Periodicity of solutions of nonhomogeneous linear difference equations
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Abstract
Firstly, sufficient conditions for nonexistence of an ωperiodic solution of the equation $x(n+1)+{a}_{0}(n)x(n)=b(n)$ are presented. Then, sufficient conditions under which every solution of the above equation is asymptotically ωperiodic are given. Next, the results obtained for the firstorder difference equation are generalized for the higherorder nonhomogeneous linear difference equation
Finally, the periodic and asymptotically periodic solutions of this equation are investigated. Many examples illustrate the results given.
MSC:39A11, 39A10.
Keywords
nonhomogeneous linear difference equation asymptotically periodic solution1 Introduction
where ${a}_{0}(n)\ne 0$, ${a}_{k}(n)\ne 0$ for each $n\in \mathbf{N}$.
For the reader’s convenience, we note that the background for difference equations theory can be found, e.g., in the wellknown monograph by Agarwal [1] as well as in those by Elaydi [2], Kelley and Peterson [3] or Kocić and Ladas [4].
The investigation of linear difference equations attracted the attention of many mathematicians. Agarwal and Popenda, in [5], set together various basic statements on the periodicity of the solutions of firstorder linear difference equations. In [6], the same authors studied periodic oscillation of solutions of nonhomogeneous higherorder difference equations. Popenda and Schmeidel (see [7]) considered the linear difference equation ${c}_{n}^{r}{y}_{n+r}+\cdots +{c}_{n}^{1}{y}_{n+1}+{c}_{n}^{0}{y}_{n}={d}_{n}$ and presented sufficient conditions for the existence of an asymptotically constant solution of the above equation. In [8], the conditions which guarantee that the linear difference equation ${x}_{n+1}{a}_{n}{x}_{n}={\sum}_{i=0}^{r}{a}_{n}^{(i)}{x}_{n+i}$ possesses an asymptotically periodic solution were given by the same authors. In [9], Popenda and Schmeidel studied the linear difference equation, where one of the coefficients is periodic or constant and the others asymptotically approach zero, and obtained sufficient conditions for the existence of asymptotically periodic solutions. Smith (see [10]) investigated oscillatory and asymptotic behavior of solutions of linear thirdorder difference equations. In [11], asymptotic behavior of solutions of a linear secondorder difference equation was studied by Trench.
For convenience, we adopt the notation for sequences $b=(b(n))$ and ${a}_{i}=({a}_{i}(n))$, where $i=0,1,2,\dots ,k$. Throughout this paper, we assume that ${\sum}_{n=k}^{l}a(n)=0$ and ${\prod}_{n=k}^{l}a(n)=1$ for $l<k$.
We begin with the following basic wellknown definition.
Definition 1 The sequence $y:\mathbf{N}\to \mathbf{R}$ is called ωperiodic if $y(n+\omega )=y(n)$ for all $n\in \mathbf{N}$. The sequence y is called asymptotically ωperiodic if there exist two sequences $u,v:\mathbf{N}\to \mathbf{R}$ such that u is ωperiodic, ${lim}_{n\to \mathrm{\infty}}v(n)=0$, and $y(n)=u(n)+v(n)$ for all $n\in \mathbf{N}$.
It is clear that every constant function is 1periodic.
If a sequence ${a}_{0}$ is ${\omega}_{1}$periodic and b is ${\omega}_{2}$periodic in (1), then throughout this paper, ω is the least common multiple of ${\omega}_{1}$ and ${\omega}_{2}$ ($\omega =lcm({\omega}_{1},{\omega}_{1})$).
In the paper, we are looking for the periodic solutions of (1) with the period less than or equal to ω. We are not interested in the solutions of (1) with the period greater than ω, but such solutions can exist.
Here, sequences ${a}_{0}\equiv 1$ and $b\equiv 0$ are 1periodic, but there are 4periodic solutions.
2 Firstorder difference equations
Periodicity of solutions of firstorder linear nonhomogeneous difference equations was considered by Agarwal and Popenda in [5]. The authors contemplate the class of equations which have the same periodic solutions.
If ${a}_{0}\equiv 1$, $b\equiv 0$, then the general solution of (2) is a constant function, then it is 1periodic.
If these conditions are satisfied, then all the solutions of the homogeneous equation are ωperiodic. We also note that if ${a}_{0}(n)=0$, for some $n\in \mathbf{N}$, then $x\equiv 0$ for large enough n, and this solution is eventually a 1periodic solution.
From (2) we see that if ${a}_{0}$ is ωperiodic, then the necessary condition for the existence of an ωperiodic solution is ωperiodicity of the sequence b.
Sequences ${a}_{0}(n)=2+{(1)}^{n}$ and $b(n)=5+{(1)}^{n}$ are 2periodic. The solution $x(n)=2+{(1)}^{n+1}$ of the above equation is 2periodic, too. Notice that there are not 2periodic solutions of the associated homogeneous equation.
The following example shows us that in the case ${a}_{0}$ is ωperiodic, ωperiodicity of the sequence b is not sufficient for the existence of an ωperiodic solution of (2).
is not a periodic sequence.
 (i)If${(1)}^{\omega}\prod _{i=0}^{\omega 1}{a}_{0}(i)\ne 1,$(4)
 (ii)If${(1)}^{\omega}\prod _{i=0}^{\omega 1}{a}_{0}(i)=1,\phantom{\rule{2em}{0ex}}\sum _{j=0}^{\omega 1}({(1)}^{\omega j1}\prod _{i=j+1}^{\omega 1}{a}_{0}(i))b(j)=0,$
 (iii)If${(1)}^{\omega}\prod _{i=0}^{\omega 1}{a}_{0}(i)=1,\phantom{\rule{2em}{0ex}}\sum _{j=0}^{\omega 1}({(1)}^{\omega j1}\prod _{i=j+1}^{\omega 1}{a}_{0}(i))b(j)\ne 0,$
then there is no ωperiodic solution of (2).
From the above, the result follows immediately. □
has not any nontrivial ωperiodic solution.
In [5] Agarwal and Popenda proved that if ${a}_{0}$ is not periodic, then equation (2) can have at most one periodic solution.
of the associated homogeneous equation has not any nontrivial periodic solution.
The following example shows us that there exists a class of equations (2) which have the same ωperiodic solutions (each of them differs on the subsequence $({a}_{0}(3n1))$).
Example 5 Let ${a}_{0}=(3,1,a,3,1,a,\dots )={(3,1,a)}_{3}$, $b={(2,4,2)}_{3}$. It is easy to check that the sequence $x={(2,4,0)}_{3}$ is a 3periodic solution of (2) independently of the values taken for a.
This leads to the problem of defining the class of equations which have the same periodic solutions.
Theorem 2 Assume that in equation (2) sequences ${a}_{0}$ and b are ωperiodic and condition (4) holds.
has the same ωperiodic solution x as equation (2) independently on ${a}_{0}^{\ast}(p)$ term.
Proof Let x and ${x}^{\ast}$ be the solutions of equations (2) and (9) respectively. The assumptions of Theorem 1 hold for equations (2) and (9), then by (5) and (7), we get that $x(0)={x}^{\ast}(0)$. Because ${a}_{0}(i)={a}_{0}^{\ast}(i)$ for $i=0,1,2,\dots ,p1$, we get $x(i)={x}^{\ast}(i)$ for $i=0,1,2,\dots ,p1$. From (6), (8), and $x(0)={x}^{\ast}(0)$, we have $x(p)={x}^{\ast}(p)=0$. So, $x(i)={x}^{\ast}(i)$ for $i=p,p+1,\dots ,\omega 1$. By ωperiodicity of x and ${x}^{\ast}$, $x={x}^{\ast}$. □
Now, we turn our attention to asymptotical periodicity of the solutions of (2).
From above, we get sufficient conditions for asymptotical periodicity of the solutions of (2) which are presented in the following theorem.
hold, then every solution of equation (2) is asymptotically ωperiodic.

