LINEAR INTEGRAL EQUATIONS OF VOLTERRA CONCERNING HENSTOCK INTEGRALS

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1 Real Analyss Exchange Vol. (),, pp M. Federson and R. Bancon, Insttute of Mathematcs and Statstcs, Unversty of São Paulo, CP 66281, e-mal: and LINEAR INTEGRAL EQUATIONS OF VOLTERRA CONCERNING HENSTOCK INTEGRALS Abstract We establsh condtons for the exstence of solutons of the lnear ntegral equaton of Volterra Z x (t) + α(s)x(s) ds = f(t), t [a, b], (V ) where the functons are Banach space-valued and R denotes ether the Bochner-Lebesgue or the Henstock ntegral. In some cases t s possble to calculate the soluton of (V ) explctly. We gve several examples. 1 Introducton In the lterature, the study of Integral Equatons deals manly wth the ntegrals of Remann, Lebesgue or Dushnk, (the latter s also called the nteror ntegral - see [9] or [10]). However these ntegrals have some defcences. The Remann ntegral s weak, the classcal defnton of the Lebesgue ntegral may be dffcult to deal wth, and the vector ntegral of Dushnk, though more general than the Remann-Steltjes ntegral, may not concde wth the Kurzwel-Henstock vector ntegral whch, n turn, s more general than both the Remann-Steltjes and the Lebesgue-Steltjes ntegrals. On the other hand, when we consder ntegral equatons n the sense of the Kurzwel-Henstock ntegrals, we beneft from ts easy to handle Remannan defnton and well-known good propertes. The am of ths paper s to gve condtons for the exstence of a soluton of the lnear ntegral equaton of Volterra x(t) + α(s)x(s) ds = f(t), t [a, b], (V ) Key Words: Volterra, Henstock ntegral, Bochner ntegral, ntegral equaton Mathematcal Revews subject classfcaton: 45D05, 34A12, 26A39 Receved by the edtors June 10,

2 390 M. Federson and R. Bancon n the sense of the varatonal ntegral of Henstock whch concdes wth the ntegral of Kurzwel n the real-valued case. We also obtan specal results for the equaton (V ) when the ntegral s that of Bochner-Lebesgue. We work n a general Banach space-valued context. Let [a, b] be a compact nterval of R, X be a Banach space and L(X) = L(X, X) be the space of lnear contnuous functons from X to X. Let x and f be functons from [a, b] to X and let α be a functon from [a, b] to L(X). As we ntend that the kernel α of (V ) s weak enough so that dscontnutes, sngulartes, nfnte varaton or nonabsolute ntegrablty can be taken nto account, we consder α Henstock ntegrable. We consder the functons x, f : [a, b] X where X s a Banach space n order to use fxed pont theorems and we prove that f ether x s a contnuous functon or x s of bounded varaton, then α x : [a, b] X s Henstock ntegrable (see Theorem 2.5 and [6]). It s a well-known result that f α s Henstock ntegrable, then there exsts a sequence of closed sets {X n } n N such that X n [a, b], (.e., X n X n+1 [a, b], for every n N and X n = [a, b]), and the restrcton of α to X n, (we wrte α Xn ), s Bochner-Lebesgue ntegrable for every n N, (see, for nstance, [16], Th. 2.10). However, t s a recent result, (see [7]), that L lm α = K α (1.1) n X n unformly for every t [a, b], where L and K denote respectvely the ntegrals of Bochner-Lebesgue and of Henstock, (we use K for Kurzwel). From the Contracton Prncple we can deduce the exstence and unqueness of a soluton of (V ) n the sense of the Bochner-Lebesgue ntegral, provded L α ( ) < 1. Then we obtan conclusons about the exstence of a soluton of (V ) n the sense of the Henstock ntegral by applyng a fxed pont theorem for sequences of mappngs correspondng to the sequence of equatons (V ) n the sense of the Bochner-Lebesgue ntegral obtaned through (1.1) and such that L X n α( ) < 1, for every n N. Snce the space of Bochner-Lebesgue ntegrable functons s a subspace of the space of Henstock ntegrable ones, then we obtan smlar but stronger results when we suppose that α s Bochner-Lebesgue ntegrable. In Secton 1 we gve the basc defntons and the ntroductory results. The man theorems le on Secton 2. Sectons 3 and 4 consst of consequences and applcatons of the man results.

