Regulted functions nd the Perron-Stieltjes integrl Miln Tvrd Dedicted to Professor Otkr Boruvk on the occsion of his ninetieth birthdy Mthemticl Institute Acdemy of Sciences of the Czech Republic 115 67 PRAHA 1, itn 25, Czech Republic (e-mil: tvrdy@mth.cs.cz) Abstrct. Properties of the Perron-Stieltjes integrl with respect to regulted functions re investigted. It is shown tht liner continuous functionls on the spce G L ( b) of functions regulted on [ b] nd left-continuous on ( b) my be represented in the form F (x) =qx()+ Z b p(t)d[x(t)] where p 2 R nd q(t) is function of bounded vrition on [ b]: Some bsic theorems (e.g. integrtion-by-prts formul, substitution theorem) known for the Perron-Stieltjes integrl with respect to functions of bounded vrition re estblished. AMS Subject Clssiction. 26 A 42, 26 A 45, 28 A 25, 46 E 99. Keywords. regulted function, function of bounded vrition, Perron-Stieltjes integrl, Kurzweil integrl, left-continuous function, liner continuous functionl. 0. Introduction This pper dels with the spce G ( b) of regulted functions on compct intervl [ b]: It is known tht when equipped with the supreml norm G ( b) becomes Bnch spce, nd liner bounded functionls on its subspce G L ( b) of functions regulted on [ b] nd left-continuous on ( b) cn be represented by mens of the Dushnik-Stieltjes (interior) integrl. This result is due to H. S. Kltenborn [7], cf. lso Ch. S. H nig [5, Theorem 5.1]. Together with the known reltionship between the Dushnik-Stieltjes integrl, the -Young-Stieltjes integrl nd the Perron-Stieltjes integrl (cf. Ch. S. H nig [6] nd. Schwbik [11],[12]) this enbles us to see tht F 1
2 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 is liner bounded functionl on G L ( b) if nd only if there exists rel number q nd function p(t) of bounded vrition on [ b] such tht F (x) =qx()+ p(t)d[x(t)] for ny x 2 G L ( b) where the integrl is the Perron-Stieltjes integrl. We will give here the proof of this fct bsed only on the properties of the Perron-Stieltjes integrl. To this im, the proof of the existence of the integrl f(t)d[g(t)] for ny function f of bounded vrition on [ b]ndny function g regulted on [ b] is crucil. Furthermore, we will prove extensions of some theorems (e.g. integrtionby-prts nd substitution theorems) needed for deling with generlized dierentil equtions nd Volterr-Stieltjes integrl equtions in the spce G ( b): 1. Preliminries Throughout the pper R n denotes the spce of rel n-vectors, R 1 = R: Given x 2 R n its rguments re denoted by x 1 x 2 ::: x n (x = (x 1 x 2 ::: x n )). N stnds for the set of ll nturl numbers (N = f1 2 :::g). Given M R M denotes its chrcteristic function ( M (t) =1ift 2 M nd M (t) =0ift 62 M:) Let ;1 < < b < 1: The sets d = ft 0 t 1 ::: t m g of points in the closed intervl [ b] such tht = t 0 <t 1 < <t m = b re clled divisions of [ b]: Given division d of [ b] its elements re usully denoted by t 0 t 1 ::: t m : The couples D =(d ) where d = ft 0 t 1 ::: t m g is divisionof[ b] nd =( 1 2 ::: m ) 2 R m is such tht t j;1 j t j for ll j =1 2 ::: m re clled prtitions of [ b]: A function f :[ b] 7! R which possesses nite limits f(t+) = lim f() nd f(s;) = lim f()!t+!s; for ll t 2 [ b) nd ll s 2 ( b] is sid to be regulted on [ b]: The set of ll regulted functions on [ b] is denoted by G ( b): Given f 2 G ( b) we dene f(;) =f() f(b+) = f(b) nd + f(t) =f(t+) ; f(t) ; f(t) =f(t) ; f(t;)
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 3 f(t) =f(t+) ; f(t;): for ny t 2 [ b]: (In prticulr, we hve ; f() = + f(b) =0 f() = + f() nd f(b) = ; f(b).) It is known (cf. [5, Corollry 3.2]) tht if f 2 G ( b) then for ny " > 0 the set of points t 2 [ b] such tht j + f(t)j >" or j ; f(t)j >" is nite. Consequently, for ny f 2 G ( b) the set of its discontinuities in [ b] is countble. The subset of G ( b) consisting of ll functions regulted on [ b] nd left-continuous on ( b) will be denoted by G L ( b): A function f : [ b] 7! R is clled nite step function on [ b] if there exists division ft 0 t 1 ::: t m g of [ b] such tht f is constnt on every open intervl (t j;1 t j ) j =1 2 ::: m: The set of ll nite step functions on [ b] is denoted by S( b): A function f :[ b] 7! R is clled brek function on [ b] if there exist sequences ft k g 1 [ b] fc ; k g1 R nd fc + k g1 R such tht t k 6= t j for k 6= j c ; =0 if k t k = c + =0 if k t k = b 1X c ; nd (1.1) k f(t) = X tkt + c + < 1 k c ; k + X tk<t c + k for t 2 [ b] or equivlently f(t) = 1X c ; k [tk b](t)+c + k (tk b](t) for t 2 [ b]: Clerly, if f is given by (1.1), then + f(t k ) = c + k nd ; f(t k ) = c ; k for ny k 2 N nd f(t;) = f(t) = f(t+) if t 2 [ b] nft k g 1 : Furthermore, we hve f() =0 nd X 1 vr b f = c ; + c + k k for ny such function. The set of ll brek functions on [ b] is denoted by B ( b): BV( b) denotes the set of ll functions with bounded vrition on [ b] nd kfk BV = jf()j +vr b f for f 2 BV( b):
