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Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus

Abstract

In this study, we discuss the existence of positive solutions for the system of m-singular sum fractional q-differential equations

$$ \begin{gathered} D_{q}^{\alpha_{i}} x_{i} + g_{i} \bigl(t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma _{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr) \\ \quad{} +h_{i} \bigl(t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} \bigr)=0 \end{gathered} $$

with boundary conditions \(x_{i}(0) = x_{i}' (1) = 0\) and \(x_{i}^{(k)}(t) = 0\) whenever \(t=0\), here \(2\leq k \leq n-1\), where \(n= [\alpha_{i}]+ 1\), \(\alpha_{i} \geq2\), \(\gamma_{i} \in(0,1)\), \(D_{q}^{\alpha}\) is the Caputo fractional q-derivative of order α, here \(q \in(0,1)\), function \(g_{i}\) is of Carathéodory type, \(h_{i}\) satisfy the Lipschitz condition and \(g_{i} (t , x_{1}, \ldots, x_{2m})\) is singular at \(t=0\), for \(1 \leq i \leq m\). By means of Krasnoselskii’s fixed point theorem, the Arzelà-Ascoli theorem, Lebesgue dominated theorem and some norms, the existence of positive solutions is obtained. Also, we give an example to illustrate the primary effects.

Introduction

Fractional calculus and q-calculus belong to the significant branches in mathematical analysis. In 1910, Jackson introduced the subject of q-difference equations [1]. Later, many researchers studied q-difference equations [212]. On the other hand, there appeared recently much work on q-differential equations by using different views and fractional derivatives; young researchers could use the main idea in their work (see, for example, [1330]).

In 2010, the singular Dirichlet problem \(D^{\alpha} x(t) + g(t, x(t), D^{\gamma} x(t) )=0\) under conditions \(x(0) = x(1) = 0\) was investigated by Agarwal et al., where α, γ belong to \((1,2)\), \((0, \alpha- 1 )\), respectively, the function g is of Carathéodory type on \([0,1] \times(0 , \infty) \times\mathbb{R}\) and \(D^{\alpha}\) is the Riemann–Liouville fractional derivative [23]. In 2012, the fractional differential equation \({}^{c}D^{\alpha} y(t) + w(t,y(t))=0\), under boundary conditions \(y(0)=y''(0)=0\) and \(y(1)= \lambda\int_{0}^{1} y(s) \,ds\), was investigated, where \(t, \alpha, \lambda \in(0,1), (2, 3), (0, 2)\), respectively, and the function \(w: J \times[0,\infty) \rightarrow [0,\infty)\) is continuous [24]. Also, in the same year, Ahmad et al., 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 \(\alpha\in(1, 2]\), \(\alpha_{i}, \beta _{i}, \gamma_{i}, \eta_{i} \in\mathbb{R}\), for \(i=1,2\) and \(T \in C([0,1] \times\mathbb{R}, \mathbb{R})\) [6]. In 2013, Zhao et al. reviewed the q-integral problem \((D_{q}^{\alpha}u)(t) + f(t, u(t) )=0\), with conditions \(u(1)\), \(u(0)\) being 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 a positive real number, \(D_{q}^{\alpha}\) is the q-derivative of Riemann–Liouville and we have the real-valued continuous map u defined on \(I \times[0, \infty) \) [10].

In 2014, the singular fractional problem \({}^{c}D_{0^{+}}^{\alpha} x(t)+ f(t , x(t), {}^{c}D_{0^{+}}^{\sigma} x(t))=0\) with boundary conditions \(x(0)=x'(0)=0\) and \(x'(1)= {}^{c}D_{0^{+}}^{\sigma} x(1)\) investigated, where t, α, σ belong to \((0,1)\), \((2, 3)\), \((0, 1)\), respectively, \(f: (0,1] \times\mathbb{R}^{2} \to\mathbb{R} \) is continuous with \(f(t,x,y)\) may be singular at \(t=0\) and \({}^{c}D_{0^{+}}^{\alpha}\) is the Caputo derivative [31]. In 2017, Aydogan et al. and Shabibi et al. studied sum-type singular fractional integro-differential equation with k-point boundary conditions together some properties and sum fractional differential system with some conditions, respectively [21, 22]. Also, in the same year, Zhou et al., provided existence criteria for the solutions of p-Laplacian fractional Langevin differential equations with anti-periodic boundary conditions:

$$\left \{ \textstyle\begin{array}{l} D_{0^{+}}^{\beta}\phi_{p} [ ( D_{0^{+}}^{\alpha}+ \lambda) x(t) ] = f(t, x(t), D_{0^{+}}^{\alpha}x(t)), \\ x(0) = -x(1), \qquad D_{0^{+}}^{\alpha}x(0) = - D_{0^{+}}^{\alpha}x(1), \end{array}\displaystyle \right . $$

and

$$\left \{ \textstyle\begin{array}{l} {}_{q}D_{0^{+}}^{\beta}\phi_{p} [ ( D_{0^{+}}^{\alpha}+ \lambda) x(t) ] = g(t, x(t), {}_{q}D_{0^{+}}^{\alpha}x(t)), \\ x(0) = -x(1), \qquad{}_{q}D_{0^{+}}^{\alpha}x(0) = - {}_{q}D_{0^{+}}^{\alpha}x(1), \end{array}\displaystyle \right . $$

for \(t \in[0,1]\), where \(0 < \alpha, \beta\leq1\), λ is larger than or equal to zero, \(1 < \alpha+ \beta<2\), \(q \in(0,1)\), and \(\phi(p) (s) = |s|^{p-2} s\), with \(p \in(1, 2]\) [15]. In 2019, Samei et al., investigated existence of solutions for equations and inclusions of multi-term fractional q-integro-differential equations with non-separated and initial boundary conditions [9].

In this article, motivated by main idea of this work and by these achievements, we are working to address the positive solutions for system of singular sum fractional q-differential equations

$$ \begin{aligned} \left \{ \textstyle\begin{array}{l} D_{q}^{\alpha_{1}} x_{1} + g_{1} (t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma _{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} ) \\ \quad{} + h_{1} (t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} )=0, \\ D_{q}^{\alpha_{2}} x_{2} + g_{2} ( t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma _{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} ) \\ \quad{}+ h_{2} (t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} )=0, \\ \vdots\\ D_{q}^{\alpha_{m}} x_{m} + g_{m} (t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} ) \\ \quad{}+ h_{m} (t, x_{1}, \ldots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \ldots, D_{q}^{\gamma_{m}} x_{m} )=0, \end{array}\displaystyle \right . \end{aligned} $$
(1)

under some conditions \(x_{i}(0) = 0\), \(x_{i}' (1) = 0\) and \(\frac{ d^{k} x_{i}(t)}{d t^{k}} |_{t=0} = 0\), for \(i \in N_{m}\) and \(k \in N_{n-1} \setminus\{1\}\), where \(\alpha_{i} \geq2\), \([\alpha_{i}]= n - 1\), \(\gamma_{i} \in J=(0,1)\), \(D_{q}^{\alpha}\) is the Caputo fractional q-derivative of order α, the function \(g_{i}\) is of Carathéodory type, \(h_{i}\) satisfy Lipschitz condition and \(g_{i} (t , x_{1}, \ldots, x_{2m})\) is singular at \(t=0\) for \(i \in N_{m} \), where \(N_{\kappa}= \{1,2, \dots, \kappa\}\).

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.

Preliminaries

First, we point out some of the materials on the fractional q-calculus and fundamental results of it which are needed in the next sections (for more information, refer to [1, 32, 33]). Then, some well-known theorems as regards the fixed point theorem and definitions are presented.

Algorithm 1
figurea

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

Algorithm 2
figureb

The proposed method for calculated \(\varGamma_{q}(x)\)

Algorithm 3
figurec

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

Algorithm 4
figured

The proposed method for calculated \((I_{q}^{\alpha}f)(x)\)

Algorithm 5
figuree

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

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 are real numbers and \(\mathbb{N}_{0} := \{ 0\} \cup\mathbb{N}\) [32]. Also, for real number α and \(a \neq0\), 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 \(z \in\mathbb{R} \setminus\{0, -1, -2, \ldots\}\) [1]. Note that \(\varGamma_{q} (z+1) = [z]_{q} \varGamma_{q} (z)\). The value of the q-Gamma function, \(\varGamma_{q}(z)\), holds for input values q and z with counting the number of sentences n in summation by simplifying analysis. For this design, we prepare a pseudo-code description of the technique for estimating the q-Gamma function of order n which we 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 \((D_{q} f)(0) = \lim_{x \to0} (D_{q} f)(x)\), which is shown in Algorithm 3 [2]. 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 \(n \geq1\), where \((D_{q}^{0} f)(x) = f(x)\) [2]. The operator

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

for \(0 \leq x \leq b\), is called the q-integral of a function f, whenever the series is absolutely converges [2]. If a 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. The operator \(I_{q}^{n}\) is given by \((I_{q}^{0} f)(x) = f(x) \) and

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

for \(n \geq1\) [2]. 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 \(x =0\) [2]. The fractional Riemann–Liouville type q-integral of the function f on \([0,1]\), for \(\alpha\geq0\), 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 \(\alpha>0\) [4, 7]. Also, the fractional Caputo type q-derivative of the function f is given by

$$ \begin{aligned}[b] \bigl( {}^{c}D_{q}^{\alpha}f \bigr) (t) & = \bigl( I_{q}^{[\alpha]-\alpha}\bigl( D_{q}^{[\alpha]} f\bigr) \bigr) (t) \\ & = \frac{1}{\varGamma_{q} ([\alpha]-\alpha )} \int_{0}^{t} (t- qs)^{ ([\alpha]-\alpha-1 )} \bigl( D_{q}^{[\alpha]} f \bigr) (s) \,d_{q}s, \end{aligned} $$
(2)

for \(t \in[0,1]\) and \(\alpha>0\) [4, 7]. 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 \(\alpha, \beta\geq0\) [7]. By using Algorithm 2, we can calculate \((I_{q}^{\alpha}f)(x)\) which is shown in Algorithm 4. One can find more details of fractional differential and q-differential equations in [3437].

