Existence and uniqueness of solutions for linear Fredholm-Stieltjes integral equations via Henstock-Kurzweil integral

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1 Existence nd uniqueness of solutions for liner Fredholm-Stieltjes integrl equtions vi Henstock-Kurzweil integrl M. Federson nd R. Binconi Astrct We consider the liner Fredholm-Stieltjes integrl eqution x (t) α (t, s) x (s) dg(s) = f (t), t [, ], in the frme of the Henstock-Kurzweil integrl nd we prove Fredholm Alterntive type result for this eqution. The functions α, x nd f re rel-vlued with x nd f eing Henstock-Kurzweil integrle. Keywords nd phrses: Liner integrl equtions, Fredholm integrl equtions, Fredholm Alterntive, Henstock-Kurzweil integrl Mthemticl Suject Clssifiction: 45A05, 45B05, 45C05, 26A39. 1 Introduction The purpose of this pper is to prove Fredholm Alterntive type result for the following liner Fredholm-Stieltjes integrl eqution x (t) α (t, s) x (s) dg(s) = f (t), t [, ], (1) in the frme of the Henstock-Kurzweil integrl. Let g : [, ] R e n element of certin suspce of the spce of continuous functions from [, ] to R. Let K g ([, ], R) denote the spce of ll functions f : [, ] R such tht Instituto de Ciêncis Mtemátics e de Computção, Universidde de São Pulo, CP 688, São Crlos, SP , Brzil. E-mil: federson@icmc.usp.r Instituto de Mtemátic e Esttstic, Universidde de São Pulo, CP 66281, São Pulo, SP , Brzil. E-mil: inconi@ime.usp.r 1

2 the integrl f(s)dg(s) exists in the Henstock-Kurzweil sense. It is known tht even when g(s) = s, n element of K g ([, ], R) cn hve not only mny points of discontinuity, ut it cn lso e of unounded vrition. The Henstock-Kurzweil integrl encompsses the Leesgue, Riemnn nd Newton integrls s well s its improper integrls. When (1) is considered in the sense of Henstock-Kurzweil, f cn hve singulrities, for instnce, nd yet it my e possile to find solution x f in K g ([, ], R). In [4], the uthors considered the liner Fredholm integrl eqution x (t) α (t, s) x (s) ds = f (t), t [, ], nd estlished Fredholm Alterntive for such eqution in the frme of the Henstock- Kurzweil integrl (see [4], Theorem 3.10). On tht occsion, the functions α, x nd f were considered s eing Bnch-spce vlued nd x nd f were Kurzweil integrle. The ltter could lso e considered in ny of the spces of Henstock, Bochner or improper Riemnn integrle functions (see [4], Remrks 3.2 nd 3.3). Also, n exmple ws given showing tht, under the hypothesis we considered, the existing results were not enough. Suppose the function α stisfies certin conditions. In the present pper, we prove tht: if for every f K g ([, ], R), eqution (1) hs unique solution x f K g ([, ], R), then the ppliction f x f is icontinuous nd eqution (1) hs resolvent with similr integrl representtion; if esides α(t, s) = 0 for m-lmost every s > t, then f x f is cusl nd the resolvent of (1) is given y the Neumnn series (see Theorem 5.1 in the sequel). We lso stte the following Fredholm Alterntive for eqution (1): for every f K g ([, ], R), either eqution (1) hs unique solution x f K g ([, ], R) nd (1) hs resolvent with similr integrl representtion, or the corresponding homogeneous eqution dmits non-trivil solutions in K g ([, ], R) (see Theorem 5.2 in the sequel). Corollry 5.1 in the sequel mixes the two results. Although the ove results re proved in the cse where α, x nd f re rel-vlued, the uxiliry theory developed throughout the pper is plced in generl strct context. As in [6], the reson for considering functions tking vlues in finite dimensionl spce is ecuse some results of the Kurzweil-Henstock integrtion theory either hold for one integrl or for the other. For instnce, in Bnch-spce vlued context, the spces of Henstock nd of Kurzweil integrls my not coincide; Riemnn integrle function my not e Henstock integrle; the Fundmentl Theorem of Clculus my fil for the Kurzweil integrl. See [7], [8] nd [3] for exmples. See lso [18]. Another ostcle we encountered is the fct tht the normed spce of Henstock-Kurzweil integrle functions is not complete (it is ultrornologicl though). Therefore one cn not 2

3 pply usul fixed point theorems in order to get existence results, for instnce. This difficulty ws lso fced y the uthors in [4] nd in [6] nd y Hönig in [11], ut it cn e overcome when some ides due to Hönig re pplied. Such ides concern pplying representtion theorems, n integrtion y prts formul nd the fundmentl theorem of clculus in order to trnsform integrl equtions in the sense of the Henstock-Kurzweil integrl into integrl equtions in the sense of the Riemnn-Stieltjes integrl. As we did in [6] for Volterr-Stieltjes type integrl eqution, we dpt, in the present pper, the ides mentioned ove to the Stieltjes cse, tht is, y mens of representtion theorems, integrtion y prts nd sustitution formuls nd the fundmentl theorem of clculus for Henstock-Kurzweil vector integrls, we trnsform the Fredholm-Stieltjes integrl eqution in the sense of Henstock-Kurzweil integrl into n ordinry Fredholm-Stieltjes integrl eqution. In prticulr, we pply the Fredholm Alterntive for the Riemnn-Stieltjes integrl (proved y the uthors in [4], Theorems 2.4 nd 2.5) to otin Fredholm Alterntive for our eqution (1). This result is presented in Section 5. The other sections re orgnized s follows. Section 2 is devoted to the fundmentl theory of the Riemnn-Stieltjes integrl in Bnch spces, where we present sic results, representtion theorems nd the Fredholm Alterntive. In Section 3, we give some sic definitions of the Henstock-Kurzweil integrtion theory. In Section 4, we present uxiliry results for the Henstock-Kurzweil integrl such s the fundmentl theorem of clculus nd sustitution formul. 2 The Riemnn-Stieltjes integrl in Bnch spces In this section we introduce the definitions nd terminology tht we re going to use throughout the pper for Riemnn-Stieltjes integrls nd we mention some properties of these integrls. We strt y defining functions of ounded B-vrition, of ounded semi-vrition nd of ounded vrition nd giving some results concerning these spces. 2.1 Functions of ounded B-vrition, of ounded semi-vrition nd of ounded vrition A iliner triple (we write BT ) is set of three vector spces E, F nd G, where F nd G re normed spces with iliner mpping B : E F G. For x E nd y F, we write xy = B (x, y) nd we denote the BT y (E, F, G) B or simply y (E, F, G). A topologicl BT is BT (E, F, G) where E is lso normed spce nd B is continuous. We suppose tht B 1. We denote y L (E, F ) the spce of ll liner continuous functions from E to F, where E nd F re normed spces. We write L (E) = L (E, E) nd E = L (E, R), where R denotes the rel line. Throughout this pper, X, Y nd Z will lwys denote Bnch spces. 3

4 Exmple 2.1. As n exmple of BT we cn consider E = L (X, Y ), F = L (Z, X), G = L (Z, Y ) nd B (v, u) = v u. In prticulr, when Z = R, we hve E = L (X, Y ), F = X, G = Y nd B (u, x) = u (x); when X = R, we hve E = Y, F = Y, G = R nd B (y, y ) = y, y ; when X = Z = R, we hve E = G = Y, F = R nd B (y, λ) = λy. Let [, ] e compct intervl of R. Any finite set of closed non-overlpping suintervls [t i 1, t i ] of [, ] such tht the union of ll intervls [t i 1, t i ] equls [, ] is clled division of [, ]. In this cse we write d = (t i ) D [,], where D [,] denotes the set of ll divisions of [, ]. By we men the numer of suintervls in which [, ] is divided through given d D [,]. Given BT (E, F, G) B nd function α : [, ] E, for every division d = (t i ) D [,] we define SB d (α) = SB [,],d (α) = sup [α (t i ) α (t i 1 )] y i ; y i F, y i 1 nd SB (α) = SB [,] (α) = sup { SB d (α) ; d D [,] }. Then SB (α) is the B-vrition of α on [, ]. We sy tht α is function of ounded B-vrition whenever SB (α) <. When this is the cse, we write α SB ([, ], E). The following properties re not difficult to prove: (SB1) SB ([, ], E) is vector spce nd the mpping α SB ([, ], E) SB (α) R + is seminorm; (SB2) for α SB ([, ], E), the function t [, ] SB [,t] (α) R + incresing; is monotoniclly (SB3) for α SB ([, ], E) nd c [, ], SB [,] (α) SB [,c] (α) + SB [c,] (α) Consider the BT (L (X, Y ), X, Y ). In this cse, we write SV ([, ], L (X, Y )) nd SV (α) respectively y SB ([, ], L (X, Y )) nd SB (α), nd ny element of SV ([, ], L (X, Y )) is clled function of ounded semi-vrition. Given function α : [, ] E, normed spce E nd division d = (t i ) D [,], we define V d (α) = V d,[,] (α) = α (t i ) α (t i 1 ) nd the vrition of α is given y V (α) = V [,] (α) = sup { V d (α) ; d D [,] }. 4

