Evolution Inclusions and Variation Inequalities for Earth Data Processing I: Operator Inclusions and Variation Inequalities for Earth Data Processing

276 Pages·2011·2.137 MB·English

by  Mikhail Z. Zgurovsky, Valery S. Mel'nik, Pavlo O. Kasyanov (auth.)

Advances in Mechanics and Mathematics
Volume 24
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Mikhail Z. Zgurovsky • Valery S. Mel’nik
Pavlo O. Kasyanov
Evolution Inclusions
and Variation Inequalities
for Earth Data Processing I
Operator Inclusions and Variation
Inequalities for Earth Data Processing
(cid:65)(cid:66)(cid:67)
Dr. Mikhail Z. Zgurovsky
Pavlo O. Kasyanov
Valery S. Mel’nik
National Technical University of Ukraine “Kyiv Polytechnic Institute”
Institute for Applied System Analysis
National Academy of Sciences of Ukraine
37, Peremogy Ave.
03056 Kyiv
Ukraine
[email protected]
ISSN1571-8689 e-ISSN1876-9896
ISBN978-3-642-13836-2 e-ISBN978-3-642-13837-9
DOI10.1007/978-3-642-13837-9
SpringerHeidelberg Dordrecht London New York
LibraryofCongressControlNumber: 2010936816
(cid:2)c Springer-VerlagBerlinHeidelberg2011
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Preface
The necessity of taking into account non-linear effects, memory effects, sharp-
ening conditions, semipenetration etc has arisen in recent years. It is caused by
intensification of processes in appliedchemistry, petrochemistry,transportationof
energy carriers, physics, energetics, mechanics, economics and in other fields of
technologyandindustry.Whenmodelingsuchphenomenawearefacedwithnon-
linear boundaryvalue problemsfor partial differentialequationswith multivalued
ordiscontinuousright-handside,variationalinequalities(evolutionalaswellassta-
tionary), an evolutional problem on manifolds (either with or without boundary),
pairedequations,cascadingsystemsetc. Interpretingaconceptofderivativeprop-
erly we can treat all these objects as operator or differential-operator inclusions
in Banach spaces and study them by the help of theory of multivalued maps of
pseudo-monotonetype.
The given book arose from seminars and lecture courses on multi-valued and
non-linear analysis and their geophysical application. These courses were deliv-
ered for rather differentcategoriesof learners in NationalTechnicalUniversity of
Ukraine“KievPolytechnicInstitute”,KievNationalTarasShevchenkoUniversity,
SecondUniversityofNaples,UniversityofSalernoetc.during10years.Thebook
isaddressedtoawidecircleofmathematicalaswellasengineeringreaders.
It is unnecessary to tell that the pioneering works of such authors as V.I.
Ivanenko, J.-L. Lions, V.V. Obukhovskii, N. Panagiotopoulos, N.S. Papageoriou,
whocreatedanddevelopedthetheoryofmentionedproblems,exertedthepowerful
influenceonthisbook.
We are thankful to V.O. Gorban, I.N. Gorban, N.V. Gorban, V.I. Ivanenko,
A.N.Novikov,V.I.Obuchovskii,N.A.Perestyuk,A.E.Shishkovforusefulremarks.
We are gratefulto the many students who have attended our lectures while we
weredevelopingthenotesforthisvolume.
We want to express our gratitude to O.P. Kogut and N.V. Zadoyanchukfor the
exceptionaldiligencewhenpreparingtheelectronicversionofthebookandfriendly
helpinlinguisticissues.
WewanttoexpressthespecialgratitudetoKathleenCassandOlenaL.Poptsova
foratechnicalsupportofourbook.
v
vi Preface
Finally,we expressourgratitudetoeditorsofthe “Springer”PublishingHouse
who worked with our book and everybody who took part in preparation of the
manuscript.
Beforehand we apologize to people whose works were missed inadvertently
whenmakingthereferences.
Wewillbegratefultoreadersforanyremarksandcorrections.
Kyiv,Ukraine MilkhailZ.Zgurovsky
August2010 ValeryS.Mel’nik
PavloO.Kasyanov
Contents
1 PreliminaryResults............................................................ 1
1.1 TheMainResultsfromMultivaluedMappingTheory.................. 2
1.2 ClassesofMultivaluedMaps............................................. 29
1.3 SubdifferentialsinInfinite-DimensionalSpaces ........................ 95
1.4 MinimaxInequalitiesinFinite-DimensionalSpaces....................125
References.......................................................................136
2 Operator Inclusions and Variation Inequalities
inInfinite-DimensionalSpaces ...............................................139
2.1 StrongSolutionsforParameterizedOperatorInclusions................139
2.2 ParameterizedOperatorInequalities.....................................145
2.3 VariationInequalitiesinBanachandFrechetSpaces ...................158
2.4 ThePenaltyMethodforMultivariationInequalities ....................162
2.5 NonlinearOperatorsEquationsoftheHammersteinType.
SystemofOperatorsEquations...........................................190
2.6 NonlinearNon-coerciveOperatorEquations
andTheirNormalization..................................................201
2.7 SomeExampleConnectedwithMembranesTheory....................238
References.......................................................................242
Index.................................................................................245
vii
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Acronyms
fora.e. Foralmosteach
l.s.c. Lowersemicontinuous
N-s.b.v. N-semiboundedvariation
N-sub-b.v. N-subboundedvariation
r.c. Radialcontinuous
r.l.s.c. Radiallowersemicontinuous
r.s.c. Radialsemicontinuous
r.u.s.c. Radialuppersemicontinuous
s.b.v. Semiboundedvariation
s.m. Semimonotone
sub-m. Submonotone
sub-b.v. Subboundedvariation
u.h.c. Upperhemicontinuous
u.s.c. Uppersemicontinuous
u.s.b.v. Uniformsemiboundedvariation
V-s.b.v. V-semiboundedvariation
w.l.s.c. Weaklylowersemicontinuous
ix
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