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10
JeanMawhin
thecaseofsystemsofsuchequations,whichcontainsinparticularboundary
valueproblemsforsystemsoffirstorderequationsoftheform
(2.4)
x
!
1=f1(tjx1j...jxn)j...jx
!
n=fn(tjx1j...jxn)j
x1(t1)=a1j...xn(tn)=anj
nowusuallyreferredasNiccoletti’sproblem.UsingEmilePicard’smethod
ofsuccessiveapproximations,Niccolettiprovedexistenceanduniqueness,for
globallyLipschitzianfunctionsf,whenb−aissufficientlysmall.
Thespecialcasewheren=2isthewellknownDirichletorPicard
boundary-valueproblem
(2.5)
x
!!=f(tjxjx!)jx(a)=a1jx(b)=a2j
alreadyconsideredbyPicard[83]in1893bythesamemethod.
Incontrasttoproblems(2.1)and(2.2)whicharealwaysuniquelysolvable,
eitherexistenceoruniquenessmayfailforproblem(2.3).Forexample,the
specialcaseof(2.5)
x
!!=−xjx(0)=0jx(π)=1j
hasnosolution,andtheotherspecialcase
x
!!=−xjx(0)=0jx(π)=0j
hasinfinitelymanysolutions.Indeed,theproblem
(2.6)
x
!!=−xjx(a)=a1jx(b)=a2j
hasasolutionifandonlyifonecanfindrealnumbersAandBsuchthat
Acosa+Bsina=a1j
Acosb+Bsinb=a2j
whichrequiresthatcosasinb−cosbsina/=0,i.e.,thatb−a/=0(modπ).
Inparticular,existence(andindeeduniqueness)isinsuredifb−a<π.
Thequestionoffindingestimatesforb−ainsuringtheexistenceofa
solutionforthetwo-pointboundaryvalueproblem
x
!!=f(tjxjx!)jx(a)=a1jx(b)=a2j
wasconsideredbyPicard[84]in1896whenfisLipschitzianwithrespectto
thelasttwovariables.Asaspecialcase,heshowedthatthelinearhomoge-
neousproblem
