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FASCICULI
MATHEMATICI
Nr40
2008
ElsayedMohammedElsayed
ONTHESOLUTIONOFRECURSIVESEQUENCE
OFORDERTWO
Abstract.Weobtaininthispaperthesolutionofthefollowing
differenceequation
xn+1=
xn1(xn±1)
xn
j
n=0j1j...
wheretheinitialconditionsx1,x0arearbitraryrealnumbers.
Keywords:differenceequations,recursivesequence,periodic
solution.
AMSMathematicsSubjectClassification:39A10.
1.Introduction
Inthispaperweobtainthesolutionofthefollowingrecursivesequence
(1)
xn+1=
xn11(xn±1)
xn
j
n=0j1j...
wheretheinitialconditionsx11,xoarearbitraryrealnumbers.
Recentlytherehasbeenalotofinterestinstudyingtheglobalattrac-
tivity,boundednesscharactertheperiodicnature,andgivingthesolution
ofnonlineardifferenceequations.Forsomeresultsinthisarea,seeforex-
ample[1-11].SinceCinar[1,2,3]investigatedthesolutionsofthefollowing
differenceequations
xn+1=
1+axnxn11
xn11
j
xn+1=
1+axnxn11
xn11
jxn+1=
1+bxnxn11
axn11
.
Elabbasyetal.[4]investigatedtheglobalstability,periodicitycharacter
andgivethesolutionofspecialcaseofthefollowingrecursivesequence
xn+1=axn
Cxndxn11
bxn
.