The Kurzweil integral and hysteresis

Transcription

1 Journl of Phyic: Conference Serie The Kurzweil integrl nd hyterei To cite thi rticle: P Krejcí 2006 J. Phy.: Conf. Ser View the rticle online for updte nd enhncement. Relted content - Outwrd pointing propertie for vectoril hyterei opertor nd ome ppliction O Klein - On prbolic eqution with hyterei nd convection: A uniquene reult M Eleuteri nd J Kopfová - Extending vector hyterei opertor V Recupero Recent cittion - Continuity propertie of Prndtl-Ihlinkii opertor in the pce of regulted function Guoju Ye et l - Exitence theorem for nonliner econd-order ditributionl differentil eqution Wei Liu et l - Anlyticl olution for cl of network dynmic with mechnicl nd finncil ppliction P. Krejí et l Thi content w downloded from IP ddre on 01/01/2019 t 00:02

2 Intitute of Phyic Publihing Journl of Phyic: Conference Serie 55 (2006) doi: / /55/1/014 Interntionl Workhop on Multi-Rte Procee nd Hyterei The Kurzweil integrl nd hyterei PKrejčí Weiertr Intitute for Applied Anlyi nd Stochtic, Mohrentr. 39, D Berlin, Germny, nd Mthemticl Intitute, Czech Acdemy of Science, Žitná 25, CZ Prh 1, Czech Republic E-mil: nd Abtrct. A hyterei opertor, clled the ply, with vrible (poibly degenerte) chrcteritic, i conidered in the pce of right-continuou regulted function. The Lipchitz continuity of the input-output mpping i proved by men of new technique bed on the Kurzweil integrl. Introduction Thi pper i devoted to the Kurzweil integrl decription of clr rte-independent evolution vritionl inequlity with moving contrint in the pce G R (0,T) of right-continuou regulted function. In the hyterei literture, thi vritionl inequlity define the o-clled ply opertor with vrible chrcteritic, cf. [3]. We prove here tht the input-output opertor i Lipchitz continuou with repect to the up-norm in G R (0,T). Thi fct in principle cn be derived from the generl Lipchitz continuity reult for polyhedrl weeping procee in [7]. Our objective here i to how tht the Kurzweil formlim, going bck to [8], ee lo [9, 10], provide imple nd trightforwrd proof. A our min tool, we derive Kurzweil integrl chrcteriztion of monotone function (Propoition 1.9), which we believe to be of independent interet in rel nlyi. The pper i divided into two ection. In Section 1 we recll ome bic fct bout Kurzweil integrtion nd give detiled proof of Propoition 1.9; the hyterei problem itelf i treted in Section The Kurzweil integrl In thi ection, we recll the definition nd ome bic propertie of the Kurzweil integrl introduced in [8] frmework for olving ODE with ingulr right-hnd ide. We cite mot of the reult without proof, nd n intereted reder cn find more informtion in [5, 9, 10]. Propoition 1.9, however, which ply n importnt role in the theory of Kurzweil integrl vritionl inequlitie, eem to be new in thi etting nd it detiled proof i given t the end of thi ection. The originl definition in [8] i not uitble for n integrl formultion of dicontinuou evolution vritionl inequlitie, nd thi i why the Young integrl w ued inted in [2, 6]. The extenion of the Kurzweil integrl in [5] contin, however, the Young integrl pecil ce, preerving the dvntge of the ey Kurzweil formlim. Here, we del only with rightcontinuou evolution procee, nd Definition 1.1 below, which goe bck to [9], turn out to be ufficient for our purpoe IOP Publihing Ltd 144

