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An increasing variables singular system of fractional q-differential equations via numerical calculations
Advances in Difference Equations volume 2020, Article number: 452 (2020)
Abstract
We investigate the existence of solutions for an increasing variables singular m-dimensional system of fractional q-differential equations on a time scale. In this singular system, the first equation has two variables and the number of variables increases permanently. By using some fixed point results, we study the singular system under some different conditions. Also, we provide two examples involving practical algorithms, numerical tables, and some figures to illustrate our main results.
1 Introduction
The subject of q-difference equations was introduced by Jackson in the first decade of the last century [1]. The fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. It is known that working on quantum calculus is equivalent to traditional infinitesimal calculus without the notion of limits. In last decades, some researchers studied q-fractional difference equations [2–5]. Later, q-fractional boundary value problems have been considered by many researchers (see, for example, [6–13]). Nowadays many researchers focus on applications of fractional calculus [14–25] or analytical studies [26–36].
In 2013, Baleanu et al. investigated the coupled system of multi-term singular fractional integro-differential boundary value problem
via boundary conditions \(k^{(i)} (0) =l^{(i)} (0)= 0 \) for \(0 \leq i \leq n-2\), \(\mathcal{D}^{\delta _{1}}_{0^{+}} [k](1)=0\) for \(2 < \delta _{1} < n-1\), \(\sigma _{1} - \delta _{1} \geq 1\), and \(\mathcal{D}^{\delta _{2}}_{0^{+}} [l](1)=0\) for \(2 < \delta _{2} < n-1\), \(\sigma _{2} - \delta _{2} \geq 1\), where \(n \geq 4\), \(n-1 < \sigma _{i}< n\), \(0 < \alpha _{i}< 1\), \(1< \beta _{ij} < 2\) for \(i=1,2\) and \(j = 1, 2, \ldots , m\), \(\gamma _{ij}\) is positive-valued continuous functions on \([0,1]\times [0,1]\) (\(i, j=1,2\)), \(\psi {ij}[k](t)= \int _{0}^{t} \gamma _{ij} (t, r)k(r) \,\mathrm{d}r\), \(w_{1}\), \(w_{2}\) satisfy the local Caratheodory condition on \([0,1]\times D(w_{1}, w_{2} \in \operatorname{Car} ([0,1] \times D))\), where \(D \subset \mathbb{R}^{m+5}\) and \(w_{i}\) may be singular at the value zero of all its variables [37]. In 2016, Taieb et al. reviewed the fractional coupled system of nonlinear differential equations
with boundary conditions \(k(0)=k_{0}^{*}\), \(l(0)=l_{0}^{*}\), \(k'(0)= k''(0)=l'(0)=l''(0)=0\), \(k'''(0) = \mathcal{J}^{\alpha _{1}}[k](a_{1})\), and \(l'''(0) = \mathcal{J}^{\alpha _{2}}[k](a_{2})\), where \(t \in [0,1]\), \(m \in \mathbb{N}^{*}\), \(\alpha _{j} >0\), \(\sigma _{j} \in (3, 4)\), \(a_{j} \in (0,1)\), \(\mathcal{D}^{\sigma _{j}}\), \(\mathcal{D}^{\beta _{j}}\) are the Caputo derivatives and \(\mathcal{J}^{\alpha _{j}}\) are the Riemann–Liouville fractional integrals [38]. In 2017, El Abidine studied the coupled system of nonlinear fractional equations
with boundary conditions \(k(0) = k^{(j)}(0)=0 \) and \(l(0) = l^{(j)}( 0)=0 \) for \(1 \leq j\leq m-2\) with \(m \geq 2\), where \(t \in \mathbb{R}^{+} = (0, \infty )\), \(m-1 < \sigma _{i} \leq m\), \(\beta _{i}\in (0,3)\) for \(i=1,2\), \(0 < \beta _{1}\leq \sigma _{2}-1\), \(0 < \beta _{2} \leq \sigma _{1}-1\), the differential operator is in the Riemann–Liouville sense and \(w_{i}\) are Borel measurable functions in \({\mathbb{R}^{+}}^{3}\) satisfying some conditions [39].