Now, we present some necessary notions. Throughout this article, we denote \(L^{1}(0,1)\), \(L^{1}[0,1]\), \(C_{\mathbb{R}}(0,1)\), \(C_{\mathbb {R}}[0,1]\), \(C_{\mathbb{R}}^{1}[0,1]\) by \(\mathcal{L}\), \(\overline {\mathcal{L}}\), \(\mathcal{A}\), \(\overline{\mathcal{A}}\), \(\overline {\mathcal{B}}\), respectively. We say that a map \(\theta: \overline{J} \times\mathcal{S} \to\mathbb{R}^{n}\) is of Carathéodory type whenever the function \(t \mapsto\theta( t, r_{1}, \dots, r_{n})\) is measurable for all \((r_{1},\dots, r_{n}) \in\mathcal{S}\) and \((r_{1}, \dots, r_{n})\mapsto\theta(t, r_{1}, \dots, r_{n})\) is continuous for \(t \in \overline{J}\) and for each compact \(C \subseteq\mathcal{S}\) there exists \(\psi_{C} \in\overline{\mathcal{L}}\) such that \(|\theta( t, r_{1}, \dots, r_{n}) |\leq\psi_{C} (t)\) for each \(t \in\overline{J}\) and \((r_{1}, \dots, e_{n}) \in C\), here \(\mathcal{S} = (0,\infty)^{2m}\). At present, we consider four norms which will be used in the sequel: \(\| x \| := \sup\{ |x(t)|: t \in\overline{J}\}\), \(\| x \|_{1} := \int_{0}^{1} |x(t)|\, dt\), \(\| (x_{1}, x_{2}, \dots, x_{n}) \|_{*} := \max\{ \| x_{i} \| : i\in N_{n}\}\) and

$$\bigl\Vert (x_{1}, x_{2}, \dots, x_{n}) \bigr\Vert _{**} := \max \bigl\{ \Vert x_{1} \Vert , \Vert x_{2} \Vert , \dots, \Vert x_{n} \Vert , \bigl\Vert x'_{1} \bigr\Vert , \bigl\Vert x'_{2} \bigr\Vert , \dots, \bigl\Vert x'_{n} \bigr\Vert \bigr\} . $$

The following lemmas can be found in [34, 3638].

Lemma 1

If \(x \in\overline{\mathcal{A}} \cap\overline{\mathcal{L}}\)with \(D_{q}^{\alpha} x\in\mathcal{A} \cap\mathcal{L}\), then \(I_{q}^{\alpha} D_{q}^{\alpha} x(t) =x(t) + \sum_{i=1}^{n} c_{i} t^{\alpha- i}\), wherenis the smallest integer greater than or equal toαand \(c_{i}\)is some real number.

Lemma 2

Assume that a nonempty subsetCof a Banach space \(\mathcal{X}\)be a closed, convex. Then, there exists \(c \in C\)such that \(c=\mathcal{O}_{1} (c) + \mathcal{O}_{2} (c)\)whenever the operators \(\mathcal{O}_{1}\)and \(\mathcal{O}_{2}\)are compact and continuous, or a contraction, respectively.

Lemma 3

The unique solution for \(D_{q}^{\alpha}x(t) + v(t) = 0\)under conditions \(x'(1)= x(0) =x''(0) = \cdots= x^{n-1}(0) =0\), here \(v \in\overline {\mathcal{L}}\), \(\alpha\in[2, \infty)\)and \(n = [\alpha] +1\), is \(x(t)= \int^{1}_{0} G_{\alpha}(t,qs) v(s) \,d_{q}s\), where

$$G_{\alpha} (t,qs) = \frac{t ( 1- qs)^{(\alpha-2)}}{ \varGamma_{q}( \alpha- 1) }, $$

whenever \(t \leq s\)and

$$G_{\alpha} (t,qs) = \frac{t (1-qs)^{(\alpha- 2)}}{ \varGamma_{q} (\alpha- 1)} - \frac{ (t - qs)^{(\alpha-1)}}{ \varGamma_{q} ( \alpha) }, $$

whenever \(s \leq t\), for all \(t, s \in\overline{J}\).

Proof

At first, by applying Lemma 1 and the boundary conditions, we conclude that \(x(t)= -I_{q}^{\alpha}v(t) + c_{1} t\) and so \(x'(1) = -I_{q}^{\alpha-1} v(1) + c_{1}\). Since \(x'(1) = 0\), \(c_{1}= I_{q}^{\alpha-1} v(1)\). Thus, \(x(t) = - I_{q}^{ \alpha} v(t) + tI_{q}^{\alpha-1} v(1)\). Hence, we obtain

$$x(t) = \int^{1}_{0} G_{\alpha} (t,qs) v(s) \,ds, $$

where

$$ G_{\alpha} (t,qs) \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{t ( 1 - qs)^{(\alpha-2)}}{ \varGamma_{q} ( \alpha-1)}, & t \leq s,\\ \frac{t (1 - qs)^{(\alpha- 2)}}{\varGamma_{q} (\alpha- 1 ) } - \frac{ ( t -qs)^{(\alpha-1)}}{ \varGamma_{q} (\alpha)}, & s \leq t, \end{array}\displaystyle \right . $$
(3)

for each \(t, s \in\overline{J}\). □

Remark 1

Consider a q-Green function as in (3). It can be seen that \(G_{\alpha} (t,qs) >0 \) if \(t \leq s\) for each \(t, s \in J\). Also, \(G_{\alpha} (t,qs) >0\) whenever \(s < t\) if and only if \((t -qs)^{(\alpha-1)} < t (\alpha-1) ( t -qs)^{(\alpha-2)}\) for all \(t, s \in J\). In addition

$$ \frac{t^{\alpha} (\alpha- 1)}{\varGamma_{q}( \alpha+1 )} \leq\frac {t}{\varGamma_{q}( \alpha)} - \frac{t^{ \alpha}}{\varGamma_{q}( \alpha+1 )} \leq \int^{1}_{0} G_{\alpha}(t,qs) \,d_{q}s $$
(4)

and \(\frac{\partial}{\partial t} G_{\alpha}(t,qs) > 0\) for \(t,s \in J\). Moreover, \(G _{\alpha}\), \(\frac{ \partial}{ \partial t} G_{\alpha} \in C_{\mathbb{R}}(\overline{J} \times\overline{J})\), \(\frac{ \partial}{ \partial t} G_{\alpha}(t,qs) \leq\frac{1}{ \varGamma _{q}( \alpha-1)}\) and \(\int^{1}_{0} \frac{ \partial}{ \partial t} G_{\alpha}(t,s) \geq\frac{1-t^{\alpha-1}}{\varGamma( \alpha)}\), for almost all \(t, s \in\overline{J}\).

Remark 2

Let \(u \in C_{\mathbb{R}}^{1} (\overline{J})\) and \(\gamma\in J\). Since

$$D_{q}^{\gamma} u(t) = \frac{1}{\varGamma_{q}( 2 - \gamma)} \int^{t}_{0} (t - qs)^{-\gamma} u'(s) \, d_{q}s, $$

for all \(t \in\overline{J}\),

$$ \bigl\vert D_{q}^{\gamma} u \bigr\vert \leq \frac{ \Vert u' \Vert }{ \varGamma_{q}( 1 - \gamma )} \int^{t}_{0} (t -qs)^{-\gamma}\, d_{q}s = \frac{ \Vert u' \Vert }{\varGamma_{q}( 2 -\gamma)} t^{ 1 -\gamma} $$
(5)

and so \(\varGamma_{q}( 2 -\gamma) |D_{q}^{\gamma} u| \leq \Vert u' \Vert \) and \(D_{q}^{\gamma} u \in C_{\mathbb{R}}(\overline{J})\).

Main results

Now, we consider the following assumptions for the problem (1):

  1. (A1)

    The maps \(g_{i}\) are Carathéodory functions on \(\overline {J} \times\mathcal{S}\) and there exist positive constants \(\ell_{i}\) such that \(g_{i} ( t, u_{1}, \dots, u_{2m} ) \geq\ell_{i}\) for each \(t \in\overline{J}\) and all \(( u_{1}, \dots, u_{2m}) \in\mathcal {S}\) where \(i \in N_{m}\).

  2. (A2)

    The maps \(h_{i}\) are nonnegative and

    $$ \bigl\vert h_{i} ( t, u_{1}, \dots, u_{2m} ) - h_{i} (t, v_{1}, \dots, v_{2m} ) \bigr\vert \leq\sum_{k=1}^{2m} {}_{i}M_{k} \vert u_{k} - v_{k} \vert , $$
    (6)

    for each t belonging to and for almost all \((u_{1}, \dots, u_{2m}), (v_{1}, \dots, v_{2m}) \in\mathcal{S}\), where \({}_{i}M_{j}\) in \([0,\infty)\), for \(i \in N_{m}\) and \(j \in N_{2m}\), are constants such that

    $$ \sum_{k=1}^{m} \biggl( {}_{i}M_{k} + \frac{ {}_{i}M_{m + k} }{ \varGamma _{q}(2 - \gamma_{i})} \biggr) < \varGamma_{q}( \alpha_{i} - 1). $$
    (7)
  3. (A3)

    There exist some maps \(\mu_{1}, \dots, \mu_{m} \in\overline {\mathcal{L}}\), some nonincreasing maps \(r_{1}, \dots, r_{m} \in C_{\mathbb {R}}(\mathcal{S})\) with

    $$\int_{0}^{1} r_{i} \biggl( L_{1} t^{\alpha_{1}}, \dots, L_{m} t^{ \alpha _{m}}, \frac{ L_{1} (1 - \gamma_{1})}{ 2 } t^{ 1 - \gamma_{1}}, \dots, \frac{ L_{m} (1- \gamma_{m})}{2} t^{1 - \gamma_{m}} \biggr) \, dt < \infty $$

    and some functions \(w_{1}, \dots, w_{m} \in C_{\mathbb{R}}( \mathcal{S} )\) such that \(w_{i}\) is nondecreasing in all components, \(\lim_{x \to \infty} \frac{ w_{i} (x, \dots, x)}{x} = 0\) and

    $$ \begin{aligned}[b] &g_{i} (t, u_{1}, \dots, u_{2m}) + h_{i} (t, u_{1}, \dots, u_{2m} ) \\ &\quad \leq r_{i}( u_{1}, \dots, u_{2m}) + \mu_{i} (t) w_{i} (u_{1}, \dots, u_{2m}), \end{aligned} $$
    (8)

    for almost all \(t \in\overline{J}\) and all \(( u_{1}, \dots, u_{2m}) \in \mathcal{S}\) where \(L_{i} \varGamma_{q}(\alpha_{i} + 1) = \ell_{i}(\alpha _{i} - 1)\) for all \(i \in N_{m}\).

Now, we prove the following lemma.

