Abstract
In the present paper, by introducing the concept of equipotentially almost automorphic sequence, the concept of weighted piecewise pseudo almost automorphic functions on time scales is proposed. Some first results about their basic properties are obtained and some composition theorems are established. Then we apply these to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive ∇-dynamic equations on time scales. In addition, the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, the obtained results are applied to the study of a class of ∇-partial differential equations on time scales.
MSC:34N05, 35B15, 43A60, 12H20, 35R12.
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1 Introduction
Almost automorphic functions, which are more general than almost periodic functions, were introduced by Bochner in relation to some aspects of differential geometry (see [1–3]). For more details as regards this topic we refer to the recent books [4–6], where the authors gave important overviews about the theory of almost automorphic functions and their applications to differential equations. Almost automorphic and pseudo almost automorphic solutions in the context of differential equations had been studied by several authors [7–21]. N’Guérékata [13] and Xiao [15, 21] with their collaborators established the existence and uniqueness theorems of pseudo almost automorphic solutions to some semilinear abstract differential equations. Recently, Blot et al. [22] introduced the concept of weighted pseudo almost automorphic functions, which generalizes the concept of weighted pseudo almost periodicity [23–26], and the author proved some interesting properties of the space of weighted pseudo almost automorphic functions like the completeness and the composition theorem, which have many applications in the context of differential equations. For other contributions to the study of weighted pseudo almost automorphy, we refer the reader to [27–30] and references therein.
On the other hand, the theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his PhD thesis in 1988 [31] in order to unify continuous and discrete analysis. This theory represents a powerful tool for applications to economics, population models, and quantum physics among others. In fact, the progressive field of dynamic equations on time scales contains links to and extends the classical theory of differential and difference equations. For instance, by choosing the time scale to be the set of real numbers, the general result yields a result for differential equations. In a similar way, by choosing the time scale to be the set of integers, the same general result yields a result for difference equations. However, since there are many other time scales than just the set of real numbers or the set of integers, one has a much more general result. For these reasons, based on the concept of almost periodic time scales proposed in [32, 33], the concept of weighted pseudo almost automorphic functions on almost periodic time scales was formally introduced by Wang and Li (2013) in [34]. Moreover, some first results were proven which concern the weighted pseudo almost automorphic mild solution to abstract Δ-dynamic equations on time scales. In addition, by using the results obtained in [32, 33], Lizama and Mesquita [35] presented some new results about basic properties of almost automorphic functions on time scales and proved the existence and uniqueness of an almost automorphic solution to a class of Δ-dynamic equations.
For another thing, many phenomena in nature are characterized by the fact that their states are subject to sudden changes at certain moments and therefore can be described by impulsive system (see [36, 37]). Many evolution processes, optimal control models in economics, stimulated neural networks, population models, artificial intelligence, and robotics are characterized by the fact that at certain moments of time they undergo abrupt changes of state. The existence of almost periodic solutions of abstract impulsive differential equations has been considered by many authors; see [38–41].
However, to the best of our knowledge, the concept of weighted piecewise pseudo almost automorphic functions on time scales has not been introduced in any literature until now, so there was no work on discussing weighted piecewise pseudo almost automorphic problems of impulsive dynamic equations on time scales before. Therefore, in this paper, by introducing the concept of equipotentially almost automorphic sequence, the concept of weighted piecewise pseudo almost automorphic functions on time scales is proposed. The first results about their basic properties are obtained and some composition theorems are established. Then we apply these composition theorems to investigate the existence of weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive ∇-dynamic equations as follows:
where is a linear operator in the Banach space and , . f, , satisfy suitable conditions that will be established later and is an almost periodic time scale. In addition, the notations and represent the right-hand and the left-hand side limits of at , respectively. In addition, some useful lemmas are obtained and the exponential stability of weighted piecewise pseudo almost automorphic mild solutions is also considered. Finally, we apply these obtained results to study a class of ∇-partial differential equations on time scales.
2 Preliminaries
In the following, we will introduce some basic knowledge of time scales which is very useful to the proof of our relative results.
