E})$$ is called λ-contraction whenever there exists $$\lambda \in (0,1)$$ such that $$P_{\rho } ( \mathcal{T}( e_{1} ), \varTheta (e_{2})) \leq \lambda \rho (e_{1}, e_{2})$$ for all $$e_{1}, e_{2}\in \mathcal{E}$$.

In 1970, Covitz and Nadler proved that each closed-valued contractive multifunction on a complete metric space has a fixed point [37]. We say that $$\mathcal{T}: \overline{J}\times \mathbb{R}^{2m} \to 2^{\mathbb{R}}$$ is a Carathéodory multifunction whenever $$t \mapsto \mathcal{T} (t, r_{1}, \ldots , r_{2m} )$$ is measurable for all $$r_{1}, \ldots , r_{2m}\in \mathbb{R}$$ and $$(r_{1}, \ldots , r_{2m}) \mapsto \mathcal{T} (t, r_{1}, \ldots , r_{2m} )$$ is an upper semi-continuous map for almost all $$t \in \overline{J}$$ [27, 29, 38]. Also, a Carathéodory multifunction $$\mathcal{T}: \overline{J} \times \mathbb{R}^{2m} \to 2^{ \mathbb{R}}$$ is called $$L^{1}$$-Carathéodory whenever, for each $$\eta >0$$, there exists $$\varUpsilon _{\eta } \in L^{1} (\overline{J}, \mathbb{R}^{+})$$ such that

$$\bigl\Vert \mathcal{T} ( t, r_{1}, \ldots , r_{2m} ) \bigr\Vert = \sup \bigl\{ \vert k \vert : k \in \mathcal{T} (t, r_{1}, \ldots , r_{2m}) \bigr\} \leq \varUpsilon _{ \eta }(t)$$

for all $$|r_{1}|, \ldots , |r_{2m}| \leq \eta$$ and for almost all $$t\in \overline{J}$$ [27, 29, 38]. For each i, define the space $$E_{i} = \{ k(t) : k(t), {}^{c}\mathcal{D}_{q}^{ \zeta _{i}} [k](t) \in \mathcal{A} \}$$ endowed with the norm

$$\Vert k \Vert _{i} = \max_{t\in \overline{J}} \bigl\vert k(t) \bigr\vert + \max_{t\in \overline{J}} \bigl\vert {}^{c}\mathcal{D}_{q}^{ \zeta _{i}} [k](t) \bigr\vert ,$$

where $$\mathcal{A}= C(\overline{J}, \mathbb{R})$$. Also, consider the product space $$\mathcal{E} = E_{1} \times \cdots \times E_{m}$$ endowed with the norm $$\|( k_{1}, \ldots , k_{m})\| = \sum_{i=1}^{m} \|k_{i}\|_{i}$$. Then $$(\mathcal{E}, \|. \|)$$ is a Banach space [39]. By using the idea of some works such as [26], define the set of the selections of $$\mathcal{S}_{1i}$$, $$\mathcal{S}_{2i}$$ at k by

\begin{aligned} &S_{\mathcal{T}_{1i}, k} = \bigl\{ p \in L^{1} (\overline{J}) : p(t) \in \mathcal{T}_{1i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{ \zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr) \bigr\} , \\ &S_{\mathcal{T}_{2i},k} = \bigl\{ p \in L^{1}(\overline{J}) : p(t) \in \mathcal{T}_{2i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), {}^{c}\mathcal{D}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , {}^{c} \mathcal{D}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr) \bigr\} \end{aligned}

for all $$t\in \overline{J}$$, $$k=(k_{1}, \dots ,k_{m}) \in \mathcal{E}$$ and $$1\leq i\leq m$$. One can check that $$S_{\mathcal{T}_{1i},k} \neq \emptyset$$ for all $$k\in \mathcal{E}$$ whenever $$\dim \mathcal{E} < \infty$$ [40]. We need the following results.

### Lemma 1

([38])

If$$\mathcal{T} : \mathcal{E} \to \mathcal{P}_{\mathrm{cl}}(\mathcal{F})$$is upper semicontinuous, then$$\operatorname{Gr}(\mathcal{T})$$is a closed subset of$$\mathcal{E}\times \mathcal{F}$$. Conversely, if$$\mathcal{T}$$is completely continuous and has a closed graph, then it is upper semicontinuous.

### Lemma 2

([40])

Let$$\mathcal{E}$$be a Banach space, $$\mathcal{T} : \overline{J}\times \mathcal{E}\to \mathcal{P}_{\mathrm{cp},\mathrm{cv}}( \mathcal{E})$$be an$$L^{1}$$-Carathéodory multifunction, and$$\mathscr{L}$$be a linear continuous mapping from$$L^{1}( \overline{J}, \mathcal{E})$$to$$C(\overline{J}, \mathcal{E})$$. Then the operator$$\mathscr{L} o S_{\mathcal{T}}: C( \overline{J}, \mathcal{E}) \to \overline{P}_{\mathrm{cp},\mathrm{cv}} ( C(\overline{J}), \mathcal{E})$$defined by$$(\mathscr{L} o S_{\mathcal{T}})(k) = \mathscr{L}( S_{\mathcal{T}, k} )$$is a closed graph operator in$$C(\overline{J},\mathcal{E}) \times C(\overline{J}, \mathcal{E})$$.

### Lemma 3

([41])

Let$$\mathcal{E}$$be a Banach space, $$\mathcal{F} \in \mathcal{P}_{\mathrm{bd}, \mathrm{cl}, \mathrm{cv}} ( \mathcal{E} )$$and$$\mathcal{M}, \mathcal{N} : \mathcal{F}\to \mathcal{P}_{\mathrm{cp}, \mathrm{cv}}( \mathcal{E})$$be two multi-valued operators. If$$\mathcal{M}(k) + \mathcal{N}(k) \subset \mathcal{F}$$for all$$k\in \mathcal{F}$$m, $$\mathcal{M}$$is a contraction and$$\mathcal{N}$$is upper semicontinuous and compact, then there exists$$k\in \mathcal{F}$$such that$$k \in \mathcal{M}(k) + \mathcal{N}(k)$$.

## 3 Main results

Now, we are ready to provide our main results.

### Lemma 4

Let$$z \in \mathcal{A}$$, $$\sigma \in (1,2]$$and$$\zeta \in (0,1)$$with$$\sigma - \zeta >1$$. Then the unique solution of the fractional problem$${}^{c}\mathcal{D}_{q}^{\sigma }[k](t) = z(t)$$with the boundary value conditions

\begin{aligned} & k(0) + \mathcal{I}_{q}^{\zeta } [k](0) + {}^{c}\mathcal{D}_{q}^{ \zeta } [k](0) = -k(1), \\ &k(1) + \mathcal{I}_{q}^{\zeta }[k](1) + {}^{c}\mathcal{D}_{q}^{\zeta }[k](1) = - k(0), \end{aligned}

is given by

\begin{aligned} k(t) ={}& \mathcal{I}_{q}^{\sigma }[z](t) + A_{1}(t, q, \zeta ) \mathcal{I}_{q}^{\sigma } [z](1) \\ &{} + A_{2}(t, q, \zeta ) \mathcal{I}_{q}^{ \sigma +\zeta } [z](1) + A_{3}(t, q, \zeta ) \mathcal{I}_{q}^{ \sigma -\zeta } [z](1) \\ ={}& \int _{0}^{1} G_{q}(t, r,\sigma , \zeta ) z(r) \,\mathrm{d}_{q}r, \end{aligned}

where

\begin{aligned}& \begin{aligned}&A_{1}(t, q, \zeta ) = \varSigma \bigl[ t \varGamma _{q}( \zeta + 2) \varGamma _{q}( 2 - \zeta ) + \varGamma _{q}( \zeta + 1 ) \varGamma _{q}( \zeta + 2 ) + \varGamma _{q}( \zeta + 1 ) \varGamma _{q}( 2 - \zeta )\bigr], \\ &A_{2}(t, q, \zeta ) = \varSigma \bigl[ ( 2 t - 1 ) \varGamma _{q}( \zeta + 1 ) \varGamma _{q}( \zeta + 2 ) \varGamma _{q} ( 2 - \zeta )\bigr], \\ &A_{3}(t, q, \zeta ) = \varSigma \bigl[ ( 2 t - 1 )\varGamma _{q} (\zeta + 1 ) \varGamma _{q} (\zeta + 2 ) \varGamma _{q}( 2 - \zeta )\bigr], \\ &\varSigma = \bigl[ \varGamma _{q}( \zeta + 2) \varGamma _{q} ( 2 - \zeta ) - 2 \varGamma _{q} (\zeta + 1 ) \varGamma _{q}( 2 + \zeta ) - 2 \varGamma _{q}( \zeta + 1 ) \varGamma _{q} ( 2 - \zeta ) \bigr]^{-1} \end{aligned} \end{aligned}
(6)

and

$$G_{q}(t,r, \sigma , \zeta ) = \textstyle\begin{cases} \frac{(1-qr)^{(\sigma -1)}}{\varGamma _{q}(\sigma )} [1+ A_{1}(t, q, \zeta ) ] & \\ \quad {} + \frac{(1-qr)^{(\sigma + \zeta - 1 )}}{\varGamma _{q}(\sigma + \zeta )} A_{2}(t, q, \zeta ) & \\ \quad {} + \frac{(1-qr)^{(\sigma -\zeta -1)}}{ \varGamma _{q}( \sigma -\zeta )} A_{3}(t, q, \zeta ), & 0< r < t< 1, \\ \frac{(1-qr)^{(\sigma -1)}}{\varGamma _{q}(\sigma )} A_{1}(t, q, \zeta ) &\\ \quad {} + \frac{(1-qr)^{(\sigma + \zeta - 1 )}}{\varGamma _{q}(\sigma + \zeta )} A_{2}(t, q, \zeta ) & \\ \quad {} + \frac{(1-qr)^{(\sigma -\zeta -1)}}{ \varGamma _{q}( \sigma -\zeta )} A_{3}(t, q, \zeta ),& 0< t< r< 1. \end{cases}$$

### Proof

It is known that the general solution of the equation $${}^{c}\mathcal{D}_{q}^{\sigma }[k](t)=z(t)$$ is given by

$$k(t) =\mathcal{I}_{q}^{\sigma }[z](t) + d_{0} + d_{1} t = \frac{1}{\varGamma _{q}(\sigma )} \int _{0}^{t} ( t - qs)^{ (\sigma -1)} z(r) \,\mathrm{d}_{q}r + d_{0} + d_{1} t,$$

where $$d_{0}$$, $$d_{1}$$ are real constants and $$t\in \overline{J}$$ (see [42]). Thus,

\begin{aligned} {}^{c}\mathcal{D}_{q}^{\zeta }[k](t) & = \mathcal{I}_{q}^{\sigma - \zeta } [z](t) + \frac{ t^{1 - \zeta } d_{1}}{ \varGamma _{q}( 2 - \zeta )} \\ & = \frac{1}{ \varGamma _{q}(\sigma - \zeta )} \int _{0}^{t} ( t - qr)^{ ( \sigma -\zeta -1)} z(r) \,\mathrm{d}_{q}r + \frac{t^{ 1 - \zeta } d_{1}}{ \varGamma _{q}( 2 -\zeta )} \end{aligned}

and

\begin{aligned} \mathcal{I}_{q}^{\zeta }[k](t) ={}& \frac{1}{ \varGamma _{q}(\sigma + \zeta )} \int _{0}^{t} ( t -qr)^{ ( \sigma + \zeta - 1)} z(r) \, \mathrm{d}_{q}r + \frac{ d_{0} t^{\zeta }}{ \varGamma _{q}( \zeta + 1)} + \frac{ d_{1} t^{\zeta +1 } }{ \varGamma _{q}( \zeta + 2)}. \end{aligned}

