Initial time value problem solutions for evolution inclusions with Sk type operators

For a large class of operator inclusions, including those generated by maps of Sk type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of Faedo-Galerkin. Одержано загальну теорему про існування розв’язк...

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Дата:	2009
Автори:	Kasyanov, P.O., Mel'nik, V.S., Toscano, S.
Формат:	Стаття
Мова:	English
Опубліковано:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2009
Теми:	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/12392
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Initial time value problem solutions for evolution inclusions with Sk type operators / P.O. Kasyanov, V.S. Mel'nik, S. Toscano // Систем. дослідж. та інформ. технології. — 2009. — № 1. — С. 116-130. — Бібліогр.: 31 назв. — англ.

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spelling	Kasyanov, P.O. Mel'nik, V.S. Toscano, S. 2010-10-07T08:25:37Z 2010-10-07T08:25:37Z 2009 Initial time value problem solutions for evolution inclusions with Sk type operators / P.O. Kasyanov, V.S. Mel'nik, S. Toscano // Систем. дослідж. та інформ. технології. — 2009. — № 1. — С. 116-130. — Бібліогр.: 31 назв. — англ. 1681–6048 https://nasplib.isofts.kiev.ua/handle/123456789/12392 517.9 For a large class of operator inclusions, including those generated by maps of Sk type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of Faedo-Galerkin. Одержано загальну теорему про існування розв’язків для широкого класу операторних включень, у тому числі ті, які породжуються відображеннями типу Sk . Результат застосовано до деякого окремого прикладу. Теорему доведено за допомогою методу Фаедо-Гальоркіна. Получена общая теорема о существовании решений для широкого класса операторних включений, в том числе те, которые порождаются отображениями типа Sk . Результат применен к некоторому частному примеру. Теорема доказана с помощью метода Фаэдо-Галeркина. en Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України Нові методи в системному аналізі, інформатиці та теорії прийняття рішень Initial time value problem solutions for evolution inclusions with Sk type operators Задача Коші для еволюційних включень з відображеннями типу Sk Задача Коши для эволюционных включений с отображениями типа Sk Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Initial time value problem solutions for evolution inclusions with Sk type operators
spellingShingle	Initial time value problem solutions for evolution inclusions with Sk type operators Kasyanov, P.O. Mel'nik, V.S. Toscano, S. Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
title_short	Initial time value problem solutions for evolution inclusions with Sk type operators
title_full	Initial time value problem solutions for evolution inclusions with Sk type operators
title_fullStr	Initial time value problem solutions for evolution inclusions with Sk type operators
title_full_unstemmed	Initial time value problem solutions for evolution inclusions with Sk type operators
title_sort	initial time value problem solutions for evolution inclusions with sk type operators
author	Kasyanov, P.O. Mel'nik, V.S. Toscano, S.
author_facet	Kasyanov, P.O. Mel'nik, V.S. Toscano, S.
topic	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
topic_facet	Нові методи в системному аналізі, інформатиці та теорії прийняття рішень
publishDate	2009
language	English
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
format	Article
title_alt	Задача Коші для еволюційних включень з відображеннями типу Sk Задача Коши для эволюционных включений с отображениями типа Sk
description	For a large class of operator inclusions, including those generated by maps of Sk type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of Faedo-Galerkin. Одержано загальну теорему про існування розв’язків для широкого класу операторних включень, у тому числі ті, які породжуються відображеннями типу Sk . Результат застосовано до деякого окремого прикладу. Теорему доведено за допомогою методу Фаедо-Гальоркіна. Получена общая теорема о существовании решений для широкого класса операторних включений, в том числе те, которые порождаются отображениями типа Sk . Результат применен к некоторому частному примеру. Теорема доказана с помощью метода Фаэдо-Галeркина.
issn	1681–6048
url	https://nasplib.isofts.kiev.ua/handle/123456789/12392
citation_txt	Initial time value problem solutions for evolution inclusions with Sk type operators / P.O. Kasyanov, V.S. Mel'nik, S. Toscano // Систем. дослідж. та інформ. технології. — 2009. — № 1. — С. 116-130. — Бібліогр.: 31 назв. — англ.
