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14
NuriyeAtaseverand˙
IbrahimYalcinkaya
wherex11jxo∈(0j∞).Also,thisresultwasgeneralizedtotheequationof
thefollowingform:
xn+11
g(xn)
xn11
forn10j1j2j...
wherex11jxo∈(0j∞).
Moreoverin[6],Simseketal.solvedthefollowingproblemforthepositive
initialvalues
xn+11
1+xn11xn13
xn15
forn10j1j2j....
Alsoin[7],Simseketal.studiedthebehaviorofthepositivesolutionsof
thedifferenceequation
xn+11
1+xn14xn19...xn1(5k+4)
xn1(5k+9)
j
n10j1j2j...
wherex1(5k+9)jx1(5k+8)j...jx11jxo∈(0j∞).
Inthispaperwestudythebehaviorofthepositivesolutionsofthefol-
lowingnonlineardifferenceequation
(1)
xn+11
1+xn1kxn1(2k+1)
xn1(3k+2)
j
n10j1j2j...
wheretheinitialvaluesx1(3k+2)jx1(3k+1)j...jx11jxo∈(0j∞)andk1
0j1j2....
Tothebestofourknowledge,theEq.(1)hasnotbeeninvestigatedsofar.
Therefore,itismeaningfultostudythequalitativepropertiesoftheEq.(1).
2.Mainresult
InthissectionweconsiderthebehaviorofthepositivesolutionsofEq.(1).
Thefollowingtheoremdescribesthebehaviorofthepositivesolutionsof
Eq.(1).
Theorem1.ConsiderthedifferenceEq.(1).Thenthefollowingstate-
mentsaretrue:
(I)Thesequences(x(3k+3)n1(3k+2)),(x(3k+3)n1(3k+1)),
...,(x(3k+3)n)are
decreasingandthereexistI1jI2j...jI3k+3>0suchthat
n→∞
n→∞
lim
lim
x(3k+3)n1(3k+2)1I1j
x(3k+3)n1(3k+1)1I2j
...j
n→∞
lim
x(3k+3)n1I3k+3