Hence, we get $$k(0) = - d_{0} - d_{1} - \frac{1}{ \varGamma _{q}( \sigma )} \int _{0}^{1} ( 1-qr)^{ \sigma - 1} z(r) \,\mathrm{d}_{q}r$$ and

\begin{aligned} &k(1) + {}^{c}\mathcal{D}_{q}^{\zeta }[k](1) + \mathcal{I}_{q}^{\zeta }[k](1) \\ &\quad = \frac{1}{\varGamma _{q}( \sigma )} \int _{0}^{1} ( 1 - qr)^{ (\sigma - 1)} z(r) \, \mathrm{d}_{q}r \\ &\qquad {} + \frac{1}{ \varGamma _{q}( \sigma + \zeta )} \int _{0}^{1} ( 1 - qr)^{(\sigma + \zeta - 1)} z(r) \, \mathrm{d}_{q}r \\ &\qquad {} + \frac{1}{ \varGamma _{q}( \sigma - \zeta )} \int _{0}^{1} ( 1 - qr)^{(\sigma - \zeta -1)} z(r) \, \mathrm{d}_{q}r \\ &\qquad {} + d_{0} \biggl[ \frac{ 1 + \varGamma _{q}( \zeta + 1)}{ \varGamma _{q}( \zeta + 1)} \biggr] \\ &\qquad {} + d_{1} \biggl[ \frac{ \varGamma _{q}( \zeta + 2 ) \varGamma _{q}( 2 - \zeta ) + \varGamma _{q}( \zeta + 2) + \varGamma _{q}( 2 - \zeta ) }{ \varGamma _{q}( \zeta + 2) \varGamma _{q}( 2 - \zeta )} \biggr]. \end{aligned}

By using the boundary conditions, we obtain

\begin{aligned} 2d_{0} + d_{1} ={}& - \frac{1}{\varGamma _{q}(\sigma )} \int _{0}^{1} ( 1 -qr)^{( \sigma -1 )} z(r) \, \mathrm{d}_{q}r \\ &{} + d_{0} \biggl[ \frac{ 1 + 2 \varGamma _{q}( \zeta + 1 )}{ \varGamma _{q} (\zeta +1) } \biggr] \\ & {} + d_{1} \biggl[ \frac{ \varGamma _{q}( \zeta + 2 ) \varGamma _{q} ( 2 - \zeta ) + \varGamma _{q} ( \zeta + 2 ) + \varGamma _{q} ( 2 - \zeta ) }{ \varGamma _{q}( \zeta + 2 ) \varGamma _{q} ( 2 - \zeta )} \biggr] \\ ={}& - \frac{1}{\varGamma _{q}(\sigma ) } \int _{0}^{1} ( 1 - qr)^{ ( \sigma -1)} z(r) \,\mathrm{d}_{q}r \\ &{} - \frac{1}{ \varGamma _{q}( \sigma - \zeta ) } \int _{0}^{1} ( 1 -qr)^{( \sigma - \zeta -1)} z(r) \, \mathrm{d}_{q}r \\ & {} - \frac{1}{ \varGamma _{q}( \sigma + \zeta )} \int _{0}^{1} ( 1 -qr)^{( \sigma + \zeta -1)} z(r) \, \mathrm{d}_{q}r. \end{aligned}

Thus,

\begin{aligned} d_{0} ={}& \varSigma \bigl[ \varGamma _{q}( \zeta + 1) \bigl( \varGamma _{q}( \zeta + 2) + \varGamma _{q}( 2 - \zeta )\bigr) \bigr] \mathcal{I}_{q}^{\alpha }[z](1) \\ & {} - \varSigma \varGamma _{q}( \zeta + 1 ) \varGamma _{q}( \zeta + 2 ) \varGamma _{q} ( 2 - \zeta ) \mathcal{I}_{q}^{\sigma + \zeta } [z](1) \\ & {} -\varSigma \varGamma _{q} ( \zeta +1) \varGamma _{q}( \zeta +2) \varGamma _{q}( 2 - \zeta ) \mathcal{I}_{q}^{ \sigma -\zeta } [z](1) \end{aligned}

and

\begin{aligned} d_{1} ={}& \varSigma \varGamma _{q}( \zeta +2) \varGamma _{q}(2-\zeta )) \mathcal{I}_{q}^{ \sigma } [z](1) \\ &{} +\varSigma 2 \varGamma _{q}(\zeta +1) \varGamma _{q}(\zeta +2) \varGamma _{q}( 2 - \zeta ) \mathcal{I}_{q}^{ \sigma + \zeta } [z](1) \\ &{} + \varSigma 2 \varGamma _{q}(\zeta +1) \varGamma _{q}(\zeta +2) \varGamma _{q}( 2 - \zeta ) \mathcal{I}_{q}^{ \sigma - \zeta } [z](1). \end{aligned}

Hence,

\begin{aligned} k(t) = {}&\frac{1}{ \varGamma _{q}(\sigma )} \int _{0}^{t} (t - qr)^{( \sigma -1)} z(r) \,\mathrm{d}_{q}r+ A_{1}(t, q, \zeta ) \mathcal{I}_{q}^{ \sigma } [z](1) \\ & {} + A_{2}(t, q, \zeta ) \mathcal{I}_{q}^{ \sigma + \zeta } [z](1)+ A_{3}(t, q, \zeta ) \mathcal{I}_{q}^{ \sigma - \zeta } [z](1) \\ = {}& \int _{0}^{1} G_{q}(t,r,\sigma , \zeta ) z(r) \,\mathrm{d}_{q}r. \end{aligned}

The converse part concludes with some straight calculation. This completes the proof. □

### Definition 5

A function $$(k_{1}, k_{2}, \ldots , k_{m})\in \prod_{i=1}^{m} \operatorname{AC}^{1} ( \overline{J})$$ is a solution for the system of fractional inclusions whenever it satisfies the boundary conditions and there exists a function $$(z_{1}, z_{2}, \ldots , z_{m})$$, $$(z'_{1}, z'_{2}, \dots , z'_{m}) \in \prod_{i=1}^{m} L^{1}( \overline{J})$$ such that

\begin{aligned} &z_{i}(t) \in \mathcal{T}_{1i} \bigl( t, k_{1}(t), \ldots , k_{m}(t),\mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr), \\ &z'_{i}(t) \in \mathcal{T}_{2i} \bigl( t, k_{1}(t), \ldots , k_{m}(t), {}^{c} \mathcal{D}_{q}^{\zeta _{1}} [k_{1}] (t), \ldots , {}^{c}\mathcal{D}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr) \end{aligned}

and

\begin{aligned} k_{i}(t) ={}& \frac{1}{ \varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t- qr )^{( \sigma _{i}-1)} z_{i}(r) \,\mathrm{d}_{q}r + A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} [z_{i}](1) \\ & {} + A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}+ \zeta _{i}} [z_{i}](1)+ A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} [z_{i}](1) \\ & {} + \frac{1}{ \varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t - qr)^{( \sigma _{i}-1)} z'_{i}(r) \,\mathrm{d}_{q}r + A_{1} (t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[z'_{i}\bigr](1) \\ & {} + A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}+ \zeta _{i}} \bigl[z'_{i}\bigr](1)+ A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} \bigl[z'_{i}\bigr](1) \\ ={}& \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z_{i}(r) \, \mathrm{d}_{q}r + \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z'_{i}(r) \,\mathrm{d}_{q}r \end{aligned}

for $$t\in \overline{J}$$ and $$1\leq i\leq m$$.

### Theorem 6

Let$$\mathcal{T}_{1i}: \overline{J}\times \mathbb{R}^{2m}\to \mathcal{P}_{\mathrm{cp},\mathrm{cv}}( \mathbb{R})$$be a set-valued map and, for each$$1\leq i\leq m$$, $$\mathcal{T}_{2i} : \overline{J}\times \mathbb{R}^{2m} \to \mathcal{P}_{\mathrm{cp},\mathrm{cv}} (\mathbb{R})$$be a Caratheodory multifunction. Assume that there exist continuous functions$$h_{1i}, h_{2i}, \gamma _{i} : \overline{J} \to (0,\infty )$$ ($$i=1, \ldots , m$$) such that$$t\vdash \mathcal{T}_{1i} (t, u_{1}, \ldots , u_{m}, v_{1}, \ldots , v_{m} )$$is measurable,

\begin{aligned} & \bigl\Vert \mathcal{T}_{1i} \bigl( t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr) \bigr\Vert \\ &\quad = \sup \bigl\{ \vert z \vert : v\in \mathcal{T}_{1i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{\zeta _{m}} [k_{m}](t)\bigr) \bigr\} \\ &\quad \leq h_{1i}(t), \\ & \bigl\Vert \mathcal{T}_{2i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), {}^{c} \mathcal{D}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , {}^{c}\mathcal{D}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr) \bigr\Vert \\ &\quad =\sup \bigl\{ \vert z \vert : z\in \mathcal{T}_{2i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), {}^{c} \mathcal{D}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , {}^{c}\mathcal{D}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr)\bigr\} \\ &\quad \leq h_{2i}(t), \end{aligned}

and

\begin{aligned} &P_{\rho } \bigl( \mathcal{T}_{1i} ( t, u_{1}, \dots , u_{k}, v_{1}, \ldots , v_{m} ), \mathcal{T}_{1i} \bigl(t, u'_{1}, \ldots , u'_{k}, v'_{1}, \ldots , v'_{m}\bigr) \bigr) \\ &\quad \leq \sum_{i=1}^{m} \gamma _{i} (t) \bigl( \bigl\vert u_{i} - u'_{i} \bigr\vert + \bigl\vert v_{i} - v'_{i} \bigr\vert \bigr) \end{aligned}

for all$$t \in \overline{J}$$, $$( k_{1}, \ldots , k_{m})\in \mathcal{E}$$, $$u_{i}$$, $$v_{i}$$, $$u'_{i}$$and$$v'_{i} \in \mathbb{R}$$and$$1 \leq i\leq m$$. If

$$\Delta = \sum_{i=1}^{m} \Vert \gamma _{i} \Vert _{\infty } \biggl( \frac{ 1 + \varGamma _{q}( \zeta _{i} + 1) }{ \varGamma _{q}(\zeta _{i} + 1) } \biggr) ( \varLambda _{1i} + \varLambda _{2i} ) < 1,$$

then the system of fractional inclusions has a solution, where

$$\Vert \gamma _{i} \Vert _{\infty }= \max _{t\in \overline{J}} \bigl\vert \gamma _{i}(t) \bigr\vert$$

and

\begin{aligned} &\varLambda _{1i} = \frac{1}{ \varGamma _{q}(\sigma _{i} + 1) } \bigl( 1 + \vert \varSigma _{i} \vert \bigl[ \varGamma _{q}(\zeta _{i} + 2 ) \varGamma _{q}( 2 - \zeta _{i} ) \\ & \hphantom{\varLambda _{1i} =}{} + \varGamma _{q}(\zeta _{i} + 1) \varGamma _{q}(\zeta _{i} + 2)+ \varGamma _{q}( \zeta _{i} + 1 ) \varGamma _{q}( 2 - \zeta _{i} ) \bigr] \bigr) \\ & \hphantom{\varLambda _{1i} =}{} + \frac{ \vert \varSigma _{i} \vert \varGamma _{q}( \zeta _{i} + 1 ) \varGamma _{q}(\zeta _{i} + 2 ) \varGamma _{q}( 2 - \zeta _{i})}{ \varGamma _{q}( \sigma _{i} + \zeta _{i} + 1 )} \\ & \hphantom{\varLambda _{1i} =}{} + \frac{ \vert \varSigma _{i} \vert \varGamma _{q}( \zeta _{i} + 1 ) \varGamma _{q}( \zeta _{i} + 2 ) \varGamma _{q}( 2 - \zeta _{i})}{ \varGamma _{q}( \sigma _{i} - \zeta _{i}+1 ) }, \end{aligned}
(7)
\begin{aligned} &\varLambda _{2i} = \frac{1}{ \varGamma _{q}( \sigma _{i} - \zeta _{i} + 1 ) } + \frac{ \vert \varSigma _{i} \vert \varGamma _{q}( \zeta _{i} + 2 )}{ \varGamma _{q}(\sigma _{i} + 1 )} \\ &\hphantom{\varLambda _{2i} =}{} + \frac{2 \vert \varSigma _{i} \vert ( \varGamma _{q} ( \zeta _{i} + 1 ) + \varGamma _{q}( \zeta _{i} + 2 )) }{ \varGamma _{q}(\sigma _{i} + \zeta _{i} + 1)} + \frac{ \vert \varSigma _{i} \vert ( 2 \varGamma _{q}( \zeta _{i}+1 ) + \varGamma _{q}( \zeta _{i} + 2 ))}{ \varGamma _{q}( \sigma _{i} - \zeta _{i} + 1 ) } \end{aligned}
(8)

