Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus

Abstract

In this investigation, by applying the definition of the fractional q-derivative of the Caputo type and the fractional q-integral of the Riemann–Liouville type, we study the existence and uniqueness of solutions for a multi-term nonlinear fractional q-integro-differential equations under some boundary conditions ${}^{c}D_{q}^{\alpha} x(t) = w ( t, x(t), (\varphi_{1} x)(t), (\varphi_{2} x)(t), {}^{c}D_{q} ^{ \beta_{1}} x(t), {}^{c}D_{q}^{\beta_{2}} x(t), \ldots, {}^{c}D _{q}^{ \beta_{n}}x(t) )$. Our results are based on some classical fixed point techniques, as Schauder’s fixed point theorem and Banach contraction mapping principle. Besides, some instances are exhibited to illustrate our results and we report all algorithms required along with the numerical result obtained.

1 Introduction

The subjects of fractional calculus and q-calculus are one of the significant branches in mathematical analysis. In 1910, the subject of q-difference equations was introduced by Jackson [1]. After that, at the beginning of the last century, studies on the q-difference equation appeared in much work, especially in [2,3,4,5,6]. For some earlier work on the topic, we refer to [7, 8], whereas the preliminary concepts on q-fractional calculus can be found in [9], as indicated: Perhaps Leibniz did not expect this number of applications when he sent a letter in 1695 to L’Hopital asking about the meaning of the derivative of order half. For countless applications on the q-fractional calculus, see for example [10,11,12,13,14,15,16,17].

In the recent years, there have appeared many papers about differential and integro-differential equations and inclusions which are valuable tools in the modeling of many phenomena in various fields of science [18,19,20,21,22,23,24,25]. In 2012, Ahmad et al. [26] discussed the existence and uniqueness of solutions for the fractional q-difference equations ${}^{c}D_{q}^{\alpha }u(t)= T ( t, u(t) )$, $\alpha _{1} u(0) - \beta _{1} D_{q} u(0) = \gamma _{1} u( \eta _{1})$ and $\alpha _{2} u(1) - \beta _{2} D_{q} u(1) = \gamma _{2} u( \eta _{2})$, for $t \in I$, where $\alpha \in (1, 2]$, $\alpha _{i}$, $\beta _{i}$, $\gamma _{i}$, $\eta _{i}$ are real numbers, for $i=1,2$ and $T \in C(J \times \mathbb{R}, \mathbb{R})$. In 2013, Zhao el al. [27] reviewed the q-integral problem $(D_{q}^{\alpha }u)(t) + f(t, u(t) )=0$, with the conditions that $u(1)$, $u(0)$ are equal to $\mu I_{q}^{\beta }u(\eta )$, 0, respectively, for almost all $t \in (0,1)$, where $q \in (0,1)$ and α, β, η belong to $(1, 2]$, $(0, 2]$, $(0,1)$, respectively, μ is positive real number, $D_{q}^{\alpha }$ is the q-derivative of Riemann–Liouville and we have a real-valued continuous map u defined on $I \times [0, \infty )$. In 2014, Ahmad et al. [28] considered the problem

$$ \textstyle\begin{cases} {}^{c}D^{\beta }_{q} ( {}^{c}D^{\gamma }_{q} + \lambda ) u(t) = p f(t, u(t)) + k I_{q}^{\xi }g(t, u(t)), \\ \alpha _{1} u(0) - \beta _{1} (t^{(1-\gamma )} D_{q} u(0))|_{t=0}= \sigma _{1} u(\eta _{1}), \qquad \alpha _{2} u(1) + \beta _{2} D_{q} u(1)= \sigma _{2} u(\eta _{2}), \end{cases} $$

for $t, q \in [0,1]$, where ${}^{c}D_{q}^{\beta }$ and ${}^{c}D_{q} ^{\gamma }$ denote the fractional q-derivative of the Caputo type, $0 < \beta$, $\gamma \leq 1$, $I_{q}^{\xi }(.)$ denotes the Riemann–Liouville integral with $\xi \in (0, 1)$, f, g are given continuous functions, λ and p, k are real constants and $\alpha _{i}, \beta _{i}, \sigma _{i}\in \mathbb{R}$, $\eta _{i} \in (0, 1)$, $i=1,2$. Also, one may refer to some research of Ahmad et al., in the recent years in [12, 14, 29,30,31]. In 2016, Abdeljawad et al. [32] stated and proved a new discrete q-fractional version of Gronwall inequality, ${}_{q}C_{a}^{\alpha }u(t) = T ( t, u(t) )$, where $u(a)=\gamma$, such that $\alpha \in (0, 1]$, $a \in \mathbb{T}_{q}= \{q^{n}: n \in \mathbb{Z} \}$, t belongs to $\mathbb{T}_{a}= [0, \infty )_{q} = \{ q^{-i} a: i=0, 1, 2, \ldots \}$, ${}_{q}C_{a}^{\alpha }$ means the Caputo fractional difference of order α and $T(t, x)$ fulfills a Lipschitz condition for all t and x. In 2019, Samei et al. [25] investigated the existence of solutions for equations and inclusions of multi-term fractional q-integro-differential with non-separated and initial boundary conditions

In this article, motivated by these achievements and the following results, we are working to stretch out solutions for the multi-term nonlinear fractional q-integro-differential equation with boundary conditions,

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

(1)

under conditions $x(0) + a x(1)=0$ and $x'(0) + bx'(1)=0$, for $t \in J: =[0,1]$ and all $q \in (0,1)$, where $1 < \alpha < 2$, $\beta _{i} \in (0,1)$ with $i=1, 2,\dots , n$, $a, b\ne -1$, $w : J {\times } \mathbb{R}^{n+3} \to \mathbb{R}$ is continuous for all variables and the mappings $\gamma _{j}$ map $J\times J$ to $\mathbb{R}^{+}$ such that $\sup_{t\in J} \vert \int _{0}^{t} \gamma _{j} (t,s) \,d_{q}s \vert$, where $j=1,2$, are finite, the maps $\varphi _{j}$, where $j=1,2$, are defined by $(\varphi _{j} u)(t) = \int _{0}^{t} \gamma _{j}(t,s) u(s) \,d_{q}s$.

The rest of the paper is arranged as follows: in Sect. 2, we recall some preliminary concepts and fundamental results of q-calculus. Section 3 is devoted to the main results, while examples illustrating the obtained results and algorithm for the problems are presented in Sect. 4.

2 Preliminaries

First of all, we point out some of the materials on the fractional q-calculus and fundamental results of it which needed in the next sections (for more information, consider [1, 9,10,11, 33]). Then, some well-known theorems of fixed point theorems are presented.

Assume that $q \in (0,1)$ and $a \in \mathbb{R}$. Define $[a]_{q}=\frac{1-q ^{a}}{1-q}$ [1]. The power function $(x-y)_{q} ^{n}$ with $n \in \mathbb{N}_{0}$ is $(x-y)_{q}^{(n)}= \prod_{k=0} ^{n-1} (x - yq^{k})$ and $(x-y)_{q}^{(0)}=1$ where $x, y \in \mathbb{R}$ and $\mathbb{N}_{0} := \{ 0\} \cup \mathbb{N}$ [10]. Also, for $\alpha \in \mathbb{R}$ and $a \neq 0$, we have $(x-y)_{q}^{(\alpha )}= x^{\alpha }\prod_{k=0} ^{\infty }(x-yq^{k}) / (x - yq^{\alpha + k})$. If $y=0$, then it is clear that $x^{(\alpha )}= x^{\alpha }$ (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 $\varGamma _{q} (z+1) = [z]_{q} \varGamma _{q} (z)$. The value of the q-Gamma function, $\varGamma _{q}(z)$, for input values q and z with counting the number of sentences n in summation is addressed by a simplifying analysis. For this design, we present a pseudo-code description of the technique for estimating q-Gamma function of order n which show in Algorithm 2. The q-derivative of the function f is defined by $(D_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}$ and $(D_{q} f)(0) = \lim_{x \to 0} (D _{q} f)(x)$, which is shown in Algorithm 3 [4]. Also, 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 all $n \geq 1$, where $(D_{q}^{0} f)(x) = f(x)$ [4]. The q-integral of a function f defined on $[0,b]$ is defined by

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

for $0 \leq x \leq b$, provided that the series absolutely converges [4]. For any positive number α and β, the q-Beta function is defined by [33]

$$ B_{q}(\alpha , \beta ) = \int _{0}^{1} (1- qs)_{q}^{(\alpha -1)} s^{ \beta -1} \,d_{q}s. $$

(2)

