Advances in Mechanics and MathematicsSeries Editors:David Y. Gao, Virginia Polytechnic Institute and State UniversityRay W. Ogden, University of GlasgowRomesh C. Batra, Virginia Polytechnic Institute and State UniversityAdvisory Board:Ivar Ekeland, University of British ColumbiaTim Healey, Cornell UniversityKumbakonom Rajagopal, Texas A&M UniversityTudor Ratiu,´Ecole Polytechnique F´ ed´ eraleDavid J. Steigmann, University of California, BerkeleyFor more titles in this series, go tohttp://www.springer.com/series/5613Volume 24
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ABC• Valery S. Mel’nikMikhail Z. ZgurovskyPavlo O. Kasyanovand Variation Inequalitiesfor Earth Data Processing IEvolution InclusionsOperator Inclusions and VariationInequalities for Earth Data Processing
cThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of aspecific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.Printed on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)Springer Heidelberg Dordrecht London New York ISBN 978-3-642-13836-2DOI 10.1007/978-3-642-13837-9e-ISBN 978-3-642-13837-9? Springer-Verlag Berlin Heidelberg 2011Cover design: deblik, BerlinDr. Mikhail Z. ZgurovskyPavlo O. KasyanovValery S. Mel’nikNational Technical University of Ukraine “Kyiv Polytechnic Institute”Institute for Applied System AnalysisNational Academy of Sciences of Ukraine37, Peremogy Ave.03056 KyivUkrainezgurovsm@hotmail.comLibrary of Congress Control Number: 2010936816ISSN 1571-8689e-ISSN 1876-9896
PrefaceThe necessity of taking into account non-linear effects, memory effects, sharp-ening conditions, semipenetration etc has arisen in recent years. It is caused byintensification of processes in applied chemistry, petrochemistry, transportation ofenergy carriers, physics, energetics, mechanics, economics and in other fields oftechnology and industry. When modeling such phenomena we are faced with non-linear boundary value problems for partial differential equations with multivaluedor discontinuous right-hand side, variational inequalities (evolutional as well as sta-tionary), an evolutional problem on manifolds (either with or without boundary),paired equations, cascading systems etc. Interpreting a concept of derivative prop-erly we can treat all these objects as operator or differential-operator inclusionsin Banach spaces and study them by the help of theory of multivalued maps ofpseudo-monotonetype.The given book arose from seminars and lecture courses on multi-valued andnon-linear analysis and their geophysical application. These courses were deliv-ered for rather different categories of learners in National Technical University ofUkraine “Kiev Polytechnic Institute”, Kiev National Taras Shevchenko University,Second University of Naples, University of Salerno etc. during 10 years. The bookis addressed to a wide circle of mathematical as well as engineering readers.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,who created and developed the theory of mentioned problems, exerted the powerfulinfluence on this book.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. Shishkov for useful remarks.We are grateful to the many students who have attended our lectures while wewere developing the notes for this volume.We want to express our gratitude to O.P. Kogut and N.V. Zadoyanchuk for theexceptionaldiligencewhenpreparingtheelectronicversionofthe bookandfriendlyhelp in linguistic issues.We want to express the special gratitude to Kathleen Cass and Olena L. Poptsovafor a technical support of our book.v
viPrefaceFinally, we express our gratitude to editors of the “Springer” Publishing Housewho worked with our book and everybody who took part in preparation of themanuscript.Beforehand we apologize to people whose works were missed inadvertentlywhen making the references.We will be grateful to readers for any remarks and corrections.Kyiv, UkraineAugust 2010Milkhail Z. ZgurovskyValery S. Mel’nikPavlo O. Kasyanov
