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6
MohamedAkkouchi
Definition2.([6])Let(X,d)beametricspaceandT:X→Xa
mapping.Forx,y∈X,wedenote
(1)
M(x,y):=max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}
Letφ∈D.ThemappingTiscalledaφ-max-contractionifthefollowing
inequality
(2)
d(Tx,Ty)≤φ(M(x,y))
holdstrueforallx,y∈X.
UsingthisconceptJ.Daneˇ
shasprovedsomefixedpointtheoremsin[6].
Theaimofthispaperistostudythewell-posedness(seeDefinition3
below)ofthefixedpointproblemfortheφ-max-contractionsoforbitally
completemetricspaces.Morepreciselyweprovidenaturalconditionson
thefunctionsφwhichensurethewell-posednessofthefixedpointproblem
fortheassociatedφ-max-contractions.
Thenotionofwell-posednesofafixedpointproblemhasevokedmuch
interesttoaseveralmathematicians,forexamples,F.S.DeBlasiandJ.My-
jak(see[1]),S.ReichandA.J.Zaslavski(see[12]),B.K.LahiriandP.Das
(see[8])andV.Popa(see[10]and[11]).
Definition3.Let(X,d)beametricspaceandT:(X,d)→(X,d)a
mapping.ThefixedpointproblemofTissaidtobewellposedif:
(a)ThasauniquefixedpointzinX;
(b)foranysequence{xn}ofpointsinXsuchthatlim
n→∞
d(Txn,xn)=0,
wehavelim
n→∞
d(xn,z)=0.
2.Mainresult
Foranyarbitraryfunctionφ:[0,fl)→[0,fl)andforeachrealnumber
t∈[0,fl),weset
(3)
Jφ(t):={s∈[0,fl):s−φ(s)≤t}.
Infact,foreachnon-negativenumbert,wehaveJφ(t)=(Id−φ)11([0,t]).
Weintroducethefollowingdefinition.
Definition4.WedenoteAthesetoffunctionsφ:[0,fl)→[0,fl)
satisfyingthefollowingconditions:
(A1):φisrightuppersemi-continuouson[0,fl).
(A2):φisnon-decreasingon[0,fl).
(A3):φ(t)<tforallt∈(0,fl).
(A4):(a)Foreacht∈[0,fl),thesetJφ(t)isbounded,andwehave
(b)lim
t→o
sup{s:s∈Jφ(t)}=0.
