Overview of V.A. Plotnikov s research on averaging of differential inclusions

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1 Overvew of V.A. Plonkov s research on averagng of dfferenal nclusons S. Klymchuk a, A. Plonkov b,c, N. Skrpnk c, a Auckland Unversy of echnology, -4 Wakefeld sr., 4, Auckland, New Zealand b Odessa Naonal Unversy named afer I.I. Mechnkov, Dvoryanskaya sr.,, 656, Odessa, Ukrane c Odessa Sae Academy of Cvl Engneerng and Archecure, Ddrhsona sr., 4, 659, Odessa, Ukrane Absrac In hs revew we wll frs look n deal a V.A. Plonkov s resuls on he subsanaon of full and paral schemes of averagng for dfferenal nclusons n he sandard form on fnal and nfne nerval. hen we wll consder he algorhms where here s no average, bu here s a possbly o fnd s esmaon from below and from above. Such approach s also used when he deecon of an average s approxmae. hs suaon s especally ypcal a consderaon of dfferenal nclusons wh fas and slow varables. In he las par we wll gve he resuls concernng he subsanaon of he full and paral averagng mehod for mpulsve dfferenal nclusons on fnal and nfne nervals. Keywords: dfferenal ncluson dfferenal nclusons wh mpulses averagng mehod Inroducon Many mporan problems of analycal dynamcs are descrbed by he nonlnear mahemacal models ha as a rule are presened by he nonlnear dfferenal or negro - dfferenal equaons. he absence of exac unversal research mehods for nonlnear sysems has caused he developmen of numerous approxmae analyc and numercally-analyc mehods ha can be realzed n effecve compuer algorhms. All hese mehods are consruced by an erave prncple,.e. eher consecuve approxmaons or chans of consecuve ransformaons of phase varables or funconal seres wh members decreasng on sze, ec. are used. I means ha frs somehow he nal approxmaon s chosen hen he addves of varous order are found usng he eraons o approach he rue soluon. hs rule s especally effecve a research of he mahemacal models descrbed by regular on small parameers nonlnear equaons. Also here exs varous mehods of he nal approxmaon choce: solvng of some lnear problem he lnearzaon mehod or solvng of some nonlnear bu essenally more smple sysem ofen he averagng mehod. Recenly, he averagng mehods combned wh he asympoc represenaons n Poncare sense began o be appled as he basc consrucve ool for solvng he complcaed problems of analycal dynamcs descrbed by he dfferenal equaons. I became possble due o he works of N.N. Bogolyubov, Yu.A. Mropolskj, A.M. Samojlenko, V.M. Volosov, E.A. Grebennkov, M.A. Krasnoselsky, S.G. Kren, A.N. Flaov, ec. he applcaon of he averagng mehod n opmal conrol prob- Overvew of V.A. Plonkov s research Correspondng auhor Emal addresses: sergy.klmchuk@au.ac.nz S. Klymchuk, a-plonkov@ukr.ne A. Plonkov, ale@ukr.ne N. Skrpnk lems conans n he works of N.N. Moseev, V.N. ebedev, F.. Chernousko,.D. Akulenko, V.A. Plonkov, ec. he developmen of he heory of dfferenal nclusons began from he works of. Wazewsk and A.F. Flppov n whch he basc resuls on exsence and properes of he soluons of he dfferenal nclusons have been receved. he dfferenal nclusons are valuable no only as he generalzaon of he heory of he dfferenal equaons, bu also for her numerous applcaons o he research of opmal conrol problems, he game heory and economcs. he possbly of he applcaon of he averagng mehod n he heory of dfferenal nclusons was consdered by V.A. Plonkov. Vcor Aleksandrovch Plonkov was born on January 5, 938 n enngrad nowadays S. Peersburg. Durng he World war II he was he nhaban of blockade enngrad. hen n 944 he famly moved o Odessa. In 96 V.A. Plonkov graduaed from Odessa Sae Unversy named afer I.I. Mechnkov, where aferwards worked n posons of he asssan, assocae professor, deparmen chef and he dean up o hs deah on Sepember 4, 6. In 969 V.A. Plonkov defended he kandda hess Research of a class of opmal conrol problems for sysems wh wo degrees of freedom n Odessa Sae Unversy and n 98 defended he docoral hess Asympocal mehods n opmal conrol problems n enngrad Sae Unversy. V.A. Plonkov s scenfc works cover a wde range of complex and acual problems n he heory of dfferenal equaons and opmal conrol ha concern a new drecon of hese heores - he dfferenal equaons wh mulvalued and dsconnuous rgh-hand sde, he quasdfferenal equaons n he merc spaces. V.A. Plonkov developed he algorhms of asympoc solvng for que a wde class of dfferenal nclusons and proved deep heorems by N.N. Bogolyubov and A.N. khonov on a subsanaon of he asympoc mehods for he dfferenal equaons wh he mulvalued and dscon- Preprn submed o Physca D Augus 9,

2 nuous rgh-hand sde and he quasdfferenal equaons, developed algorhms of numercally asympocal solvng of he conrol problems, proved he heorems of exsence and unqueness of soluons of he quasdfferenal equaons n locally compac and full merc spaces. he achevemens n hs drecon naed he mahemacal researches of asympocal mehods n he heory of he dfferenal nclusons n Russa, Belarus, Bulgara, Poland, France, he USA, ec. V.A. Plonkov publshed over 5 scenfc nworks, ncludng 6 monographes [,, 3, 4, 5, 6]. In hs revew we wll frs look n deal a V.A. Plonkov s resuls on he subsanaon of full and paral schemes of averagng for dfferenal nclusons n he sandard form on fnal and nfne nerval. hen we wll consder he algorhms where here s no average, bu here s a possbly o fnd s esmaon from below and from above. Such approach s also used when he deecon of an average s approxmae. hs suaon s especally ypcal a consderaon of dfferenal nclusons wh fas and slow varables. In he las par we wll gve he resuls concernng he subsanaon of he full and paral averagng mehod for mpulsve dfferenal nclusons on fnal and nfne nervals.. he averagng of dfferenal nclusons For dfferenal nclusons he heorem whch s he analogue of he frs N.N. Bogolyubov s heorem has been proved by V.A. Plonkov n [3, 7, 8]. I became a push for he furher developmen of he gven mehod for hs ype of he equaons... he full averagng scheme... he averagng on he fne nerval Consder he dfferenal ncluson ẋ εx, x, x = x, where R + s me, x R n s a phase vecor, ε > s a small parameer, X : R + R n compr n s a mulvalued mappng, compr n convr n s he se of all nonempy compac and convex subses of R n wh Hausdorff merc: ha, B = mn{r : A B + S r, B A + S r }, S r a s he ball n R n wh radus r and cener n he pon a R n. e us assocae wh he ncluson he followng averaged dfferenal ncluson where ξ εxξ, ξ = x, Xx = lm X, x d. 3 Here he negral of he mulvalued mappng s undersood n Aumann sense [9] and he convergence - n sense of he Hausdorff merc. heorem. [3, 7]. en he doman Q = {, x D R n } he followng hold: he mappng X, x s connuous, unformly bounded wh consan M, sasfes he pschz condon n x wh consan λ; unformly wh respec o x n he doman D he lm 3 exss; 3 for any x D D and he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any soluon ξ of he ncluson here exss a soluon x of he ncluson such ha x ξ η; 4 for any soluon x of he ncluson here exss a soluon ξ of he ncluson such ha he nequaly 4 holds. hereby, hr, clr η, 5 where R s he secon of he famly of he soluons of he averaged ncluson, clr s he closure of he secon of he famly of he soluons of he nal ncluson. Proof. Usng he condons, and he properes of Aumann s negral we oban ha he se Xx s convex and compac. Besdes Xx = lm cox, x d, so he dfferenal ncluson s also averaged o he dfferenal ncluson ẋ εcox, x, x = x. 6 From he condons, follows ha he mulvalued mappng Xx s unformly bounded wh consan M and sasfes he pschz condon wh consan λ. Really n vew of he condon of he heorem for any δ > s possble o fnd δ > such ha for all > δ he esmae s far: h Xx, X, xd < δ. hen choosng > δ we oban Xx = h Xx, {} h Xx, X, xd + h X, xd, {} <

