Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach
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Abstract
In this article, we discuss the existence and uniqueness of solution of a delay Caputo q-fractional difference system. Based on the q-fractional Gronwall inequality, we analyze the Ulam–Hyers stability and the Ulam–Hyers–Rassias stability. An example is provided to support the theoretical results.
Keywords
Delay difference equation q-fractional Gronwall inequality Existence Uniqueness Ulam–Hyers stability Ulam–Hyers–Rassias stability1 Introduction
In the theory of differential equations, Gronwall’s inequality is one of the most important tools. In 1919, for the first time, Gronwall worked on this type of inequality [1]. As time passed, many extensions of the Gronwall inequality have started to take part of the literature on mathematical inequalities. In 1935, Mikeladze published about the discrete fractional Gronwall inequality for the first time [2]. Gronwall’s inequality is useful in the analysis of qualitative and quantitative properties of the ordinary and fractional dynamical systems. That is why it attracted many researchers to work on it. Haiping Ye et al. [3] presented a generalized Gronwall inequality and studied the dependence of the solution on the order and the initial condition of a fractional differential equation. Very recently, the authors in [4], proved a Gronwall inequality for the generalized proportional fractional operators. A class of stochastic Gronwall inequalities has been studied by Wang et al. in [5]. Luo et al. in [6] studied the uniqueness and novel finite-time stability of solutions of delay difference equations using the Gronwall inequality approach. Recently, Almeida et al. in [7] and Yassine et al. in [8] presented an extension of the fractional Gronwall inequality and used it in the qualitative analysis of the solutions to generalized fractional differential equations. For the category of fractional operators with nonsingular Mittag-Leffler kernels, a recent version and its application have been reported in [9].
Difference equations have appeared in mathematical modeling to describe many real life problems, e.g., queueing problems, electrical networks, economics, etc. For that reason, many researchers have proved discrete versions of Gronwall-type inequalities in fractional calculus and applied them to study the qualitative and quantitative properties of fractional difference equations [10, 11, 12, 13, 14, 15]. Moreover, Gronwall’s inequality is widely used for the analysis of stability of fractional differential as well as fractional difference equations. In one of the most recent works, Ameen et al. discussed the Ulam stability of delay fractional differential equations with a generalized Caputo derivative using a Gronwall inequality approach [16]. Kui Liu et al. [17] also presented Ulam–Hyers stability of solutions for differential equations with Caputo–Fabrizio fractional derivative with the help of Gronwall’s inequality. For further assistance in stability analysis using the Gronwall inequality approach, one can follow the articles cited in [18, 19, 20]. As for the stability results without any Gronwall approach for the q-fractional systems, we refer to the first two works [21, 22]. Since stability, and specially Ulam–Hyers stability, is of high priority for researchers and has been studied for applied as well as mathematical problems, one can follow the most recent articles on stability cited in [23, 24, 25]. For recent operator and mathematical models, whose stability analysis is an open dilemma, we refer to [26, 27, 28].
In q-fractional calculus most probably the first article on a q-fractional Gronwall inequality was presented by Abdeljawad et al. in [29]. Later on, another new Gronwall inequality in q-fractional calculus was proved in [30], where the authors considered a nonlinear delay Caputo q-fractional difference system and discussed the uniqueness and estimates for the solutions of the system under consideration.
For \(0 < q < 1\), time scale \(\mathbb{T}_{q}\) is defined as \(\mathbb{T}_{q}= \lbrace q^{n} : n\in \mathbb{Z} \rbrace \cup \lbrace 0 \rbrace \), where \(\mathbb{Z}\) is the set of integers. If \(n_{0}\in \mathbb{Z}\) and \(a = q^{n_{0}}\), \(\mathbb{T} _{a}\) can be written as \(\mathbb{T}_{a} = [a,\infty )_{q} = \lbrace q ^{-i}a : i = 0, 1, 2,\ldots \rbrace \). Further define \(\mathbb{I} _{\tau }= \lbrace \tau a, q^{-1}\tau a, q^{-2}\tau a,\ldots,a \rbrace \) and \(\mathbb{T}_{\tau a} = [\tau a, \infty )_{q}= \lbrace \tau a, q^{-1}\tau a, q^{-2}\tau a,\ldots \rbrace \), where \(\tau =q^{d}\in \mathbb{T} _{q}\), \(d\in \mathbb{N}_{0}\), where \(\mathbb{N}_{0}= \lbrace 0,1,2,3,\ldots \rbrace \) and \(\mathbb{I}_{\tau }= \lbrace a \rbrace \) with \(d = 0\) is the non-delay case. The objective of this paper is to study the existence, uniqueness and then analyze the Ulam–Hyers and Ulam–Hyers–Rassias stabilities of Caputo q-fractional difference equation with delay of the form
where \(u:\mathbb{T}_{\tau a}\rightarrow \mathbb{R}\), \(\mathcal{F}: \mathbb{T}_{a}\times \mathbb{R}^{2}\rightarrow \mathbb{R}\), \(\mathscr{G}:\mathbb{T}_{a}\rightarrow \mathbb{T}_{a}\), \(\varPhi : \mathbb{I}_{\tau }\rightarrow \mathbb{R}\), and \({}^{c}_{q}\nabla _{a} ^{\xi }\) denotes the Caputo q-fractional difference operator of order \(0<\xi <1\). To prove our main results, we use the q-fractional Gronwall inequality proved in [30].
$$ \textstyle\begin{cases} {}^{c}_{q}\nabla _{a}^{\xi }u(t) =\mathcal{F}(t,u(t),u(\mathscr{G}( \tau t))), & t\in \mathbb{T}_{a}, \\ u(t) =\varPhi (t), & t\in \mathbb{I}_{\tau }, \end{cases} $$
(1)
This article is composed as follows: In Sect. 2, we present some basic definitions, notations, lemmas, and remarks that are important for proving our main results. In Sect. 3, we discuss the existence and uniqueness of solution of problem (1). Then, we discuss the Ulam–Hyers and Ulam–Hyers–Rassias stabilities of the above-mentioned problem. In the last section, examples are provided.
2 Essential preliminaries and definitions
In this section, we provide some basic concepts of q-fractional calculus that are essential to proving our main results. For more details on the theory of q-calculus and q-fractional calculus, we refer to [31, 32, 33, 34] (and the references therein). For more remarkable basic articles in q-fractional calculus, we refer to [35, 36, 37, 38, 39]. The book [40] and [41] are also recommended for readers.