By using the main idea of the above works, we investigate the increasing variables m-dimensional singular system of fractional q-differential equations
with boundary conditions \(k_{1} (0) = {}_{1}b_{0}\), \(k_{i}^{(j)} (0) = {}_{i}b_{j}\) for \(j=0,1, \ldots i-2\) and \(2\leq i \leq m\), \({}^{c}\mathcal{D}_{q}^{\zeta _{i-1}} k_{i} (1) = 0\) for \(\zeta _{i-1} \in [i-2, i-1]\) and \(2\leq i \leq m\), where \(t\in J:=(0,1]\), \(m \geq 2\), \(\sigma _{i} \in (i-1, i)\) for \(1\leq i\leq m\), \({}^{c}\mathcal{D}_{q}^{\sigma _{i}}\) denotes the Caputo fractional q-derivative of order \(\sigma _{i}\), \(w_{i} : J \times \mathbb{R}^{i} \to \mathbb{R}\) are continuous, \(w_{i}(t, k_{1}, k_{2},\ldots , k_{i})\) may be singular at \(t =0\) of its space variables, \(\lim_{t \to 0^{+}} w_{i} (t, k_{1}, k_{2},\ldots , k_{i})= \infty \), and there exists \(0 < \alpha _{1},\dots , \alpha _{m} <1 \) such that \(t^{\alpha _{1}} w_{1},\dots ,t^{\alpha _{m}} w_{m}\) are continuous on \(\overline{J} :=[0,1]\).
2 Essential preliminaries
Throughout this article, we apply the time scales calculus notation [40]. In fact, we consider the fractional q-calculus on the time scale \(\mathbb{T}_{t_{0}} = \{0 \} \cup \{ t: t=t_{0}q^{n} \}\), where \(n\geq 0\), \(t_{0} \in \mathbb{R}\), and \(q \in (0,1)\). Let \(a \in \mathbb{R}\). Define \([a]_{q}=\frac{1-q^{a}}{1-q}\) [1]. The power function \((x-y)_{q}^{(n)}\) with \(n \in \mathbb{N}_{0} \) is defined by \((x-y)_{q}^{(n)}= \prod_{k=0}^{n-1} (x - yq^{k})\) for \(n \geq 1\) and \((x-y)_{q}^{(0)}=1\), where x and y are real numbers and \(\mathbb{N}_{0} := \{ 0\} \cup \mathbb{N}\) [1, 2]. Also, \((x-y)_{q}^{(\alpha )}= x^{\alpha }\prod_{k=0}^{\infty }(x-yq^{k}) / (x - yq^{\alpha + k})\) for \(\alpha \in \mathbb{R}\) and \(q \neq 0\). If \(y=0\), then it is clear that \(x^{(\alpha )}= x^{\alpha }\) [6] (see Algorithm 1). The q-gamma function is given by \(\varGamma _{q}(z) = (1-q)^{(z-1)} / (1-q)^{z -1}\), where \(z \in \mathbb{R} \backslash \{0, -1, -2, \ldots \}\) [1]. Note that \(\varGamma _{q} (z+1) = [z]_{q} \varGamma _{q} (z)\). Algorithm 2 shows a pseudo-code description of the technique for estimating q-gamma function of order n. The q-derivative of function f is defined by \((\mathcal{D}_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}\) and \((\mathcal{D}_{q} f)(0) = \lim_{x \to 0} (\mathcal{D}_{q} f)(x)\), which is shown in Algorithm 3 [2, 3]. Furthermore, the higher order q-derivative of a function f is defined by \((D_{q}^{n} f)(x) = D_{q}(D_{q}^{n-1} f)(x)\) for \(n \geq 1\), where \((D_{q}^{0} f)(x) = f(x)\) [2, 3]. The q-integral of a function f is defined on \([0,b]\) by \(I_{q} f(x) = \int _{0}^{x} f(s) \,\mathrm{d}_{q} s = x(1- q) \sum_{k=0}^{\infty } q^{k} f(x q^{k})\) for \(0 \leq x \leq b\), provided the series absolutely converges [2, 3]. If x in \([0, T]\), then
whenever the series exists. In addition, we can interchange the order of double q-integral by \(\int _{0}^{t} \int _{0}^{s} h(r) \,\mathrm{d}_{q} r \,\mathrm{d}_{q} s= \int _{0}^{t} \int _{qr}^{t} h(r) \,\mathrm{d}_{q} s \,\mathrm{d}_{q} r\) [41]. Actually, the interchange of order is true since
In addition the left-hand side can be written as follows:
The operator \(I_{q}^{n}\) is given by \((I_{q}^{0} h)(x) = h(x) \) and \((I_{q}^{n} h)(x) = (I_{q} (I_{q}^{n-1} h)) (x)\) for all \(n \geq 1\) and \(h \in C([0,T])\) [2, 3]. It has been proved that \((D_{q} (I_{q} h))(x) = h(x) \) and \((I_{q} (D_{q} h))(x) = h(x) - h(0)\) whenever h is continuous at \(x =0\) [2, 3]. The fractional Riemann–Liouville type q-integral of the function h on \(J=(0,1)\) for \(\sigma \geq 0\) is defined by \(\mathcal{I}_{q}^{0} [h](t) = h(t) \) and
for \(t \in J\) [42]. Also, the Caputo fractional q-derivative of a function h is defined by
where \(t \in J\) and \(\sigma >0\) [42]. It has been proved that \(\mathcal{I}_{q}^{\beta }(\mathcal{I}_{q}^{\alpha } [h]) (x) = \mathcal{I}_{q}^{\alpha + \beta } [h] (x)\) and \(\mathcal{D}_{q}^{\alpha } [\mathcal{I}_{q}^{\alpha } [h]](x)= h(x)\), where \(\alpha , \beta \geq 0\) [42]. Algorithm 5 shows MATLAB lines for \(\mathcal{I}_{q}^{\alpha }[h](x)\).