Lemma 4

Suppose that \(\mathcal{P}\)is the set of all \(( u_{1}, \dots, u_{m} ) \)belonging to \(\overline{\mathcal{B}}^{m}\)such that \(u_{i}(t) \)and \(u'_{i}(t) \)are larger than or equal to zero for \(t \in\overline {J}\)and \(i\in\{0\} \cup N_{m}\). Also, for each natural numbernand \(i \in N_{m}\), we define the maps

$$ \begin{aligned}[b] H_{i} ( u_{1}, \dots, u_{m} ) (t) & = \int^{1}_{0} G_{\alpha_{i}} (t, qs) \\ & \quad\times h_{i} \bigl(s , u_{1} (s), \dots, u_{m}(s), D_{q}^{\gamma _{1} } u_{1} (s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr) \, d_{q}s \end{aligned} $$
(9)

and

$$ H ( u_{1} , \dots, u_{m} ) (t) =\left ( \textstyle\begin{array}{c} H_{1} ( u_{1} , \ldots, u_{m} )(t)\\ H_{2} ( u_{1} , \ldots, u_{m} )(t)\\ \vdots\\ H_{m} ( u_{1}, \dots, u_{m})(t) \end{array}\displaystyle \right ), $$
(10)

for all \(( u_{1} , \dots, u_{m})\in\mathcal{P}\). Then the self-mapHdefine on \(\mathcal{P}\)is a contraction.

Proof

First, by simple review, we can check that \(H_{i} ( u_{1} , \dots, h_{m}) (t) \geq0\) and

$$\begin{aligned} H'_{i} ( u_{1} , \dots, u_{m} ) (t)& = \int^{1}_{0} \frac{ \partial}{ \partial t} G_{\alpha_{i}} (t,qs) \\ & \quad\times h_{i} \bigl(s , u_{1}(s) , \dots, u_{m} (s) , D_{q}^{\gamma_{1}} u_{1} (s), \dots, D_{q}^{\gamma_{m} } u_{m}(s) \bigr)\, d_{q}s \\&\geq0 \end{aligned}$$

for all \(t \in\overline{J}\), \((u_{1}, \dots, u_{m}) \in\mathcal{P}\) and \(i \in \{0\} \cup N_{m}\). On the other hand,

$$\begin{aligned} & \bigl\Vert H_{i} ( u_{1} , \dots, u_{m} ) - H _{i} ( v_{1} , \dots, v_{m} ) \bigr\Vert \\ & \quad= \sup_{t \in\overline {J} } \biggl\vert \int^{1}_{0} G_{\alpha_{i}} (t,qs) \bigl[ h_{i} \bigl(s , u_{1}(s) , \dots,u_{m}(s) , D_{q}^{\gamma_{1} } u_{1} (s), \dots, D_{q}^{\gamma_{m}} u_{m} (s) \bigr) \\ & \qquad - h_{i} \bigl(s , v_{1}(s) , \ldots, v_{m}(s) , D_{q}^{\gamma _{1}} v_{1} (s), \dots, D_{q}^{\gamma_{m}} v_{m} (s) \bigr) \bigr] \, d_{q}s \biggr\vert \\ &\quad \leq \biggl\vert \int^{1}_{0} G_{\alpha_{i}} (t, qs) \, d_{q}s \biggr\vert \\ & \qquad\times \sum_{k=1}^{m} \bigl( {}_{i}M_{k} \Vert u_{k} - v_{k} \Vert + {}_{i}M_{m+k} \bigl\Vert D_{q}^{\gamma_{k} } u_{k} - D_{q}^{\gamma_{k} } v_{k} \bigr\Vert \bigr). \end{aligned}$$

By using Remark 2, we can conclude that

$$\begin{aligned} & \bigl\Vert H_{i} ( u_{1} , \dots, u_{m} ) - H _{i} ( v_{1} , \dots, v_{m} ) \bigr\Vert \\ &\quad\leq\frac{1}{\varGamma_{q} (\alpha_{i})} \sum_{k=1}^{m} \biggl( {}_{i}M_{k} \Vert u_{k} - v_{k} \Vert + \frac{ {}_{i}M_{m+k} }{ \varGamma_{q}(2- \gamma_{k}) } \bigl\Vert u'_{k} - v'_{k} \bigr\Vert \biggr) \\ & \quad\leq\frac{ \Vert ( u_{1} , \ldots, u_{m}) - ( v_{1} , \ldots, v_{m}) \Vert _{**}}{ \varGamma_{q}( \alpha_{i})} \sum_{k=1}^{m} \biggl( {}_{i}M_{k} + \frac{ {}_{i}M_{m+k} }{ \varGamma_{q}(2 - \gamma_{i})} \biggr) \\ &\quad \leq\frac{ \Vert ( u_{1} , \dots, u_{m}) - ( v_{1} , \dots, v_{m}) \Vert _{**}}{\varGamma_{q}( \alpha_{i} - 1 )} \sum_{k=1}^{m} \biggl( {}_{i}M_{k} + \frac{ {}_{i}M_{m+k} }{ \varGamma_{q}(2 - \gamma_{i})} \biggr) \end{aligned}$$

for \(i \in N_{m} \cup\{0\} \). Hence,

$$\begin{aligned} &\bigl\Vert H ( u_{1} , \dots, u_{m}) - H ( v_{1} , \dots, v_{m}) \bigr\Vert _{*} \\ &\quad = \max_{i \in N_{m}} \bigl\Vert H_{i} ( u_{1} , \dots, u_{m}) - H_{i} ( v_{1} , \dots, v_{m}) \bigr\Vert \\ & \quad\leq\max_{ i \in N_{m}} \Biggl\{ \frac{1}{\varGamma_{q}(\alpha_{i} - 1)} \sum _{k=1}^{m} \biggl( {}_{i}M_{k} + \frac{{}_{i}M_{m+k} }{ \varGamma_{q}( 2- \gamma _{i})} \biggr) \Biggr\} \\ & \quad\quad\times \bigl\Vert ( u_{1} , \ldots, u_{m}) - ( v_{1} , \dots, v_{m}) \bigr\Vert _{**}. \end{aligned}$$

In a similar way, we obtain

$$\begin{aligned} &\bigl\Vert H' ( u_{1} , \dots, u_{m}) - H' ( v_{1} , \dots, v_{m}) \bigr\Vert _{*} \\ &\quad\leq\max_{i \in N_{m}} \Biggl\{ \frac{1}{\varGamma_{q} ( \alpha_{i} - 1)} \sum _{k=1}^{m} \biggl( {}_{i}M_{k} + \frac{ {}_{i}M_{m + k}}{ \varGamma_{q}(2 - \gamma_{i})} \biggr) \Biggr\} \\ & \qquad\times \bigl\Vert ( u_{1} , \ldots, u_{m}) - ( v_{1} , \dots, v_{m} ) \bigr\Vert _{**}. \end{aligned}$$

Thus, we have

$$\begin{aligned} &\bigl\Vert H(x_{1} , \dots, u_{m}) - H ( v_{1} , \dots, v_{m}) \bigr\Vert _{**} \\ &\quad \leq\max_{i \in N_{m}} \Biggl\{ \frac{1}{\varGamma_{q} ( \alpha_{i} - 1)} \sum _{k=1}^{m} \biggl( {}_{i}M_{k} + \frac{ {}_{i}L_{m+k} }{\varGamma_{q}(2- \gamma _{i})} \biggr) \Biggr\} \\ & \qquad\times \bigl\Vert (u_{1} , \dots, u_{m}) - ( v_{1} , \dots, v_{m}) \bigr\Vert _{**}.\end{aligned} $$

By assumption (A2) and inequality (7), we conclude that H is a contraction mapping. □

At present, for \(i \in N_{m}\) and \(n \in\mathbb{N}\), we take

$$F_{i , n}( t , u_{1} , \dots, u_{2m} ) = g_{i}\bigl( t , \chi_{1}( u_{1}) , \dots, \chi_{n} (u_{2m}) \bigr), $$

where \(\chi_{n}(x)=x\) whenever \(x\geq\frac{1}{n}\) and \(\chi_{n} (x) = x\) whenever \(x < \frac{1}{n}\). By simple review, we can check that

$$\begin{aligned} &F_{i,n} ( t , u_{1} , \dots, u_{2m} ) + h_{i}( t , u_{1} , \dots, u_{2m} ) \\ &\quad \leq r_{i} \biggl( \frac{1}{n} , \dots, \frac{1}{n} \biggr) + \mu_{i} (t) w_{i} \biggl( u_{1} + \frac{1}{n} , \dots, u_{2m} + \frac{1}{n} \biggr), \end{aligned}$$

\(F_{i,n} (t, u_{1} , \dots, u_{2m} ) \geq\ell_{i}\) and

$$\begin{aligned} &F_{i,n} ( t , u_{1} , \dots, u_{2m} ) + h_{i} ( t , u_{1} , \dots, u_{2m} ) \\ &\quad \leq r_{i} ( u_{1} , \dots, u_{2m} ) + \mu_{i} (t) w_{i} \biggl( u_{1} + \frac{1}{n} , \dots, u_{2m} + \frac{1}{n} \biggr), \end{aligned}$$

for all \((u_{1}, \dots, u_{n}) \in\mathcal{S}\), \(i \in N_{m}\) and each \(t \in\overline{J}\).

First, we investigate the system of regular fractional q-differential equations

$$ \left\{ \textstyle\begin{array}{l} D_{q}^{\alpha_{1}} x_{1} + F_{1,n} (t , x_{1} , \dots, x_{m}, D_{q}^{\gamma_{1}} x_{1}, \dots, D_{q}^{\gamma_{m}} x_{m} )=0, \\ D_{q}^{\alpha_{2}} x_{2} + F_{2,n} (t , x_{1} , \dots, x_{m}, D_{q}^{\gamma_{2}} x_{1}, \dots, D_{q}^{\gamma_{m}} x_{m} )=0, \\ \vdots\\ D_{q}^{\alpha_{m}} x_{m} + F_{m,n} (t , x_{1} , \dots, x_{m} , D_{q}^{\gamma_{1}} x_{1}, \dots, D_{q}^{\gamma_{m}} x_{m} )=0, \end{array}\displaystyle \right . $$
(11)

with the same boundary conditions as in (1).

Lemma 5

Suppose that \(\mathcal{P}\)is the set which is defined in Lemma 4and \(i\in\{0\} \cup N_{m} \). Also, let us, for each natural numbernand \(i \in N_{m}\), define the maps

$$ \begin{aligned}[b] &T_{i,n} ( u_{1}, \ldots, u_{m} ) (t) \\ &\quad = \int^{1}_{0} G_{\alpha_{i}} (t,qs) F_{i,n} \bigl( s , u_{1}(s) , \dots , u_{m} (s) , D_{q}^{\gamma_{1}} u_{1} (s), \dots, D_{q}^{\gamma_{m}} u_{m} (s) \bigr) \, d_{q}s \end{aligned} $$
(12)

and

$$ \varOmega_{n} ( u_{1} , \dots, u_{m}) (t) = \left ( \textstyle\begin{array}{c} T_{1,n} ( u_{1} , \dots, u_{m} )(t) \\ T_{2,n} ( u_{1} , \dots, u_{m} )(t) \\ \vdots\\ T_{m,n} ( u_{1} , \dots, u_{m} )(t) \end{array}\displaystyle \right ), $$
(13)

for all \(( u_{1} , \dots, u_{m})\in\mathcal{P}\). Then \(\varOmega_{n}\)is a completely continuous operator on \(\mathcal{P}\)for each natural numbern.