3 < δ + h X, x, {} d δ + M, hxx, Xx h Xx, +h +h < δ + δ + X, x d, X, x d, Xx < X, x d + h X, x, X, x d X, x d + λ x x d δ + λ x x. As he value δ s chosen arbrarly, n a lm we wll receve: Xx M, hxx, Xx λ x x. he soluons of he nclusons,, 6 exs and are connuable on an nerval [, ε ]. Accordng o [] he famly of soluons H x of he ncluson s everywhere dense n he compac se Hx of he famly of soluons of he ncluson 6. Hence, s enough o prove he heorem for he nclusons wh he convex rgh-hand sde. he famles of he soluons of he nclusons and 6, and also her secons R and clr accordngly, are compac ses []. e us prove he frs saemen of he heorem and hence he valdy of he ncluson R S η clr. 7 Dvde he nerval [, ε ] on he paral nervals wh he pons = mε, =, m, m N. e ξ be a soluon of he ncluson. hen here exss a measurable selecor v Xξ such ha ξ = ξ + ε Consder he funcon vτdτ, [, + ], ξ = x. 8 ξ = ξ + εv, [, + ], ξ = x, 9 where vecor v sasfes he condon + mε v vd = mn v Xξ mε v + vd. 3 he vecor v exss and s unque n vew of he compacness and convexy of he se Xξ and he srong convexy of he funcon beng mnmzed. Se δ = ξ ξ. As hen As ξ ξ = ε vτdτ εm M m, ξ ξ ξ ξ + ξ ξ δ + εm, hxξ, Xξ λ[δ + εm ], [, + ]. From and follows ha + [v v ]d + h Xξd, + hxξ, Xξ d + Xξ d λ [δ + + εm + ] = [ ] = λ δ εm + M. 3 εm hen accordng o 8, 9 and 3 we ge δ + = ξ+ ξ + + ξ ξ + ε [v v ]d [ ] δ + ελ δ εm + M εm = λm + + λ δ m λm m m λ m = λm + + λ m m + λ + δ + λm m m k= δ... + λ k m M [ + λ + ] M m m m eλ, 4 =, m. ξ ξ = ε v M m, 5

4 hen from and 4 follows ha ξ ξ ξ ξ + ξ ξ + + ξ ξ M m eλ From he condon of he heorem follows ha for any η > and fxed m he nequaly holds h εm + X, ξ d, Xξ η. 7 Hence, here exss such measurable selecor v X, ξ, =, m ha + εm [v v ]d η. 8 Consder he famly of funcons x = x +ε v τ dτ, [, + ], x = x. 9 From 8,9 and 9 follows ha x ξ x ξ + +ε [v v ]d x ξ + η m... η, =, m. As x x = ε v τdτ M m, [, + ] hen from 5 and we have x ξ M m + η and hx, x, X, ξ hx, x, X, x + +hx, x, X, ξ λ M m + λη. akng no consderaon he choce of he funcon v and we have M ρẋ, εx, x ελ m + η. 3 Accordng o [] here exss such a soluon x of he ncluson ha he A.F. Flppov s heorem x x M ελ m + η e ελ τ dτ 4 M m + η e λ. 4 From he esmaes 6, and 4 follows ha Choosng ξ x ξ ξ + x x + ξ x M m eλ M M m + η + m + η e λ = = M m 3eλ e λ η. 5 m > M η η 3eλ + 5, η < e λ from 5 we ge he frs saemen of he heorem. he proof of he second par of he heorem s smlar o he proof of he frs one. Remark. If he condon 3 doesn hold can be replaced by he followng condon: 3 for any x D D he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D for τ [, ], where τ = ε. hen for any η, ρ] and, ] here exss such ε η, > ha for all ε, ε ] and [, ε ] he saemens and of he heorem fulfll. In case when here s no unform convergence n 3, V.A. Plonkov proved he followng heorem, whch s he generalzaon of A.N. Flaov s resul [3] on a case of dfferenal nclusons: heorem. [3, 7]. en he doman Q he followng hold: he mappng X, x s connuous, locally sasfes he pschz condon n x; n every pon x D he lm 3 exss; 3 for any x D D and he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss such ε η,, x > ha for all ε, ε ] and [, ε ] he saemens and of he heorem fulfll. Proof. Consder he se D, x = S ρ R. he se D, x D s compac. Hence he lm 3 exss unformly wh respec o x D, x. As a he proof of he heorem s enough o consder he doman Q, x = {, x D, x } he saemens of he heorem follow from he jusce of he heorem for he doman Q, x.

5 Remark. he esmaes receved n he heorem qualavely dffer from he correspondng esmaes of he heorem. he exernal concdence of he saemens of heorems and leads somemes o her wrong undersandng. Really, he heorem affrms ha he nequaly 4 holds unformly for all famly of rajecores x and ξ wh concdennal condons,.e. he exsence of εη, s affrmed. he esmae receved n he heorem s far only for soluons x and ξ begnnng n he fxed nal pon x,.e. he exsence of εx, η, s affrmed. Example. Consder he dfferenal ncluson ẋ {εax sn, a [, ]}, x = x. 6 he averaged sysem wll be ξ =, ξ = x. herefore x ξ = x e εa sn sds x = = x e εa cos. 7 Is easy o check ha for he sysem 6 he condons of he heorem do no fulfll and he condons of he heorem fulfll. Really he rgh-hand sde s no unformly bounded and x h X, xd, Xx sn d = x cos 8 does no exceed 4 x and converges o when, bu he value δ depends on x, hough δ converges o nfny when x. So he condon of he unform convergence n 8 s no far. From 7 follows ha here exss ε η,, x > such ha for all ε, ε ] and, ε ] he esmae x ξ < ε s far. For example one can ake ε = ln + η x. Bu for fxed η and he funcon ε η,, x when x, so here s no unform esmae 7 wh respec o x R. If he mappng X, x s perodc n, one can receve he more exac esmae. heorem 3. en he doman Q he condons, 3 of he heorem fulfll and besdes he mappng X, x s π perodc n. hen for any > here exs ε > and C > such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any soluon ξ of he ncluson here exss a soluon x of he ncluson such ha x ξ Cε; 9 for any soluon x of he ncluson here exss a soluon ξ of he ncluson such ha he nequaly 9 holds. Proof. If he mulvalued mappng s π perodc n and unformly bounded hen Xx = π π X, xd s he unform average for X, x. Really for kπ < k + π we have = h = π h k = π+ π X, xd, Xx = X, xd + k π + Xxd + k h = π+ π X, xd, kπ kπ π+ π X, xd, Xxd Xxd + + 4Mπ kπ X, x + Xx. Hence for > 4Mπ δ we have h X, xd, Xx < δ for all x D. e us prove he frs saemen of he heorem. Dvde he nerval [, ε ] on he paral nervals wh he pons = π, =,... e x be a soluon of he ncluson. hen here exss a measurable selecor v of he mulvalued mappng X, x such ha x = x + ε Consder he mappng x = x + ε vτdτ, [, + ], x = x. 3 v τdτ, [, + ], x = x.3 where v s he measurable selecor of he mulvalued mappng X, x such ha v v = mn v v. 3 v X,x Denoe by δ = x x, hen we have v v hx, x, X, x 5