Definition 1
([33])
For \(u:\mathbb{T}_{q}\rightarrow \mathbb{R}\), the nabla q-derivative of u is defined as
$$ \nabla _{q}u(t)=\frac{u(t)-u(qt)}{(1-q)t}, \quad t\in \mathbb{T}_{q}- \lbrace 0 \rbrace . $$
Definition 2
([33])
For \(u:\mathbb{T}_{q}\rightarrow \mathbb{R}\), the nabla q-integral of u is defined as
For \(0\leq a\in \mathbb{T}_{q}\),
$$ \int _{0}^{t}u(s)\nabla _{q}s=(1-q)t \sum_{i=0}^{\infty }q^{i}u \bigl(tq^{i}\bigr). $$
$$ \int _{a}^{t}u(s)\nabla _{q}s= \int _{0}^{t}u(s)\nabla _{q}s- \int _{0}^{a}u(s) \nabla _{q}s. $$
Definition 3
For \(\xi \in \mathbb{R}\), we have:
- (i)The nabla q-derivative of the q-factorial function with respect to t is given by$$ \nabla _{q}(t-s)_{q}^{\xi }= \frac{1-q^{\xi }}{1-q}(t-s)^{\xi -1}_{q}. $$
- (ii)The nabla q-derivative of the q-factorial function with respect to s is given by$$ \nabla _{q}(t-s)_{q}^{\xi }=- \frac{1-q^{\xi }}{1-q}(t-qs)^{\xi -1} _{q}. $$
Definition 4
Let \(u:\mathbb{T}_{q}\rightarrow \mathbb{R}\), then left q-fractional integral of order \(\xi \neq 0,-1,-2,-3,\ldots \) and starting at \(0< a\in \mathbb{T}_{q}\) is defined as
where
$$ _{q}\nabla _{a}^{-\xi }u(t)= \frac{1}{\varGamma _{q}(\xi )} \int _{a}^{t}(t-qs)^{ \xi -1}_{q}u(s) \nabla _{q} s, $$
$$ \varGamma _{q}(\xi +1)=\frac{1-q^{\xi }}{1-q}\varGamma _{q}( \xi ), \qquad \varGamma _{q}(1)=1, \quad \xi >0. $$
Definition 5
Let \(u:\mathbb{T}_{q}\rightarrow \mathbb{R}\), where \(0<\xi \notin \mathbb{N}\). Then the Caputo left q-fractional derivative of order ξ is defined as
where \(n=\lceil \xi \rceil +1\).
$$ {}_{q}^{c}\nabla _{a}^{\xi } u(t)={}_{q}\nabla _{a}^{-(n-\xi )}\nabla _{q} ^{n}u(t)=\frac{1}{\varGamma _{q}(n-\xi )} \int _{a}^{t}(t-qs)^{n-\xi -1} _{q}\nabla ^{n}_{q}u(s)\nabla _{q}s, $$
Lemma 1
([33])
For\(\xi >0\)andudefined on suitable domain, we haveIn particular, for\(\xi \in (0,1]\), we have
$$ _{q}\nabla _{a}^{-\xi }{}_{q}^{c} \nabla _{a}^{\xi }u(t)=u(t)-\sum _{k=0} ^{n-1}\frac{(t-a)^{k}_{q}}{\varGamma _{q}(k+1)}\nabla _{q}^{k}u(a). $$
$$ _{q}\nabla _{a}^{-\xi }{}_{q}^{c} \nabla _{a}^{\xi }u(t)=u(t)-u(a). $$
Lemma 2
([33])
For\(\xi \in \mathbb{R}^{+}\)and\(\mu \in (-1,\infty )\), we have
$$ {}_{q}\nabla _{a}^{-\xi }(u-a)_{q}^{\mu }= \frac{\varGamma _{q}(\mu +1)}{ \varGamma _{q}(\xi +\mu +1)}(u-a)^{\mu +\xi }_{q}, \quad 0< a< u< b. $$
Definition 6
Eq. (1) is said to be Ulam–Hyers stable if there exists a real number c such that, for all \(\epsilon >0\) and for each \(v(t)\) defined on \(\mathbb{T}_{\tau a}\), satisfying the inequality
there exists a solution \(u(t)\) defined on \(\mathbb{T}_{\tau a}\) of Eq. (1) satisfying
$$ \bigl\vert {}^{c}_{q} \nabla _{a}^{\xi }v(t)-\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}( \tau t)\bigr)\bigr) \bigr\vert \leq \epsilon , \quad t\in \mathbb{T}_{a}, $$
(2)
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq c\epsilon , \quad t\in \mathbb{T}_{\tau a}. $$
Definition 7
Eq. (1) is said to be Ulam–Hyers–Rassias stable with respect to \(\psi (t)\), \(\psi :\mathbb{T}_{a}\rightarrow \mathbb{R} ^{+}\) if there exists a real number c such that, for all \(\epsilon >0\) and for each \(v(t)\) defined on \(\mathbb{T}_{\tau a}\) satisfying the inequality
there exists a solution \(u(t)\) defined on \(\mathbb{T}_{\tau a}\) of Eq. (1) satisfying
$$ \bigl\vert {}^{c}_{q} \nabla _{a}^{\xi }v(t)-\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}( \tau t)\bigr)\bigr) \bigr\vert \leq \epsilon \psi (t), \quad t\in \mathbb{T}_{a}, $$
(3)
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq c\epsilon \psi (t), \quad t\in \mathbb{T}_{\tau a}. $$
The following lemma is the key to proceeding.
Lemma 3
([30])
Let\(\xi >0\), \(u(t)\)and\(v(t)\)be nonnegative functions, \(w(t)\)be a nonnegative and nondecreasing function for\(t\in \mathbb{T}_{a}\)such that\(w(t)\leq M\), whereMis a constant. Ifthen
$$ u(t)\leq v(t)+w(t) {}_{q}\nabla _{a}^{-\xi }u(t), $$
$$ u(t)\leq v(t)+\sum_{k=1}^{\infty }\bigl(w(t) \varGamma _{q}(\xi )\bigr)^{k}{}_{q} \nabla _{a}^{-k\xi } v(t). $$
Corollary 1
([30])
Under the hypothesis of Lemma 3, assume further that\(v(t)\)is a nondecreasing function for\(t\in \mathbb{T}_{a}\), thenwhere\({}_{q}E_{\xi }(\lambda ,t-a)=\sum_{k=0}^{\infty }\lambda ^{k} \frac{ (t-a)_{q}^{k \xi}}{\varGamma _{q}(k \xi +1)}\)is the q-Mittag-Leffler function.
$$ u(t)\leq v(t) {}_{q}E_{\xi }\bigl(w(t)\varGamma _{q}(\xi ),t-a\bigr), \quad t\in \mathbb{T}_{a}, $$
3 Main results
In this section, we present our main results. First, we start by proving the existence and uniqueness of the solution of Eq. (1). Then, we proceed to analyzing the Ulam–Hyers and Ulam–Hyers–Rassias stabilities.