3 145 The bic concept in the Kurzweil integrtion theory i tht of δ -fine prtition. Conider nondegenerte cloed intervl [, b] R, nd denote by D,b the et of ll diviion of the form d = {t 0,...,t m }, = t 0 <t 1 <...<t m = b. (1.1) With diviion d = {t 0,...,t m } D,b we ocite prtition D defined We define the et D = {(τ j, [t j 1,t j ]) ; j =1,...,m} ; τ j [t j 1,t j ] j =1,...,m. (1.2) Γ(, b) := {δ :[, b] R ; δ(t) > 0 for every t [, b]}. (1.3) An element δ Γ(, b) i clled guge. Fort [, b] ndδ Γ(, b) we denote I δ (t) := ]t δ(t),t+ δ(t)[. (1.4) Definition 1.1 Let δ Γ(, b) be given guge. A prtition D of the form (1.2) i id to be δ -fine if for every j =1,...,m we hve nd the following impliction hold: τ j [t j 1,t j ] I δ (τ j ), τ j = t j 1 j =1, τ j = t j j = m. The et of ll δ -fine prtition i denoted by F δ (, b). It i ey to ee tht F δ (, b) i nonempty for every δ Γ(, b) ; thi follow e. g. from [4, Lemm 1.2]. For given function f,g :[, b] R nd prtition D of the form (1.2) we define the Kurzweil integrl um K D (f,g) by the formul K D (f,g) = f(τ j )(g(t j ) g(t j 1 )). (1.5) j=1 Definition 1.2 Let f,g :[, b] R be given. We y tht J R i the Kurzweil integrl over [, b] of f with repect to g nd denote J = f(t) dg(t), (1.6) if for every ε>0 there exit δ Γ(, b) uch tht for every D F δ (, b) we hve Uing the fct tht the impliction J K D (f,g) ε. (1.7) δ min{δ 1,δ 2 } F δ (, b) F δ1 (, b) F δ2 (, b) (1.8) hold for every δ, δ 1,δ 2 Γ(, b), we eily check tht the vlue of J in Definition 1.2 i uniquely determined. We lit below in Propoition 1.3, 1.4 ome tndrd propertie common to mot integrl concept.

4 146 Propoition 1.3 Let f,f 1,f 2,g,g 1,g 2 : [, b] R be ny function. Then the following impliction hold. (i) If f 1(t) dg(t), f 2(t) dg(t) exit, then (f 1 + f 2 )(t) dg(t) exit nd (f 1 + f 2 )(t) dg(t) = f 1 (t) dg(t) + f 2 (t) dg(t). (1.9) (ii) If f(t) dg 1(t), f(t) dg 2(t) exit, then f(t) d(g 1 + g 2 )(t) exit nd f(t) d(g 1 + g 2 )(t) = f(t) dg 1 (t) + f(t) dg 2 (t). (1.10) (iii) If f(t) dg(t) exit, then λf(t) dg(t), f(t) d(λg)(t) exit for every contnt λ R, nd λf(t) dg(t) = f(t) d(λg)(t) = λ f(t) dg(t). (1.11) Propoition 1.4 Let f,g :[, b] R be given function nd let ], b[ be given. (i) Aume tht f(t) dg(t) exit. Then f(t) dg(t), f(t) dg(t) exit. (ii) Aume tht f(t) dg(t), f(t) dg(t) exit. Then f(t) dg(t) exit nd f(t) dg(t) = f(t) dg(t) + f(t) dg(t). (1.12) In order to preerve the conitency of (1.12) lo in the limit ce = nd = b,weet f(t) dg(t) = 0 [, b], f,g :[, b] R. (1.13) Recll tht function f :[, b] R i id to be regulted if for every t [, b] there exit both one-ided limit f(t+),f(t ) R with the convention f( ) =f(), f(b+) = f(b), ee [1]. Obviouly, the et of dicontinuity point of every regulted function i t mot countble. In greement with [10], we denote by G(, b) the et of ll regulted function f :[, b] R. Let u introduce in G(, b) ytem of eminorm f [,t] := up{ f(τ) ; τ [, t]} (1.14) for ny ubintervl [, t] [, b]. Indeed, [,b] i norm. With thi norm, G(, b) become Bnch pce. Let u note tht the pce C[, b] ofcontinuou function f :[, b] R i cloed ubpce of G(, b) with repect to the norm [,b]. Moreover, every regulted function cn be uniformly pproximted by tep function of the form w(t) = ĉ k χ {tk } (t) + k=0 m c k χ ]tk 1,t k [ (t), t [, b], (1.15) where d = {t 0,...,t m } D,b i given diviion, χ A for A [, b] i the chrcteritic function of the et A,ndĉ 0,...,ĉ m,c 1,...,c m re given rel number. We ee in prticulr tht the pce BV (, b) of ll function of bounded vrition on [, b] i contined dene ubet in G(, b). In the next ection, we will retrict ourelve to the pce G R (, b), BV R (, b) of right-continuou function from G(, b), BV (, b), repectively.