6 λ x x λ [ x x + x x ] λ ε vτ dτ + δ λ[δ + εm ]. herefore from 3, 3 and 3 follows δ + = x+ x + = = + + x + ε vτdτ x ε v τdτ + δ + ε vτ v τ dτ + δ + ελ [δ + εmτ ]dτ = = δ + πελ + π ε λm. Hence, as π + ε, we ge δ + + πελδ + π ε λm + ελπ + πελδ + π ε λm + ε λmπ + πελ + δ + π ε λm + πελ k = = π ε λm + πελ k = k= = εmπ + ελπ + εmπe λ,.e. δ Mπe λ ε, =,, akng no accoun ha for [, + ] he followng nequales hold x x ε vτdτ εm πmε, 34 x x ε v τdτ εm πmε, usng 33 we oban k= Calculae he value of he mappng x n he pons + : x + = x + ε + v d = x + εv π, 36 + where v π X, x d = π π X, x d = Xx. Consder he mappng hen ξ = ξ + εv, [, + ], ξ = x. 37 Is obvously ha x = ξ, =,,.... From 34, 37 we have x ξ 4πMε. 38 As for [, + ], =,,... ξ ξ πmε, hxξ, Xξ λπmε, ρ ξ, εxξ h εxξ, εxξ λπmε. 39 Accordng o [] from he nequaly 39 follows ha here exss such a soluon ξ of he ncluson, ha ε πλm ξ ξ e λε τ dτ επme λ. 4 From 35, 38 and 4 follows ha x ξ x x + x ξ + ξ ξ πmε e λ πMε + επm e λ = = πmε 3e λ + 5. Denoe by C = πm 3e λ + 5, hen Dx, ξ C ε. 4 he frs par of he heorem s proved. akng any soluon ξ of he ncluson and makng he calculaons smlar o he prevous, s possble o fnd a soluon x of he ncluson such hanequaly smlarly o 4 wh some consan C s far. Choosng C = maxc, C we wll receve he jusce of all saemens of he heorem. x x = πme λ + 3ε. 35 6

7 ... he averagng on he nfne nerval For generalzaon of he heorem on an nfne nerval V.A. Plonkov has exended he concep of sably of soluons of he dfferenal equaons on a case of dfferenal nclusons [4, 4]. In addon he concep of R-soluon of he dfferenal ncluson nroduced n [5, 6] was used. Defnon. [5, 6]. he absoluely connuous mulvalued mapng R : R compr n, R = X, s called he R soluon of he dfferenal ncluson ẋ X, x, x = x X compr n 4 f for almos every + lm h R +, x + Xs, xds =. 43 x R Defnon. [5] R soluon R, [, + of he dfferenal ncluson ẋ cox, x 44 s called sable f for any ε > here exss such δε > ha all R soluons R of he ncluson 44, sasfyng he nal condon h R, R < δ 45 are defned for all > and h R, R < ε. Defnon 3. [5] he R soluon R, [, + of he dfferenal ncluson 44 s called asympocally sable f s sable and for any R soluon R of he ncluson 44, sasfyng he nal condon 45 lm h R, R =. heorem 4. [4, 4]. en he doman Q he followng hold: he mappng X, x s connuous, unformly bounded, sasfes he pschz condon n x; unformly wh respec o and x n he doman Q he lm Xx = lm + X, xd 46 exss; 3 for any x D D and he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D; 4 he R soluon of he dfferenal ncluson s asympocally sable. hen for any η, ρ] here exss ε η > such ha for all ε, ε ] and he followng saemens fulfll: for any soluon ξ of he ncluson here exss a soluon x of he ncluson such ha he nequaly 4 fulflls; 7 for any soluon x of he ncluson here exss a soluon ξ of he ncluson such ha he nequaly 4 holds. hereby, hr, R η, 47 where R, R are he R soluons of he dfferenal nclusons and accordngly, R = R D. he proof of he heorem s carred on smlarly o he proof of he Banfy s heorem [7] wh changng references o he frs N.N. Bogolyubov s heorem wh references o he heorem. Example. Consder he followng dfferenal ncluson ẋ ε x + [, ] + cos, x [, 3], where x D = [ 6, 6]. he averaged ncluson s ξ ε ξ + [, ], ξ [, 3]. he R soluon of he averaged ncluson R = [3e ε, e ε + ] s asympocally sable. he fulfllmen of all oher condons of he heorem 4 s checked evdenly. he R soluon of he nal ncluson s R = [3 ε e ε + ε + ε + ε cos + ε sn ; + ε ] ε e ε + ε + ε + ε cos + ε + ε sn +. herefore hr, R ε + ε e ε + ε + ε + ε + ε < 3 ε. hus when ε = η 3 he concluson of he heorem 4 holds. In V.A. Plonkov s works he possbly of averagng of he dfferenal nclusons on he nfne nerval usng he sably of separae rajecores was also consdered. Defnon 4. [8]. he soluon ψ, [, + of he dfferenal ncluson 4 s called sable f for any ε > here exss δ > such ha for all x : x ψ < δ any soluon x wh he nal condon x = x exss for all [, + and sasfes he nequaly x ψ < ε. Defnon 5. [8]. he soluon ψ, [, + of he dfferenal ncluson 4 s called weakly sable f for any ε > here exss δ > such ha for all x : x ψ < δ some soluon x wh he nal condon x = x exss for all [, + and sasfes he nequaly x ψ < ε.

8 Defnon 6. [8]. he soluon ψ, [, + of he dfferenal ncluson 4 s called asympocally sable f s sable and lm x ψ =. Defnon 7. [8]. he soluon ψ, [, + weakly asympocally sable f s weakly sable and lm x ψ =. heorem 5. [9]. en he doman Q he condons 3 of he heorem 4 hold and besdes 4 he soluon ξ of he ncluson s weakly asympocally sable. hen for any η, ρ] here exss ε > such ha for all ε, ε ] here exss a soluon x of he ncluson such ha he nequaly 4 holds for. Proof. e ξ be a weakly asympocally sable soluon of he ncluson. I means ha for any > and η here exs ρ < η and he soluon ξ of he ncluson such haf a he momen we have ξ ξ ρ, hen for any > he nequaly holds ξ ξ < η. For ξ and ξ s possble o fnd he consan such ha for any > + ε he nequaly holds ξ ξ ρ. e us consder ha ρ s he maxmum ponn whch he equaly 49 fulflls. hen for any > ρ we have mn ξ x > ρ. 5 x R As s possble o ake ρ as he momen here exss a soluon ξ of he averaged ncluson such ha ρ = x ρ ξ ρ = ξ ρ ξ ρ, ξ ξ η for > ρ and ξ ξ ρ for any > ρ + ε. ρ Now s possble o fnd ε,, ε ] such ha for any ρ < ρ + ε and ε, ε ] here exss a soluon x of he dfferenal ncluson ha sasfes he followng nequaly ξ x ρ. From he oher sde f [ ρ, ρ + ε ] hen x ξ x ξ + ξ ξ ρ + η < η. hus, we receve ha ρ + ε <. Bu for = ρ + ε > ρ s possble o wre down he followng esmae x ξ x ξ + ξ ξ < < ρ + ρ = ρ. From he heorem follows ha for he gven ρ and s possble o choose ε ρ, > such ha here exss a soluon x of he dfferenal ncluson such ha for any ε, ε ] and [, ε ] he nequaly s rue ξ x < ρ < η. 48 e us prove ha he nequaly 4 fulflls on an nfne nerval. We wll assume ha he heorem saemens ncorrec and on he nfny an nequaly 4 s no rue,.e. here exss a momen of me > ε such ha and for any < we have mn x R ξ x = η mn ξ x < η, x R where R s he secon of he se of soluons of he dfferenal ncluson and s he frs momen of me n whch he nequaly 4 fals. hen from he nequaly 48 and our assumpon follows ha here s a momen ρ when he followng equaly holds mn ξ ρ x ρ = ρ. 49 x ρ R ρ 8 he receved esmae conradcs he nequaly 5. Hence, our assumpon s ncorrec. Remark 3. he concluson of he heorem 5 concerns no o all soluons of dfferenal ncluson, bu only o he soluon ξ. herefore he dfferenal ncluson can have non - connuable soluons for and he soluons whch are no weakly asympocally sable. hus, from hs heorem he closeness of he R soluons of he nal and he averaged nclusons does no follow. Example 3. Consder he followng dfferenal ncluson ẋ ε[, ]x + e, x [, ]. 5 he averaged ncluson s ξ ε[, ]ξ, ξ [, ]. he R soluon of he averaged ncluson R = [e ε, e ε ] s no asympocally sable as hr, R = hr, R e ε, where R s he R soluon of he averaged dfferenal ncluson wh he nal se R.

9 hus for he soluon ξ = e ε of he averaged ncluson he closes soluon of he nal ncluson s x = e ε + and ε + ε ε + ε e ε lm ξ x = lm + ε eε e =. A he same me, for example, he soluon ξ =.5e εs weakly asympocally sable and s drecly checked ha hs soluon s also he soluon of he ncluson 5. Remark 4. In he heorem 5 s possble o replace he condon 4 wh he followng: 4 he soluon ξ of he ncluson s asympocally sable. hen he concluson of he heorem wll be he followng: for any η, ρ] here exs ε > and σ > such ha for all ε, ε ] for all soluons x of he ncluson wh he nal condons x x σ he nequaly 4 holds for all. When he condon 4 holds he closeness of he R soluons of he nal and he averaged nclusons follows from he heorem. Example 4. Consder he followng dfferenal ncluson ẋ ε [, ] x + cos, x = x. 5 For any soluon ξ of he averaged ncluson he followng esmae fulflls: ξ ε [, ] ξ, ξ = x x e ε ξ x e ε. 53 Accordng o 53 he soluon ξ of he averaged ncluson s asympocally sable. For he soluons of he nal ncluson 5 he followng nequaly holds: x 4ε e ε + 4ε + 4ε + 4ε cos + ε sn x + 4ε x ε e ε + ε ε cos + sn. + ε + ε + ε hus for all soluons x of he nal ncluson we have x ξ η for ε η Dfferenal nclusons wh semconnuous rgh-hand sdes In V.A. Plonkov s works he ofen meeng n he applcaons case, when he rgh-hand sde s no connuous bu only upper semconnuous n a phase varable was consdered. heorem 6. [5]. en he doman Q he followng hold: he mappng X : Q convr n s measurable n, upper semconnuous n x, unformly bounded by a summable funcon M such ha for all > he nequaly holds M d M ; he mappng Xx sasfes he pschz condon; 3 for any x D D and he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] for any soluon x of he ncluson here exss a soluon ξ of he ncluson such ha ξ x η. 54 Remark 5. he heorem affrms ha R, ε R, + S η, where R, ε s he R-soluon of he ncluson correspondng o he parameer ε. he valdy of he ncluson R, R, ε+s η s no affrmed,.e. only he upper semconnuy n ε of he mulvalued mappng R, ε a he pon ε = s proved...4. he approxmaon of he soluon bunches n case when he average does no exs In [5, 9] V.A. Plonkov consdered he case when he lm 3 does no exs bu here exs mulvalued mappngs X, X + : D convr n such ha lm β X x, lm β X, xd =, 55 X, xd, X + x =, 56 where β, s he semdevaon of he ses n he sense of Hausdorff: βa, B = sup nf a b. a A b B Along wh he dfferenal ncluson we wll consder he followng dfferenal nclusons: ẋ εx x, x = x, 57 ẋ + εx + x +, x + = x. 58 9

10 heorem 7. [9]. en he doman Q he followng hold: he mappng X, x s unformly bounded wh consan M, measurable n, sasfes he pschz condon n x wh consan λ; he mappng X x s unformly bounded wh consan M, sasfes he pschz condon n x wh consan λ; 3 unformly wh respec o x n he doman D he lm 55 exss; 4 for any x D D and he soluons of he ncluson 57 ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] for any soluon x of he ncluson 57 here exss a soluon x of he ncluson such ha x x η. 59 Proof. Dvde he nerval [, ε ] on he paral nervals wh he pons = mε, =, m, m N. e x be a soluon of he ncluson 57. hen here exss a measurable selecor u X ξ such ha x = x + ε Consder he followng funcon where uτdτ, [, + ], x = x.6 ξ = ξ + εu, [, + ], ξ = x, 6 mε u + ud = mn u X ξ mε u + ud.6 As n 6 he funcon beng mnmzed s srongly convex and he se X x s compac and convex hen here exss he unque vecor u. e δ = x ξ, hen for [, + ] we have x ξ x ξ + x ξ δ + εm ; 63 h X x, X ξ λ[δ + εm ]. 64 From 6,64 and he properes of he suppor funcon [] follow ha + + ud u d + + h X x d, X ξ d = = max ψ= = max ψ= C X x d, ψ C X ξ d, ψ = + [ CX x, ψ CX ξ, ψ ] d max ψ= = CX x, ψ CX ξ, ψ d = + hx x, X ξ d λ [δ + + εm+ ] = [ ] = λ δ εm + M. 65 εm akng no accoun 6,6 and 65 we ge he followng esmae: [ ] δ + δ + ελ δ εm + M = λm + + λ δ εm m m As M [ + λ + ] M e λ. 66 m m m x x = ε uτdτ ξ ξ M m, so hen usng 66 we oban M m, x ξ M m + M m + M m eλ = = M m eλ From he condon of he heorem follows ha for any η > here exss ε, η > such ha for all ε ε he ncluson holds X ξ εm + Xτ, ξ dτ + S η. 68 So here exss a measurable funcon u X, ξ, [, + ] such ha + εm [u u ]d η.

