1. Introduction and Preliminaries
In this paper, we investigate the existence and uniqueness of solutions for noninstantaneous impulsive fractional quantum Hahn integro-difference equation of the form:
subject to integral boundary condition
where
,
are the fractional quantum Hahn difference and integral operators of orders
,
, respectively, acting to the function on
with quantum numbers
,
,
and
,
,
,
are real constants. The functions
,
, are continuous, where
,
,
. The given impulsive points in
J satisfy
Prompted by the application of fractional order derivatives and integrals to applied mathematics, analytical chemistry, neuron modeling and biological sciences, the theory of fractional calculus has attracted great interest from the mathematical science research community. For examples and recent development of the topic, see ([
1,
2,
3,
4,
5,
6,
7]) and references cited therein.
Impulsive differential equations have become more important in some mathematical models of real phenomena, especially in control, biological, medical, and informational models. There are two types of impulses:
instantaneous impulses in which the duration of these changes is relatively short, and
non-instantaneous impulses in which an impulsive action, starting abruptly at a fixed point and continues on a finite time interval. Some examples of such processes can be found in physics, biology, population dynamics, ecology, pharmacokinetics, and others. For results with instantaneous impulses see, e.g., the monographs [
8,
9,
10], the papers [
11,
12,
13], and the references therein. Noninstantaneous impulsive differential equation was introduced by Hernández and O’Regan [
14]. For some recent works, we refer the reader to [
15,
16,
17,
18] and references therein.
The notion of
q-derivative was introduced in 1910 by Jackson [
19] as
provided that
exists. The
q-calculus appeared as a connection between mathematics and physics. The fundamental aspects of quantum calculus can be found in book [
20]. For some recent results in quantum calculus we refer to the papers [
21,
22,
23] and the references cited therein.
Hahn [
24] established the difference operator
in 1949,
provided that
f is differentiable at
where
and
are fixed constants. The Hahn difference operator unifies the Jackson
q-difference derivative
, where
defined by (
3), for
and the forward difference
for
defined by
where
is a fixed constant. The Hahn difference operator is used for constructing families of orthogonal polynomials and investigating some approximation problems, (cf. [
25,
26,
27]).
In 2013, Tariboon and Ntouyas [
28], presented the new concepts of quantum calculus on
by defining
With the help of definition (
4), a series of quantum initial and boundary value problems contain impulses were studied. We refer the interested reader to the recent monograph [
29] for details.
In 2016, Tariboon et al. [
30], gave the generalization of Hahn difference operator on
by
provided that
f is differentiable at
.
Let
,
be quantum constants and
be the point defined by
such that
,
. The quantum shifting operator
is defined by
Please note that if
and
then (
1) is reduced to
and
respectively. The power function
,
, is defined by
where
,
. If we put (
5) and (
6) in (
7), then we obtain
and
respectively. The Gamma function in quantum calculus is defined by
where
is defined by Formula (
8). Indeed,
, where
,
, is the quantum number or
q-number. However, in our work, the number
q will be replaced by
on an interval
,
.
In the following definitions, we give the Riemann–Liouville fractional derivative and integral in Hahn calculus as well as the Caputo fractional derivative which can be found in [
31,
32,
33].
Definition 1. The fractional quantum Hahn difference operator of a Riemann–Liouville type of order on interval is defined by andwhere n is the smallest integer greater than or equal to . Definition 2. Let and f be a function defined on . The fractional quantum Hahn integral operator of Riemann–Liouville type is given by and Definition 3. The fractional quantum Hahn difference operator of Caputo type on interval is defined by andwhere n is the smallest integer greater than or equal to . In our work, the problem (
1) and (
2) is based on fractional quantum Hahn calculus in Definitions 2 and 3. If
, that is
, then Definitions 1–3 are reduced to quantum calculus on the finite interval in the framework of Tariboon and Ntouyas [
28] (see [
29,
30] for more details). Now we present some properties of fractional calculus on any interval
,
.
Theorem 1 ([
31]).
Let ,
and be given constants. The following formulas hold:;
.
Theorem 2 ([
31]).
Let be a function defined on an interval and constants ,
and ,
.
Then, we have:;
;
;
for some, , .
The rest of the paper is organized as follows. In
Section 2, we first prove a basic lemma helping us to convert the boundary value problem (
1) and (
2) into an equivalent integral equation. Then we prove the main results, one existence and uniqueness result, via Banach contraction mapping principle and one existence result by using Leray–Schauder nonlinear alternative. Some special cases are discussed. Examples are also constructed to illustrate the main results in
Section 3. The paper closes with a conclusion
Section 4.
2. Main Results
In this section, we present our main results. The next lemma deals with a linear variant of the boundary value problem (
1) and (
2).
Lemma 1. Let , , , , , , β, γ and λ be given constants which satisfy the problem (1) and (2). Assume that Then the following linear noninstantaneous impulsive fractional quantum Hahn difference equations with integral boundary conditions has a unique solution , , of the formwith , if Proof. Firstly, we take the quantum fractional Hahn integral of order
to the first equation of (
9) on an interval
and set
, then
On the first jump interval,
, calling the second equation of (
9) with (
11), we have
Next consecutive subinterval
, again by applying the quantum fractional Hahn integral of order
, we get
and then
by (
12).
