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Functionalanalysisandnonlinearboundaryvalueproblems
(2.7)
x
!!+p1(t)x!+p2(t)x=0jx(a)=0=x(b)j
onlyhasthetrivialsolutionwhen
"p2"ż
(b−a)2
2
+"p1"ż(b−a)<1.
11
In1929,CharlesJ.deLaValléePoussin[15]generalizedPicard’sunique-
nessresultfor(2.7)byshowingthattheproblem
(2.8)
x(n)+p1(t)x(n11)+...+pn11(t)x!+pn(t)x=0j
x(to)=x(t1)=...=x(tn)=0j
onlyhasthetrivialsolutionif
Σ
j=1
n
"pj"ż
(b−a)j
j!
<1j
andextendedtheresulttoproblem(2.3)withfsuchthat
(2.9)
and
n
|f(tjx1j...jxn)−f(tjy1j...jyn)|≤
Σ
Lj|xj−yj|
j=1
Σ
j=1
n
Lj
(b−a)j
j!
<1.
InPoland,linearandnonlinearboundaryvalueofinterpolationtypehad
beenconsideredin1946–47byJanMikusiński[80]andMieczysławBier-
nacki[5],
2.2.Theintroductionoffunctionalanalysis
Thedevelopmentoflinearfunctionalanalysisinthefirstquarterofthe
XXthcenturyaswellasStefanBanach’sfixedpointtheoremof1922[2]–an
abstractversionofthemethodofsuccessiveapproximations–madepossible
toexpresstheaboveresultsinafunctionalanalyticway.Butamoreessential
stepwasmadethesameyear1922byGeorgeD.BirkhoffandOliverD.Kel-
logg[6],whentheyextendedBrouwer’sfixedpointtheorem(anycontinuous
selfmapofaclosedn-ballhasatleastonefixedpoint)tocontinuousselfmaps