Proof

Assume that \(( u_{1} , \dots, u_{m}) \in\mathcal{P}\). We choose a positive constant \(\ell_{i}\) such that

$$F_{i,n} \bigl(t , u_{1}(t) , \dots, u_{m}(t) , D_{q}^{\gamma_{1}} u_{1}(t), \dots, D_{q}^{\gamma_{m}} u_{m}(t) \bigr) \geq\ell_{i}, $$

for almost all \(t \in\overline{J}\). Since \(G_{\alpha_{i}}\) and \(\frac {\partial}{\partial t} G_{\alpha_{i}}\) are nonnegative and continuous on \(\overline{J}^{2}\) for each \(i \in N_{m}\), we conclude that \(T_{i,n} ( u_{1}, \dots, u_{m} )(t) \) and \(( T_{i,n} ( u_{1} , \dots, u_{m} ) )'(t)\) larger than or equal to zero, for all \(t\in\overline{J}\) and \(i \in N_{m}\). Indeed, \(\varOmega_{n}\) maps P into P. Consider a convergent sequence \(\{ ( u_{1,k} , \dots, u_{m,k}) \} \subseteq\mathcal{P}\) with \(\lim_{k \to\infty} ( u_{1,k} , \dots, u_{m,k}) = (u_{1} , \ldots, u_{m})\). In this case, we get \(\lim_{k \to \infty} u_{i,k} = u_{i}\) and \(\lim_{k \to\infty} u'_{i,k} = u'_{i}\) uniformly on (\(i \in N_{m}\)). But

$$\varGamma_{q}( 2 -\gamma_{i}) \bigl\vert D_{q}^{\gamma_{i}} u_{i,k}(t) - D_{q}^{\gamma_{i}} u_{i}(t) \bigr\vert \leq \bigl\Vert u'_{i,k} - u'_{i} \bigr\Vert , $$

for each t in and \(i \in N_{m}\). Hence, \(\lim_{k \to \infty} D^{\mu_{i}} x_{i,k}(t) = D^{\mu_{i}} x_{i}(t)\) uniformly on . Hence,

$$\begin{aligned} &\lim_{k \to\infty} F_{i,n} \bigl(t , u_{1, k}( t) , \dots, u_{m,k}(t), D_{q}^{\gamma_{1}} u_{1,k} (t), \ldots, D_{q}^{\gamma_{m}} u_{m,k}(t) \bigr) \\ &\quad = F_{i,n} \bigl(t , u_{1}(t) , \dots, u_{m}(t) , D_{q}^{\gamma_{1}} u_{1}(t), \dots, D_{q}^{\gamma_{m}} u_{m}(t) \bigr). \end{aligned}$$

Since \(F_{i,n} \in C ( \overline{J} \times\mathbb{R}^{2m} )\), the sequence \(\{(u_{1,k} , \dots, u_{m,k}) \} \subseteq \overline{\mathcal{B}}^{m}\) is bounded, there exists a map \(\mu_{i} \in \overline{\mathcal{L}}\) such that

$$ \ell_{i} \leq F_{i,n} \bigl(t , u_{1, k}(t) , \dots , u_{m,k}(t) , D_{q}^{\gamma_{1}} u_{1,k} (t), \dots, D_{q}^{\gamma_{m}} u_{m,k}(t) \bigr) \leq\mu_{i}(t), $$
(14)

for almost all \(t \in\overline{J}\), \(i \in N_{m}\) and \(k \in\mathbb {N}\). By using the dominated convergence theorem of Lebesgue, we conclude that

$$\begin{aligned} &\bigl\vert T_{i,n} ( u_{1,k} , \dots, u_{m,k} ) (t) - T_{i,n} ( u_{1} , \dots, u_{m} ) (t) \bigr\vert \\ & \quad\leq\frac{1}{\varGamma_{q} (\alpha_{i})} \int^{1}_{0} \bigl\vert F_{i,n} \bigl(s , u_{1, k} (s) , \dots, u_{m,k}(s) , D_{q}^{ \gamma_{1}} u_{1,k}(s), \dots, D_{q}^{ \gamma_{m}} u_{m,k} (s) \bigr) \\ &\quad \quad- F_{i,n} \bigl(s , u_{1}(s) , \dots, u_{m}(s) , D_{q}^{\gamma _{1}} u_{1} (s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr) \bigr\vert \,ds \end{aligned}$$

and

$$\begin{aligned} &\bigl\vert \bigl( T_{i,n} (u_{1,k} , \dots, u_{m,k} ) \bigr)'(t) - \bigl( T_{i,n} ( u_{1} , \dots, u_{m} ) \bigr)'(t) \bigr\vert \\ & \quad \leq\frac{1}{\varGamma_{q}(\alpha_{i}-1)} \\ & \qquad\times \int^{1}_{0} \bigl\vert F_{i,n} \bigl(s , u_{1, k}(s) , \dots, u_{m,k} ( s) , D_{q}^{\gamma_{1}} u_{1,k} (s), \dots, D_{q}^{\gamma_{m}} u_{m,k} (s) \bigr) \\ & \quad\quad- F_{i,n} \bigl(s , u_{1}(s) , \dots, u_{m}(s) , D_{q}^{\gamma_{1} } u_{1} (s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr) \bigr\vert \,ds. \end{aligned}$$

Hence,

$$\lim_{k \to\infty} \bigl\vert \bigl( T_{i,n} ( u_{1,k} , \dots, u_{m,k}) \bigr)^{j} (t) - \bigl( T_{i,n} ( u_{1} , \dots, u_{m}) \bigr)^{j}(t) \bigr\vert = 0, $$

uniformly on for \(j=0,1\). Thus,

$$\bigl\Vert \varOmega_{n} (u_{1,k} , \dots, u_{m,k}) (t) - \varOmega_{n} ( u_{1} , \dots, u_{m}) (t) \bigr\Vert _{**} \to0 $$

and so \(\varOmega_{n}\) is continuous. Let \(\{ ( u_{1,k} , \dots, u_{m,k}) \} \subseteq\mathcal{P}\) be a bounded sequence. We choose a positive number M such that \(\Vert u_{i,k} \Vert \) and \(\Vert u'_{i,k} \Vert \) are smaller than or equal to M for all \(i \in N_{m}\) and \(k \geq1\). Since \(\Vert D_{q}^{\gamma_{i}} u_{i,k} \Vert \varGamma_{q}( 2 - \gamma_{i}) \leq1\) for each \(i \in N_{m}\), there exists a map \(\mu_{i} \in\overline{\mathcal {L}}\) such that inequality (14) holds for almost all \(t \in \overline{J}\), \(i \in N_{m}\) and \(k \geq1\). On the other hand,

$$\begin{aligned} 0 &\leq T_{i,n} ( u_{1,k} , \dots, u_{m,k}) (t) \\ & = \int^{1}_{0} G_{\alpha_{i}} (t,qs) \\ & \quad\times F_{i,n} \bigl(s , u_{1, k}(s) , \dots, u_{m,k} (s) , D_{q}^{\gamma_{1}} u_{1,k} (s), \dots, D_{q}^{\gamma_{m}} u_{m,k} (s) \bigr)\, ds \\ & \leq\frac{1}{ \varGamma_{q}(\alpha_{i})} \int^{1}_{0} \mu_{i} (s)\, d_{q}s = \frac{ \Vert \mu_{i} \Vert _{1}}{ \varGamma_{q}(\alpha_{i})} \end{aligned}$$

and

$$\begin{aligned} 0 & \leq\bigl(T_{i,n} ( u_{1,k} , \dots, u_{m,k}) \bigr)'(t) \\ & = \int^{1}_{0} \frac{ \partial}{\partial t} G_{ \alpha_{i}} (t,qs) \\ & \quad\times F_{i,n}\bigl(s , u_{1, k}(s) , \dots, u_{m,k}(s) , D_{q}^{\gamma_{1}} u_{1,k}(s), \dots, D_{q}^{\gamma_{m}} x_{m,k} (s) \bigr) \, d_{q}s \\ & \leq\frac{1}{ \varGamma_{q}(\alpha_{i} - 1)} \int^{1}_{0} \mu_{i} (s) \, d_{q}s = \frac{ \Vert \mu_{i} \Vert _{1}}{ \varGamma_{q}(\alpha_{i} -1) } \end{aligned}$$

for all \(i \in N_{m}\). Hence,

$$\bigl\Vert \varOmega_{n} ( u_{1,k} , \dots, u_{m,k}) (t) \bigr\Vert _{**} \leq \max _{i \in N_{m}} \frac{ \Vert \mu_{i} \Vert _{1}}{ \varGamma_{q}(\alpha _{i} -1)}. $$

Indeed, \(\{ \varOmega_{n} ( u_{1,k} , \dots, u_{m,k}) \}\) is bounded in \(\overline{\mathcal{B}}^{m}\). Assume that \(t_{1}, t_{2} \in\overline {J}\) such that \(t_{1} \leq t_{2} \) and \(i \in N_{m}\). Then, we have