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first_indexed	2025-11-26T00:18:47Z
last_indexed	2025-11-26T00:18:47Z
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fulltext	© P.O. Kasyanov, V.S. Mel'nik , S. Toscano, 2009 116 ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 TIДC НОВІ МЕТОДИ В СИСТЕМНОМУ АНАЛІЗІ, ІНФОРМАТИЦІ ТА ТЕОРІЇ ПРИЙНЯТТЯ РІШЕНЬ УДК 517.9 INITIAL TIME VALUE PROBLEM SOLUTIONS FOR EVOLUTION INCLUSIONS WITH kS TYPE OPERATORS P.O. KASYANOV, V.S. MEL'NIK, S. TOSCANO For a large class of operator inclusions, including those generated by maps of kS type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of Faedo- Galerkin. INTRODUCTION One of the most effective approach to investigate nonlinear problems, represented by partial differential equations, inclusions and inequalities with boundary values, consists in the reduction of them into differential-operator inclusions in infinite- dimensional spaces governed by nonlinear operators. In order to study these objects the modern methods of nonlinear analysis have been used [7, 8, 17, 28]. Convergence of approximate solutions to an exact solution of the differential- operator equation or inclusion is frequently proved on the basis of a monotony or a pseudomonotony of corresponding operator. In applications, as a pseudomonotone operator the sum of radially continuous monotone bounded operator and strongly continuous operator was considered [8]. Concrete examples of pseudomonotone operators were obtained by extension of elliptic differential operators when only their summands complying with highest derivatives satisfied the monotony property [17]. The papers of F. Browder and P. Hess [3, 4] became classical in the given direction of investigations. In particular in F. Browder and P. Hess work [4] the class of generalized pseudomonotone operators was introduced. Let W be real Banach space continuously embedded in real reflexive Banach space Y with dual space Y , R→×⋅〉〈⋅ YYY :, be the pairing. Further, as )( YCv we consider the family of all nonempty closed convex bounded subsets of the space Y . Multi-valued map )(: YCYA v→ refers to be generalized pseudomonotone on W if for each pair of sequences Wy nn ⊂≥1}{ and 1}{ Yd nn ⊂≥ such that )( nn yAd ∈ , yyn → weakly in W , ddn → weakly in Y , from the inequality YYnnn ydyd 〉〈≤〉〈 ∞→ ,,lim Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 117 it follows that )(yAd ∈ and YYnn ydyd 〉〈→〉〈 ,, . I.V. Skrypnik's idea of passing to subsequences in classical definitions [26], realized for stationary and evolution inclusions in M.Z. Zgurovsky, P.O. Kasyanov, V.S. Mel'nik and J. Valero papers (see [12–16], [18–21] and citations there) enabled to consider the class of 0λ w - pseudomonotone maps which includes, in particular, a class of generalized pseudomonotone on W multi-valued operators and it is closed within summing. Let us remark that any multi-valued map )(: YCYA v→ naturally generates upper and, accordingly, lower form: XywdyAwdyA YyAdY yAd ∈〉〈〉〈 ∈∈ + ωωω ,,,inf=]),([,,sup=]),([ )(_ )( . Properties of the given objects have been investigated by M.Z. Zgurovsky and V.S. Mel'nik (see [16, 18, 21]). Thus, together with the classical coercivity condition for singlevalued maps +∞→ 〉〈 Y Y y yyA ),( as +∞→Yy which ensures the important a priori estimations, arises +-coercivity (and, accordingly, –-coercivity) for