Abstract
In this paper, we introduce new concepts of Hahn difference operator, the \(q_{k},\omega_{k}\)-Hahn difference operator. We aim to establish a calculus of differences based on the \(q_{k},\omega_{k}\)-Hahn difference operator. We construct a right inverse of the \(q_{k},\omega_{k}\)-Hahn operator and study some of its properties. As applications, we establish existence and uniqueness results for first- and second-order impulsive \(q_{k},\omega_{k}\)-Hahn difference equations.
Similar content being viewed by others
1 Introduction and preliminaries
Many physical phenomena are described by equations involving nondifferentiable functions, e.g., generic trajectories of quantum mechanics [1]. Several different approaches to deal with nondifferentiable functions are followed in the literature, including the time scale approach, the fractional approach, and the quantum approach.
Quantum difference operators are receiving an increase of interest due to their applications see, e.g., [2–10]. Roughly speaking, a quantum calculus substitutes the classical derivative by a difference operator, which allows one to deal with sets of nondifferentiable functions.
In [11], Hahn introduced the quantum difference operator \(D_{q,\omega}\), where \(q\in (0,1)\) and \(\omega>0\) are fixed. The Hahn operator unifies (in the limit) the two best-known and most-used quantum difference operators: the Jackson q-difference derivative \(D_{q}\), where \(q\in(0,1)\) (cf. [6, 12, 13]); and the forward difference \(D_{\omega}\) where \(\omega>0\) (cf. [14–16]). The Hahn difference operator is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems (cf. [17–19]).
The aim of this paper is to introduce new concepts of Hahn’s difference operator, the \(q_{k},\omega_{k}\)-Hahn difference operator, to establish a calculus based on this operator and to construct the associated integral. The steps are parallel to [20]. While some properties are straightforward extensions of classical results, some others need special treatments. As applications of the \(q_{k},\omega_{k}\)-Hahn difference operator we establish existence and uniqueness results for first- and second-order impulsive fractional differential equations.
Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. Recent development in this field has been motivated by many applied problems, such as control theory, population dynamics, and medicine. For some recent works on the theory of impulsive differential equations, we refer the interested reader to the monographs [21–23]. Impulsive quantum difference equations have been established by Tariboon and Ntouyas in [24] by improving the classical quantum calculus which does not work when there exists at least one impulsive point appearing between two different points in the definition of q-derivative. For recent results on the topics of initial and boundary value problems of impulsive quantum difference equations, we refer the reader to [7].
We organize this paper as follows. In Section 2, some basic formulas of Hahn’s difference operator and the associated Jackson-Nörlund integral calculus are briefly reviewed. Our results are formulated and proved in Section 3. Applications to impulsive fractional difference equations are given in Section 4.
2 Preliminaries
Let \(q\in(0,1)\) and \(\omega>0\). Define
and let I be a real interval containing \(\omega_{0}\).
Definition 2.1
(Hahn’s difference operator [11])
Let \(f: I\to{\mathbb{R}}\). The Hahn difference operator of f is defined by
provided that f is differentiable at \(\omega_{0}\).
The function f is called \(q,\omega\)-differentiable on I, if \(D_{q,\omega}f(t)\) exists for all \(t\in I\).
Note that when \(q\to1\) we obtain the forward ω-difference operator
and when \(\omega= 0\) we obtain the Jackson q-difference operator
provided that \(f'(0)\) exists. Here f is supposed to be defined on a q-geometric set \(A\subset{\mathbb{R}}\), for which \(qt\in A\) whenever \(t\in A\).
Hence, we can state that the \(D_{q,\omega}\) operator generalizes (in the limit) the forward ω-difference and the Jackson q-difference operators [6, 25].
Notice also that, under appropriate conditions,
The Hahn difference operator has the following properties.
Lemma 2.2
([20])
Let \(f,g:I\to\mathbb{R}\) be \(q,\omega \)-differentiable at \(t\in I\). Then the following statements are true:
-
(i)
\(D_{q,\omega}(f+g)(t)=D_{q,\omega}f(t)+D_{q,\omega}g(t)\),
-
(ii)
\(D_{q,\omega}fg(t)=g(t)D_{q,\omega}f(t)+f(qt+\omega )D_{q,\omega}g(t)\),
-
(iii)
\(D_{q,\omega}cf(t)=cD_{q,\omega}f(t)\), for any constant \(c\in\mathbb{R}\),
-
(iv)
\(D_{q,\omega} (\frac{f}{g} )(t)=\frac{g(t)D_{q,\omega}f(t)-f(t)D_{q,\omega }g(t)}{g(t)g(qt+\omega)}\), for \(g(t)g(qt+\omega)\neq0\),
-
(v)
\(f(tq+\omega)=f(t)+((qt + \omega)-t)D_{q,\omega}f(t)\), \(t\in I\).
