1. Introduction
Almost automorphic functions, which are more general than the almost periodic functions, were introduced by Bochner (see [
1,
2,
3]) in relation to some aspects of differential geometry. Almost automorphic solutions in the context of differential equations have been studied by several researchers. For instance, pseudo and weighted pseudo almost automorphic mild solutions to (fractional) evolution equations were investigated by Chang et al. [
4,
5,
6], Ding et al. [
7,
8], Diagana [
9,
10]. Subsequently some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem were reported in [
11,
12] by N’Guérékata which have many applications in the context of differential equations. For more details about this topic we refer to the recent books (see [
10,
11]), where the authors gave important overviews about the theory of almost automorphic functions and their applications to differential equations.
Since time-scale calculus was proposed by Hilger [
13], Bohner and Guseinov have extensively developed this theory on the aspect of integral and dynamic equations (see [
14,
15]). To study the approximation properties of time scales, some new concepts such as almost periodic time scales and changing-periodic time scales were proposed and studied by Agarwal et al. (see [
16,
17]). In addition to these fundamental results, there have been many works on different types of dynamic equations on time scales. For example, the concept of variable time scales was introduced and a novel idea of the mutual transformation between impulsive dynamic equations and dynamic equations on variable time scales was initiated by Akhmet et al. [
18,
19,
20]. In the literature [
21], Bohner et al. established an SIR model on the general time scales and derived its exact solution. In [
22], the existing ideas of the univariate case of the time-scale calculus was generalized to the bivariate case and applied to partial dynamic equations. In the stability analysis, Martynyuk and Stamova investigated the sets of dynamic equations and hybrid dynamic systems on time scales (see [
23,
24]). In [
25,
26], two types of new high order derivations were introduced and the existence of solutions for the type high order fractional integro-differential equations was studied by Aydogan and Baleanu et al., and these types of fractional corresponding derivatives were generalized to time scales by Mozyrska, Ortigueira and Torres et al. (see [
27,
28]). In the field of studying functions and applications, it is a hot topic to study the almost automorphic and almost periodic functions and applications to dynamic equations based on time scales. For example, Hong investigated the almost periodic set-valued functions and almost periodic set dynamic equations on time scales (see [
29]). On almost automorphic functions and its related problems, Kéré, Mophou, N’Guérékata et al. investigated (
n-order ) almost automorphic and asymptotically almost automorphic functions of
n-order, some basic results were obtained and applied to abstract dynamic equations (see [
30,
31,
32]). In 2020, based on the concepts the authors introduced on translation time scales, Wang et al. established a theory of closedness of translation time scales and their applications to evolution equations and dynamical models (see the monograph [
33]). In addition, a new concept of periodic time scales and the notion of shift operators of time scales were proposed and studied under the background of studying periodic functions (see Adıvar et al. [
34,
35]). It is easy to observe that periodic time scales under translations have a nice closedness and their graininess function
is bounded.
However, some classical and important time scales are irregular and they have the unbounded graininess function
. For example,
, where
is the set of natural numbers or
or quantum-like time scale
(which has applications in quantum theory) and other types of time scales like
and
the space of the harmonic numbers (it is of interest to study almost automorphic dynamic behavior of solutions for
q-difference-like dynamic equations among others, see Wang et al. [
36,
37,
38]). It is impossible to introduce almost automorphic functions on such a type of time scale since the translation approximation of functions will never be reached for the reason that the graininess function
is a strictly increasing function for time scales. In addition, many natural phenomena must be modeled as a process in which continuous evolution is usually interrupted by an event (impulses, catastrophe, etc., see Stamova [
39,
40] and Wang et al. [
41,
42,
43]), which motivates us to investigate general evolution equations with impulses on irregular hybrid domains.
In the present paper, for the first time, we study the existence of weighted piecewise pseudo
S-almost automorphic mild solutions for the impulsive evolution dynamic equations
where
is a bounded linear operator in the Banach space
and
.
satisfy suitable conditions that will be established later and
is a complete-closed time scale attached with shift direction under non-translational shifts (S-CCTS). In addition, the notations
and
represent the right-hand and the left-hand side limits of
at
, respectively. In addition, some Lemmas are obtained and the exponential stability of weighted piecewise pseudo
S-almost automorphic mild solutions is also studied. Finally, we apply these obtained results to study a class of
-partial differential equations on
S-CCTS. The obtained results in this paper are feasible and effective on
q-difference partial dynamic equations and more.