multivalued maps .as, ]),([ )( +∞→+∞→−+ Y Y y y yyA +-coercivity is weaker condition than –-coercivity. Recent development of the monotony method in the theory of differential- operator inclusions and evolutionary variational inequalities ensures resolvability of the given objects under the conditions of coercivity, quasiboundedness and the generalized pseudomonotony (see for example [5–6, 9–10, 23–25, 27] and citations there). V.S. Mel'nik's results [22] allows to consider evolution inclusions with + -coercive 0λ w -pseudomonotone quasibounded multimappings (see [12]– [16], [31] and citations there). In this paper we introduce the differential-operator scheme for investigation nonlinear boundary-value problems with summands complying with highest derivatives are not satisfied monotone condition. A multi-valued map →YA : )( YCv→ satisfies the property kS on W , if for any sequence Wy nn ⊂≥0}{ such that 0yyn → weakly in W , 0ddn → weakly in Y as +∞→n , where )( nn yAd ∈ 1≥∀n , from 0=,lim 0 Ynn n yyd 〉−〈 ∞→ , it follows that )( 00 yAd ∈ . Now we consider the simple example of kS type operator. Let (0,1)=Ω , )(= 1 0 ΩHY be the real Sobolev space with dual space )(= 1* Ω−HY (see for details [8]). Let 1,1][: YYA →−× defined by the rule ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− y dx d dx dyA αα =),( . P.O. Kasyanov, V.S. Mel'nik , S. Toscano ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 118 Then the multivalued map YyyAy ∈−∈ },1,1][\|),({=)( ααA satisfies the property kS , it is +-coercive, but it is not –-coercive, it is not general- ized pseudomonotone and )( A− is not generalized pseudomonotone too (see [11] for detailes). We remark that stationary inclusions for multimaps with kS properties were considered by V.O. Kapustyan, P.O. Kasyanov, O.P. Kogut [11], the evolution inclusions for + -coercive 0λ w -pseudomonotone quasibounded maps by V.S. Mel'nik, P.O. Kasyanov, J. Valero (see [12]–[16], [31] and citations there). The obtained in this paper results are new results for evolution equations too. PROBLEM DEFINITION Let ),( 11 VV ⋅ and ),( 22 VV ⋅ be some reflexive separable Banach spaces, continuously embedded in the Hilbert space )),(,( ⋅⋅H such that 21=: VVV ∩ is dense in spaces 21,VV and H (1) After the identification HH ≡ we get ,, * 22 * 11 VHVVHV ⊂⊂⊂⊂ (2) with continuous and dense embeddings [8], where ),( * * iViV ⋅ is the topologically conjugate of iV space with respect to the canonical bilinear form R→×⋅〉〈⋅ iiiV VV :, ( 21,=i ) which coincides on VH × with the inner product ),( ⋅⋅ on H. Let us consider the functional spaces ),;();(= iipiri VSLHSLX ∩ where ][0,= TS , +∞<<0 T , +∞≤ <<1 ii rp 1,2)=(i . The spaces iX are Banach spaces with the norms );();(= HSyVSyy ir LiipLiX + . Moreover, iX is a reflexive space. Let us also consider the Banach space 21= XXX ∩ with the norm 21 = XXX yyy + . Since the spaces );();( HSLVSL iriiq ′ + and * iX are isometrically isomorphic, we identify them. Analogously, ),;();();();(== 21 * 22 * 11 * 2 * 1 * HSLHSLVSLVSLXXX rrqq ′′ ++++ where 1== 1111 −−− ′ − ++ iiii qprr . Let us define the duality form on XX ×* +〉〈++〉〈 ∫∫∫ τττττττττ dyfdyfdyfyf V S H S H S 1211211 )(),())(),(())(),((=, ,))(),((=)(),( 222 ττττττ dyfdyf S V S ∫∫ 〉〈+ Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 119 where 22211211= fffff +++ , );(1 HSLf iri ′ ∈ , );( * 