5 If V (f) <, then α is clled function of ounded vrition. α BV ([, ], E). We lso hve In this cse, we write BV ([, ], L (E, F )) SV ([, ], L (E, F )) nd SV ([, ], L (E, R)) = BV ([, ], E ). Remrk 2.1. Consider BT (E, F, G). The definition of vrition of function α : [, ] E, where E is normed spce, cn lso e considered s prticulr cse of the B-vririon of α in two different wys. Let E = F, G = R or G = C nd B(x, x) = x, x. By the definition of the norm in E = F, we hve V d (α) = α (t i ) α (t i 1 ) = sup x i, α (t i ) α (t i 1 ) ; x i F, x i 1 = SB d(α). Thus when we consider the BT (Y, Y, R), we write BV (α) nd BV ([, ], Y ) insted of SB (α) nd SB ([, ], Y ) respectively. Let F = E, G = R or G = C nd B(x, x ) = x, x. By the Hhn-Bnch Theorem, we hve nd hence α (t i ) α (t i 1 ) = sup { α (t i ) α (t i 1 ), x i ; x i E, x i 1} V d (α) = α (t i ) α (t i 1 ) = sup α (t i ) α (t i 1 ), x i ; x i E, x i 1 = SB d(α). Given c [, ], we define the spces BV c ([, ], X) = {f BV ([, ], X) ; f (c) = 0} nd SV c ([, ], L (X, Y )) = {α SV ([, ], L (X, Y )) ; α (c) = 0}. 5

6 Such spces re complete when endowed, respectively, with the norm given y the vrition V (f) nd the norm given y the semi-vrition SV (α). See [19], for instnce. The next results re orrowed from [15]. We include the profs here, since this reference is not esily ville. Lemms 2.1 nd 2.2 elow re respectively Theorems I.2.7 nd I.2.8 from [15]. Lemm 2.1. Let α BV ([, ], X). Then (i) For ll t ], ], there exists α(t ) = lim ε 0 α(t ε). (ii) For ll t [, [, there exists α(t+) = lim ε 0 α(t + ε). Proof. We will prove (i). The proof of (ii) follows nlogously. Consider strictly incresing sequence {t n } in [, t[ converging to t. Then Hence n α(t i ) α(t i 1 ) V [,t] (α), for ll n. α(t i ) α(t i 1 ) V [,t] (α). Then {α(t n )} is Cuchy sequence, since α(t m ) α(t n ) m i=n+1 α(t i ) α(t i 1 ) ε, for sufficiently lrge m, n. The limit α(t ) of {α(t n )} is independent of the choice of {t n } nd we finish the proof. Lemm 2.2. Let α BV ([, ], X). For every t [, ], let v(t) = V [,t] (α). Then (i) v(t+) v(t) = α(t+) α(t), t [, [. (ii) v(t) v(t ) = α(t) α(t ), t ], ]. Proof. By property (SB2), v is monotoniclly incresing nd hence v(t+) nd v(t ) exist. By Lemm 2.1, α(t+) nd α(t ) lso exist. We will prove (i). The proof of (ii) follows nlogously. Suppose s > t. Then property (SB3) implies V [,s] (α) V [,t] (α) + V [ t,s] (α). Therefore α(s) α(t) V [t,s] (α) = V [,s] (α) V [,t] (α) nd hence α(t+) α(t) v(t+) v(t). 6

7 Conversely, given d D [,t], let v d (t) = V d (α). Then for every ε > 0, there exists δ > 0 such tht v(t + σ) v(t+) ε nd α(t + σ) α(t+) ε nd there exists d : = t 0 < t 1 <... < t n = t < t n+1 = t + σ such tht v(t + σ) v d (t + σ) ε whenever 0 < σ δ. Thus v(t + σ) v(t) v d (t + σ) + ε v d (t) = α(t + σ) α(t) + ε α(t+) α(t) + 2ε nd hence v(t+) v(t) α(t+) α(t). This completes the proof. Let α BV ([, ], X). Since α(t) α() + V [,t], then Lemm 2.2 implies tht the sets {t [, [; α(t+) α(t) ε} nd {t ], ]; α(t) α(t ) ε} re finite for every ε > 0. Thus we hve the next result which cn e found in [15], for instnce (Proposition I.2.10 there). Proposition 2.1. Let α BV ([, ], X). Then the set of points of discontinuity of α is countle (nd ll discontinuities re of the first kind). Let us define BV + ([, ], X) = {α BV ([, ], X); α(t+) = α(t), t ], [}. A proof tht BV + ([, ], X) with the vrition norm is complete cn e found in [15], Theorem I We reproduce it next. Theorem 2.1. BV + ([, ], X) is Bnch spce when endowed with the vrition norm. Proof. For every t ], [, we hve α(t) α() + V (α) = V (α). Hence the mppings T t : α BV ([, ], X) α(t) X nd T t+ : α BV ([, ], X) α(t+) X re continuous. Therefore BV + ([, ], X) is closed suspce of BV ([, ], X), since it is given y the continuous mppings T t nd T t+, t ], [, nd the result follows from the fct tht BV ([, ], X) is Bnch spce with the vrition norm. Given u L(X, Z ) nd z Z, we denote n element of X y z u which is given y z u, x = z, u(x), x X. We hve z u, x = z, u(x) z u(x) z u x. We denote y u L (Y, X ) the djoint or trnsposed opertor of u L(X, Y ) which is defined y x, u (y ) = u (x), y, x X, y Y. 7

8 Then y u = u (y ), y Y since (y u) (x) = y, u(x) = u (y ), x for every x X. Next we hve [15], Proposition I.3.5 nd the corollry tht follows it. Proposition 2.2. Given function α : [, ] L(X, Z ). Then (i) SV (α) = sup{v (z α); z Z, z 1}; (ii) α SV ([, ], L(X, Z )) if nd only if z α BV ([, ], X ), for every z Z. Proof. In order to prove (i), it is enough to oserve tht, if d = (t i ) D [,], then SV d (α) = sup [α(t i ) α(t i 1 )]x i ; x i X, x i 1 = sup sup z, [α(t i ) α(t i 1 )]x i z 1 ; x i X, x i 1 = sup sup x i, z α(t i ) z α(t i 1 ) ; x i X, x i 1 z 1 = sup {V d (z α); z Z, z 1}. Now we will prove (ii). By (i), if α SV ([, ], L(X, Z )), then z α BV ([, ], X ), z Z. The converse follows y the uniform oundedness principle. Indeed. Let us define D = { (d, x); d D [,], x = (x 1,..., x ), x i X, x i 1 }. Then for ech (d, x) D, we define F d,z Z y F d,z (z) = z, [α(t i ) α(t i 1 )]x i, z Z. Then the set {F d,z ; (d, z) D} Z is simply ounded on Z, since F d,z (z) V d (z α) V (z α), for ll (d, z) D, nd ll z Z. Therefore y uniform oundedness principle, there exists M 0 such tht F d,x M, (d, z) D, tht is, for ll (d, z) D, we hve sup { F d,z (z) ; z Z, z 1} M. Hence SV (α) M. Corollry 2.1. Suppose α SV ([, ], L(X, Z )). Then (i) For every t ], ], there exists α(t ) L(X, Z ) in the sense tht lim (z α)(t ε) = z α(t ), z Z; ε 0 + 8

