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Periodicity of solutions of nonhomogeneous linear difference equations
© Janglajew and Schmeidel; licensee Springer 2012
Received: 17 July 2012
Accepted: 3 November 2012
Published: 14 November 2012
Firstly, sufficient conditions for nonexistence of an ω-periodic solution of the equation are presented. Then, sufficient conditions under which every solution of the above equation is asymptotically ω-periodic are given. Next, the results obtained for the first-order difference equation are generalized for the higher-order nonhomogeneous linear difference equation
Finally, the periodic and asymptotically periodic solutions of this equation are investigated. Many examples illustrate the results given.
where , for each .
For the reader’s convenience, we note that the background for difference equations theory can be found, e.g., in the well-known monograph by Agarwal  as well as in those by Elaydi , Kelley and Peterson  or Kocić and Ladas .
The investigation of linear difference equations attracted the attention of many mathematicians. Agarwal and Popenda, in , set together various basic statements on the periodicity of the solutions of first-order linear difference equations. In , the same authors studied periodic oscillation of solutions of nonhomogeneous higher-order difference equations. Popenda and Schmeidel (see ) considered the linear difference equation and presented sufficient conditions for the existence of an asymptotically constant solution of the above equation. In , the conditions which guarantee that the linear difference equation possesses an asymptotically periodic solution were given by the same authors. In , 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 ) investigated oscillatory and asymptotic behavior of solutions of linear third-order difference equations. In , asymptotic behavior of solutions of a linear second-order difference equation was studied by Trench.
For convenience, we adopt the notation for sequences and , where . Throughout this paper, we assume that and for .
We begin with the following basic well-known definition.
Definition 1 The sequence is called ω-periodic if for all . The sequence y is called asymptotically ω-periodic if there exist two sequences such that u is ω-periodic, , and for all .
It is clear that every constant function is 1-periodic.
If a sequence is -periodic and b is -periodic in (1), then throughout this paper, ω is the least common multiple of and ().
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 and are 1-periodic, but there are 4-periodic solutions.
2 First-order difference equations
Periodicity of solutions of first-order linear nonhomogeneous difference equations was considered by Agarwal and Popenda in . The authors contemplate the class of equations which have the same periodic solutions.
If , , then the general solution of (2) is a constant function, then it is 1-periodic.
If these conditions are satisfied, then all the solutions of the homogeneous equation are ω-periodic. We also note that if , for some , then for large enough n, and this solution is eventually a 1-periodic solution.
From (2) we see that if is ω-periodic, then the necessary condition for the existence of an ω-periodic solution is ω-periodicity of the sequence b.
Sequences and are 2-periodic. The solution of the above equation is 2-periodic, too. Notice that there are not 2-periodic solutions of the associated homogeneous equation.
The following example shows us that in the case is ω-periodic, ω-periodicity of the sequence b is not sufficient for the existence of an ω-periodic solution of (2).
is not a periodic sequence.
then there is no ω-periodic solution of (2).
From the above, the result follows immediately. □
has not any nontrivial ω-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 ω-periodic solutions (each of them differs on the subsequence ).
Example 5 Let , . It is easy to check that the sequence is a 3-periodic 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 and b are ω-periodic and condition (4) holds.
has the same ω-periodic solution x as equation (2) independently on term.
Proof Let x and 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 . Because for , we get for . From (6), (8), and , we have . So, for . By ω-periodicity of x and , . □
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 ω-periodic.
in the first one sequence is constant;
in the second sequence is 3-periodic;
in the third sequence is not periodic.
The assumptions of Theorem 2 hold (, ). The general solution of the equation is an asymptotically 2-periodic sequence.
tends to zero. The 3-periodic solution of (2) is . Therefore, every solution of (2) is asymptotically 3-periodic.
3 Some results for higher-order equations
Theorem 4 Assume that there exists such that and for , . Let the sequence b be bounded, too. Then equation (1) has not any asymptotically periodic solution .
is unbounded, while b is bounded. This contradiction completes the proof. □
The sufficient conditions for the existence of an asymptotically ω-periodic solution of equation (1) are given in the following theorem.
holds for each and the sequence is asymptotically ω-periodic. Then there exists an asymptotically ω-periodic solution of equation (1).
It means that the set S is convex.
hence it is a solution of equation (1). Because , then x is an asymptotically periodic sequence. This completes the proof. □
is such a solution.
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