Similar to the above method, we have for
as
and for
as
Generally, we compute that
with
, if
. From this, we have
Now, taking the quantum Hahn fractional integral of order
to (
13) over
, multiplying constant
and summing for
, we have
From (
14) and (
15) and boundary integral condition in (
9), we get
Substituting (
16) into (
13) we get (
10). The converse follows by direct computation. The proof is completed. □
Now, we establish the existence and uniqueness results for the boundary value problem (
1) and (
2) on
. We first define the Banach space
equipped with the norm
Based on the Lemma 1, we define the operator
by
where the notation
,
is used, that is,
which is convenient in our computations. The first result concerns the existence of a unique solution of the problem (
1) and (
2), and will be proved by using the Banach contraction mapping principle, involving the following constants:
Theorem 3. Let and , , be given continuous functions such thatwhere , and . Ifthen the boundary value problem of the noninstantaneous impulsive fractional quantum Hahn integro-difference in Equations (1) and (2) has a unique solution in J. Proof. First of all, we will show that
,where
, and the radius
r is defined by
Setting
and
, and using (
18) we have
Then, for any
, we obtain
which holds from (
19) and (
20). This shows that
. To show that
is a contraction operator, we let
and
,
, then
since
Therefore, the operator
satisfies
. As,
, we can conclude that the operator
is a contraction mapping which has a unique fixed point in
. Hence the problem (
1) and (
2) has a unique solution on
J. This completes the proof. □
If we set
,
,
, and
in (
2), then we have
which leads to the initial value problem (
1)–(
21). In addition, some constants are reduced to
,
and
. Then we get the following corollary.
Corollary 1. Suppose that f and g satisfy the conditions of Theorem 3. Ifthen the initial value problem of the noninstantaneous impulsive fractional quantum Hahn integro-difference Equations (1)–(21) has a unique solution on J. The next lemma is called the nonlinear alternative for single valued maps [
34] which will be used to prove the next result.
Lemma 2. Let X be a Banach space, U a closed, convex subset of V an open subset of U and Suppose that is a continuous, compact (that is, is a relatively compact subset of U) map. Then either
- (i)
B has a fixed point in or
- (ii)
there is a (the boundary of V in U) and with
Theorem 4. Assume that and , , are continuous functions. In addition, we suppose that:
there exist a continuous nondecreasing function and three continuous functions such that for all and ,
there exists a constant such that with
Then the boundary value problem of the noninstantaneous impulsive fractional quantum Hahn integro-difference Equations (1) and (2) has at least one solution on J. Proof. Let
be the operator defined in (
17) and now we are going to prove that the operator
is compact on a bounded ball
, where
:
. For any
, we have
which yields
. Then the set
is uniformly bounded. In compactness, we need to prove that
is an equicontinuous set. Let
and
be two points such that
. Then, for any
, we can compute that
for
,
. Therefore, we have
as
. For the case
,
, we have
Then is an equicontinuous set. Therefore, we can conclude that the set is relative compact. Thus, applying Arzelá-Ascoli theorem, the operator is completely continuous.
Finally, we will show that there exists an open set
and
such that
, where
and
. Let
and
for some
. Then for any
, using the computation in the first step, we obtain
which can be written as
From
, there exists a positive constant
K such that
. Then we define
:
. From above, the operator
is continuous and completely continuous. Therefore, there is no
such that
with
. Applying Lemma 2 we get that the operator
has a fixed point
, which obviously is a solution of problem (
1)–(
2) on
J. The proof is completed. □
Please note that the nonlinear condition (
22) of functions
f and
g is a very general condition. However, we can reduce it to be a linear one by
by choosing
,
,
and
, where constants
and
.
Corollary 2. If f and , , satisfy (24) and if , then the problem (1) and (2) has at least one solution on J. If the function
f is bounded by linear integro-term, i.e.,
by setting
,
,
then we have the corollary.
Corollary 3. Let be bounded, f satisfies condition (25) and . Then the problem (1) and (2) has at least one solution on J. Finally, we state the corresponding existence result for the initial value problem discussed in Corollary 1.
Corollary 4. Assume that f and g satisfy condition (22). If there exists a positive constant K satisfying with then the initial value problem (1)–(21) has at least one solution on J. 3. Examples
In this section, we give some examples to illustrate the usefulness of our main results.
Example 1. Consider the following noninstantaneous impulsive fractional quantum Hahn integro-difference equation with integral boundary conditions of the form:
Setting , , , , , , , , , , , , , and , . From above information, we can compute that , , , , , , , , , and . Please note that , .
Let the nonlinear function
f be defined by
Now we see that
which is Lipschitz with Lipschitz constants
and
. Then we obtain
which implies by the conclusion of Theorem 3 that the problem (
26) and (
27) has a unique solution on
.
Consider now the function
f defines as
It is easy to observe that
Thus, we set
,
,
. From (
26), we also set
. Then we get
,
and
which can be computed that there exists a positive constant
satisfying inequality in (
23). Therefore the problem (
26)–(
28) satisfies all conditions of Theorem 4 which leads to conclude that there exists at least one solution on
.
If
f is defined by
then we get
and
. By setting
,
and
, we obtain
which implies by Corollary 2 that the problem (
26)–(
29) has least one solution on
.
Put the function
f by
From (
25), we set
and
. Since
, we deduce by Corollary 3 that the problem (
26)–(
30) has least one solution on
.