9 (ii) For every t [, [, there exists α(t+) L(X, Z ) in the sense tht lim (z α)(t + ε) = z α(t+), z Z. ε 0 + Proof. We will prove (i). The proof of (ii) follows nlogously. We my suppose, without loss of generlity, tht α() = 0. Proposition 2.2 (ii), given z Z, there exists By Lemm 2.1 (i) nd T z = lim ε 0+ (z α)(t ε) X. Then the mpping T : z Z T z X is liner. It is lso continuous, since z α(t) z α() + V [,t] (z α). Besides, Proposition 2.2 (i) implies (z α)(t ε) V (z α) z SV (α). Hence T SV (α). Let t T e the trnsposed mpping of T, tht is, t T : x X t T x Z is defined y z, t T x = T z, x, where z Z. Then t T L(X, Z ) nd t T = T SV (α). Also t T = α(t ) in the sense of (i), since for every x X, we hve z t T, x = z, t T x = T z, x = lim ε 0+ (z α)(t ε), x = lim ε 0+ (z α)(t ε), x nd we finished the proof. In view of Corollry 2.1, we define the spce SV + ([, ], L(X, Z )) = { α SV ([, ], L(X, Z )); z α BV + ([, ], X ), z Z } which is complete when equipped with the semi-vrition norm. This result cn e found in [15], Theorem I.3.7. We include it here. Theorem 2.2. SV + ([, ], L(X, Z )) is Bnch spce with the semi-vrition norm. Proof. By Theorem 2.2 (i), for every z Z, the mpping F z : α SV ([, ], L(X, Z )) z α BV ([, ], X ) is continuous. By Theorem 2.1, BV + ([, ], X ) is closed suspce of BV ([, ], X ) nd therefore SV + ([, ], L(X, Z )) = {(F z ) 1 (BV ([, ], X )); z Z} is closed suspce of the Bnch spce SV ([, ], L(X, Z )) which implies the result. 2.2 Riemnn-Stieltjes integrtion For the next results we need the concept of the Riemnn-Stieltjes integrl which we define y mens of tgged divisions. 9

10 A tgged division of [, ] is ny set of pirs (ξ i, t i ) such tht (t i ) D [,] nd ξ i [t i 1, t i ] for every i. In this cse we write d = (ξ i, t i ) T D [,], where T D [,] denotes the set of ll tgged divisions of [, ]. Any suset of tgged division of [, ] is tgged prtil division of [, ] nd, in this cse, we write d T P D [,]. A guge of set E [, ] is ny function δ : E ]0, [. Given guge δ of [, ], we sy tht d = (ξ i, t i ) T P D [,] is δ-fine, if [t i 1, t i ] {t [, ] ; t ξ i < δ (ξ i )} for every i, tht is, ξ i [t i 1, t i ] ]ξ i δ(ξ i ), ξ i + δ(ξ i )[, i = 1, 2,.... Now we will define the Riemnn-Stieltjes integrls y mens of tgged divisions d = (ξ i, t i ) of [, ] nd constnt guges δ (i.e., there is δ 0 > 0 such tht δ (ξ) = δ 0 for every ξ [, ]). Let (E, F, G) e BT. Any function α : [, ] E is sid to e Riemnn integrle with respect to function f : [, ] F if there exists n I G such tht for every ε > 0, there is constnt guge δ of [, ] such tht for every δ-fine d = (ξ i, t i ) T D [,], [α (t i ) α (t i 1 )] f (ξ i ) I < ε. In this cse, we write I = dα (t) f (t). By R f ([, ], E) we men the spce of ll functions α : [, ] E which re Riemnn integrle with respect to f : [, ] F. Anlogously we sy tht f : [, ] F is Riemnn integrle with respect to α : [, ] E if there exists n I G such tht for every ε > 0, there is constnt guge δ of [, ] such tht for every δ-fine d = (ξ i, t i ) T D [,], α (ξ i ) [f (t i ) f (t i 1 )] I < ε. Then R α ([, ], F ) denotes the spce of ll functions f : [, ] F which re Riemnn integrle with respect to given α : [, ] E with integrl I = α (t) df (t). The integrls dα (t) f (t) nd α (t) df (t) defined ove re known s Riemnn Stieltjes integrls. Consider the BT (E, F, G) with E = Y = L (Y, R), F = L (X, Y ), G = X = L (X, R) nd B (y, u) = y u, for ll y Y nd ll u L (X, Y ). We will use the identifiction d c dy (t) K (t, s) = d c K (t, s) dy (t), where y : [, ] Y, K : [c, d] [, ] L (X, Y ) nd K (t, s) denotes the djoint of K (t, s) L (X, Y ). 10

11 2.3 Some properties Let E e normed spce. By C ([, ], E) we men the spce of ll continuous functions from [, ] to E endowed with the usul supremum norm which we denote y. We define C ([, ], E) = {f C ([, ], E) ; f () = 0}. The next result is well-known. It gives the Integrtion y Prts Formul for the Riemnn- Stieltjes integrls. For proof of it, see for instnce [14] or [8], Theorem 2.5. Theorem 2.3 (Integrtion y Prts). Let (E, F, G) e BT. If either α SB ([, ], E) nd f C ([, ], F ), or α C ([, ], E) nd f BV ([, ], F ), then α R f ([, ], E), f R α ([, ], F ) nd the Integrtion y Prts Formul holds. dα (t) f (t) = α () f () α () f () α (t) df (t) The ssertions in the next remrk follow y the Integrtion y Prts Formul nd some esy computtion. Remrk 2.2. Suppose (E, F, G) is BT nd α SB ([, ], E). If we define F α (f) = dα(t)f(t), f C ([, ], F ), then F α L(C([, ], F ), G) nd F α SB(α). In prticulr we hve If E = F nd G = C or R s in Remrk 2.1, then given α BV ([, ], F ), there exists F α (f) = f(t), dα(t) = lim f(ξ i ), α(t i ) α(t i 1 ), f C([, ], F ), d 0 where d = (ξ i, t i ) T D [,] nd d = mx{t i t i 1 ; i = 1, 2,..., }. Also F α C([, ], F ) nd F α V (α). If α SV ([, ], L(X, Y )), then for every f C ([, ], X) there exists the Riemnn- Stieltjes integrl dα (t) f (t). Furthermore F α (f) = dα (t) f (t), f C ([, ], X), is such tht F α L (C ([, ], X), Y ) nd F α SV (α). The next theorem sys tht ll opertors in L(C([, ], X), Y )) cn e represented y functions of ounded semi-vrition. The version we present here is specil cse of [14], Theorem I.5.1. In prticulr, it will e shown lter tht if Y = Z, then L(C([, ], X), Y )) cn e represented y functions of ounded semi-vrition which re right continuous. 11

12 Theorem 2.4. The mpping α SV ([, ], L(X, Y )) F α L(C([, ], X), Y ) where F α (f) = dα(t)f(t), for f C([, ], Y ), is n isometry (i.e., F α = SV (α)) of the first Bnch spce onto the second. We lso hve α(t)x = F α (χ ],t] x), x X. We proceed with the presenttion of results orrowed from [15] with their proofs. In the sequel, we ssume tht c is some point in the intervl [, ]. Given α BV ([, ], Y ), let us define n uxiliry function α : [, ] Y y 0, t =, α(t) = α(t+) α(), t ], [, α() α(), t =. (2) The next result will e useful to prove tht the opertors of C([, ], Y ) cn e represented y elements of BV c ([, ], Y ). It cn e found in [15], Theorem I Lemm 2.3. Let α BV ([, ], Y ). Then (i) α BV c + ([, ], Y ) nd V (α) V (α); (ii) For every f C([, ], Y ), F α (f) = F α (f). Proof. Let us prove (i). By the definition of α, α() = 0 nd α is right continuous t t ], [. Hence α BV c + ([, ], Y ). It remins to prove tht V (α) V (α). We cn suppose, without loss of generlity, tht α() = 0. Then given ε > 0 nd d = (t i ) D [,], there exist s i ]t i 1, t i [, i = 1, 2,..., 1, such tht α(s i +) = α(s i ) (y Proposition 2.1) nd α(t i +) α(s i ) ε. Therefore α(t i +) α(t i 1 ) α(t i +) α(s i ) + α(s i ) α(s i 1 ) + α(t i 1 +) α(s i 1 ) α(s i ) α(s i 1 ) + 2ε. If we consider d = (s i ) with = s 0 < s 1 <... < s 1 < s =, then α(t i +) α(t i 1 ) α(s i ) α(s i 1 ) + 2ε nd hence V d (α) V d (α)+2ε V (α)+2ε. Thus V d (α) V (α) (since ε is independent of the choice of d) nd then V (α) V (α). Now we will prove (ii). By definition, we hve F α (f) = f(t), dα(t) = lim f(ξ i ), α(t i ) α(t i 1 ), d 0 12