For instance, in (
1), by using the shift operators
in
Section 2,
- (i)
if we let
and
,
, then (
1) turns into
- (ii)
if we let
and
,
, then (
1) turns into
- (iii)
if we let
and
,
, then (
1) turns into
- (iv)
if we let
and
,
, then (
1) turns into
where
if
and
if
, it is a classical
q-dynamic system on quantum-like hybrid domains.
We provide four types of impulsive evolution dynamic equations in the above, in fact, (
1) will turn into other different types of dynamic equations on different types of complete-closed time scales attached with shift direction under translational or non-translational shifts.
The highlights of the paper can be summarized as follows
We introduce the concept of S-equipotentially almost automorphic sequences under S-CCTS.
We establish a theory of discontinuous S-almost automorphic functions and weighted piecewise pseudo S-almost automorphic functions. Some new results about their basic properties and some related theorems are obtained.
The existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains is studied.
The obtained results in this paper are effective for q-difference heat equations and other dynamic equations on more general hybrid domains.
2. S-Equipotentially Almost Automorphic Sequence Under S-CCTS
In this section, we will introduce some knowledge of complete-closed time scales under non-translational shifts (or S-CCTS for short) and then define
S-equipotentially almost automorphic sequence and study its properties. For more details about time-scale calculus and
S-CCTS, one may refer to the book [
14,
17].
For convenience, we introduce the notations. Let
be the largest open subset of
, i.e.,
.
Definition 1 ([
36])
. Let be a time scale with the shift operators associated with the initial point . The time scale is said to be bi-direction S-CCTS in shifts if Remark 1. Note that from (
2)
and (
3)
, we obtain that (
4)
can be written into the equivalent form Example 1. According to Definition 1, we provide the following examples of S-CCTS. Let . We can obtain that For such a time scale, for any , take , we attach the shift operatorsHence, there exists such that for all , i.e., . From Definition 1, is a S-CCTS with bi-direction. Remark 2. Note that if is a periodic time scales under translations and , then the shift operators will fulfill with the initial point . Hence, if , then T-CCTS is included in S-CCTS.
If
is a bi-direction S-CCTS and
is the initial point, then for any
, we define a function
,
which will be used later. Note that
and
.
Remark 3. If and , let , then one can easily obtain .
In what follows, we will demonstrate some examples to show the almost automorphic phenomena for functions on S-CCTS, which are completely different from the cases on periodic time scales under translations.
Example 2. Let and , and we define the following operators:and Step 1. Periodic function construction. We know that the set of reals is periodic under shifts . The functionis periodic under shifts with the period since Step 2. Almost periodic function construction. Based on Step 1, consider the functionwhere and . One can observe that is almost periodic under shifts . From Step 1, letwe obtain that , and note that and are periodic with different periods , respectively. Step 3.S-almost automorphic function construction. According to the above, consider the functionwhere and . One can observe that is almost automorphic under shifts . From Step 2, we can obtain that . Example 3. The time scale is periodic with period τ under the shift operator .
Step 1. Periodic function construction. The piecewise periodic function defined byLet and , we have for all . Hence, is a periodic function with period . Similarly, let , we can obtain . Step 2. Almost periodic function construction.
Through Step 1, we can obtain an almost periodic piecewise functionon , where are periodic piecewise functions on , respectively. Note that the periods of and are completely different. Step 3. Almost automorphic function construction.
According to the above, letwhich is an almost automorphic function on . Let
with
.
Next, based on Definition 1, we will introduce the concept of S-equipotentially almost automorphic sequence and study its properties.
Definition 2. Assume is S-CCTS with shifts . Let . We say is a S-derivative sequence of and Remark 4. If is a periodic time scale under translations, then one can obtain the classical derivative sequence of satisfying by letting . Particularly, if , one can obtain the derivative sequence of from [40] (pp. 191–194) immediately. Lemma 1 ([
37])
. If be the largest subset of and including a fixed number such that there exist operators , then- (1)
- (2)
- (3)
Lemma 2 ([
37])
. If and , then Based on the S-derivative sequence and its properties, we will propose the following definition.
Definition 3. Let be a S-CCTS under shifts and . The sequence is said to be S-equipotentially almost automorphic if for any sequence , there exists a subsequence such thatis well defined for each andfor each . Remark 5. In Definition 3, note that is a closed subset of , thus, for each , one has .
3. S-Almost Automorphic Functions and Weighted Pseudo S-Almost Automorphic Functions
Let be a Banach space endowed with the norm . denotes the Banach space of all bounded linear operators from to . This is simply denoted as when . is the space of bounded continuous function from to equipped with the supremum norm defined by
In the following, we will give the definition of -piecewise continuous functions on time scales.