in the first one sequence ${a}_{0}$ is constant;

in the second sequence ${a}_{0}$ is 3periodic;

in the third sequence ${a}_{0}$ is not periodic.
The assumptions of Theorem 2 hold ($c(n)=2{(1)}^{n}$, $d(n)=0$). The general solution of the equation $x(n)=c+{(1)}^{n+1}$ is an asymptotically 2periodic sequence.
tends to zero. The 3periodic solution of (2) is $x=({(1,2,3)}_{3})$. Therefore, every solution of (2) is asymptotically 3periodic.
3 Some results for higherorder equations
In this part, we study equation (1). In the following theorem, sufficient conditions under which equation (1) has no asymptotically periodic solution are given.
Theorem 4 Assume that there exists ${i}_{0}\in \{0,1,2,\dots ,k\}$ such that ${sup}_{n\in \mathbf{N}}{a}_{{i}_{0}}(n)=\mathrm{\infty}$ and ${sup}_{n\in \mathbf{N}}{a}_{j}(n)<\mathrm{\infty}$ for $j\ne {i}_{0}$, $j\in \{0,1,2,\dots ,k\}$. Let the sequence b be bounded, too. Then equation (1) has not any asymptotically periodic solution $x:\mathbf{N}\to \mathbf{R}\setminus \{0\}$.
is unbounded, while b is bounded. This contradiction completes the proof. □
The sufficient conditions for the existence of an asymptotically ωperiodic solution of equation (1) are given in the following theorem.
holds for each $i\in \{1,2,\dots ,k\}$ and the sequence $(\frac{b(n)}{{a}_{0}(n)})$ is asymptotically ωperiodic. Then there exists an asymptotically ωperiodic solution of equation (1).
It means that the set S is convex.
hence it is a solution of equation (1). Because $x\in S$, then x is an asymptotically periodic sequence. This completes the proof. □
is such a solution.
Notes
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.