Research Article | Open Access
Discrete Weighted Pseudo Asymptotic Periodicity of Second Order Difference Equations
We define the concept of discrete weighted pseudo--asymptotically periodic function and prove some basic results including composition theorem. We investigate the existence, and uniqueness of discrete weighted pseudo--asymptotically periodic solution to nonautonomous semilinear difference equations. Furthermore, an application to scalar second order difference equations is given. The working tools are based on the exponential dichotomy theory and fixed point theorem.
The concept of -asymptotic periodicity [1, 2] introduced by Henríquez et al. is natural generalization of asymptotic periodicity . Since then, many contributions on the existence of -asymptotically periodic solutions for differential equations have been made; one can see [4–10] for more details. Further Pierri and Rolnik  introduced the notion of pseudo--asymptotically periodic function, explored its properties, and investigated pseudo--asymptotic periodicity of neutral differential equations with finite delay in Banach space. With the help of weighted function, weighted pseudo--asymptotically periodic function is introduced and the applications to fractional integrodifferential equations are investigated in .
All the above-mentioned concepts are introduced in the continuous case, but it is rarely for the discrete type. For (discrete) -asymptotic periodicity, the subject has been studied in the recent paper , where the authors discussed the existence of (discrete) -asymptotically periodic solutions of semilinear difference equations with infinite delay. Motivated by the above literatures, it is natural to consider the discrete version of weighted pseudo--asymptotically periodic function, which we will discuss in the present paper.
The rapid development of the theory of difference equations has been strongly promoted by the large number of applications in physics, engineering, biology, and other subjects. The asymptotic behaviour of solutions of difference equations is at present an active of research. Many researchers have made important contributions to these topics, for example, almost periodicity [14, 15], asymptotic almost periodicity [16, 17], almost automorphy [18–20], -boundedness , stability [22, 23], and -asymptotic periodicity . However, to the best of our knowledge, (discrete) weighted pseudo--asymptotic periodicity of difference equations is quite new and an untreated topic. This is one of the key motivations of this study.
The principal aim of this paper is to introduce the concept of discrete weighted pseudo--asymptotically periodic function, which is much more general to generalize discrete -asymptotically periodic function and explore its properties and applications in difference equations. The paper is organized as follows. In Section 2, first, some notations and preliminary results are presented. Next, we propose a new class of functions called discrete weighted pseudo--asymptotically -periodic function, explore its properties, and establish the composition theorem. Section 3 is devoted to the existence and uniqueness of discrete weighted pseudo--asymptotically -periodic solution of nonautonomous semilinear difference equations. In Section 4, discrete weighted pseudo--asymptotic -periodicity of the scalar second order difference equations is investigated.
2. Preliminaries and Basic Results
Let , be two Banach spaces and let , , , , and stand for the set of natural numbers, integers, nonnegative integers, real numbers, nonnegative real numbers, and complex numbers, respectively. Let be bounded linear operator; denotes the point spectrum of .
In order to facilitate the discussion below, we further introduce the following notations.(i).(ii). (iii). (iv) -, where .(v) is the Banach space of bounded linear operators from to endowed with the operator topology. In particular, we write when .(vi) is the set of all functions satisfying that , such that
for all and with .
Theorem 1 (Matkowski’s fixed point theorem ). Let be a complete metric space and let be a map such that where is a nondecreasing function such that for all . Then has a unique fixed point .
Given a sequence of invertible operators, define where is the identity operator in .
For the first order difference equation
Definition 2 (see ). Equation (4) is said to have an exponential dichotomy if there exist projections for all and positive constants such that(i), (ii), (iii), where is the complementary projection of .
Next, we propose a new class of functions called discrete weighted pseudo--asymptotically -periodic function and explore its properties including composition result.
Let denote the collection of functions (weights) . For and , set Denote
Definition 3. Let . is said to be equivalent to (i.e., ) if .