13 where ξ i [t i 1, t i ]. We my suppose, without loss of generlity, tht α() = 0 nd then Proposition 2.1 implies α(t) = α(t), for ll t [, ] ut countle suset. Then if we tke the points t i of the division d = (t i ) D [,] in the complement of tht countle suset, we otin (ii). The next representtion theorem cn e found in [15], Theorem I Theorem 2.5. The mpping α BV c ([, ], Y ) F α C([, ], Y ) is n isometry (i.e., F α = V (α)) of the first Bnch spce onto the second. Proof. It is cler tht the mpping is liner nd F α V (α). We will prove tht the mpping is one-to-one, tht is, α 0 implies F α 0. It is enough to show tht there exists f C([, ], Y ) such tht F α (f) 0. If α() 0, then there exists y 0 Y such tht y 0, α() = 1. If we tke f(t) y 0, then F α (f) = y 0, dα(t) = y 0, α() = 1. If α() 0, let t 0 ], [ e such tht α(t 0 ) 0 nd consider y 0 Y such tht y 0, α(t 0 ) = 1. Define 0, t t 0 f n (t) = n(t t 0 )y 0, y 0, t 0 t t n t t. n Then f n C([, ], Y ) nd f n = y 0, n N, which implies F α (f n ) = t0 + 1 n t 0 n(t t 0 ) y 0, dα(t) + t n y 0, dα(t), where the second integrl equls y 0, α(t ) which converges to y n 0, α(t 0 +) = 1 nd the norm of the first integrl is ounded y y 0 V [t0,t ](α) which converges to 0. n Now we will prove tht the mpping is onto nd it is n isometry. It suffices to show tht for every F C([, ], Y ), there exists n α BV ([, ], Y ) such tht F = F α nd V (α) F. Then y Lemm 2.3, the sme pplies to α given y (2). Since C([, ], Y ) is suspce of the spce B([, ], Y ) of ounded functions from [, ] to Y, then the Hhn-Bnch Theorem implies F C([, ], Y ) dmits liner continuous extension F B([, ], Y ) such tht F = F. If for ll s [, ], s c, we define ψ s (t) = 1, if t s, ψ s (t) = 0, if s < t, nd ψ c = 0, then for every y Y nd t [, ], we hve yψ t B([, ], Y ) nd this function tkes only two vlues: 0 nd y. The mpping y Y F (yψ t ) R is liner nd continuous 13

14 ecuse F (yψ t ) F y ). Therefore there exists one nd only one element α(t) Y such tht F (yψ t ) = y, α(t), y Y. We ssert tht α BV ([, ], Y ) nd V (α) F. Indeed, given d = (t i ) D [,], we hve V d (α) = α(t i ) α(t i 1 = sup y i, α(t i ) α(t i 1 ; y i Y, y i 1 = sup y i 1 F y i (ψ ti ψ ti 1 ) F, since y i(ψ ti ψ ti 1 ) 1. Also, given f C([, ], Y ), we hve F (f) = f(t), dα(t), tht is, F α = F. Indeed, ecuse for ξ i [t i 1, t i ], we hve f(t), dα(t) = lim f(ξ i ), α(t i ) α(t i 1 ) d 0 = lim d 0 F f(ξ i )(ψ ti ψ ti 1 ) = F (f), since lim d 0 f(ξ i)(ψ ti ψ ti 1 ) = f, where the limit is tken in B([, ], Y ). Also f f(ξ i )(ψ ti ψ ti 1 ) = sup s f(s) f(ξ i )[ψ ti (s) ψ ti 1 (s)] ω d(f), where ω d (f) is the oscilltion of f with respect to d nd converges to 0 s d 0. Remrk 2.3. Theorem 2.5 lso holds for BV c ([, ], Y ), c [, ], insted of BV ([, ], Y ). The next two results re lso orrowed from [15] (see respectively Theorem I.3.8 nd Corollry I.3.9 there). Theorem 2.6. The mpping α SV + c ([, ], L(X, Z )) F α L(C([, ], X), Z ) is liner isometry (i.e., F α = SV (α)) of the first Bnch spce onto the second. 14

15 Proof. The mpping is clerly liner. Also F α SV (α) y Remrk 2.2. Let us prove tht the mpping is injective. If α 0, there exists t 0 ], ] such tht α(t 0 ) 0. Hence there exists z Z such tht z α(t 0 ) 0, where z α(t) X. Therefore z α BV c + ([, ], X ) nd z α 0. By Theorem 2.5, F z α 0, where F z α is the element of C([, ], X) defined y z α. Thus there exists f C([, ], X) such tht F z α (f) 0. On the other hnd, F z α (f) = f(t), d(z α)(t) = z, dα(t) f(t) = z, F α (f) nd hence F α (f) 0, tht is, F α 0. Now we will show tht given F L(C([, ], X), Z ), there exists α SV c + ([, ], L(X, Z )) such tht F = F α nd SV (α) F. For every z Z, we hve z F C([, ], X) nd z F z F. Then Theorem 2.5 implies tht there is one nd only one element α z BV c + ([, ], X ) such tht z F = F αz, tht is, for every f C([, ], X), we hve (z F )(f) = f(t), dα z (t) nd V (α z ) = z F. We ssert tht α z1 +z 2 (t) = α z1 (t) + α z2 (t), t [, ]. Indeed, we hve (z 1 + z 2 ) F = z 1 F + z 2 F nd hence for every f C([, ], X), we hve f(t), dα z1 +z 2 (t) = f(t), d(α z1 + α z2 )(t). Then the uniqueness of the representtion in Theorem 2.5 implies the ssertion. In similr wy, one proves tht α λz (t) = λα z (t). We hve α(t)x Z. Indeed. Given t [, ] nd x X, if we define α(t)x R Z y (α(t)x)z = x, α z (t), then the ove computtion implies tht the mpping z Z (α(t)x)z = x, α z (t) R is liner. The mpping is lso continuous ecuse α(t)x = sup { x, α z (t) ; z Z, z 1} nd x, α z (t) x α z (t) x V (α z ) = x z F x F z nd hence α(t)x x F. We hve α L(X, Z ), since the mpping x X α(t)x Z is liner nd α(t) F (y the lst inequlity). We lso ssert tht α SV + c of α, we hve z α = α z BV + c ([, ], L(X, Z )) nd SV (α) F. Indeed, y the definition ([, ], X ) for every z Z. Hence y Proposition 2.2 (i), SV (α) = sup {V (z α); z Z, z 1} = sup {V (α z ); z Z, z 1} F, since V (α z ) = z F z F. Finlly F = F α, since for every z Z, we hve z F = F αz z F = z F α. = F z α = z F α nd then 15