2 iiqi VSLf ∈ . Remark, that for each Xf ∈ ⎩ ⎨ ⎧ +++ ∈ ′ ∈ ;);(' max : inf= 1 11 1,2)=();(2),;(1 22211211= * HSf fff f rL iiVS iqLifHS ir Lif ffX .);(;);(;);(' * 22 22* 11 21 2 12 ⎭ ⎬ ⎫ VSfVSfHSf qLqLrL Following by [17], we may assume that there is a separable Hilbert space σV such that 1VV ⊂σ , 2VV ⊂σ with continuous and dense embedding, HV ⊂σ with compact and dense embedding. Then 22 11 , σσσσ VVHVVVVHVV ⊂⊂⊂⊂⊂⊂⊂⊂ with continuous and dense embedding. For 1,2=i let us set ,=),;();(= 2,1,, σσσσσ XXXVSLHSLX ipiri ∩∩ * 2, * 1, *** , =),;();(= σσσσσ XXXVSLHSLX iqiri ++ ′ , σσσσσ 2,1, * ,, =},\|{= WWWXyXyW iii ∩∈′∈ . For multivalued (in general) map : XXA ⇒ let us consider such problem: ⎩ ⎨ ⎧ ⊂∈ ∋+′ ),;(,=(0) ,)( HSCWuau fuAu (3) where Ha∈ and Xf ∈ are arbitrary fixed elements. The goal of this work is to prove the solvability for the given problem by the Faedo-Galerkin method. THE CLASS )( XH Let us note that )( XB H∈ , if for an arbitrary measurable set SE ⊂ and for arbitrary Bvu ∈, the inclusion Buvu E ∈−+ χ)( is true. Here and further for Xd ∈ )()(=))(( τχττχ EE dd for a.e. ⎩ ⎨ ⎧ ∈ ∈ .else,0 ,,1 =)(, E S E τ τχτ Lemma 1 [30]. )( XB H∈ if and only if 1≥∀n , Bd n ii ⊂∀ 1=}{ and for arbitrary measurable pairwise disjoint subsets n jjE 1=}{ of the set S : SE j n j =1=∪ the following Bd jEj n j ∈∑ χ1= is true. Let us remark, that )(, ** XX H∈∅ ; Xf ∈∀ )(}{ Xf H∈ ; if : VSK ⇒ is an arbitrary multi-valued map, then P.O. Kasyanov, V.S. Mel'nik , S. Toscano ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 120 ).(}a.e.for)()(\|{ * XSttKtfXf H∈∈∈∈ At the same time for an arbitrary 0\Vv∈ that is not equal to 0 the closed convex set ),([0,1]},\|{= * XvfXfB H∉∈≡∈ αα as Bvg T ∉⋅⋅⋅ )(=)( /2][0;χ . CLASSES OF MULTI-VALUED MAPS Let us consider now the main classes of multi-valued maps. Let Y be some reflexive Banach space, Y be its topologically adjoint, R→×⋅〉〈⋅ YYY :, be the pairing, : YYA ⇒ be the strict multi-valued map, i.e. ∅≠)(yA .Xy∈∀ For this map let us define the upper )( sup=)( Xyd dyA A∈ + and the lower )(_ inf=)( Xyd dyA A∈ norms, where Yy∈ . Let us consider the next maps which are connected with :A :co YYA ⇒ and ,:co YYA ⇒ which are defined by the next relations ))((co=))(co( yAyA and ))((co=))(co( yAyA respectively, where ))((co yA is the weak closeness of the convex hull of the set )(yA in the space Y . It is known that strict multi-valued maps :, YYBA ⇒ have such properties [16, 18, 20]: 1) +++ +≤+ ]),([]),([]),([ 2121 vyAvyAvvyA , −−− +≥+ ]),([]),([]),([ 2121 vyAvyAvvyA y∀ , 1v , Yv ∈2 ; 2) −+ −− ]),([=]),([ vyAvyA , )()()( ]),([]),([=]),()([ −+−+−+ ++ vyBvyAvyByA Yvy ∈∀ , ; 3) )()( ]),(c[=]),([ −+−+ vyAovyA Yvy ∈∀ , ; 4) YvyAvyA )()( )(]),([ −+−+ ≤ , +++ +≤+ )()()()( yByAyByA , partially the inclusions )(c yAod ∈ is true if and only if .,]),([ YvvdvyA Y ∈∀〉〈≥+ Let .YD ⊂ If R→×⋅⋅ YDa :),( , then for every Dy∈ the functional ),( wyawY ∋ is positively homogeneous convex and lower semi-continuous if and only if there exists the multi-valued map : YYA ⇒ with the definition domain DAD =)( such, that .),(]),([=),( YwADywyAwya ∈∀∈∀+ Further, yyn in Y will mean, that ny converges weakly to y in Y . Let W be some normalized space that continuously embedded into Y . Let us consider multi-valued map : YYA ⇒ . Definition 1. The strict multi-valued map : YYA ⇒ is called: Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 