Let \((\mathcal{E}, \rho )\) be a metric space. Denote by \(\mathcal{P}( \mathcal{E})\) and \(2^{\mathcal{E}}\) the class of all subsets and the class of all nonempty subsets of \(\mathcal{E}\), respectively. Thus, \(\mathcal{P}_{cl}( \mathcal{E})\), \(\mathcal{P}_{bd}( \mathcal{E})\), \(\mathcal{P}_{cv}( \mathcal{E})\), and \(\mathcal{P}_{cp}( \mathcal{E})\) denote the class of all closed, bounded, convex, and compact subsets of \(\mathcal{E}\), respectively. For each i, consider the space \(E_{i} = \{ k_{i}(t) : k_{i}(t)\in \mathcal{A} \} \) endowed with the norm \(\|k_{i}\|_{\infty }= \max_{t\in \overline{J}} |k_{i}(t)|\), where \(\mathcal{A}= C(\overline{J}, \mathbb{R})\). Also, define the product space \(\mathcal{E} = E_{1} \times \cdots \times E_{m}\) endowed with the norm \(\|( k_{1}, \ldots , k_{m})\| = \max_{1 \leq i \leq m } \|k_{i}\|_{\infty }\). Then \((\mathcal{E}, \|. \|)\) is a Banach space [43]. Similar to the idea of the works [44, 45], define the set of the selections of \(\mathcal{S}\) at k by
for all \(t\in \overline{J}\) and \(k=(k_{1}, \dots ,k_{m}) \in \mathcal{E}\). One can check that \(S \neq \emptyset \) for all \(k\in \mathcal{E}\) whenever \(\dim \mathcal{E} < \infty \) [46]. We need next results.
Lemma 1
The general solution of the q-fractional equation \({}^{c}\mathcal{D}_{q}^{\sigma } [k](t) =0\)is given by \(k(t) = d_{0} + d_{1} t + d_{2} t^{2} + \cdots + d_{m-1} t^{m-1}\)for \(\sigma >0\), where \(d_{i} \in \mathbb{R}\)for \(i=0, 1, \ldots , m-1\)and \(m = [\sigma ] + 1\).
Theorem 2
([43], Schauder’s fixed point)
Assume that \((\mathcal{E}, \rho ) \)is a complete metric space, S is a closed convex subset of \(\mathcal{E}\), and \(\mathcal{N}: \mathcal{E} \to \mathcal{E}\)is a map such that the set \(K=\{ \mathcal{N}(k) : k \in S\}\)is relatively compact in \(\mathcal{E}\). Then \(\mathcal{N}\)has at least one fixed point.
3 Main results
Now, we are ready to provide our results about the m-dimensional system of singular fractional q-differential equations. First, we prove next basic result to give the integral representation of problem (1).
Lemma 3
Let \(m \geq 2\)for \(i \in \{ 1,2, \ldots , m\}\), \(\sigma _{i}\in (i-1, i)\), \(\varrho _{1}, \dots , \varrho _{m}\in \mathcal{A}\), and \(t\in J\). Then the m-dimensional system
under the conditions
has a unique solution \(k=(k_{1}, k_{2}, \ldots , k_{m})\), where
Proof
By using Lemma 1, we obtain the fractional q-integral equation
for \(1 \leq i \leq m\). Let
By using the assumptions, we find \(k_{1} (0) = -{}_{1}d_{0} = {}_{1}b_{0}\), \(k_{i}^{(j)}(0) = - j! {}_{i}d_{j}= {}_{i}b_{j}\) for \(j=0, 1, 2, \ldots , i-2\) and
for \(i-2 < \zeta _{i-1} < i-1\), where \(2 \leq i \leq m\). Thus, \({}_{1}d_{0}= - {}_{1}b_{0}\) and
for \(2 \leq i \leq m\). By substituting these constants and (7) in (6), we find (5). □
Now, define the nonlinear operator \(\mathcal{N}: S \to S\) by
where
for \(t \in \overline{J}\).