11 Consder he funcon x = x + ε hen from 68,69 follows ha u τdτ, [, + ], x = x.69 x ξ η. As x x M m, we oban he followng nequales: x ξ M m + η, 7 h X, x, X, ξ λm m + λη = M = λ m + η. 7 From he nequaly 7 and he way of choosng he funcon u we ge ρ ẋ, εx, x ελ M m + η. hen for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] for any soluon x of he ncluson here exss a soluon x + of he ncluson 58 such ha x + x η. 73 he proof of he heorem s carred on smlarly o he proof of he heorem 7. Remark 6. If R, R, R + are he secons of he famles of he soluons of he nclusons, 57 and 58 accordngly hen R R + S η, R R + + S η. 74 Remark 7. In he capacy of he mappngs X x and X + x one can use he superor and nferor lm of he sequence of ses []: X x = lm X, xd, X + x = lm X, xd. he ses X x and X + x are he maxmum and he mnmum wh respec o he ncluson among he ses X x and X + x, has for any X x and X + x he nclusons hold Accordng o [] here exss such a soluon x of he ncluson ha x x ελ M m + η e ελ τ dτ M m + η e λ. 7 From 67, 7, 7 follows ha x x 3e λ + 5 M m + η e λ. Choosng m 3e λ + 5 M η and η η e λ, we ge and he heorem s proved. x x η heorem 8. [9]. en he doman Q he followng hold: he mappng X, x s unformly bounded, measurable n, sasfes he pschz condon n x; he mappng X + x s unformly bounded, sasfes he pschz condon n x; 3 unformly wh respec o x n he doman D he lm 56 exss; 4 for any x D D and he soluons of he ncluson 58 ogeher wh a ρ neghborhood belong o he doman D. X x X x, X + x X + x. Remark 8. If he lm 3 exss hen X x = X + x = Xx and from heorems 7,8 he heorem follows. Remark 9. If he lm 3 exss, s calculaon s usually carred ou by means of suppor funcon. hus s ofen mpossble o calculae he exac value of he suppor funcon and he ses X x and X + x appear as he resul of he approxmae calculaon of he se Xx. Example 5. Consder he lnear dfferenal ncluson [ ] cos ẋ ε sn x + S r, x = x,75 where r = + e +.5 snln +. Is obvous ha he marx cos A = lm sn d = e us average he mulvalued mappng U = S r. As S r d = [ = S e snln cosln + +, 4.

12 hen he average U n hs case does no exs, bu obvously here exs such ses U = S r, r.5 and U + = S r, r.5 ha he followng hold lm β U, lm β Ud =, 76 Ud, U + =. 77 hen he nclusons 57 and 58 assume he form: ẋ ε [ x + S r ], x = x, ẋ + ε [ x + + S r ], x + = x. e us fnd he R soluons of hese nclusons wh he help of he Cauchy formula R = e ε x + ε where Smlarly Is obvous ha hen e ε s S r ds = e ε x + e ε S r. R + = e ε x + e ε S r. U = lm U d = S.5, U + = lm U d = S.5. R R, R + R +, R = e ε x + e ε S.5, R + = e ε x + e ε S.5. For he nal ncluson 75 all he condons of he heorems 7,8 hold. So for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] he nclusons 74 are rue... he paral averagng scheme Is also possble o use he paral averagng of he dfferenal nclusons,.e. o average only some summands or facors. Such varan of he averagng mehod also leads o he smplfcaon of he nal ncluson and happens o be useful when he average of some funcons does no exs or her presence n he sysem does no complcae s research. Along wh he dfferenal ncluson we wll consder he parally averaged dfferenal ncluson ξ εx, ξ, ξ = x, 78 where lm h X, x d, X, x d =. 79 heorem 9. [8]. en he doman Q he followng hold: he mappngs X, x, X, x are connuous, unformly bounded wh consan M, sasfy he pschz condon n x wh consan λ; unformly wh respec o x n he doman D he lm 79 exss; 3 for any x D D, ε, σ] and he soluons of he ncluson 78 ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η,, σ] such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any soluon ξ of he ncluson 78 here exss a soluon x of he ncluson such ha x ξ η; 8 for any soluon x of he ncluson here exss a soluon ξ of he ncluson 78 such ha he nequaly 8 holds. hereby, hr, R η, where R, R are he closures of he secons of he famles of he soluons of he nal and he averaged nclusons. Proof. Whou loss of generaly when provng he heorem we can suppose ha he ses X, x and X, x are convex. Really f s no rue we wll consder o he nclusons ẋ εcox, x, x = x, 8 ξ εcox, ξ, ξ = x. 8 Accordng o [] he famles of soluons of he nclusons 78, 79 are everywhere dense n he compac ses of he famles of soluons of he ncluson 8,8. Hence, s enough o prove he heorem for he nclusons wh he convex rghhand sde. e us prove he second saemen of he heorem and herefore he valdy of he ncluson R S η R. 83 Dvde he nerval [, ε ] on he paral nervals wh he pons = mε, =, m, m N. e x be a soluon of he ncluson 78. hen here exss a measurable selecor v X, x such ha x = x + ε vτ dτ, [, + ], x = x. 84

13 Consder he funcon y = y + ε where v z = z τ dτ, [, + ], y = x,85 mn v z. 86 z X,y he measurable funcon z exss [] and s unque n vew of he compacness and convexy of he se X, y and he srong convexy of he funcon beng mnmzed. From we have x y x x + +x y δ + εm, 87 v z hx, x, X, y λδ + εm, 88 where δ = x y, [, + ], =, m. From follows he esmae + δ + δ + ε v z d δ +ελ [δ + + εm + ] = λm + + λ δ m. m herefore δ + M m As for [, + ] x x M m, hen usng 89 we oban [ + λ + ] M m m eλ. 89 y y M m, 9 x y x x + x y + +y y M m eλ Consder he funcon y = y + ε z τdτ, [, + ], y = x,9 where z z = mn z z. 93 z X,y From he condon of he heorem follows ha for any η > here exss ε, η > such ha for all ε ε he nequaly holds + εm h X, y d, + X, y d η. 3 and Hence + z z d y + y + η εm + ε z z d + y y y y + η m... η, =, m. 94 As for [, + ] y y M m, 95 hen akng no accoun 9 and 94 we ge y y η + M m. 96 Accordng o he condon of he heorem and he nequales 95, 96 we have h X, y, X, y h X, y, X, y + +h X, y, X, y M λ m + η and herefore usng 93 we ge ρẏ, εx, y ελ M m + η. 97 Accordng o [] from 97 follows he exsence of such a soluon ξ of he ncluson 78 ha y ξ ελ M m + η M m + η e ελ τ dτ e λ. 98 From he esmaes 9, 96, 98 we ge x ξ M m 3eλ e λ η. Choosng m M3eλ + 5 η η and η e λ we ge he second saemen of he heorem. he proof of he frs par of he heorem s smlar o he proof of he second one. Remark. If one of he ses X, x or X, x degeneraes no a pon hen he correspondng ncluson becomes he dfferenal equaon whch has he unque soluon defned for. In hs case he whole famly of soluons of he second ncluson belongs o he η- neghborhood of he gven soluon.