Let \(h(t)=qt+\omega\), \(t\in I\). Note that h is a contraction, \(h(I)\subseteq I\), \(h(t)< t\) for \(t>\omega_{0}\), \(h(t)>t\) for \(t<\omega _{0}\), and \(h(\omega_{0})=\omega_{0}\).
We use the standard notation of the q-number as \([\alpha]_{q}=\frac{1-q^{\alpha}}{1-q}\) for \(\alpha\in{\mathbb{R}}\).
Lemma 2.3
([20])
Let \(k\in{\mathbb{N}}\) and \(t\in I\). Then
Next, we define the notion of a \(q,\omega\)-integral, known as the Jackson-Nörlund integral.
Definition 2.4
([20])
Let \(f:I\to\mathbb{R}\) be a function and \(a,b,\omega_{0}\in I\). The \(q,\omega\)-integral of f from a to b is defined by
where
provided that the series converges at \(t=a\) and \(t=b\).
The function f is \(q,\omega\)-integrable over I if it is \(q,\omega \)-integrable over \([a,b]\), for all \(a,b\in I\).
Note that in the integral formulas (2.6) and (2.7), when \(\omega\to0\), we obtain the Jackson q-integral
where
(see, e.g., [26]); while if \(q\to1\) we obtain the Nörlund sum,
where
The following properties of Jackson-Nörlund integration can be found in [20].
Lemma 2.5
Let \(f,g:I\to\mathbb{R}\) be \(q,\omega\)-integrable on I, \(K\in\mathbb{R}\), and \(a,b,c\in I\). Then the following formulas hold:
-
(i)
\(\int_{a}^{a}f(t)\,d_{q,\omega}t=0\),
-
(ii)
\(\int_{a}^{b}Kf(t)\,d_{q,\omega}t=K\int_{a}^{b}f(t)\,d_{q,\omega}t\),
-
(iii)
\(\int_{a}^{b}f(t)\,d_{q,\omega}t=-\int_{b}^{a}f(t)\,d_{q,\omega}t\),
-
(iv)
\(\int_{a}^{b}f(t)\,d_{q,\omega}t=\int_{c}^{b}f(t)\,d_{q,\omega }t+\int_{a}^{c}f(t)\,d_{q,\omega}t\),
-
(v)
\(\int_{a}^{b} (f(t)+g(t) )\,d_{q,\omega}t=\int _{a}^{b}f(t)\,d_{q,\omega}t+\int_{a}^{b} g(t)\,d_{q,\omega}t\),
-
(vi)
\(\int_{a}^{b}f(t)D_{q,\omega}g(t)\,d_{q,\omega}t= [f(t)g(t) ]_{a}^{b}-\int_{a}^{b}D_{q,\omega}f(t)g(qt+\omega)\,d_{q,\omega}t\).
Property (vi) of the above lemma is known as \(q,\omega\)-integration by parts.
The next result is the fundamental theorem of Hahn calculus.
Lemma 2.6
([20])
Let \(f:I\to\mathbb{R}\) be continuous at \(\omega_{0}\) and define \(F(t):=\int_{\omega_{0}}^{t}f(s)\,d_{q,\omega}s\). Then F is continuous at \(\omega_{0}\). In addition, \(D_{q,\omega}F(t)\) exists for every \(t\in I\) and
On the other hand,
Existence and uniqueness results for first-order abstract Hahn difference equations were studied in [29], by using the method of successive approximation.
3 New concepts of Hahn calculus
Let there be a dense interval \(J_{k}=[t_{k},t_{k+1}]\subseteq\mathbb{R}\) and given constants \(0< q_{k}<1\), \(\omega_{k}>0\) and
Note that if \(t_{k}=0\), \(q_{k}=q\), and \(\omega_{k}=\omega\), then \(\theta _{k}=\omega_{0}\), where \(\omega_{0}\) is defined in (2.1).
Definition 3.1
Let f be a function defined on \(J_{k}\). The \(q_{k},\omega_{k}\)-Hahn difference operator is given by
and \({{}_{t_{k}}}D_{q_{k},\omega_{k}}f(\theta_{k})=f'(\theta_{k})\) provided that f is differentiable at \(\theta_{k}\).