Lemma 4
Let \(m \geq 2\), \(\sigma _{1} \in (0, 1)\), \(\sigma _{1} > \alpha _{1}\), \(\sigma _{i} \in (i-1, i)\)for \(i=2, \ldots , m\), \(\alpha _{i} \in (0,1)\)for \(i=1,2, \ldots , m\), \(f_{i} : J \to \mathbb{R}\)be a function with \(\lim_{t \to 0^{+}} f_{i}(t)= \infty \), and the maps \(t^{\alpha _{i}} f_{i} (t)\)be continuous on JÌ…. Then the maps
are continuous on JÌ….
Proof
By using the definition of the maps \(k_{i}(t)\), we have
and by the continuity of the maps \(t^{\alpha _{i}}f_{i}(t)\), we get \(k_{i}(0) = {}_{i}b_{0}\) for \(i=1,2,\ldots , m\). Now, we consider some cases.
- (I):
-
Let \(t_{0}=0\) and \(t \in J\). Since \(t^{\alpha _{i}}f_{i}(t)\) is continuous, there exist \(M_{1}, \ldots , M_{n}>0\) such that \(|t^{\alpha _{i}}f_{i} (t)| \leq M_{i}\) for all \(t \in \overline{J}\). Thus,
$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(0) \bigr\vert \\ &\quad = \textstyle\begin{cases} \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert , & i=1, \\ \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} t^{j} - \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} t^{i-1} & \\ \quad {}\times \int _{0}^{1} (t-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert ,& 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & i=1, \\ \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=1}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } t^{j} + \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i}}{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} t^{i-1} & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r.& 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$Hence, by using the q-beta function, we get
$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(0) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i} t^{\sigma _{i} - \alpha _{i} }}{ \varGamma _{q}(\sigma _{i})} \int _{0}^{1} (1-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r, & i=1, \\ \frac{M_{i} t^{\sigma _{i} - \alpha _{i} }}{\varGamma _{q}(\sigma _{i})} \int _{0}^{1} (1-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}+ \sum_{j=1}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } t^{j} & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1})} t^{i-1}, & 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i}B_{q}(\sigma _{i}, 1- \alpha _{i}) t^{\sigma _{i} - \alpha _{i} } }{ \varGamma _{q}(\sigma _{i})}, & i=1, \\ \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) t^{\sigma _{i} - \alpha _{i} }}{\varGamma _{q}(\sigma _{i})} + \sum_{j=1}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } t^{j} & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } t^{i-1}, & 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$which, by assumption \(\sigma _{1} >\alpha _{1}\) and the fact \(\sigma _{i}> \alpha _{i}\), tend to zero as \(t\to 0\) for \(i=1,2,\ldots , m\).
- (II):
-
Let \(t_{0} \in (0,1)\) and \(t \in (t_{0}, 1]\). Then we have
$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(t_{0}) \bigr\vert \\ &\quad = \textstyle\begin{cases} \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}- \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t_{0}} (t_{0} - qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert ,& i=1, \\ \vert \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t} (t-qr)^{( \sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r & \\ \quad {}- \frac{1}{\varGamma _{q}(\sigma _{i})} \int _{0}^{t_{0}} (t_{0}-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{{}_{i}b_{j}}{j!} ( t^{j} -t_{0}^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) }{(i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} ( t^{i-1} - t_{0}^{i-1} ) & \\ \quad {}\times \vert \int _{0}^{1} (1-qr)^{(\sigma _{i}- \zeta _{i-1}-1)} r^{-\alpha _{i}} r^{\alpha _{i}} f_{i}(r) \,\mathrm{d}_{q}r \vert ,& 2 \leq i \leq m, \end{cases}\displaystyle \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} [ \int _{0}^{t} (t-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r &\\ \quad {}- \int _{0}^{t_{0}} (t_{0} - qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r ],& i=1, \\ \frac{M_{i}}{\varGamma _{q}(\sigma _{i})} [ \int _{0}^{t} (t-qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r & \\ \quad {}- \int _{0}^{t_{0}} (t_{0} - qr)^{(\sigma _{i}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r ]& \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } ( t^{j} - t_{0}^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i}}{ (i-1)! \varGamma _{q}(\sigma _{i}-\zeta _{i-1})} ( t^{i-1}- t_{0}^{i-1} ) & \\ \quad {}\times \int _{0}^{1} (1-qr)^{(\sigma _{i} - \zeta _{i-1}-1)} r^{-\alpha _{i}} \,\mathrm{d}_{q}r. & 2 \leq i \leq m. \end{cases}\displaystyle \end{aligned}$$Hence,
$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(t_{0}) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{ \varGamma _{q}(\sigma _{i})} ( t^{\sigma _{i} - \alpha _{i} } - t_{0}^{\sigma _{i} - \alpha _{i}} ), & i=1, \\ \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{\varGamma _{q}(\sigma _{i})} ( t^{\sigma _{i} - \alpha _{i} } - t_{0}^{\sigma _{i} - \alpha _{i}} ) & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } ( t^{j} - t_{0}^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1}) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } & \\ \quad {}\times ( t^{i-1} - t_{0}^{i-1} ), & 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$which similar to case I tends to zero as \(t\to 0\) for \(i=1,2,\ldots , m\).