$$\begin{aligned} & \bigl\vert \bigl( T_{i,n} ( u_{1,k} , \dots, u_{m,k}) \bigr)' (t_{2}) - \bigl( T_{i,n} ( u_{1,k} , \dots, u_{m,k}) \bigr)' (t_{1}) \bigr\vert \\ & \quad\leq\frac{ t_{2} -t_{1} }{ \varGamma_{q}( \alpha_{i} -1)} \int^{1}_{0} (1 -qs)^{(\alpha_{i} - 2)} \\ & \qquad\times F_{i,n} \bigl(s , u_{1, k}(s) , \dots, u_{m,k} (s) , D_{q}^{\gamma_{1}} u_{1,k} (s), \dots, D_{q}^{\gamma_{m}} x_{m,k} (s) \bigr) \, d_{q}s \\ & \qquad+ \frac{1}{ \varGamma_{q}( \alpha_{i} )} \biggl\vert \int^{t_{2}}_{0} ( t_{2} - qs)^{(\alpha_{i} - 1)} \\ & \qquad\times F_{i,n} \bigl(s , u_{1, k} (s) , \dots, u_{m,k} (s), D_{q}^{\gamma_{1}} u_{1,k} (s), \dots, D_{q}^{\gamma_{m}} u_{m,k} (s) \bigr) \, d_{q}s \\ & \qquad- \int^{ t_{1}} _{0} (t_{1} -qs)^{(\alpha_{i} - 1)} \\ & \qquad\times F_{i,n} \bigl( s , u_{1, k}(s) , \dots, u_{m,k}(s) , D_{q}^{\gamma_{1}} u_{1,k} (s), \dots, D_{q}^{\gamma_{m}} u_{m,k} (s) \bigr)\, d_{q}s \biggr\vert \\ &\quad \leq\frac{ \Vert F_{i,n} \Vert _{1}}{ \varGamma_{q}( \alpha_{i} -1)} ( t_{2} - t_{1} ) + \frac{1}{ \varGamma_{q}( \alpha_{i} )} \biggl[ \int^{t_{1}}_{0} \bigl( (t_{2} - qs)^{(\alpha_{i} - 1)} - (t_{1} - s)^{ \alpha_{i} - 1} \bigr) \\ & \qquad\times F_{i,n}\bigl(s , u_{1, k}(s) , \dots, u_{m,k}(s) , D_{q}^{\gamma_{1}} u_{1,k}(s), \dots, D_{q}^{\gamma_{m}} u_{m,k}(s) \bigr)\, d_{q}s \\ & \qquad+ \int^{t_{2}} _{t_{1}} (t_{2} - qs)^{(\alpha_{i} - 2)} \\ & \qquad\times F_{i,n} \bigl(s , u_{1, k}(s) , \dots, u_{m,k}(s) , D_{q}^{\gamma_{1}} u_{1,k}(s), \dots, D_{q}^{\gamma_{m}} u_{m,k}(s) \bigr) \, d_{q}s \biggr] \\ & \quad\leq\frac{ \Vert \mu_{i} \Vert _{1}}{ \varGamma_{q}(\alpha_{i} -1)} (t_{2} - t_{1}) \\ & \qquad+ \frac{1}{ \varGamma_{q}( \alpha_{i})} \biggl[ \int^{t_{1}}_{0} \bigl( (t_{2} - qs)^{(\alpha_{i} - 1)} - (t_{1} - qs)^{(\alpha_{i} - 1)} \bigr) \mu_{i} (s)\, d_{q}s + (t_{2} - t_{1})^{\alpha_{i} -1} \Vert \mu_{i} \Vert _{1} \biggr]. \end{aligned}$$

Since the function \(|t - qs|^{(\alpha_{i} -1)}\) is uniformly continuous on \(\overline{J}^{2} \), there exists \(\delta> 0\) such that \(( t_{2} - qs)^{(\alpha_{i} -1)} - ( t_{1} - qs)^{(\alpha_{i} -1)} < \varepsilon\) for all \(t_{1}, t_{2} \in\overline{J}\) with \(t_{1} \leq t_{2}\), \(t_{2} - t_{1} < \delta\) and \(s \in[0, t_{1}]\), where \(\varepsilon> 0\) be given. Take \(t_{2} - t_{1} < \min\{ \delta, \varepsilon\}\), then we have

$$\bigl\vert \bigl( T_{i,n} ( u_{1,k} , \dots, u_{m,k}) \bigr)'( t_{2}) - \bigl( T_{i,n} ( u_{1,k} , \dots, u_{m,k}) \bigr)' (t_{1}) \bigr\vert < \frac{ 3 \varepsilon \Vert \mu_{i} \Vert _{1} }{ \varGamma_{q}(\alpha _{i}) }. $$

Thus,

$$\bigl\Vert \varOmega'_{n} ( u_{1,k} , \dots, u_{m,k}) (t_{2}) - \varOmega'_{n} ( u_{1,k} , \dots, u_{m,k}) (t_{1}) \bigr\Vert _{*} < \max_{i \in N_{m}} \frac{ 3 \varepsilon \Vert \mu_{i} \Vert _{1}}{ \varGamma_{q}( \alpha_{i})}. $$

This implies that \(\{ \varOmega'_{n} ( u_{1,k}, \dots, u_{m,k}) \}\) is equi-continuous on . At present, by using the Arzelà-Ascoli theorem, \(\{ \varOmega_{n} ( u_{1,k} , \dots, u_{m,k}) \}\) is relatively compact. Therefore \(\varOmega_{n}\) is completely continuous. □

Now, we are ready to provide our main results about the problem (1).

Theorem 6

The problem (11) under boundary conditions in (1) has a solution \(( u_{1,n} , \dots, u_{m,n}) \)belongs to \(\mathcal{P}\)such that \(u_{i,n}(t) \varGamma (\alpha_{i} + 1) \geq\ell_{i} t^{\alpha_{i}} (\alpha_{i} -1)\), for all \(t \in\overline{J}\)and \(i \in N_{m}\), whenever assumptions (A1) and (A2) hold.

Proof

The mapping \(H: P \to P\) is a contraction and the operator \(\varOmega_{n} : P \to P\) is completely continuous, by employing Lemma 4 and Lemma 5, respectively. Now, by applying Lemma 2, there exists \(( u_{1,n} , \dots, u_{m,n}) \in\mathcal{P}\) such that \(( u_{1,n} , \dots, u_{m,n}) = \varOmega_{n} ( u_{1,n} , \dots , u_{m,n}) + H ( u_{1,n} , \dots, u_{m,n})\). Therefore, \(u_{i,n}= T_{i,n} ( u_{1,n} , \dots, u_{m,n}) + H_{i} ( u_{1,n} , \dots, u_{m,n})\) for all \(i \in N_{m}\). Hence,

$$\begin{aligned} u_{i,n}(t) &= \int^{1}_{0} G_{\alpha_{i}}(t,qs) F_{i,n} \bigl(s , u_{1}(s) , \dots, u_{m}(s) , D^{\mu_{1}} u_{1}(s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr) \, d_{q}s \\ & \quad+ \int^{1}_{0} G_{\alpha_{i}}(t, qs) h_{i} \bigl(s , u_{1}(s) , \dots, u_{m}(s) , D_{q}^{\gamma_{1}} u_{1}(s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr) \, d_{q}s \end{aligned}$$

for all \(i \in N_{m}\). By applying the hypothesis, we obtain \(u_{i,n}(t) \varGamma(\alpha_{i} + 1) \geq\ell_{i} t^{\alpha_{i}} (\alpha_{i} -1)\) for all \(t \in\overline{J}\) and \(i \in N_{m} \). By simple review, we can see that the element \(( u_{1,n} , \dots, u_{m,n}) \in\mathcal{P}\) is a solution of the problem (11) under the boundary conditions in (1). □

Lemma 7

Let the element \((u_{1,n} , \dots, u_{m,n}) \)be a solution for the problem (11) under the boundary conditions in (1). Then \(\{ ( u_{1,n} , \dots, u_{m,n} ) \}_{n\geq1}\)is relatively compact in \(\mathcal{P}\)whenever assumptions (A1), (A2) and (A3) hold.

Proof

As we found in Theorem 6,

$$\begin{aligned} u_{i,n}(t) & = \int^{1}_{0} G_{\alpha_{i}}(t,qs) \\ & \quad\times F_{i,n} \bigl(s , u_{1,n}(s) , \dots, u_{m,n}(s) , D_{q}^{\gamma_{1}} u_{1,n} (s), \dots, D_{q}^{\gamma_{m}} u_{m,n}(s) \bigr) \, d_{q}s \\ & \quad + \int^{1}_{0} G_{\alpha_{i}}(t,qs) \\ & \quad\times h_{i} \bigl(s , u_{1,n}(s) , \dots, u_{m,n}(s) , D_{q}^{\gamma_{1}} u_{1,n} (s), \dots, D_{q}^{\gamma_{m}} u_{m,n} (s) \bigr) \, d_{q}s \end{aligned}$$

for all \(n \in\mathbb{N}\), \(t \in\overline{J}\) and \(i \in N_{m}\). Hence,

$$\frac{ m_{i} (1 - t^{\alpha{i}-1})}{\varGamma_{q}(\alpha_{i})} \leq\ell _{i} \int^{1}_{0} \frac{\partial}{\partial t} G_{\alpha_{i}}(t , qs)\, d_{q}s \leq u'_{i,n} (t), $$

for \(t \in\overline{J}\) and so,

$$\begin{aligned} D_{q}^{\gamma_{i}} u_{i,n} (t) & = \frac{1}{\varGamma_{q} (1- \gamma_{i})} \int ^{t}_{0} (t- qs)^{(-\gamma_{i})} u'_{i,n} (s)\, d_{q}s \\ & \geq\frac{ \ell_{i}}{ \varGamma_{q}(\alpha_{i}) \varGamma_{q}(1- \gamma_{i}) } \int^{t}_{0} ( t - qs)^{(-\gamma_{i})} ( 1 - qs)^{(\alpha_{i} - 1) } \, d_{q}s \\ & > \frac{ \ell_{i}}{\varGamma_{q}(\alpha_{i}) \varGamma_{q}(1- \gamma_{i})} \int ^{t}_{0} ( t - qs)^{(-\gamma_{i})} ( 1 - qs) \, d_{q}s. \end{aligned}$$

Thus,

$$\begin{aligned} D_{q}^{\gamma_{i}} u_{i,n} (t) & > \frac{ \ell_{i} t^{ 1 - \gamma_{i}}}{ \varGamma_{q}(\alpha_{i}) \varGamma_{q}(2 - \gamma_{i})} - \frac{ \ell_{i} t^{2 - \gamma_{i}}}{ \varGamma_{q}(\alpha_{i}) \varGamma_{q}(3 - \gamma_{i})} \\ & = \frac{ \ell_{i} t^{ 1 -\gamma_{i}}}{\varGamma_{q}(\alpha_{i}) } \biggl(\frac{ \varGamma_{q} (3- \gamma_{i}) - t \varGamma_{q}( 2 - \gamma_{i}) }{ \varGamma _{q}( 2 - \gamma_{i}) \varGamma_{q}( 3- \gamma_{i})} \biggr) \\ & = \frac{ \ell_{i} t^{1 -\gamma_{i}}}{ \varGamma_{q}(\alpha_{i}) } \biggl(\frac{ 2 - \gamma_{i} - t }{ \varGamma_{q}( 3 - \gamma_{i}) } \biggr) \\ & \geq\frac{ \ell_{i} t^{ 1 - \gamma_{i}} (1 -\gamma_{i})}{\varGamma _{q}(\alpha_{i}) \varGamma_{q}( 3 - \gamma_{i})} \end{aligned}$$

for \(t \in\overline{J}\). Since \(\varGamma_{q}(3- \gamma_{i}) \leq2\), we have \(2 \varGamma_{q}(\alpha_{i}) D_{q}^{\gamma_{i}} u_{i,n} (t) \geq \ell _{i} t^{1-\gamma_{i}}(1 - \gamma_{i})\). Now, put

$$L_{i} = \ell_{i} \min \biggl\{ \frac{1}{ \varGamma_{q}( \alpha_{i})} , \frac{ \alpha_{i} - 1}{ \varGamma_{q}( \alpha_{i} + 1)} \biggr\} . $$