4 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 It is known tht for ny f 2 BV( b) there exist uniquely determined functions f C 2 BV( b) nd f B 2 BV( b) such tht f C is continuous on [ b] f B is brek function on [ b] nd f(t) = f C (t) +f B (t) on [ b] (the Jordn decomposition of f 2 BV( b)). In prticulr, if W = fw k g k2n is the set of discontinuities of f in [ b] then (1.2) f B (t) = 1X ; f(w k ) [w k b](t)+ + f(w k ) (w k b](t) 2 [ b]: Moreover, if we put (1.3) f B n (t) = nx ; f(w k ) [w k b](t)+ + f(w k ) (w k b](t) on [ b] for ny n 2 N then (1.4) lim kf B n ; f B k BV =0 (cf. e.g. [14, the proof of Lemm I.4.23]). Obviously, Given f 2 G ( b) we dene S( b) B ( b) BV( b) G ( b): kf k = sup jf(t)j: t2[ b] Clerly, kfk < 1 for ny f 2 G ( b) nd when endowed with this norm, G ( b) becomes Bnch spce (cf. [5, Theorem 3.6]). It is known tht S( b) isdense in G ( b) (cf. [5, Theorem 3.1]). It mens tht f :[ b] 7! R is regulted if nd only if it is uniform limit on [ b] of sequence of nite step functions. Obviously, G L ( b) is closed in G ( b) nd hence it is lso Bnch spce. (Neither S( b) nor B ( b) re closed in G ( b) of course.) For some more detils concerning regulted functions see the monogrphs by Ch. S. H nig [5] nd by G. Aumnn [1] nd the ppers by D. Fr kov [2] nd [3]. The integrls which occur in this pper re the Perron-Stieltjes integrls. We will work with the following denition which is specil cse of the denition due to J. Kurzweil [8]. Let ;1 <<b<1: An rbitrry positive vlued function :[ b] 7! (0 1) is clled guge on [ b]: Given guge on [ b] the prtition (d ) of [ b] is sid to be -ne if [t j;1 t j ] ( j ; ( j ) j + ( j )) for ny j =1 2 ::: m:
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 5 Given functions f g : [ b] 7! R nd prtition D =(d ) of[ b] let us dene S D (f g) = mx f( j )[g(t j ) ; g(t j;1 )]: We sy tht I 2 R is the Kurzweil integrl of f with respect to g from to b nd denote I = f(t)d[g(t)] or I = if for ny ">0 there exists guge on [ b] such tht I ; SD (f g) <" for ll -ne prtitions D of [ b]: The Perron-Stieltjes integrl with respect to function not necessrily of bounded vrition ws dened by A. J. Wrd [15] (cf. lso S. Sks [10, Chpter VI]). It cn be shown tht the Kurzweil integrl is equivlent totheperron-stieltjes integrl (cf. [11, Theorem 2.1], where the ssumption g 2 BV( b) is not used in the proof nd my be omitted). Consequently, the Riemnn-Stieltjes integrl (both of the norm type nd of the -type, cf. [4]) is its specil cse. The reltionship between the Kurzweil integrl, the -Young-Stieltjes integrl nd the Perron-Stieltjes integrl ws described by. Schwbik (cf. [11] nd [12]). Since we will mke use of some of the properties of the -Riemnn-Stieltjes integrl, let us indicte here the proof tht this integrl is included in the Kurzweil integrl. (For the denition of the -Riemnn-Stieltjes integrl, see e.g. [4, Sec. II.9].) 1.1. Proposition. R Let f g : [ b] 7! R nd I 2 e such tht the -Riemnnb Stieltjes integrl f dg exists nd equls I. Then the Perron-Stieltjes integrl R b f dg exists nd equls I s well. Proof. Let f dg = I 2 R i.e. for ny ">0 there is division d 0 = fs 0 s 1 ::: s m0 g of [ b] such tht for ny division d = ft 0 t 1 ::: t m g which is its renement (d 0 d) nd ny 2 R m such tht D =(d ) is prtition of [ b] the inequlity S D (f g) ; I <" f dg
6 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 is stised. Let us dene ( 1 2 " (t) = minfjt ; s jj j =0 1 ::: m 0 g if t 62 d 0 " if t 2 d 0 : Then prtition D =(d ) of [ b] is " -ne only if for ny j = 1 2 ::: m 0 there is n index i j such tht s j = i j : Furthermore, S D (f g) = mx h i f( j )[g(t j ) ; g( j )] + f( j )[g( j ) ; g(t j;1 )] for ny prtition D =(d )of[ b]: Consequently, forny " -ne prtition D =(d ) of [ b] the corresponding integrl sum S D (f g) equls the integrl sum S D 0(f g) corresponding to prtition D 0 = (d 0 0 ) where d 0 is division of [ b] such tht d 0 d 0 nd hence S D 0(f g) ; I <": This mens tht the Kurzweil integrl f dg exists nd holds. f dg = f dg = I To prove the existence of the Perron-Stieltjes integrl f dg for ny f 2 BV( b) nd ny g 2 G ( b) in Theorem 2.8 the following ssertion is helpful. 1.2. Proposition. Let f 2 BV( b) be continuous on [ b] nd let g 2 G ( b) then both the -Riemnn-Stieltjes integrls exist. f dg nd Proof. Let f 2 BV( b) which is continuous on [ b] nd g 2 G ( b) be given.according to the integrtion-by-prts formul [4, II.11.7] for -Riemnn-Stieltjes integrls to prove the lemm it is sucient toshow tht the integrl gdf exists. First, let us ssume tht n rbitrry 2 [ b] isgiven nd g = [ ] : Let us put gdf f bg d 0 = f bg if = or = b if 2 ( b):
