On the Difference Equation
We show that the difference equation , where , the parameters , and initial values , are real numbers, can be solved in closed form considerably extending the results in the literature. By using obtained formulae, we investigate asymptotic behavior of well-defined solutions of the equation.
Recently, there has been some reestablished interest in difference equations which can be solved, as well as in their applications, see, for example, [1–16]. For some old results in the topic see, for example, the classical book .
In recently accepted paper  are given formulae for the solutions of the following four difference equations: and some of these formulae are proved by induction.
Moreover, we will consider here the following more general equation: where and the parameters as well as initial values are real numbers, and describe the behaviour of all well-defined solutions of the equation.
For a solution , of the difference equation is said that it is eventually periodic with period , if there is an such that If , then it is said that the solution is periodic with period . For some results in this area see, for example, [19–26] and the references therein.
2. Solutions of Equation (1.2)
By using the change of variables equation (1.2) is transformed into the following linear first-order difference equation: for which it is known (and easy to see) that if , and if .
From (2.1), we have for , from which it follows that for every and .
Remark 2.1. It is easy to check that the formulae in Theorems 2.1 and 4.1 from  are direct consequences of formula (2.8), whereas formulae in Theorem 3.1 and Theorem 5.1 from  are direct consequences of formula (2.7).
Remark 2.2. Note that from formula (2.8), it follows that in the case , , all well-defined solutions of equation (1.2) are periodic with period . This can be also obtained from (1.2), without knowing explicit formulae for its solutions. Namely, in this case, (1.2) can be written as follows: since we assume , for all , from which it follows that the sequence is constant, that is, for some . Hence, as claimed.
Remark 2.3. Note also that in the case , , from (2.7), we have for every and , from which the behaviour of the solutions in the case easily follows.
3. Asymptotic Behavior of Well-Defined Solutions of Equation (1.2)
In this section, we derive some results on asymptotic behavior of well-defined solutions of equation (1.2). We will use well-known asymptotic formulae as follows: for , where is the Landau “big-oh” symbol.
Before we formulate and prove our next result, we will prove an auxiliary result which is incorporated in the lemma that follows.
Lemma 3.2. If , then equation (1.2) has -periodic solutions.
Remark 3.3. Note that -periodic solutions in the previous lemma could be prime -periodic. For this it is enough to choose initial conditions , such that the string is not periodic with a period less then .
Proof. First note that by Lemma 3.2, in this case, there are -periodic solutions of the equation. We know that in this case well-defined solutions of the equation are given by formula (2.7). From this and by using asymptotic formulae (3.1), we obtain that for sufficiently large
From (3.12) and since , it easily follows that the sequences are convergent for each , from which the theorem follows.
Proof. In this case, well-defined solutions of equation (1.2) are also given by formula (2.7). Further note that for each holds
Now note that , due to the assumption . Using this fact and (3.14), it follows that for sufficiently large , say we have
From this, we have as , where from which the theorem follows.
Proof. From (2.2), in this case, we have which means that the sequence is two-periodic, and consequently the sequence is two-periodic. Hence from which it follows that that is, the sequences and are -periodic from which the result easily follows.
Theorem 3.7. Let , , be odd in (1.2), and be a well-defined solution of equation (1.2). Then the following statements are true.(a)If , then the solution is -periodic.(b)If , then and , as .(c)If , then and , as .
Proof. As in Theorem 3.6, we obtain (3.18) and consequently (3.19) holds. If for some , then from (1.2) and (3.19) we have
From (3.21) we obtain From relation (3.22), the statements in this theorem easily follow.
The second author is supported by Grant no. P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government Grant no. MSM 00 216 30519. The fourth author is supported by Grant no. FEKT-S-11-2-921 of the Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science Projects III 41025, III 44006, and OI 171007.
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