Definition 4. We say is -piecewise continuous with respect to a sequence which satisfy , if is continuous on and -continuous on . Furthermore, are called intervals of continuity of the function .
For convenience,
denotes the set of all
-piecewise continuous functions with respect to a sequence
. For
, let
let
be a subset of
and
Remark 6. is uniformly rd-continuous on the interval if and only if for any , there exists such that for all right dense points and implies (see Definition 2.1 of [29]). Now, let
and let
be a map such that the set
forms a strictly increasing sequence. For
and
, we introduce the notations
,
. Denote by
the element from the space
. For every sequence of real numbers
with
, we shall consider the sets
, where
Definition 5. The sequence , is convergent to pointwise, , , if and only if for any there exists such that impliesfor , is an arbitrary distance in . For convenience, consider the metric space
with the metric
where
and
,
.
Theorem 1. The metric space is complete.
Proof. For any given Cauchy sequence
, we can obtain that the sequences
and
be the Cauchy sequences in the metric space
and
respectively. Hence, for any
, there exists some
such that
implies
which yields that
for any
. Thus there exists
such that
implies
. Therefore, there exists
such that
implies
. Moreover, for
, we can obtain
where
and
, thus there exists some
such that
implies
so we obtain
, which indicates
is complete.
For any fixed
,
is a Cauchy sequence in the Banach space
, hence for any
, there exists some
such that
implies
We claim that
is also bounded on
. In fact, there exists some
such that
implies
, so for all
we have
To complete the proof, it is sufficient to show
in norm on
, i.e.,
as
. According to (
5), there exists some
such that
implies
for all
, so
. This completes the proof. □
Definition 6. Let be a bi-direction S-CCTS. A function is said to be -piecewise S-almost automorphic if the following conditions are fulfilled:
- (i)
Let be a S-equipotentially almost automorphic sequence.
- (ii)
Let be a bounded function with respect to a sequence . Then φ is said to be piecewise S-almost automorphic if from every sequence , we can extract a subsequence such thatis well defined for each andfor each . Denote by the set of all such functions. - (iii)
A bounded function with respect to a sequence is said to be piecewise S-almost automorphic if is piecewise S-automorphic in uniformly for , where B is any bounded subset of . Denote by the set of all such functions.
Similarly, we can also introduce the concept of piecewise S-almost automorphic functions that belong to .
Let
U be the set of all functions
which are positive and locally
-integrable over
. For a given
and
, set
for each
.
Remark 7. Particularly, if we let and , then (
6)
will turn into the integral on the quantum time scale: Moreover, let and , then the integral is It is clear that
. Now for
define
We are now ready to introduce the sets
and
of weighted pseudo
S-almost automorphic functions:
Lemma 3. Let be bi-direction S-CCTS under shifts and . Then where if and only if for every ,where , is the Δ-measurability function on the time scale and Proof. - (a)
Necessity. For contradiction, suppose that there exists
such that
Then there exists
such that for every
,
So we get
where
. This contradicts the assumption.
- (b)
Sufficiency. Assume that . Then for every , there exists such that for every , where and .
Therefore, , that is, . This completes the proof. □
Lemma 4. Let be bi-direction S-CCTS under shifts . Then is a shift invariant set of with respect to if , i.e., for any , one has if .
Proof. For any
,
, we have
Since
, then by Lemma 3, we have
Furthermore,
, thus
Again, using Lemma 3, one can get for any . This completes the proof. □
By Definition 6, the following two Lemmas are obvious.
Lemma 5. Let be bi-direction S-CCTS under shifts and , then the range of ϕ, , is a relatively compact subset of .
Lemma 6. Let be bi-direction S-CCTS under shifts . If with , and , where , then .
Lemma 7. Let be bi-direction S-CCTS under shifts . The decomposition of a weighted piecewise pseudo S-almost automorphic function according to is unique for any .
Proof. Let
, if
, then we have
. Hence, we obtain that there exists some positive constant
c such that
so
, i.e.,
and
. This completes the proof. □
Lemma 8. Let be a sequence of piecewise S-almost automorphic functions such that uniformly, then φ is piecewise S-almost automorphic.
Proof. From Definition 6, denote
and
. Let
be an arbitrary sequence of real numbers. Then we can extract a subsequence
of
such that
for each
pointwise.
We claim that the sequence of functions
is a Cauchy sequence. In fact, we can obtain
Let
. By the uniform convergence of
there exists a positive integer
N such that for all
implies
. By using (
7), (
8) and the completeness of the space
, we can deduce the pointwise convergence of the sequence
, say to a function
.