3 Lnear Integral Equatons of Volterra The Integrals of Kurzwel and Henstock Let [a, b] be a compact nterval of R. We say that d = (ξ, t ) s a tagged dvson of [a, b] whenever (t ) s a dvson of [a, b] (.e., a = t 0 < t 1 <... < t n = b) and x [t 1, t ], for every. We denote by T D the set of all tagged dvsons of [a, b]. A gauge of [a, b] s a functon δ : [a, b] (0, ) and d = (t ) T D s δ-fne f for every, [t 1, t ] B δ(ξ) (ξ ) = {t [a, b]; t ξ < δ (ξ )}. In what follows X and Y are Banach spaces, L (X, Y ) s the Banach space of lnear contnuous functons from X to Y, L(X) = L(X, X) and X = L(X, R). Gven functons f : [a, b] X and α : [a, b] L (X, Y ), we say that f s Kurzwel α-ntegrable, (we wrte f K α ([a, b], X)), and that I Y s ts ntegral, (we wrte I = K dα f = K dα(t)f(t) ), f gven ε > 0, there s a gauge δ of [a, b] such that for every δ-fne d = (t ) T D, [α (t ) α (t 1 )]f (ξ ) I < ε. We say that f s Henstock α-ntegrable or varatonal α-ntegrable, (we wrte f H α ([a, b], X)), f there s a functon F α : [a, b] Y, (called the assocated functon of f wth respect to α), such that for every ε > 0, there s a gauge δ of [a, b] such that for every δ-fne d = (t ) T D, [α(t ) α(t 1 )]f(ξ ) [F α (t ) F α (t 1 )] < ε. Clearly H α ([a, b], X) K α ([a, b], X) and f X s of fnte dmenson, then H α ([a, b], X) = K α ([a, b], X). We denote by f α the ndefnte ntegral of f K α ([a, b], X) that s, f α (t) = K dα f for every t [a, b]. When α (t) = t, then we replace K α ([a, b], X), H α ([a, b], X) and f α respectvely by K([a, b], X), H([a, b], X) and f. Gven A [a, b], let χ A denote the characterstc functon of A and f A denote the restrcton of f to A. Let α : [a, b] L (X, Y ) be a functon and [c, d] [a, b]. The followng propertes are not dffcult to prove. ) If f K α ([a, b], X) (resp. f H α ([a, b], X)), then f K α ([c, b], X) (resp. f H α ([c, d], X)). ) If f K α ([a, b], X), then K dα (t) χ [c,d] (t) f (t) = K dα (t) f (t). [c,d] ) Gven f K α ([c, d], X), let f : [a, b] X be such that f [c,d] = f and f (t) = 0 otherwse. Then K dα (t) f (t) = K dα (t) f (t). [c,d]

4 392 M. Federson and R. Bancon Let C([a, b], X) and G([a, b], X) be respectvely the Banach spaces of contnuous and of regulated functons from [a, b] to X endowed wth the supremum norm whch we denote by. We denote respectvely by f (ξ+) and by f (ξ ) the rght and left lmts of f : [a, b] X at ξ [a, b] when they are defned and exst. Let C σ ([a, b], L (X, Y )) be the set of all functons α : [a, b] L (X, Y ) that are weakly contnuous, (.e., for every x X, the functon t [a, b] α (t) x Y s contnuous), and let and G σ ([a, b], L (X, Y )) be the set of all weakly regulated functons α : [a, b] L (X, Y ) (.e., for every x X, the functon t [a, b] α (t) x Y s regulated). Gven x X, let α ( ξ ˆ+ ) x = lm (αx) (ξ+), for every ξ [a, b), and let α ( ξ ˆ ) x = ρ 0 lm (αx) (ξ ), for every ξ (a, b]. By the Banach-Stenhauss Theorem, α ( ξ ˆ+ ) ρ 0 and α ( ξ ˆ ) exst and belong to L (X, Y ). Then by the Unform Boundedness Prncple t follows that G σ ([a, b], L (X, Y )) s a Banach space when equpped wth the supremum norm. Let SV ([a, b], L (X, Y )) be the space of all functons α : [a, b] L (X, Y ) of bounded semvaraton (also called of bounded B- varaton - see [19]) wth semvaraton denoted by SV (α) and let BV ([a, b], X) be the space of all functons f : [a, b] X of bounded varaton wth varaton denoted by V (f). Then BV ([a, b], L (X, Y )) SV ([a, b], L (X, Y )) and SV ([a, b], L (X, R)) = BV ([a, b], X ). Moreover, SV ([a, b], L (X)) = BV ([a, b], L (X)) f X s of fnte dmenson. When endowed wth the norm gven by the varaton, the space BV a ([a, b], X) = {f BV ([a, b], X); f (a) = 0} s complete. For more nformaton about the above spaces see [8], [9] or [10]. The followng result s the analogous of Saks-Henstock Lemma for the Steltjes case. Its proof follows the standard steps (see, for nstance, [19]) Lemma 2.1. (Saks-Henstock Lemma) Let α : [a, b] L(X, Y ) and f K α ([a, b], X). If for ε > 0, the gauge δ of [a, b] s such that for every δ-fne d = (t ) T D, d [α (t ) α (t 1 )]f (ξ ) K =1 dα (t) f (t) < ε, then for a c 1 η 1 d 1 c 2 η 2 d 2... c k η k d k b wth [c j, d j ] (η j δ(η j ), η j + δ(η j )) for every j, k {[α (d j ) α (c j )]f (η j ) K dα (t) f (t)} ε. [c j,d j] j=1