16 The next result is consequence of Theorem 2.6 with Z = Y. Corollry 2.2. For every F α L(C([, ], X), Y ), there exists one nd only one α SV c + ([, ], L(X, Y )) such tht F = F α. Remrk 2.4. Under the hypotheses of Corollry 2.2, we write α F = α. Note tht the mpping F α F my not e onto if Y Y. By Corollry 2.1, if α SV ([, ], L(X, Z )), then for every t [, [, there exists one nd only one element α(t+) L(X, Z ) such tht for every x X nd every z Z, we hve lim α(t + ε)x, z = α(t+)x, z. ε 0 If we define α + (t) = α(t+), < t <, nd α + () = α(), then α + is function of ounded semi-vrition nd we write α + SV + ( ], ], L(X, Z )). Moreover for every f C([, ], X), we hve dα+ (t)f(t) = dα(t)f(t) nd F α = SV (α + ). The next result follows from [2], Stz 10 nd from Theorem 2.6. Theorem 2.7. The mpping α SV + c ( ], ], L(X, Z )) F α L(C([, ], X), Z ) is liner isometry (i.e., F α = SV (α)) of the first Bnch spce onto the second. Similrly s in Corollry 2.2, we hve Corollry 2.3. For every F α L(C([, ], X), Y ), there exists one nd only one α SV c + ( ], ], L(X, Y )) such tht F = F α. Let SV + + ( ], ], L(X, Ẏ )) denote the spce of functions α SV ( ], ], L(X, Y )) such tht α() L(X, Y ) nd t α(s)xds Y, for ll t [, ] nd ll x X. Let χ A e the chrcteristic function of set A [, ]. The next theorem completes Corollry 2.3 nd chrcterizes the imge of the mpping F α F. It is orrowed from [11] (Theorem 1.4 there) nd its proof cn lso e found in [6], Theorem 10. Theorem 2.8. The mpping α SV + ( ], ], L(X, Ẏ )) F α L(C([, ], X), Y ), where F α (f) = dα(t)f(t), is n isometry (i.e., F α = SV (α)) of the first Bnch spce onto the second. Furthermore t α(s)xds = F α(g t,x ) nd α()x = F α (χ [,] x), where for t [, ] nd x X, we define g t,x (s) = (s )x, if s t, nd g t,x (s) = (t )x, if t s. 16

17 Let SV + + ([, ], L(X, Ẏ )) = {α SV ( ], ], L(X, Ẏ )); α(+) = α() insted of α() L(X, Y )}. The next two representtion theorems re respectively Theorems 1.5 nd 1.6 from [11]. Their proofs cn lso e found in [6], Theorems 11 nd 12. Theorem 2.8 is used to prove Theorem 2.9 which, in turn, is pplied in the proof of Theorem Theorem 2.9. The mpping α SV + ([, ], L(X, Ẏ )) F α L(C ([, ], X), Y ) is n isometry of the first Bnch spce onto the second. The nottion elow is going to e used in the next theorem. Given function α : [c, d] [, ] L(X, Y ), we write α t (s) = α s (t) = α(t, s) nd we consider the following properties: (C σ ) for s nd x X, the function t [c, d] α s (t)x Y is continuous, ( C σ ) for < s nd x X, the function t [c, d] s α(t, σ)xdσ Y is continuous, nd for x X (nd s = ), the function t [c, d] α (t)x Y is continuous, (SV u ) For ll t [c, d], α t SV ([, ], L(X, Y )) nd SV u (α t ) := sup SV (α t ) <. c t d moreover α(t, ) = 0 for ll t [c, d], then we write (SV u) ). ) For ll t [c, d], α t SV + ([, ], L(X, Ẏ )) nd SV u (α) := sup SV (α t ) <. c t d (SV + u If We write α C σ SV u ([c, d] [, ], L(X, Y )) if α stisfies (C σ ) nd (SV u ). Anlogously, α C σ SV u([c, d] [, ], L(X, Y )) if α stisfies ( C σ ) nd (SV u). We write α C σ SV + u ([c, d] [, ], L(X, Ẏ )) if α stisfies ( C σ ) nd (SV + u ). Theorem The mpping α C σ SV + u ([c, d] [, ], L(X, Ẏ )) F α L(C([, ], X), C([c, d], Y )), where (F α f)(t) = d sα(t, s)f(s), c t d, is n isometry (i.e., F α = SV u (α)) of the first Bnch spce onto the second. Besides, s α(t, σ)xdσ = F α(g s,x )(t), s nd α(t, )x = F α (χ [,] x)(t). We denote y K (X, Y ) the suspce of L (X, Y ) of compct liner opertors. In prticulr, we write K (X) = K (X, X). We conclude this section of uxiliry results mentioning Fredholm Alterntive for Riemnn-Siteltjes integrls. For proof of it, see [13] nd [4], Theorems 2.4 nd

18 Theorem Suppose K C σ (SV ) u ([, ] [, ], L (X)). Given t [, ], let K (t, s 0 ) x = lim s s0 K (t, s) x for every s 0 ], [ nd every x X. Suppose the mpping K : t [, ] K (t) = K t SV ([, ], L (X)) elongs to C ([, ], SV ([, ], K (X))). Given λ R, λ 0, consider the integrl equtions λx (t) λu (t) λy (s) λz (s) d s K (t, s) x (s) = f (t), t [, ], (3) d s K (t, s) u (s) = 0, t [, ], (4) K (t, s) dy (t) = g (s), s [, ], (5) K (t, s) dz (t) = 0, s [, ]. (6) Then the Fredholm Alterntive holds for these equtions, tht is, (i) either for every f C ([, ], X), eqution (3) hs one nd only one solution nd the sme pplies to eqution (5), (ii) or eqution (4) hs non-trivil solutions nd the sme pplies to eqution (6). If (i) holds, then eqution (3) dmits solution if nd only if f (t) dz (t) = 0 for every solution z of eqution (6). Similrly, if (i) holds, then eqution (5) dmits solution if nd only if u (t) dg (t) = 0 for every solution u of eqution (4). Also the spce of solutions of (4) hs finite dimension equl to tht of the spce of solutions of (6) which equls the codimension of (λi F K )C([, ], X) in C([, ], X) nd the codimension of (λi (F K ) )BV + ([, ], X ) in BV + ([, ], X ). 3 Guge integrls in Bnch spces 3.1 Definitions nd terminology In this section, we consider functions α : [, ] L (X, Y ) nd f : [, ] X. 18

19 We sy tht α is Kurzweil f-integrle or Kurzweil integrle with respect to f, if there exists I Y such tht for every ε > 0, there is guge δ of [, ] such tht for every δ-fine d = (ξ i, t i ) T D [,], α (ξ i ) [f (t i ) f (t i 1 )] I < ε. In this cse, we write I = (K) α (t) df (t) nd α K f ([, ], L (X, Y )). Anlogously, we sy tht f is Kurzweil α integrle or Kurzweil integrle with respect to α, if there exists I Y such tht given ε > 0, there is guge δ of [, ] such tht [α (t i ) α (t i 1 )] f (ξ i ) I < ε, whenever d = (ξ i, t i ) T D [,] is δ-fine. In this cse, we write I = (K) dα (t) f (t) nd f K α ([, ], X). If the guge δ in the definition of α K f ([, ], L (X, Y )) is constnt function, then we get the Riemnn-Stieltjes integrl α (t) df (t) nd we write α R f ([, ], L (X, Y )). Similrly, when we consider only constnt guges δ in the definition of f K α ([, ], X), we otin the Riemnn-Stieltjes integrl dα (t) f (t) nd we write f Rα ([, ], X). The vector integrl of Henstock is more restrictive thn tht of Kurzweil. We define it in the sequel. We sy tht α is Henstock f-integrle or Kurzweil integrle with respect to f, if there exists function A f : [, ] Y (clled the ssocite function of α) such tht for every ε > 0, there is guge δ of [, ] such tht for every δ-fine d = (ξ i, t i ) T D [,], α (ξ i ) [f (t i ) f (t i 1 )] [A f (t i ) A f (t i 1 )] < ε. We write α H f ([, ], L (X, Y )) in this cse. In n nlogous wy we define the Henstock α-integrility of f : [, ] X or Henstock integrility of f with respect to α nd we write f H α ([, ], X) (see [7]). Clerly H f ([, ], L (X, Y )) K f ([, ], L (X, Y )) nd H α ([, ], X) K α ([, ], X). If we identify the isomorphic spces L(R, R) nd R, then ll the spces K f ([, ], L (R)), K f ([, ], R), H f ([, ], L (R)) nd H f ([, ], R) cn lso e identified, since K f ([, ], R) = H f ([, ], R) (see, for instnce, [17]). Given f : [, ] X nd α K f ([, ], L (X, Y )), we define the indefinite integrl α f : [, ] Y of α with respect to f y α f (t) = (K) t α (s) df (s), t [, ]. 19