- (III):
-
Let \(t_{0}=1\) and \(t \in [0, t_{0})\). By using similar arguments as in the previous case, one can obtain
$$\begin{aligned} & \bigl\vert k_{i}(t) - k_{i}(t_{0}) \bigr\vert \\ &\quad \leq \textstyle\begin{cases} \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{ \varGamma _{q}(\sigma _{i})} ( t_{0}^{\sigma _{i} - \alpha _{i} } - t^{\sigma _{i} - \alpha _{i}} ), & i=1, \\ \frac{M_{i} B_{q}(\sigma _{i}, 1- \alpha _{i}) }{\varGamma _{q}(\sigma _{i})} ( t_{0}^{\sigma _{i} - \alpha _{i} } - t^{\sigma _{i} - \alpha _{i}} ) & \\ \quad {}+ \sum_{j=0}^{i-2} \frac{ \vert {}_{i}b_{j} \vert }{ j! } ( t_{0}^{j} - t^{j} ) & \\ \quad {}+ \frac{\varGamma _{q} (i - \zeta _{i-1} ) M_{i} B_{q}(\sigma _{i} - \zeta _{i-1} , 1- \alpha _{i}) }{(i-1)! \varGamma _{q}(\sigma _{i} - \zeta _{i-1}) } & \\ \quad {}\times ( t_{0}^{i-1} - t^{i-1} ), & 2 \leq i \leq m, \end{cases}\displaystyle \end{aligned}$$
which similar to the previous case tends to zero as \(t\to 1\) for \(i=1,2,\ldots , m\). This completes the proof. □
Lemma 5
Let \(m \geq 2\), \(\sigma _{1} \in (0, 1)\), \(\sigma _{1} > \alpha _{1}\), \(\sigma _{i} \in (i-1, i)\)for \(i=2, \ldots , m\), \(\alpha _{i} \in (0,1)\)for \(i=1,2, \ldots , m\), \(w_{i} : J \times \mathbb{R}^{i} \to \mathbb{R}\)be a function with \(\lim_{t \to 0^{+}} w_{i}(t, \ldots )= \infty \), and \(t^{\alpha _{i}} w_{i} (t)\)be continuous on \(\overline{J} \times \mathbb{R}^{i}\). Then the operator \(\mathcal{N}: S \to S\)defined by Eq. (8) is completely continuous.
Proof
Let \(( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \in S\) with
and \(\| ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \|=l_{0}\) for all \(( k_{1}, k_{2}, \ldots , k_{m}) \in S\). Hence,
By using the continuity of the map \(t^{\alpha _{i}} \varrho _{i} (t, k_{1}, k_{2}, \ldots , k_{m})\), we get the map
is uniformly continuous on \(\overline{J} \times [-l, l]^{i}\). For each \(\varepsilon > 0\), choose \(\lambda \in (0,1)\) such that
for all \(t \in \overline{J}\) whenever \(\| ( k_{1}, k_{2}, \ldots , k_{m}) - ( {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}) \|< \lambda \). Thus,
and
Now, by using (9), we obtain
where \(\varLambda _{i}= \frac{ \varGamma _{q}(1- \alpha _{i})}{ \varGamma _{q}(\sigma _{i}+ 1- \alpha _{i}) }\) whenever \(i=1\) and
whenever \(2 \leq i \leq m\). Now, by applying last result and (11), we get
Also, (10) and (11) imply that
for all \(t \in \overline{J}\). Hence, \(\| \mathcal{N} [ k_{1}, k_{2}, \ldots , k_{m}] (t) - \mathcal{N} [ {}_{0}k_{1}, {}_{0}k_{2}, \ldots , {}_{0}k_{m}] (t) \| \to 0\) as
Thus, the operator \(\mathcal{N}\) is continuous. Now consider a bounded subset \(K \subset S\). Then there exists a positive constant δ such that \(\|( k_{1}, k_{2}, \ldots , k_{m}) \| \leq \delta \) for all \(( k_{1}, k_{2}, \ldots , k_{m}) \in K\). Since the maps \(t^{\alpha _{i}} w_{i}(t, k_{1}, k_{2}, \ldots , k_{i} )\) are continuous on \(\overline{J} \times [-\delta , \delta ]^{i}\) for \(i = 1, 2, \ldots , m\), there exist positive constants \(L_{i}\) such that
for all \(t \in \overline{J}\) and \(( k_{1}, k_{2}, \ldots , k_{m} ) \in K\). Consider the norm
Note that
Now, by using (13), we get