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

### Proof

Consider the subset $$\mathcal{F} = \{(k_{1}, \ldots , k_{m})\in \mathcal{E} : \| (k_{1}, \ldots , k_{m})\| \leq M \}$$ of $$\mathcal{E}$$, where

$$M = \sum_{i=1}^{m} \bigl( \Vert h_{1i} \Vert _{\infty }+ \Vert h_{2i} \Vert _{\infty } \bigr) (\varLambda _{1i} + \varLambda _{2i} ).$$

It is easy to see that $$\mathcal{F}$$ is a closed, bounded, and convex subset of the Banach space $$\mathcal{E}$$. Now, define the multi-valued operators $$\mathcal{M}, \mathcal{N} : \mathcal{F} \to \mathcal{P}( \mathcal{E})$$ by

$\begin{array}{rl}& \mathcal{M}\left({k}_{1},\dots ,{k}_{m}\right)=\left(\begin{array}{c}{M}_{1}\left({k}_{1},\dots ,{k}_{m}\right)\\ {M}_{2}\left({k}_{1},\dots ,{k}_{m}\right)\\ ⋮\\ {M}_{m}\left({k}_{1},\dots ,{k}_{m}\right)\end{array}\right),\\ & \mathcal{N}\left({k}_{1},\dots ,{k}_{m}\right)=\left(\begin{array}{c}{N}_{1}\left({k}_{1},\dots ,{k}_{m}\right)\\ {N}_{2}\left({k}_{1},\dots ,{k}_{m}\right)\\ ⋮\\ {N}_{m}\left({k}_{1},\dots ,{k}_{m}\right)\end{array}\right),\end{array}$

where the multifunctions $$M_{i} (k_{1}, \ldots , k_{m})$$ and $$N_{i} (k_{1}, \ldots , k_{m})$$ are the set of all $$\theta \in \mathcal{E}_{i}$$ with the feature that there exist $$z \in S_{\mathcal{T}_{1i}, ( k_{1}, \ldots , k_{m})}$$ and $$\theta \in S_{\mathcal{T}_{2i}, ( k_{1}, \ldots , k_{m})}$$, respectively, such that

$$\theta (t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z(r) { \,\mathrm{d}}_{q}s$$

for all $$t\in \overline{J}$$ and $$1\leq i \leq m$$. Thus, the system of fractional q-differential inclusions is equivalent to the inclusion problem $$k \in \mathcal{M} (k) + \mathcal{N}(k)$$. We show that the set-valued maps $$\mathcal{M}$$ and $$\mathcal{N}$$ satisfy the conditions of Lemma 3 on $$\mathcal{F}$$. First, we show that $$\mathcal{M}$$ is compact-valued on $$\mathcal{F}$$. Note that $$M_{i}=\mathscr{L}_{i} \circ S_{\mathcal{T}_{1i}}$$, where $$\mathscr{L}_{i}$$ is the continuous linear operator on $$L^{1}(\overline{J}, \mathbb{R})$$ into $$E_{i}$$ defined by $$\mathscr{L}_{i} [z](t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z(r) \,\mathrm{d}_{q}s$$. Let $$(k_{1}, \ldots , k_{m})\in \mathcal{F}$$ and $$\{z_{n}\}$$ be a sequence in $$S_{\mathcal{T}_{1i}, (k_{1}, \ldots , k_{m})}$$. Then, by the definition of $$S_{\mathcal{T}_{1i}, ( k_{1}, \ldots , k_{m})}$$, we have

$$z_{n} (t) \in \mathcal{T}_{1i} \bigl( t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t) \bigr)$$

for almost $$t\in \overline{J}$$. Since $$\mathcal{T}_{1i} (t, k_{1} (t), \ldots , k_{m}(t), \mathcal{I}_{q}^{ \zeta _{1} } [k_{1}](t), \ldots , \mathcal{I}_{q}^{\zeta _{m}} [k_{m}](t) )$$ is compact for all $$t\in \overline{J}$$, there is a convergent subsequence of $$\{z_{n}(t) \}$$, call it again $$\{z_{n}(t)\}$$), that converges in measure to some $$z(t) \in S_{\mathcal{T}_{1i} , (k_{1}, \ldots , k_{m})}$$ for almost all $$t\in \overline{J}$$. Since $$\mathscr{L}_{i}$$ is continuous, we conclude that $$\mathscr{L}_{i} [z_{n}](t)\to \mathscr{L}_{i} [z](t)$$ pointwise on . In order to show that the convergence is uniform, we have to show that $$\{\mathscr{L}_{i} [z_{n}]\}$$ is an equicontinuous sequence. Let $$t_{1} < t \in \overline{J}$$. Then we have

\begin{aligned} & \bigl\vert \mathscr{L}_{i} [z_{n}] (t) - \mathscr{L}_{i} [z_{n}] (t_{1}) \bigr\vert \\ &\quad \leq \int _{0}^{1} \bigl\vert G_{q}(t, r, \sigma _{i}, \zeta _{i}) - G_{q}(t_{1}, r, \sigma _{i}, \zeta _{i}) \bigr\vert \bigl\vert z_{n}(r) \bigr\vert \,\mathrm{d}_{q}r \\ &\quad \leq \Vert h_{1i} \Vert _{\infty } \biggl[ \frac{ 1}{ \varGamma _{q}(\sigma _{i} + 1 ) } \bigl( t^{\sigma _{i}} - t_{1}^{ \sigma _{i}}+ \vert \varSigma _{i} \vert ( t -t_{1}) \varGamma _{q}(\zeta _{i} + 2 ) \varGamma _{q}( \zeta _{i} + 1 ) \bigr) \\ & \qquad {} + \frac{2 \vert \varSigma _{i} \vert ( t - t_{1}) \varGamma _{q}(\zeta _{i} + 2) \varGamma _{q}(\zeta _{i} + 1 ) \varGamma _{q}( 2 - \zeta _{i} ) }{ \varGamma _{q}( \sigma _{i} + \zeta _{i} + 1 )} \\ & \qquad {} + \frac{2 \vert \varSigma _{i} \vert ( t - t_{1}) \varGamma _{q} ( \zeta _{i} + 2 ) \varGamma _{q}( \zeta _{i} + 1 ) \varGamma _{q}( 2 - \zeta _{i})}{ \varGamma _{q}( \sigma _{i} - \zeta _{i} + 1 )} \biggr] \end{aligned}

and

\begin{aligned} & \bigl\vert {}^{c}\mathcal{D}_{q}^{\zeta _{i}} \bigl(\mathscr{L}_{i} [z_{n}](t)\bigr) - {}^{c}\mathcal{D}_{q}^{ \zeta _{i}}\bigl( \mathscr{L}_{i} [z_{n}](t_{1})\bigr) \bigr\vert \\ &\quad \leq \Vert h_{1i} \Vert _{\infty } \biggl[ \frac{ t^{\sigma _{i} - \zeta _{i} } - t_{1}^{\sigma _{i} - \zeta _{i}} }{ \varGamma _{q}(\sigma _{i} - \zeta _{i} + 1 )} + \frac{ \vert \varSigma _{i} \vert ( t^{ 1 - \zeta _{i} } - t_{1}^{1 - \zeta _{i} }) \varGamma _{q}( 2 + \zeta _{i} ) }{ \varGamma _{q}( \sigma _{i} + 1 )} \\ &\qquad {} + \frac{2 \vert \varSigma _{i} \vert ( t^{ 1 - \zeta _{i} } - t_{1}^{ 1- \zeta _{i}} ) \varGamma _{q}( 2 + \zeta _{i} ) \varGamma _{q} (1 + \zeta _{i} )}{ \varGamma _{q}(\sigma _{i} + \zeta _{i} + 1 ) } \\ &\qquad {} + \frac{2 \vert \varSigma _{i} \vert ( t^{ 1 - \zeta _{i} } - t_{1}^{1 - \zeta _{i}} ) \varGamma _{q}( 2 + \zeta _{i}) \varGamma _{q}( 1 + \zeta _{i})}{\varGamma _{q}( \sigma _{i} - \zeta _{i} + 1 )} \biggr]. \end{aligned}

Hence, the right-hand side of the inequalities tends to 0 as $$t\to t_{1}$$, and so the sequence $$\{\mathscr{L}_{i} [z_{n}]\}$$ is equicontinuous. Now, by using the Arzela–Ascoli theorem we deduce that there is a uniformly convergent subsequence. Thus, there is a subsequence of $$\{z_{n}\}$$, we show it again by $$\{z_{n}\}$$, such that $$\mathscr{L}_{i} [z_{n}](t) \to \mathscr{L}_{i} [z](t)$$ for each $$t \in \overline{J}$$. Note that $$\mathscr{L}_{i} [z] \in \ell _{i} (S_{\mathcal{T}_{1i},( k_{1}, \ldots , k_{m})} )$$. Hence,

$$M_{i} (k_{1}, \ldots , k_{m} ) = \mathscr{L}_{i} (S_{\mathcal{T}_{1i}, (k_{1}, \ldots , k_{m}) } )$$

is compact for all $$(k_{1}, \ldots , k_{m}) \in \mathcal{F}$$ and $$i=1,\dots ,m$$, and so $$\mathcal{M} (k_{1}, \ldots , k_{m})$$ is compact. Now, we show that $$\mathcal{M} (k_{1}, \ldots , k_{m})$$ is convex for all $$( k_{1}, \ldots , k_{m}) \in \mathcal{F}$$. Let $$( x_{1}, \ldots , x_{m})$$, $$( x'_{1}, \ldots , x'_{m} ) \in \mathcal{M}(k)$$. Choose $$z_{i}$$, $$z'_{i} \in S_{\mathcal{T}_{1i}, (k_{1}, \ldots , k_{m})}$$ such that

\begin{aligned} &x_{i} (t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z_{i} (r) \, \mathrm{d}_{q}r, \\ &x'_{i} (t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z'_{i}(r) \,\mathrm{d}_{q}r \end{aligned}

for almost all $$t\in \overline{J}$$ and $$1 \leq i \leq m$$. Let $$0\leq \lambda \leq 1$$. Then we have

$$\bigl[\lambda x_{i} + (1-\lambda ) x'_{i} \bigr] (t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) \bigl[ \lambda z_{i}(r) + ( 1 - \lambda ) z'_{i}(r)\bigr] \, \mathrm{d}_{q}r.$$

Since $$\mathcal{T}_{1i}$$ is convex-valued for all $$1 \leq i \leq m$$,

$$\bigl[ \lambda x_{i} + ( 1 - \lambda ) x'_{i} \bigr] \in M_{i} (k_{1}, \ldots , k_{m}).$$

Thus,

\begin{aligned}& \lambda (x_{1}, \ldots , x_{m}) + ( 1 -\lambda ) \bigl( x'_{1}, \ldots , x'_{m} \bigr) = \bigl(\lambda x_{1} + ( 1 - \lambda ) x'_{1}, \ldots , \lambda x_{m} + ( 1 - \lambda ) x'_{m} \bigr) \in \mathcal{M}(k). \end{aligned}