Contents1Preliminary Results............................................................1.1The Main Results from Multivalued Mapping Theory..................1.2Classes of Multivalued Maps............................................. 291.3Subdifferentials in Infinite-Dimensional Spaces ........................ 951.4Minimax Inequalities in Finite-Dimensional Spaces ....................125References .......................................................................136122Operator Inclusions and Variation Inequalitiesin Infinite-Dimensional Spaces ...............................................1392.1Strong Solutions for Parameterized Operator Inclusions................1392.2Parameterized Operator Inequalities .....................................1452.3Variation Inequalities in Banach and Frechet Spaces ...................1582.4The Penalty Method for Multivariation Inequalities ....................1622.5Nonlinear Operators Equations of the Hammerstein Type.System of Operators Equations...........................................1902.6Nonlinear Non-coercive Operator Equationsand Their Normalization..................................................2012.7Some Example Connected with Membranes Theory....................238References .......................................................................242Index.................................................................................245vii
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Acronymsfor a.e.l.s.c.N-s.b.v.N-sub-b.v.r.c.r.l.s.c.r.s.c.r.u.s.c.s.b.v.s.m.sub-m.sub-b.v.u.h.c.u.s.c.u.s.b.v.V -s.b.v.w.l.s.c.For almost eachLower semicontinuousN-semibounded variationN-subbounded variationRadial continuousRadial lower semicontinuousRadial semicontinuousRadial upper semicontinuousSemibounded variationSemimonotoneSubmonotoneSubbounded variationUpper hemicontinuousUpper semicontinuousUniform semibounded variationV -semibounded variationWeakly lower semicontinuousix
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IntroductionSystem analysis is a unique ground where almost all mathematical disciplines arecloselyrelatedandwidelyapplied.Besides, whilesolvingdifficultsystemproblems,necessity to create new mathematical directions which in their turn simulate thedevelopment of classical chapters of mathematics arises. Here, approaches devel-oped to solve system problems quite often give new methods to grasp the problemsof classical mathematics.Motivatedby variousapplicationsthe theoryof partial dif-ferential equations and inclusions is now a well-developed branch of mathematicsattracting both mathematicians and experts in other fields. Having used some addi-tional a priori estimations we establish the smoothness properties of such solutions.The book deals with the solvability for nonlinearoperator problems and with differ-ential operator inclusions in infinite dimensional spaces with multivalued maps ofw?0-pseudomonotonetype.When investigating partial differential equations, inclusions and evolution varia-tion inequalities describing different processes in hydromechanics, geophysics andotherfields ofcontinuumandquantummechanicsandphysicsthe followingschemeis frequently used: at first we construct approximate solutions of the problem, afterthat we establish a priori estimations for the given solutions and then we prove thatthere exists such convergent sequence of the approximate solutions which limit issome exact solution of the original problem.Let us cite someexamples.At first we considersome problemconnectedwiththeanalysis of ecological state of the atmospherein physical anomalies conditions. Thecharacterofdiffusionand transferof harmfulimpuritiesin the atmosphereare deter-mined by factors of natural and anthropogeniccharacter. The natural factors includethe stability level of the atmosphere, its meteorological state, such as wind direc-tion and velocity,temperature, humidity,air pressure and others. The anthropogenicfactors are connected with the impact on the atmosphere due to industrial or otheractivities of a man. Such factors are the intensity of atmospheric pollution deter-mined by the volume, physical, and chemical properties of impurities dischargedinto the atmosphere. The anthropogenic factor is also the human impact on theatmospheric meteorological state, which may even lead to climatic changes.We will divide the factors determining the ecological state of the atmosphereinto ordinary or classical, and anomalous or nonclassical. The influence of ordinaryfactors results in monotonous (nonjump, two-directional, recoverable) changes inxi
