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Functionalanalysisandnonlinearboundaryvalueproblems
2.Ordinarydifferentialequationswithinterpolationconditions
9
2.1.Earlyhistory
Polynomialinterpolationseemstohavebeenthemotivationofapaper
ofOnoratoNiccolettiof1898[82],devotedtononlinearordinarydifferential
equationswithsomelinearboundaryconditions.
Theinterpolationbyapolynomialofdegreen−1(n≥2aninteger)of
thevaluesa1ja2j...janofarealfunctiongivenatnpoints
a=t1<t2<...<tn=bj
canofcoursebewrittenintheformofa‘boundaryvalueproblem’on[ajb]=
[t1jtn]
(2.1)
x(n)=0jx(tk)=ak(k=1j2j...jn).
Itsuniquesolution(apolynomialofdegreen−1whichvanishesatnpoints
isidenticallyzero),isgivenbytheLagrangeinterpolationpolynomial
x(t)=
Σ
k=1
n
(tk−t1)...1
(t−t1)...1
(tk−tk)...(tk−tn)
(t−tk)...(t−tn)
where^
·meansthatthecorrespondingfactorismissing.Thequotationsmarks
areusedbecausethedataarenotonlygivenattheboundaryof[ajb]when
n≥3.Buttheterminologymultipointboundaryvalueproblemisstandard.
Naturalgeneralizationsarethelinearnon-homogeneousmultipointbound-
aryvalueproblem
(2.2)
x(n)=h(t)jx(tk)=ak(k=1j2j...jn)j
wheretheintegrablefunctionh:[ajb]→Risgiven,andthenonlinearnon-
homogeneousmultipointboundaryvalueproblem
(2.3)
x(n)=f(tjxjx!j...jx(n11))jx(tk)=ak(k=1j2j...jn)j
wherethenonlinearCarathéodoryfunctionf:[ajb]×Rn→Risgiven.
SuchproblemsarespecialcasesofthoseconsideredalreadybyNiccoletti
in1898,whereamoregeneralclassofboundaryconditionsinvolvingalsothe
valuesofsomederivativesatsomepointsisconsidered.Niccolettialsotreated