Similarly, $$\mathcal{N}$$ is compact and convex-valued. Here, we show that $$\mathcal{M} (f)+ \mathcal{N}(f) \subset \mathcal{F}$$ for all $$f \in \mathcal{F}$$. Let $$f\in \mathcal{F}$$ and $$( x_{1}, \ldots , x_{m}) \in \mathcal{M}(f)$$ and $$( x'_{1}, \ldots , x'_{m}) \in \mathcal{N}(f)$$. Then we can choose $$(z_{1}, \ldots , z_{m})\in S_{\mathcal{T}_{11},f } \times \cdots \times S_{\mathcal{T}_{1m},f}$$ and $$(z'_{1}, \ldots , z'_{m})\in S_{\mathcal{T}_{21},f} \times \cdots \times S_{\mathcal{T}_{2m},f}$$ such that $$x_{i}(t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z_{i}(r) \,\mathrm{d}_{q}r$$ and $$x'_{i}(t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z'_{i}(r) \,\mathrm{d}_{q}r$$ for almost all $$t\in \overline{J}$$ and $$1\leq i\leq m$$. Hence, we get

\begin{aligned} \bigl\vert x_{i}(t) + x'_{i}(t) \bigr\vert \leq{}& \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](t) + A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](1) \\ & {} + A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i} + \zeta _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](1) \\ & {} + A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](1) \end{aligned}

and

\begin{aligned} \bigl\vert {}^{c}\mathcal{D}_{q}^{\zeta _{i}} [x_{i}](t) + {}^{c}\mathcal{D}_{q}^{ \zeta _{i}} \bigl[x'_{i}\bigr](t) \bigr\vert \leq{}& \mathcal{I}_{q}^{\sigma _{i}-\zeta _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](t) \\ &{} + \frac{1}{t^{\zeta _{i}}} A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](1) \\ &{} + \frac{t}{(t-1)t^{\zeta _{i}}} A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i} + \zeta _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](1) \\ & {} + \frac{t}{(t-1)t^{\zeta _{i}}} A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} \bigl[ \vert z_{i} \vert + \bigl\vert z'_{i} \bigr\vert \bigr](1). \end{aligned}

Hence, $$\max_{t \in \overline{J} } |x_{i}(t) + x'_{i}(t)| \leq ( \|h_{1i} \|_{\infty }+ \|h_{2i}\|_{\infty } )\varLambda _{1i}$$ and

$$\max_{t\in \overline{J}} \bigl\vert {}^{c} \mathcal{D}_{q}^{\zeta _{i}} [z_{i}](t) + {}^{c}\mathcal{D}_{q}^{\zeta _{i}} \bigl[x'_{i}\bigr] (t) \bigr\vert \leq \bigl( \Vert h_{1i} \Vert _{\infty }+ \Vert h_{2i} \Vert _{\infty } \bigr) \varLambda _{2i}$$

for $$1\leq i\leq m$$, and so

\begin{aligned} \bigl\Vert (x_{1},\ldots , x_{m}) + \bigl( x'_{1}, \ldots , x'_{m} \bigr) \bigr\Vert &= \sum_{i=1}^{m} \bigl\Vert x_{i} + x'_{i} \bigr\Vert _{i} \\ & \leq \sum_{i=1}^{m} \bigl( \Vert h_{1i} \Vert _{\infty }+ \Vert h_{2i} \Vert _{\infty } \bigr) (\varLambda _{1i} + \varLambda _{2i}) = M. \end{aligned}

In this step, we show that the operator $$\mathcal{N}$$ is compact on $$\mathcal{F}$$. To do this, it is enough to prove that $$\mathcal{N}(\mathcal{F})$$ is uniformly bounded and equicontinuous. Let $$(x_{1},\ldots , x_{m})\in \mathcal{N}(\mathcal{F})$$. Choose $$(z_{1},\ldots ,z_{m}) \in S_{\mathcal{T}_{21}, k} \times \cdots \times S_{\mathcal{T}_{2m},k}$$ such that $$x_{i}(t) = \int _{0}^{1} G_{q}(t,r,\sigma _{i}, \zeta _{i}) z_{i}(r) \,\mathrm{d}_{q}r$$ for some $$k\in \mathcal{F}$$ and all $$1\leq i \leq m$$. Hence,

\begin{aligned} \bigl\vert x_{i}(t) \bigr\vert \leq{}& \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \vert z_{i} \vert \bigr](t) + A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \vert z_{i} \vert \bigr](1)+ A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i} + \zeta _{i}} \bigl[ \vert z_{i} \vert \bigr](1) \\ &{} + A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} \bigl[ \vert z_{i} \vert \bigr](1) \end{aligned}

and

\begin{aligned} \bigl\vert {}^{c}\mathcal{D}_{q}^{\zeta _{i}} [x_{i}](t) \bigr\vert \leq{}& \mathcal{I}_{q}^{ \sigma _{i}-\zeta _{i}} \bigl[ \vert z_{i} \vert \bigr](t) + \frac{1}{t^{\zeta _{i}}} A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \vert z_{i} \vert \bigr](1) \\ & {} + \frac{t}{(t-1)t^{\zeta _{i}}} A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i} + \zeta _{i}} \bigl[ \vert z_{i} \vert \bigr](1) \\ & {} +\frac{t}{(t-1)t^{\zeta _{i}}} A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} \bigl[ \vert z_{i} \vert \bigr](1). \end{aligned}

Thus, $$\max_{t\in \overline{J}} |x_{i}(t)| \leq \|h_{1i}\|_{\infty }\varLambda _{2i}$$ and $$\max_{t\in \overline{J}} | {}^{c}\mathcal{D}_{q}^{ \zeta _{i}} [x_{i}](t)| \leq \|h_{2i}\|_{\infty }\varLambda _{2i}$$ for $$1\leq i\leq m$$, and so

$$\bigl\Vert (x_{1}, \ldots , x_{m}) \bigr\Vert = \sum_{i=1}^{m} \Vert x_{i} \Vert _{\infty }\leq \sum _{i=1}^{m} \Vert h_{2i} \Vert _{\infty }( \varLambda _{1i} + \varLambda _{2i}).$$

Now, we show that $$\mathcal{N}$$ maps $$\mathcal{F}$$ to equicontinuous subsets of $$\mathcal{E}$$. Let $$t, t_{1} \in \overline{J}$$ with $$t_{1} < t$$, $$k\in \mathcal{F}$$, and $$(x_{1}, \ldots , x_{m}) \in \mathcal{N}(k)$$. Choose $$(z_{1}, \ldots ,z_{m}) \in S_{\mathcal{T}_{21}, k} \times \cdots \times S_{\mathcal{T}_{2k}, k}$$ such that $$x_{i}(t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) z_{i}(r) \,\mathrm{d}_{q}r$$ for all $$1\leq i \leq m$$. Then we have

\begin{aligned} \bigl\vert x_{i}(t) - x_{i}(t_{1}) \bigr\vert \leq {}& \Vert h_{2i} \Vert _{\infty } \\ & {} \times \biggl[ \frac{ 1}{ \varGamma _{q}(\sigma _{i} + 1 ) } \bigl( t^{ \sigma _{i}} - t_{1}^{\sigma _{i}}+ \vert \varSigma _{i} \vert ( t -t_{1}) \varGamma _{q}( \zeta _{i} + 2 ) \varGamma _{q}(\zeta _{i} + 1 ) \bigr) \\ & {} + \frac{2 \vert \varSigma _{i} \vert ( t - t_{1} ) \varGamma _{q}(\zeta _{i} + 2) \varGamma _{q}(\zeta _{i} + 1 ) \varGamma _{q}( 2 - \zeta _{i} ) }{ \varGamma _{q}( \sigma _{i} + \zeta _{i} + 1 )} \\ &{} + \frac{2 \vert \varSigma _{i} \vert ( t - t_{1}) \varGamma _{q} ( \zeta _{i} + 2 ) \varGamma _{q}( \zeta _{i} + 1 ) \varGamma _{q}( 2 - \zeta _{i}) }{ \varGamma _{q}( \sigma _{i} - \zeta _{i} + 1 )} \biggr] \end{aligned}

and

\begin{aligned} \bigl\vert {}^{c}\mathcal{D}_{q}^{\zeta _{i}} [x_{i}] (t) - {}^{c} \mathcal{D}_{q}^{ \zeta _{i}} [x_{i}] (t_{1}) \bigr\vert \leq{}& \Vert h_{2i} \Vert _{\infty } \biggl[ \frac{ t^{\sigma _{i} - \zeta _{i} } - t_{1}^{\sigma _{i} - \zeta _{i}} }{ \varGamma _{q}(\sigma _{i} - \zeta _{i} + 1 )} \\ & {} + \frac{ \vert \varSigma _{i} \vert ( t^{ 1 - \zeta _{i} } - t_{1}^{1 - \zeta _{i} } ) \varGamma _{q}( 2 + \zeta _{i} ) }{ \varGamma _{q}( \sigma _{i} + 1 )} \\ & {} + \frac{2 \vert \varSigma _{i} \vert ( t^{ 1 - \zeta _{i} } - t_{1}^{ 1- \zeta _{i}} ) \varGamma _{q}( 2 + \zeta _{i} ) \varGamma _{q} (1 + \zeta _{i} )}{ \varGamma _{q}(\sigma _{i} + \zeta _{i} + 1 ) } \\ & {} + \frac{2 \vert \varSigma _{i} \vert ( t^{ 1 - \zeta _{i} } - t_{1}^{1 - \zeta _{i}} ) \varGamma _{q}( 2 + \zeta _{i}) \varGamma _{q}( 1 + \zeta _{i})}{\varGamma _{q}( \sigma _{i} - \zeta _{i} + 1 )} \biggr]. \end{aligned}

Note that the right-hand side of these inequalities tends to 0 as $$t\to t_{1}$$. By using the Arzela–Ascoli theorem, $$\mathcal{N}$$ is compact. Here, we show that $$\mathcal{N}$$ has a closed graph. Let $$(k_{1}^{n}, \ldots , k_{m}^{n})\in \mathcal{F}$$ and $$(x_{1}^{n}, \ldots , x_{m}^{n}) \in \mathcal{N}(k_{1}^{n}, \ldots , k_{m}^{n})$$ be such that $$(k_{1}^{n}, \ldots , k_{m}^{n})\to (k_{1}^{0}, \ldots , k_{m}^{0})$$ and also $$(x_{1}^{n}, \ldots , x_{m}^{n}) \to (x_{1}^{0}, \ldots , x_{m}^{0})$$ for all n. We show that $$(x_{1}^{0}, \ldots , x_{m}^{0}) \in \mathcal{N}( k_{1}^{0}, \ldots , k_{m}^{0})$$. For each natural number n, choose $$(z_{1}^{n}, \ldots , z^{n}_{m}) \in S_{\mathcal{T}_{21}, (k^{n}_{1}, \ldots , k^{n}_{m})} \times \cdots \times S_{\mathcal{T}_{2m}, (k^{n}_{1}, \ldots , k^{n}_{m})}$$ such that

$$x^{n}_{i} (t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) z_{i}^{n}(r) \,\mathrm{d}_{q}r$$

for all $$t\in \overline{J}$$ and $$1\leq i\leq m$$. Again, consider the continuous linear operator $$\mathscr{L}_{i} : L^{1}(\overline{J}, \mathbb{R}) \to E_{i}$$ by $$\mathscr{L}_{i}[z](t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) z(r) \,\mathrm{d}_{q}r$$. By using Lemma 2, $$\mathscr{L}_{i} \circ S_{\mathcal{T}_{2i}}$$ is a closed graph operator. Since $$x_{i}^{n} \in \mathscr{L}_{i}(S_{\mathcal{T}_{2i},(k^{n}_{1}, \ldots , k^{n}_{m})} )$$ for all n, $$1\leq i\leq m$$ and $$(k_{1}^{n}, \ldots , k_{m}^{n}) \to (k_{1}^{0}, \ldots , k_{m}^{0})$$, there exists $$z_{i}^{0} \in S_{\mathcal{T}_{2i}, (k_{1}^{0}, \ldots , k_{m}^{0}) }$$ such that $$x_{i}^{0} (t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) z_{i}^{0} (r) \,\mathrm{d}_{q}r$$. This implies that $$x_{i}^{0} \in N_{i} (k_{1}^{0}, \ldots , k_{m}^{0})$$ for all $$1\leq i\leq m$$. Thus, $$N_{i}$$ has a closed graph for all $$1\leq i\leq m$$, and so $$\mathcal{N}$$ has a closed graph. This shows that the operator $$\mathcal{N}$$ is upper semi-continuous. Now, we show that $$\mathcal{M}$$ is a contractive multifunction. Let $$k = (k_{1}, \ldots , k_{m})$$, $$f = (f_{1}, \ldots , f_{m})\in \mathcal{E}$$, and $$(x_{1}, \ldots , x_{m}) \in \mathcal{M} (f)$$. Then we can choose $$(z_{1}, \ldots , z_{m}) \in S_{\mathcal{T}_{11},f} \times S_{ \mathcal{T}_{12},f} \times \cdots \times S_{\mathcal{T}_{1m},f}$$ such that $$x_{i}(t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) z_{i} (r) \,\mathrm{d}_{q}r$$ for all $$t\in \overline{J}$$ and $$i= 1,\ldots , m$$. Since