We say that f is \(q_{k},\omega_{k}\)-differentiable on \(J_{k}\) provided \({{}_{t_{k}}}D_{q_{k},\omega_{k}}f(t)\) exists for all \(t\in J_{k}\). Note that if \(\omega_{k}= 0\) in (3.2), then \({{}_{t_{k}}}D_{q_{k},0}f={{}_{t_{k}}}D_{q_{k}}f\), where \({{}_{t_{k}}}D_{q_{k}}\) is the \(q_{k}\)-derivative of the function \(f(t)\) which was first established in [24] by
It is easy to see that if \(t_{k}=0\) and \(q_{k}=q\), then (3.3) is reduced to the Jackson q-difference operator in (2.4).
Example 3.2
Let \(f(t)=t^{2}\) for \(t\in J_{k}=[2,16]\) and constants \(q_{k}=1/2\), \(\omega_{k}=3\). Then \(\theta_{k}=8\) and the \(q_{k},\omega _{k}\)-Hahn derivative on \(J_{k}\) is given by
and \({{}_{2}}D_{\frac{1}{2},3}f(8)=64\).
It is easy to prove the following results.
Theorem 3.3
Let \(f,g:J_{k}\to\mathbb{R}\) be \(q_{k},\omega _{k}\)-differentiable at \(t\in J_{k}\). Then the following formulas hold:
-
(i)
\({{}_{t_{k}}}D_{q_{k},\omega_{k}}(f+g)(t)={{}_{t_{k}}}D_{q_{k},\omega _{k}}f(t)+{{}_{t_{k}}}D_{q_{k},\omega_{k}}g(t)\),
-
(ii)
\({{}_{t_{k}}}D_{q_{k},\omega_{k}}fg(t)=g(t){{}_{t_{k}}}D_{q_{k},\omega _{k}}f(t)+f(q_{k}t+(1-q_{k})t_{k}+\omega_{k}){{}_{t_{k}}}D_{q_{k},\omega_{k}}g(t)\),
-
(iii)
\({{}_{t_{k}}}D_{q_{k},\omega_{k}}cf(t)=c{{}_{t_{k}}}D_{q_{k},\omega _{k}}f(t)\), for any constant \(c\in\mathbb{R}\),
-
(iv)
\({{}_{t_{k}}}D_{q_{k},\omega_{k}} (\frac {f}{g} )(t)=\frac{g(t){{}_{t_{k}}}D_{q_{k},\omega _{k}}f(t)-f(t){{}_{t_{k}}}D_{q_{k},\omega_{k}}g(t)}{g(t)g(q_{k}t+(1-q_{k})t_{k}+\omega _{k})}\), for \(g(t)g(q_{k}t+(1-q_{k})t_{k}+\omega_{k})\neq0\).
Next, we define the higher-order \(q_{k},\omega_{k}\)-derivative of functions.
Definition 3.4
Let f be a function defined on \(J_{k}\). We define the second-order \(q_{k},\omega_{k}\)-derivative \({{}_{t_{k}}}D_{q_{k},\omega_{k}}^{2}f\) provided \({{}_{t_{k}}}D_{q_{k},\omega_{k}}f\) is \(q_{k},\omega_{k}\)-differentiable on \(J_{k}\) with \({{}_{t_{k}}}D^{2}_{q_{k},\omega_{k}}f={{}_{t_{k}}}D_{q_{k},\omega _{k}}({{}_{t_{k}}}D_{q_{k},\omega_{k}}f):J_{k}\rightarrow\mathbb{R}\). In addition, we define the higher-order \(q_{k},\omega_{k}\)-derivative \({{}_{t_{k}}}D_{q_{k},\omega_{k}}^{n}f:J_{k}\rightarrow\mathbb{R}\), with \({{}_{t_{k}}}D_{q_{k},\omega_{k}}^{n}f={{}_{t_{k}}}D_{q_{k},\omega _{k}}({{}_{t_{k}}}D_{q_{k},\omega_{k}}^{n-1}f)\) and \({{}_{t_{k}}}D_{q_{k},\omega_{k}}^{0}f=f\).
The new definition of \(q_{k},\omega_{k}\)-integral is given as follows.
Definition 3.5
Assume \(f:J_{k}\rightarrow\mathbb{R}\) is a function and \(a,b\in J_{k}\). We define the \(q_{k},\omega_{k}\)-integral of f from a to b by
where
for \(t\in J_{k}\), provided that the series converge at \(t=a\) and \(t=b\). The function f is called \(q_{k},\omega_{k}\)-integrable on \(J_{k}\) and we say that f is \(q_{k},\omega_{k}\)-integrable over \([a,b]\) for all \(a,b\in J_{k}\).
Note that if \(t_{k}=0\), \(q_{k}=q\), and \(\omega_{k}=\omega\), then (3.4) and (3.5) are reduced to (2.6) and (2.7), respectively.