Now, we claim that and pointwise on .
Let
, there exists some positive integer
M such that
and
pointwise so that
since for each
M, there exists some positive integer
such that
, then we can obtain
for
, where
is some positive integer depending on
t and
.
Similarly, the same step can be applied for pointwise on , thus, we can obtain the desired result. This completes the proof. □
Theorem 2. Let be bi-direction S-CCTS under shifts . For , is a Banach space.
Proof. For any convergent sequence
with
uniformly for
, we can obtain
by letting
we have
which indicates
is a closed subspace of
. Therefore,
is itself a Banach space. Then by Lemmas 7 and 8, we have
is a Banach space. The proof is completed. □
Definition 7. Let . One says that equivalent , written if .
Theorem 3. Let . If , then .
Proof. Assume that
. There exist
such that
. So
where
, and
This completes the proof. □
Lemma 9. Let be bi-direction S-CCTS under shifts . If and , then .
Proof. Let
, from every sequence
, we can extract a subsequence
such that
uniformly exists on
. Since
, one can extract
such that
Hence,
. This completes the proof. □
By Lemmas 3 and 9, one can get the following theorem immediately without proof.
Theorem 4. Let be bi-direction S-CCTS under shifts and , where , and the following conditions hold:
- (i)
is bounded for every bounded subset .
- (ii)
are uniformly continuous in each bounded subset of Ω uniformly in .
Then if and .
Theorem 4 has the following consequence:
Corollary 1. Let , where . Assume that f and g are Lipschitzian in uniformly in . Then if .
Next, we will show the following two Lemmas, which are useful in the proof of our results.
Lemma 10. Let be bi-direction S-CCTS under shifts . If is an S-almost automorphic function with respect to the sequence T and is S-equipotentially almost automorphic satisfying , where is an initial point, then is an S-almost automorphic sequence in .
Proof. Let
. Obviously, from the definition of
, it is easy to know that
. Since
is an
S-almost automorphic function and
is
S-equipotentially almost automorphic, from Definitions 3 and 6, for any sequence
, there exists a subsequence
such that
and from
for each
k, we obtain
Hence,
is an
S-almost automorphic sequence in
. This completes the proof. □
Denote and .
To prove the following basic Lemma, we introduce notations , , , , and .
Lemma 11. Let be bi-direction S-CCTS under shifts and be Δ-differentiable to its second argument with , where is a positive number. A necessary and sufficient condition for a bounded sequence to be in is that there exists a uniformly continuous function such that , where Proof. Necessity. We define a function
Since
has the bounded
-derivative, it is uniformly continuous on
. For each
, note that
where
.
We have
since
where
.
Sufficiency. Let
, there exists
, for
, such that
Without loss of generality, let
and
there exists
such that
. Let
. Therefore,
so one can obtain
it is easy to see that
is increasing with respect to
, one can find some
such that
from (
9) and (
10), we have
noting that
implies
, since
, it follows from the inequality (
11) that
. This completes the proof. □
By Lemma 11, we can straightly get the following theorem:
Theorem 5. Let be bi-direction S-CCTS under shifts and be Δ-differentiable to its second argument with , where is a positive number. A necessary and sufficient condition for a bounded sequence to be in is that there exists a uniformly continuous function such that .
Theorem 6. Let be bi-direction S-CCTS under shifts and be Δ-differentiable to its second argument with , where is a positive number. Assume that and the sequence of vector-valued functions is weighted pseudo S-almost automorphic, i.e., for any is weighted pseudo S-almost automorphic sequence. Suppose is bounded for every bounded subset , is uniformly continuous in uniformly in . If such that , then is a weighted pseudo S-almost automorphic sequence.
Proof. Fix
, first we show
is weighted pseudo
S-almost automorphic. Since
, where
,
. It follows from Lemma 10 that the sequence
is
S-almost automorphic. To show
is weighted pseudo
S-almost automorphic, we need to show that
. By the assumption,
, so is
. Let
, there exists
such that for
, we have
Without loss of generality, let , there exists such that . Let . Therefore, repeating the proof of Lemma 11, we can obtain is weighted pseudo S-almost automorphic.