The q-derivative of the function f is defined by $(D_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}$ and $(D_{q} f)(0) = \lim_{x \to 0} (D _{q} f)(x)$, which is shown in Algorithm 3 [4, 11, 33]. If a is in $[0, b]$, then

$$ \int _{a}^{b} f(u) \,d_{q} u = I_{q} f(b) - I_{q} f(a) = (1-q) \sum _{k=0} ^{\infty } q^{k} \bigl[ b f\bigl(b q^{k}\bigr) - a f\bigl(a q^{k}\bigr) \bigr], $$

whenever the series exists, which is shown in Algorithm 4. The operator $I_{q}^{n}$ is given by $(I_{q}^{0} h)(x) = h(x)$ and

$$ \bigl(I_{q}^{n} h\bigr) (x) = \bigl(I_{q} \bigl(I_{q}^{n-1} h\bigr)\bigr) (x), $$

for $n \geq 1$ and $g \in C([0,b])$ [4]. It has been proved that $(D_{q} (I_{q} f))(x) = f(x)$ and $(I_{q} (D_{q} f))(x) = f(x) - f(0)$ whenever f is continuous at $x =0$ [4]. The fractional Riemann–Liouville type q-integral of the function f on J, of $\alpha \geq 0$ is given by $(I_{q}^{0} f)(t) = f(t)$ and

$$ \bigl(I_{q}^{\alpha }f\bigr) (t) = \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t} (t- qs)^{(\alpha - 1)} f(s) \,d_{q}s, $$

for $t \in J$ and $\alpha >0$ [31, 34]. Also, the fractional Caputo type q-derivative of the function f is given by

$$ \begin{aligned} \bigl( {}^{c}D_{q}^{\alpha }f \bigr) (t) & = \bigl( I_{q}^{[\alpha ]-\alpha }\bigl( D_{q}^{[\alpha ]} f\bigr) \bigr) (t) \\ & = \frac{1}{\varGamma _{q} ([\alpha ]-\alpha )} \int _{0} ^{t} (t- qs)^{ ([\alpha ]-\alpha -1 )} \bigl( D_{q}^{[ \alpha ]} f \bigr) (s) \,d_{q}s, \end{aligned} $$

(3)

for $t \in J$ and $\alpha >0$ [31, 34]. It has been proved that $( I_{q}^{\beta } (I_{q}^{\alpha } f)) (x) = ( I_{q} ^{\alpha + \beta } f) (x)$ and $(D_{q}^{\alpha } (I_{q}^{\alpha } f) ) (x) = f(x)$, where $\alpha , \beta \geq 0$ [34]. By using Algorithm 2, we can calculate $(I_{q}^{\alpha }f)(x)$ which is shown in Algorithm 5.

Algorithm 1

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

Algorithm 2

The proposed method for calculation of $\varGamma_{q}(x)$

Algorithm 3

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

Algorithm 4

The proposed method for calculation of $\int_{a}^{b} f(r) \,d_{q} r$

Algorithm 5

The proposed method for calculation of $(I_{q}^{\alpha}f)(x)$

Theorem 1

(Schauder’s fixed point theorem [35])

LetEbe a closed, convex and bounded subset of a Banach spaceXand self-mapTdefined onEbe continuous. ThenThas a fixed point inEwhenever $T(E)$is a relatively compact subset ofX.

3 Main results

Here, we investigate the inclusion of fractional q-derivative (1). First, we recall the following key result.

Lemma 2

([17])

Let $\alpha >0$and $n=[\alpha ]+1$. Then ${I_{q}^{\alpha }}^{c}D_{q} ^{\alpha } x(t) = x(t) + c_{0} + c_{1} t + c_{2} t + \cdots + c_{n-1} t^{n-1}$, where $c_{0}, c_{1}, \ldots , c_{n-1}$belong to $\mathbb{R}$.

Let us define the set X of all $f \in C(I)$, such that ${}^{c}D_{q} ^{{\beta }_{i}} x$ belongs to $C(I)$ ($i=1, 2, \dots , n$) and $q\in (0,1)$, where $0<\beta _{i}<1$. It is known that $(X,\| \cdot \|)$ with the norm $\Vert x \Vert = \max_{t\in J} \vert x(t) \vert + \sum_{i=1}^{n} \max_{t\in J} \vert {}^{c}D_{q}^{ \beta _{i}} x(t) \vert$, is a Banach space.

Lemma 3

Suppose thatfin $C(J)$and $\alpha \in (1,2)$. Then the boundary value problem

$$ \textstyle\begin{cases} {}^{c}D_{q}^{\alpha } x(t)=f(t), \quad t\in J, \\ x(0) + a x(1) = 0, \qquad x'(0) + bx'(1) =0, \end{cases} $$

is equivalent to the followingq-integral equation:

$$ x(t) = I_{q}^{\alpha }f(t) - I_{q}^{\alpha }f(1) + \frac{ab - b(1+a) t}{(1+a)(1+b) } I_{q}^{\alpha -1} f(1). $$

(4)

Proof

First of all, we see that Lemma 2 implies that

$$ x(t)= \int _{0}^{t}\frac{(t-qs)^{(\alpha -1)}}{\varGamma _{q}(\alpha )}f(s) \,d_{q}s+c_{1}t+c_{2}, $$

(5)

where $c_{1}$, $c_{2}$ are arbitrary constants. By applying the boundary conditions we find

$$\begin{aligned}& c_{1} =- \frac{b}{1+b} I_{q}^{\alpha -1} f(1), \\& c_{2} = -\frac{a}{1+a} I_{q}^{\alpha }f(1) +\frac{ab}{(1+a)(1+b)} I _{q}^{\alpha -1} f(1). \end{aligned}$$

Substituting $c_{1}$ and $c_{2}$ in (5) we get (4). The converse follows by direct computation. The proof is completed. □

Theorem 4

Let $\ell \in L^{\frac{1}{\kappa }}(J,\mathbb{R}^{+})$, $0<\kappa < \alpha -1$such that

$$ \vert F_{t,x_{i},u_{i}} - F_{t,x'_{i},v_{i}} \vert \leq \ell (t) \Biggl( \sum_{i=1}^{3} \bigl\vert x_{i} - x'_{i} \bigr\vert + \sum _{i=1}^{n} \vert u_{i} - v _{i} \vert \Biggr), $$

for each $t\in J$, $x_{i}$, $x'_{i}$, with $i=1,2,3$and $u_{1}, u_{2}, \dots , u_{n}$, $v_{1}, v_{2}, \dots , v_{n} \in \mathbb{R}$, where $F_{t,x_{i},u_{i}} = w(t, x_{1}, x_{2}, x_{3}, u_{1}, u_{2}, \dots , u _{n})$and $F_{t,x'_{i},v_{i}} =w (t, x'_{1}, x'_{2}, x'_{3}, v_{1}, v _{2}, \dots , v_{n})$. Then the problem (1) has a unique solution provided

$$\begin{aligned} \Delta =& (1 + {}_{0} \lambda _{1} + {}_{0}\lambda _{2}) \Biggl[ \frac{( 1 + 2a) \ell ^{\ast } k_{1}}{ (1+a) \varGamma _{q}(\alpha )} + \frac{b \ell ^{\ast } k_{2}}{(1 + a)(1+b) \varGamma _{q}(\alpha -1)} \\ &{} + \sum_{i=1}^{n} \biggl( \frac{ \varGamma _{q}(\alpha - \kappa ) \ell ^{\ast } k_{2}}{ \varGamma _{q}( \alpha -1) \varGamma _{q}( \alpha -\beta _{i} - \kappa +1)} + \frac{b \ell ^{\ast } k_{2} }{(1+b) \varGamma _{q}(2- \beta _{i}) \varGamma _{q}(\alpha -1)} \biggr) \Biggr] \\ < &1, \end{aligned}$$

(6)

where

$$ {}_{0}\lambda _{i} = \sup_{t\in J} \biggl\vert \int _{0}^{t} \gamma _{i} (t,s) \,d_{q}s \biggr\vert ,\quad i=1,2, \qquad \ell ^{ \ast } = \biggl( \int _{0}^{1} \bigl(\ell (s) \bigr)^{\frac{1}{ \kappa }} \,d_{q}s \biggr)^{\kappa }, $$

$k_{1} = ( \frac{1-\kappa }{ \alpha - \kappa } )^{1- \kappa }$and $k_{2}= ( \frac{1- \kappa }{ \alpha -\kappa -1} ) ^{ 1-\kappa }$.

Proof

Briefly, we put

$$ F_{u(s)}= w \bigl( s, u(s),(\varphi _{1} u) (s), ( \varphi _{2} u) (s), {}^{c}D_{q}^{\beta _{1}} u(s), {}^{c}D_{q}^{\beta _{2}} u(s), \dots , {}^{c}D_{q}^{\beta _{n}} u(s) \bigr), $$

and using Lemma 3, we define a self-map T on X by

$$ (Tu) (t) = I_{q}{\alpha } F_{u(t)} - I_{q}^{\alpha } F_{u(1)} + g(t)I _{q}^{\alpha -1} F_{u(1)}, $$

where $a_{0} = \frac{a}{1+a}$ and $g(t) = \frac{ab-b(1+a)t }{(1+a)(1+b) }$ is a real-valued function on J. At present, by using the Hölder inequality, for each $u, v\in X$ and $t\in J$, we get