On the other hand, by using (14) and (15), we get
Thus \(\mathcal{N} (K)\) is bounded. Let \(( k_{1}, k_{2}, \ldots , k_{m}) \in K\) and \(t_{1}, t_{2} \in \overline{J}\) with \(t_{1} < t_{2}\). Then we have
and
Hence,
Now, by using (16) and (17), we obtain
The right-hand side of (18) is independent of \((k_{1}, k_{2}, \ldots , k_{m})\) and, by assumption \(\sigma _{1} >\alpha _{1}\) and the fact \(\sigma _{i}> \alpha _{i}\), tends to zero as \(t_{1} \to t_{2}\). This implies that \(\mathcal{N}(K)\) is equicontinuous. Now, by using the Arzelà –Ascoli theorem, we conclude that \(\mathcal{N}\) is completely continuous. □
Theorem 6
The m-dimensional system of singular fractional q-differential equations (1) has a unique solution on JÌ… whenever there exist nonnegative constants \({}_{i}\eta _{j}\) (\(j=1, 2, \ldots , i\), \(i=1, 2, \ldots , m\), \(m \geq 2\)) satisfying
for all \(t \in \overline{J}\)and \((k_{1}, \ldots , k_{i})\), \((l_{1}, \ldots , l_{i}) \in \mathbb{R}^{i}\), and also
where the constants \(\varLambda _{i}\)are defined by (11).
Proof
We prove that \(\mathcal{N}\) is a contractive operator on S. Assume that \((k_{1}, k_{2}, \ldots , k_{m}) \in S\) and \((l_{1},l_{2}, \ldots , l_{m}) \in S\). Then we have
for almost all \(t \in \overline{J}\). Hence,
Now, by using (19), we obtain
If we apply (21) and (22), then we get
Now, by using (20), we have
Hence, \(\mathcal{N}\) is a contraction. By using the Banach contraction principle, \(\mathcal{N}\) has a unique fixed point which is the unique solution for system (1). □
Now, we consider different conditions on system (1).
Theorem 7
Let \(m \geq 2\), \(\sigma _{1} \in (0, 1)\), \(\sigma _{1} > \alpha _{1}\), \(\sigma _{i} \in (i-1, i)\)for \(i=2, \ldots , m\), \(\alpha _{i} \in (0,1)\)for \(i=1,2, \ldots , m\), \(w_{i} : J \times \mathbb{R}^{i} \to \mathbb{R}\)be functions with \(\lim_{t \to 0^{+}} w_{i}(t, \ldots ) = \infty \), and \(t^{\alpha _{i}} w_{i} (t, \ldots )\)be continuous maps on \(\overline{J} \times \mathbb{R}^{i}\). Then system (1) has a solution on JÌ….
Proof
Assume that
and define the set \(K_{r} \subset S\) by
where
We show that \(\mathcal{N}\) maps \(K_{r}\) into \(K_{r}\). For \((k_{1}, k_{2}, \ldots , k_{m}) \in K_{r}\) and \(t \in \overline{J}\), put
Thus, we have
Hence,
Now, by using (25) and (26), we conclude that
and so \(\| \mathcal{N} [k_{1}, k_{2}, \ldots , k_{m}](t) \| \leq r\). By using Lemma 4, we get
Moreover, \(\mathcal{N}[ k_{1}, k_{2}, \ldots , k_{m}](t) \in K_{r}\) for \((k_{1}, k_{2}, \ldots , k_{m}) \in K_{r}\). Thus \(\mathcal{N} (K_{r}) \subset K_{r}\), and so \(\mathcal{N}\) maps \(K_{r}\) into \(K_{r}\). On the other hand, by using Lemma 5, \(\mathcal{N}\) is completely continuous. Now, by using Lemma 2, the map \(\mathcal{N}\) has a fixed point which is a solution for system (1). □
Now, we provide two examples to illustrate our main results. In this way, we give a computational technique for checking the m-dimensional system (1). We need to present a simplified analysis which is able to execute the values of the q-gamma function. For this purpose, we provide a pseudo-code description of the method for calculation of the q-gamma function of order n in Algorithms 2, 3, 5, and 4.