20 If in ddition α H f ([, ], L (X, Y )), then α f (t) = A f (t) A f () for every t [, ]. In n nlogous wy, given α : [, ] L (X, Y ), we define the indefinite integrl f α : [, ] Y of f with respect to α y f α (t) = (K) t dα (s) f (s), t [, ], for every f K α ([, ], X). In prticulr, when α (t) = t, then insted of K α ([, ], X), R α ([, ], X), H α ([, ], X) nd f α we write, respectively, K ([, ], X), R ([, ], X), H ([, ], X) nd f, tht is, f (t) = (K) t f (s) ds, for every t [, ]. We proceed s to define the equivlence clsses of Kurzweil nd of Henstock integrle functions. Let m denote the Leesgue mesure. A function f : [, ] X stisfies the Strong Lusin Condition nd we write f SL ([, ], X) if given ε > 0 nd B [, ] with m (B) = 0, there is guge δ of B such tht for every δ-fine d = (ξ i, t i ) T P D [,] with ξ i B for ll i, we hve f (t i ) f (t i 1 ) < ε. If we denote y AC ([, ], X) the spce of ll solutely continuous functions from [, ] to X, then we hve AC ([, ], X) SL ([, ], X) C ([, ], X). In SL ([, ], X), we consider the usul supremum norm,, induced from C ([, ], X). Given f SL ([, ], X) nd α H f ([, ], L (X, Y )), let β : [, ] L (X, Y ) e such tht β = α m-lmost everywhere. Then β H f ([, ], L (X, Y )) nd β f (t) = α f (t), for every t [, ]. See [7] (the corollry fter Theorem 5 there) for proof of this fct. An nlogous result holds when we replce H f ([, ], L (X, Y )) y K f ([, ], L (X, Y )). Suppose f SL ([, ], X). Two functions β, α K f ([, ], L (X, Y )) re clled equivlent if nd only if β f = α f. We denote y K f ([, ], L (X, Y )) nd y H f ([, ], L (X, Y )) respectively the spces of ll equivlence clsses of functions of K f ([, ], L (X, Y )) nd of H f ([, ], L (X, Y )) nd we equip these spces with the Alexiewicz norm { t } α A,f = sup (K) α (s) df (s) ; t [, ] = α f. 3.2 Some properties As it should e expected, the vector integrls of Kurzweil nd of Henstock re liner, dditive over non-overlpping intervls nd invrint over sets of Leesgue mesure zero. In this section, me mention severl other properties of these integrls. We strt with the Fundmentl Theorem of Clculus which holds for the Henstock integrl. Its proof follows stndrd steps (see [17], p. 43, for instnce) dpted to Bnch spce-vlued functions. See lso [18]. 20

21 Theorem 3.1 (Fundmentl Theorem of Clculus). If F C ([, ], X) nd there exists the derivtive F (t) = f (t), for every t [, ], then f H ([, ], X) nd (K) t f (s) ds = F (t) F (), t [, ]. The next two versions of the Fundmentl Theorem of Clculus for the vector Henstock integrl cn e found in [7], Theorems 1 nd 2 respectively. Theorem 3.2. If f SL ([, ], X) nd A SL ([, ], Y ) re oth differentile nd α : [, ] L (X, Y ) is such tht A (t) = α (t) f (t) for m-lmost every t [, ], then α H f ([, ], L (X, Y )) nd A = α f. Theorem 3.3. If f SL ([, ], X) is differentile nd α H f ([, ], L (X, Y )) is ounded, then α f SL ([, ], Y ) nd there exists the derivtive ( α f ) (t) = α (t) f (t) for m-lmost every t [, ]. Corollry 3.1. Suppose f SL ([, ], X) is differentile nd non-constnt on ny nondegenerte suintervl of [, ] nd α H f ([, ], L (X, Y )) is ounded nd such tht α f = 0. Then α = 0 m-lmost everywhere. The next result is prticulr cse of [8], Theorem 2.2. Theorem 3.4. If f C ([, ], X) nd α K f ([, ], L (X, Y )), then α f C ([, ], Y ). For the vector Henstock integrl we hve the following nlogue which cn lso e found in [7], Theorem 7. Theorem 3.5. If f SL ([, ], X) nd α H f ([, ], L (X, Y )), then α f SL ([, ], Y ). The next result follows from Theorem 3.5. A proof of it cn e found in [6], Theorem 5. Theorem 3.6. Suppose f SL ([, ], X) is non-constnt on ny non-degenerte suintervl of [, ]. Then the mpping α H f ([, ], L (X, Y )) α f C ([, ], X) is n isometry (i.e., α f = α A,f ) onto dense suspce of C ([, ], X). The next result gives sustitution formul for the vector Kurzweil integrl. It lso holds for the Riemnn-Stieltjes integrl insted. For proof of it, see [5], Theorem 11. Theorem 3.7. Let α SV ([, ], L (X, Y )), f : [, ] Z, β K f ([, ], L (Z, X)) nd g(t) = β f (t) = t β(s)df(s), t [, ]. Then α K g([, ], L(X, Y ) if nd only if αβ K f ([, ], L (Z, Y )). In this cse, we hve (K) α (t) β (t) df (t) = 21 α (t) dg (t) (7)

22 nd (K) α (t) β (t) df (t) [SV (α) + α () ] β A,f. (8) Using Theorems 3.4 nd 2.3, we hve the next corollry. See [5], Corollry 8. Corollry 3.2. If α SV ([, ], L (X, Y )), f C ([, ], W ), β K f ([, ], L (W, X)) nd g(t) = β f (t) = t β(s)df(s), t [, ], then αβ K f ([, ], L (W, Y )) nd (7) nd (8) hold. By E ([, ], L (X, Y )) we men the spce of ll step functions from [, ] to L (X, Y ), tht is, α elongs to E ([, ], L (X, Y )) if nd only if α is ounded, there is division d = (t i ) D [,] nd there re numers α 1, α 2,..., α such tht α(t) = α iχ [ti 1,t i [(t), for every t [, [. For proof of the next proposition, see [6], Theorem 8. Proposition 3.1. Let f SL([, ], X) e differentile nd non-constnt on ny nondegenerte suintervl of [, ]. Then the spces C ([, ], L (X, Y )) nd E ([, ], L (X, Y )) re dense in K f ([, ], L (X, Y )) in the Alexiewicz norm A,f. 4 Auxiliry results In this section we prove uxiliry results concerning vector guge integrls which will e useful in the next section. We strt y giving representtion theorem which sys tht given certin g SL([, ], R), the elements of K g ([, ], L(R, X)) cn e represented y functions of ounded vrition which re continuous to the right. Theorem 4.1. Given g SL([, ], R) differentile nd non-constnt in ny non-degenerte suintervl of [, ], then the mpping α BV + ([, ], X ) H α,g K g ([, ], L(R, X)), where H α,g (f) = (K) α(s)f(s)dg(s), is n isometry (i.e., H α,g = V (α)) onto nd, for every t [, ] nd every x L(R, X), we hve t α(s)xdg(s) = H α,g(χ [,t] x). Proof. The mpping is clerly liner. We ssert tht the mpping is one-to-one. Indeed. Given ρ [, ] nd x L(R, X), we define function f ρ,x : [, ] L(R, X) y f ρ,x (s) = x if s ], ρ], nd y f ρ,x (s) = 0 otherwise. Then Theorem 2.3 implies f ρ,x R g ([, ], L(R, X)). Also H α,g (f ρ,x ) = ρ α(s)xdg(s). Since for ech x L(R, X), the function α( )x : [, ] L(R) is such tht α( )x R g ([, ], L(R)) K g ([, ], L(R)) = H g ([, ], L(R)), then y the Fundmentl Theorem of Clculus (Theorem 3.2), there exists d ρ ( (K) ρ α(s)xdg(s)) = α(ρ)xg (ρ) for m-lmost every ρ [, ]. 22