$$\begin{aligned} \bigl\vert (Tu) (t) - (Tv) (t) \bigr\vert ={}& \bigl\vert I_{q}^{\alpha } ( F _{u(t)}- F_{v(t)} ) - a_{0} I_{q}^{\alpha } ( F_{u(1)} - F _{v(1)} ) \\ & {} + g(t) I_{q}^{\alpha -1} ( F_{u(1)} - F_{v(1)} ) \bigr\vert \\ \leq{}& I_{q}^{\alpha } \vert F_{u(t)} - F_{v(t)} \vert + a_{1} I _{q}^{\alpha } \vert F_{u(1)} - F_{v(1)} \vert \\ & {} + \bigl\vert g(t) \bigr\vert I_{q}^{\alpha -1} \vert F_{u(1)} - F_{v(1)} \vert \\ \leq{}& I_{q}^{\alpha } \Biggl(\ell (t) \Biggl( \bigl\vert u(t)-v(t) \bigr\vert + \sum_{i=1} ^{2} \bigl\vert (\varphi _{i} u) (t) - (\varphi _{i} v) (t) \bigr\vert \\ & {} + \sum_{i=1}^{n} \bigl\vert {}^{c}D_{q}^{\beta _{i}} u(t) - {} ^{c}D_{q}^{\beta _{i}} v(t) \bigr\vert \Biggr) \Biggr) \\ & {} + a_{1} I_{q}^{\alpha } \Biggl( \ell (1) \Biggl( \bigl\vert u(1) - v(1) \bigr\vert \\ & {} + \sum_{i=1}^{2} \bigl\vert ( \varphi _{i} u) (1) - (\varphi _{i} v) (1) \bigr\vert \\ & {} + \sum_{i=1}^{n} \bigl\vert {}^{c}D_{q}^{\beta _{i}} u(1) - {} ^{c}D_{q}^{\beta _{i}} v(1) \bigr\vert \Biggr) \Biggr) \\ & {} + a_{2} \bigl(1+ 2 \vert a \vert \bigr) I_{q}^{\alpha -1} \Biggl( \ell (1) \Biggl( \bigl\vert u(1)-v(1) \bigr\vert \\ & {} + \sum_{i=1}^{2} \bigl\vert ( \varphi _{i} u) (1) - (\varphi _{i} v) (1) \bigr\vert \\ & {} + \sum_{i=1}^{n} \bigl\vert {}^{c}D_{q}^{ \beta _{i}} u(1) - {}^{c}D_{q}^{ \beta _{i}} v(1) \bigr\vert \Biggr) \Biggr) \\ \leq{}& b_{1} d \int _{0}^{t} (t-qs)^{ (\alpha -1)} \ell (s) \,d_{q}s \\ & {} + a_{1} b_{1} d \int _{0}^{1} (1-qs)^{(\alpha -1)} \ell (s) \,d_{q}s \\ &{} + a_{2} b_{2} \bigl(1 +2 \vert a \vert \bigr) d \int _{0}^{1} (1-qs)^{(\alpha - 2)} \ell (s) \,d_{q}s \\ \leq{}& b_{1} d \biggl( \int _{0}^{t} \bigl( (t-qs)^{(\alpha -1)} \bigr) ^{\frac{1}{1- \kappa }} \,d_{q}s \biggr)^{1-\kappa } \biggl( \int _{0} ^{t} \bigl( \ell (s) \bigr)^{\frac{1}{ \kappa }} \,d_{q}s \biggr) ^{\kappa } \\ & {} + a_{1} b_{1} d \biggl( \int _{0}^{1} \bigl( (1-qs)^{ ( \alpha -1)} \bigr)^{ \frac{1}{1 - \kappa }} \,d_{q}s \biggr)^{1 - \kappa } \\ & {} \times \biggl( \int _{0}^{1} \bigl( \ell (s) \bigr)^{\frac{1}{ \kappa }} \,d_{q}s \biggr)^{\kappa } \\ & {} + a_{2} b_{2} \bigl(1 +2 \vert a \vert \bigr) d \biggl( \int _{0}^{1} \bigl((1-qs)^{( \alpha - 2)} \bigr)^{\frac{1}{ 1 - \kappa }} \,d_{q}s \biggr)^{1- \kappa } \\ & {} \times \biggl( \int _{0}^{1} \bigl( \ell (s) \bigr)^{ \frac{1}{ \kappa }} \,d_{q}s \biggr)^{ \kappa } \\ \leq{}& b_{1} \ell ^{\ast } d k_{1} + a_{1} b_{1} \ell ^{\ast }d k_{1} + a_{2} b_{2} \bigl( 1 + 2 \vert a \vert \bigr) \ell ^{ \ast } d k_{2} \\ \leq{}& \biggl[ \frac{(1 +2 \vert a \vert )b_{1} \ell ^{ \ast }}{ \vert 1+a \vert } k_{1} + a _{2} \bigl(1+2 \vert a \vert \bigr)b_{2} \ell ^{\ast } k_{2} \biggr] d, \end{aligned}$$

where $d=\|u-v\|$, $a_{1} =\frac{|a|}{|1+a|}$, $a_{2} = \frac{|b|}{|1 + a||1+b|}$, $b_{1} =\frac{1+ {}_{0}\lambda _{1} + {}_{0}\lambda _{2}}{ \varGamma _{q}( \alpha )}$ and $b_{2} =\frac{ 1+ {}_{0}\lambda _{1} + {}_{0}\lambda _{2}}{ \varGamma _{q}( \alpha -1)}$. Also, we have

$$\begin{aligned} \bigl\vert {}^{c}D_{q}^{\beta _{i}}(Tu) (t) - {}^{c}D_{q}^{ \beta _{i}}(Tv) (t) \bigr\vert ={}& \bigl\vert I_{q}^{1-\beta _{i}} (Tu)'(t) - I_{q}^{1-\beta _{i}}(Tv)'(t) \bigr\vert \\ ={}& \biggl\vert I_{q}^{1-\beta _{i}} \biggl( I_{q}^{\alpha -1} F_{u(s)} - \frac{b}{1+b} I_{q}^{\alpha -1} F_{u(1)} \biggr) \\ & {}- I_{q}^{1 - \beta _{i}} \biggl( I_{q}^{\alpha -1} F _{ v(s)} - \frac{b}{1+b} I_{q}^{\alpha -1} F_{v(1)} \biggr) \biggr\vert \\ \leq{}& I_{q}^{1 - \beta _{i}} \bigl(I_{q}^{\alpha -1} \vert F_{u(s)} - F_{v(s)} \vert \bigr) \\ & {} + a_{3} I_{q}^{1 -\beta _{i}} \bigl( I_{q}^{\alpha -1} \vert F_{u(s)} - F_{v(s)} \vert \bigr) \\ \leq{}& b_{1} d I_{q}^{1-\beta _{i}} \biggl( \int _{0}^{s} (s-q\tau )^{( \alpha -2)} \ell (\tau ) \,d_{q}\tau \biggr) \,d_{q}s \\ &{} + a_{3} b_{2} d I_{q}^{1-\beta _{i}} \biggl( \int _{0}^{1} (1-q \tau )^{(\alpha -2)} \ell (\tau ) \,d_{q}\tau \biggr) \,d_{q}s \\ \leq{}& \frac{b_{2} \ell ^{\ast } d }{ \varGamma _{q}(1-\beta _{i})} k_{2} \int _{0}^{t} (t-qs)^{(-\beta _{i})} s^{\alpha -\kappa -1} \,d_{q}s \\ &{} + \frac{a_{3} b_{2} \ell ^{\ast } d }{ \varGamma _{q}(1-\beta _{i})} k_{2} \int _{0}^{t} (t-qs)^{(-\beta _{i})} \,d_{q} s \\ \leq{}& \frac{b_{2} \ell ^{\ast }d }{ \varGamma _{q}(1-\beta _{i}) } k_{2} \int _{0}^{1} (1-qs )^{(-\beta _{i})} s^{\alpha -\kappa -1} \,d_{q} s \\ & {} + \frac{a_{3} b_{2} \ell ^{\ast }d}{ \vert 1+b \vert ) \varGamma _{q}(2- \beta _{i}) } k_{2}, \end{aligned}$$

where $a_{3} =\frac{|b|}{|b+1|}$. Since

$$ B_{q}(\alpha -\kappa , 1-\beta _{i}) = \int _{0}^{1} (1-qs)^{(-\beta _{i})} s^{ \alpha - \kappa -1} \,d_{q}s = \frac{ \varGamma _{q} (\alpha - \kappa ) \varGamma _{q} (1-\beta _{i})}{ \varGamma _{q}( \alpha -\beta _{i} - \kappa + 1)}, $$

we obtain

$$\begin{aligned} \bigl\vert {}^{c}D_{q}^{\beta _{i}} (Tu) (t) - {}^{c}D_{q}^{\beta _{i}} (T v) (t) \bigr\vert & \leq \biggl[\frac{b_{2} \varGamma (\alpha -\kappa ) \ell ^{\ast }}{ \varGamma _{q}( \alpha - \beta _{i}-\kappa +1)} k_{2} + \frac{a _{3} b_{2} \ell ^{\ast }}{ \varGamma _{q}( 2 -\beta _{i}) } k_{2} \biggr]d, \end{aligned}$$

for all $i=1,2,\dots ,n$. Hence, we get

$$\begin{aligned} \Vert Tu - Tv \Vert \leq{}& \Biggl[ \frac{b_{1}(1+2 \vert a \vert ) \ell ^{\ast }}{ \vert 1+a \vert } k_{1} + a_{2} b_{2} \ell ^{\ast } k_{2} \\ & {} + \sum_{i=1}^{n} \biggl( \frac{ b_{2} \ell ^{\ast } \varGamma _{q}( \alpha - \kappa )}{ \varGamma _{q}( \alpha - \beta _{i} - \kappa + 1)} k _{2} + \frac{a_{3} b_{2} \ell ^{\ast }}{\varGamma _{q}( 2-\beta _{i})} k _{2} \biggr) \Biggr] d \\ = {}&\Delta d. \end{aligned}$$