3.1 Existence and uniqueness results
Consider the space \(X=l_{\infty }(\mathbb{T}_{\tau a})\) of bounded functions (sequences) on \(\mathbb{T}_{\tau a}\), where \(\mathbb{T}_{ \tau a}=[\tau a,\infty )_{q}\). The space X is a Banach space with the norm defined by \(\Vert z \Vert _{X}= \sup_{t\in \mathbb{T}_{\tau a}}|z(t)|\). In the following lemma we present the solution representation.
Lemma 4
\(u(t)\)satisfies Eq. (1) if and only if it satisfies the following q-sum equation:
$$ u(t)= \textstyle\begin{cases} \varPhi (t) & t\in \mathbb{I}_{\tau }, \\ \varPhi (a)+{}_{q}\nabla _{a}^{-\xi }\mathcal{F}(t,u(t),u(\mathscr{G}( \tau t))), & t\in \mathbb{T}_{a}. \end{cases} $$
(4)
Proof
For \(t\in \mathbb{I}_{\tau }\), it is clear that \(u(t)=\varPhi (t)\) is the solution of Eq. (1). Now, for \(t\in \mathbb{T}_{a}\), applying \({}^{c}_{q}\nabla _{a}^{\xi }\) on both sides of Eq. (4), we get
Using the facts that \({}^{c}_{q}\nabla _{a}^{\xi } (\mathit{constant})=0\) and \({}^{c}_{q}\nabla _{a}^{\xi }({}_{q}\nabla _{a}^{-\xi })u(t)=u(t)\), which can be seen by Theorem 7 in [33], we have
On the other hand, from Eq. (4), for \(t\in \mathbb{I} _{\tau }\), we have \(u(t)=\varPhi (t)\). Also, by applying \({}_{q}\nabla _{a} ^{-\xi }\) on both sides of Eq. (1) and making use of Lemma 1, we get
hence we get
□
$$ {}^{c}_{q}\nabla _{a}^{\xi }u(t)={}^{c}_{q} \nabla _{a}^{\xi }\varPhi (a)+ {}^{c}_{q} \nabla _{a}^{\xi }\bigl({}_{q}\nabla _{a}^{-\xi }\bigr)\mathcal{F}\bigl(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr)\bigr). $$
$$ {}^{c}_{q}\nabla _{a}^{\xi }u(t)= \mathcal{F}\bigl(t,u(t),u\bigl(\mathscr{G}(\tau t)\bigr)\bigr). $$
$$ u(t)=u(a)+{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,u(t),u\bigl(\mathscr{G}( \tau t)\bigr)\bigr), $$
$$ u(t)=\varPhi (a)+{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,u(t),u\bigl(\mathscr{G}( \tau t)\bigr)\bigr). $$
Now we present the following uniqueness theorem.
Theorem 1
Assume the following:
- \((A_{1})\)
\(\mathcal{F}\)andΦare continuous functions defined as\(\mathcal{F}:\mathbb{T}_{a}\times \mathbb{R}^{2}\rightarrow \mathbb{R}\)and\(\varPhi :\mathbb{I}_{\tau }\rightarrow \mathbb{R}\);
- \((A_{2}) \)
- \(\mathcal{F}\)satisfies the Lipschitz condition with\(\mathscr{L}>0\)such that, for\(t\in \mathbb{T}_{a}\),$$ \bigl\Vert \mathcal{F}(t,u_{1},u_{2})- \mathcal{F}(t,v_{1},v_{2}) \bigr\Vert \leq \mathscr{L} \bigl( \Vert u_{1}-v_{1} \Vert + \Vert u_{2}-v _{2} \Vert \bigr); $$
- \((A_{3})\)
The map\(\mathscr{G}\)preserves the delay interval\(\mathbb{I}_{\tau }\).
If\(u(t)\)and\(v(t)\)satisfy problem (1), then\(u(t)=v(t)\).
Proof
Let \(z(t)=u(t)-v(t)\), then we have to show that \(z(t)=0\). It is immediate that \(z(t)=0\) for \(t\in \mathbb{I}_{\tau }\). For \(t\in \mathbb{T}_{a}\), we have
If \(t\in \mathbb{I}_{\tau ^{-1}}= \lbrace a,\ldots,\tau ^{-1}a \rbrace \), then (A3) forces \(z(\mathscr{G}(\tau t))=0\). Then, together with the other assumptions, it will imply that
An application of Corollary 1 will imply
and hence \(z(t)=0\) for \(t\in \mathbb{I}_{\tau ^{-1}}\). Next, for \(t\in [\tau ^{-1}a,\infty )_{q}\), we have
$$ z(t)={}_{q}\nabla _{a}^{-\xi }\mathcal{F} \bigl(t,u(t),u\bigl(\mathscr{G}(\tau t)\bigr)\bigr)-{}_{q}\nabla _{a}^{-\xi }\mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr). $$
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert &= \bigl\Vert {}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,u(t),u\bigl(\mathscr{G}( \tau t)\bigr)\bigr)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr) \bigr\Vert \\ &\leq {}_{q}\nabla _{a}^{-\xi } \bigl\Vert \mathcal{F}\bigl(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr)\bigr)-\mathcal{F} \bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr) \bigr\Vert \\ &\leq {}_{q}\nabla _{a}^{-\xi }\mathscr{L} \bigl( \bigl\Vert u(t)-v(t) \bigr\Vert + \bigl\Vert u\bigl(\mathscr{G}(\tau t)\bigr)-v\bigl(\mathscr{G}( \tau t)\bigr) \bigr\Vert \bigr) \\ &\leq {}_{q}\nabla _{a}^{-\xi }\mathscr{L} \bigl( \bigl\Vert z(t) \bigr\Vert + \bigl\Vert z\bigl(\mathscr{G}(\tau t) \bigr) \bigr\Vert \bigr) \\ &\leq \frac{\mathscr{L}}{\varGamma _{q}(\xi )} \int _{a}^{t}(t-qs)^{ \xi -1}_{q} \bigl\Vert z(s) \bigr\Vert \nabla _{q}s. \end{aligned}$$