Therefore, \(u_{i,n} (t) \geq L_{i} t^{\alpha_{i}}\) and \(2 D_{q}^{\gamma _{i}} u_{i,n} (t) \geq L_{i} (1-\gamma_{i}) t^{1-\gamma_{i}}\) for all \(n \geq 1\), \(t \in\overline{J}\) and \(i \in N_{m}\). Indeed,

$$\begin{aligned} &r_{i} \bigl( u_{1,n}(t) , \dots, u_{m,n}(t) , D_{q}^{\gamma_{1}} u_{1,n} (t), \dots, D_{q}^{\gamma_{m}} u_{m,n}(t) \bigr) \\ & \quad\leq r_{i} \biggl( L_{1} t^{\alpha_{1}} , \dots, L_{m} t^{\alpha _{m}}, \frac{ L_{1} (1 - \gamma_{1}) }{ 2} t^{ 1 - \gamma_{1} } ,\dots, \frac{ L_{m} (1 - \gamma_{m})}{2} t^{1 - \gamma_{m}} \biggr) \end{aligned}$$

for \(n \in\mathbb{N}\), \(t \in\overline{J}\) and \(i \in N_{m}\). Hence, we conclude that

$$\begin{aligned} 0 &\leq u'_{i,n}(t)\\&= \int^{1}_{0} \frac{ \partial}{ \partial t} G_{ \alpha _{i}}( t, qs) \\ & \quad\times F_{i,n} \bigl(s , u_{1,n}(s) , \dots, u_{m,n}(s) , D_{q}^{\gamma_{1}} u_{1,n}(s), \dots, D_{q}^{ \gamma_{m}} u_{m,n} (s) \bigr)\, d_{q}s \\ & \quad+ \int^{1}_{0} \frac{ \partial}{ \partial t} G_{\alpha_{i}}( t, qs) \\ & \quad\times h_{i} \bigl(s , u_{1,n}(s) , \dots, u_{m,n}(s) , D_{q}^{\gamma_{1}} u_{1,n}( s), \dots, D_{q}^{\gamma_{m}} u_{m,n}(s) \bigr) \, d_{q}s \\ & \leq\frac{1}{\varGamma_{q}( \alpha_{i} -1 )} \int^{1}_{0} r_{i} \biggl( K_{1} s^{ \alpha_{1}} , \dots, L_{m} s^{ \alpha_{m}}, \frac{ L_{1} (1 - \gamma_{1}) }{2} s^{1 - \gamma_{1}} ,\dots, \frac{ L_{m} (1 - \gamma_{m})}{ 2} s^{1 - \gamma_{m}} \biggr) \, d_{q}s \\ &\quad + \frac{1}{\varGamma_{q}( \alpha_{i} - 1 )} \int^{1}_{0} \gamma_{i} (s) \\ & \quad\times w_{i} \bigl( u_{1,n}(s) , \dots, u_{m,n}(s) , D_{q}^{\gamma_{1}} u_{1,n} (s), \dots, D_{q}^{\gamma_{m}} u_{m,n} (s) \bigr)\, d_{q}s \end{aligned}$$

for all \(t \in\overline{J}\), \(n\geq1\) and \(i \in N_{m}\). Also, for all \(i \in N_{m}\), we obtain

$$\begin{aligned} \varLambda_{i}= & \int^{1}_{0} r_{i} \biggl( L_{1} s^{\alpha_{1}} , \dots, L_{m} s^{\alpha_{m}}, \frac{ L_{1} (1 - \gamma_{1})}{2} s^{ 1 - \gamma_{1}}, \dots, \frac{ L_{m} ( 1 - \gamma_{m})}{2} s^{ 1 - \gamma_{m}} \biggr) \, d_{q}s < \infty. \end{aligned}$$

Assume that \(\lambda_{n} = \Vert ( u_{1,n} , \dots, u_{m,n} ) \Vert _{**}\). Then, for all i and n, we have \(\| u_{i,n} \| \) and \(\Vert u'_{i,n} \Vert \) smaller than or equal to \(\lambda_{n}\). Thus, \(\varGamma_{q}(2 - \gamma _{i}) |D_{q}^{\gamma_{i}} u_{i,n}(t)| \leq \lambda_{n} \) for each \(n \in\mathbb{N}\), \(t \in\overline{J}\) and \(i \in N_{m}\). Hence

$$\begin{aligned} 0& \leq u'_{i,n}(t)\\ & \leq \frac{1}{ \varGamma_{q}( \alpha_{i} -1 )} \biggl( \varLambda_{i} + w_{i} \biggl( 1 + \lambda_{n} , \dots, 1 + \lambda_{n} ,\\&\quad 1 + \frac{ \lambda_{n}}{ \varGamma_{q}(2 - \gamma_{1})} , \dots, 1 + \frac{ \lambda_{n}}{ \varGamma_{q}( 2 - \gamma_{m})} \biggr) \biggr) \Vert \mu_{i} \Vert _{1}\end{aligned} $$

and \(0 \leq u_{i,n}(t) = \int^{t}_{0} u'_{i,n}(s) \,ds \) for \(n \in\mathbb {N}\), \(t \in\overline{J}\) and \(i \in N_{m}\). By a similar method, we get

$$\begin{aligned}& \begin{aligned} 0 &\leq u_{i,n}(t) \\& \leq \frac{1}{ \varGamma_{q}( \alpha_{i} -1 )} \biggl( \varLambda_{i} + w_{i} \biggl( 1 + \lambda_{n} , \dots, 1 + \lambda_{n} , \\ & \quad1 + \frac{ \lambda_{n}}{ \varGamma_{q}(2- \gamma_{1})} , \dots, 1 + \frac{ \lambda_{n}}{\varGamma_{q}( 2 - \gamma_{m})} \biggr) \biggr) \Vert \mu _{i} \Vert _{1},\end{aligned} \\& \begin{aligned}\lambda_{n} & \leq \frac{1}{\varGamma_{q}( \alpha_{i} -1 )} \biggl( \varLambda _{i} + w_{i} \biggl( 1 + \lambda_{n} , \dots, 1 + \lambda_{n}, \\ & \quad1+ \frac{ \lambda_{n}}{ \varGamma_{q}(2- \gamma_{1})} , \dots, 1 + \frac{ \lambda_{n}}{ \varGamma_{q}(2 - \gamma_{m})} \biggr) \biggr) \Vert \mu _{i} \Vert _{1},\end{aligned} \end{aligned}$$

for all i. Since \(\lim_{x \to\infty} \frac{w_{i} (x, \dots, x)}{x} = 0\) for all \(i \in N_{m}\), there exists \(M_{i} > 0\) such that

$$\begin{aligned} &\frac{1}{\varGamma_{q}( \alpha_{i} -1 )} \biggl( \varLambda_{i} + w_{i} \biggl( 1 + \gamma_{i} , \dots, 1 + \nu_{i}, \\ & \quad1 + \frac{ \nu_{i}}{ \varGamma_{q}( 2 - \gamma_{1})} , \dots, 1 + \frac{ \nu_{i}}{ \varGamma_{q}( 2 - \gamma_{m})} \biggr) \biggr) \Vert \mu _{i} \Vert _{1} < \nu_{i}, \end{aligned}$$

for all \(\nu_{i}> M_{i}\). We take \(M = \max\{M_{1}, \dots, M_{m}\}\). Then we have

$$\begin{aligned} &\frac{1}{ \varGamma_{q}( \alpha_{i} -1 )} \biggl( \varLambda_{i} + w_{i} \biggl( 1 + \nu, \dots, 1 + \nu, \\ & \quad1 + \frac{ \nu}{ \varGamma_{q}( 2 - \gamma_{1})} , \dots, 1 + \frac { \nu}{ \varGamma_{q}( 2 - \gamma_{m})} \biggr) \biggr) \Vert \mu_{i} \Vert _{1} < \nu, \end{aligned}$$

for all \(\nu> M\). Thus,

$$\lambda_{n} = \bigl\Vert (u_{1,n} , \dots, u_{m,n}) \bigr\Vert _{**} = \max_{i \in N_{m}} \bigl\{ \Vert u_{i,n} \Vert , \bigl\Vert u'_{i,n} \bigr\Vert \bigr\} < M, $$

which implies \(\{ \Vert ( u_{1,n} , \dots, u_{m,n}) \Vert _{**} \} \) is bounded in \(\overline{ \mathcal{B}}_{m}\). Now, take

$$\begin{aligned} C_{i} & = w_{i} \biggl( 1+ M , \dots, 1 + M, 1 + \frac{M}{ \varGamma_{q}( 2 - \gamma_{1})} , \dots, 1+ \frac{M}{ \varGamma_{q}( 2 - \gamma_{m}) } \biggr) \end{aligned}$$

and

$$U_{i}(t) = r_{i} \biggl( L_{1} t^{\alpha_{1}} , \dots, L_{m} t^{\alpha _{m}}, \frac{ L_{1} ( 1 - \gamma_{1})}{2} t^{1 - \gamma_{1}} \dots, \frac{ L_{m} ( 1 - \gamma_{m})}{ 2 } t^{ 1 - \gamma_{m}} \biggr), $$

for all i and each \(t\in\overline{J}\). Then, we have \(\varLambda_{i} = \int^{1}_{0} U_{i} (t) \,dt\) and

$$\begin{aligned} & F_{i,n} \bigl( t , u_{1,n} (t) , \dots, u_{m,n}(t) , D_{q}^{\gamma_{1}} u_{1,n}(t), \dots, D_{q}^{\gamma_{m}} u_{m,n}(t) \bigr) \\ & \qquad+ h_{i} \bigl( t , u_{1,n}(t) , \dots, u_{m,n} (t) , D_{q}^{ \gamma_{1,n}} u_{1} (t), \dots, D_{q}^{\gamma_{m}} u_{m,n}(t) \bigr) \\ &\quad \leq U_{i}(t) + C_{i} \mu_{i}(t). \end{aligned}$$

