Almost Automorphic Functions of Order and Applications to Dynamic Equations on Time Scales

Abstract

We revisit the notion on almost automorphic functions on time scales given by Lizama and Mesquita (2013). Then we present the notion of almost automorphic functions of order . Finally, we apply this notion to study the existence and uniqueness and the global stability of almost automorphic solution of first order to a dynamical equation with finite time varying delay.

1. Introduction

The concept of time scales was initiated in 1988 by Hilger in his outstanding Ph.D. thesis [1]. The purpose of such theory was to unify both continuous and discrete analysis. Consequently, using time scales in studying dynamic systems prevents from proving results separately for differential equations and difference equations. Since then, several papers were devoted to dynamical systems on time scales [2–8]. We refer also readers to the excellent book by Bohner and Peterson [9] and their edited book [10] which contains high quality contributions to the theory.

It was natural to study almost periodic time scales as well as almost periodic differential equations on almost periodic time scales [3]. Our initial motivation for the current study comes from [4] where the authors studied the existence and exponential stability of almost periodic solutions of a neutral-type BAM neural network with delays on time scales, using exponential dichotomy of linear dynamical systems.

Recently Lizama and Mesquita introduced the notion of almost automorphic functions on time scales in their work [11]. The purpose of this paper is twofold. First we would like to revisit Lizama and Mesquita’s paper in light of the following remarks.

Let be a time scale. It is said to be invariant under translations if We prove in Lemma 25 that .

However, we observe that the inclusion may be strict. Indeed, let us consider the time scale , where ; it is obviously invariant under translations, and it contains and but not . Then . This also proves that the invariant under translations time scales is not symmetric.

For this reason, several results in [11] hold only if the time scale is symmetric.

Secondly we would like to study the existence and stability of almost automorphic solutions of the following linear dynamic system with finite delay: where is an appropriate time scale, is the number of neurons in the network, denote the activation of the th neuron at time , and represents the rate with which the th neurons will rest their potential to the resting state in isolation when they are disconnected from the network and the external inputs at time . The matrix represents the connection strengths between neurons at time ; is an element of feedback templates at time ; is the activation function. is the transmission delay at time and satisfies ; denote the bias of the th neuron at time and .

We organized our paper as follows. In Section 2, we recall some definitions and recent results on time scales. In Section 3, we present properties of almost automorphic functions on symmetric time scales along with a composition theorem. In Section 4, we introduce and present elementary properties of almost automorphic functions on time scales of order and study differentiation and integration of such functions in Section 5 and superposition of operators on the space of such functions in Section 6. Finally in Section 7, we study the existence, uniqueness, and global stability of system (2).

2. Preliminaries

In this section we recall some definitions and recent results on time scales.

Definition 1. A time scale is an arbitrary nonempty closed subset of real numbers.

Definition 2. Let be a time scale. The forward and backward jump operators and the graininess are defined, respectively, by

In Definition 5, we put and .

Definition 3. Let be time scales.(i)A point is called right-dense if and .(ii)A point is called left-dense if and .

Definition 4 (see [9]). A function is called regulated provided its right-sided limits exist (finite) at all right-dense points in and its left-sided limits exist (finite) at all left-dense points in .

Definition 5 (see [9]). A function is called -continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in .

We will denote the set of -continuous function by . From now on, we define the set by

Definition 6 (see [9]). Let be a function and . We define to be the number (provided it exists) with the property that, given any , there exists a neighborhood of such that We call the delta (or Hilger) derivative of at .
Moreover, we say that is delta (or Hilger) differentiable (or in short differentiable) on provided exists for all . The function is called the (delta) derivative of on .

Next we recall some easy and useful relationships concerning the delta derivative.

Theorem 7 (see [9]). Assume is a function and let . Then one has the following. (i)If is differentiable at , then is continuous at .(ii)If is continuous at and is right-scattered, then is differentiable at with (iii)If   is right-dense, then is differentiable at if and only if the limit exists as a finite number. In this case (iv)If is differentiable at , then

Theorem 8 (see [9]). Assume are differentiable at . Then (i)the sum is differentiable at with (ii)for any constant , is differentiable at with (iii)the product is differentiable at with

We define higher order derivatives of a function on time scale in the usual way.