xiiIntroductionthe atmospheric ecological state, see for example [Ma82]. Under the influence ofanomalous factors, the atmosphere behavior is characterized by jumps or unilateral,nonrecoverablestate changes, see for example [DuLi76].There are known natural and anthropogenic anomalies which result in the atmo-sphere’s anomalous behavior, such as inverse stratification of the atmosphere, heatisland, processes with restrictions of inequalities typical in the upper or lower levelof impurity concentration. Let us characterize these anomalies, see for example[ZgMel04,ZgNov96].This is a meteorological factor of natural origin. It occurs when the temperaturerises with the altitude (see Fig.1a). Inversion holds the impurity in the vertical layerandweakenstheturbulentexchangewithinit(seeFig.1b).Thus,inversionpromotesan increased accumulation of the impurities near to the groundlayer. Such phenom-ena are especially dangerous in valleys, low lands and other places with weak aircirculation.This heat anomaly occurs over large industrial centers. It is of anthropogenicori-gin. The essence of this anomaly is in an excess of the atmospheric air temperaturewithin industrial centers over the air temperature in their vicinities. The heat islandweakens turbulentexchangein the horizontallayer andholds an impurityin the areaof the center. The heat island does not impede penetration of impurities from thebordering vicinities (see Fig.2). Thus, this class of anomalies promotes the accu-mulation of impurities within the boundaries of industrial centers. It is especiallydangerous during summer at night in places with weak air circulation.Restrictions on the upper level of impurity concentration in the atmosphere aredetermined by physical and chemical properties within the impurity. For example,when some threshold concentration level of a finely disperse impurity in the atmo-sphere exceeded– u ? umax, its particles are crystallized and precipitated by gravity(see Fig.3). Thus, here is a nonlinear effect of the atmospheric purification. Thisprocess is activated when some threshold concentration level is overcome.Restrictions on the lower level of impurity concentration in the atmosphere arealso determined by physical and chemical properties of this impurity. An exampleof this anomaly is the process of washing-out a gaseous impurity by precipita-tion. There exists the lower threshold value of gaseous impurity concentration –u ? uminat which the action of precipitation on the concentration level becomesindistinguishable (see Fig.3).Classical problemsof atmosphericdiffusionand transfer are stated in the form ofvariation principles. If there are no additional requirements to these problems solu-tion,theyarereducedto theboundaryproblemsformathematicalphysicsequations,see for example [OmSe89].Anomalous problems of atmospheric diffusion differ from the ordinary or clas-sical problems by the presence of additional restrictions on the states. These restric-tions are written in the form of inequalities and characterize nonlinear effectsof unilateral conductivity of the boundary and other peculiarities. Thus, accord-ing for these peculiarities at the stage of a variation statement of a mathematicalphysics problem results in variation inequalities introduced and developed, see forexample [DuLi76].