By assumption, $\Delta < 1$, thus the mapping F is a contraction and so by using the Banach contraction mapping principle, F has a unique fixed point which is the unique solution of the problem (1). This completes the proof. □

Corollary 1

Assume that there exists $M>0$such that

$$ \vert F_{t,x_{i},u_{i}} - F_{t, x'_{i}, v_{i}} \vert \leq M \Biggl[ \sum _{i=1}^{3} \bigl\vert x_{i} -x'_{i} \bigr\vert + \sum _{i=1}^{n} \vert u_{i}-v_{i} \vert \Biggr], $$

for each $t\in J$and real numbers $x_{i}$, $x'_{i}$for $i=1, 2, 3$, $u_{i}$, $v_{i}$for $i=1,2,\dots , n$, where $F_{t,x_{i}, u_{i}} = f(t, x_{1}, x_{2}, x_{3}, u_{1}, u_{2}, \dots ,u_{n})$, and $F_{t, x'_{i}, v_{i}} =f (t, x'_{1}, x'_{2}, x'_{3}, v_{1}, v_{2}, \dots , v_{n})$. Then the problem (1) has a unique solution whenever

$$\begin{aligned} &(1+ {}_{0}\lambda _{1} + {}_{0}\lambda _{2}) \Biggl[ \frac{ [(1+2a)(1+b)+ b \alpha ] M }{(1+a)(1+b)\varGamma _{q} ( \alpha +1)} \\ &\quad {} + \sum_{i=1}^{n} \biggl( \frac{M}{\varGamma _{q}(\alpha -\beta _{i}+1)}+ \frac{b M}{(1+b) \varGamma _{q}( 2-\beta _{i}) \varGamma _{q}(\alpha )} \biggr) \Biggr] < 1, \end{aligned}$$

where ${}_{0}\lambda _{i} = \sup_{t\in J} \vert \int _{0}^{t} \gamma _{i}(t,s) \,d_{q}s \vert$, $i=1,2$.

Theorem 5

Let $f : J {\times }\mathbb{R}^{n+3}\to \mathbb{R}$be a continuous function. In addition, we assume that there exist a positive constant $\kappa < \alpha - 1$and a function $\ell \in L^{\frac{1}{\kappa }}( J, \mathbb{R}^{+})$. Then problem (1) has a solution whenever

$$ \vert F_{t, x_{j}, u_{i}} \vert \leq \ell (t) + \sum _{j=1}^{3} c _{j} \vert x_{j} \vert ^{\nu _{j}} + \sum _{i=1}^{n} d_{i} \vert u_{i} \vert ^{\eta _{i}}, $$

(7)

where $c_{j}$, $\nu _{j}$belong to $[0, \infty )$, $(0,1)$, respectively, for $j=1,2,3$and $d_{i}$, $\eta _{i}$belong to $[0, \infty )$, $(0,1)$, respectively, for $i=1,2, \dots , n$, or whenever

$$ \vert F_{t, x_{j}, u_{i}} \vert \leq \sum _{i=1}^{3} c_{j} \vert x_{j} \vert ^{ \nu _{j}} + \sum _{i=1}^{n} d_{i} \vert u_{i} \vert ^{\eta _{i}}, $$

(8)

where $c_{j}$, $\nu _{j}$belong to $(0, \infty )$, $(1,\infty )$, respectively, for $j=1,2,3$and $d_{i}$, $\eta _{i}$belong to $(0, \infty )$, $(1,\infty )$, respectively, for $i=1,2, \dots , n$.

Proof

First, we assume that the condition (7) is satisfied. Recall that $k_{1}= ( \frac{1-\kappa }{\alpha - \kappa } )^{1- \kappa }$ and $k_{2}= ( \frac{1-\kappa }{\alpha - \kappa - 1} )^{1- \kappa }$. Let $B_{r}$ is the set of all $u \in X$ such that $\|u\|$ less than or equal to r; here

$$\begin{aligned}& r \geq \max \Bigl\{ \bigl( ( n + 4) A_{0} c_{1} \bigr)^{ \frac{1}{1 - \nu _{1}}}, \bigl( (n + 4 ) A_{0} c_{2} {}_{0}\lambda _{1}^{\nu _{2}} \bigr)^{ \frac{1}{ 1 - {\nu _{2}}}}, \\& \hphantom{r \geq}{} \bigl( (n+4) A_{0} c_{3} {}_{0}\lambda _{2}^{\nu _{3}} \bigr) ^{ \frac{1}{ 1 - \nu _{3}}}, \max_{i} \bigl( (n + 4) A_{0}d_{i} \bigr) ^{ \frac{1}{ 1 - \eta _{i}}}, (n+4)K_{0} \Bigr\} , \\& A_{0} = \frac{(1 + 2 \vert a \vert )[1+(1+a) \vert b \vert ] }{ (1+a)(1+b) \varGamma _{q}( \alpha +1)} \\& \hphantom{A_{0} =}{} + \sum_{i=1}^{n} \biggl(\frac{1}{ \varGamma _{q}( \alpha -\beta _{i}+1)} + \frac{ \vert b \vert }{ \vert 1+b \vert \varGamma _{q}(\alpha ) \varGamma _{q}( 2-\beta _{i})} \biggr), \\& K_{0} = \frac{( 1+ 2 \vert a \vert )\ell ^{*}}{ \vert 1+a \vert \varGamma _{q}( \alpha )} k _{1} + \frac{ \vert b \vert (1+2 \vert a \vert )\ell ^{*}}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}( \alpha -1)} k_{2} \\& \hphantom{K_{0} =}{} + \frac{1}{\varGamma _{q}(\alpha -1)} \sum_{i=1}^{n} \biggl(\frac{ \varGamma _{q}( \alpha -l) \ell ^{*}}{ \varGamma _{q}(\alpha -\beta _{i}-\kappa +1)} k_{2} + \frac{ \vert b \vert \ell ^{*}}{ \vert 1+b \vert \varGamma _{q} (2-\beta _{i}) } k _{2} \biggr), \end{aligned}$$

and $\ell ^{*} = ( \int _{0}^{1} (\ell (t) )^{ \frac{1}{ \kappa }} \,d_{q}s )^{\kappa }$. Note that $B_{r}$ is a closed, bounded and convex subset of the Banach space X. For each $u \in B_{r}$, we obtain

$$\begin{aligned} \bigl\vert (Tu) (t) \bigr\vert = {}& \bigl\vert I_{q}^{\alpha } F_{u(t)} - a_{0} I_{q}^{\alpha }F_{u(1)} + g(t)I_{q}^{\alpha -1} F_{u(1)} \bigr\vert \\ \leq{}& I_{q}^{\alpha } \vert F_{u(t)} \vert + \frac{ \vert a \vert }{ \vert 1+a \vert } I _{q}^{\alpha } \vert F_{u(1)} \vert + \frac{ \vert b \vert (1+2 \vert a \vert )}{ \vert 1+a \vert \vert 1+b \vert } I_{q}^{\alpha - 1} \vert F_{u(1)} \vert \\ \leq{}& I_{q}^{\alpha }\ell (t) + A_{r} I_{q}^{\alpha }(1) + \frac{ \vert a \vert }{ \vert 1+a \vert } I_{q}^{\alpha }\ell (1)+ \frac{ \vert a \vert }{ \vert 1+a \vert } A_{r} I _{q}^{\alpha }(1) \\ & {} + \frac{ \vert b \vert (1+2 \vert a \vert )}{ \vert 1+a \vert \vert 1+b \vert } I_{q}^{\alpha -1} \ell (1) + \frac{ \vert b \vert (1+2 \vert a \vert )}{ \vert 1+a \vert \vert 1+b \vert } A_{r} I_{q}^{\alpha -1} (1) \\ \leq{}& \frac{1}{\varGamma _{q}( \alpha )} \biggl( \int _{0}^{t} \bigl((t-qs)^{( \alpha -1)} \bigr)^{\frac{1}{1-l}} \,d_{q}s \biggr)^{1-l} \\ &{} \times \biggl( \int _{0}^{t} \bigl(\ell (s) \bigr)^{ \frac{1}{l}} \,d_{q}s \biggr)^{l} \\ &{} + \frac{ \vert a \vert }{ \vert 1+a \vert \varGamma _{q} (\alpha )} \biggl( \int _{0} ^{1} \bigl((1-qs)^{ (\alpha -1)} \bigr)^{ \frac{1}{1-l}} \,d_{q}s \biggr) ^{1-l} \\ & {} \times \biggl( \int _{0}^{1} \bigl( m(s) \bigr)^{ \frac{1}{l}} \,d_{q}s \biggr)^{l} \\ & {} + \frac{ \vert b \vert (1+2 \vert a \vert )}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}(\alpha -1)} \biggl( \int _{0}^{1} \bigl((1-qs)^{(\alpha -2)} \bigr)^{ \frac{1}{1-l}} \,d_{q}s \biggr)^{1-l} \\ & {} \times \biggl( \int _{0}^{1} \bigl(m(s) \bigr)^{\frac{1}{l}} \,d_{q} s \biggr)^{l} \\ & {} + \frac{(1+2 \vert a \vert )(1+(1+\alpha ) \vert b \vert )}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}( \alpha + 1)} A_{r} \\ \leq{}& \frac{(1+2 \vert a \vert ) \ell ^{*}}{ \vert 1+a \vert \varGamma _{q}(\alpha )} k_{1} + \frac{ \vert b \vert (1+2 \vert a \vert ) \ell ^{*}}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}( \alpha -1)} k_{2} \\ & {} + \frac{(1+2 \vert a \vert )(1+(1+\alpha ) \vert b \vert )}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}( \alpha +1)} A_{r}, \end{aligned}$$

