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FASCICULI
MATHEMATICI
Nr44
2010
E.Camouzis,E.DrymonisandG.Ladas
PATTERNSOFBOUNDEDNESSOFTHERATIONAL
SYSTEM
xn+1=
A1+C1yn
o1+β1xn
andyn+1=
A2+B2xn+C2yn
o2+β2xn+γ2yn
Abstract.Weestablishtheboundednesscharacterofsolutions
oftherationalsysteminthetitle,withtheparameters01,β1
positiveandtheremainingeightparametersnonnegativeandwith
arbitrarynonnegativeinitialconditionssuchthatthedenomina-
torsarealwayspositive.Wepresenteasilyverifiablenecessaryand
sufficientconditions,explicitlystatedintermsoftheparameters,
whichdeterminetheboundednesscharacterofthesystem.
Keywords:boundedness,patternsofboundedness,rational
equations,rationalsystems.
AMSMathematicsSubjectClassification:39A10.
1.Introduction
Weestablishtheboundednesscharacterofsolutionsoftherationalsys-
temintheplane,
(1)
(
I
4
I
l
xn+1=
yn+1=
A2+B2xn+C2ynj
A1+C1ynj
o2+β2xn+γ2yn
o1+β1xn
n=0j1j...
withtheparameters01jβ1positiveandtheremainingeightparameters
nonnegativeandwitharbitrarynonnegativeinitialconditionssuchthatthe
denominatorsarealwayspositive.
System(1)contains
(22−1)×(23−1)×(23−1)=147
specialcasesofsystems,eachwithpositiveparameters.
Weestablisheasilyverifiablenecessaryandsufficientconditions,explic-
itlystatedintermsoftheparameters,underwhichtheboundednesschar-
acterizationofthesystemis:
(B,B)j(B,U)j(U,B)or(U,U).
