Periodic Solutions of a System of Nonlinear Difference Equations with Periodic Coefficients

Abstract

This paper is dealt with the following system of difference equations where , the initial values are the positive real numbers, and the sequences , , , and are two-periodic and positive. The system is an extension of a system where every positive solution is two-periodic or converges to a two-periodic solution. Here, the long-term behavior of positive solutions of the system is examined by using a new method to solve the system.

1. Introduction

Studying concrete nonlinear difference equations and systems has attracted a great recent interest. Many studies have been published on this topic in the last twenty years (see, e.g., [1–15]). Particularly, there has been a renewed interest in solvable nonlinear difference equations and systems for fifteen years (see, e.g., [5, 16–21], and the related references therein). Solvable difference equations are not interesting for themselves only, but they can be also applied in other areas of mathematics, as well as other areas of science (see, e.g., [22, 23]).

One of the first examples of solvable nonlinear difference equations is presented in note [24] where Brand solves the nonlinear difference equation:where the initial value is a real number and the parameters , , , and are the real numbers with the restrictions and and studies long-term behavior of solutions of the equation. The note presents a transformation which transforms the nonlinear equation into a linear one. The idea has been used many times in showing solvability of some difference equations, as well as of some systems of difference equations (see, e.g., [5, 18, 20, 21, 25, 26]). Another example of solvable nonlinear difference equations is the following system of nonlinear difference equation:where the initial values and are the positive real numbers and the parameters , , , and are the positive real numbers. System (2) can be transformed into an equation of form (1) by dividing the first equation of (2) by its second one. So, the results on equation (1) can be used to obtain the results on system (2). System (2) was studied for the first time in [17] by using the method described above. Also, in [17], it is shown that every positive solution of the system (2) is two-periodic or converges to a two-periodic solution. For more results on system (2), see [19, 22, 27].

System (2) can be extended by interchanging the parameters , , , and with the sequences , , , and . More concretely, another extension of (2) is the following system of difference equations:where the initial values are the positive real numbers and , , , and are the two-periodic sequences of positive real numbers. For extensions with periodic sequences of some difference equations and systems, see [2, 28, 29].

Our main purpose in this paper is to determine the long-term behavior of positive solutions of system (3). We also use a new method to solve the system without needing some other nonlinear difference equations such as (1). Throughout this paper, we assume that , ; , ; , ; and , with , , , and . We also adopt the conventions:where is any sequence and .

Definition 1. A solution of the systemis eventually periodic with period , if there is such that for . If , then the solution is periodic with period .
The following result is extracted from [30].

Remark 1. A product with positive terms is convergent if and only if converges.

2. Main Results

In this section, we formulate and prove our main results.

Theorem 1. Assume that and , , , and are the two-periodic sequences of positive real numbers. Then, system (3) can be solved in closed form.

Proof. First, it is easy to show by induction that , , for all . Multiplying both equations in (3) by the following positive product:we obtainfor all . Note that the equalities (7) and (8) constitute a linear system with respect to the following products:Therefore, we can write this system in the vector form:where and , which is simpler, for all . LetThen, since the sequences , , , and are two-periodic, the matrix becomesNow, we decompose (10) with respect to even-subscript and odd-subscript terms as follows:for all . From which (13) and (14) follows thatorLet . Then, we consider two cases of the matrix as the following: Case 1: . In this case, the first row in the matrix is linearly dependent on the second one. Without loss of generality, we may assume that where is a positive constant such that Using (17) in system (16), we have which implies the relation for all . By the last three relations, we have from which it follows that for all . Using (22) and (23) in (13), we obtain for all . On the other hand, the changes of variables in (9) yield for all . Hence, from (25) and (26), we obtain By employing (13) in (27)–(30), we have the following closed formulas: which is valid for all , respectively. Consequently, in the case , by using the formulas (22) and (23) in (31)–(34), we have the general solution of (3) as follows: where for all . Case 2: . In this case, both rows in the matrix are linearly independent of each other. This case also implies that has two different eigenvalues denoted by and . Since these eigenvalues will correspond to two linear independent eigenvectors, we may write the matrix as follows:whereTherefore, we may write system (16) as the following:wherefor all . From (42), we havefrom which it follows thatfor all . Multiplying both sides of (45) by the matrix , we haveor after some computationswherefor all . From the last vectorial equality, we havefor all . From (13), (52), and (53), we have the formulasfor all . Also, we can write the formulas (31)–(34) as the following:for all . Finally, by employing (52)–(55) in (56)–(59), we have the general solution of (3) as the following:for all .

The following theorem determines and characterizes the long-term behavior of positive solutions of (3) according to the parameters in the case .

Theorem 2. Assume that , and , , , and are the two-periodic sequences of positive real numbers. Ifthen, for the solutions of system (3), the following is true:(i)If , then , , , and as (ii)If , then , , , and as (iii)If , then every solution of (3) is two-periodic, where is given by (39)

Proof. The proof follows directly from formulas (35)–(38). That is to say, it is clearly seen from these formulas that ifthen , , , and as . Ifthen , , , and as . Ifthen every solution of (3) is two-periodic such that , , , and .

The following theorem determines and characterizes the long-term behavior of positive solutions of (3) according to the parameters in the case .

Theorem 3. Assume that , and , , , and are the two-periodic sequences of positive real numbers. Ifthen, for the solutions of system (3), the following is true:(i)If , then , , , and as (ii)If , then , , , and as (iii)If , then every solution of (3) converges to a two-periodic positive solution of the system, where and are given by (48) and (50)

Proof.  (i)-(ii). Let Then, from (56)–(59), we have We assume without loss of generality that . Then, we have the limits and so where Note that . Hereby, convergence characters of the infinite series are the same. We can say from a well-known fundamental result about infinite series that (74) and (75) are divergent, if . So, the proofs of items (i)-(ii) follow from (67)–(70) and Remark 1.(iii)From (74), for sufficiently large , we havewhereNote that if , then as . That is to say, (74) is convergent. Since , (75) is convergent too. In this case, the proof of item (iii) follows from (67)–(70) and Remark 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Authors’ Contributions

The author has contributed solely to the writing of this paper and has read and approved the manuscript.