where $a_{0}$ and $g(t)$ as defined in Theorem 4 (i.e. $a_{0} = \frac{a}{1+a}$ and $g(t) = \frac{ab-b(1+a)t }{(1+a)(1+b) }$, $t\in J$),

$$\begin{aligned} F_{u(s)} &= f \bigl(s, u(s),(\varphi _{1} u) (s),( \varphi _{2} u) (s), {}^{c}D^{\beta _{1}} u(s), {}^{c}D^{\beta _{2}} u(s), \dots , {}^{c}D ^{\beta _{n}} u(s) \bigr) \end{aligned}$$

and $A_{r} = c_{1}r^{\nu }_{1} + c_{2} {}_{0}\lambda _{1}^{\nu _{2}} r ^{\nu _{2}} + c_{3} {}_{0}\lambda _{2}^{\nu _{3}} r^{\nu _{3}} + \sum_{i=1}^{n} d_{i} r^{\eta _{i}}$. Also, we have

$$\begin{aligned} \bigl\vert {}^{c}D_{q}^{\beta _{i}}(Tu) (t) \bigr\vert ={}& \bigl\vert I_{q}^{1- \beta _{i}} (Tu)'(t) \bigr\vert \\ ={}& \biggl\vert I_{q}^{1-\beta _{i}} \biggl( I_{q}^{\alpha -1} F_{u(s)} - \frac{b}{1+b} I_{q}^{\alpha -1} F_{u(1)} \biggr) \biggr\vert \\ \leq{}& \int _{0}^{t} \frac{(t-qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{s} \frac{(s-q\tau )^{(\alpha -2)}}{\varGamma _{q}( \alpha -1)} \vert F_{u(\tau )} \vert \,d_{q}\tau \biggr) \,d_{q}s \\ & {} + \frac{ \vert b \vert }{ \vert 1+b \vert } \int _{0}^{t} \frac{(t-qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{1} \frac{(1-q\tau )^{(\alpha -2)}}{ \varGamma _{q}( \alpha -1)} \vert F_{u(\tau )} \vert \,d_{q}\tau \biggr) \,d_{q}s \\ \leq{}& \int _{0}^{t} \frac{(t-qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{s} \frac{(s-q\tau )^{(\alpha -2)}}{ \varGamma _{q}(\alpha -1)} \ell (\tau ) \,d_{q}\tau \biggr) \,d_{q}s \\ &{} + A_{r} \int _{0}^{t} \frac{(t-qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{s} \frac{(s-q\tau )^{(\alpha -2)}}{ \varGamma _{q}(\alpha -1)} \,d_{q}\tau \biggr) \,d_{q}s \\ & {} + \frac{ \vert b \vert }{ \vert 1+b \vert } \int _{0}^{t} \frac{(t-qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{1} \frac{(1-q\tau )^{(\alpha -2)}}{ \varGamma _{q}( \alpha -1)} \ell (\tau ) \,d_{q} \tau \biggr) \,d_{q}s \\ &{} +\frac{ \vert b \vert }{ \vert 1+b \vert } A_{r} \int _{0}^{t} \frac{(t-qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{1} \frac{(1-q\tau )^{(\alpha -2)}}{ \varGamma _{q}(\alpha -1)} \,d_{q}\tau \biggr) \,d_{q}s \\ \leq {}&\frac{1}{\varGamma _{q}(\alpha -1) \varGamma _{q}(1-\beta _{i})} \int _{0}^{t} (t-qs)^{(-\beta _{i})} \\ & {} \times \biggl[ \biggl( \int _{0}^{s} \bigl( (s-q\tau )^{(\alpha -2)} \bigr)^{\frac{1}{1-l}} \,d_{q}\tau \biggr)^{1-l} \biggl( \int _{0}^{s} \bigl( \ell (\tau ) \bigr)^{\frac{1}{l}} \,d_{q}\tau \biggr) ^{l} \biggr] \,d_{q} s \\ & {} + \frac{ A_{r}}{\varGamma _{q}(\alpha ) \varGamma _{q}(1-\beta _{i})} \int _{0}^{t} (t-qs)^{(-\beta _{i})} s^{\alpha -1} \,d_{q}s \\ & {} + \frac{ \vert b \vert }{ \vert 1+b \vert \varGamma _{q}(\alpha -1) \varGamma _{q}(1-\beta _{i})} \int _{0}^{t} (t-qs)^{(-\beta _{i})} \\ & {} \times \biggl[ \biggl( \int _{0}^{1} \bigl( (1-q\tau )^{(\alpha -2)} \bigr)^{\frac{1}{1-l}} \,d_{q}\tau \biggr)^{1-l} \biggl( \int _{0}^{1} \bigl( \ell (\tau ) \bigr)^{\frac{1}{l}} \,d_{q}\tau \biggr) ^{l} \biggr] \,d_{q}s \\ & {} + \frac{ \vert b \vert }{ \vert 1+b \vert \varGamma _{q}(\alpha ) \varGamma _{q}(2-\beta _{i})} A_{r} \\ \leq{}& \frac{\ell ^{*}}{ \varGamma _{q}(\alpha -1) \varGamma _{q}(1-\beta _{i})} k_{2} \int _{0}^{t} (t-qs)^{(-\beta _{i})} s^{\alpha -l-1} \,d_{q}s \\ & {} + \frac{1}{\varGamma _{q}(\alpha ) \varGamma _{q}(1-\beta _{i})} A _{r} \int _{0}^{t} (t-qs)^{(-\beta _{i})} s^{\alpha -1} \,d_{q}s \\ & {} +\frac{ \vert b \vert \ell ^{*}}{ \vert 1+b \vert \varGamma _{q}(\alpha -1) \varGamma _{q}(1- \beta _{i})} k_{2} \int _{0}^{t} (t-qs)^{(-\beta _{i})} \,d_{q}s \\ & {} + \frac{ \vert b \vert }{ \vert 1+b \vert \varGamma _{q}( \alpha ) \varGamma _{q}( 2-\beta _{i})} A_{r} \\ \leq{}& \frac{\ell ^{*}}{\varGamma _{q}(\alpha -1) \varGamma _{q}(1-\beta _{i})} k_{2} \int _{0}^{1} (1-qs)^{(-\beta _{i})} s^{\alpha -l-1} \,d_{q}s \\ & {} + \frac{1}{\varGamma _{q}(\alpha ) \varGamma _{q}(1-\beta _{i})} A_{r} \int _{0}^{1} (1-qs)^{(-\beta _{i})} s^{\alpha -1} \,d_{q}s \\ & {} + \frac{ \vert b \vert \ell ^{*}}{ \vert 1+b \vert \varGamma _{q}(\alpha -1) \varGamma _{q}( 2-\beta _{i})} k_{2} + \frac{ \vert b \vert }{ \vert 1+b \vert \varGamma _{q} (\alpha ) \varGamma _{q}(2-\beta _{i})} A_{r}. \end{aligned}$$

Since, by considering Eq. (2),

$$ B_{q} (\alpha -l, 1-\beta _{i}) = \int _{0}^{1} (1-qs)^{(-\beta _{i})} s^{\alpha -\kappa -1} \,d_{q}s = \frac{ \varGamma _{q}(\alpha -l) \varGamma _{q}(1-\beta _{i})}{ \varGamma _{q}(\alpha - \beta _{i}-\kappa +1)} $$

and on the other hand

$$ B_{q} (\alpha ,1-\beta _{i}) = \int _{0}^{1} (1-q \xi )^{(-\beta _{i})} \xi ^{\alpha -1} \,d_{q}\xi = \frac{ \varGamma _{q}(\alpha ) \varGamma _{q}( 1-\beta _{i})}{ \varGamma _{q}( \alpha - \beta _{i}+1)}, $$

we conclude that

$$\begin{aligned} \bigl\vert {}^{c}D_{q}^{\beta _{i}} (Tu) (t) \bigr\vert \leq{}& \frac{ \varGamma _{q}( \alpha - l) \ell ^{*} }{\varGamma _{q}( \alpha -1) \varGamma _{q}(\alpha -\beta _{i}-\kappa +1)} k_{2} \\ & {} + \frac{ \vert b \vert \ell ^{*}}{ \vert 1+b \vert \varGamma _{q} (\alpha -1) \varGamma _{q}( 2-\beta _{i})} k_{2} + \frac{ 1}{ \varGamma _{q}(\alpha -\beta _{i}+1)} A_{r} \\ & {} + \frac{ \vert b \vert }{ \vert 1+b \vert A_{r} \varGamma _{q}(\alpha ) \varGamma _{q}(2- \beta _{i})} A_{r} \end{aligned}$$

for each $i=1, 2,\dots , n$. Hence,

$$\begin{aligned} \Vert Tu \Vert \leq{}& \frac{(1+2 \vert a \vert ) \ell ^{*}}{ \vert 1+a \vert ) \varGamma _{q}(\alpha )} k _{1} + \frac{ \vert b \vert (1+2 \vert a \vert ) \ell ^{*}}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}(\alpha -1)} k _{2} \\ &{} + \frac{1}{\varGamma _{q} (\alpha -1) } \sum_{i=1}^{n} \biggl[ \frac{ \varGamma _{q}(\alpha -l) \ell ^{*} }{\varGamma _{q}( \alpha -\beta _{i}-l+1)} k_{2} \\ &{} + \frac{ \vert b \vert \ell ^{*}}{ \vert 1+b \vert \varGamma _{q}( \alpha -1) \varGamma _{q}( 2-\beta _{i})} k_{2} \biggr] \\ & {} + A_{r} \Biggl( \frac{(1+2 \vert a \vert )(1+ (1+ \alpha ) \vert b \vert )}{ \vert 1+a \vert \vert 1+b \vert \varGamma _{q}(\alpha +1)} \\ & {} + \sum_{i=1}^{n} \biggl[ \frac{1}{\varGamma _{q}( \alpha -\beta _{i}+1)} + \frac{ \vert b \vert }{ \vert 1+b \vert \varGamma _{q} (\alpha ) \varGamma _{q} (2-\beta _{i})} \biggr] \Biggr) \\ ={}& K_{0} + A_{r} A_{0} \leq \frac{r}{ n + 4}(n+4) = r . \end{aligned}$$

