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Well-posednessofafixedpoint...
Tosimplifynotations,ifφ∈A,weset
ψ(t):=sup{s:s∈Jφ(t)},
foreveryt≥0.
WegiveexamplesofelementsoftheclassA.
Examples.
(1)φ(t)=qt,forallt∈[0,fl),where0≤q<1.
(2)φ(t)=t
1+t,forallt∈[0,fl).
Werecallthefollowingelementaryandclassicalresult.
7
Lemma1.Letf:[0,fl)→[0,fl)beafunctionsatisfyingtheconditions
(A1),(A2)and(A3),thenfsatisfies
n→∞
lim
fn(t)=0∀t≥0,
wherefn:f◦f...◦fn-times.(Bydefinitionfo=Id).
Beforegivingthemainresult,weneedtorecallthefollowinglemmaof[6].
Let(X,d)beametricspaceandT:X→Xamapping.ForxinXand
nandinteger,let
OT(x,n):={x,Tx,...,T
nx}.
Thenwehave
OT(x)=Un≥1OT(x,n).
Lemma2([6]).Let(X,d)beametricspace.Letφ:[0,fl)→[0,fl)
beafunctionsatisfyingthecondition(D2).SupposethatT:X→Xisa
φ-max-contraction.i.e.,Tsatisfiestheinequality
d(Tx,Ty)≤φ(max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)})
forallx,y∈X.
Letxbeanarbitraryx∈X.Then
(i)foranynon-negativeintegersnandm,wehave
(4)
(5)
diam(OT(T
mx,n))≤φm(diam(OT(x,n+m))).
(ii)Foranynon-negativeintegerm,wehave
diam(OT(T
mx))≤φm(diam(OT(x))),
providedthatdiam(OT(x))isfinite.
Themainresultofthispaperreadsasfollows.