IntroductionxiiiFig. 1 Temperature profile (a) and anomalous behavior effect of atmosphere (b)Let u be impurity concentration in atmosphere, determined on the open set ˝ ofspace R3with the smooth bound B D BL[ BR[ BU[ BD[ BF[ BTand timeinterval .0;tf/ for tf < 1, Q D ˝ ? .0;tf/, ˙ D B ? .0;tf/, is the solution ofthe inequality, see for example [DuLi76].@u? .f;v ? u/;@t;v ? u/ C .A.?/u;v ? u/ C ˚j.v/ ? ˚j.u/j D 1;2;ujtD0D u08v 2 H1.˝/ D V;in˝(1)
xivIntroductionFig. 2 Accumulation of impurities as a result of heat island effectFig. 3 Restrictions on the upper and lower level of impurity concentration in the atmospherewhere .f;g/ is the action of functional f 2 .H1.˝//?on the element g 2 H1.˝/,operator A.?/ W V ! V?, is defined by bilinear form:Z˝.A.?/u;?/ D3XiD1hk.z/@u@zi@?@zi? ci.z/@u@zi?i@z ?Z˝d.z/u?dz; 8? 2 H1.˝/
Introductionxvif .f;g/ 2 L2.˝/, operation .f;g/ coincides with inner product in L2.˝/, i.e.,.f;g/ DRci.z/ (for i D 1;2;3) be the wind auxiliary parameters z1;z2;z3, respectively;d.z/ be the impurity absorption coefficient. Variable f.t;z/ Dis the force function of the process; qi.t/ be the sources function, operating inthe sub-spaces ˝i 2 ˝, i D 1;2;:::;K, K – the number of external actionpoints; ı.z ? zi/ – characteristic function. Let us set up that parameters ? Dfk.z/;ci.z/;d.z/g, functional ˚j, and the force function of the process f havethe technological restrictions. So, we obtain the solvability problem for evolutionvariation inequality (1)Let us point out that in mathematical theory that there occur such objects istoo little investigated or nearly uninvestigated in mathematics. These are operatorinclusions, variation inequalities with multivaluedmaps, differential-operatorinclu-sions, evolution variation inequalities with multimaps, systems containing operatorequalities, inequalities and inclusions of different types etc.Foralltheseclassesofobjectstheoriginalmathematicalapparatusisbeingdevel-oped. Thereupon all results obtained are new ones even for those cases that havebeen rather actively investigated in mathematical literature. Specifically, severalJ.-L. Lions’s problems from his nonlinear boundary problems theory were solved.So in the present book the new mathematical apparatus is developed, whichmakes it possible to investigate problems for a wide class of nonlinear objects withdistributedparameters,alsohereauthorsofferanaxiomaticstudyofdifferentclassesof nonlinear maps in Cartesian product of spaces (fcontrol spaceg ? fstate spaceg),examine their relations and cite a great number of examples.Using these constructions authors prove Theorems on solvability and aboutdependence on parameter for solutions of nonlinear operator equations, operatorinclusions, variation inequalities, different-type system of equations and inclusions,differentialoperatorequations,evolutioninclusionsand parabolicvariationinequal-ities. These results are of great significance in their own right and have not beenilluminated in mathematical literature before. To illustrate the distinction featuresof the main object of our consideration let us cite one more motivational example.Let ˝ be a bounded domain in Rnwith a regular boundary @˝. Let us considerthe following optimization problem for a quasilinear elliptic system:˝f.z/g.z/dzIk.z/ – coefficient of turbulent diffusion; z D .z1;z2;z3/IK PiD1qi.t/ı.z ? zi/,nXiD1@@xi u.x/ˇˇˇˇ@y@xiˇˇˇˇp?2@y@xi!D f .x/; x 2 ˝;(2)y j@˝D 0(3)Z˝jy .x/jpdx C ? kukL1.˝/! inf(4)