5 Lnear Integral Equatons of Volterra 393 Theorem 2.2. ([2], Th ) If α G σ ([a, b], L(X, Y )) (respectvely α C σ ([a, b], L(X, Y ))) and f K α ([a, b], X), then f α G([a, b], X) (resp. f α C([a, b], X)). Proof. In what follows we prove that f α (ξ+) f α (ξ) = [α ( ξ ˆ+ ) α (ξ)]f (ξ), for every ξ [a, b). The proof that f α (ξ) f α (ξ ) = [α (ξ) α ( ξ ˆ ) ]f (ξ), for every ξ (a, b], follows n an analogous way. By hypothess, f K α ([a, b], X). Hence, gven ε > 0, there s a gauge δ of [a, b] such that for every δ-fne d = (t ) T D, [α(t ) α(t 1 )]f(ξ ) K dαf < ε 2. Let ξ [a, b). Snce α G σ ([a, b], L(X, Y )), there exsts (αx) (ξ+), for every x X. In partcular, there exsts µ > 0, such that for every 0 < ρ < µ, [α(ξ + ρ) α(ξ ˆ+)]f(ξ) < ε 2. If δ(ξ) < µ and 0 < ρ < δ(ξ), then Lemma 2.1 mples that [α(ξ + ρ) α(ξ)]f(ξ) K dα f ε 2. [ξ,ξ+ρ] Thus, f α (ξ+) f α (ξ) [α ( ξ ˆ+ ) α (ξ)]f (ξ) = K dα f [α ( ξ ˆ+ ) α (ξ)]f (ξ) [ξ,ξ+ρ] K dα f [α (ξ + ρ) α (ξ)]f (ξ) [ξ,ξ+ρ] + [α (ξ + ρ) α (ξ)]f (ξ) [α ( ξ ˆ+ ) α (ξ)]f (ξ) < ε. Gven a dvson d = (t ) of [a, b], let d = max {t t 1 }. Then the Remann-Steltjes ntegrals are gven by dα f = dα (t) f (t) = lm [α (t ) α (t 1 ) f (ξ )] d 0

6 394 M. Federson and R. Bancon and α df = α (t) df (t) = lm α (ξ ) [f (t ) f (t 1 )]. d 0 The followng asserton s well known. Theorem 2.3. ([8], I.3.4 and I.4.5 or [9], I.4.6, I.4.12, I.4.19 and I.4.20) ) If α SV ([a, b], L(X, Y )) and f C([a, b], X), then dα f exsts and dα f SV (α) f. ) If α C ([a, b], L(X, Y )) and f BV ([a, b], X), then α df exsts and α df α V (f). Theorem 2.4. ([2], Th ) dα f exsts f and only f α df exsts and, n ths case, the Integraton by Parts Formula dα f = α (b) f (b) α (a) f (a) α df holds. Proof. Suppose that the Remann-Steltjes ntegral, dα f, exsts. Then, for every ε > 0, there s a δ > 0 such that for every d = (t ) T D wth d = max {t t 1 } < δ, [α(t ) α(t 1 )]f(ξ ) dα f < ε. Hence, { α (b) f (b) α (a) f (a) dα f } α (ξ ) [f (t ) f (t 1 )] = [α (t ) f (t ) α (t 1 ) f (t 1 )] dα f α (ξ ) [f (t ) f (t 1 )] = [α (t ) α (ξ )]f (t ) + [α (ξ ) α (t 1 )]f (t 1 ) dα f α (ξ ) [f (t ) f (t 1 )] < ε. Analogously, f α df exsts, then dα f exsts wth dα f = α (b) f (b) α (a) f (a) α df and the proof s complete. The next asserton follows from Theorems 2.3 and 2.4.