It is trivial to show that “” is a binary equivalence relation on . The equivalence class of a given weight is denoted by . It is clear that .
Let ; define by for and
Definition 4. A function is called discrete asymptotically -periodic if there exist such that . The collection of those functions is denoted by .
Definition 5. A function is called discrete -asymptotically -periodic if there exists such that . The collection of those functions is denoted by .
Definition 6. A function is called discrete pseudo--asymptotically -periodic if there exists such that Denote by the set of such functions.
Definition 7. Let . A function is called discrete weighted pseudo--asymptotically -periodic if there exists such that Denote by the set of such functions.
Next, we will show some properties of including composition theorem.
Lemma 8. Let ; then the following properties hold: (i) if ;(ii) if , ;(iii);(iv) is a Banach space when endowed with the sup norm
Proof. The proof is straightforward, so the details are omitted here.
Lemma 9. Assume that . If , then (i);(ii); (iii); (iv)if , then .
Proof. (i) Since , there exist and such that ; then
Let ; then
The fact that implies that
that is, ; hence
Proceeding in a similar manner, we have . Hence holds.
From the proof of , it is not difficult to see that , , and hold, so the details are omitted here. The proof is completed.
Lemma 10. Let , ; then for all .
Proof. Without loss of generality, we may assume that , since implies that there exists such that , . For then Note that ; therefore .
We will establish composition theorem for discrete weighted pseudo--asymptotically -periodic function.
Lemma 11. Let ; then , , if and only if, for any , where .
Proof. Sufficiency: it is clear that and ; there exists such that
Then, for ,
That is, .
Necessity: suppose the contrary that there exists , such that does not converge to as . That is, there exists , such that, for each , Then for which contradicts the fact that Thus (19) holds.
Theorem 12. Let . Assume that ; then if .
Proof. Since , for any , there exists such that
for all and . For the above , since , there exists such that, for ,
then by Lemma 11. So
For , one has Due to the arbitrariness of , one has which implies that .
Corollary 13. Let . Assume that and there exists a constant such that Then if .
3. Nonautonomous Semilinear Difference Equations
In this section, consider the following nonautonomous semilinear difference equations: Its associated homogeneous linear difference equation is given by
To establish our results, we introduce the following conditions.(H1) Let, .(H2) There exists a constant such that ()There exists a linear nondecreasing function such that (H3)Equation (34) admits an exponential dichotomy on with projection , positive constants .(H4) For all , .
Theorem 14. Assume that hold and , where then (33) has a unique solution which is given by
Proof. Similarly as the proof of [26, 27], it can be shown that given by (38) is the solution of (33). Define the operator as follows:
Since and holds, then by Corollary 13.
Next, we show that . In fact, let where
then, Since , , then for each by Lemma 10; hence
By Lebesgue dominated convergence theorem, one has hence . Similarly, one can prove . So is well defined.
For , by and exponential dichotomy, one has hence is a contraction. By the Banach contraction mapping principle, has a unique fixed point , which is the unique solution of (33). The proof is completed.
Theorem 15. Assume that , , , and hold; then (33) has a unique solution if as for each .
Proof. Define the operator as in (39), so is well defined. For , one has Since as for each , by Matkowski fixed point theorem (Theorem 1), has a unique fixed point , which is the unique solution of (33).
Example 16. Consider the system
where is a nonsingular matrix such that , , and and there exists a constant such that
Since , the system admits an exponential dichotomy with positive constants  and holds with . By Theorem 14, if we suppose that , then (47) has a unique discrete weighted pseudo--asymptotically -periodic solution.
4. Scalar Second Order Difference Equations
Let ; we study the existence and uniqueness of discrete weighted pseudo--asymptotically -periodic solutions to a scalar second order difference equation given by where and and satisfies the following.(A1)The function is Lipschitz in uniformly in ; that is, there exists a constant such that (A2) are periodic functions, in the sense that there exists such that