\begin{aligned} &P_{\rho } \bigl( \mathcal{T}_{1i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](t)\bigr), \\ &\qquad {} \mathcal{T}_{1i} \bigl(t, f_{1} (t), \ldots , f_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [f_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{1}} [f_{m}](t) \bigr) \bigr) \\ & \quad \leq \gamma _{i} (t) \sum_{i=1}^{m} ( \bigl\vert k_{i} (t) - f_{i}(t) \bigr\vert + \bigl\vert \mathcal{I}_{q}^{\zeta _{i}} [k_{i}](t) - \mathcal{I}_{q}^{\zeta _{i}} [k_{i}](t) \bigr\vert , \end{aligned}

for almost all $$t\in \overline{J}$$ and $$i=1, \ldots , m$$, by using (5), there exists

$$u_{i} \in \mathcal{T}_{1i} \bigl(t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{k}} [k_{m}](t) \bigr)$$

such that

$$\bigl\vert z_{i}(t) - u_{i} \bigr\vert \leq \gamma _{i}(t) \sum_{i=1}^{m} \bigl( \bigl\vert k_{i}(t) - f_{i}(t) \bigr\vert + \bigl\vert \mathcal{I}_{q}^{ \zeta _{i}} [k_{i}] (t) - \mathcal{I}_{q}^{\zeta _{i}} [f_{i}](t) \bigr\vert \bigr)$$

for almost all $$t\in \overline{J}$$ and $$i=1,\ldots , m$$. Consider the set-valued mapping $$\varOmega _{i} : \overline{J} \to 2^{\mathbb{R}}$$ defined by

$$\varOmega _{i} (t) = \bigl\{ y \in \mathbb{R} : \bigl\vert z_{i} (t) - y \bigr\vert \leq \gamma _{i} (t) g(t) \text{ for almost all } t\in \overline{J} \bigr\} ,$$

where $$g(t) = \sum_{i=1}^{m} ( |k_{i}(t) - f_{i}(t) | + | \mathcal{I}_{q}^{ \zeta _{i}} [k_{i}] (t) - \mathcal{I}_{q}^{\zeta _{i}} [f_{i}](t) |$$. Put

$$\varrho _{i} = \gamma _{i} \sum _{i=1}^{m} \bigl( \vert k_{i} - f_{i} \vert + \bigl\vert \mathcal{I}_{q}^{\zeta _{i}} [k_{i}] - \mathcal{I}_{q}^{\zeta _{i}} [f_{i}] \bigr\vert \bigr).$$

Since $$z_{i}$$ and $$\varrho _{i}$$ are measurable for all i,

$$\varOmega _{i}(\cdot) \cap \mathcal{T}_{1i} \bigl( t, k_{1}(\cdot), \ldots , k_{m}(\cdot), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](\cdot), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}](\cdot) \bigr)$$

is a measurable multifunction. Thus, we can choose

$$z'_{i}(t) \in \mathcal{T}_{1i} \bigl( t, k_{1}(t), \ldots , k_{m}(t), \mathcal{I}_{q}^{\zeta _{1}} [k_{1}](t), \ldots , \mathcal{I}_{q}^{ \zeta _{m}} [k_{m}] (t) \bigr)$$

such that

\begin{aligned} \bigl\vert z_{i}(t) - z'_{i}(t) \bigr\vert & \leq \gamma _{i} (t) \sum _{i=1}^{m} \bigl( \bigl\vert k_{i}(t) - f_{i}(t) \bigr\vert + \bigl\vert \mathcal{I}_{q}^{\zeta _{i}} [k_{i}](t) - \mathcal{I}_{q}^{ \zeta _{i}} [f_{i}](t) \bigr\vert \bigr) \\ & \leq \gamma _{i}(t) \sum_{i=1}^{m} \biggl( \bigl\vert k_{i}(t) - f_{i}(t) \bigr\vert + \frac{ \Vert k_{i} - f_{i} \Vert _{i}}{ \varGamma _{q}(\zeta _{i} + 1 ) } \biggr) \end{aligned}

and

$$x'_{i}(t) = \int _{0}^{1} G_{q}(t,r, \sigma _{i}, \zeta _{i}) z'_{i}(r) \,\mathrm{d}_{q}r$$

for all $$t\in \overline{J}$$ and $$i=1, \ldots , m$$. Since

\begin{aligned} \bigl\vert x_{i}(t) - x'_{i}(t) \bigr\vert \leq{}& \mathcal{I}_{q}^{\sigma _{i}} \bigl[ \bigl\vert z_{i} - z'_{i} \bigr\vert \bigr](t) + A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \bigl\vert z_{i} - z'_{i} \bigr\vert \bigr](1) \\ & {} + A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i} + \zeta _{i}} \bigl[ \bigl\vert z_{i} - z'_{i} \bigr\vert \bigr](1) \\ & {} + A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}-\zeta _{i}} \bigl[ \bigl\vert z_{i} - z'_{i} \bigr\vert \bigr](1) \end{aligned}

and

\begin{aligned} \bigl\vert {}^{c}\mathcal{D}^{\zeta _{i}} [x_{i}](t) + {}^{c}\mathcal{D}^{ \zeta _{i}} \bigl[x'_{i} \bigr](t) \bigr\vert \leq{}& \mathcal{I}_{q}^{\sigma _{i}-\zeta _{i}} \bigl[ \bigl\vert z_{i} -z'_{i} \bigr\vert \bigr](t) \\ & {} + \frac{1}{t^{\zeta _{i}}} A_{1}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}} \bigl[ \bigl\vert z_{i} -z'_{i} \bigr\vert \bigr](1) \\ & {} + \frac{t}{ (t-1) t^{\zeta _{i}}} A_{2}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i} + \zeta _{i}} \bigl[ \bigl\vert z_{i} -z'_{i} \bigr\vert \bigr](1) \\ & {} + \frac{t}{(t-1)t^{\zeta _{i}}} A_{3}(t, q, \zeta _{i}) \mathcal{I}_{q}^{ \sigma _{i}- \zeta _{i}} \bigl[ \bigl\vert z_{i} -z'_{i} \bigr\vert \bigr](1), \end{aligned}

we get $$\max_{t\in \overline{J}} |x_{i}(t) - x'_{i}(t)| \leq \|\gamma _{i} \|_{\infty } ( \frac{ 1 + \varGamma _{q}(\zeta _{i} + 1 ) }{ \varGamma _{q}(\zeta _{i} + 1 ) } ) \varLambda _{1i} \|k - f\|$$ and

$$\max_{t\in \overline{J}} \bigl\vert {}^{c} \mathcal{D}_{q}^{\zeta _{i}} x_{i}(t) -{}^{c}\mathcal{D}_{q}^{\zeta _{i}} z'_{i}(t) \bigr\vert \leq \Vert \gamma _{i} \Vert _{\infty } \biggl( \frac{ 1 + \varGamma _{q}(\zeta _{i} + 1 ) }{ \varGamma _{q}( \zeta _{i} + 1 )} \biggr) \varLambda _{2i} \Vert k - f \Vert$$

for each $$1\leq i\leq m$$. Thus,

\begin{aligned} & \bigl\Vert (x_{1}, \ldots , x_{m}) - \bigl(x'_{1}, \ldots , x'_{m} \bigr) \bigr\Vert \\ & \quad = \sum_{i=1}^{m} \bigl\Vert x_{i} - x'_{i} \bigr\Vert _{i} \\ &\quad \leq \sum_{i=1}^{m} \Vert \gamma _{i} \Vert _{\infty } \biggl( \frac{ 1 + \varGamma _{q}(\zeta _{i} + 1 ) }{ \varGamma _{q}( \zeta _{i} + 1 ) } \biggr) ( \varLambda _{1i} + \varLambda _{2i} ) \Vert k-f \Vert . \end{aligned}

This implies that $$P_{\rho }(\mathcal{M}(k), \mathcal{M}(f))\leq \lambda \|k-f\|$$. Now, by using Lemma 3, the operator inclusion $$k\in \mathcal{M}(k) + \mathcal{N}(k)$$ has a solution which is a solution for the system of q-fractional inclusions. This completes the proof. □

Now, we give an example to illustrate our main result. In this way, we give a computational technique for checking the system. We need to present a simplified analysis that 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, 4, 5, and 6.

### Example 1

Consider the three-dimensional system of fractional q-differential inclusions

$$\textstyle\begin{cases} {}^{c}\mathcal{D}_{q}^{\frac{3}{2}} [k_{1}](t) \in \mathcal{T}_{1i} (t, k_{1}(t), k_{2}(t), k_{3}(t), \mathcal{I}_{q}^{\frac{1}{4}} [k_{1}](t), \mathcal{I}_{q}^{\frac{1}{2}} [k_{2}](t), \mathcal{I}_{q}^{ \frac{3}{5}} [k_{3}](t) ) \\ \hphantom{{}^{c}\mathcal{D}_{q}^{\frac{3}{2}} [k_{1}](t) \in}{} + \mathcal{T}_{21} ( t, k_{1}(t), k_{2}(t), k_{3}(t), {}^{c} \mathcal{D}_{q}^{\frac{1}{4}} [k_{1} ](t), {}^{c}\mathcal{D}_{q}^{ \frac{1}{2}} [k_{2}](t), {}^{c}\mathcal{D}_{q}^{\frac{3}{5}} [k_{3}] (t) ), \\ {}^{c}\mathcal{D}_{q}^{\frac{7}{4}} [k_{2}](t) \in \mathcal{T}_{12} ( t, k_{1} (t), k_{2} (t), k_{3}(t), \mathcal{I}_{q}^{\frac{1}{4}} [k_{1}](t), \mathcal{I}_{q}^{\frac{1}{2}} [k_{2}](t), \mathcal{I}_{q}^{ \frac{3}{5}} [k_{3}](t) ) \\ \hphantom{{}^{c}\mathcal{D}_{q}^{\frac{7}{4}} [k_{2}](t) \in}{} + \mathcal{T}_{22} ( t, k_{1}(t), k_{2}(t), k_{3}(t), {}^{c} \mathcal{D}_{q}^{\frac{1}{4}} [k_{1}](t), {}^{c}\mathcal{D}_{q}^{ \frac{1}{2}} [k_{2}](t), {}^{c}\mathcal{D}_{q}^{ \frac{3}{5}} [k_{3}](t) ), \\ {}^{c}\mathcal{D}_{q}^{\frac{9}{5}} [k_{3}](t) \in \mathcal{T}_{13} ( t, k_{1} (t), k_{2} (t), k_{3}(t), \mathcal{I}_{q}^{\frac{1}{4}} [k_{1}](t), \mathcal{I}_{q}^{\frac{1}{2}} [k_{2}](t), \mathcal{I}_{q}^{ \frac{3}{5}} [k_{3}](t) ) \\ \hphantom{{}^{c}\mathcal{D}_{q}^{\frac{9}{5}} [k_{3}](t) \in}{} + \mathcal{T}_{23} ( t, k_{1}(t), k_{2}(t), k_{3}(t), {}^{c} \mathcal{D}_{q}^{\frac{1}{4}} [k_{1}](t), {}^{c}\mathcal{D}_{q}^{ \frac{1}{2}} [k_{2}](t), {}^{c}\mathcal{D}_{q}^{ \frac{3}{5}} [k_{3}](t) ) \end{cases}$$
(9)