5 147 Remrk 1.5 Propoition 1.4 need ome comment. Whenever we integrte function f,g defined in [, b] over n intervl [r, ] [, b], we implicitly conider their retriction f [r,],g [r,]. In prticulr, in ce of regulted function, we hve e. g. f [r,] (+) = f(), f [r,] (r ) =f(r). Note tht we del here with function tht re defined for ll t [, b]. The concept of lmot everywhere i meningle here. The following explicit formul cn eily be derived from the definition. Propoition 1.6 For every f :[, b] R, g G(, b), r b, we hve (i) (ii) (iii) (iv) χ {r} (t) dg(t) =g(r+) g(r ), ( ) f(t) d χ {r} (t) = 0 if r ], b[, f() if r =, f(b) if r = b, χ ]r,[ (t) dg(t) =g( ) g(r+) ]r, b], ( ) f(t) d χ ]r,[ (t) =f(r) f() ]r, b]. We ee in prticulr tht the integrl f(t) dg(t) exit whenever one of the function f,g i tep function nd the other one i regulted. By denity rgument, we obtin the following reult. Theorem 1.7 (Propertie of the Kurzweil integrl) (i) If f G(, b) nd g BV (, b), then f(t) dg(t) exit nd tifie the etimte f(t) dg(t) f [,b] Vr g. (1.16) [,b] (ii) If f BV (, b) nd g G(, b), then f(t) dg(t) exit nd tifie the etimte ( ) b f() g()+ f(t) dg(t) f(b) +Vr f g [,b]. (1.17) [,b] (iii) For every f G(, b), g BV (, b) we hve the integrtion-by-prt formul f(t) dg(t)+ g(t) df (t) = f(b) g(b) f() g() (1.18) + ( ) (f(t) f(t )) (g(t) g(t )) (f(t+) f(t)) (g(t+) g(t)). t [,b] (iv) If f n G(, b) nd g n BV (, b) re uch tht f n f [,b] 0, g n g [,b] 0 n,nd Vr [,b] g n C independently of n, then lim f n (t) dg n (t) = n f(t) dg(t). (1.19)

6 148 Corollry 1.8 For every g BV (, b) we hve g(t+) dg(t) = 1 2 ( g(b) 2 g() 2) t [,b] g(t+) g(t ) 2. (1.20) We conclude thi ection by Kurzweil integrl chrcteriztion of monotone function, which will be referred to everl time in the next ection in the context of Kurzweil integrl vritionl inequlitie. Propoition 1.9 Let f G(, b) nd g BV (, b) be uch tht (i) f(t) > 0 for every t [, b], (ii) f(τ) dg(τ) 0 for every <t b. Then g i nondecreing in [, b]. The proof of thi ttement will be divided into everl tep. convergence reult. Let u trt with n ey Lemm 1.10 For every f G(, b) nd g BV (, b) we hve +h lim f(τ) dg(τ) = f(t)(g(t+) g(t)) t [, b[, (1.21) h 0+ t lim h 0+ t h f(τ) dg(τ) = f(t)(g(t) g(t )) t ], b]. (1.22) Proof. Let t [, b[ be fixed. For τ [, b] put f 1 (τ) =f(t+), f 2 (τ) =(f(t) f(t+)) χ {t} (τ), f 3 (τ) =f(τ) f 1 (τ) f 2 (τ). Let ε>0 be rbitrry, nd let h ]0,b t[ be uch tht f(τ) f(t+) ε, g(τ) g(t+) ε for τ ]t, t + h]. Then f 3 (τ) ε for ll τ [t, t + h], hence, by (1.16), +h f 3 (τ) dg(τ) ε Vr g. (1.23) t Furthemore, we hve by Remrk 1.5 nd Propoition 1.6 tht +h f 1 (τ) dg(τ) = f(t+) (g(t + h) g(t)), (1.24) t +h t [,b] f 2 (τ) dg(τ) = (f(t) f(t+)) (g(t+) g(t)), (1.25) hence, conequence of (1.24) (1.25), +h (f 1 (τ)+f 2 (τ)) dg(τ) f(t)(g(t+) g(t)) t = f(t+) (g(t + h) g(t+)) ε f [,b], (1.26) nd (1.21) follow from (1.23) nd (1.26). The proof of (1.22) i imilr. Under the hypothee of Propoition 1.9, we thu hve in prticulr A next tep, we prove the following lemm. g(t ) g(t) g(t+) t [, b]. (1.27)

7 149 Lemm 1.11 Let the hypothee of Propoition 1.9 hold. Then for every non-negtive function w G(, b) we hve w(τ) f(τ) dg(τ) 0. (1.28) Proof. It uffice to ume tht w i tep function of the form (1.15). Indeed, for n rbitrry function w G(, b) we find equence {w n } of tep function uch tht w w n [,b] 0 n, nd ue Propoition 1.7 (iv). Hence, let w be in (1.15), nd let u conider ny h>0 uch tht t k 1 + h<t k h for ll k =1,...,m. We then hve where w(τ) f(τ) dg(τ) = k h t k 1 +h ( k 1 +h t k 1 k h t k 1 +h k t k h ) + + w(τ) f(τ) dg(τ), w(τ) f(τ) dg(τ) =c k k h t k 1 +h f(τ) dg(τ) 0 for ll h nd k by hypothei. Letting h tend to 0+ we obtin from Lemm 1.10 nd formul (1.27) tht w(τ) f(τ) dg(τ) (ĉ k 1 f(t k 1 )(g(t k 1 +) g(t k 1 )) + ĉ k f(t k )(g(t k ) g(t k ))) 0. Lemm 1.12 Let the hypothee of Propoition 1.9 hold, nd let in ddition f(t+) > 0, f(t ) > 0 for ll t [, b]. Then g i nondecreing in [, b]. Proof. There exit ome r>0 uch tht f(t) r for ll t [, b]. Hence, we my ue Lemm 1.11 with the function w(τ) = 1 f(τ) χ ],t[ (τ) for ny choice of ], t[ [, b], nd the deired inequlity 0 w(τ) f(τ) dg(τ) = 1 dg(τ) = g(t) g() (g(+) g()) (g(t) g(t )) g(t) g(), χ {} dg(τ) χ {t} dg(τ) follow eily. We now p to the proof of Propoition 1.9. Proof of Propoition 1.9. We define the et N + = {t [, b]; f(t+) = 0}, N = {t [, b]; f(t ) =0}, N = N + N. For every t N,wehveeither f(t) >f(t ) orf(t) >f(t+), hence N i t mot countble. Furthermore, N i cloed. Indeed, if for intnce {t i } i equence in N, t i t,thenin neighborhood of ech t i there exit ˆt i uch tht f(ˆt i ) < 1/i, ˆt i t, hence f(t ) =0. A imilr rgument work for t i t.

