On Two Systems of Difference Equations

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Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2010, Article ID 405121,4pages doi:10.1155/2010/405121

Research Article

On Two Systems of Difference Equations

Bratislav Iri ˇcanin

¹

and Stevo Stevi ´c

²

1Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11120 Belgrade, Serbia

2Mathematical Institute, Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia

Correspondence should be addressed to Stevo Stevi´c,[email protected] Received 11 January 2010; Accepted 9 March 2010

Academic Editor: Leonid Berezansky

Copyrightq2010 B. Iriˇcanin and S. Stevi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We give very short and elegant proofs of the main results in the work of Yalcinkaya et al.2008.

1. Introduction and a Proof of Some Resent Results

Motivated by our paper1, the authors of2studied the following two systems of diﬀerence equations:

xⁱ_n1 x^i1mod_n ^k

xⁱ¹_n ^mod^k−1, i1, . . . , k, n∈N0, 1.1 xⁱ_n1 x^i−1mod_n ^k

xⁱ⁻¹_n ^mod^k−1, i1, . . . , k, n∈N0, 1.2

where we regard that 0modk kmodk k.

Following line by line the proofs of the main results in1they proved the following resultsee Theorems 2.1 and 2.4 in2

Theorem A. Assumek∈N,then the following statements are true.

aIfk 0mod 2,then every (well-defined) solution of systems1.1and1.2is periodic with periodk.

bIfk 1mod 2, then every (well-defined) solution of systems1.1and1.2is periodic with period 2k.

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2 Discrete Dynamics in Nature and Society Here we give a very short and elegant proof of Theorem A.

Proof of Theorem A. By using the changeynⁱxⁱn −1,i1, . . . , k,system1.1becomes yⁱ_n

y^i1mod_n−1 ^k₋₁

, i1, . . . , k, n∈N, 1.3

while system1.2becomes yⁱ_n

y^i−1mod_n−1 ^k₋₁

, i1, . . . , k, n∈N. 1.4

From1.3and1.4, for eachi∈ {1, . . . , k},andn≥k, we obtain correspondingly that

yⁱ_n

y^i1k−1_n−k ^mod^k₋₁^k

y_n−kⁱ ₋₁^k , yⁱ_n

y^{i−1−k−1}_n−k ^mod^k₋₁^k

yⁱ_n−k₋₁^k .

1.5

From1.5, withk0mod 2,it follows that

yⁱ_n yⁱ_n−k, i1, . . . , k, 1.6

from which the statement inaeasily follows.

Ifk1mod 2,we have that y_nⁱ

yⁱ_n−k₋₁

, i1, . . . , k, 1.7

from which it follows that

y_nⁱy_n−2kⁱ , i1, . . . , k, 1.8 n≥2k, implying the statement inb, as desired.

2. An Extension on Theorem A

Here we extend Theorem A in a natural way. Let gcdk, ldenote the greatest common divisor of the integers k and l, lcmk, l the least common multiple of k and l, and for r ∈ Nlet f^rx ff^r−1x,wheref¹x fx.

Theorem 2.1. Assume thatfis a real function such thatf^rx≡xon its domain of definition, for somer∈N,then all well-defined solutions of the system of diﬀerence equations

x¹_n f x_n−1²

, x²_n f x³_n−1

, . . . , x_n^kf x¹_n−1

, n∈N0, 2.1

are periodic with periodT lcmk, r.

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Discrete Dynamics in Nature and Society 3 Proof. We use our method of “prolongation” described in1. Note that for eachs∈N,system 2.1is equivalent to a system ofksdiﬀerence equations of the same form, where

x_nⁱx^jki_n , 2.2

for everyn∈N0,i∈ {1, . . . , k}andj1, . . . , s.

From2.1and sincef^rx≡x, forn≥r−1 we have

x_nⁱ¹ f xⁱ²_n−1

f² xⁱ³_n−2

· · ·f^r

x^ir_n−r¹

x^ir1_n−r . 2.3

for eachi∈ {1,2, . . . , k},and everyn≥r−1.

It is clear that

Tk·r1k1·r, 2.4

wherer1, k1 ∈Nare such that gcdk, r1 1 and gcdk1, r 1.

From2.3we have

xⁱ¹_n xîr1_n−r · · ·x_n−kîk₁¹_r^r¹xîkr_n−kr₁¹¹x_n−Tⁱ¹, 2.5

for eachi0,1, . . . , k−1,andn≥T−1, from which the result follows.

The following result is proved similarly. Hence we omit its proof.

Theorem 2.2. Assume thatfis a real function such thatf^rx≡xon its domain of definition, for somer∈N,then all well-defined solutions of the system of diﬀerence equations

x_n²f x¹_n−1

, . . . , x_n^kf x^k−1_n−1

, x_n¹f x^k_n−1

, n∈N0, 2.6

are periodic with periodT lcmk, r.

Remark 2.3. The proof of Theorem A follows from Theorems2.1and2.2. Indeed, note that the functionfx x/x−1satisfies the conditionf²x ≡ xon its domain of definition. By Theorems2.1and2.2we know that all well-defined solutions of systems1.1and1.2are periodic with periodT lcmk,2,from which the result follows.

Remark 2.4. We also have to say that the main result in3is a trivial consequence of a result in1 see Remark 5 therein. Just note that the simple change of variablesx_nⁱ ay_nⁱ,i ∈ {1, . . . , k},transforms their system1.3satisfying conditionsa1 a2 · · · a_k aand b1b2 · · ·b_kba², into system4in1.

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4 Discrete Dynamics in Nature and Society

Acknowledgment

The results in this note were presented at the talk: S. Stević, on a class of max-type difference equations and some of our old results, Progress on Difference Equations 2009, Bedlewo, Poland, May 25–29, 2009.

References

1 B. D. Iriˇcanin and S. Stevi´c, “Some systems of nonlinear diﬀerence equations of higher order with periodic solutions,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 13, no. 3-4, pp.

499–507, 2006.

2 ˙I. Yalçinkaya, C. Çinar, and M. Atalay, “On the solutions of systems of difference equations,” Advances in Difference Equations, vol. 2008, Article ID 143943, 9 pages, 2008.

3 G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On ak-order system of Lyness-type diﬀerence equations,” Advances in Diﬀerence Equations, vol. 2007, Article ID 31272, 13 pages, 2007.