7 Lnear Integral Equatons of Volterra 395 Theorem 2.5. The Remann-Steltjes ntegrals dα f and α df exsts and the Integraton by Parts Formula holds f one of the followng condtons s satsfed: ) α SV ([a, b], L(X, Y )) and f C([a, b], X); ) α C ([a, b], L(X, Y )) and f BV ([a, b], X). Theorem 2.6. ([19], Th. 15) If α SV ([a, b], L(X, Y )) G σ ([a, b], L(X, Y )) and f G([a, b], X), then f K α ([a, b], X). The next result comes from the defntons. Theorem 2.7. Let α H ([a, b], L(X, Y )) and f K eα ([a, b], X) (respectvely f H eα ([a, b], X)). If f s bounded, then α f K ([a, b], Y ) (resp. α f H ([a, b], Y )) and K α f = K d α f, where α denotes the ndefnte ntegral of α. Corollary 2.8. Let α H ([a, b], L(X, Y )) wth α SV ([a, b], L(X, Y )) and f G([a, b], X) (respectvely f C([a, b], X)). Then α f K ([a, b], Y ) wth K α f = K d α f (resp. K α f = d α f ). Proof. Suppose that f G([a, b], X). Then f s bounded. From Theorem 2.2, α C ([a, b], L(X, Y )). Hence by Theorem 2.6, f K eα ([a, b], X). Suppose now that f C([a, b], X). By Theorems 2.2 and 2.5, f R eα ([a, b], X). But R eα ([a, b], X) K eα ([a, b], X) (see the comments bellow) and the proof s complete. If we take constant gauges n the defnton of K α ([a, b], X), then we obtan R α ([a, b], X) and ths fact that the Remann-Steltjes ntegrals are partcular cases of the Kurzwel vector ntegrals was essental n the prevous proof. If X s of fnte dmenson, then the Remann-Steltjes ntegrals are also Henstock vector ntegrals, once n ths case the spaces of vector ntegrable functons of Kurzwel and of Henstock concde. However, when the dmenson of X s nfnte, then we may have f R([a, b], X) \ H([a, b], X) as shown by the next example. Example 2.1. Let X = l 2 ([a, b]) and f : [a, b] X be defned by f (t) = e t (.e., e t (s) = 1 f s = t, and e t (s) = 0 f s t). Gven ε > 0, there exsts δ > 0 wth δ 1/2 ε < (b a) such that for every d = (t ) T D 1/2 wth d = max {t t 1 } < δ, f (ξ ) (t t 1 ) 0 = [ e ξ (t t 1 ) = (t t 1 ) 2] 1/2 2 2

8 396 M. Federson and R. Bancon where we appled the Bessel s equalty. Hence, [ (t t 1 ) 2] 1/2 < δ 1/2 (t t 1 ) 1/2 = [ δ (b a) ] 1/2 < ε and t follows that f R([a, b], X) wth f = 0. Now, suppose that f H([a, b], X) and let F be the assocated functon of f. Then F (t) F (r) = [r,t] f (s) ds = f (t) f (r) = 0, for every [r, t] [a, b]. Hence F s constant and, therefore, for every d = (t ) T D, F (t ) F (t 1 ) f (ξ ) (t t 1 ) = e ξ (t t 1 ) = and we have a contradcton. Thus, f / H([a, b], X). t t 1 = b a Theorem 2.9. Let α H ([a, b], L(X, Y )) wth α BV ([a, b], L(X, Y )) and f C([a, b], X). Then α f H ([a, b], Y ) wth K α f = d α f. Proof. Snce α H ([a, b], L(X, Y )), for every ε > 0, there s a gauge δ of [a, b] such that for every δ-fne d = (t ) T D, α (ξ ) (t t 1 ) K α < ε. [t 1,t ] From the Corollary 2.8, α f K ([a, b], Y ) wth K α f = d α f. Hence, α (ξ ) f (ξ ) (t t 1 ) K α f [t 1,t ] < α (ξ ) f (ξ ) (t t 1 ) K α (t) f (ξ ) dt [t 1,t ] + K α f K α (t) f (ξ ) dt [t 1,t ] [t 1,t ] f α (ξ ) (t t 1 ) K α [t 1,t ] + K α (t) [f (t) f (ξ )]dt [t 1,t ] < f ε + d α (t) [f (t) f (ξ )]dt. [t 1,t ]

9 Lnear Integral Equatons of Volterra 397 Now applyng the Integraton by Parts Formula and then the Fundamental Theorem of Calculus for the Remann ntegral we have that [t 1,t ] d α(t)[f(t) f(ξ )] = α(t )[f(t ) f(ξ )] α(t 1 )[f(t 1 ) f(ξ )] α(t) d[f(t) f(ξ )] [t 1,t ] = α(t )[f(t t) f(ξ )] α(t)df(t) α(t 1 )[f(t 1 ) f (ξ )] [ξ,t ] α (t) df (t) [t 1,ξ ] = α (t ) df (t) α (t) df (t) [ξ,t ] [ξ,t ] + α (t 1 ) df (t) α (t) df (t) [t 1,ξ ] [t 1,ξ ] = [ α (t ) α (t)]df (t) + [ α (t 1 ) α (t)]df (t) [ξ,t ] [t 1,ξ ] <V ( α) where ω (f) denotes the oscllaton of f on [a, b]. Snce f s contnuous and α s of bounded varaton, the proof s complete. We denote by 1 ([a, b], X) the space of all functons f : [a, b] X whch are Bochner-Lebesgue ntegrable wth fnte ntegral. The ntegral of f 1 ([a, b], X) s denoted by L f = L f (t) dt and we wrte f 1 = L f (). From the Remannan defnton of 1([a, b], X) (see [18] and [13]), t follows that 1 ([a, b], X) H([a, b], X). Besdes, f X = R, then the postve functons whch are Kurzwel-Henstock ntegrable are also Lebesgue ntegrable. Thus, f f H ([a, b], R) s absolutely ntegrable (.e., f () 1 ([a, b], R)), then f s Lebesgue ntegrable. From Example 1.1 before Theorem 2.9, we also observe that when the dmenson of X s nfnte, then t may happen that f R([a, b], X) \ 1 ([a, b], X). Theorem ([13], 9) Let f H([a, b], X). Then f s absolutely ntegrable f and only f f BV ([a, b], X). In any case, f 1 = V ( f).

