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FASCICULI
MATHEMATICI
Nr43
2010
NuriyeAtaseverand˙
IbrahimYalcinkaya
ONACLASSOFRATIONALDIFFERENCE
EQUATIONS
Abstract.Inthispaperwestudythebehaviorofthepositive
solutionsofthefollowingnonlineardifferenceequation
xn+11
1+xn1kxn1(2k+1)
xn1(3k+2)
j
n10j1j2j...
wheretheinitialvaluesx1(3k+2),x1(3k+1),
...,x11,x0∈(0j∞)
andk10j1j2j....
Keywords:differenceequation,positivesolutions,period(3k+
3)solution.
AMSMathematicsSubjectClassification:39A10.
1.Introduction
Differenceequationsappearnaturallyasdiscreteanaloguesandasnu-
mericalsolutionsofdifferentialanddelaydifferentialequationshavingap-
plicationsinbiology,ecology,economy,physics,etc.So,recentlytherehas
beenanincreasinginterestinthestudyofqualitativeanalysisofrational
differenceequations.Forsomerecentresultsconcerningamongotherprob-
lems,theperiodicnatureofscalarnonlineardifferenceequationssee,for
examples[1-10].
In[1-5]Cinarstudiedthesolutionsofthedifferenceequations
xn+11
1+xnxn11
xn11
jxn+11
1+Ixnxn11
xn11
j
xn+11
−1+Ixnxn11
xn11
j
xn+11
1+bxnxn11
Ixn11
j
xn+11
In+bnxn
cnxn11
wherethepositiveinitialvalues.
In[9]Stevicstudiedthefollowingproblem
xn+11
1+xn
xn11
forn10j1j2j...