8 150 All element of N cn be ordered into equence { j ; j N}. Let ε>0 be given. For every j N we find h j > 0 uch tht g(τ) g( j +) ε 2 j 1 for τ ] j, j + h j ], g(τ) g( j ) ε 2 j 1 for τ [ j h j, j [. The et N i compct, we cn therefore find finite covering } (1.29) N m ] jk h jk, jk + h jk [, (1.30) k=0 with j0 < j1 <...< jm. We my ume tht jk 1 h jk 1 < jk h jk, jk 1 + h jk 1 < jk + h jk for ll k = 1,...,m; otherwie we remove the redundnt intervl from the covering (1.30). We now et 0 =, b m = b, nd for k =1,...,m chooe } b k 1 = jk 1 + h jk 1, k = jk h jk if jk 1 + h jk 1 jk h jk, b k 1 = k ] jk h jk, jk 1 + h jk 1 [ ] jk 1, jk [ \N if jk 1 + h jk 1 > jk h jk. (1.31) Then m N [ 0,b 0 [ ] k,b k [ ] m,b m ]. For k <t b k, k =0,...,m, we hve by (1.27) nd (1.29) tht g() g(t) +ε 2 j k. On the other hnd, for b k 1 <t k, k =1,...,m, it follow from Lemm 1.12 tht g() g(t). Conequently, for ll <t b we hve g() g(t)+ε 2 j k g(t)+ε. k=0 Since ε>0 h been choen rbitrrily, we obtin the ertion. 2. A vritionl inequlity We now conider fixed intervl [0,T] nd two (input) function u, r G R (0,T) uch tht r(t) 0 for ll t [0,T], nd n initil condition x 0 [ r(0),r(0)]. The problem conit in finding the output ξ G R (0,T) uch tht u(t) ξ(t) r(t) t [0,T], T (u(t) ξ(t) y(t)) dξ(t) 0 y G(0,T), y(t) r(t) t [0,T], 0 u(0) ξ(0) = x 0. Thi extend the model conidered in [3] in two repect: we dmit dicontinuou input, nd r(t) i llowed to vnih t ny point. The etting of [3] however contin dditionl nonlineritie tht we neglect here for implicity. We firt contruct olution to Problem (2.1) if the input re tep function of the form (2.1) u(t) = u k 1 χ [tk 1,t k [ (t)+u m χ {T } (t), r(t) = r k 1 χ [tk 1,t k [ (t)+r m χ {T } (t), (2.2)

9 151 where 0 = t 0 <t 1 <... < t m = T i given diviion, nd u k,r k for k =0,...m re given rel number. To thi end, we ue the projection Q c (x) =mx{ c, min{x, c}} for c 0 nd it complement P c (x) =x Q c (x) for x R, nd clim tht the olution ξ i given by the formul ξ(t) = ξ k 1 χ [tk 1,t k [ (t)+ξ m χ {T } (t) (2.3) with ξ 0 = u 0 x 0 nd ξ k = ξ k 1 + P rk (u k ξ k 1 ) for k =1,...m. (2.4) Indeed, we hve u k ξ k = Q rk (u k ξ k 1 ) [ r k,r k ]. Uing Propoition 1.6, we cn evlute the integrl in (2.1) explicitly nd obtin T (u(t) ξ(t) y(t)) dξ(t) = (ξ k ξ k 1 )(u k ξ k y(t k )) (2.5) 0 for every regulted function y uch tht y(t) r(t) for ll t. For ll x R nd y c we hve P c (x)(q c (x) y) 0. Putting z k = u k ξ k 1, we thu cn rewrite (2.5) T (u(t) ξ(t) y(t)) dξ(t) = P rk (z k )(Q rk (z k ) y(t k )) 0, (2.6) 0 hence (2.1) i tified. To extend the et dmiible input, we tte nd prove the min reult of thi ection on Lipchitz