Now, we show
is weighted pseudo
S-almost automorphic. Let
Since
are two weighted pseudo
S-almost automorphic, by Lemma 11 and Theorem 5, we know that
. For every
, there exists a number
such that
,
Since
is bounded for every bounded set
is bounded for every bounded set
. For every
, we have
Noting that is uniformly continuous in uniformly in , we then get that is uniformly continuous in for . Then by Theorem 4, . Again, using Lemma 11 and Theorem 5, we have that is a weighted pseudo S-almost autmorphic sequence, that is, is weighted pseudo S-almost automorphic. This completes the proof. □
From Theorem 6, one can easily get the following corollary:
Corollary 2. Let be bi-direction S-CCTS under shifts and be Δ-differentiable to its second argument with , where is a positive number. Assume the sequence of vector-valued functions is weighted pseudo S-almost automorphic, , if there is a number such thatfor all and such that , then is a weighted pseudo S-almost automorphic sequence. 4. Weighted Piecewise Pseudo S-Almost Automorphic Mild Solutions to the Impulsive -Evolution Equations
In this section, we investigate the existence and exponential stability of a piecewise weighted pseudo
S-almost automorphic mild solution to Equation (
1). For this, we will provide a Lemma that will be used in our main results.
Lemma 12. Let be bi-direction S-CCTS under shifts and be Δ-differentiable to its second argument with , where is a positive number. Assume that , for all , there exist constants such that Then there exists positive constants and such that Proof. Obviously, if
,
, the result holds. Assume that
. Since
, one has
Hence, by the Inequality (
12), we can obtain
Therefore, there exists a positive constant
such that
where
. This completes the proof. □
Remark 8. It is easy to observe that if is bounded, then there exists a sufficiently small constant and a sufficiently large constant such that is valid. Therefore, Lemma 12 holds when is an almost periodic time scale from [33]. For the time scale from in Example 1, we can obtain thatHence, for all , we can obtain Consider the impulsive linear
-evolution equation
where
is a linear operator in the Banach space
. We denote by
the Banach space of all bounded linear operators from
to
. This is simply denoted as
when
.
Definition 8 ([
11])
. is called the linear evolution operator associated to (13) if satisfies the following conditions:- (1)
, where denotes the identity operator in ;
- (2)
- (3)
the mapping is continuous for any fixed
Definition 9. A function is called a mild solution of Equation (
1)
if for any , one has In the following, consider (
1) with the following assumptions:
- (H1)
Let be bi-direction S-CCTS under shifts and be -differentiable to its second argument with , where is a positive number.
- (H2)
The family
of operators in
generates an
S-exponentially stable evolution system
, i.e., there exist
and
such that
- (H3)
, where and is uniformly continuous in each bounded subset of uniformly in ; is a weighted pseudo S-almost periodic sequence, is uniformly continuous in uniformly in , , where is the initial point.
Remark 9. In the assumption , let , Equation (
14)
turns into Let and , it becomes Moreover, let the time scale be the quantum time scale , then Let
and
. To investigate the existence and uniqueness of a weighted piecewise pseudo
S-almost automorphic mild solution to Equation (
1), we need the following Lemma:
Lemma 13. Assume and are satisfied. If is defined bythen . Proof. Let . Since v is almost automorphic, there exists a subsequence such that is well defined for each .
Now, we consider
where
.
Since
, one can choose sufficiently small constant
such that
is also positive regressive. Further, noting that
, by
and Lemma 12, we have
Therefore, by the condition , we have Furthermore, it is easy to see that as , and for any , by Lebesgue’s dominated convergence theorem, we get
Moreover, we consider
where
. By Lemma 12, we can get
where
.
Since , . Hence, for any , by Lebesgue’s dominated convergence theorem, we get So we have is well defined for each . Therefore, . This completes the proof. □
Theorem 7. Assume that are satisfied. Let , where and is exponentially stable, . Then Proof. Fix
, then we have
, where
,
, so
and
By Lemma 13, we can easily see that .
Then
and
Since , we have . Hence, .
It remains to show
. For any
, there exist
such that
Since
, noting that for
,
, then we can obtain
Since
, for
, we have
Clearly, as
, one has
Thus, , then . This completes the proof. □
Theorem 8. Assume are satisfied and the following conditions hold:
- (A1)
The family of operators in generates an δ-exponentially stable evolution system , i.e., there exist and such that - (A2)
, and f satisfies the Lipschitz condition with respect to the second argument, i.e., - (A3)
is a weighted pseudo S-almost periodic sequence, and there exists a number such thatfor all .
Assume thatthen Equation (
1)
has a unique weighted piecewise pseudo S-almost automorphic mild solution. Proof. Consider the nonlinear operator
given by
By Theorem 7, we see that maps into .
It suffices now to show that the operator
has a fixed point in
. For
, one has the following:
Since
,
is a contradiction. Hence,
has a fixed point in
, then Equation (
1) has a unique weighted piecewise pseudo
S-almost automorphic mild solution. This completes the proof. □