If \(t_{1} , t_{2} \in\overline{J}\) such that \(t_{1} \leq t_{2}\), then we obtain

$$\begin{aligned} \bigl\vert u'_{i,n} (t_{2}) - u'_{i,n} (t_{1}) \bigr\vert & = \biggl\vert \int^{1}_{0} \biggl( \frac{ \partial}{\partial t} G_{ \alpha_{i}} (t_{2} , qs) - \frac { \partial}{ \partial t} G_{ \alpha_{i} } (t_{2} , qs) \biggr) \\ & \quad\times \bigl[ F_{i,n} \bigl( s , u_{1,n}(s) , \dots, u_{m,n} (s) , D_{q}^{\gamma_{1}} u_{1,n} (s), \dots, D_{q}^{\gamma_{m}} u_{m,n} (s) \bigr) \\ & \quad+ h_{i} \bigl(s , u_{1,n} (s) , \dots, u_{m,n} (s) , D_{q}^{\gamma _{1}} u_{1,n} (s), \dots, D_{q}^{\gamma_{ m}} u_{m,n} (s) \bigr) \bigr] \, d_{q}s \biggr\vert \\ & \leq\frac{1}{ \varGamma_{q}( \alpha_{i} -1 )} \biggl[ ( t_{2} - t_{1} ) \int^{1}_{0} U_{i} (s) + C_{i} \mu_{i} (s) \, d_{q}s \\ & \quad+ \int_{0}^{t_{1}} \bigl( (t_{2} - qs )^{( \alpha_{i} -2)} - (t_{1} - qs )^{(\alpha_{i} -2)} \bigr) \\ & \quad\times \bigl( U_{i}(s) + C_{i} \mu_{i} (s) \bigr) \, d_{q}s \\ & \quad+ \int_{t_{1}}^{t_{2}} (t_{2} - qs)^{(\alpha_{i} -2)} \bigl( U_{i} (s) + C_{i} \mu_{i} (s) \bigr) \, d_{q}s \biggr] \\ & \leq\frac{1}{\varGamma_{q}( \alpha_{i} -1 )} \biggl[ ( t_{2} - t_{1} ) \bigl( \varLambda_{i} + C_{i} \Vert \mu_{i} \Vert _{1} \bigr) \\ & \quad+ \int_{0}^{t_{1}} \bigl( (t_{2} - qs )^{(\alpha_{i} -2)} - ( t_{1} - qs)^{(\alpha_{i} -2)} \bigr) \\ & \quad\times \bigl( U_{i}(s) + C_{i} \mu_{i} (s) \bigr) \, d_{q}s \\ & \quad+ (t_{2} - t_{1})^{\alpha_{i} -2} \bigl( \varLambda_{i} + C_{i} \Vert \mu_{i} \Vert _{1} \bigr) \biggr]. \end{aligned}$$

Let \(\varepsilon_{i} > 0\) be given. Choose \(\delta( \varepsilon_{i} ) > 0\) such that

$$(t_{2} - qs )^{(\alpha_{i} -2)} - (t_{1} - qs)^{(\alpha_{i} -2)} < \varepsilon_{i}, $$

for all \(0 \leq t_{1} < t_{2} \leq1\) with \(t_{2} - t_{1} < \delta( \varepsilon_{i} )\) and \(s \in(0, t]\). Take

$$\delta< \min \bigl\{ \delta( \varepsilon_{1}), \dots, \delta( \varepsilon_{m} ), \sqrt[ \alpha_{1} - 2]{ \varepsilon_{1}}, \dots, \sqrt[ \alpha_{m} - 2]{ \varepsilon_{m}} \bigr\} , $$

then \(\varGamma_{q}( \alpha_{i} -1) \vert u'_{i,n} ( t_{2}) - u'_{i,n} (t_{1} ) \vert \leq 3 \varepsilon_{i} ( \varLambda_{i} + C_{i} \Vert \mu _{i} \Vert _{1} )\), for all \(i \in N_{m}\). Hence, \(\{( u_{1,n} , \dots, u_{m,n})' \}\) is equi-continuous. Indeed, \(\{(u_{1,n} , \dots, u_{m,n}) \}_{n \geq1} \subseteq\overline{\mathcal{B}}^{m}\) is relatively compact. □

Theorem 8

The system (1) has a solution \(( u_{1} , \ldots, u_{m}) \in \mathcal{P}\)such that \(2 D_{q}^{ \gamma_{i}} u_{i} (t) \geq L_{i} (1- \gamma_{i}) t^{1 - \gamma_{i}}\)and \(u_{i} (t) \geq L_{i} t^{\alpha _{i}}\)for all \(t \in\overline{J}\)and \(i \in N_{m}\)whenever the assumptions (A1), (A2) and (A3) hold.

Proof

As we found in Theorem 6, for each natural number n the system (11) under the boundary conditions in (1) has a solution \(( u_{1,n} , \dots, u_{m,n} ) \) in \(\mathcal{P}\). By applying Lemma 7, we have \(\{ ( u_{1,n}, \dots, u_{m,n}) \}_{n\geq1}\) is relatively compact in \(\overline{\mathcal{B}}^{m}\). Also, by employing the Arzelá–Ascoli theorem, \(( u_{1} , \dots, u_{m})\) exists such that \(\lim_{n \to\infty} ( u_{1,n} , \dots, u_{m,n}) = ( u_{1} , \dots, u_{m})\). It is obvious that \(( u_{1} , \dots, u_{m})\) satisfies the boundary conditions of the problem (1), \(D_{q}^{\gamma_{i}} u_{i,n} \to D_{q}^{\gamma_{i}} u_{i}\) and

$$\begin{gathered} \lim_{n \to\infty} F_{i,n} \bigl(t , u_{1,n}(t) , \dots, u_{m,n}(t) , D_{q}^{\gamma_{1}} u_{1,n}(t), \dots, D_{q}^{\gamma_{m}} u_{m,n}(t) \bigr) \\ \qquad{}+ h_{i} \bigl(t , u_{1,n}(t) , \dots, u_{m,n}(t) , D_{q}^{\gamma _{1}} u_{1,n}(t), \dots, D_{q}^{\gamma_{m} } u_{m,n} (t) \bigr) \\ \quad = g_{i} \bigl(t , u_{1}(t) , \dots, u_{m}(t) , D_{q}^{\mu_{1}} u_{1}(t), \dots, D_{q}^{\gamma_{m}} u_{m}(t) \bigr) \\ \qquad{}+ h_{i} \bigl(t , u_{1}(t) , \dots, u_{m}(t) , D_{q}^{\mu_{1}} u_{1}(t), \dots, D_{q}^{\gamma_{m}} u_{m}(t) \bigr)\end{gathered} $$

for each t belonging to and \(i \in N_{m}\). Thus \(( u_{1} , \dots, u_{m}) \in\mathcal{P}\). At present, suppose that \(K = \sup_{n\geq1} \Vert ( u_{1,n} , \dots, u_{m,n}) \Vert _{**}\). Then we have \(\Vert D_{q}^{\gamma_{i}} u_{i,n} \Vert \leq\frac{K}{ \varGamma _{q}(2 - \gamma_{i})}\) for all n and \(i \in N_{m}\). Hence,

$$\begin{aligned} 0 & \leq G_{\alpha_{i}} (t,qs) \bigl[ F_{i,n} \bigl(s, u_{1,n}(s), \dots , u_{m,n}(s) , D_{q}^{\gamma_{1}} u_{1,n}(s), \dots, D_{q}^{\gamma_{m}} u_{m,n} (s) \bigr) \\ & \quad+ h_{i} \bigl(s , u_{1,n}(s) , \dots, u_{m,n}(s), D_{q}^{\gamma _{1}} u_{1,n} (s), \dots, D_{q}^{\gamma_{m}} u_{m,n}(s) \bigr) \bigr] \\ & \leq\frac{1}{\varGamma_{q}( \alpha_{i} - 1)} \biggl[ U_{i}(s) \\ & \quad+ w_{i} \biggl( 1 + K , \dots, 1+ K , 1+ \frac{K}{ \varGamma_{q}(2 - \gamma_{i})} , \dots, 1 + \frac{K}{ \varGamma_{q}(2 - \gamma_{i})} \biggr) \mu_{i} (s) \biggr] \end{aligned}$$

for almost all \((t,qs) \in\overline{J}^{2} \), \(n \geq1\) and \(i \in N_{m}\). At present, the dominated theorem of Lebesgue implies that

$$\begin{aligned} u_{i} (t) & = \int^{1}_{0} G_{\alpha_{i}} (t,qs) g_{i} \bigl(s , u_{1}(s) , \dots, u_{m}(s) , D_{q}^{\gamma_{1}} u_{1}(s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr)\, d_{q}s \\ & \quad+ \int^{1}_{0} G_{\alpha_{i}} (t,qs) g_{i} \bigl(s , u_{1}(s) , \dots , u_{m}(s) , D_{q}^{\gamma_{1}} x_{1}(s), \dots, D_{q}^{\gamma_{m}} u_{m}(s) \bigr) \, d_{q}s \end{aligned}$$

for all \(i \in N_{m}\) and \(t \in\overline{J}\). This completes the proof. □

Example and numerical check technique for the problems

In this part, we give complete computational techniques for illustrating of the problem (1), in Theorems 8, such that it covers all the problems, and present numerical examples which solve the problems perfectly. Foremost, we present a simplified analysis that can be executed to calculate the value of q-Gamma function, \(\varGamma_{q} (x)\), for input values q and x by counting the number of sentences n in summation. To this aim, we consider a pseudo-code description of the method for the calculated q-Gamma function of order n in Algorithm 2 (for more details, see the following link: https://en.wikipedia.org/wiki/Q-gamma_function). Table 1 shows that when q is constant, the q-Gamma function is an increasing function. Also, for smaller values of x, an approximate result is obtained with smaller values of n. It is shown by underlined rows. Table 2 shows that the q-Gamma function for values q close to 1 is obtained with higher values of n in comparison with other columns. They have been underlined in line 8 of the first column, line 17 of the second column and line 29 of third columns of Table 2. Also, Table 3 is the same as Table 2, but x values increase in 3. Similarly, the q-Gamma function for values q near to one is obtained with more values of n in comparison with other columns. Furthermore, we provide Algorithm 3, which calculates \((D_{q}^{\alpha}f) (x)\).

Table 1 Some numerical results for calculation of \(\varGamma_{q}(x)\) with \(q=\frac{1}{3}\) that is constant, \(x=4.5, 8.4, 12.7\) and \(n=1, 2, \ldots, 15\) of Algorithm 2
Table 2 Some numerical results for calculation of \(\varGamma_{q}(x)\) with \(q=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}\), \(x=5\) and \(n=1, 2, \ldots, 35\) of Algorithm 2
Table 3 Some numerical results for calculation of \(\varGamma_{q}(x)\) with \(x=8.4\), \(q=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}\) and \(n=1, 2, \ldots, 40\) of Algorithm 2

Here, we provide an example to illustrate the results of Theorem 8.