121 • 0λ -pseudomonotone on W , if for any sequence Wy nn ⊂≥0}{ such, that 0yyn in W , 0ddn in Y as +∞→n , where )(co nn yAd ∈ 1≥∀n , from the inequality 0,lim 0 ≤〉−〈 ∞→ Ynnn yyd (4) it follows the existence of subsequence 1},{ ≥kknkn dy from 1},{ ≥nnn dy , for that YwwyyAwyd Yknkn k ∈∀−≥〉−〈 − ∞→ ]),([,lim 00 (5) is fulfilled; • bounded, if for every 0>L there exists such 0>l , that LyYy Y ≤∈∀ : , it follows that lyA ≤+)( . Definition 2. The strict multi-valued map : XXA ⇒ is called: • the operator of the Volterra type , if for arbitrary Xvu ∈, , St∈ from the equality )(=)( svsu for a.e. ][0,ts∈ , it follows, that ++ ]),([=]),([ tt vAuA ξξ Xt ∈∀ξ : 0=)(stξ for a.e. ];[0,\ tSs∈ • +(-)-coercive, if there exists the real function RR →+:γ such, that +∞→)(sγ as +∞→s and ;)(]),([ )( YyyyyyA YY ∈∀≥−+ γ • demi-closed, if from that fact, that yyn → in Y , ddn in Y , where )( nn yAd ∈ , 1≥n , it follows, that )(yAd ∈ . Let us consider multi-valued maps, that act from mX into mX , 1≥m . Let us remark, that embeddings * mmm XYX ⊂⊂ are continuous, and the embedding mW into mX is compact [17]. Definition 3. The multi-valued map )(: * mvm XCX →A is called ),( * mm XW - weakly closed, if from that fact, that yyn in ,mW ddn in ,* mX )( nn yd A∈ 1≥∀n it follows, that ).(yd A∈ Lemma 2. The multi-valued map )(: * mvm XCX →A satisfies the property kS on mW if and only if )(: * mvm XCX →A is ),( * mm XW -weakly closed. Proof. Let us prove the necessity. Let yyn in ,mW ddn in ,* mX where )( nn yd A∈ 1.≥∀n Then yyn → in mX and 0, →〉−〈 mXnn yyd as .+∞→n Therefore, in virtue of A satisfies the kS property on ,mW we obtain, that )(yd A∈ . Let us prove sufficiency. Let yyn in ,mW ddn in ,* mX 0, ≤〉−〈 mXnn yyd as +∞→n , where )( nn yd A∈ 1.≥∀n Then yyn → in mX and ).(yd A∈ P.O. Kasyanov, V.S. Mel'nik , S. Toscano ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 122 The lemma is proved. Corollary 1. If the multi-valued map )(: * mvm XCX →A satisfies the property kS on mW , then A is 0λ -pseudomonotone on mW . THE MAIN RESULTS In the next theorem we will prove the solvability and justify the Faedo-Galerkin method for the problem (3). Theorem 1. Let 0=a , )()(: ** XXCXA v H∩→ be + -coercive bounded map of the Volterra type, that satisfies the property kS on σW . Then for arbitrary Xf ∈ there exists at least one solution of the problem (3), that can be obtained by the Faedo-Galerkin method. Proof. From +-coercivity for : XXA ⇒ it follows, that Xy∈∀ .)(]),([ XX yyyyA γ≥+ So, :0>0r∃ 0.>)( 0 ≥ X frγ Therefore, 0.],)([=: 0 ≥−∈∀ +yfyAryXy X (6) T h e s o l v a b i l i t y o f a p p r o x i m a t e p r o b l e m s. Let us consider the complete vectors system Vh ii ⊂≥1}{ such that 1α ) 1}{ ≥iih orthonormal in H ; 2α ) 1}{ ≥iih orthogonal in V ; 3α ) Vvvhvhi iiVi ∈∀≥∀ ),(=),(1 λ , where ∞→≤≤ jλλλ ,...,0 21 as ∞→j , V),( ⋅⋅ is the natural inner product in V , i.e. 1}{ ≥iih is a special basis [29]. Let for each 1≥m m iim hH 1=}{span= , on which we consider the inner product induced from H that we again denote by ),( ⋅⋅ . Due to the equivalence of H and H it follows that mm HH ≡* ; );(= 0 mpm HSLX , );(= 0 * mqm HSLX , },{max= 210 rrp , 1>0q : 1=1/1/ 00 qp + , =⋅〉〈⋅ mX, mmXX X×⋅〉〈⋅= \|, , }\|{:= mmm XyXyW ∈′∈ , where y′ is the derivative of an element mXy∈ is considered in the sense of ),(* mHSD . For any 1≥m let );( XXI mm L∈ be the canonical embedding of mX in X , * mI be the adjoint operator to mI . Then 1.