As customary, the following properties should be to stated. However, the proof is easy and we omit it.
Theorem 3.6
Let \(f,g:J_{k}\to\mathbb{R}\) be \(q_{k},\omega_{k}\)-integrable on \(J_{k}\), \(K\in\mathbb{R}\), and \(a,b,c\in J_{k}\). Then the following formulas hold:
-
(i)
\(\int_{a}^{a}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=0\),
-
(ii)
\(\int_{a}^{b}Kf(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=K\int _{a}^{b}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s\),
-
(iii)
\(\int_{a}^{b}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=-\int _{b}^{a}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s\),
-
(iv)
\(\int_{a}^{b}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=\int _{c}^{b}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s+\int_{a}^{c}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s\),
-
(v)
\(\int_{a}^{b} (f(s)+g(s) ){_{t_{k}}}\,d_{q_{k},\omega _{k}}s=\int_{a}^{b}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s+\int_{a}^{b} g(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s\).
Lemma 3.7
Let h be the transformation
and \(\theta_{k}\in J_{k}\) is defined by (3.1). Then the ith-order iteration of h is given by
In addition, the sequence \(\{h^{i}(t)\}_{i=1}^{\infty}\) is an increasing (a decreasing) sequence in i when \(t<\theta_{k}\) (\(\theta_{k}< t\)) with
Proof
By directly computation, it is easy to show that (3.7) holds. For \(t\in J_{k}\) and \(i\in\mathbb{N}\), we have
If \(t<\theta_{k}\) or \(\theta_{k}< t\), then we see that the sequence \(\{ h^{i}(t)\}_{i=1}^{\infty}\) is increasing or decreasing, respectively. Therefore, equation (3.8) is true for all \(t\in J_{k}\). □
Now, we will state and prove the fundamental theorem of \(q_{k},\omega _{k}\)-Hahn calculus.
Theorem 3.8
Suppose that the function \(f:J_{k}\to \mathbb{R}\) is continuous at \(\theta_{k}\in J_{k}\). We define
Then we have, for \(t,a,b\in J_{k}\),
-
(i)
\({{}_{t_{k}}}D_{q_{k},\omega_{k}}F(t)=f(t)\),
-
(ii)
\(\int_{\theta_{k}}^{t}{{}_{t_{k}}}D_{q_{k},\omega _{k}}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=f(t)-f(\theta_{k})\),
-
(iii)
\(\int_{a}^{b}{{}_{t_{k}}}D_{q_{k},\omega _{k}}f(s){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=f(b)-f(a)\).
Proof
From (3.9), we observe that
Then, by (3.2), we have
This shows that (i) holds.
To prove (ii), by Definitions 3.1, 3.5, and Lemma 3.7, we get
Now, we show that (iii) holds. From (ii) for any \(a,b\in J_{k}\), we obtain
This completes the proof. □
Lemma 3.9
Let \(f, g:J_{k}\to\mathbb{R}\) be \(q_{k},\omega_{k}\)-integrable on \(J_{k}\). Then the following integration by parts formula holds:
Proof
By Theorem 3.8 we have
On the other hand, by (ii) of Theorem 3.3 and (v) of Theorem 3.6,
Combining these two equalities we get the desired formula. □
Lemma 3.10
Let \(\theta_{k}\in J_{k}\), \(\alpha\in\mathbb{R}\), and \(\beta\in\mathbb{R}\setminus\{-1\}\). Then for \(t\in J_{k}\) the following formulas hold:
-
(i)
\({{}_{t_{k}}}D_{q_{k}}(t-\theta_{k})^{\alpha}= [\alpha]_{q_{k}}(t-\theta_{k})^{\alpha-1}\),
-
(ii)
\(\int_{\theta_{k}}^{t}(s-\theta_{k})^{\beta }{_{t_{k}}}\,d_{q_{k},\omega_{k}}s= (\frac{1-q_{k}}{1-q_{k}^{\beta +1}} )(t-\theta_{k})^{\beta+1}\).
Proof
From Definition 3.1, for \(t\neq\theta _{k}\), we have
For \(t=\theta_{k}\), we obtain \({{}_{t_{k}}}D_{q_{k},\omega_{k}}0=0\). Therefore the formula (i) holds.
Now, we are going to prove (ii). For \(\beta\in\mathbb{R}\setminus \{-1\}\), Definition 3.5 implies
The proof is completed. □
Corollary 3.11
For \(a,b\in J_{k}\), the following formula holds:
Example 3.12
From Corollary 3.11 for \(a,b\in J_{k}\), we have the following cases:
-
(i)
If \(\beta=0\), then \(\int_{a}^{b}1 {_{t_{k}}}\,d_{q_{k},\omega_{k}}s=b-a\).