xviIntroductionwhere p ? 2, ? > 0, f 2 Lq.˝/,1pC1qD 1,u 2 U D fv 2 L2.˝/j0 < ˛ ? v.x/ ? ˇ for a.e. x 2 ˝g:Forafixedu 2 U theboundaryproblem(2),(3)hasauniquegeneralizedsolution?which continuously (within the correspondent topology) depends on u 2 U andf 2 Lq.˝/. However problem (2)–(4) has no solution, namely a good problem ofthe differential equations theory is bad one for the optimization theory.On the other hand let us consider ˇˇˇˇ@y@vA@˝wherey 2 W1p.˝/ Dy 2 Lp.˝/ˇˇˇˇ@y@xi2 Lp.˝/; i D 1;:::;n?;nXiD1@@xi@y@xiˇˇˇˇp?2@y@xi!D f .x/; x 2 ˝;(5)ˇˇˇˇD u.?/; ? 2 @˝;(6)f 2 Lq.˝/;u 2 U ? Lq.@˝/;@y@?ADnXiD1ˇˇˇˇ@y@xiˇˇˇˇp?2@y@xi?i:It is required to minimize a functionalZ˝jy .x/jpdx C ı kukLq.@˝/! inf;ı > 0(7)on the solutions of (5), (6). The pair .uIy/ 2 U ? W1refers to be admissible, and a problem (5)–(7) refers to be regular if the set ofadmissible pairs is nonempty. We remark that from the point of view of the partialdifferential equations theory problem (5), (6) is a “bad” one since it is not coer-cive. At the same time the optimization problem (5)–(7) is a “good” one since itis regular, coercive and also has a solution. To prove its regularity it is sufficientto consider (5) with the boundary condition (3). Thus (3), (5) has a unique solu-tion y02 W1u0Dexamples show basic distinctions of the purposes of the partial differential equa-tions theory and the optimization theory for the objects described by the partialdifferential equations.p.˝/, satisfying (5), (6),p.˝/, and acting on it by the operator@vAj@˝2 Lq.@˝/. Hence, the pair .u0Iy0/ is admissible for (5)–(7). These@@vAj@˝we receive an element@y0
IntroductionxviiThe boundary problem (5), (6) generates a nonlinear map A W U ? X ! X?bythe rulehA.u;y/;wi DnXiD1Z˝ˇˇˇˇ@y@xiˇˇˇˇp?2@y@xi@w@xidxCZ@˝u.?/w.?/d?; u 2 Lq.@˝/;y;w 2 W1p.˝/ D X:Let us consider the parameterized multivalued mapB W X ! 2X?; B .y/ D fA.u;y/ju 2 U g:With its help (5), (6) can be represented as the operator inclusionB .y/ 3 f:(8)Thus, to prove regularity of the optimization problem (5)–(7) it is sufficient toestablish a resolvability of the operator inclusion (8).When describing series of real processes in chemistry, physics, biology, ecology,economics or geophysics there may arise a necessity to investigate the situationwhen an equation or a system modeling the correspondentprocess has series of fea-tures,namely:instability,breakphenomenon,noncoercivity,nonuniquenessofsolu-tionsandbifurcationphenomena,changesofmodel’sstructuredependingoncontrolfunctions, etc. Such distributed systems are called nonlinear singular ones. Wheninvestigating singular systems, difficulties in trans-calculating arise and approxima-tion solution algorithm must contain regularization procedure.Using optimal control methods for nonlinear infinite-dimension systems wedevelop a new fundamental approach to study singular boundary problems forpartial differential equations. This approach is based on replacement of initial ill-posed problem by auxiliary (associated) optimal control problem [ZgMel04]. Onthis way authors managed to construct stable solving algorithms for Navier–Stokesequations, Benard systems, etc. Therefore the methods of optimal control theoryrepresent a mathematical apparatus for effective research of ill-posed equations ofmathematical physics.Let us cite some more examples. Let ˝ ? Rnbe a bounded domain with asmoothboundaryv > 0.Fory W .0;T/?˝ ! Rnletusconsiderthen-dimensionalNavier–Stokes problem@y@t? v?y CnXiD1yi@y@xiD f.t;x/ ? rp;(9)divy D 0;XyjtD0D y0.x/;(10)y j˙D 0;D .0;T/ ? @˝;(11)(12)
xviiiIntroductionwhere f is a inhomogeneity function, p is a pressure.Auxiliary extremal problem:@y@t? v?y CnXiD1ui@y@xiD f.t;x/ ? rp:(13)Here the function u W .0;T/ ? ˝ ! Rnbelongs to the class of solutions of (9)–(12) W . For each u 2 W problem (10)–(13) has a unique generalized solution. Onsolutions of (13) the following extremal problem is posedJ.u/ D ku ? y.u/k2W! inf:(14)Under natural conditions existence of a solution of (10)–(14) is proved usingmethods of optimal control theory. If for some u J.u/ D 0 , u D y there exists asolution for (9)–(12).The next example concerns evolution inequalities on cones. Let us consider apositive solution problem for evolution inequalitiesy0C A.y/ ? f; y.0/ D y0? 0;(15)y ? 