Definition 9 (see [9]). For a function one will talk about the second derivative provided is differentiable on with derivative . Similarly one defines higher order derivatives . Finally, for , one denotes and , and and are defined accordingly. For convenience one also puts and .

Theorem 10 (see [9] (Leibniz formula)). Let be the set consisting of all possible strings of length , containing exactly times and times . If exists for all , then holds for all .

The following results on chain rule can be found in [9].

Theorem 11 (chain rule). Let be continuously differentiable and suppose is delta differentiable on . Then is delta differentiable and the formula holds for all .

Theorem 12 (chain rule). Assume that is strictly increasing and is a time scale. Let , where is a Banach space. If and exist for , then where denote the in .

Definition 13. If , , and is a function on , then the improper integral of is defined by provided this limit exists. In this case, the improper integral is said to converge.

Lemma 14 (see [4]). Let , , and assume that is continuous at , where with . Assume also that is -continuous on . Suppose that, for each , there exists a neighborhood of such that where denotes the derivative of with respect to the first variable. Then

We now present some definitions and results useful for the study of some dynamical systems.

Definition 15 (see [9]). One says that a function is regressive provided
The set of all regressive functions will be denoted by .

Definition 16. One defines the set of all positively regressive elements of by

Definition 17 (see [9]). If , then one defines the generalized exponential function by where the cylinder transformation is given by where is the principal logarithm function. For , we define for all .

The generalized exponential functions have the following properties.

Lemma 18 (see [9]). Assume that are two regressive functions. Then(i) and ;(ii);(iii); (iv).

Lemma 19 (see [9]). Assume that . Then(i), for all ;(ii)if for all , , then for all .

Lemma 20 (see [9]). If and , then (i), (ii) for all .

Proposition 21 (see [9]). Let be -continuous and regressive, , and . Then the unique solution of the initial value problem is given by

We now present some definitions about matrix-valued functions on .

Definition 22. Let be an matrix-valued function on . One says that is -continuous on if each entry of is -continuous on . One denotes by the class of all -continuous matrix-valued functions on .
We say that is delta differentiable on if each entry of is delta differentiable on . And in this case, we have

Definition 23. An matrix-valued function is called regressive if The class of all such regressive -continuous functions is denoted by

3. Almost Automorphic Functions of Order on Time Scales

From now on, is a (real or complex) Banach space.

Definition 24 (see [11]). A time scale is called invariant under translations if

Lemma 25. Let be an invariant under translations time scale. Then one has (i);(ii).

Proof. (i)In view of the definition of , if , then for all we have . Thus, . Conversely, assume that . Then, for any , we have . Thus .(ii)It is clear that if then . Now assume that . If , then we have for any ; particularly, for , we have . This contradicts the fact that . Thus .

We have the following properties of the points in , forward jump operator, and the graininess function when the time scales are invariant under translations.

Lemma 26 (see [2]). Let be an invariant under translation time scale. If is right-dense (resp., right-scattered), then for every , is right-dense (resp., right-scattered).

Lemma 27 (see [2]). Let be an invariant under translations time scale and . Then (i) and , for every ;(ii), for every .

Remark 28. As we pointed out in Section 1 time scales invariant under translations are not automatically symmetric. Since almost automorphic functions are defined on symmetric domains, some definitions and results in [11] on these functions will be given with additional assumption on the time scale. More precisely we will assume that the time scale is symmetric and invariant under translations.

Definition 29. Let be Banach space and let be a symmetric time scale which is invariant under translations. Then the -continuous function is called almost automorphic on if for every sequence on , there exists a subsequence such that is well defined for each and for each .

We denote by the space of all almost automorphic functions on time scales .

Remark 30. In view of Lemma 27, if is a symmetric time scale which is invariant under translations then the graininess function is an almost automorphic function.

We have the following properties.

Theorem 31. Let be a symmetric time scale which is invariant under translation. Assume that the -continuous functions are almost automorphic on . Then the following assertions hold: (i) is almost automorphic on time scales;(ii) is almost automorphic on for every scalar ;(iii)for each , the function defined by is almost automorphic on time scales;(iv) defined by is almost automorphic on time scales;(v); that is, is a bounded function;(vi), where

Proof. See [11].