continuity of the input-output mpping defined by (2.1). Theorem 2.1 Let u i,r i G R (0,T) nd initil condition x 0 i [ r i (0),r i (0)] be given, i =1, 2. Let ξ 1,ξ 2 be correponding olution to (2.1), nd ume tht ξ i BV R (0,T) for i =1, 2. Then for every t [0,T] we hve ξ 1 (t) ξ 2 (t) mx{ ξ 1 (0) ξ 2 (0), u 1 u 2 [0,t] + r 1 r 2 [0,t] }. (2.7) Before ping to the proof of Theorem 2.1, we derive n ey uxiliry reult. Lemm 2.2 Let u, r G R (0,T) nd n initil condition x 0 [ r(0),r(0)] be given. Let ξ be olution to (2.1), nd ume tht ξ BV R (0,T). Then for every 0 <t T nd every regulted function y uch tht y(τ) r(τ) for ll τ, we hve (u(τ) ξ(τ) y(τ)) dξ(τ) 0. (2.8) Furthermore, if both u nd r re contnt in n intervl [t 0,t 1 ] [0,T], then ξ i contnt in [t 0,t 1 ]. Proof. Set y (τ) =y(τ) χ ],t] (τ) +(u(τ) ξ(τ)) ( χ [0,] + χ ]t,t ] )(τ) for τ [0,T]. Uing Propoition 1.4, 1.6, nd Remrk 1.5 (note tht ξ i right-continuou!), we obtin tht 0 = = T 0 T + (u(τ) ξ(τ) y (τ)) dξ(τ) (u(τ) ξ(τ) y(τ)) dξ(τ) t (u(τ) ξ(τ) y(τ)) χ {t} τ) dξ(τ) (u(τ) ξ(τ) y(τ)) χ {} τ) dξ(τ) (u(τ) ξ(τ) y(τ)) dξ(τ). (2.9)

10 152 Aume now tht u nd r re contnt in [t 0,t 1 ]. For ll t ]t 0,t 1 ] we then hve (u(t 0 ) ξ(τ) y(τ)) dξ(τ) 0. t 0 We my chooe y(τ) =u(t 0 ) ξ(t 0 ) nd ue (1.20) for g(τ) =u(t 0 ) ξ(τ) to obtin 0 t 0 (ξ(t 0 ) ξ(τ)) dξ(τ) = 1 2 ξ(t 0) ξ(t) hence ξ(t) =ξ(t 0 ) for ll t [t 0,t 1 ]. τ [t 0,t] ξ(τ) ξ(τ ) 2, The bove proof how why it w convenient to chooe the tet function y in Problem (2.1) in G(0,T) nd not in G R (0,T). More refined reult in [2] however how tht even the choice y BV R (0,T) i ufficient. Proof of Theorem 2.1. The proce defined by (2.1) i rte-independent, hence it uffice to prove tht ξ 1 (t) ξ 2 (t) mx{ ξ 1 (0) ξ 2 (0), u 1 u 2 [0,T ] + r 1 r 2 [0,T ] } (2.10) for ll t [0,T]. Set ū = u 1 u 2, r = r 1 r 2, ξ = ξ1 ξ 2, nd ume tht there exit t 1 ]0,T] uch tht ξ(t 1 ) > ū [0,T ] + r [0,T ]. Putting y(τ) =Q r1 (τ)(u 2 (τ) ξ 2 (τ)) nd y(τ) = Q r2 (τ)(u 1 (τ) ξ 1 (τ)) in the vritionl inequlitie for ξ 1 nd ξ 2, repectively, we obtin for ll 0 <t t 1 tht (ū(τ) ξ(τ)+p r1 (τ)(u 2 (τ) ξ 2 (τ))) dξ 1 (τ) 0, (2.11) ( ū(τ)+ ξ(τ)+p r2 (τ)(u 1 (τ) ξ 1 (τ))) dξ 2 (τ) 0. (2.12) We hve for ll c, c > 0ndx R the impliction x c P c (x) c c, hence, P r1 (τ)(u 2 (τ) ξ 2 (τ)) r [0,T ], P r2 (τ)(u 1 (τ) ξ 1 (τ)) r [0,T ] for ll τ [0,T]. Letting t t 1 in (2.11) (2.12), nd uing (1.22), we obtin ξ 1 (t 1 ) ξ 1 (t 1 ), ξ 2 (t 1 ) ξ 2 (t 1 ), hence ξ(t 1 ) ξ(t 1 ) > ū [0,T ] + r [0,T ]. There exit therefore n intervl [t 0,t 1 ] [0,T] uch tht ξ(t) > ū [0,T ] + r [0,T ] for ll t [t 0,t 1 ]. Applying Propoition 1.9 to (2.11) (2.12), we conclude tht