A note: Every homogeneous difference equation of degree one admits a reduction in order

Sedaghat, Hassan

doi:10.1080/10236190802201453

Cited by 13 publications

(29 citation statements)

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“…Homogeneous difference equations are considered in [1][2][3][4][5][6][7][8], where these equations are equivalent to lower order equations using semiconjugate factors. In [2], we considered second and third order homogeneous rational difference equations, and then, we obtained monotonic, non-monotonic and oscillated solutions.…”

Section: Introductionmentioning

confidence: 99%

Planar homogeneous systems of difference equations

Mazrooei‐Sebdani

2014

Math. Meth. Appl. Sci.

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This paper is concerned to additive and multiplicative systems of homogeneous difference equations of non‐negative degree. We apply a reduction in order for both additive and multiplicative systems. Then, we consider convergence and monotony of positive solutions. In fact, using convergence results on factor maps, we obtain convergence results on homogeneous systems. We will conclude that monotonic behaviour on the invariant ray (i.e. xMathClass-rel=falsemml-overliner¯y for multiplicative systems and xMathClass-rel=falsemml-overlines¯MathClass-bin+y for additive systems) may or may not be the representative of other solutions. To illustrate our results, some examples are presented by multiplicative and additive homogeneous systems of rational equations. Copyright © 2014 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

Planar homogeneous systems of difference equations

Mazrooei‐Sebdani

2014

Math. Meth. Appl. Sci.

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“…Homogeneous difference equations considered in . Now, consider the difference equation of order k + 1

x_{n MathClass-bin+ 1} MathClass-rel= f_{n} (x_{n} MathClass-punc, x_{n MathClass-bin- 1} MathClass-punc, MathClass-op\dots MathClass-punc, x_{n MathClass-bin- k}) MathClass-punc, n MathClass-rel= 0 MathClass-punc, 1 MathClass-punc, 2 MathClass-punc, MathClass-op\dots MathClass-punc,

where

f_{n} MathClass-punc: D MathClass-rel\to double-struckR

and

D MathClass-rel\subseteq {double-struckR}^{k MathClass-bin+ 1}

.…”

Section: Introductionmentioning

confidence: 99%

“…This equality also shows that if

\overset{r}{falsemml-overline} MathClass-rel> 1

, then every orbit in

\overset{r}{falsemml-overline} x

goes to infinity monotonically and if

\overset{r}{falsemml-overline} MathClass-rel= 1

, then every orbit in

\overset{r}{falsemml-overline} x

is stationary (a point). The invariant ray

\overset{r}{falsemml-overline} x

is analogous to a fixed point for Equation , in the sense that by taking the quotient of (0, ∞ ) 2 modulo

\overset{r}{falsemml-overline} x

, Equation is transformed into a topological conjugate of Equation , and the ray

\overset{r}{falsemml-overline} x

into the point

\overset{r}{falsemml-overline}

; on the space of rays through the origin . Monotonic behaviour on the invariant ray may or may not be the representative of other solutions.…”

Section: Introductionmentioning

confidence: 99%

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Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Mazrooei‐Sebdani

2013

Math. Meth. Appl. Sci.

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Consider the non‐autonomous equations: xnMathClass-bin+1MathClass-rel=xn()AnxnMathClass-bin+BnxnMathClass-bin−1αnxnMathClass-bin+βnxnMathClass-bin−1MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbsp1emnbspαnxnMathClass-bin+βnxnMathClass-bin−1MathClass-rel>0MathClass-punc, xnMathClass-bin+1MathClass-rel=xnMathClass-bin−1()BnxnMathClass-bin+DnxnMathClass-bin−2αnxnMathClass-bin+γnxnMathClass-bin−2MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbsp1emnbspαnxnMathClass-bin+γnxnMathClass-bin−2MathClass-rel>0MathClass-punc, xnMathClass-bin+1MathClass-rel=xn()BnxnMathClass-bin−1MathClass-bin+CnxnMathClass-bin−2βnxnMathClass-bin−1MathClass-bin+γnxnMathClass-bin−2MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbsp1emnbspβnxnMathClass-bin−1MathClass-bin+γnxnMathClass-bin−2MathClass-rel>0MathClass-punc, where AnMathClass-rel≥01emnbsp1emnbspMathClass-punc,BnMathClass-rel≥01emnbsp1emnbspMathClass-punc,CnMathClass-rel≥01emnbsp1emnbspMathClass-punc,αnMathClass-rel≥01emnbsp1emnbspMathClass-punc,βnMathClass-rel≥01emnbsp1emnbspMathClass-punc,γnMathClass-rel≥0MathClass-punc,1emnbsp1emnbsp1emnbsp1emnbspnMathClass-rel=0MathClass-punc,1MathClass-punc,2MathClass-punc,MathClass-op…MathClass-punc, and also msubnormallimnMathClass-rel→MathClass-rel∞AnMathClass-rel=AMathClass-rel≥01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞BnMathClass-rel=BMathClass-rel≥01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞CnMathClass-rel=CMathClass-rel≥01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞αnMathClass-rel=αMathClass-rel>01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞βnMathClass-rel=βMathClass-rel>01emnbsp1emnbspMathClass-punc,msubnormallimnMathClass-rel→MathClass-rel∞γnMathClass-rel=γMathClass-rel>0MathClass-punc.These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.

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Homogeneous rational difference equations of degree 1: convergence, monotone and oscillatory solutions for second- and third-order cases

Mazrooei‐Sebdani

2012

Journal of Difference Equations and Applications

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A note: Every homogeneous difference equation of degree one admits a reduction in order

Cited by 13 publications

References 2 publications

Planar homogeneous systems of difference equations

Planar homogeneous systems of difference equations

Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Homogeneous rational difference equations of degree 1: convergence, monotone and oscillatory solutions for second- and third-order cases

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