We have the following remark on the property given in [11].

Remark 32. Notice that (i)in order to give a sense to , we consider as an element of instead of as in [11];(ii)we need the symmetry of the time scale to obtain that if , that is, to give a sense to the definition of in (iv).

Remark 33. The space equipped with the norm is a Banach space (see [11] pp. 2280).

Lemma 34. If , the range is relatively compact in .

Proof. Let be fixed and let be a sequence in . Then, for any , there is such that . By invariance under translations of , for each , we can find such that . Hence, for all , we have . Since is almost automorphic on time scale, there exists a subsequence of such that Thus, the subsequence converges to . Therefore, is relatively compact in .

Theorem 35 (see [11]). Let be a symmetric time scale which is invariant under translations. Let also and be two almost automorphic functions on time scales. Then the function defined by is almost automorphic on time scales.

Theorem 36 (see [11]). Let be a symmetric time scale which is invariant under translations and let be a sequence of almost automorphic functions such that converges uniformly for each . Then, is an almost automorphic function.

Theorem 37 (see [11]). Let be a symmetric time scale which is invariant under translations and let and be Banach spaces. Suppose is an almost automorphic function on time scales and is a continuous function; then the composite function is almost automorphic on time scales.

Definition 38 (see [11]). Let be a (real or complex) Banach space and let be a symmetric time scale which is invariant under translations. Then a -continuous function is called almost automorphic on for each if for every sequence on , there exists a subsequence such that is well defined for each , and for each and .

Theorem 39 (see [11]). Let be a symmetric time scale which is invariant under translations and let be almost automorphic functions on time scale in for each in . Then the following assertions hold: (i) is almost automorphic on time scale in for each in ;(ii) is almost automorphic function on time scale in for each in , where is an arbitrary scalar;(iii), for each ;(iv), for each , where is the function in Definition 38.

Theorem 40. Let be a symmetric time scale which is invariant under translations and let be almost automorphic functions on time scale in for each in and satisfy Lipschitz condition in uniformly in ; that is, for all . Assume that almost automorphic on time scale. Then the function defined by is almost automorphic on time scale.

We can now introduce the notion of almost automorphic functions of order on time scales.

4. Almost Automorphic Functions of Order on Time Scales

We denote by the linear space of all functions that are th differentiable on and is -continuous. We denote by the subspace of consisting of such function for which where denotes the th derivative of , , , and . In the space we introduce the norm Then we have the following.

Proposition 41. equipped with the norm defined above is a Banach space.

Proof. It’s clear that is a linear space and that (37) is a norm on . Now, let be a Cauchy sequence in . Then for any , there exists such that, for all , , we have In other words, given , there is such that, for all , , In particular, for all , , , are Cauchy sequences in which is a Banach space. Thus if we denote by the limit of , we have that
Since for , passing to the limit in these above relations as , we obtain for , This means that, for all , which on the one hand proves that since for any , and on the other hand shows that

Definition 42. Let be a (real or complex) Banach space and let be a symmetric time scale which is invariant under translations. Then a -continuous function is called almost automorphic if is almost automorphic in uniformly for each , where is any bounded subset of .

We denote by the space of all almost automorphic functions on time scales .

Definition 43. A function is said to be -almost automorphic (briefly .), if belong to for all .

Denote by the set of . functions.

Directly from the above definitions it follows that . Moreover, putting , we have .

Lemma 44. We have .

Proof. It is straightforward from the definition of an almost automorphic function on time scales (see Theorem 31).

Proposition 45. A linear combination of -. functions is a -. function. Moreover, let be a Banach space over the field . Let , , and . Assume that exists for all and is almost automorphic on time scale. Then the following functions are also in : (i), (ii), (iii), (iv), where is fixed.

Proof. For the proof of (i) and (ii), one proceeds as in [11]. To prove (iii), we use the Leibnitz formula on time scales, the definition, and the properties of an almost automorphic function; we get the result easily.
Now, let us prove (iv). For any , if we consider the function defined by , then we have , for all in . It is clear that is strictly increasing, , and , for all . Using Theorem 12, we obtain for each . Hence, for being an almost automorphic function on time scale, we deduce that is almost also automorphic on time scale. Similarly, we prove that is almost automorphic for . Thus, .