ξ 1 i nonincreing, ξ 2 i nondecreing, hence ξ i nonincreing in [t 0,t 1 ]. Let t be the infimum of ll t 0 [0,t 1 ] uch tht ξ(t) > ū [0,T ] + r [0,T ] for every t [t 0,t 1 ]. The bove rgument yield tht t = 0 nd tht ξ i nonincreing in [0,t 1 ], which we wnted to prove. The ce ξ(t 1 ) < ū [0,T ] r [0,T ] i imilr. The proof of Theorem 2.1 i complete. Let now u, r G R (0,T)begiven,r(t) 0 for ll t, nd for ome fixed intervl [t 0,t 1 ] [0,T] et u 1 (t) = u 2 (t) = u(t), r 1 (t) = r 2 (t) = r(t) for t [0,t 0 ], u 1 (t) = u(t), r 1 (t) = r(t) u 2 (t) =u(t 0 ), r 2 (t) =r(t 0 )fort [t 0,t 1 ]. Then Theorem 2.1 nd Lemm 2.2 yield ξ(t 1 ) ξ(t 0 ) u u(t 0 ) [t0,t 1 ] + r r(t 0) [t0,t 1 ]. (2.13) In prticulr, if both u nd r belong to BV R (0,T), then o doe ξ nd we hve Vr ξ Vr u +Vr r. (2.14) [0,T ] [0,T ] [0,T ]

11 153 We hve een tht Problem (2.1) h unique olution ξ whenever u nd r re tep function. Since every BV -function cn be uniformly pproximted by tep function with uniformly bounded vrition, we obtin, immedite conequence of Theorem 2.1 nd 1.7 (iv), the following reult. Corollry 2.3 Problem (2.1) h unique olution ξ BV R (0,T) for every u, r BV R (0,T), r(t) 0 for ll t [0,T], nd for every x 0 [ r(0),r(0)]. Moreover, the olution opertor P : (u, r, x 0 ) ξ dmit Lipchitz continuou extenion to P : D G R (0,T), where D = {(u, r, x 0 ) G R (0,T) G R (0,T) R ; r(t) 0 for t [0,T], x 0 [ r(0),r(0)]},nd inequlity (2.7) hold for ll (u i,r i,x 0 i ) D, ξ i = P(u i,r i,x 0 i ), i =1, 2. We do not know if the extended opertor P : D G R (0,T) till dmit the Kurzweil integrl repreenttion (2.1), becue in generl, f(t) dg(t) i not well defined if both f nd g re only regulted. There re ome indiction tht (2.1) i meningful in thi itution, too, but no proof i yet vilble. Thing re different if r i bounded from below by poitive contnt. Then, imilrly in [6], ξ h bounded vrition nd the integrl formul hold. To conclude thi pper, we give n elementry proof of lightly tronger verion of thi fct. Propoition 2.4 Let r 0 > 0 be given, nd let (u, r, x 0 ) D be uch tht r(t) r 0 for ll t [0,T]. Then ξ = P(u, r, x 0 ) i piecewie monotone nd, in prticulr, belong to BV R (0,T). Proof. Since both u nd r re regulted, there exit n integer n with the property tht for every equence 1 <b 1 2 <b 2... n <b n of point in [0,T], the following impliction hold: u(b k ) u( k ) + r(b k ) r( k ) 2 3 r 0 k =1,...,n = n n. (2.15) We clim tht ξ h t mot (3n + 1) monotonicity intervl. To prove thi conjecture, we tcitly conider uniformly convergent BV -pproximtion of u nd r preerving the property (2.15), o tht the vritionl inequlity (2.1) i meningful. Since the number of monotonicity intervl (hence the totl vrition) of ξ remin bounded independently of the pproximtion, we my p to the limit uing Theorem 1.7 nd obtin the ertion. Aume tht there exit m>n nd point 0 t 0 <t 1 <...