Example 1
Consider the increasing variables singular 5-dimensional system of fractional q-differential equations
under the boundary value conditions \(k_{1}(0) = \frac{7}{9}\), \(k_{2}(0)=\frac{3}{5}\),
and \({}^{c}\mathcal{D}_{q}^{ \frac{1}{2}} [k_{2}](1) = {}^{c}\mathcal{D}_{q}^{\frac{4}{3}} [k_{3}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{5}{2}} [k_{4}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{11}{3}} [k_{5}](1) =0\), where \(t\in (0,1]\). Put
\(m=5\), \(\sigma _{1} = \frac{7}{9} \in (0,1)\), \(\sigma _{2} = \frac{8}{7} \in (1,2)\), \(\sigma _{3} = \frac{11}{4} \in (2,3)\), \(\sigma _{4}= \frac{16}{5} \in (3, 4)\), \(\sigma _{5} = \frac{31}{7} \in (4,5)\), \(\zeta _{1} = \frac{1}{2} \in [0,1]\), \(\zeta _{2} = \frac{4}{3} \in [1,2]\), \(\zeta _{3} = \frac{5}{2} \in [2,3]\), \(\zeta _{4} = \frac{11}{3} \in [3,4]\), \({}_{1}b_{0} = \frac{7}{9}\), \({}_{2}b_{0} = \frac{3}{5}\), \({}_{3}b_{0} = \frac{1}{2}\), \({}_{4}b_{0} = \sqrt{5}\), \({}_{5}b_{0} = \frac{ \sqrt{3}}{3}\), \({}_{3}b_{1} = 2\sqrt{3}\), \({}_{4}b_{1} = \frac{ \sqrt{5}}{3}\), \({}_{5}b_{1} = 1\), \({}_{4}b_{2} = \frac{15}{7}\), \({}_{5}b_{2} = 0\), and \({}_{5}b_{3} = \frac{13}{4}\). Now, we check inequalities (19) and (20). For each \(t \in \overline{J}\), \((k_{1}, k_{2}, \ldots , k_{5})\), and \((l_{1}, l_{2}, \ldots , l_{5}) \in \mathbb{R}^{5}\), we have
\(\alpha _{1} = \frac{4}{7}\), \({}_{1}\eta _{1}=\frac{3}{10\pi }\),
\(\alpha _{2} = \frac{2}{5}\), \({}_{2}\eta _{1}={}_{2}\eta _{2} = \frac{2}{25\pi }\),
\(\alpha _{3} = \frac{5}{8}\), \({}_{3}\eta _{1}={}_{3}\eta _{2} = {}_{3}\eta _{3} = \frac{1}{20\pi }\),
\(\alpha _{4} = \frac{7}{9}\), \({}_{4}\eta _{1} = {}_{4}\eta _{2} = {}_{4}\eta _{3} = {}_{4}\eta _{4} =\frac{2}{25\pi }\),
and \(\alpha _{5} = \frac{10}{11}\), \({}_{5}\eta _{1} = {}_{5}\eta _{2} = {}_{5}\eta _{3} = {}_{5}\eta _{4} = {}_{5}\eta _{5} =\frac{1}{20\pi }\). On the other hand, by using (11), we obtain
Tables 1, 2, and 3 show \(\varLambda _{i} \approx 1.4269\), 6.1292, 2.1068, 2.2574, 3.8301, \(\varLambda _{i} \approx 1.9041\), 9.5549, 2.2455, 2.2349, 2.4713, \(\varLambda _{i} \approx 2.1668\), 11.5144, 2.2172, 2.0036, 1.4726 for \(1 \leq i \leq 5\) and for \(q= \frac{1}{10}\), \(\frac{1}{2}\), \(\frac{6}{7}\), respectively. It is clear that \(\sum_{j=1}^{2} {}_{2}\eta _{j}= \frac{4}{25\pi } \), \(\sum_{j=1}^{3} {}_{3}\eta _{j}= \frac{3}{20\pi }\), \(\sum_{j=1}^{4} {}_{2}\eta _{j}= \frac{8}{25\pi }\), and \(\sum_{j=1}^{5} {}_{2}\eta _{j}= \frac{1}{4\pi }\). In Tables 4, 5, and 6, we can see that \(\varSigma =0.3122\), 0.4866, and 0.5864, indeed
for \(q=\frac{1}{10}\), \(\frac{1}{2}\), and \(\frac{6}{7}\), respectively (Fig. 1). Thus, the assumptions and conditions of Theorem 6 hold. Hence the singular 5-dimensional system of fractional q-differential equations (28) has a unique solution on \((0,1]\). Note that Algorithm 6 shows us how we can obtain the parameters of Example 1.