with boundary conditions $$k(0) + \mathcal{I}_{q}^{\frac{1}{4}} [k_{1}](0) + {}^{c}\mathcal{D}_{q}^{ \frac{1}{4}} [k_{1}](0) = -k_{1}(1)$$, $$k_{1}(1) + \mathcal{I}_{q}^{\frac{1}{4}} [k_{1}](1) + {}^{c} \mathcal{D}_{q}^{\frac{1}{4}} [k_{1}](1)=-k_{1}(0)$$, $$k_{2}(0) + \mathcal{I}_{q}^{\frac{1}{2}} [k_{2}](0) + {}^{c} \mathcal{D}_{q}^{ \frac{1}{2}} [k_{2}] (0) = - k_{2}(1)$$, $$k_{2}(1) + \mathcal{I}_{q}^{\frac{1}{2}} [k_{2}](1) + {}^{c} \mathcal{D}_{q}^{ \frac{1}{2} } [k_{2}] (1) = -k_{2}(0)$$, $$k_{3}(0) + \mathcal{I}_{q}^{\frac{3}{5}} [k_{3}](0) + {}^{c} \mathcal{D}_{q}^{ \frac{3}{5}} [k_{3}] (0) = - k_{3}(1)$$, and $$k_{3}(1) + \mathcal{I}_{q}^{\frac{3}{5}} [k_{3}](1) + {}^{c} \mathcal{D}_{q}^{ \frac{3}{5} } [k_{3}] (1) = -k_{3}(0)$$, where $$\mathcal{T}_{ij} : \overline{J} \times \mathbb{R}^{6} \to \mathcal{P}_{\mathrm{cp}, \mathrm{cv}} (\mathbb{R})$$ is such that

\begin{aligned} &\mathcal{T}_{11}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = \biggl[0, \frac{ \sin k_{1}}{ 270 ( 1 + t^{2} ) } + \frac{ t \vert k_{2} \vert }{ 135( 1 + \vert k_{2} \vert ) } \\ &\hphantom{\mathcal{T}_{11}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =}{} + \frac{1}{ 270} \cos k_{3} + \frac{ \vert k_{4} \vert }{ 270 ( 1 + \vert k_{4} \vert ) } \\ &\hphantom{\mathcal{T}_{11}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =} {} + \frac{t k_{5}^{2}}{ 270 ( 1 + k_{5}^{2} ) } + \frac{t}{ 270 ( 1 + \vert k_{6} \vert ) }+ t^{2} + t \biggr], \\ &\mathcal{T}_{12}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = \biggl[0, \frac{ t^{2} \vert k_{1} \vert }{ 330( 1 + \vert k_{1} \vert ) } + \frac{ \vert k_{2} \vert }{330( 1 + \vert k_{2} \vert ) } \\ & \hphantom{\mathcal{T}_{12}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =}{} + \frac{1}{330} \sin k_{3} + \frac{1}{330} \cos k_{4} + e^{t} \\ & \hphantom{\mathcal{T}_{12}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =} {} + \frac{t \vert k_{5} \vert }{ 330 ( 2 + \vert k_{5} \vert ) } + \frac{ \vert k_{6} \vert e^{t}}{ 330 ( 1 + e^{t} \vert k_{6} \vert ) } + 2 \biggr], \\ &\mathcal{T}_{13}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = \biggl[0, \frac{ t}{ 190( 1 + \vert k_{1} \vert ) } + \frac{t \vert k_{2} \vert }{190( 1 + \vert k_{2} \vert ) } \\ &\hphantom{\mathcal{T}_{13}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =} {} + \frac{1}{190} \cos k_{3} + \frac{1}{190} \sin k_{4} + e^{t} \\ & \hphantom{\mathcal{T}_{13}(t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =}{} + \frac{e^{t} \vert k_{5} \vert }{ 190 ( 1 + e^{t} \vert k_{5} \vert ) } + \frac{ \vert k_{6} \vert }{ 190 ( 1 + k_{6}| ) } + \frac{1}{95} \biggr], \\ &\mathcal{T}_{21} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = \biggl[ e^{- \vert k_{1} \vert } - \frac{ \vert k_{2} \vert }{ 1 + \vert k_{2} \vert } + e^{ - \vert k_{3} \vert } + \sin k_{4} \\ & \hphantom{\mathcal{T}_{21} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =}{} - \frac{ \cos (k_{5}k_{6}) }{ 1 + \cos (k_{5}k_{6}) } + t^{2}, \cos k_{1} + \cos k_{2} \\ & \hphantom{\mathcal{T}_{21} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =}{} + \cos (k_{3} k_{4})+ \frac{ k_{5}^{2}k_{6}^{2} }{ 1 + k_{5}^{2}k_{6}^{2} } + t + \sin t \biggr], \\ &\mathcal{T}_{22} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = \biggl[-1 , \frac{ \sin ^{2} k_{1}}{ 1 +t } + t \cos ^{2} k_{2} + e^{t} \sin (k_{3}) \\ &\hphantom{\mathcal{T}_{22} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =} {} + e^{t}\cos ( k_{4}) + \frac{ e^{k_{5}k_{6}}}{ 1 + e^{k_{5}k_{6}} } \biggr], \\ &\mathcal{T}_{23} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) = \biggl[\frac{k_{1}}{4(1+k_{1})}+ \frac{k_{3}}{1+k_{3}} + \frac{e^{- \vert k_{6} \vert }}{4(1+k_{6})}, t \cos ^{2} k_{1} \\ & \hphantom{\mathcal{T}_{23} (t, k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}) =}{} + \sin k_{3}+ \frac{\cos ^{3}k_{6}}{1 + t^{3}} \biggr]. \end{aligned}

Define $$h_{11}(t) = t^{2} + \frac{46}{45} t + \frac{1}{ 270 ( 1 + t^{2}) } + \frac{1}{135}$$, $$h_{12}(t) = \frac{1}{330} t^{2} +\frac{1}{330}t+ e^{t} + \frac{332}{165}$$, $$h_{13} = \frac{1}{95} t +\frac{3}{95} + e^{t}$$, $$h_{21}(t) = \sin t+t+ 4$$, $$h_{22} (t) = 2e^{t} + t +\frac{1}{1+t}+1$$, $$h_{23} = \frac{1}{1+t^{3}}+t + \frac{5}{2}$$, $$\gamma _{1} (t) = \frac{1}{270( 1 + t^{2} ) } + \frac{2}{135} t + \frac{1}{135}$$, $$\gamma _{2} (t) = \frac{1}{330} t^{2} + \frac{1}{330} t+ \frac{4}{330}$$, and $$\gamma _{3} (t) =\frac{1}{95} t+ \frac{2}{95}$$. Put $$\sigma _{1} =\frac{3}{2}$$, $$\sigma _{2} =\frac{7}{4}$$, $$\sigma _{3} =\frac{9}{5}$$, $$\zeta _{1} =\frac{1}{4}$$, $$\zeta _{2} =\frac{1}{2}$$, and $$\zeta _{2} =\frac{3}{5}$$. It is easy to check that $$\|\mathcal{T}_{ij} (t,k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6})\| \leq h_{ij}(t)$$ and

$$P_{\rho }\bigl( \mathcal{T}_{1j} (t,k_{1}, k_{2}, k_{3}, k_{4}, k_{5}, k_{6}), \mathcal{T}_{1j} \bigl(t,k'_{1}, k'_{2}, k'_{3}, k'_{4}, k'_{5}, k'_{6}\bigr) \bigr) \leq \gamma _{j} (t) \Biggl( \sum_{i=1}^{6} \bigl\vert k_{i} - k'_{i} \bigr\vert \Biggr)$$