Hence, T maps $B_{r}$ into $B_{r}$. Now, suppose that T satisfy the condition (8). In this case, choose

$$\begin{aligned} 0< {}& r \\ \leq{}& \min \biggl\{ \biggl( \frac{1}{(n+3) A_{0} c_{1}} \biggr) ^{\frac{1}{\nu _{1}-1}}, \biggl( \frac{1}{(n+3) A_{0} c_{2} {}_{0} \lambda _{1}^{\nu _{2}}} \biggr)^{\frac{1}{\nu _{2} -1}}, \\ &{} \biggl( \frac{1}{(n+3) A_{0} c_{3} {}_{0}\lambda _{2}^{\nu _{2}}} \biggr)^{\frac{1}{\nu _{2} -1}}, \max _{i} \biggl( \frac{1}{(n+3) A_{0} d_{i}} \biggr)^{ \frac{1}{\eta _{i} - 1}} \biggr\} . \end{aligned}$$

By applying a similar argument, one can prove that $\|Tu\| \leq r$ and so T is a self-map on $B_{r}$. Also, one can easy to check that T is continuous, because w is continuous. For each $u\in B_{r}$, take

$$ N = \max_{t\in J} \bigl\vert w \bigl( t, u(t), (\varphi _{1} u) (t), ( \varphi _{2} u) (t), {}^{c}D_{q}^{\beta _{1}} u(t), {}^{c}D_{q}^{\beta _{2}} u(t), \dots , {}^{c}D_{q}^{\beta _{n}} u(t) \bigr) \bigr\vert + 1. $$

Thus, for each $0< t_{1}< t_{2}< 1$, we have

$$\begin{aligned} \bigl\vert (Tu) (t_{2}) - (Tu) (t_{1}) \bigr\vert ={}& \biggl\vert I_{q}^{\alpha }F_{u(t_{2})} - I_{q}^{\alpha }F_{u(t_{1})} + \frac{b (t _{1}-t_{2})}{1+b} I_{q}^{\alpha -1} F_{u(1)} \biggr\vert \\ \leq {}& \int _{0}^{t_{1}} \frac{(t_{2} - qs)^{ (\alpha -1)} - (t_{1} - qs)^{( \alpha -1)}}{ \varGamma _{q}(\alpha )} \vert F_{u(s)} \vert \,d_{q}s \\ & {} + \int _{t_{1}}^{t_{2}} \frac{(t_{2} - qs)^{(\alpha -1)}}{\varGamma _{q}( \alpha )} \vert F_{u(s)} \vert \,d_{q}s \\ & {} + \frac{ \vert b \vert (t_{2} - t_{1} )}{ \vert 1+b \vert } \int _{0}^{1} \frac{(1-qs)^{ (\alpha -2)}}{ \varGamma _{q}(\alpha -1)} \vert F_{u(s)} \vert \,d_{q}s \\ \leq{}& N \int _{0}^{t_{1}} \frac{(t_{2}-qs)^{(\alpha -1)} - (t_{1} - qs)^{( \alpha -1)}}{ \varGamma _{q}(\alpha )} \,d_{q}s \\ & {} + N \int _{t_{1}}^{t_{2}} \frac{(t_{2} -qs)^{(\alpha -1)}}{ \varGamma _{q}( \alpha )} \,d_{q}s \\ & {} + \frac{ N \vert b \vert (t_{2} - t_{1})}{ \vert 1+b \vert } \int _{0}^{1} \frac{(1-qs)^{(\alpha -2)}}{ \varGamma _{q}(\alpha -1)} \,d_{q}s \\ ={}& \frac{N}{ \varGamma _{q}( \alpha +1)}\bigl( t_{2}^{\alpha }-t_{1}^{\alpha } \bigr) + \frac{ N \vert b \vert }{ \vert 1+b \vert \varGamma _{q}( \alpha )}(t_{2} - t_{1}). \end{aligned}$$

Furthermore, for all $i= 1,2, \dots , n$, we obtain

$$\begin{aligned} & \bigl\vert {}^{c}D_{q}^{\beta _{i}} (Tu) (t_{2}) -{}^{c}D_{q}^{\beta _{i}}(Tu) (t _{1}) \bigr\vert \\ &\quad = \biggl\vert \int _{0}^{t_{2}} \frac{(t_{2}- qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})}( Tu)'(s) \,d_{q}s - \int _{0}^{t_{1}} \frac{(t_{1} -qs)^{(-\beta _{i})}}{\varGamma _{q}( 1-\beta _{i})}(Tu)'(s) \,d_{q}s \biggr\vert \\ &\quad \leq \int _{0}^{t_{1}} \frac{(t_{1}- qs)^{(-\beta _{i})} - (t_{2} - qs)^{(- \beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{s} \frac{(s-q\tau )^{(\alpha -2)}}{ \varGamma _{q}(\alpha -1)} \vert F_{u(\tau )} \vert \,d_{q}\tau \biggr) \,d_{q}s \\ &\qquad {} + \frac{ \vert b \vert }{ \vert 1+b \vert } \int _{0}^{t_{1}} \frac{(t_{1}-qs)^{(-\beta _{i})} - (t_{2}- qs)^{(- \beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \\ & \qquad {} \times \biggl( \int _{0}^{1} \frac{(1-q\tau )^{(\alpha -2)}}{ \varGamma _{q}( \alpha -1)} \vert F_{u(\tau )} \vert \,d_{q}\tau \biggr) \,d_{q}s \\ & \qquad {} + \int _{t_{1}}^{t_{2}} \frac{(t_{2} - qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \biggl( \int _{0}^{s} \frac{(s-q\tau )^{(\alpha -2)}}{ \varGamma _{q}( \alpha -1)} \vert F_{u(\tau )} \vert \,d_{q}\tau \biggr) \,d_{q}s \\ & \qquad {} + \frac{ \vert b \vert }{ \vert 1+b \vert } \int _{t_{1}}^{t_{2}} \frac{(t_{2}- qs)^{(-\beta _{i})}}{ \varGamma _{q}(1- \beta _{i})} \biggl( \int _{0}^{1} \frac{(1-q\tau )^{(\alpha -2)}}{ \varGamma _{q} ( \alpha -1)} \vert F _{u(\tau )} \vert \,d_{q}\tau \biggr) \,d_{q}s \\ &\quad \leq \frac{N}{\varGamma _{q}(\alpha )} \int _{0}^{t_{1}} \frac{(t_{1}- qs)^{(- \beta _{i})} - (t_{2}- qs)^{(-\beta _{i})}}{ \varGamma _{q}( 1 - \beta _{i})} s^{\alpha -1} \,d_{q}s \\ &\qquad {} + \frac{N \vert b \vert }{ \vert 1+b \vert \varGamma _{q}(\alpha )} \int _{0}^{t_{1}} \frac{(t _{1}- qs)^{(-\beta _{i})}- (t_{2} - qs)^{(-\beta _{i})}}{ \varGamma _{q}( 1- \beta _{i})} \,d_{q}s \\ & \qquad {} + \frac{N}{ \varGamma _{q}(\alpha )} \int _{t_{1}}^{t_{2}} \frac{(t_{2} - qs)^{(-\beta _{i})}}{ \varGamma _{q}(1- \beta _{i})} s^{\alpha -1} \,d_{q}s \\ & \qquad {} + \frac{N \vert b \vert }{ \vert 1+b \vert \varGamma _{q}(\alpha )} \int _{t_{1}}^{t _{2}} \frac{(t_{2}- qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \,d_{q}s \\ &\quad \leq \frac{(1+2 \vert b \vert ) N}{ \vert 1+b \vert \varGamma _{q}( \alpha )} \int _{0}^{t_{1}} \frac{(t_{1}- qs)^{(-\beta _{i})} - (t_{2} - qs)^{(-\beta _{i})}}{ \varGamma _{q}( 1- \beta _{i})} \,d_{q}s \\ & \qquad {} + \frac{(1+2 \vert b \vert )N}{ \vert 1+b \vert \varGamma _{q}(\alpha )} \int _{t_{1}} ^{t_{2}} \frac{(t_{2} - qs)^{(-\beta _{i})}}{ \varGamma _{q}(1-\beta _{i})} \,d_{q}s \\ &\quad \leq \frac{(1+2 \vert b \vert )N}{ \vert 1+b \vert \varGamma _{q}(\alpha ) \varGamma _{q}(2-\beta _{i})} \bigl[ \bigl(t_{2}^{1 - \beta _{i}} - t_{1}^{1-\beta _{i}} \bigr) + 2 (t_{2} - t_{1} )^{1-\beta _{i}} \bigr]. \end{aligned}$$