$$ \bigl\Vert z(t) \bigr\Vert \leq 0. _{q}E_{\xi }\bigl[ \mathscr{L}\varGamma _{q}( \xi ),t-a\bigr], $$
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert &= \bigl\Vert {}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,u(t),u\bigl(\mathscr{G}( \tau t)\bigr)\bigr)-_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr) \bigr\Vert \\ &\leq {}_{q}\nabla _{a}^{-\xi } \bigl\Vert \mathcal{F}\bigl(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr)\bigr)-\mathcal{F} \bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr) \bigr\Vert \\ &\leq {}_{q}\nabla _{a}^{-\xi }\mathscr{L} \bigl( \bigl\Vert u(t)-v(t) \bigr\Vert + \bigl\Vert u\bigl(\mathscr{G}(\tau t)\bigr)-v\bigl(\mathscr{G}( \tau t)\bigr) \bigr\Vert \bigr) \\ &\leq \frac{\mathscr{L}}{\varGamma _{q}(\xi )} \int _{a}^{t}(t-qs)^{ \xi -1}_{q} \bigl( \bigl\Vert z(s) \bigr\Vert + \bigl\Vert z\bigl(\mathscr{G}(\tau s) \bigr) \bigr\Vert \bigr) \nabla _{q}s. \end{aligned}$$
If we let \(\hat{z}(t)=\sup_{\beta \in \mathbb{I}_{\tau }} \Vert z( \mathscr{G}(\beta t)) \Vert \), then we have
Finally, Corollary 1 implies that
Hence again we have \(z(t)=0\), i.e., \(u(t)=v(t)\) for \(t\in \mathbb{T} _{\tau a}\). □
$$\begin{aligned} \hat{z}(t) &\leq \frac{\mathscr{L}}{\varGamma _{q}(\xi )} \int _{a}^{t}(t-qs)^{ \xi -1}_{q} \bigl(\hat{z}(s)+ \hat{z}(s)\bigr) \nabla _{q}s \\ &\leq \frac{2\mathscr{L}}{\varGamma _{q}(\xi )} \int _{a}^{t}(t-qs)^{ \xi -1}_{q} \hat{z}(s)\nabla _{q}s. \end{aligned}$$
$$ \bigl\Vert z(t) \bigr\Vert \leq \hat{z}(t)\leq 0.{}_{q}E_{\xi } \bigl[2 \mathscr{L}\varGamma _{q}(\xi ),t-a\bigr]. $$
Now we present the following existence and uniqueness theorem for problem (1).
Theorem 2
In addition to assumptions\((A_{1})\), \((A_{2})\), and\((A_{3})\), let us assume that
- \((A_{4})\)
\(\frac{2\mathscr{L}(T-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}<1\)for some\(T>a\).
Then problem (1) has a unique solution in\(\mathbb{T} _{\tau a}\).
Proof
On the Banach space \(X=l_{\infty }(\mathbb{T}_{\tau a})\) define an operator as follows:
$$ \mathscr{T}u(t)= \textstyle\begin{cases} \varPhi (t), & t\in \mathbb{I}_{\tau }, \\ \varPhi (a)+{}_{q}\nabla ^{-\xi }_{a}\mathcal{F}(t,u(t),u(\mathscr{G}( \tau t)), & t\in \mathbb{T}_{a}. \end{cases} $$
For \(t\in \mathbb{I}_{\tau }\), we have \(|\mathscr{T}v(t)-\mathscr{T}u(t)|=0\), \(u,v\in X\). Now, for \(t\in \mathbb{T}_{a}\), we have
Since \(\frac{2\mathscr{L}(T-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}<1\), then the operator \(\mathscr{T}\) is a contraction and by the Banach fixed point theorem there exists a unique fixed point. Clearly this unique fixed point is the unique solution of problem (1). □
$$\begin{aligned} \bigl\vert \mathscr{T}v(t)-\mathscr{T}u(t) \bigr\vert &= \bigl\vert {}_{q}\nabla ^{-\xi }_{a}\mathcal{F}(t,v(t),v \bigl( \mathscr{G}(\tau t)\bigr)-{}_{q}\nabla ^{-\xi }_{a} \mathcal{F}(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr) \bigr\vert \\ &\leq {}_{q}\nabla ^{-\xi }_{a} \bigl\vert \mathcal{F}(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)- \mathcal{F}(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr) \bigr\vert \\ &\leq \mathscr{L}\Bigl(\max_{t\in \mathbb{T}_{\tau a}} \bigl\vert v(t)-u(t) \bigr\vert + \max_{t\in \mathbb{T}_{\tau a}} \bigl\vert v\bigl(\mathscr{G}(\tau t)\bigr)-u\bigl(\mathscr{G}( \tau t)\bigr) \bigr\vert \Bigr){}_{q} \nabla ^{-\xi }_{a}(1) \\ &\leq \frac{2\mathscr{L}(t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)} \Vert v-u \Vert _{X} \\ &\leq \frac{2\mathscr{L}(T-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)} \Vert v-u \Vert _{X}, \quad t< T. \end{aligned}$$
3.2 Ulam–Hyers stability
In the rest of what follows, we prove the Ulam–Hyers stability of problem (1).