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 7 It is esy to see tht then for ny prtition D =(d ) such tht d 0 d = ft 0 t 1 ::: t m g we hve = t k for some k 2f0 1 ::: mg nd ( f() ; f() if k+1 > S D (g f) = f(t k+1 ) ; f() if k+1 = : Since f is ssumed to be continuous, it is esy to show tht for given ">0 there exists division d of [ b] such tht d 0 d nd SD (g f) ; [f() ; f()] <" holds for ny prtition D =(d ) of[ b] with d d i.e. nd [ ] df = f() ; f() for ll 2 [ b]: By similr rgument we could show the following reltions: s well. It follows tht the integrl [ ) df = f() ; f() for ll 2 ( b] [ b] df = f(b) ; f() for ll 2 [ b] ( b] df = f(b) ; f() for ll 2 [ b) gdf exists for ny f 2 BV( b) continuous on [ b] nd ny g 2 S( b)(cf. Remrk 2.2). Now, if g 2 G ( b) is rbitrry, then there exists sequence fg n g 1 S( b) such tht lim kg n ; gk =0: Since by the preceding prt of the proof of the lemm ll the integrls g ndf hve nite vlue, by mens of the convergence theorem [4, Theorem II.15.1] vlid
8 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 for -Riemnn-Stieltjes integrls we obtin tht the integrl gdf the reltion exists nd holds. This completes the proof. lim g n df = gdf 2 R A direct corollry of Proposition 1.2 nd of [4, Theorem II.13.17] is the following ssertion which will be helpful for the proof of the integrtion-by-prts formul Theorem 2.15. (Of course, we could prove it by n rgument similr to tht used in the proof of Proposition 1.2, s well.) 1.3. Corollry. Let f 2 BV( b) nd g 2 G ( b): Let + f(t) + g(t) = ; f(t) ; g(t) =0 for ll t 2 ( b): Then both the -Riemnn-Stieltjes integrls exist. f dg nd gdf It is well known (cf. e.g. [14, Theorems I.4.17, I.4.19 nd Corollry I.4.27] tht if f 2 G ( b) nd g 2 BV( b) then the integrl f dg exists nd the inequlity f dg ; kfk vr b g holds. The Kurzweil integrl is n dditive function of intervls nd possesses the usul linerity properties. For the proofs of these ssertions nd some more detils concerning the Kurzweil integrl with respect to functions of bounded vrition see e.g. [8], [9], [13] nd [14]. 2. Perron-Stieltjes integrl with respect to regulted functions In this section we del with the integrls f(t)d[g(t)] nd g(t)d[f(t)]
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 9 where f 2 BV( b) nd g 2 G ( b): we prove some bsic theorems (integrtion - by - prts formul, convergence theorems, substitution theorem nd unsymmetric Fubini theorem) needed in the theory of Stieltjes integrl equtions in the spce G ( b): However, our rst tsk is the proof of existence of the integrl f dg for ny f 2 BV( b) nd ny g 2 G ( b): First, we will consider some simple specil cses. 2.1. Proposition. Let g 2 G ( b) be rbitrry. Then for ny 2 [ b] we hve (2.1) (2.2) (2.3) [ ] dg = g(+) ; g() [ ) dg = g(;) ; g() [ b] dg = g(b) ; g(;) (2.4) nd (2.5) ( b] dg = g(b) ; g(+) [ ] dg = g(+) ; g(;) where [) (t) (b] (t) 0 nd the convention g(;) =g() g(b+) = g(b) is used. Proof. Let g 2 G ( b) nd 2 [ b] be given. ) Let f = [ ] : It follows immeditely from the denition tht Z f dg = g() ; g(): In prticulr, 2.1 holds in the cse = b: Let 2 [ b) let ">0begiven nd let ( 1 j ; tj if <t b 2 " (t) = " if t = : It is esy to see tht ny " -ne prtition D =(d ) of[ b] must stisfy Consequently, Z b 1 = t 0 = t 1 <+ " nd S D (f g) =g(t 1 ) ; g(): nd f dg = g(+) ; g()
10 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 f dg = Z f dg + f dg = g() ; g()+g(+) ; g() =g(+) ; g() i.e. the reltion (2.1) is true for every 2 [ b]: b) Let f = [ ) : If = then f 0 g(;) ; g() = 0 nd (2.2) is trivil. Let 2 ( b]: For given ">0 let us dene guge " on [ ] by ( 1 j ; tj if t< 2 " (t) = " if t = : Then for ny " -ne prtition D =(d ) of[ ] we hve t m = m = t m;1 <; " nd S D (f g) =g(t m;1 ) ; g(): It follows immeditely tht Z nd in view of the obvious identity f dg = g(;) ; g() f dg =0 this implies (2.2). c) The remining reltions follow from 2.1, 2.2 nd the equlities nd [ b] = [ b] ; [ ) ( b] = [ b] ; [ ] [ ] = [ ] ; [ ) : 2.2. Remrk. Since ny nite step function is liner combintion of functions [ b] ( b) nd ( b] ( <b), it follows immeditely from Proposition 2.1 tht the integrl f dg exists for ny f 2 S( b) ndny g 2 G ( b): Other simple cses re covered by
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 11 2.3. Proposition. Let 2 [ b]: Then for ny function f :[ b] 7! R the following reltions re true (2.6) (2.7) (2.8) (2.9) nd (2.10) ( ;f() if <b f d [ ] = 0 if = b ( ;f() if > f d [ ) = 0 if = ( f() if > f d [ b] = 0 if = ( f() if <b f d ( b] = 0 if = b f d [ ] = 8 >< >: ;f() if = 0 if <<b f(b) if = b where [) (t) (b] (t) 0 nd the convention g(;) =g() g(b+) = g(b) is used. For the proof see [14, I.4.21 nd I.4.22]. 2.4. Corollry. Let W = fw 1 w 2 ::: w n g [ b] c 2 R nd h : [ b] 7! e such tht (2.11) Then (2.12) h(t) =c for ll t 2 [ b] n W: f dh = f(b)[h(b) ; c] ; f()[h() ; c] holds for ny function f :[ b] 7! R: Proof. A function h :[ b] 7! R fulls (2.11) if nd only if h(t) =c + nx [h(w j ) ; c] [w j ](t) on [ b]: Thus the formul (2.12) follows from (2.6) (with = b) nd from (2.10) in Proposition 2.3.