Hence,

$$\begin{aligned} \bigl\Vert Tu(t_{2})- Tu(t_{1}) \bigr\Vert \leq{}& \frac{N}{\varGamma _{q}(\alpha + 1)}\bigl( t _{2}^{\alpha }- t_{1}^{ \alpha }\bigr) + \frac{N \vert b \vert }{ \vert 1+b \vert \varGamma _{q}( \alpha )}(t_{2}- t_{1}) \\ & {} + \sum_{i=1}^{n} \frac{(1+2 \vert b \vert )N}{ \vert 1+b \vert \varGamma _{q}(\alpha ) \varGamma _{q}(2-\beta _{i})} \bigl[ \bigl(t_{2}^{ 1 - \beta _{i}} - t_{1} ^{1- \beta _{i}} \bigr) \\ & {} + 2 ( t_{2} - t_{1} )^{1-\beta _{i}} \bigr], \end{aligned}$$

which implies that $\|Tu (t_{2}) - Tu(t_{1}) \| \to 0$ as $t_{1} \to t_{2}$. Thus, T is uniformly bounded and equicontinuous and so the theorem of Arzelá–Ascoli implies that T is completely continuous. At present, from Theorem 1, T has a fixed point in $B_{r}$. Finally, the problem (1) has a solution. □

Corollary 2

Assume that a real-valued functionfdefined on $J {\times } \mathbb{R}^{n+3}$is continuous. Then the problem (1) has at least one solution whenever there exist a positive constant $l< \alpha -1$and a real-valued function $\ell \in L^{ \frac{1}{l}} (J, \mathbb{R}^{+})$such that $\vert w ( t , x_{1}, x_{2}, x_{3}, u_{1}, u_{2}, \ldots , u_{n} ) \vert \leq \ell (t)$, for eachtinJ, and $x_{j}$, with $j=1,2,3$, $u_{i}$, with $1\leq i \leq n$, in $\mathbb{R}$.

4 Examples illustrative for the problems with algorithms

In this part, we give complete computational techniques for checking working to illustrate of the problem (1), in our theorems, such that it covers all the problems and we present numerical examples which entail perfect solutions. Foremost, we present a simplified analysis that can be executed to calculate the value of the q-Gamma function, $\varGamma _{q} (x)$, for input q, x and different values of n. To this aim, we consider a pseudo-code description of the method for calculating the q-Gamma function of order n in Algorithm 2 (for more details, see the link https://en.wikipedia.org/wiki/Q-gamma_function). Now we give the following examples to illustrate our results.

Example 1

Consider the multi-term nonlinear fractional q-integro-differential equation

$$ \textstyle\begin{cases} {}^{c}D_{q}^{\frac{8}{5}} u(t) = \frac{ e^{-\pi t }}{30 \sqrt{\pi } + e^{- \pi t}} [ \frac{ \cos t + e^{t}}{1 + t^{4}} + \frac{ 2 \vert u(t) \vert }{1+ 2 \vert u(t) \vert } \\ \hphantom{{}^{c}D_{q}^{\frac{8}{5}} u(t) =}{}+ \frac{ e^{-\pi t} \sin \pi t}{1+ t^{3}} (1+ \frac{ 2 \vert ( \varphi _{1} u)(t) + {}^{c}D_{q}^{ \frac{1}{3}} u(t) \vert }{ 1+ 2 \vert (\varphi _{1} u)(t) + {}^{c}D_{q}^{ \frac{1}{3}} u(t) \vert } ) \\ \hphantom{{}^{c}D_{q}^{\frac{8}{5}} u(t) =}{} + \frac{1+ \cos ^{2}\pi t}{ 3(t^{\frac{4}{3}} + 6)} ( 3(\varphi _{2} u)(t) + \frac{ 2 \vert {}^{c}D_{q}^{ \frac{2}{5}} u(t) \vert }{ 1+ 2 \vert {}^{c}D_{q}^{\frac{2}{5}} u(t) \vert } ) ], \end{cases} $$

(9)

under boundary conditions $u(0) + u(1)=0$ and $u'(0) + u'(1)=0$, where $(\varphi _{1} u)(t)$ and $(\varphi _{2} u)(t)$ are defined by $\frac{1}{10} \int _{0}^{t} e^{-2(s-t)} u(s) \,d_{q}s$ and $\frac{1}{10} \int _{0}^{t} e^{-(s - t)/4} u(s) \,d_{q}s$, respectively, with

$$\begin{aligned} {}_{0}\lambda _{1} & = \sup_{t\in J} \biggl\vert \int _{0}^{t} \frac{e^{-2(s-t)}}{10} \,d_{q}s \biggr\vert = \sup_{t\in J} \Biggl\vert t ( 1- q) \sum_{k=0}^{\infty }q^{k} \frac{ e^{2t(1-q^{k})}}{10} \Biggr\vert \\ & = \vert 1- q \vert \sum_{k=0}^{\infty } \biggl\vert q^{k} \frac{ e ^{2(1-q^{k})}}{10} \biggr\vert \end{aligned}$$

and

$$\begin{aligned} {}_{0}\lambda _{2} & =\sup_{t\in I} \biggl\vert \int _{0}^{t} \frac{e^{-(s-t)/4}}{10} \,d_{q}s \biggr\vert = \sup_{t\in I} \Biggl\vert t (1-q) \sum_{k=0}^{\infty }q^{k} \frac{e^{t(1- q^{k})/4}}{10} \Biggr\vert \\ & = \vert 1- q \vert \sum_{k=0}^{\infty } \biggl\vert q^{k} \frac{e ^{(1- q^{k})/4}}{10} \biggr\vert . \end{aligned}$$

Then we have

$$\begin{aligned} \vert F_{u(t)} - F_{v(t)} \vert \leq{}& \frac{1}{30 \sqrt{\pi }} \bigl( \bigl\vert u(t) - v(t) \bigr\vert \\ & {} + \bigl\vert (\varphi _{1} u) (t) - (\varphi _{1} v) (t) \bigr\vert + \bigl\vert (\varphi _{2} u) (t) - (\varphi _{2} v) (t) \bigr\vert \\ & {} + \bigl\vert {}^{c}D_{q}^{\frac{1}{3}} u(t)- {}^{c}D_{q}^{ \frac{1}{3}} v(t) \bigr\vert + \bigl\vert {}^{c}D_{q}^{\frac{2}{5}} u(t) - {}^{c}D_{q}^{\frac{2}{5}} v(t) \bigr\vert \bigr), \end{aligned}$$

where

$$\begin{aligned} &F_{u(t)} = w \bigl( t, u(t), (\varphi _{1} u) (t), ( \varphi _{2} u) (t), {}^{c}D_{q}^{\frac{1}{3}} u(t), {}^{c}D_{q}^{\frac{2}{5}} u(t) \bigr), \\ &F_{v(t)} = w \bigl(t, v(t), (\varphi _{1} v) (t), ( \varphi _{2} v) (t), {}^{c}D_{q}^{\frac{1}{3}} v(t), {}^{c}D_{q}^{ \frac{2}{5}} v(t) \bigr). \end{aligned}$$

Take $\ell (t) =\frac{1}{30 \sqrt{\pi }}$ belongs to $L^{\frac{1}{5}}( J, \mathbb{R}^{+})$, $\kappa =\frac{1}{5}$ and

$$ \ell ^{\ast }= \biggl( \int _{0}^{1} \biggl( \frac{1}{30\sqrt{\pi }} \biggr)^{5} \,d_{q}s \biggr)^{ \frac{1}{5}} = \Biggl( (1-q) \sum_{k=0}^{\infty } \frac{q ^{k}}{(30\sqrt{\pi })^{5}} \Biggr)^{ \frac{1}{5}}. $$