Example 2
Consider the singular system of fractional q-differential equations
with boundary value conditions \(k_{1}(0) = \frac{2}{3}\),
\({}^{c}\mathcal{D}_{q}^{ \frac{1}{7}} [k_{2}](1) = {}^{c}\mathcal{D}_{q}^{\frac{8}{5}} [k_{3}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{11}{4}} [k_{4}](1) = {}^{c}\mathcal{D}_{q}^{ \frac{7}{2}} [k_{5}](1) =0\), where \(t\in (0,1]\). Put
\(m=5\), \(\sigma _{1} =\frac{9}{10} \in (0,1)\), \(\sigma _{2} = \frac{9}{5} \in (1,2)\), \(\sigma _{3} = \frac{17}{6} \in (2,3)\), \(\sigma _{4}= \frac{24}{7} \in (3, 4)\), \(\sigma _{5} = \frac{13}{3} \in (4,5)\), \(\zeta _{1}= \frac{2}{11} \in [0,1]\), \(\zeta _{2}= \frac{5}{3} \in [1,2]\), \(\zeta _{3}= \frac{7}{3} \in [2,3]\), \(\zeta _{4}= \frac{13}{4} \in [3,4]\), \({}_{1}b_{0} = \frac{2}{3}\), \({}_{2}b_{0} = -1\), \({}_{3}b_{0} = 1\), \({}_{4}b_{0} = \sqrt{7}\), \({}_{5}b_{0} = \frac{2}{3}\), \({}_{3}b_{1} = \frac{2}{3}\), \({}_{4}b_{1} = \frac{ \sqrt{7}}{3}\), \({}_{5}b_{1} = 1\), \({}_{4}b_{2} = \frac{\sqrt{5}}{3}\), \({}_{5}b_{2} = \frac{3}{8}\), and \({}_{5}b_{3} = \frac{2\sqrt{2}}{5}\). Now, we check (23) and (24). For each \(t \in \overline{J}\) and \((k_{1}, k_{2}, \ldots , k_{5}) \in \mathbb{R}^{5}\), we have
for \(\alpha _{1}=\frac{3}{4}\),
for \(\alpha _{2}=\frac{2}{3}\),
for \(\alpha _{3} = \frac{4}{5}\),
for \(\alpha _{4}=\frac{1}{2}\),
for \(\alpha _{5} = \frac{8}{9}\). Now, by using (11), we get
Tables 7, 8, and 9 show \(\varLambda _{i} \approx 2.0428\), 3.2300, 3.3499, 1.2683, 3.2252, \(\varLambda _{i} \approx 3.812\), 4.3215, 4.2023, 0.8837, 2.1222, \(\varLambda _{i} \approx 3.6791\), 4.8820, 4.4534, 0.6683, 1.2984 for \(1 \leq i \leq 5\) and \(q=\frac{1}{10}\), \(\frac{1}{2}\), \(\frac{6}{7}\), respectively. Now, by using (24) and Algorithm 7, we conclude next results. According to Tables 10, 11, and 12, consider the set \(K_{r} \subset S\) as
for \(q=\frac{1}{10}\), \(\frac{1}{2}\), and \(\frac{6}{7}\), respectively. Table 10 shows that \(L_{1} \varLambda _{1} + |{}_{1}b_{0}| \approx 0.0812\), \(L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!} \approx 1.1371\), 2.6822, 5.0190, 2.0625, 5.0190. Table 11 shows \(L_{1} \varLambda _{1} + |{}_{1}b_{0}| \approx 0.1226\), \(L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!} \approx 1.1834\), 2.9406, 4.6798, 2.0235, Table 12 shows that \(L_{1} \varLambda _{1} + |{}_{1}b_{0}| \approx 0.1463\), \(L_{i}\varLambda _{i} + \sum_{j=0}^{i-2} \frac{|{}_{i}b_{j}|}{j!} \approx 1.2072\), 3.0164, 4.4898, 1.9944 for \(2 \leq i \leq 5\) and \(q=\frac{1}{10}\), \(\frac{1}{2}\), \(\frac{6}{7}\), respectively. Also, Table 13 shows us \(r \approx 5.0190\), 4.6798, 4.4898 for \(q=\frac{1}{10}\), \(\frac{1}{2}\), \(\frac{6}{7}\), respectively (Figs. 3 and 2). Now, by using Theorem 7, the singular system of fractional q-differential equations (29) has a solution.
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The first author was supported by Bu Ali Sina Uinversity. The third author was supported by Azarbaijan Shahid Madani University.
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Samei, M.E., Baleanu, D. & Rezapour, S. An increasing variables singular system of fractional q-differential equations via numerical calculations. Adv Differ Equ 2020, 452 (2020). https://doi.org/10.1186/s13662-020-02913-5
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DOI: https://doi.org/10.1186/s13662-020-02913-5