12 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 2.5. Remrk. It is well known (cf. [14, I.4.17] or [13, Theorem 1.22]) tht if g 2 BV( b) h :[ b] 7! R nd h n :[ b] 7! R R n 2 N re such tht h ndg exist b for ny n 2 N nd lim kh n ; hk =0 then hdg exists nd (2.13) lim h n dg = hdg holds. To prove n nlogous ssertion for the cse g 2 G ( b) we need the following uxiliry ssertion. 2.6. Lemm. Let f 2 BV( b) nd g 2 G ( b): The the inequlity (2.14) js D (f g)j ; jf()j + jf(b)j +vr b f kgk holds for n rbitrry prtition D of [ b]: Proof. For n rbitrry prtition D = (d ) of [ b] we hve (putting 0 = nd m+1 = b) js D (f g)j = jf(b)g(b) ; f()g() ; m+1 X [f( j ) ; f( j;1 )]g(t j;1 )j m+1 X jf(b)j + jf()j + jf( j ) ; f( j;1 )j kgk ; jf()j + jf(b)j +vr b f kgk: 2.7. Theorem. Let g 2 G ( b) nd let h n h:[ b] 7! e such tht h n dg exists for ny n 2 N nd lim kh n ; hk BV =0: Then hdg Proof. Since exists nd (2.13) holds. jf(b)j jf()j + jf(b) ; f()j jf()j +vr b f we hve by (2.14) js D ((h m ; h k )g)j 2kh m ; h k k BV kgk
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 13 for ll m k 2 N nd ll prtitions D of [ b]: Consequently, (h m ; h k )dg 2kh m ; h k k BV kgk holds for ll m k 2 N : This immeditely implies tht there is q 2 R such tht It remins to show tht lim h n dg = q: (2.15) For given ">0 let n 0 2 N (2.16) q = be such tht nd let " be such guge on [ b] tht (2.17) hdg: h n0 dg ; q <" nd kh n0 ; hk BV <" S D (h n0 g) ; h n0 dg <" for ll " -ne prtitions D of [ b]: Given n rbitrry " -ne prtition D of [ b] we hve by (2.16), (2.17) nd Lemm 2.6 jq ; S D (h g)j q ; h n0 dg + + S D (h n0 g) ; S D (h g) h n0 dg ; S D (h n0 g) 2" + js D ([h n0 ; h]g)j 2" +2kh n0 ; hk BV kgk 2" (1 + kgk) wherefrom the reltion (2.15) immeditely follows. This completes the proof of the theorem. Now we cn prove the following 2.8. Theorem. Let f 2 BV( b) nd g 2 G ( b): Then the integrl f dg
14 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 exists nd the inequlity (2.18) holds. f dg ; jf()j + jf(b)j +vr b f kgk Proof. Let f 2 BV( b) nd g 2 G ( b) be given. Let W = fw k g k2n be the set of discontinuities of f in [ b] nd let f = f C + f B be the Jordn decomposition of f (i.e. f C is continuous on [ b] nd f B is given by (1.2)). We hve lim kf B n ; f B k BV =0 for f B n n 2 N given by (1.3). By (2.3) nd (2.4), (2.19) f B n dg = nx + f(w k )(g(b) ; g(w k +)) + ; f(w k )(g(b) ; g(w k ;)) holds for ny n 2 N : Thus ccording to Theorem 2.7 the integrl nd f B dg exists (2.20) f B dg = lim f B n dg: The integrl f C dg exists s the -Riemnn-Stieltjes integrl by Proposition 1.2. This mens tht f dg exists nd f dg = f C dg + f B dg = The inequlity (2.18) follows immeditely from Lemm 2.6. f C dg + lim f B n dg: 2.9. Remrk. Since 1X + f(w k )(g(b) ; g(w k +)) + ; f(w k )(g(b) ; g(w k ;)) 2kgk 1X j + f(w k )j + j ; f(w k )j 2kgk(vr b f) < 1
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 15 we hve in virtue of (2.19) nd (2.20) (2.21) f B dg = 1X + f(w k )(g(b) ; g(w k +)) + ; f(w k )(g(b) ; g(w k ;)) : As direct consequence of Theorem 2.8 we obtin 2.10. Corollry. Let h n 2 G ( b) n 2 N nd h 2 G ( b) be such tht Then for ny f 2 BV( b) the integrls lim kh n ; hk =0: nd f dh lim nd f dh n = f dh n n 2 N f dh: exist 2.11. Lemm. Let h : [ b]! R c 2 R nd W = fw k g k2n [ b] be such tht (2.11) nd (2.22) 1X jh(w k ) ; cj < 1 hold. Given n 2 N let us put W n = fw 1 w 2 ::: w n g nd (2.23) h n (t) = ( c if t 2 [ b] n Wn h(t) if t 2 W n : Then h n 2 BV( b) for ny n 2 N h 2 BV( b) nd (2.24) lim kh n ; hk BV =0 Proof. The functions h n n2 N nd h evidently hve bounded vrition on [ b]. For given n 2 N we hve h n (t) ; h(t) = ( 0 if t 2 W n or t 2 [ b] n W c ; h n (t) if t = w k for some k>n:
16 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 Thus, (2.25) nd, moreover, lim h n(t) =h(t) on [ b] mx ; ; h n (t j ) ; h(t j ) ; h n (t j;1 ) ; h(t j;1 ) X X 1 2 k=n+1 jh(w k ) ; cj holds for ny n 2 N nd ny division ft 0 t 1 ::: t m g of [ b]: Consequently, (2.26) vr b (h n ; h) 2 1X k=n+1 jh(w k ) ; cj holds for ny n 2 N : In virtue of the ssumption (2.22) the right-hnd side of (2.26) tends to0sn!1: Hence (2.24) follows from (2.25) nd (2.26). 2.12. Proposition. Let h : [ b] 7! R c 2 R nd W = fw k g k2n be such tht (2.11) nd (2.22) hold. Then hdg = holds for ny g 2 G ( b): 1X [h(w k ) ; c]g(w k )+c [g(b) ; g()] Proof. Let g 2 G ( b) be given. Let W n = fw 1 w 2 ::: w n g for n 2 N nd let the functions h n n2n be given by (2.23). Given n rbitrry n 2 N then (2.1) (with = b) nd (2.5) from Proposition 2.1 imply Since (2.22) yields h n dg = nx [h(w k ) ; c]g(w k )+c[g(b) ; g()]: nx [h(wk ) ; c]g(w k ) X 1 2 kgk nd Lemm 2.11 implies we cn use Theorem 2.7 to prove tht hdg = lim lim kh n ; hk BV =0 h n dg = 1X jh(w k ) ; cj < 1 [h(w k ) ; c]g(w k )+c [g(b) ; g()]:
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 17 2.13. Proposition. Let h :[ b] 7! R c 2 R nd W = fw k g k2n full (2.11). Then (2.27) f dh = f(b)[h(b) ; c] ; f()[h() ; c] holds for ny f 2 BV( b): Proof. Let f 2 BV( b): For given n 2 N let W n = fw 1 w 2 ::: w n g nd let h n be given by (2.23). Then (2.28) Indeed, let ">0begiven nd let n 0 2 N lim kh n ; hk =0: be such tht k n 0 implies (2.29) jh(w k ) ; cj <": (Such n n 0 exists since jh(w k ) ; cj = j ; h(w k )j = j + h(w k )j for ny k 2 N nd the set of those k 2 N for which the inequlity (2.29) does not hold my be only nite.) Now, for ny n n 0 nd ny t 2 [ b] such tht t = w k for some k > n (t 2 W n W n )we hve jh n (t) ; h(t)j = jh n (w k ) ; h(w k )j = jc ; h(w k )j <": Since h n (t) =h(t) for ll the other t 2 [ b] (t 2 ; [ b] n W [ W n ), it follows tht jh n (t) ; h(t)j <"on [ b] i.e. kh n ; hk <": This proves the reltion (2.28). By Corollry 2.4 we hve for ny n 2 N f dh n = f(b)[h(b) ; c] ; f()[h() ; c]: Mking use of (2.28) nd Corollry 2.10 we obtin we obtin f dh = lim f dh n = f(b)[h(b) ; c] ; f()[h() ; c]: 2.14. Corollry. Let h 2 BV( b) c 2 R nd W = fw k g k2n full (2.11). Then (2.27) holds for ny f 2 G ( b): Proof. By Proposition 2.12, (2.27) holds for ny f 2 BV( b): Mking use of the density of S( b) BV( b) in G ( b) nd of the convergence theorem mentioned in Remrk 2.5 we complete the proof of our ssumption.
18 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 2.15. Theorem. (Integrtion-by-prts) If f 2 BV( b) nd g 2 G ( b) then both the integrls R f dg nd b gdf exist nd (2.30) f dg + + X gdf = f(b)g(b) ; f()g() t2[ b] ; f(t) ; g(t) ; + f(t) + g(t) : Proof. The existence of the integrl gdf is well known, while the existence of R b gdf is gurnteed by Theorem 2.8. Furthermore, = f dg + ; gdf f(t)d[g(t)+ + g(t)] + f(t)d[ + g(t)] + g(t)d[f(t) ; ; f(t)] g(t)d[ ; f(t)]: It is esy to verify tht the function h(t) = + g(t) fulls the reltion (2.11) with c = 0 nd h(b) = 0: Consequently, Proposition 2.13 yields Similrly, by Corollry 2.14 we hve Hence f(t)d[ + g(t)] = ;f() + g(): g(t)d[ ; f(t)] = ; f(b)g(b): (2.31) f dg + g(t)df = f(t)d[g(t+)] + g(t)d[f(t;)] + f() + g()+ ; f(b)g(b): The rst integrl on the right-hnd side my be modied in the following wy: (2.32) f(t)d[g(t+)] = f(t;)d[g(t+)] + ; f(t)d[g(t+)]:
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 19 Mking use of Proposition 2.12 nd tking into ccount tht g 1 (t) = g(t) on [ b] for the function g 1 dened by g 1 (t) =g(t+) on [ b] we further obtin X (2.33) ; f(t)d[g(t+)] = ; f(t)g(t): Similrly, t2[ b] (2.34) g(t)d[f(t;)] = = g(t+)df(t;) ; g(t+)d[f(t;)] ; X + g(t)d[f(t;)] t2[ b] + g(t)f(t): The function f(t;) is left-continuous on [ b] while g(t+) is right-continuous on [ b): It mens tht both the integrls f(t;)d[g(t+)] nd g(t+)d[f(t;)] exist s the -Riemnn-Stieltjes integrls (cf. Corollry 1.3), nd mking use of the integrtion-by-prts theorem for these integrls (cf. [4, Theorem II.11.7]) we get (2.35) f(t;)d[g(t+)] + Inserting (2.32) - (2.35) into (2.31) we get f dg + nd this completes the proof + X g(t+)d[f(t;)] = f(b;)g(b) ; f()g(+): gdf = f(b;)g(b) ; f()g(+) t2[ b] ; f(t)[ ; g(t)+ + g(t)] X ; [ ; f(t)+ + f(t)] + g(t) t2[ b] + f() + g()+ ; f(b)g(b) = f(b)g(b) ; f()g() X + t2[ b] ; f(t) ; g(t) ; + f(t) + g(t)