for $$i=1,2$$ and $$j=1,2,3$$. By using (6) and (7), (8), we obtain

\begin{aligned} &\varSigma _{1} = \bigl[ \varGamma _{q}( \zeta _{1} + 2) \varGamma _{q} ( 2 - \zeta _{1}) - 2 \varGamma _{q} (\zeta _{1} + 1 ) \varGamma _{q}( 2 + \zeta _{1}) - 2 \varGamma _{q}( \zeta _{1} + 1 ) \varGamma _{q} ( 2 - \zeta _{1} ) \bigr]^{-1} \\ & \hphantom{\varSigma _{1}}= \biggl[ \varGamma _{q} \biggl( \frac{1}{4} + 2 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{4} \biggr) - 2 \varGamma _{q} \biggl( \frac{1}{4} + 1 \biggr) \varGamma _{q} \biggl( 2 + \frac{1}{4} \biggr) \\ & \hphantom{\varSigma _{1}=}{} - 2 \varGamma _{q} \biggl( \frac{1}{4} + 1 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{4} \biggr) \biggr]^{-1} \\ & \hphantom{\varSigma _{1}}= \biggl[ \varGamma _{q} \biggl( \frac{9}{4} \biggr) \varGamma _{q} \biggl( \frac{7}{4} \biggr) - 2 \varGamma _{q} \biggl( \frac{5}{4} \biggr) \varGamma _{q} \biggl( \frac{9}{4} \biggr) - 2 \varGamma _{q} \biggl( \frac{5}{4} \biggr) \varGamma _{q} \biggl( \frac{7}{4} \biggr) \biggr]^{-1}, \end{aligned}
(10)
\begin{aligned} &\varLambda _{11} = \frac{1}{ \varGamma _{q}(\sigma _{1} + 1) } \bigl( 1 + \vert \varSigma _{1} \vert \bigl[ \varGamma _{q}( \zeta _{1} + 2 ) \varGamma _{q}( 2 - \zeta _{1} ) \\ &\hphantom{\varLambda _{11} =}{} + \varGamma _{q}(\zeta _{1} + 1) \varGamma _{q}(\zeta _{1} + 2)+ \varGamma _{q}( \zeta _{1} + 1 ) \varGamma _{q}( 2 - \zeta _{1}) \bigr] \bigr) \\ &\hphantom{\varLambda _{11} =}{} + \frac{ \vert \varSigma _{1} \vert \varGamma _{q}( \zeta _{1} + 1 ) \varGamma _{q}(\zeta _{1} + 2 ) \varGamma _{q}( 2 - \zeta _{1})}{ \varGamma _{q}( \sigma _{1} + \zeta _{1} + 1 )} \\ & \hphantom{\varLambda _{11} =}{} + \frac{ \vert \varSigma _{1} \vert \varGamma _{q}( \zeta _{1} + 1 ) \varGamma _{q}( \zeta _{1} + 2 ) \varGamma _{q}( 2 - \zeta _{1})}{ \varGamma _{q}( \sigma _{1} - \zeta _{1}+1 ) } \\ &\hphantom{\varLambda _{11} } = \frac{1}{ \varGamma _{q} ( \frac{3}{2} + 1 ) } \biggl( 1 + \vert \varSigma _{1} \vert \biggl[ \varGamma _{q} \biggl( \frac{1}{4} + 2 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{4} \biggr) \\ &\hphantom{\varLambda _{11} =}{} + \varGamma _{q} \biggl(\frac{1}{4} + 1 \biggr) \varGamma _{q} \biggl(\frac{1}{4} + 2 \biggr)+ \varGamma _{q} \biggl(\frac{1}{4} + 1 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{4} \biggr) \biggr] \biggr) \\ & \hphantom{\varLambda _{11} =}{} + \frac{ \vert \varSigma _{1} \vert \varGamma _{q} ( \frac{1}{4} + 1 ) \varGamma _{q} ( \frac{1}{4} + 2 ) \varGamma _{q} ( 2 -\frac{1}{4} ) }{ \varGamma _{q} ( \frac{3}{2} + \frac{1}{4}+ 1 ) } \\ &\hphantom{\varLambda _{11} =}{} + \frac{ \vert \varSigma _{1} \vert \varGamma _{q} ( \frac{1}{4} + 1 ) \varGamma _{q} ( \frac{1}{4} + 2 ) \varGamma _{q} ( 2 - \frac{1}{4} ) }{ \varGamma _{q} ( \frac{3}{2}- \frac{1}{4} + 1 ) } \\ &\hphantom{\varLambda _{11} } = \frac{1}{ \varGamma _{q} ( \frac{5}{2} ) } \biggl( 1 + \vert \varSigma _{1} \vert \biggl[ \varGamma _{q} \biggl( \frac{9}{4} \biggr) \varGamma _{q} \biggl( \frac{7}{4} \biggr) \\ & \hphantom{\varLambda _{11} =}{} + \varGamma _{q} \biggl(\frac{5}{4} \biggr) \varGamma _{q} \biggl( \frac{9}{4} \biggr)+ \varGamma _{q} \biggl(\frac{5}{4} \biggr) \varGamma _{q} \biggl( \frac{7}{4} \biggr) \biggr] \biggr) \\ & \hphantom{\varLambda _{11} =}{} + \frac{ \vert \varSigma _{1} \vert \varGamma _{q} ( \frac{5}{4} ) \varGamma _{q} ( \frac{9}{4} ) \varGamma _{q} ( \frac{7}{4} ) }{ \varGamma _{q} ( \frac{11}{4} ) } + \frac{ \vert \varSigma _{1} \vert \varGamma _{q} ( \frac{5}{4} ) \varGamma _{q} ( \frac{9}{4} ) \varGamma _{q} ( \frac{7}{4} ) }{ \varGamma _{q} ( \frac{9}{4} ) }, \end{aligned}
(11)
\begin{aligned} &\varLambda _{21} = \frac{1}{ \varGamma _{q}( \sigma _{1} - \zeta _{1} + 1 ) } + \frac{ \vert \varSigma _{1} \vert \varGamma _{q}( \zeta _{1} + 2 )}{ \varGamma _{q}(\sigma _{1} + 1 )}+ \frac{2 \vert \varSigma _{1} \vert ( \varGamma _{q} ( \zeta _{1} + 1 ) + \varGamma _{q}( \zeta _{1} + 2 )) }{ \varGamma _{q}(\sigma _{1} + \zeta _{1} + 1)} \\ & \hphantom{\varLambda _{21} =}{} + \frac{2 \vert \varSigma _{1} \vert ( 2 \varGamma _{q}( \zeta _{1}+1 ) + \varGamma _{q}( \zeta _{1} + 2 ))}{ \varGamma _{q}( \sigma _{1} - \zeta _{1} + 1 ) } \\ &\hphantom{\varLambda _{21} } = \frac{1}{ \varGamma _{q} ( \frac{3}{2} - \frac{1}{4}+ 1 ) } + \frac{ \vert \varSigma _{1} \vert \varGamma _{q} ( \frac{1}{4} + 2 ) }{ \varGamma _{q} (\frac{3}{2} + 1 )}+ \frac{2 \vert \varSigma _{1} \vert ( \varGamma _{q} ( \frac{1}{4} + 1 ) + \varGamma _{q} ( \frac{1}{4} + 2 ) ) }{ \varGamma _{q} ( \frac{3}{2}+ \frac{1}{4} + 1 ) } \\ & \hphantom{\varLambda _{21} =}{} + \frac{ 2 \vert \varSigma _{1} \vert ( 2 \varGamma _{q} ( \frac{1}{4} + 1 ) + \varGamma _{q} ( \frac{1}{4} + 2 ) ) }{ \varGamma _{q} ( \frac{3}{2} - \frac{1}{4} + 1 )} \\ & \hphantom{\varLambda _{21} }= \frac{1}{ \varGamma _{q} ( \frac{9}{4} ) } + \frac{ \vert \varSigma _{1} \vert \varGamma _{q} ( \frac{9}{4} ) }{ \varGamma _{q} (\frac{5}{2} )} + \frac{2 \vert \varSigma _{1} \vert ( \varGamma _{q} ( \frac{5}{4} ) + \varGamma _{q} ( \frac{9}{4} ) ) }{ \varGamma _{q} ( \frac{11}{4} ) } \\ & \hphantom{\varLambda _{21} =}{} + \frac{ 2 \vert \varSigma _{1} \vert ( 2 \varGamma _{q} ( \frac{5}{4} ) + \varGamma _{q} ( \frac{9}{4} ) ) }{ \varGamma _{q} ( \frac{9}{4} )}, \end{aligned}
(12)
\begin{aligned} &\varSigma _{2} = \bigl[ \varGamma _{q}( \zeta _{2} + 2) \varGamma _{q} ( 2 - \zeta _{2}) - 2 \varGamma _{q} (\zeta _{2} + 1 ) \varGamma _{q}( 2 + \zeta _{2}) - 2 \varGamma _{q}( \zeta _{2} + 1 ) \varGamma _{q} ( 2 - \zeta _{2} ) \bigr]^{-1} \\ & \hphantom{\varSigma _{2}}= \biggl[ \varGamma _{q} \biggl( \frac{1}{2} + 2 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{2} \biggr) - 2 \varGamma _{q} \biggl( \frac{1}{2} + 1 \biggr) \varGamma _{q} \biggl( 2 + \frac{1}{2} \biggr) \\ &\hphantom{\varSigma _{2} =}{} - 2 \varGamma _{q} \biggl( \frac{1}{2} + 1 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{2} \biggr) \biggr]^{-1} \\ &\hphantom{\varSigma _{2} } = \biggl[ \varGamma _{q} \biggl( \frac{5}{2} \biggr) \varGamma _{q} \biggl( \frac{3}{2} \biggr) - 2 \varGamma _{q} \biggl( \frac{3}{2} \biggr) \varGamma _{q} \biggl( \frac{5}{2} \biggr) - 2 \varGamma _{q} \biggl( \frac{3}{2} \biggr) \varGamma _{q} \biggl( \frac{3}{2} \biggr) \biggr]^{-1}, \end{aligned}
(13)
\begin{aligned} &\varLambda _{12} = \frac{1}{ \varGamma _{q}(\sigma _{2} + 1) } \bigl( 1 + \vert \varSigma _{2} \vert \bigl[ \varGamma _{q}(\zeta _{2} + 2 ) \varGamma _{q}( 2 - \zeta _{2} ) \\ & \hphantom{\varLambda _{12} =}{} + \varGamma _{q}(\zeta _{2} + 1) \varGamma _{q} (\zeta _{2} + 2)+ \varGamma _{q}( \zeta _{2} + 1 ) \varGamma _{q}( 2 - \zeta _{2} ) \bigr] \bigr) \\ & \hphantom{\varLambda _{12} =}{} + \frac{ \vert \varSigma _{2} \vert \varGamma _{q}( \zeta _{2} + 1 ) \varGamma _{q}(\zeta _{2} + 2 ) \varGamma _{q} ( 2 - \zeta _{2})}{ \varGamma _{q}( \sigma _{2} + \zeta _{2} + 1 )} \\ & \hphantom{\varLambda _{12} =}{} + \frac{ \vert \varSigma _{2} \vert \varGamma _{q}( \zeta _{2} + 1 ) \varGamma _{q}( \zeta _{2} + 2 ) \varGamma _{q} ( 2 - \zeta _{2})}{ \varGamma _{q}( \sigma _{2} - \zeta _{2} + 1 ) } \\ &\hphantom{\varLambda _{12} } = \frac{1}{ \varGamma _{q} (\frac{7}{4} + 1 ) } \biggl( 1 + \vert \varSigma _{2} \vert \biggl[ \varGamma _{q} \biggl( \frac{1}{2} + 2 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{2} \biggr) \\ &\hphantom{\varLambda _{12} =}{} + \varGamma _{q} \biggl(\frac{1}{2} + 1 \biggr) \varGamma _{q} \biggl( \frac{1}{2} + 2 \biggr) + \varGamma _{q} \biggl( \frac{1}{2} + 1 \biggr) \varGamma _{q} \biggl( 2 - \frac{1}{2} \biggr) \biggr] \biggr) \\ & \hphantom{\varLambda _{12} =}{} + \frac{ \vert \varSigma _{2} \vert \varGamma _{q} ( \frac{1}{2} + 1 ) \varGamma _{q} ( \frac{1}{2} + 2 ) \varGamma _{q} ( 2 - \frac{1}{2} ) }{ \varGamma _{q} ( \frac{7}{4}+\frac{1}{2} + 1 ) } \\ &\hphantom{\varLambda _{12} =}{} + \frac{ \vert \varSigma _{2} \vert \varGamma _{q} ( \frac{1}{2} + 1 ) \varGamma _{q} ( \frac{1}{2} + 2 ) \varGamma _{q} ( 2 - \frac{1}{2} )}{ \varGamma _{q} ( \frac{7}{4} - \frac{1}{2} + 1 ) } \\ &\hphantom{\varLambda _{12} } = \frac{1}{ \varGamma _{q} (\frac{11}{4} ) } \biggl( 1 + \vert \varSigma _{2} \vert \biggl[ \varGamma _{q} \biggl( \frac{5}{2} \biggr) \varGamma _{q} \biggl( \frac{3}{2} \biggr) \\ &\hphantom{\varLambda _{12} =}{} + \varGamma _{q} \biggl(\frac{3}{2} \biggr) \varGamma _{q} \biggl( \frac{5}{2} \biggr) + \varGamma _{q} \biggl( \frac{3}{2} \biggr) \varGamma _{q} \biggl( \frac{3}{2} \biggr) \biggr] \biggr) \\ & \hphantom{\varLambda _{12} =}{} + \frac{ \vert \varSigma _{2} \vert \varGamma _{q} ( \frac{3}{2} ) \varGamma _{q} ( \frac{5}{2} ) \varGamma _{q} ( \frac{3}{2} ) }{ \varGamma _{q} ( \frac{13}{4} ) } + \frac{ \vert \varSigma _{2} \vert \varGamma _{q} ( \frac{3}{2} ) \varGamma _{q} ( \frac{5}{2} ) \varGamma _{q} ( \frac{3}{2} )}{ \varGamma _{q} ( \frac{9}{4} ) }, \end{aligned}
(14)
\begin{aligned} &\varLambda _{22} = \frac{1}{ \varGamma _{q}( \sigma _{2} - \zeta _{2} + 1 ) } + \frac{ \vert \varSigma _{2} \vert \varGamma _{q}( \zeta _{2} + 2 )}{ \varGamma _{q}(\sigma _{2} + 1 )} \\ & \hphantom{\varLambda _{22} =}{} + \frac{2 \vert \varSigma _{2} \vert ( \varGamma _{q} ( \zeta _{2} + 1 ) + \varGamma _{q}( \zeta _{2} + 2 )) }{ \varGamma _{q}(\sigma _{2} + \zeta _{2} + 1)} \\ & \hphantom{\varLambda _{22} =}{} + \frac{ 2 \vert \varSigma _{2} \vert ( 2 \varGamma _{q} ( \zeta _{2} + 1 ) + \varGamma _{q}( \zeta _{2} + 2 ) ) }{ \varGamma _{q}( \sigma _{2} - \zeta _{2} + 1 ) } \\ & \hphantom{\varLambda _{22} }= \frac{1}{ \varGamma _{q} ( \frac{7}{4} - \frac{1}{2} + 1 ) } + \frac{ \vert \varSigma _{2} \vert \varGamma _{q} ( \frac{1}{2} + 2 ) }{ \varGamma _{q} ( \frac{1}{2} + 1 )} \\ & \hphantom{\varLambda _{22} =}{} + \frac{2 \vert \varSigma _{2} \vert ( \varGamma _{q} ( \frac{1}{2} + 1 ) + \varGamma _{q} ( \frac{1}{2} + 2 ) ) }{ \varGamma _{q} (\frac{7}{4} + \frac{1}{2} + 1 ) } \\ & \hphantom{\varLambda _{22} =}{} + \frac{2 \vert \varSigma _{2} \vert ( \varGamma _{q} ( \frac{1}{2} + 1 ) + \varGamma _{q} ( \frac{1}{2} + 2 ) ) }{ \varGamma _{q} ( \frac{7}{4} - \frac{1}{2} + 1 )} \\ &\hphantom{\varLambda _{22} } = \frac{1}{ \varGamma _{q} ( \frac{9}{4} ) } + \frac{ \vert \varSigma _{2} \vert \varGamma _{q} ( \frac{5}{2} ) }{ \varGamma _{q} ( \frac{3}{2} )} + \frac{2 \vert \varSigma _{2} \vert ( \varGamma _{q} ( \frac{3}{2} ) + \varGamma _{q} ( \frac{5}{2} ) ) }{ \varGamma _{q} (\frac{13}{4} ) } \\ & \hphantom{\varLambda _{22} =}{} + \frac{2 \vert \varSigma _{2} \vert ( \varGamma _{q} ( \frac{3}{2} ) + \varGamma _{q} ( \frac{5}{2} ) ) }{ \varGamma _{q} ( \frac{9}{4} )}, \end{aligned}
(15)
\begin{aligned} &\varSigma _{3} = \bigl[ \varGamma _{q}( \zeta _{3} + 2) \varGamma _{q} ( 2 - \zeta _{3}) - 2 \varGamma _{q} (\zeta _{3} + 1 ) \varGamma _{q}( 2 + \zeta _{3}) - 2 \varGamma _{q}( \zeta _{3} + 1 ) \varGamma _{q} ( 2 - \zeta _{3} ) \bigr]^{-1} \\ &\hphantom{\varSigma _{3} } = \biggl[ \varGamma _{q} \biggl( \frac{3}{5}+ 2 \biggr) \varGamma _{q} \biggl( 2 - \frac{3}{5} \biggr) - 2 \varGamma _{q} \biggl( \frac{3}{5} + 1 \biggr) \varGamma _{q} \biggl( 2 + \frac{3}{5} \biggr) \\ & \hphantom{\varSigma _{3} =}{} - 2 \varGamma _{q} \biggl( \frac{3}{5} + 1 \biggr) \varGamma _{q} \biggl( 2 - \frac{3}{5} \biggr) \biggr]^{-1} \\ &\hphantom{\varSigma _{3} } = \biggl[ \varGamma _{q} \biggl( \frac{13}{5} \biggr) \varGamma _{q} \biggl( \frac{7}{5} \biggr) - 2 \varGamma _{q} \biggl( \frac{8}{5} \biggr) \varGamma _{q} \biggl( \frac{13}{5} \biggr) - 2 \varGamma _{q} \biggl( \frac{8}{5} \biggr) \varGamma _{q} \biggl( \frac{7}{5} \biggr) \biggr]^{-1}, \end{aligned}
(16)
\begin{aligned} &\varLambda _{13} = \frac{1}{ \varGamma _{q}(\sigma _{3} + 1) } \bigl( 1 + \vert \varSigma _{3} \vert \bigl[ \varGamma _{q}(\zeta _{3} + 2 ) \varGamma _{q}( 2 - \zeta _{3} ) \\ & \hphantom{\varLambda _{13} =}{} + \varGamma _{q}(\zeta _{3} + 1) \varGamma _{q}(\zeta _{3} + 2)+ \varGamma _{q}( \zeta _{3} + 1 ) \varGamma _{q}( 2 - \zeta _{3} ) \bigr] \bigr) \\ &\hphantom{\varLambda _{13} =}{} + \frac{ \vert \varSigma _{3} \vert \varGamma _{q}( \zeta _{3} + 1 ) \varGamma _{q}(\zeta _{3} + 2 ) \varGamma _{q}( 2 - \zeta _{3})}{ \varGamma _{q}( \sigma _{3} + \zeta _{3} + 1 )} \\ &\hphantom{\varLambda _{13} =}{} + \frac{ \vert \varSigma _{3} \vert \varGamma _{q}( \zeta _{3} + 1 ) \varGamma _{q}( \zeta _{3} + 2 ) \varGamma _{q}( 2 - \zeta _{3})}{ \varGamma _{q}( \sigma _{3} - \zeta _{3} + 1 ) } \\ &\hphantom{\varLambda _{13} } = \frac{1}{ \varGamma _{q} (\frac{9}{5} + 1 ) } \biggl( 1 + \vert \varSigma _{3} \vert \biggl[ \varGamma _{q} \biggl( \frac{3}{5} + 2 \biggr) \varGamma _{q} \biggl( 2 - \frac{3}{5} \biggr) \\ & \hphantom{\varLambda _{13} =}{} + \varGamma _{q} \biggl( \frac{3}{5} + 1 \biggr) \varGamma _{q} \biggl( \frac{3}{5} + 2 \biggr) + \varGamma _{q} \biggl( \frac{3}{5} + 1 \biggr) \varGamma _{q} \biggl( 2 - \frac{3}{5} \biggr) \biggr] \biggr) \\ &\hphantom{\varLambda _{13} =}{} + \frac{ \vert \varSigma _{3} \vert \varGamma _{q} ( \frac{3}{5} + 1 ) \varGamma _{q} (\frac{3}{5} + 2 ) \varGamma _{q} ( 2 - \frac{3}{5} ) }{ \varGamma _{q} ( \frac{9}{5} + \frac{3}{5} + 1 ) } \\ &\hphantom{\varLambda _{13} =}{} + \frac{ \vert \varSigma _{3} \vert \varGamma _{q} ( \frac{3}{5} + 1 ) \varGamma _{q} ( \frac{3}{5} + 2 ) \varGamma _{q} ( 2 - \frac{3}{5} ) }{ \varGamma _{q} ( \frac{9}{5}- \frac{3}{5} + 1 ) } \\ & \hphantom{\varLambda _{13} }= \frac{1}{ \varGamma _{q} (\frac{14}{5} ) } \biggl( 1 + \vert \varSigma _{3} \vert \biggl[ \varGamma _{q} \biggl( \frac{13}{5} \biggr) \varGamma _{q} \biggl( \frac{7}{5} \biggr) \\ & \hphantom{\varLambda _{13} =}{} + \varGamma _{q} \biggl( \frac{8}{5} \biggr) \varGamma _{q} \biggl( \frac{13}{5} \biggr) + \varGamma _{q} \biggl( \frac{8}{5} \biggr) \varGamma _{q} \biggl( \frac{7}{5} \biggr) \biggr] \biggr) \\ & \hphantom{\varLambda _{13} =}{} + \frac{ \vert \varSigma _{3} \vert \varGamma _{q} ( \frac{8}{5} ) \varGamma _{q} (\frac{13}{5} ) \varGamma _{q} ( \frac{7}{5} ) }{ \varGamma _{q} ( \frac{17}{5} ) } + \frac{ \vert \varSigma _{3} \vert \varGamma _{q} ( \frac{8}{5} ) \varGamma _{q} ( \frac{13}{5} ) \varGamma _{q} ( \frac{7}{5} ) }{ \varGamma _{q} ( \frac{11}{5} ) }, \end{aligned}
(17)
\begin{aligned} &\varLambda _{23} = \frac{1}{ \varGamma _{q}( \sigma _{3} - \zeta _{3} + 1 ) } + \frac{ \vert \varSigma _{3} \vert \varGamma _{q}( \zeta _{3} + 2 )}{ \varGamma _{q}( \sigma _{3} + 1 )} \\ & \hphantom{\varLambda _{23} =}{}+ \frac{2 \vert \varSigma _{3} \vert ( \varGamma _{q} ( \zeta _{3} + 1 ) + \varGamma _{q}( \zeta _{3} + 2 )) }{ \varGamma _{q}(\sigma _{3} + \zeta _{3} + 1)} \\ & \hphantom{\varLambda _{23} =}{} + \frac{2 \vert \varSigma _{3} \vert ( 2 \varGamma _{q}( \zeta _{3} + 1 ) + \varGamma _{q}( \zeta _{3} + 2 ))}{ \varGamma _{q}( \sigma _{3} - \zeta _{3} + 1 )} \\ &\hphantom{\varLambda _{23} } = \frac{1}{ \varGamma _{q} ( \frac{9}{5} - \frac{3}{5}+ 1 ) } + \frac{ \vert \varSigma _{3} \vert \varGamma _{q} ( \frac{3}{5}+ 2 ) }{ \varGamma _{q} ( \frac{9}{5} + 1 ) } \\ & \hphantom{\varLambda _{23} =}{} + \frac{ 2 \vert \varSigma _{3} \vert ( \varGamma _{q} ( \frac{3}{5}+ 1 ) + \varGamma _{q} ( \frac{3}{5} + 2 ) ) }{ \varGamma _{q} ( \frac{9}{5} + \frac{3}{5} + 1 ) } \\ & \hphantom{\varLambda _{23} =}{} + \frac{2 \vert \varSigma _{3} \vert ( \varGamma _{q} ( \frac{3}{5}+ 1 ) + \varGamma _{q} ( \frac{3}{5} + 2 ) ) }{ \varGamma _{q} ( \frac{9}{5} - \frac{3}{5} + 1 ) } \\ &\hphantom{\varLambda _{23} } = \frac{1}{ \varGamma _{q} ( \frac{11}{5} ) } + \frac{ \vert \varSigma \vert \varGamma _{q} ( \frac{13}{5} ) }{ \varGamma _{q} ( \frac{14}{5} ) } + \frac{ 2 \vert \varSigma _{3} \vert ( \varGamma _{q} ( \frac{8}{5} ) + \varGamma _{q} ( \frac{13}{5} ) ) }{ \varGamma _{q} ( \frac{17}{5} ) } \\ & \hphantom{\varLambda _{23} =}{} + \frac{2 \vert \varSigma _{3} \vert ( \varGamma _{q} ( \frac{8}{5} ) + \varGamma _{q} ( \frac{13}{5} ) ) }{ \varGamma _{q} ( \frac{11}{5} )}. \end{aligned}
(18)