Lemma 5
If a function\(v(t)\)defined on\(\mathbb{T}_{\tau a}\)is a solution of inequality (2), then\(v(t)\)satisfies the following inequality:
$$ \bigl\vert v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}( \tau t)\bigr)\bigr) \bigr\vert \leq \frac{\epsilon (t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}. $$
Proof
As we know, \(v(t)\) satisfies inequality (2) if and only if there exists a function \(h(t)\) such that \(|h(t)|\leq \epsilon \) and
Now, applying \({}_{q}\nabla _{a}^{-\xi }\) on both sides of Eq. (5), we get
or
$$ {}^{c}_{q}\nabla _{a}^{\xi }v(t)-\mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}( \tau t)\bigr)\bigr)=h(t), \quad t\in \mathbb{T}_{a}. $$
(5)
$$ \bigl({}_{q}\nabla _{a}^{-\xi }\bigr) \bigl({}^{c}_{q}\nabla _{a}^{\xi } \bigr)v(t)-{}_{q}\nabla _{a}^{-\xi }\mathcal{F} \bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr)={}_{q}\nabla _{a}^{-\xi }h(t), $$
$$ v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}( \tau t)\bigr)\bigr)={}_{q} \nabla _{a}^{-\xi }h(t). $$
Hence it follows that
□
$$\begin{aligned} \bigl\vert \bigl({}_{q}\nabla _{a}^{-\xi } \bigr) \bigl({}^{c}_{q}\nabla _{a}^{\xi } \bigr)v(t)-{}_{q}\nabla _{a}^{-\xi }\mathcal{F} \bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert &\leq {}_{q} \nabla _{a}^{-\xi } \bigl\vert h(t) \bigr\vert \\ &\leq \epsilon {}_{q}\nabla _{a}^{-\xi }(1) \\ &=\frac{\epsilon (t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)} \\ &=\frac{\epsilon (T-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}, \quad t< T. \end{aligned}$$
Theorem 3
Under assumptions\((A_{1})\), \((A_{2})\), \((A_{3})\), and\((A_{4})\), Eq. (1) is Ulam–Hyers stable.
Proof
Assume that \(v(t)\) is a solution of inequality (2) and \(u(t)\) is the unique solution of Eq. (1) satisfying the condition \(u(t)=v(t)\) for \(t\in \mathbb{I}_{\tau }\). Then we have
For \(t\in \mathbb{I}_{\tau }\), we have \(|v(t)-u(t)|=0\). Now, for \(t\in \mathbb{I}_{\tau ^{-1}}= \lbrace a,\ldots,\tau ^{-1}a \rbrace \), we have
where we have used \(v(\mathscr{G}(\tau t))-u(\mathscr{G}(\tau t))=0\) for \(t\in \mathbb{I}_{\tau ^{-1}}\). Now, using Lemma 5, we conclude that
Since \(\frac{\epsilon (t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}\) is a nonnegative and nondecreasing function \(\forall t\in \mathbb{I}_{ \tau ^{-1}}\), then using Gronwall’s inequality in Corollary 1, we see that
Now, for \(t\in [\tau ^{-1}a,\infty )_{q}\), following the same steps as mentioned above, we get
Let \(\hat{z}(t)=\sup_{\beta \in \mathbb{I}_{\tau }}|v(\mathscr{G}( \beta t))-u(\mathscr{G}(\beta t))|\), so we get
Similarly, the use of Gronwall’s inequality in Corollary 1 will lead to
So we have
or
That is,
Hence completing the proof. □
$$ u(t)= \textstyle\begin{cases} v(t), & t\in \mathbb{I}_{\tau }, \\ v(a)+{}_{q}\nabla _{a}^{-\xi }\mathcal{F}(t,u(t),u(\mathscr{G}(\tau t)), & t\in \mathbb{T}_{a}. \end{cases} $$
$$\begin{aligned} \bigl\vert v(t)-u(t) \bigr\vert &= \bigl\vert v(t)-v(a)-{}_{q} \nabla _{a}^{-\xi }\mathcal{F}\bigl(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert \\ &\leq \bigl\vert v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert \\ & \quad {}+ \bigl\vert {}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr)-{}_{q} \nabla _{a}^{-\xi }\mathcal{F}\bigl(t,u(t),u\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert \\ &\leq \bigl\vert v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert \\ & \quad {}+ \bigl\vert {}_{q}\nabla _{a}^{-\xi } \bigl(\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr)-\mathcal{F} \bigl(t,u(t),u\bigl(\mathscr{G}(\tau t)\bigr)\bigr)\bigr) \bigr\vert \\ &\leq \bigl\vert v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert + \mathscr{L} {}_{q}\nabla _{a}^{-\xi }\bigl( \bigl\vert v(t)-u(t) \bigr\vert \bigr), \end{aligned}$$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \frac{\epsilon (t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}+ \mathscr{L} {}_{q}\nabla _{a}^{-\xi }\bigl( \bigl\vert v(t)-u(t) \bigr\vert \bigr). $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \biggl[ \frac{(t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}{}_{q}E_{\xi } \bigl(\mathscr{L}\varGamma _{q}(\xi ), t-a\bigr) \biggr] \epsilon , \quad \forall t\in \mathbb{I}_{\tau ^{-1}}. $$
$$\begin{aligned} \bigl\vert v(t)-u(t) \bigr\vert &\leq \bigl\vert v(t)-v(a)-{}_{q} \nabla _{a}^{-\xi }\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert \\ & \quad {}+\mathscr{L} {}_{q}\nabla _{a}^{-\xi } \bigl( \bigl\vert v(t)-u(t) \bigr\vert + \bigl\vert v\bigl( \mathscr{G}(\tau t)\bigr)-u\bigl(\mathscr{G}(\tau t)\bigr) \bigr\vert \bigr). \end{aligned}$$
$$\begin{aligned} \hat{z}(t) &\leq \frac{\epsilon (t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}+ \mathscr{L} {}_{q}\nabla _{a}^{-\xi }\bigl(\hat{z}(t)+\hat{z}(t)\bigr) \\ &\leq \frac{\epsilon (t-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}+2 \mathscr{L} {}_{q}\nabla _{a}^{-\xi }\hat{z}(t). \end{aligned}$$
$$ \hat{z}(t)\leq \biggl[\frac{ (t-a)^{\xi }_{q}}{\varGamma _{q}( \xi +1)}{}_{q}E_{\xi } \bigl(2\mathscr{L}\varGamma _{q}(\xi ),t-a\bigr) \biggr] \epsilon , \quad \forall t\in \bigl[\tau ^{-1}a,\infty \bigr)_{q}. $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \hat{z}(t)\leq \biggl[ \frac{ (t-a)^{\xi } _{q}}{\varGamma _{q}(\xi +1)}{}_{q}E_{\xi }\bigl(2\mathscr{L}\varGamma _{q}(\xi ),t-a\bigr) \biggr] \epsilon , \quad \forall t, $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \biggl[\frac{ (T-a)^{\xi }_{q}}{\varGamma _{q}( \xi +1)}{}_{q}E_{\xi } \bigl(2\mathscr{L}\varGamma _{q}(\xi ),T-a\bigr) \biggr] \epsilon , \quad \forall t< T. $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq c\epsilon . $$
3.3 Ulam–Hyers–Rassias stability
This subsection is devoted to the Ulam–Hyers–Rassias stability of problem (1).