A time scale is a closed subset of ℝ. It follows that the jump operators defined by and (supplemented by and ) are well defined. The point is left-dense, left-scattered, right-dense, right-scattered if , , , , respectively. If has a right-scattered minimum m, define ; otherwise, set . By the notations , and so on, we will denote time scale intervals
where with .
The graininess function is defined by : , for all .
Definition 2.1 ([42])
The function is called ld-continuous provided that it is continuous at each left-dense point and has a right-sided limit at each point, write . Let , the Delta derivative of f at t such that
for all , at fixed t. Let F be a function, it is called the antiderivative of provided for each . If , then we define the delta integral by
Definition 2.2 ([42])
A function is called ν-regressive provided for all . The set of all regressive and ld-continuous functions will be denoted by . We define the set .
Definition 2.3 ([42])
If r is a regressive function, then the generalized exponential function is defined by
for all , where the ν-cylinder transformation is as in
Lemma 2.1 ([42])
Assume that are two ν-regressive functions, then
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
.
Lemma 2.2 ([43])
For each in the single-point set is ∇-measurable and its ∇-measure is given by .
Lemma 2.3 ([43])
If and , then
If and , then
For more details of time scales and ∇-measurability, one is referred to [42, 43]. For more on time scales, see [44–49].
A time scale is called an almost periodic time scale if
Remark 2.4 Definition 3.1 introduced in [35] is the same as the concept of almost periodic time scales proposed in [32, 34], and is also called an invariant time scale under translations in [35].
After these preparations, in the next section, we will introduce the concept of weighted piecewise pseudo almost automorphic functions on time scales in a Banach space and some of their basic properties are investigated.
3 Weighted piecewise pseudo almost automorphic functions on time scales
In the following, we will give the definition of ld-piecewise continuous functions on time scales.
Definition 3.1 We say is ld-piecewise continuous with respect to a sequence which satisfy , , if is continuous on and ld-continuous on . Furthermore, are called intervals of continuity of the function .
For convenience, denotes the set of all ld-piecewise continuous functions with respect to a sequence , . Similar to Definition 3.1, we can also introduce the concept of functions which belong to .
Throughout the paper, we denote by a Banach space; let be the set consisting of all sequences such that . For , let be the space formed by all bounded ld-piecewise continuous functions such that is continuous at t for any and for all ; let Ω be a subset of and let be the space formed by all bounded piecewise continuous functions such that, for any , . For any , is continuous at .
Let be the space of all functions such that ϕ satisfies the condition: for any , there exists a positive number such that if the points , belong to the same interval of continuity of φ and implies .
Now, we introduce the set
which denotes all unbounded increasing sequences of real numbers. 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 3.2 Let , . We say is a derivative sequence of and
Definition 3.3 Let , . We say , , is equipotentially almost automorphic on an almost periodic time scale if, for any sequence , there exists a subsequence such that
is well defined for each and
for each .
Definition 3.4 A function is said to be ld-piecewise almost automorphic if the following conditions are fulfilled:
-
(i)
is an equipotentially almost automorphic sequence.
-
(ii)
Let be a bounded function with respect to a sequence . We say that φ is piecewise almost automorphic if from every sequence , we can extract a subsequence such that
is well defined for each and
for each . Denote by the set of all such functions.
-
(iii)
A bounded function with respect to a sequence is said to be piecewise almost automorphic if is piecewise automorphic in uniformly in , where B is any bounded subset of . Denote by the set of all such functions.
Similarly, we can also introduce the concept of piecewise almost automorphic functions which 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 3.1 In (2), if , , one can easily get
if , , one has the following:
Define
It is clear that . Now for define
Similarly, we define
We are now ready to introduce the sets and of weighted pseudo almost periodic functions:
Lemma 3.2 Let . Then where if and only if, for every ,
where 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 .
Now, we have
Therefore, , that is, . This completes the proof. □
Lemma 3.3 is a translation invariant set of with respect to Π if , i.e., for any , one has if .