Note that Tables 1 and 2 show that $$\varLambda _{11}\approx 2.5743$$, $$\varLambda _{12}\approx 2.5222$$, $$\varLambda _{13}\approx 2.5131$$, $$\varLambda _{21}\approx 3.9450$$, $$\varLambda _{22}\approx 3.9032$$, $$\varLambda _{23}\approx 3.8920$$. Table 3 shows that $$\varLambda _{11}\approx 2.3064$$, $$\varLambda _{12}\approx 2.1178$$, $$\varLambda _{13}\approx 2.0901$$, $$\varLambda _{21}\approx 3.7015$$, $$\varLambda _{22}\approx 3.5288$$, $$\varLambda _{23}\approx 3.4997$$, and Table 4 leads us to $$\varLambda _{11}\approx 2.1482$$, $$\varLambda _{12}\approx 1.8883$$, $$\varLambda _{13}\approx 1.8536$$, $$\varLambda _{21}\approx 3.5434$$, $$\varLambda _{22}\approx 3.3065$$, and $$\varLambda _{23}\approx 3.2801$$ for $$q=\frac{1}{10}$$, $$q=\frac{1}{2}$$, and $$q=\frac{6}{7}$$ respectively. By the definition of $$\gamma _{i}$$, for $$j=1,2,3$$, we get $$\|\gamma _{1}\|_{\infty }= \frac{7}{270}$$, $$\|\gamma _{2}\|_{\infty }= \frac{1}{55}$$, and $$\|\gamma _{3}\|_{\infty }=\frac{3}{95}$$. Now, by using Eqs. (7), (8), (10), (11), (12), (13), (14), (15), (16), (17), and (18), we obtain

\begin{aligned} \Delta ={}& \sum_{i=1}^{m} \Vert \gamma _{i} \Vert _{\infty } \biggl( \frac{ 1 + \varGamma _{q}( \zeta _{i} + 1) }{ \varGamma _{q}(\zeta _{i} + 1) } \biggr) ( \varLambda _{1i} + \varLambda _{2i} ) \\ ={}& \frac{7}{270} \biggl( \frac{ 1 + \varGamma _{q} ( \frac{1}{4} + 1 ) }{ \varGamma _{q} (\frac{1}{4} + 1 ) } \biggr) ( \varLambda _{11} + \varLambda _{21} ) \\ &{} + \frac{1}{55} \biggl( \frac{ 1 + \varGamma _{q} ( \frac{1}{2} + 1 ) }{ \varGamma _{q} (\frac{1}{2} + 1 ) } \biggr) ( \varLambda _{12} + \varLambda _{22} ) \\ & {} + \frac{3}{95} \biggl( \frac{ 1 + \varGamma _{q} ( \frac{3}{5} + 1 ) }{ \varGamma _{q} (\frac{3}{5} + 1 ) } \biggr) ( \varLambda _{13} + \varLambda _{23} ). \end{aligned}

By using Table 5, one finds the values of Δ for $$q = \frac{1}{10}$$, $$q = \frac{1}{2}$$, and $$q = \frac{6}{7}$$ (see Fig. 1). In fact, we get $$\Delta \approx 0.9892 <1$$, $$\Delta \approx 0.9038 <1$$, and $$\Delta \approx 0.8512 <1$$ for $$q=\frac{1}{10}, \frac{1}{2}, \frac{6}{7}$$, respectively. Now, by using Theorem 6, system (9) of fractional differential inclusions has a solution.