10 398 M. Federson and R. Bancon Proof. Suppose that f s absolutely ntegrable. Snce { } ( V f) = sup f (t ) f (t 1 ) ; a = t 0 < t 1 <... < t n = b we have that f (t ) f (t 1 ) = K f [t 1,t ] L [t 1,t ] f = f 1. Now, suppose that f BV ([a, b], X). We wll prove that there exsts K f = L f = V ( f ). Gven ε > 0, we wll fnd a gauge δ of [a, b] such that for every δ-fne d = (t ) T D, But f d = (t ) T D s δ-fne, then f(ξ t) (t t 1 ) V ( f) f(ξ ) (t t 1 ) V ( f) < ε. f(ξ ) (t t 1 ) K f + K f V [t 1,t ] [t 1,t ] f (ξ ) (t t 1 ) K By the defnton of V [t 1,t ] f + f (t ) f (t 1 ) V ( ) f ( ) f ( f), we may take the dvson of [a, b] such that the last summand s smaller than ε/2. Snce f H([a, b], X), then we may take a gauge δ of [a, b] such that for every δ-fne d = (t ) T D, the frst summand s also smaller than ε/2, and we may suppose that the ponts chosen for the second summand are ponts of the δ-fne tagged dvson d = (t ). The proof s then complete. Theorem If α 1 ([a, b], L(X, Y )) and f G([a, b], X), then α f 1 ([a, b], Y ) and L α f = K d α f. Proof. Snce f s bounded, then f <. The functon α () f () : [a, b] R s m-measurable (m for the Lebesgue measure) and α (t) f (t)

11 Lnear Integral Equatons of Volterra 399 f α (t) for every t [a, b]. From the fact that 1 ([a, b], R) s a vector lattce t follows that α () f () 1 ([a, b], R) and therefore α f 1 ([a, b], Y ). Now we prove the equalty. Let ε > 0. Snce α 1 ([a, b], L(X, Y )) H ([a, b], L(X, Y )), there s a gauge δ 1 of [a, b] such that for every δ 1 -fne d = (ζ j, s j ) T D, α (ζ j ) (s j s j 1 ) [ α (s j ) α (s j 1 )] < ε. j By Theorem 2.10, α s a functon of bounded varaton and therefore regulated and of bounded semvaraton. Then by Theorem 2.6, there exsts the Kurzwel ntegral K d α f whch means that there s a gauge δ 2 of [a, b] such that for every δ 2 -fne d = (ρ k, r k ) T D, [ α (r k ) α (r k 1 )]f (ρ k ) K k d α f < ε. Let δ be a gauge of [a, b] such that δ(ξ) δ l (ξ), for every ξ [a, b] and l = 1, 2. Then for every δ-fne d = (t ) T D, we have that α(ξ )f(ξ )(t t 1 ) K d α f α (ξ ) f (ξ ) (t t 1 ) [ α (t ) α (t 1 )]f (ξ ) + [ α (t ) α (t 1 )]f (ξ ) K d α f < ε f + ε. Corollary Let I([a, b], X) denote one of the spaces BV ([a, b], X) or C([a, b], X). If α 1 ([a, b], L(X, Y )) and f I([a, b], X), then α f 1 ([a, b], Y ) wth L α f = d α f. Proof. In any of the cases the result comes from Theorem 2.5, snce α C ([a, b], L(X, Y )) (Theorem 2.2) and α BV ([a, b], L(X, Y )) (Theorem 2.10). 3 The Volterra-Henstock Lnear Integral Equaton The frst result of ths secton gves a necessary condton for the exstence of the Henstock ntegral.