Proof For any , , , , we have
Hence
Since , by Lemma 3.2, we have
Furthermore, , thus
Again, using Lemma 3.2, one can get for any . This completes the proof. □
By Definition 3.4, one can easily get the following lemma.
Lemma 3.4 Let , then the range of ϕ, , is a relatively compact subset of .
Lemma 3.5 If with , and , where , then .
Proof (1) For any , , one has . Since , there exists a sequence such that , .
Furthermore, by Lemma 3.3, , so there exists such that , . Hence, let , and one has
i.e. for each as .
-
(2)
If , noting that Definition 3.4, the above sequence and the number is suitable for the increasing sequence , so the proof process is the same as (1). This completes the proof. □
Lemma 3.6 The decomposition of a weighted piecewise pseudo almost automorphic function according to is unique for any .
Proof Assume that and . Then . Since , and , in view of Lemma 3.5, we deduce that . Consequently, , i.e. . This completes the proof. □
Theorem 3.7 For , is a Banach space.
Proof Assume that is a Cauchy sequence in . We can write uniquely . Using Lemma 3.5, we see that , from which we deduce that is a Cauchy sequence in . Hence, is a Cauchy sequence in . We deduce that , , and finally . This completes the proof. □
Definition 3.5 Let . One says that is equivalent to , written if .
Theorem 3.8 Let . If , then .
Proof Assume that . There exist such that . So
where and
This completes the proof. □
Lemma 3.9 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. □
Theorem 3.10 Let , 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 .
Proof We have , where and and , where and . Hence, the function can be decomposed as
By Lemma 3.9, . Now, consider the function
Clearly, . For Ψ to be in , it is sufficient to show that
Let K be a bounded subset of Ω such that , . By (ii), is uniformly continuous in uniformly in , and we see that, for given , there exists such that and implies that
Thus, for each , implies for all ,
where . For and any fixed , let , we can obtain
Now since , Lemma 3.2 yields
and this implies that .
Finally, we need to show . Note that and is uniformly continuous in uniformly in . By the assumption (ii), is uniformly continuous in uniformly in , so is ϕ. Since is relatively compact in , for , there exists such that , where for some and
It is easy to see that the set is open and . Define
Then it is clear that if , . So we get
In view of (3), it follows that
Thus we get
Since and , it follows that
by Lemma 3.2, . This completes the proof. □
Theorem 3.10 has the following consequence.
Corollary 3.11 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 3.12 If is an almost automorphic function with respect to the sequence T and is equipotentially almost automorphic satisfying , , then is an almost automorphic sequence in .
Proof Let , . Obviously, from the definition of Π, it is easy to know that . Since is an almost automorphic function and is equipotentially almost automorphic, from Definition 3.3 and Definition 3.4, for any sequence , we find that there exists a subsequence such that
and
Hence, is an almost automorphic sequence in . This completes the proof. □
Lemma 3.13 A necessary and sufficient condition for a bounded sequence to be in is that there exists a uniformly continuous function such that , , , .
Proof Necessity. We define a function
where . It is obviously uniformly continuous on . since
where , .
Sufficiency. Let , there exists such that, for , , we have
Without loss of generality, let , , , there exist 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 (4) and (5), we have
noting that implies , since , it follows from the inequality (6) that . This completes the proof. □
By Lemma 3.13, we can straightforwardly get the following theorem.
Theorem 3.14 A necessary and sufficient condition for a bounded sequence to be in is that there exists a uniformly continuous function such that , , , .
Theorem 3.15 Assume that and the sequence of vector-valued functions is weighted pseudo almost automorphic, i.e., for any , is weighted pseudo almost automorphic sequence. Suppose is bounded for every bounded subset , is uniformly continuous in uniformly in . If such that , then is a weighted pseudo almost automorphic sequence.
Proof Fix , first we show is weighted pseudo almost automorphic. Since , where , . It follows from Lemma 3.12 that the sequence is almost automorphic. To show that is weighted pseudo 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,
so one can obtain
it is easy to see that is increasing with respect to , and one can find some such that
from (7) and (8), we have
noting that implies , since , it follows from the inequality (9) that . Hence, is weighted pseudo almost automorphic.