Theorem 46. If a sequence of -. functions is such that as , then is -. function.

Proof. From the assumption, it is clear that . Moreover, uniformly on for each . Thus Theorem 36 allows us to say that is a . function.

Corollary 47. considered with norm (37) turns out to be a Banach space.

Proof. In view of Proposition 45 and Lemma 44, is a linear subspace of . Let , , , be a Cauchy sequence. Then there exists such that . By Theorem 46, , so it is a Banach space.

5. Differentiation and Integration

The first result in this section gives a sufficient condition which guarantees that the derivative of a function is also a . function.

Theorem 48. Let be a symmetric time scale which is invariant under translations. Let also be a Banach space and an almost automorphic function on . Assume that is -differentiable on and is uniformly continuous. Then is also almost automorphic on time scales.

Proof. Assume that the points of are right-dense. Then for being invariant under translations, we obtain . Hence and since is uniformly continuous, it follows from Theorem   in [12] that is almost automorphic on time scale.
Now, let us suppose that has at least a right-scattered point ; then we have Given a sequence , since is almost automorphic, there is a subsequence such that since Lemma 27 holds. On the other hand, we have The proof is completed.

Theorem 49. If and is uniformly continuous, then .

Proof. In view of Theorem 48, we have . This means that is in . Then it follows that .

Similarly as in [12], we introduce some useful notations in order to facilitate the proof.

Notation 1. Let be a symmetric time scale which is invariant under translations. If is a function and a sequence is such that we will write .

Remark 50 (see [12]). Consider the following.(i) is a linear operator. Given a fixed sequence , the domain of is . is a linear set.(ii)Let us write and suppose that and . The product operator is well defined. It is also a linear operator.(iii) maps bounded functions into bounded functions, and for an almost authorphic function on time scale , we get .

Now we are ready to enunciate and prove Bohl-Bohr’s type theorem known from the literature for almost automorphic functions on time scale. The proof is inspired by the proof of Theorem   in [12].

Theorem 51. Let be a symmetric time scale containing zero and invariant under translations. Let also . One considers the function defined by . Then if and only if is relatively compact in .

Proof. In view of Lemma 34, it suffices to prove that if is relatively compact in .
Assume that is relatively compact in and let . Then there exists a subsequence of such that pointwise on , and for some .
We get, for every , Making the change of variable , we obtain If we apply the Lebesgue dominated theorem to this latter identity, we get for each . Let us observe that the range of the function is also relatively compact and so that we can extract a subsequence of such that for some . Now we can write so that Let us prove now that . Using Notation 1 we get where . Now it is easy to observe that and belong to the domain of ; therefore is also in the domain of and we deduce the equation We can continue indefinitely the process to get But we have and is a bounded function.
This leads to contradiction if . Hence, , and using Remark 50, we have , so is almost automorphic. The proof is complete.

Theorem 52. If and the range is relatively compact, then .

Proof. If and the range is relatively compact, then in view of Theorem 51, . Since , , for . Thus and, consequently, .

Theorem 52 is a special case of the following.

Theorem 53. If and the range is relatively compact then .

Proof. If and the range is relatively compact then in view of Theorem 51, . Therefore, . This means that .

Corollary 54. If and the range is relatively compact, then .

Proof. We know that , for each . In view of Theorem 53, we have .

6. Superposition Operators

In this section, and are two Banach spaces.

Proposition 55. Let be a bounded linear operator and . Then we have .

Proof. Since is a bounded linear operator, we have . Therefore, observing the fact that for each because , it follows from Theorem 37 that , for all . Hence .

For every , we define the function as follows: where is a bounded linear operator. We have the following.

Corollary 56. Let be a bounded linear operator with a relatively compact range. Then -valued function defined above is in .

Proof. According to Proposition 55, for every . Since is a bounded linear operator, the range of the operator contains the range of . Hence is relatively compact and it follows from Theorem 53 that .

Remark 57. In Corollary 56, if operator is compact (or of the finite rank), the stated result holds.

Now we will consider the superposition of operator (the autonomous case) acting on the space . Using this fact we will prove the following result with the Fréchet derivative.