<t 2m T uch tht ξ(t 0 ) <ξ(t 1 ) >ξ(t 2 ) <...>ξ(t 2m ). We mke ue of the following impliction: u(τ) ξ(τ) <r 0 τ [, t] = ξ i nonincreing in [, t], u(τ) ξ(τ) > r 0 τ [, t] = ξ i nondecreing in [, t]. } (2.16) Indeed, (2.16) follow from Propoition 1.9 provided we put in (2.8) y(τ) =±r 0. Hence, the et A 2j 1 = {τ [t 2j 2,t 2j 1 ]; u(τ) ξ(τ) r 0 }, A 2j = {τ [t 2j 1,t 2j ]; u(τ) ξ(τ) r 0 } re non-empty for ll j =1,...,m,ndwemyet k =upa k, k =1,...,2m. (2.17) By (2.16) nd by the right-continuity of ξ,wehveξ( 2j 1 ) ξ(t j 1 ), ξ( 2j ) ξ(t 2j ) for j =1,...,m, hence ξ( 1 ) >ξ( 2 ) <...>ξ( 2m ), (2.18)

12 154 nd, by definition of k,wehve u( 2j 1 ) ξ( 2j 1 ) r 0, u( 2j ) ξ( 2j ) r 0 for j =1,...,m. Furthermore, by (2.13), we hve ξ( k ) ξ( k ) u( k ) u( k ) + r( k ) r( k ) k =1,...,2m. Uing (2.18), we thu obtin for j =1,...,m tht u( 2j 1 ) u( 2j ) 2r 0 + ξ( 2j 1 ) ξ( 2j ) 2r 0 u( 2j 1 ) u( 2j 1 ) r( 2j 1 ) r( 2j 1 ) u( 2j ) u( 2j ) r( 2j ) r( 2j ), (2.19) nd imilrly, for j =2,...,m, u( 2j 1 ) u( 2j 2 ) 2r 0 u( 2j 1 ) u( 2j 1 ) r( 2j 1 ) r( 2j 1 ) u( 2j 2 ) u( 2j 2 ) r( 2j 2 ) r( 2j 2 ). (2.20) The et M := {k =1,...,2m ; u( k ) u( k ) + r( k ) r( k ) (2/3)r 0 } contin, by (2.15), t mot n element. For 2m 1 2n indice k {1,...,2m} we thu hve by (2.19) (2.20) tht u( k ) u( k 1 ) 2r r 0 = 2 3 r 0. From (2.15) we conclude tht 2m 1 2n n, tht i, 2m 3n + 1 conjectured. Thi complete the proof of Propoition 2.4. Acknowledgment Thi work h been prtilly upported by the Univerity College Cork, Irelnd. Bibliogrphy [1] Aumnn G 1954 Reelle Funktionen (in Germn) (Berlin: Springer) [2] Brokte M nd Krejčí P 2003 Dulity in the pce of regulted function nd the ply opertor Mth. Z [3] Chernorutkii V V nd Krnoel kii M A 1992 Hyterei ytem with vrible chrcteritic Nonlin. Anl. TMA [4] Krejčí P nd Kurzweil J A nonexitence reult for the Kurzweil integrl Mth. Bohem [5] Krejčí P 2003 The Kurzweil integrl with excluion of negligible et Mth. Bohem [6] Krejčí P nd Lurençot Ph 2002 Generlized vritionl inequlitie J. Convex Anl [7] Krejčí P nd Vldimirov A 2003 Polyhedrl weeping procee with oblique reflection in the pce of regulted function Set-Vlued Anl. 11 (2003) 91 [8] Kurzweil J 1957 Generlized ordinry differentil eqution nd continuou dependence on prmeter Czecholovk Mth. J. 7 (82) 418 [9] Schwbik Š 1973 On modified um integrl of Stieltje type Čopi Pět. Mt [10] Tvrdý M 1989 Regulted function nd the Perron-Stieltje integrl Čopi Pět. Mt