-
(ii)
If \(\beta=1\), then \(\int_{a}^{b}(s-\theta _{k}){_{t_{k}}}\,d_{q_{k},\omega_{k}}s=\frac{(b-a)}{1+q_{k}} [b+a-2\theta _{k} ]\).
-
(iii)
\(\int_{t_{k}}^{b}(s-t_{k}){_{t_{k}}}\,d_{q_{k},\omega _{k}}s=\frac{(b-t_{k})^{2}-\omega_{k}(b-t_{k})}{1+q_{k}}\).
(i) and (ii) are obvious. To prove (iii), from (i) and (ii) we obtain
Theorem 3.13
Let f be the \(q_{k},\omega_{k}\)-integrable function on \(J_{k}\). Then we have
Proof
By Definition 3.5, we have
Indeed,
Hence, we obtain
This completes the proof. □
4 Impulsive \(q_{k},\omega_{k}\)-Hahn difference equations
In this section, we use our results on \(q_{k},\omega_{k}\)-Hahn calculus to establish existence and uniqueness results for impulsive \(q_{k},\omega _{k}\)-Hahn difference equations of the first and second order. Let \(J_{0}=[t_{0}, t_{1}]\), \(J_{k}=(t_{k}, t_{k+1}]\) for \(k=1,2,\ldots,m\) be subintervals of \(J=[0, T]\) such that \(\theta_{k}\in J_{k}\) for \(k=0,1,2,\ldots,m\). Let \(PC(J, \mathbb{R})\) = {\(x: J\rightarrow \mathbb{R}: x(t)\) is continuous everywhere except for some \(t_{k}\) at which \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) exist and \(x(t_{k}^{-})=x(t_{k})\), \(k=1,2,\ldots,m\)}. \(PC(J, \mathbb{R})\) is a Banach space with the norm \(\|x\|_{PC}=\sup\{ |x(t)|: t\in J\}\).
4.1 First-order impulsive \(q_{k},\omega_{k}\)-Hahn difference equations
In this subsection, we study the existence and uniqueness of solutions for the following initial value problem for first-order impulsive \(q_{k},\omega_{k}\)-Hahn difference equation
where \(\alpha\in\mathbb{R}\), \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{k}<\cdots <t_{m}<t_{m+1}=T\), \(f:J\times\mathbb{R}\rightarrow\mathbb{R}\) is a continuous function, \(\varphi_{k}\in C(\mathbb{R},\mathbb{R})\), \(\Delta x(t_{k})=x(t_{k}^{+})-x(t_{k})\), \(k=1,2,\ldots,m\), and quantum numbers \(0< q_{k}<1\), \(\omega_{k}>0\) such that \(\theta_{k}\in J_{k}\) for \(k=0, 1, 2, \ldots,m\).
Lemma 4.1
Let \(x\in PC(J,\mathbb{R})\) satisfying (4.1). The impulsive \(q_{k},\omega_{k}\)-Hahn difference initial value problem (4.1) is equivalent to the integral equation
with \(\sum_{t_{0}< t_{0}}=0\).
Proof
For \(t\in J_{0}\), applying \(q_{0},\omega_{0}\)-integral from \(t_{0}\) to t in the first equation of (4.1) and using Theorem 3.8(iii), we obtain
Since \(\theta_{0}\in J_{0}\), we have \(t_{1}\geq\theta_{0}\) and also, for \(t=t_{1}\),
For \(t\in J_{1}\), taking the \(q_{1},\omega_{1}\)-integral to the first equation of (4.1) with \(k=1\) and applying Theorem 3.8(iii) again, we have
From the impulsive condition \(x(t_{1}^{+})=x(t_{1})+\varphi_{1}(x(t_{1}))\), we get
For \(t\in J_{2}\), the \(q_{2},\omega_{2}\)-integration and impulsive condition imply
From the above process, for any \(t\in J_{k}\), \(k=0,1,\ldots,m\), we obtain the desired result in (4.2).
Conversely, for any \(t\in J_{k}\), \(k=0,1,\ldots,m\), applying \(q_{k},\omega _{k}\)-derivative to (4.2) and using Theorem 3.8(i), we have
By direct computation, we have \(\Delta x(t_{k})=\varphi_{k} (x(t_{k}) )\) and also \(x(0)=\alpha\). The proof is completed. □
Now, we are in a position to prove an existence and uniqueness result for the problem (4.1), via Banach contraction mapping principle.