12 400 M. Federson and R. Bancon Lemma 3.1. If f H([a, b], X), then there exsts a sequence of closed sets {X n } n N such that X n [a, b] (.e., X n X n+1 [a, b], for every n N and X n = [a, b]) and f 1 (X n, X), for every n N. Furthermore, L X f = K n f unformly for every t [a, b]. lm n Proof. It suffces to adapt the proof gven n [7] for the Banach space-valued case. Lemma 3.2. ([14], Th ) Let {T n } n N be a sequence of mappngs from X to X such that each mappng has a fxed pont x n = T n (x n ), let T : X X be such that for some nteger m, T m s a contracton, where T m s the composton of T m tmes, and suppose that T n T unformly. Then x n x 0 = T (x 0 ). Theorem 3.3. Gven α H ([a, b], L(X, Y )) and a functon f : [a, b] X, then α f H ([a, b], Y ) f one of the followng condtons s satsfed: ) f BV ([a, b], X); ) f C([a, b], X) and α BV ([a, b], L(X, Y )). In any case, K α f = d α f. Proof. For ), see [2], Th , or [6]; for ) see Theorem 2.9. Remark 1. It s also true that f α BV ([a, b], L(X, Y )) and f H([a, b], X), then α f H ([a, b], Y ) and K α f = α d f, (see [2], Th or [6]). We consder the next Volterra-Henstock lnear ntegral equaton x(t) + K α(s)xt(s)ds = f(t), t [a, b], (V ) H n the followng two cases a) α H ([a, b], L (X)) s bounded and x, f BV a ([a, b], X); b) α H ([a, b], L (X)) wth α BV ([a, b], L (X)) and x, f C([a, b], X). In vew of Theorem 2.10, we can replace b) by b ) α H ([a, b], L (X)) s absolutely ntegrable and x, f C([a, b], X). In the real case when X = R, t follows from the consderatons before Theorem 2.10 that b ) s equvalent to a case when α 1 ([a, b], R). But ths wll be treated n a more general context (the Bochner-Lebesgue ntegral) n Secton 4. By means of Lemmas 3.1 and 3.2 we wll be able to obtan conclusons about equaton ((V ) H ) through the analyss of equatons of ((V ) L ) type n any of the cases a) and b ).

13 Lnear Integral Equatons of Volterra 401 Case a): Let α H ([a, b], L (X)) be bounded and f BV a ([a, b], X). By Lemma 3.1 and the Corollary 2.12, gven n N, we can consder the mappng T n gven by (T n x) (t) = f (t) L χ Xn (s) α (s) x (s) ds, t [a, b], where χ Xn denotes the characterstc functon of X n. By Theorem 2.10, T n takes elements from BV a ([a, b], X) to BV a ([a, b], X). Furthermore, each T n s contnuous snce χ Xn () α () 1 <, and T n s a contracton whenever χ Xn () α () 1 < 1. Consder the contnuous mappng T : BV a ([a, b], X) BV a ([a, b], X) defned by (T x) (t) = f (t) K α (s) x (s) ds, t [a, b]. Gven x BV a ([a, b], X), T x really belongs to BV a ([a, b], X). In fact, gven any dvson (t ) of [a, b], then T x (t ) T x (t 1 ) f (t ) f (t 1 ) + K α (s) x (s) ds [t 1,t ] V (f) + α x (b a). Wth the notaton and consderatons above, the proof of the next theorem follows easly. Theorem 3.4. Gven α H ([a, b], L (X)) bounded, consder the Volterra- Henstock lnear ntegral equaton x (t) + K α (s) x (s) ds = f (t), t [a, b], (V ) H the Volterra-Bochner-Lebesgue lnear ntegral equatons obtaned from Lemma 3.1 x(t) + L χ Xn (s)α(s)x(s)ds = f(t), t [a, b], and n N (V ) Ln and the mappng T : BV a ([a, b], X) BV a ([a, b], X) defned by (T x) (t) = f (t) K α (s) x (s) ds, t [a, b],