=);1 *( σσ XXmIm L ≥∀ (7) Let us consider such maps [12]: .:=),(:=: *** fIfXCXIAIA mmvmmmm → So, from (6) and corollary 1, applying analogical thoughts with [12], [14] we will obtain, that Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 123 )1j mA is 0λ -pseudomonotone on mW ; )2j mA is bounded; )3j 0],)([ ≥− +yfyA mm mXy∈∀ : 0= ry X . Let us consider the operator )(: mmmm XXLDL →⊂ with the definition domain ,=}0=(0)\|{=)( 0 mmm WyWyLD ∈ that acts by the rule: ,=0 yyLWy mm ′∈∀ where the derivative y′ we consider in the sense of the distributions space );( mHSD . From [12] for the operator mL the next properties are true: )4j mL is linear; )5j 0,0 ≥〉〈∈∀ yyLWy mm ; )6j mL is maximal monotone. Therefore, conditions 1j )– 6j ) and the theorem 3.1 from [13] guarantees the existence at least one solution )( mm LDy ∈ of the problem: ,,)()( 0ryfyAyL Xmmmmmm ≤∋+ that can be obtained by the method of singular perturbations. This means, that my is the solution of such problem: ⎪⎩ ⎪ ⎨ ⎧ ≤∈ ∋+′ ,,,0=(0) )( RyWyy fyAy Xmmmm mmmm (8) where 0= rR . P a s s i n g t o t h e l i m i t. From the inclusion from (8) it follows, that 1≥∀m :)( mm yAd ∈∃ ).(=)(= ** mmmmmmmm yAIyAyfdI ∈′− (9) 1 . The boundedness of 1}{ ≥mmd in X follows from the boundedness of A and from (8). Therefore, .1:0> 11 cdmc Xm ≤≥∀∃ (10) 2 . Let us prove the boundedness 1}{ ≥′ mmy in * σX . From (9) it follows, that 1≥∀m )(= * mmm dfIy −′ , and, taking into account (7), (8) and (10) we have: .<2* +∞≤≤′ cyy WmXm σσ (11) In virtue of (8) and the continuous embedding );( mm HSCW ⊂ we obtain (see [24]) that 0>3c∃ such, that .)(1, 3ctyStm Hm ≤∈∀≥∀ (12) P.O. Kasyanov, V.S. Mel'nik , S. Toscano ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 124 3 . In virtue of estimations from (10)–(12), due to the Banach-Alaoglu theorem, taking into account the compact embedding YW ⊂ , it follows the existence of subsequences 1111 }{}{,}{}{ ≥≥≥≥ ⊂⊂ mmkkmmmkkm ddyy and elements Wy∈ , Xd ∈ , for which the next converges take place: yy km in ddW km, in X )()( tyty km in H for each St∈ , (13) )()( tyty km → in H for a.e. St∈ , as ∞→k . From here, as 1≥∀k 0=(0) kmy , then 0=(0)y . 4 . Let us prove, that .= dfy −′ (14) Let N∈∈ nSD ),(ϕ and nHh∈ . Then 1≥∀k : nmk ≥ we have: ,,'=),))()(')((( 〉+〈+∫ ψττττϕ kmkmkmkm S dyhddy where XXh n ⊂∈⋅ )(=)( τϕτψ . Let us remark, that here we use the property of Bochner integral [8](theorem IV.1.8, c.153). Since for nmk ≥ nkm HH ⊃ , then .,=,' 〉〈〉+〈 ψψ kmkmkm fdy Therefore, 1≥∀k : nmk ≥ .,)()(=, ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 〉〈 ∫ hdff S km τττϕψ Hence, for all 1≥k : nmk ≥ →−= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′∫ ψτττϕ ,,)()( kk m S m dfhdy ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −→ ∫ S hddf ,)()(()( ττττϕ as ∞→k . (15) The last follows from the weak convergence kmd to d in X . From the convergence (13) we have: ( )hyhdy S mk ),(,)()( ϕτττϕ ′→ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′∫ as ∞→k , (16) where ττϕτϕϕϕ dyyyS S )()()()()( ′−=′−=′∈∀ ∫D . Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 125 Therefore, from (15) and (16) it follows, that .,))()()((=)),(( )( 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −′∈∀∈∀ ∫ ≥ hddfhyHhS S m m ττττϕϕϕ ∪D Since m m H∪ 1≥ is dense in V we have, that .))