Theorem 4.2
Suppose that the following assumptions are fulfilled:
- (H1):
-
the continuous function \(f:J\times \mathbb{R}\to\mathbb{R}\) satisfies
$$ \bigl\vert f(t,x)-f(t,y)\bigr\vert \leq L_{1}\vert x-y\vert , \quad L_{1}>0, \forall t\in J, x,y\in\mathbb{R}; $$ - (H2):
-
the continuous functions \(\varphi_{k}:\mathbb{R}\to \mathbb{R}\), \(k=1,2,\ldots, m\) satisfy
$$ \bigl\vert \varphi_{k}(x)-\varphi_{k}(y)\bigr\vert \leq L_{2}\vert x-y\vert , \quad L_{2}>0, \forall x,y\in \mathbb{R}. $$
If
then the impulsive \(q_{k},\omega_{k}\)-Hahn difference initial value problem (4.1) has a unique solution on J.
Proof
Let us define an operator \(\mathcal{A}:PC(J,\mathbb {R})\to PC(J,\mathbb{R})\) by
with \(\sum_{t_{0}< t_{0}}=0\). Let \(\sup_{t\in J}|f(t,0)|=M_{1}\) and \(\max\{ |\varphi_{k}(0)|: k=1,2,\ldots,m\}=M_{2}\). Choosing a positive constant r such that
and setting a ball \(B_{r}=\{x\in PC(J,\mathbb{R}) : \|x\|\leq r\}\), we will show that \(\mathcal{A}B_{r}\subset B_{r}\). For any \(x\in B_{r}\) and \(t\in J\), we have
This means that \(\|\mathcal{A}x\|\leq r\), which yields \(\mathcal {A}B_{r}\subset B_{r}\).
For \(x,y\in PC(J,\mathbb{R})\) and for each \(t\in J\), we have
which leads to \(\|\mathcal{A}x-\mathcal{A}y\|\leq(L_{1}T+mL_{2})\|x-y\| \). As \(L_{1}T+mL_{2}<1\), it follows from the Banach contraction mapping principle that \(\mathcal{A}\) is a contraction. Hence, we deduce that \(\mathcal{A}\) has a fixed point which is the unique solution of (4.1) on J. This completes the proof. □
Example 4.3
Consider the first-order impulsive \(q_{k},\omega_{k}\)-Hahn difference initial value problem of the form
Here \(J=[0,10]\), \(q_{k}=(k+1)/(k+2)\), \(\omega_{k}=1/(k+3)\), \(k=0,1,\ldots ,9\), \(m=9\), \(T=10\), \(f(t,x)=(1/(t^{2}+40))((x^{2}+2|x|)/(|x|+1))e^{-t}+(3/4)\), and \(\varphi _{k}(x)=(|x|)/((4+k)(|x|+4))\). Observe that \(\theta_{k}=\omega _{k}/(1-q_{k})+t_{k} =(k^{2}+4k+2)/(k+3)\in J_{k}\), \(k=0,1,\ldots,9\). Since \(|f(t,x)-f(t,y)|\leq(1/20)|x-y|\) and \(|\varphi_{k}(x)-\varphi _{k}(y)|\leq(1/20)|x-y|\), then (H1) and (H2) are satisfied with \(L_{1}=1/20\) and \(L_{2}=1/20\), respectively. We can show that
Therefore, by Theorem 4.2, we deduce that the problem (4.4) has a unique solution on \([0, 10]\).
4.2 Second-order impulsive \(q_{k},\omega_{k}\)-Hahn difference equations
In this subsection, we consider the second-order initial value problem of the impulsive \(q_{k},\omega_{k}\)-Hahn difference equation
where \(\alpha,\beta\in\mathbb{R}\), \(0=t_{0}< t_{1}< t_{2}<\cdots <t_{k}<\cdots<t_{m}<t_{m+1}=T\), \(f\in C(J\times\mathbb{R},\mathbb{R})\), \(\varphi_{k}, \varphi^{*}_{k}\in C(\mathbb{R},\mathbb{R})\), \(\Delta x(t_{k})=x(t_{k}^{+})-x(t_{k})\), \(k=1,2,\ldots,m\), and the numbers \(0< q_{k}<1\), \(\omega_{k}>0\) such that \(\theta_{k}\in J_{k}\) for \(k=0, 1, 2, \ldots,m\).
Lemma 4.4
A function \(x\in PC(J,\mathbb {R})\) is the solution of (4.5) if and only if x satisfies the integral equation
with \(\sum_{t_{0}< t_{0}}=0\).