0;(16)y 2 X, A W X ! X?. This problem is nontrivial even for evolution equations.In order of modeling and prognostication of large social-economic systems weremark that one of directions was construction of network models with associatememory (something like Hopfield neuronets) for a wide class of processes: social,economic, political, natural ones. Among recent achievements on this way therecan be mentioned the invention of multivalued solutions behavior for such models,openingperspectivesforbetterunderstandingoftheoccurrenceofdifferentprogressscenarios for large social systems.Multivalued solutions of models for social systems open perspectives for wideapplications. So, there arise problems studying dependence on parameter of mod-els behavior in multivalued case, occurrence of limit cycles, etc., and especiallyattractors in the case of multivalued maps which are actively investigated due tothe grounds laid before. To solve these problems we need nearly all arsenal ofsystem analysis methods: nonlinear functional analysis, dynamic systems theory,differential-operatorand discrete equations theory.One more direction of research related to modeling of movement (of transportand pedestrians) for a large group of objects. As one of approaches to solving suchkind of problems the cellular automaton method was chosen, where movement isanalyzed within its discrete approximation (where the space is divided into regularblocsofcells)andweconsidertransitionsofanobjectmovingfromonecelltoother.Mathematically this problem looks like investigation of discrete dynamic systems(determined or stochastic ones). There was made a graph modeling under certainconditions using models adapted for real geometry of traffic current and jams, andalso problemsaboutdiscountingthemovementprognosisbymembersofthe motion
Introductionxixwere set. Such problems again bring us to necessity of further mathematical investi-gations from the point of multivalued maps and also differential-operatorinclusionsand optimization theory.Motivated by various applications the theory of partial differential equationsand inclusions is now a well-developed branch of mathematics attracting bothmathematicians and experts in other fields. The book deals with the solvabilityof nonlinear operator, differential-operator problems and evolution inclusions ininfinite-dimensional spaces with multivalued maps of “pseudomonotone”type.Many important applied problems (for example for processes of diffusion of oilin porous mediums, for processes of atmospheric diffusion and transfer, etc.) can bereduced to so-called problems with one-sided boundary conditions or to variationinequalities, which also generate differential-operator inclusions. Let us considerthe simplest example of this kind [Li69]: it is required to find a solution of theequation My D f in a domain ˝ with a boundary ? , such that on ? the followingconditionsu ? 0;are satisfied. The generalized solution of such problem does not satisfy the inte-grated identity (as, for example, in a Dirichlet problem) but it satisfies someintegrated inequality which is called a variation inequality.The theory of variation inequalities as a new section of the partial differentialequations theory arose in 60-s of the last century from the Signorini problem ofthe elasticity theory, which is completely investigated in G.Fichera’s work [Fi64].In this work fundamentals of the variation inequalities theory were laid. Thenthe investigation of variation inequalities was continued in papers of J.-L. Lions[Li69], G. Stampacchia [St64] and their pupils. In particular an abstract statementof problems that could be reduced to such inequalities was considered.At the same time the Russian school of mathematicians laid the basis of thedifferential inclusions theory. A great number of applications generated the devel-opment of the theory of the differential equations with discontinuous right partswhich generate differential inclusions in finite-dimensional spaces. A great dealof problems in mechanics, electrical engineering and automatic control theorythat are described by the given objects were considered in A.A.Andronov’s work[AnViHa59]. Wide use of different switches (relays) requires the developed con-structions of the given theory. The