Theorem 4
In addition to assumptions\((A_{1})\), \((A_{2})\), \((A_{3})\), and\((A_{4})\), assume thatThen Eq. (1) is Ulam–Hyers–Rassias stable with respect toψ.
- \((A_{5})\)
- there exist a continuous function\(\psi :\mathbb{T}_{a} \rightarrow \mathbb{R}^{+}\)and\(\lambda _{\psi }\in \mathbb{R}^{+}\)such that$$ _{q}\nabla _{a}^{-\xi }\psi (t)\leq \lambda _{\psi }\psi (t). $$
Proof
Let \(v(t)\) be a solution of inequality (3). So we have
Applying \({}_{q}\nabla _{a}^{-\xi }\), we get
Now, using assumption \((A_{5})\), we get
This implies
Let us consider \(u(t)\) such that
For \(t\in \mathbb{I}_{\tau }\), we have \(|v(t)-u(t)|=0\). Now, for \(t\in \mathbb{I}_{\tau ^{-1}}\), we have
Using inequality (6) and the observation that \(v(\mathscr{G}(\tau t))-u(\mathscr{G}(\tau t))=0\) for \(t\in \mathbb{I}_{\tau ^{-1}}\), we have
Now, by using Gronwall’s inequality in Corollary 1, we conclude that
For \(t\in [\tau ^{-1}a,\infty )_{q}\), using the same steps as mentioned above, we have
Again, letting \(\hat{z}(t)=\sup_{\beta \in \mathbb{I}_{\tau }}|v( \mathscr{G}(\beta t))-u(\mathscr{G}(\beta t))|\), we see that
Finally, the use of Gronwall’s inequality as in Corollary 1 implies that
So we have
or we may write it as follows:
This implies that
Hence Eq. (1) is Ulam–Hyers–Rassias stable. □
$$ -\epsilon \psi (t)\leq {}^{c}_{q}\nabla _{a}^{\xi }v(t)-\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr)\leq \epsilon \psi (t). $$
$$ -\epsilon {}_{q}\nabla _{a}^{-\xi }\psi (t) \leq \bigl({}_{q}\nabla _{a}^{- \xi }\bigr) \bigl({}^{c}_{q}\nabla _{a}^{\xi } \bigr)v(t)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}(\tau t)\bigr)\bigr)\leq \epsilon {}_{q}\nabla _{a}^{-\xi } \psi (t). $$
$$ -\epsilon \lambda _{\psi }\psi (t)\leq v(t)-v(a)-{}_{q} \nabla _{a}^{- \xi }\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr)\leq \epsilon \lambda _{\psi } \psi (t). $$
$$ \bigl\vert v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi }\mathcal{F}\bigl(t,v(t),v\bigl(\mathscr{G}( \tau t)\bigr)\bigr) \bigr\vert \leq \epsilon \lambda _{\psi } \psi (t). $$
(6)
$$\begin{aligned} &{}^{c}_{q}\nabla _{a}^{\xi }u(t) =\mathcal{F}\bigl(t,u(t),u\bigl(\mathscr{G}( \tau t)\bigr)\bigr), \quad t\in \mathbb{T}_{a}, \\ &u(t) =v(t), \quad t\in \mathbb{I}_{\tau }. \end{aligned}$$
$$\begin{aligned} \bigl\vert v(t)-u(t) \bigr\vert &= \bigl\vert v(t)-v(a)-{}_{q} \nabla _{a}^{-\xi }\mathcal{F}(t,u(t),u\bigl( \mathscr{G}( \tau t)\bigr) \bigr\vert \\ &\leq \bigl\vert v(t)-v(a)-{}_{q}\nabla _{a}^{-\xi } \mathcal{F}(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr) \bigr\vert \\ & \quad {}+ \bigl\vert {}_{q}\nabla _{a}^{-\xi }( \mathcal{F}(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)-\mathcal{F}\bigl(t,u(t),u \bigl(\mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert . \end{aligned}$$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \epsilon \lambda _{\psi }\psi (t)+\mathscr{L} {}_{q} \nabla _{a}^{-\xi } \bigl( \bigl\vert v(t)-u(t) \bigr\vert \bigr). $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \epsilon \lambda _{\psi }\psi (t){}_{q}E_{\xi }\bigl( \mathscr{L}\varGamma _{q}(\xi ),t-a\bigr). $$
$$\begin{aligned} \bigl\vert v(t)-u(t) \bigr\vert &\leq \bigl\vert v(t)-v(a)-{}_{q} \nabla _{a}^{-\xi }\mathcal{F}\bigl(t,v(t),v\bigl( \mathscr{G}(\tau t)\bigr)\bigr) \bigr\vert \\ & \quad {}+\mathscr{L} {}_{q}\nabla _{a}^{-\xi } \bigl( \bigl\vert v(t)-u(t) \bigr\vert + \bigl\vert v\bigl( \mathscr{G}(\tau t)\bigr)-u\bigl(\mathscr{G}(\tau t)\bigr) \bigr\vert \bigr) \\ &\leq \epsilon \lambda _{\psi }\psi (t)+\mathscr{L} {}_{q} \nabla _{a}^{- \xi }\bigl( \bigl\vert v(t)-u(t) \bigr\vert + \bigl\vert v\bigl(\mathscr{G}(\tau t)\bigr)-u\bigl(\mathscr{G}(\tau t) \bigr) \bigr\vert \bigr). \end{aligned}$$
$$ \hat{z}(t)\leq \epsilon \lambda _{\psi }\psi (t)+2\mathscr{L} {}_{q} \nabla _{a}^{-\xi }\hat{z}(t). $$
$$\begin{aligned} \hat{z}(t) &\leq \epsilon \lambda _{\psi }\psi (t){}_{q}E_{\xi } \bigl(2 \mathscr{L}\varGamma _{q}(\xi ),t-a\bigr) \\ &\leq \epsilon \psi (t) \bigl[ \lambda _{\psi }{}_{q}E_{\xi } \bigl(2 \mathscr{L}\varGamma _{q}(\xi ),t-a\bigr) \bigr]. \end{aligned}$$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \hat{z}(t)\leq \epsilon \psi (t) \bigl[ \lambda _{ \psi }{}_{q}E_{\xi }\bigl(2\mathscr{L} \varGamma _{q}(\xi ),t-a\bigr) \bigr], \quad \forall t. $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq \epsilon \psi (t) \bigl[ \lambda _{\psi }{}_{q}E_{ \xi }\bigl(2\mathscr{L}\varGamma _{q}(\xi ),T-a\bigr) \bigr], \quad \forall t< T. $$
$$ \bigl\vert v(t)-u(t) \bigr\vert \leq c\epsilon \psi (t). $$
4 Example
The following example will illustrate Theorem 3.