Proof
For \(t\in J_{0}\), taking \(q_{0},\omega_{0}\)-integral for the first equation of (4.5) and using the second initial condition, we get
which leads to
For \(t\in J_{0}\), the \(q_{0},\omega_{0}\)-integration for (4.7) and the first initial condition of (4.5) imply
In particular, for \(t=t_{1}\), we have
Let us consider the interval \(J_{1}=(t_{1},t_{2}]\). By the \(q_{1},\omega _{1}\)-integration for (4.5) with respect to \(t\in J_{1}\), we have
From the second impulsive condition of (4.5), that is, \({{}_{t_{1}}}D_{q_{1},\omega_{1}}x(t_{1}^{+})={{}_{t_{0}}}D_{q_{0},\omega _{0}}x(t_{1})+\varphi_{1}^{*}(x(t_{1}))\), we obtain
For \(t\in J_{1}\), taking the \(q_{1},\omega_{1}\)-integration for (4.8) and using Example 3.12(i), we get
Applying the first impulsive condition of (4.5), that is, \(x(t_{1}^{+})=x(t_{1})+\varphi_{1}(x(t_{1}))\), we obtain
Repeating the above method, for \(t\in J\), we obtain (4.6) as desired.
Conversely, it can easily be shown by direct computation that the integral equation (4.6) satisfies the impulsive initial value problem (4.5). This completes the proof. □
From Example 3.12(iii) with \(b=t_{k+1}\), we set the notation
Also, we use the notations
where \(U\in\{L,N\}\).
Theorem 4.5
Assume that the conditions (H1) and (H2) of Theorem 4.2 are satisfied. Further, we suppose that:
- (H3):
-
The continuous functions \(\varphi_{k}^{*}:\mathbb{R}\to \mathbb{R}\), \(k=1,2,\ldots, m\), satisfies
$$ \bigl\vert \varphi_{k}^{*}(x)-\varphi_{k}^{*}(y)\bigr\vert \leq L_{3}\vert x-y\vert , \quad L_{3}>0, \forall x,y\in \mathbb{R}. $$
If
where \(\Psi(L)\) is defined by (4.9), then the impulsive \(q_{k},\omega_{k}\)-Hahn difference initial value problem (4.5) has a unique solution on J.
Proof
In view of Lemma 4.4, we define an operator \(\mathcal{Q}:PC(J,\mathbb{R})\to PC(J,\mathbb{R})\) by
with \(\sum_{t_{0}< t_{0}}=0\). By transforming the impulsive initial value problem (4.5) into a fixed point problem \(x=\mathcal{Q}x\), we will show that the operator \(\mathcal{Q}\) has a fixed point which is a unique solution of problem (4.5) via the Banach contraction mapping principle.
Setting \(\sup_{t\in J}|f(t,0)|=N_{1}\), \(\max\{|\varphi_{k}(0)|: k=1,2,\ldots,m\}=N_{2}\), and \(\max\{|\varphi_{k}^{*}(0)|: k=1,2,\ldots,m\}=N_{3}\), we will prove that \(\mathcal{Q}B_{R}\subset B_{R}\), where \(B_{R}=\{x\in PC(J,\mathbb{R}):\|x\|\leq R\}\) and the positive constant R satisfies
For \(x\in B_{R}\), taking into account Example 3.12(iii), we get
Then we have \(\|\mathcal{Q}x\|\leq R\), which implies \(\mathcal {Q}B_{R}\subset B_{R}\).
Finally, for \(x,y\in PC(J,\mathbb{R})\) and for each \(t\in J\), we get
It follows that \(\|\mathcal{Q}x-\mathcal{Q}y\|\leq\Psi(L)\|x-y\|\). As \(\Psi(L)<1\), we deduce from the Banach contraction mapping principle that \(\mathcal{Q}\) is a contraction. Therefore, we see that the operator \(\mathcal{Q}\) has a fixed point which is a unique solution of the impulsive \(q_{k},\omega_{k}\)-Hahn difference initial value problem (4.5) on J. The proof is completed. □
Example 4.6
Consider the second-order impulsive \(q_{k},\omega_{k}\)-Hahn difference initial value problem of the form
Here \(J=[0,10]\), \(q_{k}=(k+3)/(4k+6)\), \(\omega_{k}=(k+1)/(2k+5)\), \(k=0,1,\ldots,9\), \(m=9\), \(T=10\), \(\alpha=2/3\), \(\beta=5/7\), \(f(t,x)=(1/(t^{2}+10))((x^{2}+2|x|)/(1+|x|))(e^{-\cos^{2}t}/88)+(1/2)\), \(\varphi_{k}(x)=(|x|/(5(k+5)(1+|x|)))+(2/3)\), and \(\varphi _{k}^{*}(x)=(|\sin x|)/(10(\sqrt{k}+40))+(3/4)\). Observe that \(\theta _{k}=\omega_{k}/(1-q_{k})+t_{k} =(6k^{2}+19k+6)/(6k+15)\in J_{k}\), \(k=0,1,\ldots ,9\). Also, we can find that \(\sum_{k=1}^{10}\Omega(k-1)=4.720324567\).