14 402 M. Federson and R. Bancon where n all cases x, f BV a ([a, b], X). If χ Xn ()α() 1 < 1 for each n N, then gven f BV a ([a, b], X), each equaton (V ) Ln admts one and only one soluton x n BV a ([a, b], X). Consder also the followng condtons: ) {x n } n N has a convergent subsequence x nk x 0 ; ) T m s a contracton for some m > 1. If ) s satsfed, then x 0 BV a ([a, b], X) s a soluton of (V ) H. If ) s satsfed, then there exsts x = lm n x n and x BV a ([a, b], X) satsfes (V ) H. Proof. For each n N, χ Xn () α () 1 < 1. Hence each T n gven by (T n x) (t) = f (t) L χ Xn (s) α (s) x (s) ds, t [a, b], s a contracton and therefore has a unque fxed pont x n by the Contracton Prncple. Therefore (V ) Ln has one and only one soluton x n BV a ([a, b], X). If ) holds, then gven n k N, x 0 T x 0 x 0 x nk + x nk T nk x nk + T nk x nk T nk x 0 + T nk x 0 T x 0, where the frst summand tends to zero as k 0 (by the convergence of the subsequence), the second summand s equal to zero (once x nk s a fxed pont of T nk ), the thrd summand s smaller than χ Xn () α () 1 x nk x 0 whch tends to zero as k 0 (because χ Xn () α () 1 < 1 and by the convergence of the subsequence), and the fourth summand tends to zero as k 0, (by Lemma 3.1). Suppose now that ) holds. From Lemma 3.1, T n T. As a matter of fact, T n T unformly. Thus x n x = T x, (by Lemma 3.2), and we complete the proof. Lemma 3.5. (see [12], 3.3; see [17] for the real-valued case) Suppose that f H([a, b], X) and g : [a, b] X s a functon such that g = f m-almost everywhere (m for the Lebesgue measure). Then g H([a, b], X) and g = f. We say that two functons g and f of H([a, b], X) are equvalent f g = f and we wrte H([a, b], X) A to denote the space of all equvalence classes of functons of H([a, b], X) endowed wth the Alexewcz norm f H([a, b], X) f A = f. In what follows we wll wrte f H([a, b], X) A to denote that we have pcked up a functon f = f Φ Φ, where Φ H([a, b], X) A. The next asserton s a consequence of [11], Th. 3.5.

15 Lnear Integral Equatons of Volterra 403 Theorem 3.6. Gven α SV ([a, b], L (X)) wth α (b) = 0, consder the Volterra-Henstock lnear ntegral equaton x (t) + K α (s) x (s) ds = f (t), t [a, b], (V ) L where x, f H([a, b], X) A. Then for every f H([a, b], X) A there exsts one and only one soluton x H([a, b], X) A wth x (t) = f (t) K ρ (t, s) x (s) ds, t [a, b], where the kernel ρ : [a, b] [a, b] L (X) s bounded and can be gven by the Neumann seres whch converges n L (H([a, b], X) A ). Theorem 3.7. Let α H ([a, b], L (R)) = H ([a, b], R) be bounded and suppose that there exsts a sequence {X n } n N such that each X n s a fnte unon of nonoverlappng closed ntervals, X n [a, b] and α BV (X n, L (R)) = BV (X n, R), for every n N. Consder the Volterra-Kurzwel-Henstock lnear ntegral equatons x (t) + K α (s) x (s) ds = f (t), t [a, b]. (V ) H and x (t) + K χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Hn where x, f BV a ([a, b], R). Then gven n N and f BV a ([a, b], R), (V ) Hn admts one and only one soluton x n BV a ([a, b], R), there exsts x = lm x n n and x BV a ([a, b], R) satsfes (V ) H. Proof. For every n N, let X n = kn [a n, b n]. By Lemma 3.5, we may suppose that α ( b n) = 0 for every and every n. Snce α BV (Xn, L (R)) mples that α BV ( [a n, b n], L (R) ) for every, then gven n N and {1,..., k n }, t follows from Theorem 3.6 that there exsts a unque soluton x n H ( [a n, b n], R ) A of =1 x n (t) + K [a n,t] α [a n,b n ] (s) x (s) ds = f (t), t [a n, b n], (V ) Hn,

16 404 M. Federson and R. Bancon such that x n (t) = f [a n,b n ] (t) K [a n,t] ρ n (t, s) x n (s) ds, t [a n, b n], wth bounded kernel ρ n:[a n, b n] [a n, b n] L (R) gven by the Neumann seres whch converges n L ( H ( [a n, b n], R ) A). As a matter of fact, x n BV ( [a n, b n], R ) because f BV ( [a n, b n], R ) and ρ n < (see the consderatons before Theorem 3.4). Now, for every n N and every {1,..., k n }, let yn : [a, b] R be gven by yn = x n on [a n, b n] and yn = 0 otherwse, and let φ n : [a, b] [a, b] L (R) be defned by φ n = ρ n on [a n, b n] [a n, b n] and φ n = 0 otherwse. Then, x n = kn yn BV a ([a, b], R) s a (unque) soluton of (V ) Hn. Moreover, =1 k n k n ) x n (t) = yn(t) = (χ [a n,b n ] f (t) K φ =1 =1 [a n,b n ] n (t, s) yn (s) ds k n ( K ) =f (t) φ [a n,b n ] n (t, s) yn (s) ds =1 =f(t) K =f (t) K k n X n =1 X n φ n (t, s) y n (s) ds ρ n (t, s) x n (s) ds, where ρ n (t, s) = kn φ n (t, s), for each n N. =1 t [a, b], Gven n N, let T n : BV a ([a, b], R) BV a ([a, b], R) be defned by (T n x) (t) = f (t) K χ Xn (s) α (s) x (s) ds, t [a, b]. Snce the Domnated Convergence Theorem holds for the real-valued Henstock ntegral and X n [a, b], then T n T, where T : BV a ([a, b], R) BV a ([a, b], R) s gven by (T x) (t) = f (t) K α (s) x (s) ds, t [a, b], wth x, f BV a ([a, b], R). From the fact that the functons x, f, x n and χ Xn ( ) α ( ) are of bounded varaton, then T n T unformly. The rest of