Example 1

Consider the system as (1) with \(m= 2\):

$$ \left\{ \textstyle\begin{array}{l} D_{q}^{\frac{5}{2}} u_{1} + \frac{1}{ \sqrt[3]{t^{2}}} ( 2 + c_{1} u_{1}+ c_{2} u_{2} + c_{3} D_{q}^{ \frac{1}{3}} u_{1} + c_{4} D_{q}^{ \frac{1}{2}} u_{2} )\\ \quad{}+ ( 0.1 e^{ \frac{1}{1 + u_{1}}} + 0.2 e^{\frac{1}{ 1 + u_{2}}} + 0.1 e^{ \frac{1}{1 + D_{q}^{1}{3} u_{1}}} + 0.2 e^{ \frac{1}{ 1 + D_{q}^{1}{2} u_{2}}} ) = 0,\\ D_{q}^{ \frac{7}{3} } u_{2} + \frac{1}{ \sqrt{t} } ( 1 + d_{1} u_{1}+ d_{2} u_{2} + d_{3} D_{q}^{\frac{1}{3}} u_{1} + d_{4} D_{q}^{\frac{1}{2}} u_{2} ) \\ \quad{}+ ( 0.2 e^{ \frac{1}{ 1 + u_{1}}} + 0.2 e^{ \frac{1}{ 1 + u_{2}}} + 0.3 e^{ \frac{1}{ 1 + D_{q}^{1}{3} u_{1}}} + 0.1 e^{ \frac{1}{ 1 + D_{q}^{ 1}{2} u_{2}}} ) =0, \end{array}\displaystyle \right . $$
(15)

under boundary conditions \(u_{1}(0) = u_{2}(0)= 0\), \(u'_{1}(1) = u'_{2}(1)= 0\) and \(u''_{1}(0) = u''_{2}(0)= 0\), where \(c_{i}, d_{i}\) are positive constants, for \(i=1,2,3,4\). Note that \(\mathcal{S}=(0, \infty)^{4}\). We take the functions

$$\begin{aligned}& g_{1} ( t, u_{1}, u_{2}, u_{3}, u_{4} ) = \frac{1}{ \sqrt[3]{t^{2}} } ( 2 + c_{1} u_{1}+ c_{2} u_{2}+ c_{3} u_{3} + c_{4} u_{4} ), \\& g_{2} (t, u_{1}, u_{2}, u_{3}, u_{4} ) = \frac{1}{\sqrt{t}} ( 1 + d_{1} u_{1} + d_{2} u_{2} + d_{3} u_{3} + d_{4} u_{4} ), \\& \begin{aligned}h_{1} (t, u_{1}, u_{2}, u_{3}, u_{4} ) & = r_{1} ( u_{1}, u_{2}, u_{3}, u_{4}) \\ & = 0.1 e^{ \frac{1}{1 + u_{1}}} + 0.2 e^{ \frac{1}{ 1 + u_{2}}} + 0.1 e^{\frac{1}{ 1 + u_{3}}} + 0.2 e^{ \frac{1}{1 + u_{4}} },\end{aligned} \\& \begin{aligned}h_{2} ( t, u_{1}, u_{2}, u_{3}, u_{4} ) & =r_{2} (u_{1}, u_{2}, u_{3}, u_{4}) \\ & = 0.2 e^{\frac{1}{ 1 + u_{1}}} + 0.2 e^{ \frac{1}{ 1 +u_{2}}} + 0.3 e^{ \frac{1}{ 1 +u_{3}}} + 0.1 e^{ \frac{1}{ 1 +u_{4}}},\end{aligned} \\& w_{1} (u_{1}, u_{2}, u_{3}, u_{4} ) = 2+ c_{1} u_{1}+ c_{2} u_{2} + c_{3} u_{3} + c_{4} u_{4}, \\& w_{2} ( u_{1}, u_{2}, u_{3}, u_{4} ) = 1 + d_{1} u_{1}+ d_{2} u_{2}+ d_{3} u_{3} + d_{4} u_{4}, \end{aligned}$$

\(\lambda_{1}(t) = \frac{1}{ \sqrt[3]{t^{2}}}\) and \(\lambda_{2}(t)= \frac {1}{\sqrt{t}}\). Put \(m= 2\), \(\alpha_{1} = \frac{5}{2}\), \(\alpha_{2} = \frac{7}{3}\), \(\gamma_{1} = \frac{1}{2}\), \(\gamma_{2} = \frac{1}{3}\),

$${}_{i}M_{j} = \left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0.1 & 0.2 & 0.1 & 0.2\\ 0.2 & 0.2 & 0.3 & 0.1 \end{array}\displaystyle \right ], $$

\(\ell_{1} = 2\) and \(\ell_{2}= 1\). By simple review, we can see that \(g_{1}\) and \(g_{2}\) are Carathéodory functions, \(g_{1} (t, u_{1}, u_{2}, u_{3}, u_{4}) \geq2\), \(g_{2}(t, u_{1}, u_{2}, u_{3}, u_{4}) \geq1\) for all \(( u_{1}, u_{2}, u_{3}, u_{4}) \in\mathcal{S}\) and each \(t \in \overline{J}\), \(h_{1}\) and \(h_{2}\) are nonnegative and \(h_{1}\), \(h_{2}\) satisfy inequality (6):

$$\begin{aligned}& \bigl\vert h_{1}( t, u_{1}, u_{2}, u_{3}, u_{4}) - h_{1}(t, v_{1}, v_{2}, v_{3}, v_{4}) \bigr\vert \leq\sum _{i=1}^{4} {}_{1}M_{i} \vert u_{i} - v_{i} \vert , \\& \bigl\vert h_{2}(t, u_{1}, u_{2}, u_{3}, u_{4}) - h_{2} (t, v_{1}, v_{2}, v_{3}, v_{4}) \bigr\vert \leq\sum _{i=1}^{4} {}_{2}M_{i} \vert u_{i} - v_{i} \vert , \end{aligned}$$

for \((u_{1}, u_{2}, u_{3}, u_{4}), (v_{1}, v_{2}, v_{3}, v_{4}) \in(0,\infty)^{4}\) and \(t \in\overline{J}\). Also, by putting values in the problem in (7), we have

$$\begin{aligned}& \begin{aligned}[b] \eta_{1}&= \sum_{ k=1}^{2} {}_{1}M_{k} + \frac{ {}_{1}M_{2+k} }{ \varGamma_{q}( 2- \gamma_{1}) } = 0.1 + 0.2 + \frac{0.1}{ \varGamma_{q}(\frac{5}{3})} + \frac{0.2}{ \varGamma_{q}(\frac{5}{3})} \\ & < \varGamma_{q} \biggl( \frac{3}{2} \biggr)= \varGamma_{q}(\alpha_{1} - 1),\end{aligned} \end{aligned}$$
(16)
$$\begin{aligned}& \begin{aligned}[b] \eta_{2}& = \sum_{k=1}^{2} {}_{2}M_{k} + \frac{ {}_{2}M_{2+k} }{ \varGamma _{q}(2- \gamma_{2})} = 0.2 + 0.2 + \frac{0.3}{ \varGamma_{q} (\frac {3}{2} ) } + \frac{0.1}{ \varGamma_{q} (\frac{3}{2})} \\ & < \varGamma_{q} \biggl( \frac{4}{3} \biggr)= \varGamma_{q}(\alpha_{2} - 1).\end{aligned} \end{aligned}$$
(17)

Tables 4 and 5 show the values of inequalities (16) and (17), respectively. On the other hand, the maps \(r_{1}\) and \(r_{2}\) are nonincreasing with respect to all components. If

$$\begin{aligned}& L_{1} = \ell_{1} \frac{ \alpha_{1} - 1}{ \varGamma_{q} ( \alpha_{1} + 1)} = 2 \times \frac{ \frac{3}{2}}{ \varGamma_{q}( \frac{7}{2})} =\frac{3}{\varGamma_{q} ( \frac{7}{2})}, \\& L_{2} = \ell_{2} \frac{\alpha_{2} - 1}{\varGamma_{q} (\alpha_{2} + 1)} = 1 \times \frac{ \frac{4}{3}}{ \varGamma_{q} (\frac {10}{3})} = \frac{4}{ 3 \varGamma_{q} ( \frac{10}{3}) }, \end{aligned}$$

then

$$\begin{aligned}& \int_{0}^{1} r_{1} \biggl( L_{1} t^{ \alpha_{1}} , L_{2} t^{\alpha _{2}}, \frac{ L_{1} ( 1 -\gamma_{1})}{ 2} t^{1 - \gamma_{1}}, \frac{ L_{2} ( 1 -\gamma_{2})}{2} t^{ 1 - \gamma_{2}} \biggr) \, dt < \infty, \\& \int_{0}^{1} r_{2} \biggl( L_{1} t^{\alpha_{1}} , L_{2} t^{\alpha _{2}}, \frac{ L_{1} ( 1 -\gamma_{1})}{2} t^{ 1 - \gamma_{1}}, \frac{ L_{2} (1 - \gamma_{2})}{2} t^{1 - \gamma_{2}} \biggr) \, dt < \infty. \end{aligned}$$

Also, the functions \(w_{1}\) and \(w_{2}\) are nondecreasing with respect to all components and

$$\begin{aligned} \lim_{x \to\infty} \frac{w_{1} (x, \ldots, x)}{x} & = \lim_{x \to \infty} \frac{2+ c_{1} x + c_{2} x + c_{3} x + c_{4} x}{x} = 0, \\ \lim_{x \to\infty} \frac{w_{2} (x, \ldots, x)}{x} & =\lim_{x \to \infty} \frac{1+ d_{1} x+ d_{2} x+ d_{3} x + d_{4} x}{x} = 0. \end{aligned}$$

Therefore, Theorem 8 implies that the problem (15) has a positive solution.

Table 4 Some numerical results of \(\eta_{1}\) and \(\varGamma_{q}(\alpha_{1} - 1)\) from inequality (16) in Example 1 for \(q \in \{ \frac{1}{8}, \frac{1}{2}, \frac{8}{9} \}\). One can check that \(\frac{ \eta_{1}}{\varGamma_{q}(\alpha_{1} - 1)}\) by approximation is smaller than 1
Table 5 Some numerical results of \(\eta_{2}\) and \(\varGamma_{q}(\alpha_{2} - 1)\) from (17) in Example 1 for \(q \in \{ \frac {1}{8}, \frac{1}{2}, \frac{8}{9} \}\). One can heck that \(\frac {\eta_{2}}{\varGamma_{q}(\alpha_{2} - 1)}\) by approximation is smaller than 1

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Samei, M.E. Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus. Adv Differ Equ 2020, 23 (2020). https://doi.org/10.1186/s13662-019-2480-y

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MSC

  • 34A08
  • 39A12
  • 34B16

Keywords

  • Existence of solutions
  • Caputo q-derivative
  • Singularity
  • Fractional q-differential equations