23 If α 0, then there exist ρ [, ] nd x L(R, X) such tht α(ρ)x 0. Besides, we cn suppose without loss of generlity tht g (ρ) 0 y the invrince of the integrl over sets of Leesgue mesure zero. Indeed. From [7], Theorem 9 we hve (K) ρ α(s)xdg(s) = (K) ρ α(s)xg (s), for every ρ [, ]. In fct the integrl ove is in the Riemnn-Stieltjes sense. In prticulr, ρ α(s)xdg(s) = ρ α(s)xg (s) nd hence ( ρ ) d ρ α(s)xdg(s) = α(ρ)xg (ρ) 0. Thus H α,g (f ρ,x ) = ρ α(s)xdg(s) is non-constnt nd hence H α,g(f ρ,x ) 0 nd the mpping is one-to-one. Since SV ([, ], L(X, R)) = BV ([, ], X ), it follows from Corollry 3.2 tht H α,g V (α). Let f K g ([, ], L(R, X)) nd H K g ([, ], L(R, X)) nd define Ĥ( f g ) = H(f). By Theorem 3.6, there is unique continuous extension of Ĥ to C ([, ], X) which we still denote y Ĥ. This new opertor, Ĥ, elongs to C ([, ], X). If α represents Ĥ, then Theorem 2.9 implies Ĥ( f g ) = dα(s) f g (s) nd Ĥ = V (α). Moreover (SV + u H(f) = Ĥ( f g ) = α(s)d f g (s) = dα(s) f g (s) = (K) α(s)f(s)dg(s), where we pplied Theorem 2.3 nd Corollry 3.2 to otin the lst two equlities. Since y definition f g = f g, then H = Ĥ = V (α) nd the result follows. Let g SL([, ], Z). Given function α : [c, d] [, ] L(X, Y ), we consider the following properties: ( C g σ ) for < s nd x L(Z, X), the function t [c, d] s α(t, σ)xdg(σ) Y is continuous, nd for x L(Z, X) (nd s = ), the function t [c, d] α(t, )xg () Y is continuous, nd, gin, we consider the following property mentioned erlier: ) for ll t [c, d], α t SV + ([, ], L(X, Ẏ )) nd SV u (α) := sup SV (α t ) <. c t d Then we write α C g σ SV + u ([c, d] [, ], L(X, Ẏ )) if α stisfies ( C g σ ) nd (SV + u ). If, in ddition, we consider functions of ounded vrition in property (SV + u ), then we write α C g σ BV + u ([c, d] [, ], L(X, Ẏ )). Notice tht when Y = R, the spces C g σ BV + u ([c, d] [, ], L(X, Ẏ )) nd C g σ BV u([c, d] [, ], L(X, Y )) cn e identified nd hence we write simply C g σ BV u([c, d] [, ], X ) in this cse. The proof of the next result follows the steps of the proof of Theorem 4.1, however insted of Theorem 2.9, one should pply Theorem See [6], Theorem

24 Theorem 4.2. Let g SL([, ], R) e differentile nd non-constnt in ny non-degenerte suintervl of [, ]. Then the mpping α C g σ BV +u ([c, d] [, ], X ) H α,g L(K g ([, ], L(R, X)), C([c, d], R)), where (H α,g (β))(t) = (K) α(t, s)β(s)dg(s) for ech t [c, d], is n isometry (i.e., H α,g = V (α)) onto nd, for every s [, ], every t [, ] nd every x L(R, X), we hve s α(t, σ)xdg(σ) = (H α,g(χ [,s] x))(t). Using Proposition 3.1, the next result follows esily. Corollry 4.1. Under the hypotheses of Theorem 4.2, for every s [, ], every t [, ] nd every β K g ([, ], L(R, X)), we hve s α(t, σ)β(σ)dg(σ) = (H α,g(χ [,s] β))(t). Let g SL(, ], R). We sy tht n opertor H L(K g ([, ], L(R)), C([, ], R)) is cusl if given f K g ([, ], L(R)) = K g ([, ], R) nd t [, ], then f [,t] = 0 implies H(f) [,t] = 0, where h A denotes the restriction of function h to suset A of its domin. We proceed s to show tht in fct the isometry in Theorem 4.2 is onto the spce of cusl opertors, provided α(t, s) = 0 for m-lmost every s > t. The next result cn e found in [6], Theorem 14. Theorem 4.3. Let α : [, ] [, ] L(R) e such tht α(t, s) = 0, for m-lmost every s > t, nd let g SL([, ], R) e differentile nd non-constnt in ny non-degenerte suintervl of [, ]. Then the mpping α C g σ BV +u ([, ] [, ], L(R)) H α,g L(K g ([, ], L(R)), C([, ], R)), where (H α,g (f))(t) = (K) α(t, s)f(s)dg(s) for ech t [c, d], is n isometry (i.e., H α,g = V (α)) onto the suspce of cusl opertors. For proof of the next lemm, see [9] for instnce. Lemm 4.1 (Strddle Lemm). Suppose f, F : [, ] X re such tht F (ξ) = f (ξ), for ll ξ [, ]. Then given ε > 0, there exists δ (ξ) > 0 such tht whenever ξ δ (ξ) < s < ξ < t < ξ + δ (ξ). F (t) F (s) f (ξ) (t s) < ε (t s), When g(s) = s, Theorem 4.4 elow is prticulr cse of [4], Theorems 3.6 nd 3.7. The proof is strightforwrd dpttion of such results to cope with the introduction of the function g. The version we give next completes [6], Lemm 4 whose proof is included here. 24

25 Theorem 4.4. Let α C g σ BV +u ([c, d] [, ], X ), g SL([, ], R) e differentile nd non-constnt in ny non-degenerte suintervl of [, ], nd K g : [c, d] [, ] X e such tht K g (t, s)x = s α(t, σ)xdg(σ), for ech x L(R, X). Then K g C σ SV u([, ] [, ], X ) nd for ll t [c, d] nd ll y L(R) = R, the mpping s [, ] (K g (t, s)) y X is continuous, where (K g (t, s)) denotes the djoint of K g (t, s). Besides, we hve d s K g (t, s)β(s) = α(t, s)β(s)dg(s) for every function β C([, ], L(R, X)) nd ll t [c, d]. Suppose, in ddition, tht α t SV ([, ], K(X, R)), for every t [c, d]. Then the mpping is continuous. K g : t [c, d] K g (t, ) SV ([, ], X ) Proof. Since α fulfills ( C g σ ), then K g (, s) fulfills (C σ ), for ech s [, ]. Let x L(R, X) nd d = (s i ) D [,]. Then [ K t g (s i ) Kg(s t i 1 ) ] x = [ si ] α(t, ρ)xdg(ρ) s i 1 = α(t, ρ)xdg(ρ) V (α t ) x g() g() nd hence Kg t = K g (t, ) SV ([, ], X ), for ech t [c, d]. For every t [c, d] nd every x L(R, X), the function Kg( )x t is continuous on [, ], since it is n indefinite integrl (see Theorem 3.4). Therefore given t [c, d] nd y L(R) R, the function Kg( )) t y is continuous on [, ]. We ssert tht Kg C([c, d], SV ([, ], X )). Indeed. Let t 0 [c, d] nd d = (s i ) D [,]. Then ( ) K t g K t 0 g = SV K t g K t 0 g = sup = sup d, x 1 d, x 1 [( K t g K t 0 g ) (si ) ( K t g K t 0 g ) (si 1 ) ] x = [ si si ] = sup α(t, ρ)xdg(ρ) α(t 0, ρ)xdg(ρ) d, x 1 s i 1 s i 1 = α(t, ρ)xdg(ρ) α(t 0, ρ)xdg(ρ) = sup K g (t, )x K g (t 0, )x d, x 1 25

