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6
ElsayedM.Elsayed
Elabbasyetal.[5]studiedtheglobalstability,periodicitycharacterand
givethesolutionofsomespecialcasesofthedifferenceequation
xn+1=
;+γΠ
Oxn1k
k
iloxn1i
.
Elabbasyetal.[6]investigatedtheglobalstability,periodicitycharacter
andgivethesolutionofsomespecialcasesofthedifferenceequation
xn+1=
dxn1lxn1k
Cxn15−b
+a.
Karatasetal.[8]gavethatthesolutionofthedifferenceequation
xn+1=
1+xn12xn15
xn15
.
Simseketal.[11]obtainedthesolutionofthedifferenceequation
xn+1=
1+xn11
xn13
.
Here,werecallsomenotationsandresultswhichwillbeusefulinourinves-
tigation.
LetIbesomeintervalofrealnumbersandlet
f:Ik+1→I
beacontinuouslydifferentiablefunction.Thenforeverysetofinitialcon-
ditionsx1kjx1k+1j...jxo∈Ijthedifferenceequation
(2)
xn+1=f(xnjxn11j...jxn1k)j
n=0j1j...j
hasauniquesolution{xn}∞
nl1k[10].
Apointx∈IiscalledanequilibriumpointofEq(2)if
x=f(xjxj...jx).
Thatis,xn=xforn≥0,isasolutionofEq(2),orequivalently,xisafixed
pointoff.
Definition.[Periodicity]Asequence{xn}∞
nl1kissaidtobeperiodicwith
periodpifxn+p=xnforalln≥−k.
.