Since \(|f(t,x)-f(t,y)|\leq(1/440)|x-y|\), \(|\varphi_{k}(x)-\varphi _{k}(y)|\leq(1/30)|x-y|\), and \(|\varphi_{k}^{*}(x)-\varphi_{k}^{*}(y)|\leq (1/410)|x-y|\), (H1), (H2), and (H3) are satisfied with \(L_{1}=1/440\), \(L_{2}=1/30\), and \(L_{2}=1/410\), respectively. From the above information, we find that
Therefore, by Theorem 4.5, we deduce that the problem (4.12) has a unique solution on \([0, 10]\).
References
Feynman, RP, Hibbs, AR: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)
Almeida, R, Torres, DFM: Hölderian variational problems subject to integral constraints. J. Math. Anal. Appl. 359, 674-681 (2009)
Bangerezako, G: Variational q-calculus. J. Math. Anal. Appl. 289, 650-665 (2004)
Bangerezako, G: Variational calculus on q-nonuniform lattices. J. Math. Anal. Appl. 306, 161-179 (2005)
Cresson, J, Frederico, GSF, Torres, DFM: Constants of motion for non-differentiable quantum variational problems. Topol. Methods Nonlinear Anal. 33, 217-231 (2009)
Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)
Ahmad, B, Ntouyas, SK, Tariboon, J: Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities. World Scientific, Singapore (2016)
Zhang, L, Baleanu, D, Wang, G: Nonlocal boundary value problem for nonlinear impulsive \(q_{k}\)-integrodifference equation. Abstr. Appl. Anal. 2014, Article ID 478185 (2014)
Abdeljawad, T, Baleanu, D, Jarad, F, Agarwal, RP: Fractional sums and differences with binomial coefficients. Discrete Dyn. Nat. Soc. 2013, Article ID 104173 (2013)
Abdeljawad, T, Baleanu, D: Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 16, 4682-4688 (2011)
Hahn, W: Über Orthogonalpolynome, die q-Differenzengleichungen genügen. Math. Nachr. 2, 4-34 (1949)
Gasper, G, Rahman, M: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)
Jackson, FH: Basic integration. Q. J. Math. 2, 1-16 (1951)
Bird, MT: On generalizations of sum formulas of the Euler-Maclaurin type. Am. J. Math. 58, 487-503 (1936)
Jagerman, DL: Difference Equations with Applications to Queues. Dekker, New York (2000)
Jordan, C: Calculus of Finite Differences, 3rd edn. Chelsea, New York (1965). Introduction by Carver, HC
Kwon, KH, Lee, DW, Park, SB, Yoo, BH: Hahn class orthogonal polynomials. Kyungpook Math. J. 38, 259-281 (1998)
Lesky, PA: Eine Charakterisierung der klassischen kontinuierlichen, diskretenund q-Orthgonalpolynome. Shaker, Aachen (2005)
Petronilho, J: Generic formulas for the values at the singular points of some special monic classical \(H_{q,\omega}\)-orthogonal polynomials. J. Comput. Appl. Math. 205, 314-324 (2007)
Annaby, MH, Hamza, AE, Aldwoah, KA: Hahn difference operator and associated Jackson-Nörlund integrals. J. Optim. Theory Appl. 154, 133-153 (2012)
Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995)
Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, vol. 2. Hindawi Publishing Corporation, New York (2006)
Tariboon, J, Ntouyas, SK: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 282 (2013)
Ernst, T: The different tongues of q-calculus. Proc. Est. Acad. Sci. 57, 81-99 (2008)
Jackson, FH: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193-203 (1910)
Fort, T: Finite Differences and Difference Equations in the Real Domain. Clarendon, Oxford (1948)
Nörlund, N: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924)
Hamza, AE, Ahmed, SM: Existence and uniqueness of solutions of Hahn difference equations. Adv. Differ. Equ. 2013, 316 (2013)
Acknowledgements
This research was funded by Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Thailand. Contract No. 5942107.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this article. They read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Tariboon, J., Ntouyas, S.K. & Sudsutad, W. New concepts of Hahn calculus and impulsive Hahn difference equations. Adv Differ Equ 2016, 255 (2016). https://doi.org/10.1186/s13662-016-0982-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-016-0982-4