14 Remark. If he convergence n 79 akes place n every pon x D hen smlarly o he heorem one can prove he exsence of such ε η,, x > ha for all ε, ε ] he conclusons of he heorem 9 are rue. Remark. If he mappngs X, x and X, x are perodc n hen n he esmae 8 s possble o replace η wh Cε. Remark 3. If X, x Xx hen he subsanaon of he full averagng scheme heorem follows from he heorem 9. Remark 4. e he mappng X, x be X, x = x = X +ω X, xd, ω ω < + ω, =,,.... Dvdng he nerval [, ε ] on paral nervals wh he sep ω, s possble o show ha he esmae 8 holds wh η = Cε. Remark 5. Smlarly o he above he varous schemes of averagng for negro - dfferenal nclusons ẋ εx, x, φ, s, xsds, x = x, where X : R R n R m compr n, φ : R R R n R m have been consdered n [4, 3].. he averagng of mpulsve dfferenal nclusons.. Dfferenal nclusons wh mpulses n fxed momens of me In hs secon we wll dscuss V.A.Plonkov s resuls on he subsanaon of he mehod of full and paral averagng on fne and nfne nervals for he dfferenal nclusons whch are exposed o mpulse nfluence n he fxed momens of me.... he full averagng scheme he averagng on he fne nerval. Consder he dfferenal ncluson wh mulvalued mpulses ẋ εx, x, τ, x = x, 99 x =τ εi x. If for any x D here exss he lm + Yx = lm X, xd + τ <+ I x, hen n he correspondence o he ncluson 99 we wll se he followng averaged ncluson ẏ εyy, y = x. heorem. [6]. en he doman Q he followng hold: he mappngs X : Q convr n, I : D convr n are connuous, unformly bounded and sasfy he pschz condon n x; unformly wh respec o and x n he doman Q he lm exss and, + d <, where, + s he quany of pons of he sequence {τ } on he nerval, + ]; 3 for any x D D and he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any soluon y of he ncluson here exss a soluon x of he ncluson 99 such ha x y η; for any soluon x of he ncluson 99 here exss a soluon y of he ncluson such ha he nequaly holds. Proof. From he condons, follows ha he mulvalued mappng Y : D convr n s unformly bounded wh consan M = M + d and sasfes he pschz condon wh consan λ = λ + d. e y be a soluon of he ncluson. Dvde he nerval [, ε ] on he paral nervals wh he sep γε such ha γε and εγε when ε. hen here exss a measurable selecor v of he mappng Yy such ha y = y j + ε j vsds, [ j, j+ ], y = x, 3 where j = jγε, j =, m, mγε ε < m + γε. Consder he funcon y = y j + εv j j, [ j, j+ ], y = x, 4 where he vecors v j sasfy he condon j+ γεv j vsds = = mn v Yy j j j j+ γεv vsds. 5 he vecor v j exss and s unque n vew of he compacness and convexy of he se Yy j and he srong convexy of he funcon beng mnmzed. 4

15 Denoe by δ j = y j y j. For [ j, j+ ] usng 3 and 4 we have y y j M ε j M εγε, y y j M ε j M εγε. herefore for [ j, j+ ] he followng nequales hold 6 y y j y j y j + y y j δ j + εm j, hyy, Yy j λ y y j λ δ j + εm j. 7 From 5 and 7 follows ha j+ [vs v j ]ds j+ hyys, Yy j ds j j γ ε λ δ j γε + εm. 8 Consderng 3 and 4 we oban γ ε δ j+ δ j + ελ δ j γε + εm = ε γ ε = + λ εγεδ j + λ M. 9 From he nequaly 9 akng no accoun ha δ = we ge ε γ ε δ λ M, and so on δ + λ εγεδ + λ M ε γ ε ε γ ε λ M + λ εγε + ε γ ε δ j+ λ M +λ εγε ++λ εγε = = M εγε M εγε + λ εγε + + λ εγε εγε M εγε e λ. So n vew of he nequales 6 he esmae s rue: y y y y j + y j y j + y j y M εγε+ M εγε e λ M εγε e λ From he condon of he heorem follows ha for any η > exss ε η > such ha for ε ε η he nequaly holds j+ h Yy j, Xs, y j ds+ γε + γε j I y j < η. j τ < j+ Hence, here exs measurable selecor u j X, y j and vecors p j I y j such ha v j+ u j sds + p j γε j τ < j+ < η. 3 j Consder he famly of funcons x = x j + ε u j sds + ε p j, j j τ < j, j+ ], x = x. 4 From 4, 3 and 4 usng ha x = y follows ha for j =, m x j y j x j y j + η εγε... jη εγε η. 5 As for j, j+ ] we have x x j M + dεγε = M εγε, akng no accoun he nequaly 6 we ge x y η + M εγε, 6 x y j η + M εγε. e us show ha here exss a soluon x of he ncluson 99 has suffcenly close o x. e θ,..., θ p be he momens of mpulses τ, ha geno he semnerval j, j+ ]. For convenence denoe by θ = j, θ p+ = j+. e µ + k = x θ k + xθ k +, µ k = x θ k xθ k, k =, p. Usng he pschz condon we have ρ ẋ, εx, x h εx, y j, εx, x ελ x y j ελm εγε + η = η, ρ x =θk, εi x θ k h εi y j, εi x θ k ελ y j x θ k ελm εγε + η = η. Accordng o A.F. Flppovs heorem [] beween he mpulse pons here exss a soluon x of he ncluson 99 such ha for θ k, θ k+ ] he esmae holds x x µ + k eελ θ k + ε θ k e ελ s η ds.