26 which tends to zero s t t 0, since K g (, s) C σ ([c, d], X ), for ech s [, ] nd, in prticulr, K g (, ) C σ ([c, d], X ). Let β C([, ], L(R, X)), t [c, d] nd γ = β g. In ccordnce with Corollry 3.2, α t β K g ([, ], L(R)) with (K) α t (s)β(s)dg(s) = α t (s)dγ(s) = d s ( α t (s) ) γ(s), (9) where we pplied Theorem 2.3 to otin the lst equlity. Since K g (t, s)x = s αt (ρ)xdg(ρ) with K g (t, ) SV ([, ], X ) for ech t [c, d], then Theorem 2.3 gin implies the Riemnn- Stieltjes integrl d sk g (t, s)β(s) exists for ech t [c, d] nd ech β C([, ], L(R, X)). We lso ssert tht d s K g (t, s)β(s) = (K) α t (s)β(s)dg(s), (10) for every t [c, d] nd every β C([, ], L(R, X)). Indeed. It is enough to prove tht (10) holds when β is step function. Hence we need to show tht for every t [c, d] nd every x L(R, X), d s K g (t, s)x = (K) α t (s)xdg(s), (11) Since α t ( )x R g ([, ], L(R)) y Theorem 2.3 nd R g ([, ], L(R)) K g ([, ], L(R)) = H g ([, ], L(R)), ( it follows from the Fundmentl Theorem of Clculus (Theorem 3.3) tht s there exists d s αt (ρ)xdg(ρ) ) = α t (s)xg (s) m-lmost everywhere on [,(]. Therefore given s t [c, d] nd x L(R, X), we hve d s K g (t, s)x = d s (K g (t, s)x) = d s αt (ρ)xdg(ρ) ) = α t (s)xg (s) for m-lmost every s [, ]. Then from the invrince of the integrl over sets of m-mesure zero, we otin d s K g (t, s)x = (K) α t (s)xg (s)ds. (12) Now we will prove tht αt (s)xg (s)ds = αt (s)xdg(s) which, together with (12), implies (11). Given ε > 0, t [c, d] nd x L(R, X), let δ 1 nd δ 2 e constnt guges of [, ] from the definitions of αt (s)xdg(s) nd αt (s)xg (s)ds respectively. Given ξ [, ], let δ 3 (ξ) > 0 e such tht g(v) g(s) g (ξ)(v s) < ε(v s) (13) whenever ξ δ 3 (ξ) < s < ξ < v < ξ + δ 3 (ξ) (see Lemm 4.1) nd let δ(ξ) = min{δ i (ξ); i = 1, 2, 3}. Then for every δ-fine d = (ξ i, s i ) T D [,], we hve α t (s)xdg(s) α t (s)xg (s)ds 26

27 α t (s)xdg(s) α t (ξ i )x [g(s i ) g(s i 1 )] + + α t (ξ i )x [g(s i ) g(s i 1 )] α t (ξ i )xg (ξ i )(s i s i 1 ) + + α t (ξ i )xg (ξ i )(s i s i 1 ) α t (s)xg (s)ds < < ε + V (α t ) x g(s i ) g(s i 1 ) g(ξ i )(s i s i 1 ) + ε < 2ε + V (α t ) x ε( ), where we pplied y the integrility with respect to g of α t ( )x, the integrility of α t ( )xg ( ) nd (13). The result follows esily. The next result cn e found in [1] or in [12], Theorem 3.9. Theorem 4.5. Given K C σ SV u ([, ] [, ], L(X)), suppose there is division d = (s i ) D [,] such tht sup { SV [si 1,t](K t ); t [s i 1, s i ] } < 1, i = 1, 2,...,. Then K hs resolvent given y the Neumnn series. Theorem 4.6 elow is orrowed from [11], Theorem 3.1. See lso [6], Lemm 7. Theorem 4.6. Let E e normed spce nd F e Bnch spce such tht F R with continuous immersion. Let H L(E, F ) e such tht for every f E, the eqution x Hx = f dmits one nd only one solution x f E. Then the mpping f R x f R is icontinuous. If in ddition the Neumnn series I + H + H 2 + H = (I H) 1 converges in L(F ), then it lso converges in L(E). 5 The Fredholm Alterntive for the Kurzweil-Henstock- Stieltjes integrl In this section, we write BV ([, ], R) insted of BV ([, ], L(R)) nd C g σ BV +u ([, ] [, ], R) insted of C g σ BV +u ([, ] [, ], L(R)). 27

28 Theorem 5.1. Given g SL([, ], R) differentile nd non-constnt in ny non-degenerte suintervl of [, ] nd H L(K g ([, ], R), C([, ], R)), let α C g σ BV +u ([, ] [, ], R) e the corresponding kernel y Theorem 4.2 (i.e., H = H α,g ). Suppose H is such tht for ech f K g ([, ], R), the liner Fredholm-Henstock-Kurzweil-Stieltjes integrl eqution x (t) (K) α (t, s) x (s) dg(s) = f (t), t [, ], (14) (i.e., the eqution x H(x) = f) dmits unique solution x f K g ([, ], R). Then there exists unique kernel ρ C g σ BV +u ([, ] [, ], R) nd, for ech f K g ([, ], R), x f (t) = f(t) (K) ρ (t, s) f (s) dg(s), t [, ]. (15) If moreover α(t, s) = 0, for m-lmost every s > t, then H is cusl s well s the ijection f x f given y (15) with ρ(t, s) = 0, for m-lmost every s > t, nd the Neumnn series I H ρ,g = I + H α,g + (H α,g ) 2 + (H α,g ) converges in L(K g ([, ], R)). Proof. The first prt of the proof follows s in [11], Theorem 3.2 dpted for the Stieltjes cse. If in Theorem 4.6 we tke E = K g ([, ], R) nd F = C([, ], R), then (I H) 1 L(K g ([, ], R)). If we define I R = (I H) 1, then R = H(I H) 1 elongs to L(K g ([, ], R), C([, ], R)) nd it cn e represented y kernel ρ C g σ BV +u ([, ] [, ], R). Now we will prove the second prt of the theorem which follows s in [6], Theorems 15 nd 16. Suppose α(t, s) = 0 for m-lmost every s > t. Let y = x f nd h(t) = (K) α(t, s)f(s)dg(s). Both functions y nd h re continuous, since h = H α,g(χ [,t] f) nd y = H α,g (χ [,t] x) (see Corollry 4.1) with χ [,t] f, χ [,t] x K g ([, ], R) nd H α,g L(K g ([, ], R), C([, ], R)) (see Theorem 4.3). Then eqution (14) is equivlent to the following eqution y (t) (K) α (t, s) y (s) dg(s) = h (t), t [, ]. (16) Let K g : [, ] [, ] R e such tht K g (t, s)x = s t αt (σ)xdg(σ), for every x R. Then y Theorem 4.4, we hve d s K g (t, s)y(s) = (K) Thus eqution (16) is equivlent to the following eqution y (t) α t (s)y(s)dg(s). d s K g (t, s)y(s) = h (t), t [, ]. (17) 28

29 Agin y Theorem 4.4, K g C σ BV u ([, ] [, ], R) nd since α(t, s) = 0 for m-lmost every s > t, then K g (t, s) = 0 for s > t. Thus if (17) dmits unique solution, then the opertor h y h is cusl. Now we will prove tht there exists d = (s i ) D [,] such tht sup { BV [si 1,t]((K g ) t ); t [s i 1, s i ] } < 1, i = 1, 2,...,. Let us consider the guge δ of [, ] defined s ove nd let d = (ξ i, s i ) T D [,] e δ-fine. Given i {1, 2,..., } nd t [s i 1, s i ], let d i = (r j ) e division of [s i 1, t]. Then [ (Kg ) t (r j ) (K g ) t (r j 1 ) ] r j x j = α t (σ)x j dg(σ) r = t α t (σ)x j dg(σ) j 1 s i j j V u (α) x j g(t) g(s i 1 ) < V u (α) x j ε < x j 2. Hence BV [si 1,t]((K g ) t ) 1/2. Thus the ssertion out the Neumnn series for the resolvent of eqution (17) in L(C([, ], R)) follows from Theorem 4.5. Besides, the opertor F Kg given y ( F Kg y ) (t) = t d sk g (t, s)y(s) is cusl s well s ( ) n ( ) 1. F Kg nd hence I FKg By Theorem 4.6, the sme pplies to the resolvent of eqution (17) in L(K g ([, ], R)). Thus from the fct tht F Kg = H α,g (see Theorems 4.4 nd 4.3), the opertor I H ρ,g = (I H α,g ) 1 is cusl. Next we give Fredholm Altentive type result for our Fredholm-Henstock-Kurzweil- Stieltjes integrl eqution. Theorem 5.2. Let g SL([, ], R) e differentile nd non-constnt in ny non-degenerte suintervl of [, ], f K g ([, ], R) nd α C g σ BV +u ([, ] [, ], R). Consider the liner Fredholm-Henstock-Kurzweil-Stieltjes integrl eqution x (t) (K) nd its corresponding homogeneous eqution u (t) (K) α (t, s) x (s) dg(s) = f (t), t [, ], (18) α (t, s) u (s) dg(s) = 0, t [, ]. (19) Consider lso the following integrl equtions s [ ] y (s) α (t, σ) dy (t) dg(σ) = w (s), s [, ], (20) nd Then z (s) s [ ] α (t, σ) dz (t) dg(σ) = 0, s [, ]. (21) 29