Abstract
In this paper, we deal with the form and the periodicity of the solutions of the max-type system of difference equations where the initial conditions are positive two-periodic sequences.
1. Introduction
Difference equations and systems which do not stem from the differential ones have attracted some attention in the last few decades (see, e.g., [1–26]). Some of the systems that are of interest are symmetric or those retrieved from symmetric by alteration of their parameters. Recently, there has been enormous interest in various types of nonlinear difference equations and the references cited therein. One of the reasons for this is a necessity for some techniques which can be used in inspecting equations arising in mathematical models illustrating real life situations in biology, control theory, economics, physics, sociology, and so on (see, e.g., [21, 23, 24]). Among these equations, the so-called max-type difference equations also attracted some attention.
The study of the system of max-type difference equations attracted recently a considerable attention; see, for example, [1–16], and the references listed therein. This type of difference equations stemming from, for example, certain models are useful in automatic control theory (see [27]). In the beginning, on the study of these equations, experts have been focused on the investigation of the behavior of some particular cases.
In [3], Berenhaut et al. explained the boundedness nature of positive solutions of the following max-type difference equation system:where are periodic parameters. In 2020, Balibrea et al. [7] obtained in an elegant way the general solution of the following max-type system of difference equations:where with . The initial conditions and
In [8], Cunningham et al. evaluated the solution of the following max-type difference equation with period 2:where are periodic sequences with period .
In 2015, Grove et al. [13] obtained the solution of the following max-type difference equation system:where The parameter is positive real number.
In [15], Su et al. obtained the solution of the following max-type difference equation system with a period 3 parameter:
In [26], C.M. Kent and M.A. Radin explained the boundedness nature of positive solutions of the following difference equation:where are periodic parameters.
Motivated by the above study, our purpose in this paper is to evaluate the eventual periodicity of the following max-type -system of difference equations:where , and are positive periodic sequences and initial conditions .
Theorem 1. Suppose that is a solution of system (1)–(3) such that , , and . Then, the following statements hold: (1)If , , and , then 2If , , and , then 3If , , and , then
Proof. (1)From the following conditions: we have By induction, we obtained formula as follows: which are formulas of odd terms in (4)–(6). Hence, it remains only to prove the formulas for even terms in (4)–(6). Similarly, we can find for even terms. By induction,(2)Because , , and , we have Then, By induction, we obtain formulas for even terms as given in (11).(3)Because , , and , thenSo, we haveBy induction, we obtain formulas as given in (26):The proof is completed.
Theorem 2. Suppose that is a solution of systems (1)–(3) such that , , and . Then, the following statements hold:(1)If , , and , then 2If , , and , 3If,, and , 4If,, and ,
Proof. (1)From the conditions , , and , we have By induction, we obtained formula as follows: Similarly, we can find the proof for even terms. By induction, as given in (24).(2)Because , , and , then we have By induction, we obtain the formulas as stated in (25).(3)Because , , and , then By induction, we obtain formula as in (26).(4)Because , , and , thenThen, by induction, we get formulas as given in (27).
Theorem 3. Suppose that is a solution of systems (1)–(3) such thatthen we haveThen, the following system holds for odd solutions:
Theorem 4. Suppose that is a solution of systems (1)–(3) such thatthen the following statement holds:
Theorem 5. Suppose that is a solution of systems (1)–(3) such thatthen the following statement holds:
Theorem 6. Suppose that is a solution of systems (1)–(3) such thatthen the following statement holds:
2. Conclusion
We investigate the closed form solutions of an important type of difference equation. The eventual periodicity of the following max-type -system of difference equations iswhere , and are positive periodic sequences and initial conditions .
Data Availability
All data utilized in this article have been included and the sources where they were adopted were cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.