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10
E.Camouzis,E.DrymonisandG.Ladas
AspecialcaseofSystem(1)hastheboundednesscharacterization(B,B)
whenbothcomponentsofeverysolutionofthesystem,inthisspecialcase,
arebounded.
AspecialcaseofSystem(1)hastheboundednesscharacterization(B,U)
whenthefirstcomponentofeverysolutioninthisspecialcaseofthesystem
isalwaysboundedandthereexistsolutionsinwhichthesecondcomponentis
unboundedinsomerangeoftheparametersandforsomeinitialconditions.
Similarly,wedefinetheboundednesscharacterizations(U,B)and(U,U)
foraspecialcaseofSystem(1).
Theboundednesscharacterofsolutionsofasystemisoneofthemain
ingredientsinunderstandingtheglobalbehaviorofthesystemincludingits
globalstability.Boundednessisalsoessentialinthestudyofmostapplica-
tions.
System(1)isaspecialcaseofthe“full”rationalsystemintheplane,
(2)
whichcontains
(
I
4
I
l
xn+1=
yn+1=
A2+B2xn+C2ynj
A1+B1xn+C1ynj
o2+β2xn+γ2yn
o1+β1xn+γ1yn
n=0j1j...
7×7×7×7=2401
specialcaseseachwithpositiveparameters.Alargenumberofopenprob-
lemsandconjecturesaboutSystem(2)wereposedin[9]and[11].Forsome
workontheboundednesscharacterofSystem(2)see[1]-[8],and[11]-[12].
Forthenumberingsystemofthe2401specialcasescontainedinSystem(2),
seeAppendices1and2in[9].Alsoforsomebasicresultsintheareaof
differenceequationsandsystemssee[10]and[13]-[15].
System(1)hastheboundednesscharacterization(B,B),ifandonlyif:
(3)
B2=0jC1jβ2∈(0j∞)and(72=0orC2>0).
System(1),undercondition(3)isrestrictedtothegroupofthefollowing
20specialcases
(4)
(
I
(21j7)j(21j8)j(21j16)j(21j22)j(21j23)j
I
4
I
(21j26)j(21j31)j(21j34)j(21j41)j(21j46)j
(29j7)j(29j8)j(29j16)j(29j22)j(29j23)j
I
l
(29j26)j(29j31)j(29j34)j(29j41)j(29j46).
andhastheboundednesscharacterization(B,B).
System(1)hastheboundednesscharacterization(B,U),ifandonlyif:
(5)
B2=C2=0andC1jβ2j72∈(0j∞).