Example 1
Let us consider the following problem:
$$\begin{aligned} &{}_{q}^{c}\nabla ^{\frac{1}{2}}_{a}u(t) =\frac{\sin (u(t))+\sin (u( \tau t))}{200}, \quad t\in \mathbb{T}_{a}, \\ &u(t) =\cos 2t, \quad t\in \mathbb{I}_{\tau }. \end{aligned}$$
- \((A_{1})\)
\(\mathcal{F}(t,u,u^{*})=\frac{\sin u+\sin u^{*}}{200}\) and \(\varPhi (t)=\cos 2t\) are continuous functions.
- \((A_{2})\)
- \(\mathcal{F}\) satisfies the Lipschitz condition with Lipschitz constant \(\mathscr{L}=\frac{1}{200}\) as follows:$$\begin{aligned} \bigl\vert \mathcal{F}(t,u_{1},u_{2})- \mathcal{F}(t,v_{1},v_{2}) \bigr\vert &\leq \frac{1}{200}\bigl( \vert \sin u_{1}-\sin v_{1} \vert + \vert \sin u_{2}-\sin v_{2} \vert \bigr) \\ &\leq \frac{1}{200}\bigl( \vert u_{1}-v_{1} \vert + \vert u_{2}-v_{2} \vert \bigr). \end{aligned}$$
- \((A_{3})\)
\(\mathscr{G}(\tau t)=\tau t\) preserves the delay interval \(\mathbb{I}_{\tau }\),
- \((A_{4})\)
\(\frac{2\mathcal{L}(T-a)^{\xi }_{q}}{\varGamma _{q}(\xi +1)}=\frac{2( \frac{1}{200})(T-a)^{\frac{1}{2}}_{q}}{\varGamma _{q}(\frac{1}{2}+1)}<1\) for some \(T>a\).
Hence all the conditions are satisfied, so the problem under consideration is Ulam–Hyers stable.
5 Conclusion
We summarize our concluding statements by the following: We have investigated our work in the frame of the classical Caputo q-fractional operators. We plan to continue this type of study for other types of fractional operators on the quantum time scale.
- 1.
Studying and analyzing fractional dynamical systems on different time scales are essential, and they have various applications in engineering and science. We have done our investigations on the quantum time scale.
- 2.
We have proved the existence and uniqueness of solution for a delay Caputo q-fractional difference system depending on the Banach fixed point theorem and a recently proven version of q-Gronwall’s inequality.
- 3.
We have analyzed the Ulam–Hyers stability and the Ulam–Hyers–Rassias stability for q-fractional system under investigation.
- 4.
An example is provided to support the Ulam–Hyers–Rassias stability proven theoretical result.
Notes
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors have made equal contribution to this article. All authors read and approved the final manuscript.
Funding
This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.
Competing interests
There do not exist any competing interests regarding this article.
References
- 1.Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. (2) 20, 292–296 (1919) http://www.jstor.org/stable/10.2307/1967124 MathSciNetCrossRefGoogle Scholar
- 2.Mikeladze, Sh.E.: De la résolution numérique des équations intégrales. Bull. Acad. Sci. URSS VII, 255–257 (1935) (in Russian) zbMATHGoogle Scholar
- 3.Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007). https://doi.org/10.1016/j.jmaa.2006.05.061 MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Alzabut, J., Abdeljawad, T., Jarad, F., Sudsutad, W.: A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, 101, 1–12 (2019). https://doi.org/10.1186/s13660-019-2052-4 MathSciNetCrossRefGoogle Scholar
- 5.Wang, X., Fan, S.: A class of stochastic Gronwall’s inequality and its application. J. Inequal. Appl. 2018, 1 (2018). https://doi.org/10.1186/s13660-018-1932-3 MathSciNetCrossRefGoogle Scholar
- 6.Luo, D., Luo, Z.: Uniqueness and novel finite-time stability of solutions for a class of nonlinear fractional delay difference systems. Discrete Dyn. Nat. Soc. 2018, Article ID 8476285, 1–7 (2018). https://doi.org/10.1155/2018/8476285 MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Almeida, R., Malinowska, A.B., Odzijewicz, T.: An extension of the fractional Gronwall inequality. In: Advances in Non-Integer Order Calculus and Its Applications, 2019. https://doi.org/10.1007/978-3-030-17344-9_2 CrossRefzbMATHGoogle Scholar
- 8.Adjabi, Y., Jarad, F., Abdeljawad, T.: On generalized fractional operators and a Gronwall type inequality with applications. Filomat 31(17), 5457–5473 (2017). https://doi.org/10.2298/FIL1717457A MathSciNetCrossRefGoogle Scholar
- 9.Jarad, F., Abdeljawad, T., Hammouch, Z.: On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative. Chaos Solitons Fractals 117, 16–20 (2018) MathSciNetCrossRefGoogle Scholar
- 10.Atici, F.M., Eloe, P.W.: Gronwall’s inequality on discrete fractional calculus. Comput. Math. Appl. 64, 3193–3200 (2012). https://doi.org/10.1016/j.camwa.2011.11.029 MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Ferreira, R.A.C.: A discrete fractional Gronwall inequality. Proc. Am. Math. Soc. 140, 1605–1612 (2012). https://doi.org/10.1090/S0002-9939-2012-11533-3 MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Xu, R., Zhang, Y.: Generalized Gronwall fractional summation inequalities and their applications. J. Inequal. Appl., 2015, 242 1–10 (2015). https://doi.org/10.1186/s13660-015-0763-8 MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Fečkan, M., Pospisil, M.: Note on fractional difference Gronwall inequalities. Electron. J. Qual. Theory Differ. Equ. 2014, 64, 1–18 (2014) MathSciNetCrossRefGoogle Scholar
- 14.Abdeljawad, T., Al-Mdallal, Q.M.: Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality. J. Comput. Appl. Math. 339, 218–230 (2018). https://doi.org/10.1016/j.cam.2017.10.021 MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Alzabut, J., Abdeljawad, T.: A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solution and its application on the uniqueness of solutions for nonlinear delay fractional difference system. Appl. Anal. Discrete Math. 12, 036 (2018) www.jstor.org/stable/90020603 MathSciNetCrossRefGoogle Scholar