basic directions of the theory of the differentialequations with discontinuous right-hand side and theories of differential inclusionsin finite-dimensional spaces were stated in A.F.Filippov’s monograph [FiAr88].The theory of multivaluedmaps as a new section of nonlinear analysis has arisenat the intersection of two mathematical sciences, namely the variation inequalitiestheory and the theory of differential-operator inclusions, which in particular, aregenerated by variation inequalities. Fundamentals of the given theory were laid inthe papers of B.M. Pshenichniy, M.M. Vainberg, V.F. Dem’yanov, L.V. Vasiliev,A.M. Rubinov, A.D. Ioffe, V.M.Tihomirov, A.A. Chikrii, I.Ekeland, J.-P. Aubin,H.Frankowska and others in [Au79, AuFr90, DemVas81, IoTi79, AuEk84, Chi97,Li69,Psh80]. One of the main classes of multivalued maps is subdifferential maps,@u@n? 0;u@u@nD 0:
xxIntroductiontheir basic properties (for Banach spaces) were investigated in [AuEk84, Chi97,DemVas81,IoTi79].Significantprogressin study ofnonlinearboundaryproblemsfor partial differen-tial equations and inclusions is caused by profound development of methods of thenonlinear analysis which can be applied to different sections of mathematics. At thepresent time it became natural to reduce the given problems to nonlinear operatoror differential–operator equations and inclusions in functional spaces. When usingsuch approach the results for concrete systems with contributed parameters turn outas a corollary of operator existence Theorems.Convergence of approximate solutions to an exact solution of the differential-operator equation or inclusion is frequently proved on the basis of a monotony ora pseudomonotony of corresponding operator. If the given property of an initialoperator takes place then it is possible to prove convergence of the approximatedsolutions within weaker a priori estimations than it is demandedwhen using embed-ding Theorems. The monotony concept has been introduced in papers of Vainberg,Kachurovsky, Minty, Sarantonello and others. Significant generalization to mono-tonicity has given H.Brezis [Br68]. Namely, Brezis refers operator A W X ! X?tobe pseudomonotone if(a) The operator A is bounded(b) From un! u weakly in X and fromlimn!1hA.un/;un? uiX? 0it follows, thatlimn!1hA.un/;un? viX? hA.u/;u ? viXIn applications, as a pseudomonotone operator the sum of radially continuousmonotone bounded operator and strongly continuous operator was considered[GaGrZa74]. Concrete examples of pseudomonotone operators were obtained byextension of elliptic differential operators when only their summands comply-ing with highest derivatives satisfied the monotony property [Li69]. In papers ofJ.-L. Lions, H. Gajewski, K. GrR oger, K. Zacharias [Li69, GaGrZa74] the mainresults of solvability theory for abstract operator equations and differential operatorequations that are monotone or pseudomonotone in Brezis sense are set out. Alsothe application of the given Theorems to the concrete equations of mathematicalphysics, and in particular, to free boundary problems.The theory of monotone operators in reflexive Banach spaces is one of the majorareas of nonlinear functional analysis. Its basis is so-called variation methods, andsince 60-s of the last century the theory intensively has been developed in tightinteraction with the theory of convex functions and the theory of partial differentialequations. The papers of F. Browder and P. Hess [Br77,BrHe72] became classicalones in the given direction of investigations. In particular in F. Browder and P. Hesswork [BrHe72] the class of generalized pseudomonotone operators that enveloped8v 2 X:
Introductionxxia class of monotone mappings was introduced. Let W be some normalized spacecontinuously embedded in the normalized space Y. The strict multivalued map A WY ! 2Y?is called generalized pseudomonotoneon W if for each pair of sequencesfyngn?1? W and fdngn?1? Y?such that dn2 A.yn/, yn! y weakly in W ,dn! d weakly star in Y?, from the inequalitylimn!1hdn;yniY? hd;yiYit follows that d 2 A.y/ and hdn;yniY ! hd;yiY.The grave disadvantage of the given theory is the fact that in common case it isimpossible to prove a closure within the sum of pseudomonotone (in the classicalsense) maps (the given statement is problematic). This disadvantage becomes moresubstantial on investigating differential-operator inclusions and evolution variationinequalities when we necessarily consider the sum of the classical monotone map-ping and the subdifferential(on Gateaux or on Clarke) for multivaluedmap which isgeneralized pseudomonotone [CaVyLeMo04]. Here we realize I.V. Skrypnik’s ideaof passing to subsequences in classical definitions [Sk94]. This enable us to con-sider essentially wider class of ?0-pseudomonotone maps, closed within the sumof maps, that appeared problematic for classical definitions [ZgMel02]. In V.S.Mel’nik and P.O. Kasyanov papers [KaMe05a,KaMeTo06] there was introducedthe class of w?0-pseudomonotone maps which includes, in particular, a class ofgeneralized pseudomonotonemultivalued operators and also it is closed as for sum-ming. A strict multivalued map A W Y ! 2Y?with the nonempty,convex,bounded,closed valuesis called w?0-pseudomonotone(?0-pseudomonotoneon W ), if foranysequence fyngn?0? W such that yn! y0weakly in W , dn! d0weakly star inY?as n ! C1, where dn2 A.yn/ 8n ? 1, from inequalitylimn!1hdn;yn? y0iY? 0the existence of such subsequences fynkgk?1from fyngn?1and fdnkgk?1fromfdngn?1for whichlimk!1hdnk;ynk? wiY? ŒA.y0/;y0? w??8w 2 Yis follows. Here we have to prove a resolvability for differential-operatorinclusionswith ?0-pseudomonotoneon D.L/ multivalued maps in Banach spaces:Lu C A.u/ C B.u/ 3 f;1, B W X2 ! 2X?u 2 D.L/;(17)where A W X1 ! 2X?pseudomonotone type with nonempty, convex, closed, bounded values, X1, X2areBanach spaces continuously embedded in some Hausdorff linear topological space,X D X1\ X2, L W D.L/ ? X ! X?is linear, monotone, closed, densely definedoperator with a linear definitional domain D.L/.2 are multivalued maps of D.L/?0-
xxiiIntroductionLet us remarkthat any multivaluedmap A W Y ! 2Y?, naturally generatesupperand, accordingly, lower form:ŒA.y/;!?CD supd2A.y/hd;wiY;ŒA.y/;!??Dinfd2A.y/hd;wiY;y;! 2 X:Properties of the given objects have been investigated in M.Z. Zgurovsky andV.S. Mel’nik works [Me97,ZgMel99,ZgMel00,ZgMel02]. Thus, together with theclassical coercivity condition for operator A:hA.y/;yiYkykY! C1;askykY ! C1;which ensures the important a priori estimations, arises C-coercivity (and, accord-ingly, ?-coercivity):ŒA.y/;y?C.?/kykY! C1;askykY ! C1:C-coercivity is much weaker condition than ?-coercivity.When investigatingmultivaluedmaps of w?0-pseudomonotonetype it was foundout that even for subdifferentials of convex lower semicontinuous functionals theboundness condition is not natural [KaMe05b]. Thus it was necessary to introducean adequate relaxation of the boundness condition which would have enveloped atleast a class of monotone multivalued maps. In the paper [IvMel88] the followingdefinition was introduced: multivalued map A W Y ! 2Y?satisfies Property .˘/ iffor any bounded set B ? Y , any y02 Y, for some k > 0 and for some selector d(d.y/ 2 A.y/ 8y 2 B), for whichhd.y/;y ? y0iY ? kfor all y 2 B;there exists such C > 0 thatkd.y/kY? ? Cfor all y 2 B:Recent development of the monotony method in the theory of differential-operator inclusions and evolution variation inequalities [CaMo03, DeMiPa03,NaPa95, CaVyLeMo04, Pa85, Pa86, Pa87a, Pa87b, Pa88a, Pa88b, Pa94, PaYa06]ensures resolvability of the given objects under the conditions of ?-coercivity,boundness and the generalized pseudomonotony (it is necessary to notice, that theproof is not constructive). With relation to applications it would be actual to relaxsome conditions of multivalued maps in a problem (17) replacing ?-coercivity byC-coercivity, boundness by Condition .˘/ and pseudomonotony in classical senseor generalize...