20 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 The following proposition describes some properties of indenite Perron-Stieltjes integrls. 2.16. Proposition. Let f : [ b] 7! R nd g : [ b] 7! e such tht exists. Then the function f dg h(t) = Z t f dg is dened on [ b] nd (i) if g 2 G ( b) then h 2 G ( b) nd (2.36) + h(t) =f(t) + g(t) ; h(t) =f(t) ; g(t) on [ b] (ii) if g 2 BV( b) nd f is bounded on [ b] then h 2 BV( b): Proof. The former ssertion follows from [8, Theorem 1.3.5]. immeditely from the inequlity mx Z tj tj;1 f dg mx kfk (vr tj tj;1 g) = kfk(vr b g) which is vlid for ny division ft 0 t 1 ::: t m g of [ b]: (2.37) The ltter follows In the theory of generlized dierentil equtions the substitution formul h(t)d hz t f(s)d[g(s)] i = h(t)f(t)d[g(t)] is often needed. In [4, II.19.3.7] this formul is proved for the -Young-Stieltjes integrl under the ssumption tht g 2 G ( b) h is bounded on [ b] nd the integrl f dg s well s one of the integrls in (2.37) exists. In [14, Theorem I.4.25] this ssertion ws proved for the Kurzweil integrl. Though it ws ssumed there tht g 2 BV( b) this ssumption ws not used in the proof. We will givehere slightly dierent proof bsed on the Sks-Henstock lemm (cf. e.g. [13, Lemm 1.11]). 2.17. Lemm. (Sks-Henstock) Let f g : [ b] 7! e such tht the integrl f dg exists. Let ">0 be given nd let be guge on [ b] such tht S D (f g) ; f dg <"
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 21 holds for ny -ne prtition D of [ b]: Then for n rbitrry system f([ i i ] i ) i =1 2 ::: kg of intervls nd points such tht (2.38) nd the inequlity (2.39) 1 1 1 2 k k k b [ i i ] [ i ; ( i ) i + ( i )] i =1 2 ::: k kx h i=1 Z i f( i )[g( i ) ; g( i )] ; i f dgi <" holds. Mking use of Lemm 2.17 we cn prove the following useful ssertion 2.18. Lemm. If f : [ b] 7! R nd g : [ b] 7! R re such tht then for ny ">0 there exists guge on [ b] such tht mx Z t j f( j )[g(t j ) ; g(t j;1 (2.40) )] ; f dg <" tj;1 holds for ny -ne prtition (d ) of [ b]: f dg exists, Proof. Let :[ b] 7! (0 1) be such tht Z b S D (f g) ; f dg X Z = m t j f( j )[g(t j ) ; g(t j;1 )] ; tj;1 f dg < " 2 for ll -ne prtitions D =(d )of[ b]: Let us choose n rbitrry -ne prtition D = (d ) of [ b]: Let i = t p i nd i = t p i;1 i = 1 2 ::: k be ll the points of the division d such tht Z i f( p i)[g( i ) ; g( i )] ; f dg 0: i Then the system f([ i i ] i ) i = 1 2 ::: kg where i = p i fulls (2.38) nd (2.39) nd hence we cn use Lemm2.17toprove tht the inequlity kx Z i f( p i)[g( i ) ; g( i )] ; f dg < " 2 i i=1 is true. Similrly, if! i = t q i nd i = t q i;1 i = 1 2 ::: r re ll points of the division d such tht Z! i f( q i)[g(! i ) ; g( i )] ; f dg 0 i
22 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 the the inequlity rx i=1 Z! i f( q i)[g(! i ) ; g( i )] ; f dg < " 2 i follows from Lemm 2.17, s well. Summrizing, we conclude tht mx Z t j f( j )[g(t j ) ; g(t j;1 )] ; f dg tj;1 = kx i=1 + Z i f( p i)[g( i ) ; g( i )] ; f dg i Z! i f( q i)[g(! i ) ; g( i )] ; f dg i rx i=1 < " + " = ": 2 2 This completes the proof. 2.19. Theorem. Let h :[ b] 7! e bounded on [ b] nd let f g :[ b] 7! e such tht the integrl f dg exists. Then the integrl h(t)f(t)d[g(t)] exists if nd only if the integrl hz t i h(t)d f(s)d[g(s)] exists, nd in this cse the reltion (2.37) holds. Proof. Let jh(t)j C < 1 on [ b]: Let us ssume tht the integrl h(t)f(t)d[g(t)] exists nd let ">0begiven. There exists guge 1 on [ b] such tht X m h( k )f( j )[g(t j ) ; g(t j;1 )] ; h(t)f(t)d[g(t)] < " 2
