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9
Thedefinitionofintegraloperatorsisstandard.Givenafunction
k:[0,1]×[0,1]→Rby
K(x)(t):=∫
0
1
k(t,s)x(s)ds
(0<t<1)
(3)
wedenotetheintegraloperatorgeneratedby
k
.Asbefore,weareinter-
estedinconditionson
k
,possiblybothnecessaryandsufficient,under
whichtheoperator
K
mapsacertainfunctionspace
X
intoitself.In
mostcases,theoperator
K
isthenbounded,orevencompact,inthe
spaceX.
AlloperatorsstudiedinChapter2arelinear.InChapter3we
discusstwoclassesofnonlinearoperators,namelycompositionand
superpositionoperators.Thecompositionoperator
Cf
,generatedby
somefunction
f:R→R
,actingonfunctions
x:[
0,1
]→R
isdefinedby
Cf(x)(t):=f(x(t))(0<t<1).
(4)
Moregenerally,thesuperpositionoperator
Sf
,generatedbysome
function
f:[
0,1
]×R→R
,actingonfunctions
x:[
0,1
]→R
isde-
finedby
Sf(x)(t):=f(t,x(t))(0<t<1).
(5)
Inspiteoftheirsimpleform,theoperators
(4)
and
(5)
exhibit
astrangeandunexpectedbehavioureveninsuchsimplespaceslike
thoseintroducedinthefirstchapter.Forinstance,whileweobtain
boundednessoftheoperator
(4)
asa“fringebenefit”wheneveritmaps
BV
,
WBVp
,
RBVp
,or
ΛBV
intoitself,tofindcriteriaforitscontinuity
isaveryhardproblem.Wewillcollectthemostimportantresultsand
illustratethembyseveralexampleswhicharescatteredoveravast
literature.
Finally,thetheoreticalresultsofthesecondandthirdchapter
areappliedinChapter4,asmentionedabove,toseveralnonlinear
integralequationsinvolvingHammersteinandHammerstein–Volterra
operators.AHammersteinequationhastheform
x(t)=g(t)+λ∫
0
1
k(t,s)f(x(s))ds
(0<t<1),
(6)
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