16 Denoe by γ k = θ k+ θ k γε, γ γ p = γε. hen µ k+ µ+ k eελγ k + η λ e λεγε. 7 When geng over he mpulse pon we have µ + k+ µ k+ + εh I y j, I xθ k+ µ k+ +εh I x θ k+, I xθ k+ +εh I y j, I x θ k+ µ k+ + ελµ k+ + εh I y j, I x θ k+ + ελµ k+ + η. 8 From 7 and 8 follows ha µ + k+ + ελeελγ k µ + k ec. Hence + β, β = η λ + ελ e λεγε + η. µ + + ελeλεγ µ + + β + ελe λεγε µ + + β, µ + + ελeελγ µ + + β + ελ e ελγ +γ µ + + β + ελeελγ + β + ελ e λεγε µ + + β + ελe λεγε + µ + k+ + ελk+ e ελγε µ + + +β e λεγε + ελ k ελ + = = + ελ k+ e λεγε µ + + β e λεγε + ελk + ελ + ελ + ελ e λ+dεγε µ + + η e λεγε + λ e λεγε eλdεγε + ελ + = ελ = αµ + + β, where α = e λεγε+d, + ελ β = εγεm + η e λεγε + λ So e λεγε e λdεγε + ελ + ελ. δ + j+ = x j+ x j+ αδ + j + β. We oban he sequence of he nequales δ + =, δ+ β, δ + αβ + β = α + β,..., δ + j+ α j β = α j+ α β eλ+d + ελ e λ+dεγε M εγε + η e λεγε + λ 6 and hen As e λεγε e λdεγε + ελ + ελ. + ελ lm e λεγε + = ε λ e λεγε e λdεγε + ελ + ελ lm = ε e λ+dεγε λεγε eλdεγε e = lm ε λεγε + γε e λ+dεγε λεγε δ + j+ CM εγε + η = d + d, for ε ε. herefore for j, j+ ] he nequaly holds x x x x j + x j x j + x x j M + dεγε + M εγε + CM εγε + η = = M + Cεγε + Cη. 9 In vew of he nequales,6 and 9 we ge ha x y can be done less han any preassgned η by means of choosng ε ε and η. he second saemen of he heorem s proved smlarly. he corollary of he gven heorem s he followng saemen: heorem. [6, 4]. en he doman Q he condons, of he heorem hold and besdes 3 for any X D D and he R soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η, > such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any R soluon R of he ncluson here exss an R soluon X of he ncluson 99 such ha hx, R < η; for any R soluon X of he ncluson 99 here exss an R soluon R of he ncluson such ha he nequaly holds. he averagng on he nfne nerval. Consder he nal ncluson 99 and he averaged ncluson. heorem. [6, 4]. en he doman Q he condons - 3 of he heorem hold and besdes 4 he R- soluons of he ncluson are unformly asympocally sable.

17 hen for any η, ρ] here exss such ε η > ha for all ε, ε ] and he nequaly holds h R, ε, Rε η, where R, ε s he R-soluon of he ncluson 99, Rε s he R-soluon of he ncluson, R, ε = R = X. heorem 3. [6, 4]. en he doman Q he saemens, of he heorem hold and besdes 3 he ncluson has a perodc R- soluon Rτ, τ = ε, whch rajecory C s asympocally orbal sable and ogeher wh a ρ neghborhood belongs o he doman D. hen for any η, ρ] here exs such ε > and η, η] ha for all ε, ε ], η, η ] and he nequaly holds hr, C η, where R s he R- soluon of he ncluson 99, sasfyng he nal condon hr, C η. heorem 4. [6, 4]. en he doman Q he saemens, of he heorem hold and besdes 3 he ncluson has an asympocally sable equlbrum sae R, ha ogeher wh a ρ neghborhood belongs o he doman D. hen for any η, ρ] here exs such ε > and η, η] ha for all ε, ε ], η, η ] and he nequaly holds hr, R η, where R s he R-soluon of he ncluson 99, sasfyng he nal condon hr, R η.... he paral averagng scheme Along wh he mpulsve dfferenal ncluson 99 we wll consder he mpulsve dfferenal ncluson ẏ ε X, y, ν j, y = x, y =ν j εk j y, where for any, x Q he lm + lm h X, xd + exss. + X, xd + ν j <+ τ <+ I x, K j x = heorem 5. [6, 4]. en he doman Q he followng hold: he mappngs X, X : Q convr n I j, K j : D convr n are connuous, unformly bounded and sasfy he pschz condon n x; unformly wh respec o and x n he doman Q he lm exss and, + d <, j, + d <, where, + and j, + are he quanes of pons of he sequences {τ } and {s j } on he nerval, + ]; 3 for any x D D, ε, σ] and he soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η,, σ] such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any soluon y of he ncluson here exss a soluon x of he ncluson 99 such ha x y η; 3 for any soluon x of he ncluson 99 here exss a soluon y of he ncluson such ha he nequaly 3 holds. heorem 6. [6, 4]. en he doman Q he saemens, of he heorem 5 hold and besdes 3 for any X D D, ε, σ] and he R soluons of he ncluson ogeher wh a ρ neghborhood belong o he doman D. hen for any η, ρ] and > here exss ε η,, σ] such ha for all ε, ε ] and [, ε ] he followng saemens fulfll: for any R soluon Y of he ncluson here exss an R soluon X of he ncluson 99 such ha hx, Y η, 4 where X = Y = X ; for any R soluon X of he ncluson 99 here exss an R soluon Y of he ncluson such ha he nequaly 4 holds... Dfferenal nclusons wh mpulses n non-fxed momens of me Consder he dfferenal ncluson wh mpulses n nonfxed momens of me ẋ F, x, ε, x = x, ετ x, σ x, 5 x =ετ x εi x, 6 x =σ p x K px. 7 e us assgn o he ncluson 5-7 he followng dfferenal ncluson ẏ F, y, ε, y = x, ετ x, σ y, 8 y =ετ y εi y, 9 7