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 23 is stised for ny 1 -ne prtition (d ) of [ b]: By Lemm 2.18 there exists guge on [ b] such tht (t) 1 (t) on [ b] nd mx Z t j f( j )[g(t j ) ; g(t j;1 )] ; f dg < " 2C tj;1 holds for ny -ne prtition (d ) of[ b]: Let us denote k(t) = Z t f dg for t 2 [ b]: Then for ny -ne prtition D =(d ) of[ b] we hve S D (h k) ; X = m + mx X m + X m h(t)f(t)d[g(t)] Z t j mx h( j ) f dg ; h( j )f( j )[g(t j ) ; g(t j;1 )] tj;1 h( j )f( j )[g(t j ) ; g(t j;1 )] ; h( j ) hz t j tj;1 h(t)f(t)d[g(t)] f dg ; f( j )[g(t j ) ; g(t j;1 )]i h( j )f( j )[g(t j ) ; g(t j;1 )] ; This implies the existence of the integrl hdk hf dg <" nd the reltion (2.37). wy. The second impliction cn be proved in n nlogous The convergence result 2.10 enbles us to extend the known theorems on the chnge of integrtion order in iterted integrls Z d h i Z d (2.41) g(t)d h(t s)d[f(s)] g(t)d t [h(t s)] d[f(s)] c where ;1 < c < d < 1 nd h is of strongly bounded vrition on [c d] [ b] (cf. [14, Theorem I.6.20]). In wht follows v(h) denotes the Vitli vrition of the c
24 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 function h on [c d] [ b] (cf. [4, Denition III.4.1] or [14, I.6.1]). For given t 2 [c d] vr b h(t :) denotes the vrition of the function s 2 [ b] 7! h(t s) 2 R on [ b]: Similrly,fors 2 [ b] xed, vr d ch(: s) stnds for the vrition of the function t 2 [c d] 7! h(t s) 2 R on [c d]: 2.20. Theorem. Let h :[c d] [ b] 7! e such tht v(h)+vr b h(c :)+vrd ch(: ) < 1: Then for ny f 2 BV( b) nd ny g 2 G (c d) both the integrls (2.41) exist nd (2.42) Z d c g(t)d h h(t s)d[f(s)] i = Z d c g(t)d t [h(t s)] d[f(s)]: Proof. Let us notice tht by [14, Theorem I.6.20] our ssertion is true if g is lso supposed to be of bounded vrition. in the generl cse of g 2 G ( b) there exists sequence fg n g 1 n=1 S( b)such tht lim kg ; g n k =0: Then, since the function v(t) = h(t s)d[f(s)] is of bounded vrition on [c d] (cf. the rst prt of the proof of [14, Theorem I.6.20]), the integrl on left-hnd side of (2.42) exists nd by Corollry 2.10 nd [14, Theorem I.6.20] we hve (2.43) Z d c Let us denote g(t)d w n (t) = h Z d c h(t s)d[f(s)] i Z d h i = lim g n (t)d h(t s)d[f(s)] c Z d = lim c g n (t)d t [h(t s)] g n (t)d t [h(t s)] for s 2 [ b] nd n 2 N : d[f(s)]: Then w n 2 BV( b) for ny n 2 N (cf. [14, Theorem I.6.18]) nd by [14, Theorem I.4.17] mentioned here in Remrk 2.5 we obtin lim w n(s) = Z d c g n (t)d t [h(t s)] := w(s) on [ b]: As ; jw n (s) ; w(s)j kg n ; gk vr d c h(: s) ; kg n ; gk v(h)+vr d ch(: )
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 25 for ny s 2 [ b] (cf. [14, Lemm I.6.6]), we hve lim kw n ; wk =0: It mens tht w 2 G ( b) nd by Theorem 2.8 the integrl w(s)d[f(s)] = exists s well. Since obviously lim Z d c Z d g n (t)d t [h(t s)] d[f(s)] = lim = the reltion (2.42) follows from (2.43). c g(t)d t [h(t s)] w(s)d[f(s)] = w n (s)d[f(s)] Z d c d[f(s)] g(t)d t [h(t s)] d[f(s)] 3. Liner bounded functionls on G L (,b) By Theorem 2.8 the expression (3.1) F (x) =qx()+ pdq is dened for ny x 2 G ( b) nd ny =(p q) 2 BV( b) R: Moreover, for ny 2 BV( b) R the reltion (3.1) denes liner bounded functionl on G L ( b): Proposition 2.3 immeditely implies 3.1. Lemm. Let =(p q) 2 BV( b) e given. Then (3.2) F ( [ b] ) = q F ( ( b] )=p() for ny 2 [ b) F ( [b] ) = p(b): 3.2. Corollry. If = (p q) 2 BV( b) R nd F (x) = 0 for ll x 2 S( b) which re left-continuous on ( b) then p(t) 0 on [ b] nd q =0:
26 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 3.3. Lemm. Let x 2 G ( b) be given. Then for given =(p q) 2 BV( b)r (3.3) F (x) =x() if p 0 on [ b] nd q =1 F (x) =x(b) if p 1 on [ b] nd q =1 F (x) =x(;) if p = [ ) on [ b] 2 ( b] nd q =1 F (x) =x(+) if p = [ ] on [ b] 2 [ b) nd q =1: Proof follows from Proposition 2.1. 3.4. Corollry. If x 2 G ( b) nd F (x) = 0 for ll = (p q) 2 BV( b) R then (3.4) x() =x(+) = x(;) =x(+) = x(b;) =x(b) holds for ny 2 ( b): In prticulr, if x 2 G L ( b) (x is left-continuous on ( b)) nd F (x) =0for ll =(p q) 2 BV( b) R then x(t) 0 on [ b]: 3.5. Remrk. The spce BV( b) R is supposed to be equipped with the usul norm (kk BVR = jqj + kpk BV for = (p q) 2 BV( b) R). Obviously, it is Bnch spce with respect to this norm. 3.6. Proposition. The spces G L ( b) nd BV( b) R form dul pir with respect to the biliner form (3.5) x 2 G L ( b) 2 BV( b) R 7! F (x): Proof follows from Corollries 3.2 nd 3.4. On the other hnd, we hve 3.7. Lemm. If F is liner bounded functionl on G L ( b) nd (3.6) p(t) = ( F ((t b] ) if t 2 [ b) F ( [b] ) if t = b then p 2 BV( b) nd (3.7) where jp()j + jp(b)j +vr b p 2kF k kf k = sup jf (x)j: x2g L ( b) kxk1