Now, we show that is weighted pseudo almost automorphic. Let
Since , both are pseudo almost automorphic, by Lemma 3.13 and Theorem 3.14, 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 find that is uniformly in uniformly in . Then by Theorem 3.10, . Again, using Lemma 3.13 and Theorem 3.14, we find that is a weighted pseudo almost automorphic sequence, that is, is weighted pseudo almost automorphic. This completes the proof. □
From Theorem 3.15, one can easily get the following corollary.
Corollary 3.16 Assume the sequence of vector-valued functions is weighted pseudo almost automorphic, , if there is a number such that
for all , , if such that , then is a weighted pseudo almost automorphic sequence.
4 Weighted piecewise pseudo almost automorphic mild solutions to abstract impulsive ∇-dynamic equations
In this section, we investigate the existence and exponential stability of a weighted piecewise pseudo almost automorphic mild solution to Eq. (1). Before starting our investigation, we will show a lemma which will be used in our proofs.
Lemma 4.1 Let , for all , , there exist constants such that
Then there exist positive constants and such that
Proof Obviously, if , , the result holds. Assume that . Since , one has
Since is an almost periodic time scale, μ is bounded. Hence, by the inequality (10), we can obtain
Therefore, there exists a positive constant such that
where . This completes the proof. □
Remark 4.2 It is easy to see that if is almost periodic, then is bounded, so there exist a sufficiently small constant and a sufficiently large constant such that (10) is valid. Therefore, Lemma 4.1 holds when is an almost periodic time scale.
Let be an almost periodic time scale, and consider the impulsive ∇-dynamic 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 4.1 is called the linear evolution operator associated to (11) if satisfies the following conditions:
-
(1)
, where Id denotes the identity operator in ;
-
(2)
;
-
(3)
the mapping is continuous for any fixed .
Definition 4.2 An evolution system is called exponentially stable if there exist and such that
Definition 4.3 A function is called a mild solution of Eq. (1) if, for any , , , , one has
In fact, using the semigroup theory, we can easily see that
is a mild solution to
For any , we can find , , for ,
by using , we have
then we have
Repeating this procedure, we get
In the following, consider the abstract differential system (1) with the following assumptions:
(H1) The family of operators in generates an exponentially stable evolution system , i.e., there exist and such that
and for any sequence , there exists a subsequence such that
(H2) , where and is uniformly continuous in each bounded subset of Ω uniformly in ; is a weighted pseudo almost periodic sequence, is uniformly continuous in uniformly in , .
To investigate the existence and uniqueness of a weighted piecewise pseudo almost automorphic mild solution to Eq. (1), we need the following lemma.
Lemma 4.3 Let , , and (H1)-(H2) are satisfied. If is defined by
then .
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 ν-positive regressive. Further, noting that , by (H1) and Lemma 4.1, we have
where , .
Therefore, by the condition (H1), 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 4.1, we can get
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 4.4 Let , where and is exponentially stable, . Then
Proof Fix , then we have , where , , so
and
By Lemma 4.3, we can easily see that .
Moreover, we have
Then
and
Since , we have . Hence, .
It remains to show . For any , there exist , such that
Since , , and noting that, for , , we have
Since , for , , we have
Clearly, as , one has
Hence
Thus, , then . This completes the proof. □
Lemma 4.5 If satisfies the following inequality:
then
Proof Define
Consider
For , we can calculate that
This implies that . Further we have
Thus
This completes the proof. □
The following existence result is based on the contraction principle.
Theorem 4.6 Assume the following conditions hold:
(A1) The family of operators in generates an exponentially stable evolution system , i.e., there exist and such that
and for any sequence , there exists a subsequence such that
(A2) and f satisfies the Lipschitz condition with respect to the second argument, i.e.,
(A3) is a weighted pseudo almost periodic sequence, and there exists a number such that
for all , .
Assume that
then Eq. (1) has a unique weighted piecewise pseudo almost automorphic mild solution.
Proof Consider the nonlinear operator Γ given by
By Theorem 4.4, we see that Γ maps into .