For different values of q, which are shown in Tables 1, 2 and 3, by using Algorithm 6, we obtain

$$\begin{aligned} \Delta ={}& ( 1+ {}_{0}\lambda _{1} + {}_{0}\lambda _{2} ) \biggl[ \frac{3\ell ^{\ast }}{ 2\varGamma _{q}(\alpha )} k_{1} + \frac{ \ell ^{\ast }}{4 \varGamma _{q}(\alpha -1)} k_{2} \\ &{} + \frac{\varGamma _{q}(\alpha -\kappa ) \ell ^{\ast }}{ \varGamma _{q}( \alpha -1)} k_{2} \biggl( \frac{1}{ \varGamma _{q}( \alpha - \beta _{1}- \kappa +1)} + \frac{1}{\varGamma _{q}( \alpha -\beta _{2} -\kappa +1)} \biggr) \\ & {} + \frac{\ell ^{\ast }}{ 2\varGamma _{q}(\alpha -1)} k_{2} \biggl( \frac{1}{ \varGamma _{q}( 2 -\beta _{1})} + \frac{1}{\varGamma _{q}( 2-\beta _{2})} \biggr) \biggr] \\ ={}& ( 1+ {}_{0}\lambda _{1} + {}_{0} \lambda _{2} ) \biggl[ \frac{3 \ell ^{\ast }}{ 2\varGamma _{q}(\frac{8}{5})} \biggl( \frac{4}{7} \biggr) ^{\frac{4}{5}} + \frac{\ell ^{\ast }}{4 \varGamma _{q}(\frac{3}{5})} ( 2 )^{\frac{4}{5}} \\ & {} + \frac{\varGamma _{q}(\frac{7}{5}) \ell ^{\ast }}{ \varGamma _{q}( \frac{3}{5})} (2 )^{\frac{4}{5}} \biggl( \frac{1}{ \varGamma _{q}( \frac{31}{15})} + \frac{1}{\varGamma _{q}(2)} \biggr) \\ &{} + \frac{\ell ^{\ast }}{ 2\varGamma _{q}(\frac{3}{5})} (2 ) ^{\frac{4}{5}} \biggl( \frac{1}{ \varGamma _{q}( \frac{5}{3})} + \frac{1}{ \varGamma _{q}(\frac{8}{5})} \biggr) \biggr] \\ < {}& 1, \end{aligned}$$

where $k_{1}= (\frac{ 1 -\kappa }{\alpha -\kappa } )^{1-\kappa }$ and $k_{2}= (\frac{1-\kappa }{ \alpha -\kappa -1} )^{1-\kappa }$. Now, by using Algorithms 1 and 2, we calculated ${}_{0}\lambda _{1}$, ${}_{0}\lambda _{2}$, $\ell ^{\ast }$, $\varGamma _{q}( \frac{8}{5})$, $\varGamma _{q}(\frac{3}{5})$, $\varGamma _{q}(\frac{7}{5})$, $\varGamma _{q}(\frac{31}{15})$ and $\varGamma _{q}(2)$ for some values $n \in \mathbb{N}$ and $q \in (0,1)$. Table 1 shows these calculated values. So, from Theorem 4, the problem (9) has a unique solution. In Tables 1, 2 and 3, we put

$$\begin{aligned} \varOmega ={}& \frac{3\ell ^{\ast }}{ 2\varGamma _{q}(\alpha )} k_{1} + \frac{ \ell ^{\ast }}{4 \varGamma _{q}(\alpha -1)} k_{2} \\ &{} + \frac{\varGamma _{q}(\alpha -\kappa ) \ell ^{\ast }}{ \varGamma _{q}(\alpha -1)} k_{2} \biggl( \frac{1}{ \varGamma _{q}( \alpha -\beta _{1}-\kappa +1)} + \frac{1}{\varGamma _{q}( \alpha -\beta _{2} -\kappa +1)} \biggr) \\ & {} + \frac{\ell ^{\ast }}{ 2 \varGamma _{q}(\alpha -1)} k_{2} \biggl( \frac{1}{ \varGamma _{q}( 2 -\beta _{1})} + \frac{1}{\varGamma _{q}( 2-\beta _{2})} \biggr). \end{aligned}$$

Algorithm 6 shows the technique of calculation Δ which was introduced in Eq. (6). Tables 1, 2 and 3 show variables of Δ when $q=\frac{1}{3}$, $q=\frac{1}{2}$ and $q=\frac{4}{5}$, respectively. As it is seen, always $\Delta <1$ for all n and $q \in (0,1)$. In addition, when values q are close to one, Δ is obtained with more values of n in comparison with other rows. It is shown by underlined rows. They have been underlined in line 10 of Table 1, line 14 of Table 2 and line 31 of Table 3.

Algorithm 6

The proposed method for calculation of Δ

Table 1 Some numerical results for calculation of Δ with $q=\frac{1}{3}$ and $n=15$ of Algorithm 6

Table 2 Some numerical results for calculation of Δ with $q=\frac{1}{2}$ and $n=19$ of Algorithm 6

Table 3 Some numerical results for calculation of Δ with $q=\frac{4}{5}$ and $n=35$ of Algorithm 6

Example 2

Consider the multi-term nonlinear fractional q-integro-differential equation

$$ \textstyle\begin{cases} {}^{c}D_{q}^{\frac{7}{4}} u (t) = \frac{\lambda e^{-2\pi t}}{ \sqrt{1 + t^{3}}} + \frac{\sin \pi t}{ \sqrt{2\pi + \vert u(t) \vert + \vert {}^{c}D_{q}^{ \frac{1}{2}} u(t) \vert }} (u(t))^{\sigma _{1} } \\ \hphantom{{}^{c}D_{q}^{\frac{7}{4}} u (t) =}{}+ \frac{e^{-2\pi t}( 1 + \cos ^{2} u(t))}{ (t+6)^{2}} ((\varphi _{1} u)(t) )^{\sigma _{2}} \\ \hphantom{{}^{c}D_{q}^{\frac{7}{4}} u (t) =}{} + \frac{t u(t)}{( 5 + t^{2})(1+ \vert u(t) \vert )} ((\varphi _{2} u)(t) ) ^{\sigma _{3}} \\ \hphantom{{}^{c}D_{q}^{\frac{7}{4}} u (t) =}{}+ \frac{(1 + \alpha )(t - \frac{1}{2})^{2}}{ \varGamma _{q}( \alpha )(1+ \vert u(t)+ {}^{c}D_{q}^{ \frac{3}{2}} u(t) \vert )} \sum_{k=1}^{4} (\frac{ \sin k \pi t}{2^{k}} ) ({}^{c}D_{q}^{ \beta _{k}} u(t))^{ \delta _{k}}, \end{cases} $$

(10)

under boundary conditions $u(0) + \frac{1}{4} u(1) =0$ and $u'(0) + \frac{3}{4} u'(1) =0$, here $\beta _{1}=\frac{1}{3}$, $\beta _{2} = \frac{3}{5}$, $\beta _{3}=\frac{1}{2}$, $\beta _{4} =\frac{1}{6}$, $\lambda \in [0,\infty )$,

$$\begin{aligned} (\varphi _{1} u) (t) = \int _{0}^{t} \frac{s e^{-(s-t)}u(s)}{s^{2} + 4} \,d_{q}s,\qquad (\varphi _{2} u) (t) = \int _{0}^{t} \frac{16(t-s)^{4} u(s)}{\sqrt{1 + s^{2}}} \,d_{q}s. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \vert F_{u(t)} \vert \leq{}& \ell (t) + \frac{1}{\sqrt{2\pi }} \bigl\vert u(t) \bigr\vert ^{ \sigma _{1}} + \frac{1}{18} \bigl\vert (\varphi _{1} u) (t) \bigr\vert ^{\sigma _{2}} + \frac{1}{5} \bigl\vert (\varphi _{2} u) (t) \bigr\vert ^{\sigma _{3}} \\ & {} + \sum_{k=1}^{4} \frac{1 + \alpha }{ \varGamma _{q} (\alpha )2^{k+2}} \bigl\vert {}^{c}D_{q} ^{\beta _{k}} u(t) \bigr\vert ^{\delta _{k}}, \end{aligned}$$

where

$$\begin{aligned} F_{u(t)} &= w \bigl( t, u(t),(\varphi _{1} u) (t), ( \varphi _{2} u) (t), {}^{c}D_{q}^{ \beta _{1}} u(t), {}^{c}D_{q}^{\beta _{2}} u(t), {}^{c}D _{q}^{\beta _{3}} u(t), {}^{c}D_{q}^{\beta _{4}} u(t) \bigr), \end{aligned}$$

and $m(t) = \frac{\lambda e^{-\pi t}}{\sqrt{1+t^{2}}}$ for t belongs to J. Also, if $l=\frac{1}{2}$ and $\lambda =1$, then we have

$$ \ell ^{*}= \biggl( \int _{0}^{1} \bigl(\ell (t) \bigr)^{\frac{1}{\kappa }} \,d_{q} s \biggr) ^{\kappa }= \Biggl( (1-q) \sum_{k=0}^{\infty } \biggl( \frac{\lambda q ^{k} e^{-\pi q^{k}}}{\sqrt{1+q^{2k}}} \biggr)^{2} \Biggr)^{ \frac{1}{2}}. $$

Table 4 shows the variables of $\varGamma _{q}(\alpha )$, $\varGamma _{q}(\alpha -1)$, $A_{0}$, $\ell ^{*}$ and $K_{0}$ when $q =\frac{1}{3}$ and $m=1, \ldots , 40$. Since $0< \sigma _{j}$, for $j=1, 2, 3$, and $\delta _{i} < 1$, for $i=1, 2, 3, 4$, the assumption (7) holds. At present, if $\lambda = 0$, $\delta _{i} > 1$ and $\sigma _{j}> 1$ for $i=1, 2, 3, 4$ and $j=1, 2, 3$, respectively, the second condition, (8) of Theorem 5 holds. Thus, problem (10) has at least one solution. Note the features of the q-Gamma function, for values of q close to one, the results are obtained at a greater rate of m.

Table 4 Some numerical results for calculation of $\varGamma_{q}(\alpha )$, $\varGamma_{q}(\alpha-1)$, A, M and K in Theorem 5 with $q=\frac{1}{3}$ and $m=40$