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 27 Proof is nlogous to tht of prt c (i) of [5, Theorem 5.1]. Indeed, for n rbitrry division ft 0 t 1 ::: t m g of [ b] we hve sup jcjj1 cj2r = sup jcjj1 cj2r p()c 0 + p(b)c m+1 + mx [p(t j ) ; p(t j;1 )]c j m;1 X F (c 0 ( b] + c m+1 [b] ; sup jf (h)j =2kF k: khk2 h2g L ( b) c j (t j;1 tj ] + c m (t m;1 b)) In prticulr, for c 0 = sgn p() c m+1 = sgn p(b) nd c j = sgn(p(t j ) ; p(t j;1 )) j =1 2 ::: m we get jp()j + jp(b)j + mx nd the inequlity (3.7) immeditely follows. jp(t j ) ; p(t j;1 )j2kf k Using the ides from the proof of [5, Theorem 5.1] we my now prove the following representtion theorem. 3.8. Theorem. F is liner bounded functionl on G L ( b) (F 2 G L( b)) if nd only if (3.8) The mpping there is n =(p q) 2 BV( b) R such tht F (x) =F (x) is n isomorphism. := qx()+ pdx for ny : 2 BV( b) R 7! F 2 G L( b) x 2 G L ( b): Proof. Let liner bounded functionl F on G L ( b) be given nd let us put (3.9) q = F ( [ b] ) nd p(t) = ( F ((t b] ) if t 2 [ b) F ( [b] ) if t = b: Then Lemm 2.6 implies =(p q) 2 BV( b) R nd by Lemm 3.1 we hve nd F ( [ b] )=F ( [ b] ) F ( (t b] ) = F ( (t b] ) for ny t 2 [ b)
28 sopis p st. mt. 114 (1989), No. 2, pp. 187-209 F ( [b] ) = F ( [b] ): Since ll functions from S( b)\ G L ( b) obviously re nite liner combintions of the functions [ b] ( b] 2 [ b) [b] it follows tht F (x) =F (x) holds for ny x 2 S( b) \ G L ( b): Now, the density of S( b) \ G L ( b) in G L ( b) implies tht F (x) =F (x) for ll x 2 G L ( b): This completes the proof of the rst ssertion of the theorem. Given n x 2 G L ( b) then Lemm 2.6 yields nd consequently, jf (x)j ; jp()j + jp(b)j +vr b p + jqj kxk kf kjp()j + jp(b)j +vr b p + jqj 2; kpk BV + jqj =2kk BVR : On the other hnd, ccording to Lemm 3.7 we hve kpk BV ; jp()j + jp(b)j +vr b p 2kF k: Furthermore, in virtue of (3.9) we hve jqj kf k nd hence It mens tht kk BVR = kpk BV + jqj 2kF k: 1 kf kkk 2 BVR 3kF k nd this completes the proof of the theorem. References [1] Aumnn G., Reelle Funktionen, (Springer-Verlg, Berlin, 1969). [2] Fr kov D., Continuous dependence on prmeter of solutions of generlized dierentil equtions, sopis p st. mt., 114 (1989), 230{261. [3] Fr kov D., Regulted functions, Mth. Bohem. 116 (1991), 20{59.
M.Tvrd : Regulted functions nd the Perron-Stieltjes integrl 29 [4] Hildebrndt T. H., Introduction to the Theory of Integrtion, (Acdemic Press, New York- London, 1963). [5] H nig Ch. S., Volterr Stieltjes-Integrl Equtions, (North Hollnd nd Americn Elsevier, Mthemtics Studies 16, Amsterdm nd New York, 1975). [6] H nig Ch. S., Volterr-Stieltjes integrl equtions, in Functionl Dierentil Equtions nd Bifurction, Proceedings of the So Crlos Conference 1979 (Lecture Notes in Mthemtics 799, Springer-Verlg, Berlin, 1980), pp. 173{216. [7] Kltenborn H. S., Liner functionl opertions on functions hving discontinuities of the rst kind, Bulletin A. M. S. 40 (1934), 702{708. [8] Kurzweil J., Generlized ordinry dierentil equtions nd continuous dependence on prmeter, Czechoslovk Mth. J. 7(82)(1957), 418{449. [9] Kurzweil J., Nichtbsolute konvergente Integrle, (BSB B. G. Teubner Verlgsgesselschft, Leipzig, 1980). [10] Sks S., Theory of the Integrl, (Monogre Mtemtyczne, Wrszw, 1937), [11] Schwbik., On the reltion between Young's nd Kurzweil's concept of Stieltjes integrl, sopis P st. Mt. 98 (1973), 237{251. [12] Schwbik., On modied sum integrl of Stieltjes type, sopis P st. Mt. 98 (1973), 274{ 277. [13] Schwbik., Generlized Dierentil Equtions (Fundmentl Results), (Rozprvy SAV, d MPV, 95 (6)) (Acdemi, Prh, 1985). [14] Schwbik., Tvrd M., Vejvod O., Dierentil nd Integrl Equtions: Boundry Vlue Problems nd Adjoints, (Acdemi nd D. Reidel, Prh nd Dordrecht, 1979). [15] Wrd A. J., The Perron-Stieltjes integrl, Mth. Z. 41 (1936), 578{604.