Theorem 58. If and , then .

Proof. First, we observe that the result holds if for , in view of Theorem 37, we have that if and . So, it suffices to show that to complete the proof of the theorem.
By Theorem 11, for each , we have Since and we have that, for any , the function belongs to . Therefore for being continuous, Theorem 37 allows us to say that the function is also in for any and consequently the function is . It then follows from Theorem 35 that since .

7. Applications to First-Order Dynamic Equations on Time Scales

Definition 59 (see [5]). Let be -continuous matrix-valued function on . The linear system is said to admit an exponential dichotomy on if there exist positive constants , projection , and the fundamental solution matrix of (65) satisfying where is the matrix norm on . This means that if then we can take .

Consider the following system: where is an matrix-valued function which is regressive on and is -continuous.

Theorem 60 (see [11]). Let be a symmetric time scale which is invariant under translation and let be almost automorphic and nonsingular on and and are bounded. Assume that linear system (65) admits an exponential dichotomy and is an almost automorphic function on time scales. Then system (67) has an almost automorphic solution as follows: where is the fundamental solution matrix of  (65).

Lemma 61 (see [3]). Let be an almost automorphic function, , and . Then the linear system admits an exponential dichotomy on .

In view of Lemma 61 and Theorem 60 we have the following result.

Lemma 62. Let be a symmetric time scale which is invariant under translation. Let be such that the functions are almost automorphic, , and , . Assume also that is an almost automorphic function on time scales. Then system (67) has an almost automorphic solution as follows:

Lemma 63. Let be a symmetric time scale which is invariant under translation. Let be such that the functions are -almost automorphic, , and , . Assume also that is a -almost automorphic function on time scales. Then system (67) has a -almost automorphic solution as follows:

Proof. By Lemma 62, is a -almost automorphic solution to system (67). Now, since using on the one hand the fact that functions , , and are -almost automorphic and on the other hand the fact that Theorem 35 holds, we deduce that , , are also -almost automorphic. The proof of Lemma 63 is then complete.

In the following we will consider with the norm obtained by taking in (37).

Set . Then with the norm , is a Banach space.

Definition 64. Let be a -almost automorphic solution of (2) with initial value . Assume that there exists a positive constant with such that for , there exists such that for an arbitrary solution of (2) with initial value , satisfies where
Then, the solution is said to be exponentially stable.

In what follows, we will give sufficient condition for the existence of -almost automorphic solutions of (2).

Let Then it is clear that endowed with the norm , where is obtained by taking in (37), is a Banach space.

We make the following assumption.(H₁)Assume , , , , and .(H₂)There exists a positive constant , , such that (H₃), , and there exists a positive constant such that

For convenience, for a -almost automorphic function , we set and by .

Theorem 65. Assume that and hold. Assume also that Then, (2) has a unique -almost automorphic solution in , where is a positive constant satisfying

Proof. For any , we consider the following -almost automorphic system: Since holds, it follows from Lemma 61 that the system admits an exponential dichotomy on . Thus by Lemma 63, system (2) has a -almost automorphic solution:
Now, we prove that the following mapping is a contraction on : where To this end we proceed in two steps.
Step 1. We prove that if then .
Let ; then using and the fact that , we have Consequently,
Step 2. We prove that is a contraction on .
Let and be in . Then using and the fact that , we obtain Hence, Because , , is a contraction on . We then deduce by the fixed point theorem of Banach that has a unique solution in . Consequently system (2) has a unique -almost automorphic solution in :

Theorem 66. Assume that and hold. Assume also that Then, the -almost automorphic solution of (2) is globally exponentially stable.

Proof. According to Theorem 65, system (2) has a -almost automorphic solution with initial value Assume that is an arbitrary solution of (2) with initial value . Let , . Then it follows from (2) that Multiplying both sides of the first equation in (93) by and integrating on , where , we obtain, for , This means that, for , Then choose and One proves by contradiction as in [8] that This means that the -almost automorphic solution of (2) is globally exponentially stable.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his/her careful reading and valuable suggestions. Aril Milce received a Ph.D. scholarship from the French Embassy in Haiti. He is also supported by “Ecole Normale Supérieure,” an entity of the “Université d’Etat d’Haïti.”