18 y =σ p y K py, 3 where [, ] s me, x D R n s a phase vecor, ε > s a small parameer, he mpulse surfases τ j, σ j : D R, he mulvalued mappngs F j : [, ] D, ϵ] convr n, I j, K p j : D convr n, =, k, p =, r, j =,. e D x be he Boulgard conngen cone for x D,.e. { } D x = y R n : lm s nf x + sy z =. s z D heorem 7. [6, 5]. en he doman Q = [, ] D, ε] he followng condons fulfll: he mappngs F j, x, ε, I j j x and funcons τ x are connuous n, sasfy he pschz condon n x and F j, x, ε D x, x + I j x D; he mappngs F j, x, ε I j x are unformly bounded wh consan M; 3 unformly wh respec o, x Q + lm ε h F, x, ε d + ε I x, <ετ x<+ + F, x, ε d + ε I x = ; <ετ x<+ 4 he numbers J j, +, j =, of he asympocally small mpulses of he soluons of he nclusons 5, 6 and 8, 9 sasfy he followng nequales on he nerval, + ] J j, + A ε < ; 5 he surfaces = ετ j x do nonersec each oher and for all x D, z εi j, j =,, he followng nequales hold τ j x τ j j x + z, τ+ x τ j x M; 6 he mappngs K j x and funcons σ j x sasfy he pschz condon wh consan µ and x + K j x D, x D; 7 he surfaces = σ j x do nonersec each oher and for all x D, z K j, j =, he followng nequales hold σ j x σ j x + z; 8 µm <, hk x, K x ξ, σ x σ x ξ. hen for any η > here exs ξ > and δ > such ha for any soluon x of he ncluson 5-7 here exss a soluon y of he ncluson 8-3 wh he nal condon y x δ such ha x y η, [, ]\ [s p δ p, s p + δ p ] [, + ], p where s p = σ p ys p, = τ y, δ p + < Cη. p 8 Remark 6. o oban he classcal N.N. Bogolubov s negral connuy condon we converge ε o zero and o nfny n he condon 3. In hs sense hs heorem generalzes he frs N.N. Bogolyubov s heorem for he mehod of paral averagng. Remark 7. In hs revew we have no consdered he V.A. Plonkov s resuls devoed o he averagng of he dfferenal equaons wh dsconnuous rgh-hand sde n case of he sldng mode [5, 6, 7], o he applcaon of he averagng mehod and dfferenal nclusons o he research of conrol problems [,, 4, 7, 8, 9, 3, 3, 3, 33], o averagng of dscree nclusons [34], o quesons of asympocal researches of sngularly perurbed dfferenal nclusons and o he generalzaon of he A.N. khonov heorem on dfferenal nclusons [4, 35, 36, 37]. Concluson For now he quesons of he consrucon of he hgher approxmaons for R-soluons of dfferenal nclusons, he possbly of applcaon of he averagng mehod for dfferenal nclusons wh dsconnuous rgh-hand sde, dfferenal equaons and nclusons n semlnear merc spaces wh fas and slow varables, ec. are open. Is caused by he basc dffcules conneced wh nonlneary of he spaces. [] V. I. Nebesnov, V. A. Plonkov, F. Y. Kuzjushn, he opmal conrol of SRP on he dsurbance, Food ndusry, Moskow, 974, Russan. [] V. I. Nebesnov, V. A. Plonkov, Mahemacal mehods of he nvesgaon of he operang condons of he shp s sysems, Reklamnformburo MMF, Moskow, 977, Russan. [3] V. A. Plonkov, Asympoc mehods n opmal conrol problems, Odessa Sae Unversy, Odessa, 976, Russan. [4] V. A. Plonkov, he averagng mehod n conrol problems, ybd, Kev, 99, Russan. [5] V. A. Plonkov, A. V. Plonkov, A. N. Vyuk, Dfferenal equaons wh a mulvalued rgh-hand sde. Asympoc mehods, AsroPrn, Odessa, 999, Russan. [6] N. A. Peresyuk, V. A. Plonkov, A. M. Samolenko, N. V. Skrpnk, Impulsve dfferenal equaons wh a mulvalued and dsconnuous rghhand sde, Nasonalna Akademya Nauk Ukran, Insu Maemak, Kev, 7, Russan. [7] V. A. Plonkov, Averagng mehod for dfferenal nclusons and s applcaon o opmal conrol problems, Dfferensal nye Uravnenya , Russan. [8] V. A. Plonkov, Paral averagng of dfferenal nclusons, Ma. Zamek , Russan. [9] R. J. Aumann, Inegrals of se-valued funcons, J. Mah. Anal. Appl [] G. Pangan, On he fundamenal heory of mulvalued dfferenal equaons, J. Dff. Eq [] J.. Davy, Properes of he soluon se a generalzed dfferenal equaon, Bull. Ausral. Mah. Soc [] A. Plś, rajecores and quasrajecores of a orenor feld, Bull. Acad. Pol. ac. Ser. sc. mah., asron. e phys [3] A. N. Flaov, Asympoc mehods n he heory of dfferenal and negro - dfferenal nclusons, Fan, ashken, 974, Russan. [4] V. A. Plonkov, Averagng of dfferenal nclusons, Ukran. Ma. Zh , Russan. [5] A. I. Panasyuk, V. I. Panasyuk, An asympoc opmzaon of nonlnear conrol sysems, Belorussan Sae Unversy, Mnsk, 977, Russan. [6] A. I. Panasyuk, V. I. Panasyuk, Abou one equaon arzed from a dfferenal ncluson, Ma. Zameky , Russan. [7] C. Banf, Sull.approssmazone d processe non sazonar n macanca non lneare, Boll. Unon Ma. Ial

19 [8] V. I. Blagodaskkh, A. F. Flppov, Dfferenal nclusons and opmal conrol, opology, ordynary dfferenal equaons, dynamcal sysems. rudy Ma. Ins. Seklov , Russan. [9] V. A. Plonkov, V. M. Savchenko, On he averagng of dfferenal nclusons when he average of he rgh-hand sde s absen, Ukranan Mah. J [] V. I. Blagodaskkh, Inroduce n opmal conrol lnear heory, Vssh. shkola, Moscow,, Russan. [] A. F. Flppov, Classcal soluons of dfferenal equaons wh mulvalued rgh-hand sde, SIAM J. Conrol [] K. Kuraowsk, opology. V.., Academc Press, New York and ondon, 966. [3] V. A. Plonkov, O. G. Rudyk, A scheme for averagng negro-dfferenal nclusons, Sove Mah. Iz. VUZ [4] V. A. Plonkov,. I. Plonkova, Averagng of dfferenal nclusons wh mulvalued mpulses, Ukranan Mah. J [5] V. A. Plonkov, R. P. Ivanov, N. M. Kanov, Mehod of averagng for mpulsve dfferenal nclusons, Plska Sud. Mah. Bulgar [6] V. A. Plonkov, Asympoc mehods n he heory of dfferenal equaons wh a dsconnuous rgh-hand sde, Akad. Nauk Ukrany Ins. Ma. Preprn , Russan. [7] V. A. Plonkov, Asympoc mehods n he heory of dfferenal equaons wh dsconnuous and mul-valued rgh-hand sdes. dynamcal sysems,., J. Mah. Sc [8] V. I. Nebesnov, V. A. Plonkov, o he dynamcs of he energec sysems wh wo degrees of freedom, Mashnovedene AS USSR , Russan. [9] V. A. Plonkov, Asympoc nvesgaon of he equaons of conrolled moon, Sove J. Compu. Sysems Sc [3] V. A. Plonkov, N. A. Smrnova, Averagng of he equaons of moon n problems of perodc opmzaon, Akuska ulrazvukovaya ehnka , Russan. [3] V. A. Plonkov,. S. Zverkova, O. E. Slobodyanyuk, Averagng of nonperodc problems of he conrol of a dsconnuy surface, Ukranan Mah. J [3] V. A. Plonkov,. S. Zverkova, Averagng of boundary value problems n ermnal opmal conrol problems, Dfferensal nye Uravnenya , Russan. [33] V. A. Plonkov, he paral averagng mehod n ermnal conrol problems, Dfferencal nye Uravnenja , Russan. [34] V. A. Plonkov,. I. Plonkova, A.. Yarovo, An averagng mehod for dscree sysems and s applcaon o conrol problems, Nonlnear Oscl. N. Y [35] V. A. Plonkov, V. V. Barda, Averagng of boundary value problems for ordnary dfferenal equaons wh slow and fas varables, Ukranan Mah. J [36] V. A. Plonkov, M. arban, Jusfcaon of a paral averagng scheme for sysems wh slow and fas varables, Dfferenal Equaons [37] V. A. Plonkov, he averagng of dfferenal nclusons, Dep. VINII, 54-79, Russan