It suffices now to show that the operator Γ has a fixed point in . For , one has the following:
Since , Γ is a contradiction. Hence, Γ has a fixed point in , then Eq. (1) has a unique weighted piecewise pseudo almost automorphic mild solution. This completes the proof. □
Finally, we investigate the exponential stability of a weighted piecewise pseudo almost automorphic mild solution to Eq. (1). By using the Gronwall-Bellman inequality (see Lemma 4.5) and the Lipschitz condition, it can be formulated as follows.
Theorem 4.7 Suppose the conditions of Theorem 4.6 hold. Assume further that , where , . Then Eq. (1) has a weighted piecewise pseudo almost automorphic mild solution which is exponentially stable.
Proof By Theorem 4.6, we know that Eq. (1) has a weighted piecewise pseudo almost automorphic mild solution , by using integral form of Eq. (1):
where , , .
Let and be two solutions of Eq. (1), then
Hence,
Let and it follows that
from Lemma 4.5, one has
or
where
Hence, Eq. (1) has a weighted piecewise pseudo almost automorphic mild solution which is exponentially stable. This completes the proof. □
Example 4.8 Let be an almost periodic time scale with and . Consider the following problem:
where satisfies () and , and , .
Define , let , . Clearly, it follows from the same discussion as in Section 3.1 in [50] that one can easily see that the evolution system satisfies () with , . Furthermore, by Definition 3.3, it is easy to check that , , is an equipotentially almost automorphic sequence and
Hence, . Let , . Clearly, both f and satisfy the assumptions given in Theorem 4.6 and Theorem 4.7 with . Moreover,
and since , one has , so
Therefore, Eq. (12) has a weighted piecewise pseudo almost automorphic mild solution which is exponentially stable.
5 Conclusion and further discussion
In this paper, the concept of weighted piecewise pseudo almost automorphic functions on time scales is introduced and discussed. It is well known that the Δ-dynamic equations are more popular in the references, however, ∇-dynamic equations are also interesting in both theory and practice. Therefore, we choose to investigate the weighted piecewise pseudo almost automorphic mild solutions to Eq. (1). All obtained results are essentially new.
Moreover, Definition 4.3 gives the expression of mild solutions to Eq. (1). It is worth emphasizing that the methods used in this paper can also be applied to study the weighted piecewise pseudo almost automorphic mild solutions to other abstract impulsive ∇-dynamic equations and Δ-dynamic equations. Now, similar to the discussion below Definition 4.3, we can list the mild solutions to three representative classes of abstract impulsive dynamic equations on time scales.
Type (i). Consider a ∇-dynamic equation as follows:
where is a linear operator in the Banach space and . Then, for any , , , , Eq. (13) has the following mild solution:
Type (ii). Consider a Δ-dynamic equation as follows:
where is a linear operator in the Banach space and , . Then, for any , , , , Eq. (14) has the following mild solution:
Type (iii). Consider a Δ-dynamic equation as follows:
where is a linear operator in the Banach space and . Then, for any , , , , Eq. (15) has the following mild solution:
By the same discussion method in this paper, one can show that the above three mild solutions are weighted piecewise pseudo almost automorphic solutions to their corresponding abstract impulsive dynamic equations on time scales. For example, in [49], the concept of rd-piecewise continuous functions is introduced, by the same discussion method in this paper, the weighted piecewise pseudo almost automorphic mild solutions to Eq. (14) and Eq. (15) can be studied. Furthermore, all the results can be applied to study almost automorphic solutions to impulsive dynamic models such as impulsive neural networks or biological models, economical models with impulsive effects on time scales.
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The authors would like to express their sincere thanks to the referees for suggesting some corrections that help making the content of the paper more accurate. This work is supported by the foundation of Yunnan University in China (No. 2013CG020).
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Wang, C., Agarwal, R.P. Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive ∇-dynamic equations on time scales. Adv Differ Equ 2014, 153 (2014). https://doi.org/10.1186/1687-1847-2014-153
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DOI: https://doi.org/10.1186/1687-1847-2014-153