()()((=)()( ττττϕϕϕ ddfyS S −′∈∀ ∫D Therefore, = Xdfy ∈−′ . 5 . In order to prove, that y is the solution of the problem (3) it remains to show, that y satisfies the inclusion fyAy ∋+′ )( . In virtue if identity (14), it is enough to prove, that )(yAd ∈ . From (13) it follows the existence of Sll ⊂≥1}{τ such that Tlτ as +∞→l and )()(1 llkm yyl ττ →≥∀ in H as +∞→k (17) Let us show that for any 1≥l 0=)(:]),([, twXwwyAwd ∈∀≤〉〈 + for a e. ],[ Tt lτ∈ . (18) Let us fix an arbitrary 1}{ ≥∈ llττ . For 1,2=i let us set ),()(=)(),;,();,(=)( 2,1,, ττττττ σσσσσ XXXVTLHTLX ipiri ∩∩ )()(=)(),;,();,(=)( * 2, * 1, *** , ττττττ σσσσσ XXXVTLHTLX iqiri ++ ′ , )()(=)()},(\|)({=)( 2,1, * ,, ττττττ σσσσσ WWWXyXyW iii ∩∈′∈ . 1.),(=),(=0 ≥kyaya kmk ττ Similarly we introduce )(τX , )(* τX , )(τW . From (17) it follows that 0aak → in H as +∞→k . (19) For any 1≥k let )(τWzk ∈ be such that ⎩ ⎨ ⎧ ∋+′ ,=)( ,0)( kk kk az zJz τ (20) where ))(()(: * ττ XCXJ v→ be the duality (in general multivalued) mapping, i.e. ).(,)(=)(==]),([=]),([ 222 )( ττ XuuJuJuuuJuuJ X ∈−+−+ We remark that the problem (20) has a solution )(τWzk ∈ because J is monotone, coercive, bounded and demiclosed (see [1–2, 8, 13]). Let us also note that for any 1≥k P.O. Kasyanov, V.S. Mel'nik , S. Toscano ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 126 0.=2,2=)( 2 )()( 22 ττ XkXkkHkHk zzzaTz +〉′〈− Hence, . 2 1=)(1 3)(* cazzk HkXkXk ≤≤′≥∀ ττ Due to (19), similarly to [8, 13], as +∞→k , kz weakly converges in W to the unique solution Wz ∈0 of the problem (20) with initial time value condition 0=(0) az . Moreover, 0zzk → in )(τX as +∞→k (21) because 2 )(0 2 )(lim ττ XXkk zz ≤ +∞→ , 0zzk in )(τX and )(τX is a Hilbert space. For any 1≥k let us set ⎪⎩ ⎪ ⎨ ⎧ ∈ ⎪⎩ ⎪ ⎨ ⎧ ∈ ,elsewhere),(ˆ ],[0,if),( =)( ,elsewhere),( ],[0,if ),( =)( td ttd tg tz tty tu k km k k km k ττ where )(ˆ kk uAd ∈ is an arbitrary. As 1}{ ≥kku is bounded, : XXA ⇒ is bounded, then 1}ˆ{ ≥kkd is bounded in X . In virtue of (21), (13), (17) ( ) =)()(),(lim=,lim 0 dttytytduug kkkkkk −〉−〈 ∫+∞→+∞→ τ ( ) ( ) =)()(),('lim=)()(),(')(lim= 00 dttytytydttytytytf kk k kk k −−− ∫∫ +∞→+∞→ ττ ( ) ( ) =)(),('lim)((0) 2 1lim= 0 22 dttytyyy k k HkHk k ∫+∞→+∞→ +− τ τ ( ) ( ) 0.=)(),()((0) 2 1= 0 22 dttytyyy HH ′+− ∫ τ τ So, 0.=,lim 〉−〈 +∞→ uug kk k (22) Let us show that )( kk uAg ∈ 1≥∀k . For any Xw∈ let us set ⎪⎩ ⎪ ⎨ ⎧ ∈ ⎩ ⎨ ⎧ ∈ .elsewhere),( ],[0, if ,0 =)( ,elsewhere,0 ],[0,if ),( =)( tw t tw ttw tw ττ τ τ In virtue of A is the Volterra type operator we obtain that ≤〉〈+〉〈〉〈 τ τ wdwdwg kkmk ,ˆ,=, =,ˆ]),([ 〉〈+≤ + τ τ wdwyA kkm ≤〉〈++ τ τ wdwuA kk ,ˆ]),([= Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 127 .]),([]),([ ++ +≤ τ τ wuAwuA kk Due to )()( XuA k H∈ , similarly to [30], we obtain that .]),([=]),([]),([ +++ + wuAwuAwuA kkk τ τ As Xw∈ is an arbitrary, then )( kk uAg ∈ 1≥∀k . Due to 1}{ ≥kku is bounded in X , then 1}{ ≥kkg is bounded in X . Thus, up to a subsequence 11 },{},{ ≥≥ ⊂ kkkjjkjk gugu , for some Wu∈ , Xg∈ the next convergence takes place uu jk in ggW jk,σ in X as ∞→j . (23) We remark that )(=)(),(=)( tdtgtytu for a.e. ][0,. τ∈t . (24) In virtue of (22), (23), as A satisfies the property kS on σW , we obtain that )(uAg∈ . Hence, due to (24), as A is the Volterra type operator, for any Xw∈ such that 0=)(tw for a.e. ],[ Tt τ∈ we have .]),([=]),([,=, ++≤〉〈〉〈 wyAwuAwgwd . As 1}{ ≥∈ llττ is an arbitrary, we obtain (18). From (18), due to the functional +→ ]),([ wyAw is convex and lower semicontinuous on X (hence it is continuous on X ) we obtain that for any Xw∈ +≤〉〈 ]),([, wyAwd . So, )(yAd ∈ . The theorem is proved. In a standard way (see [17]), by using the results of the theorem 1, we can obtain such proposition. Corollary 2. Let )()(: ** XXCXA v H∩→ be bounded map of the Volterra type, that satisfies the property kS on σW . Moreover, let for some 0>c +∞→ − ++ Xy yAcyyA )(]),([ (25) as +∞→Xy . Then for any Ha∈ , Xf ∈ there exists at least one solution of the problem (3), that can be obtained by the Faedo-Galerkin method. Proof. Let us set 2 2 2 = c a Hε . We consider Ww∈ : ⎩ ⎨ ⎧ +′ ,=(0) ,0=)( aw wJw ε where )(: XCXJ v→ be the duality map. Hence cw X ≤ . We define )()(:ˆ ** XXCXA v H∩→ by the rule: )(=)(ˆ wzAzA + , Xz∈ . Let us set =ˆ Xwff ∈′− . If Wz∈ is the solution of the problem P.O. Kasyanov, V.S. Mel'nik , S. Toscano ISSN 1681–6048 System Research & Information Technologies, 2009, № 1 128 ⎪⎩ ⎪ ⎨ ⎧ ∋+′ ,0=(0) ,)(ˆ z fzAz then wzy += is the solution of the problem (3). It is clear that Â is a bounded map of the Volterra type, that satisfies the property kS on W . So, due to the theorem 1, it is enough to prove the + -coercivity for the map Â . This property follows from such estimates: ≥+−++≥ +++ ]),([]),([]),(ˆ[ wwzAwzwzAzzA ,)(]),([ ++ +−++≥ wzAcwzwzA cwzz XX −+≥ . The corollary is proved. Analyzing the proof of the theorem 1 we can obtain such result. Corollary 3. Let )()(: * XXCXA v H∩→ be bounded map of the Volterra type, that satisfies the property kS on σW , Ha nn ⊂≥0}{ : 0aan → in H as +∞→n , Wyn ∈ , 1≥n be the corresponding to initial data na solution of the problem (3). If 0yyn in X , as +∞→n , then Wy∈ is the solution of the problem (3) with initial data 0a . Moreover, up to a subsequence, 0yyn in );( HSCW ∩σ . EXAMPLE Let us consider the bounded domain nR⊂Ω with rather smooth boundary Ω∂ , ][0,= TS , )(0;= TQ ×Ω , )(0;= TT ×Ω∂Γ . For R∈ba, we set =],[ ba [0,1]}\|)(1{ ∈−+= ααα ba . Let )(= 1 0 ΩHV be real Sobolev space, )(= 1* Ω−HV be its dual space, )(= 2 ΩLH , Ha∈ , Xf ∈ . We consider such problem: ),()],(),,([),( txftxytxy t txy ∋∆−∆+ ∂ ∂ in Q , )(=,0)( xaxy in Ω , 0=),( txy in TΓ . (26) We consider )()(: * XXCXA v H∩→ , 1\|)(\|),(\|{=)( ≤∈⋅∆ ∞ tpSLppyyA a.e. in }S . where ∆ means the energetic extension in X of Laplacian (see [8] for details), )(),(=),)(( tptxytxpy ⋅∆⋅∆ for a.e. Qtx ∈),( . We remark that .=]),([,=)( 2 XX yyyAyyA ++ (27) Initial time value problem solutions for evolution inclusions with kS type operators Системні дослідження та інформаційні технології, 2009, № 1 129 We rewrite the problem (26) to the next one (see [8] for details): .=(0),)( ayfyAy ∋+′ (28) The solution of the problem (28) is called the generalized solution of (26). Due to the corollary 2 and (27), it is enough to check that A satisfies the property kS on W . Indeed, let yyn in W , ddn in X , where nnn ypd ∆= , )(SLpn ∞∈ , 1\|)(\| ≤tpn for a.e. St∈ . Then yyn → in Y and up to a subsequence ppn → weakly star in )(SL∞ , where 1\|)(\| ≤tp for a.e. St∈ . As 0))(;( 2 2 →−≤Ω∆−∆ − YnLnnn yyHSypyp , then ypyp nn ∆→∆ weakly in ))(;( 2 2 Ω−HSL . 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Initial time value problem solutions for evolution inclusions with Sk type operators

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