- 16.Ameen, R., Jarad, F., Abdeljawad, T.: Ulam stability for delay fractional differential equations with a generalized Caputo derivative. Filomat 32(15), 5265–5274 (2018). https://doi.org/10.2298/FIL1815265A MathSciNetCrossRefGoogle Scholar
- 17.Liu, K., Fečkan, M., ÓRegan, D., Wang, J.: Hyers–Ulam stability and existence of solutions for differential equations with Caputo–Fabrizio fractional derivative. Open Math., 7(4), 333, 1–14 (2019). https://doi.org/10.3390/math7040333 CrossRefGoogle Scholar
- 18.Wang, J., Fec̆kan, M., Zhou, Z.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012). https://doi.org/10.1016/j.jmaa.2012.05.040 MathSciNetCrossRefzbMATHGoogle Scholar
- 19.Wang, J., Zhou, Y., Fec̆an, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012). https://doi.org/10.1016/j.camwa.2012.02.021 MathSciNetCrossRefzbMATHGoogle Scholar
- 20.Wu, G.C., Baleanu, D., Zeng, S.D.: Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion. Commun. Nonlinear Sci. Numer. Simul. 57, 299–308 (2018). https://doi.org/10.1016/j.cnsns.2017.09.001 MathSciNetCrossRefGoogle Scholar
- 21.Jarad, F., Abdeljawad, T., Baleanu, D.: Stability of q-fractional non-autonomous systems. Nonlinear Anal., Real World Appl. 14(1), 780–784 (2013). https://doi.org/10.1016/j.nnorwa.2012.08.001 MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Jarad, F., Abdeljawad, T., Gundodu, E., Baleanu, D.: On the Mittag-Leffler stability of q-fractional nonlinear dynamical systems. Proc. Rom. Acad. 12(4), 309–314 (2011) MathSciNetGoogle Scholar
- 23.Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019). https://doi.org/10.1016/j.chaos.2019.07.026 MathSciNetCrossRefGoogle Scholar
- 24.Khan, H., Abdeljawad, T., Aslam, M., Khan, R.A., Khan, A.: Existence of positive solution and Hyers–Ulam stability for a nonlinear singular-delay-fractional differential equation. Adv. Differ. Equ. 2019, 104 (2019). https://doi.org/10.1186/s13662-019-2054-z MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Khan, A., Gómez-Aguilar, J.F., Khan, T.S., Khan, H.: Stability analysis and numerical solutions of fractional order HIV/AIDS model. Chaos Solitons Fractals 122, 119–128 (2019). https://doi.org/10.1016/j.chaos.2019.03.022 MathSciNetCrossRefGoogle Scholar
- 26.Khan, H., Jarad, F., Abdeljawad, T., Khan, A.: A singular ABC-fractional differential equation with p-Laplacian operator. Chaos Solitons Fractals 129, 56–61 (2019) MathSciNetCrossRefGoogle Scholar
- 27.Khan, H., Li, Y., Khan, A., Khan, A.: Existence of solution for a fractional order Lotka–Volterra reaction-diffusion model with Mittag-Leffler kernel. Math. Methods Appl. Sci. 42(9), 3377–3387 (2019). https://doi.org/10.1002/mma.5590 MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Khan, A., Khan, T.S., Syam, M.I., Khan, H.: Analytical solutions of time-fractional wave equation by double Laplace transform method. Eur. Phys. J. Plus 134(4), 163 (2019). https://doi.org/10.1140/epjp/i2019-12499-y CrossRefGoogle Scholar
- 29.Abdeljawad, T., Alzabut, J.: The q-fractional analogue for Gronwall-type inequality. J. Funct. Spaces Appl. 2013, Article ID 543839, 1–7 (2013). https://doi.org/10.1155/2013/543839 MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Abdeljawad, T., Alzabut, J., Baleanu, D.: A generalized q-fractional Gronwall inequality and its applications to nonlinear delay q-fractional difference systems. J. Inequal. Appl. 2016, 1 (2016). https://doi.org/10.1186/s13660-016-1181-2 MathSciNetCrossRefzbMATHGoogle Scholar
- 31.Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001) CrossRefGoogle Scholar
- 32.Atici, F.M., Eloe, P.W.: Fractional q-calculus on a time scale. J. Nonlinear Math. Phys. 14(3), 333–344 (2007). https://doi.org/10.2991/jnmp.2007.14.3.4 MathSciNetCrossRefzbMATHGoogle Scholar
- 33.Abdeljawad, T., Baleanu, D.: Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4682–4688 (2011) MathSciNetCrossRefGoogle Scholar
- 34.Abdeljawad, T., Alzabut, J.: On Riemann–Liouville fractional q-difference equations and their application to retarded logistic type model. Math. Methods Appl. Sci. 41, 8953–8962, 1–10 (2018). https://doi.org/10.1002/mma.4743 MathSciNetCrossRefzbMATHGoogle Scholar
- 35.Al-Salam, W.A.: q-Analogues of Cauchy’s formula. Proc. Am. Math. Soc. 17, 182–184 (1952–1953) MathSciNetGoogle Scholar
- 36.Al-Salam, W.A., Verma, A.: A fractional Leibniz q-formula. Pac. J. Math. 60, 1–9 (1975) MathSciNetCrossRefGoogle Scholar
- 37.Al-Salam, W.A.: Some fractional q-integrals and q-derivatives. Proc. Edinb. Math. Soc. 15, 135–140 (1969) MathSciNetCrossRefGoogle Scholar
- 38.Agrawal, R.P.: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philol. Soc. 66, 365–370 (1969). https://doi.org/10.1017/S0305004100045060 MathSciNetCrossRefzbMATHGoogle Scholar
- 39.Rajkovic, P.M., Marinkovi, S.D., Stankovi, M.S.: Fractional integrals and derivatives in q-calculus. Appl. Anal. Discrete Math. 1(1), 311–323 (2007) MathSciNetCrossRefGoogle Scholar
- 40.Annaby, M.H., Mansour, Z.S.: q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Heidelberg (2012) CrossRefGoogle Scholar
- 41.Rajkovic, P.M., Marinkovic, S.D., Stankovic, M.S.: On q-analogues of Caputo derivative and Mittag-Leffler function. Fract. Calc. Appl. Anal. 10, 359–373 (2007) MathSciNetzbMATHGoogle Scholar
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