17 Lnear Integral Equatons of Volterra 405 the proof follows the steps of the proof of Theorem 3.4, (see the Remark after Theorem 3.4). Remark 2. When we consder Banach space-valued functons, then nether the Domnated Convergence Theorem nor the Monotone Convergence Theorem hold for the Henstock ntegral. The next example of Brkhoff ([1]) shows us that fact. Example 3.1. Consder f : [0, 1] X = l 2 (N) defned by f = f, where f (t) = 2 e,j, f j 2 < t j , = 1, 2,..., j = 0, 1,..., 2 1. We use e 2,j to denote a doubly nfnte set of orthonormal vectors of l 2 (N). Then f = f f s such that f H ([0, 1], X) for every N, but f s nowhere dfferentable and hence f / H ([0, 1], X) (by the Fundamental Theorem of Calculus - see [12] or [5] for the Banach case or [16] for the real-valued case). We pont out however that f (t) = K f exsts n the sense of the Kurzwel [0,t] ntegral for every t [0, 1]. Indeed, because the space X = l 2 (N) fulflls the requred condtons whch make the Monotone Convergence Theorem be vald for the Kurzwel ntegral (see [3]). The next result comes drectly from the Contracton Prncple. Theorem 3.8. Let α H ([a, b], L (X)) be bounded wth (b a) α < 1, then gven f BV a ([a, b], X), there s one and only one x BV a ([a, b], X) that satsfes the lnear ntegral equaton of Volterra-Henstock x (t) + K α (s) x (s) ds = f (t), t [a, b]. (V ) H Case b ): Let α H ([a, b], L (X)) be absolutely ntegrable and suppose f C([a, b], X). By Theorems 2.5 and 2.6 we can consder the mappng T : C([a, b], X) C([a, b], X) defned by (T x) (t) = f (t) K α (s) x (s) ds, t [a, b]. In addton, by Lemma 3.1, for every n N, we can consder the contnuous mappng T n : C([a, b], X) C([a, b], X) gven by (T n x) (t) = f (t) L χ Xn (s) α (s) x (s) ds, t [a, b]. Lkewse case a), f χ Xn () α () 1 < 1, then T n s a contracton. =1

18 406 M. Federson and R. Bancon Theorem 3.9. Gven α H ([a, b], L (X)) absolutely ntegrable, consder the lnear ntegral equaton of Volterra-Henstock x (t) + K α (s) x (s) ds = f (t), t [a, b], (V ) H the Volterra-Bochner-Lebesgue lnear ntegral equatons obtaned through Lemma 3.1 x (t) + L χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Ln and the mappng T : C([a, b], X) C([a, b], X) defned by (T x) (t) = f (t) K α (s) x (s) ds, t [a, b], where x, f C([a, b], X). If χ Xn () α () 1 < 1 for each n N, then gven f C([a, b], X), each equaton (V ) Ln admts one and only one soluton x n C([a, b], X). Consder also the followng condtons: ) {x n } n N has a convergent subsequence x nk x 0 ; ) α s bounded and T m s a contracton for some m > 1. If ) s satsfed, then x 0 C([a, b], X) s a soluton of (V ) H. If ) s satsfed, then there exsts x = lm n x n and x C([a, b], X) satsfes (V ) H. Proof. The proof s analogous to the that of Theorem 3.4. Theorem Let α H ([a, b], L (X)) be absolutely ntegrable and bounded wth (b a) α < 1. Then for every f C([a, b], X), there s one and only one x C([a, b], X) that satsfes the Volterra-Henstock lnear ntegral equaton x (t) + K α (s) x (s) ds = f (t), t [a, b]. (V ) H Proof. The asserton follows drectly from the Contracton Prncple. 4 The Volterra-Bochner-Lebesgue Lnear Integral Equaton Let I([a, b], X) denote one of the Banach spaces G([a, b], X), BV a ([a, b], X) or C([a, b], X). From Theorem 2.11 and ts Corollary, we can consder the

19 Lnear Integral Equatons of Volterra 407 Volterra-Bochner-Lebesgue lnear ntegral equaton x (t) + L α (s) x (s) ds = f (t), t [a, b] (V ) L where α 1 ([a, b], L (X)) and x, f I([a, b], X). Prncple mples the followng. Then the Contracton Theorem 4.1. Suppose α 1 ([a, b], L (X)) wth α 1 < 1. Then gven f I([a, b], X), there s one and only one x I([a, b], X) that satsfes the lnear ntegral equaton of Volterra-Bochner-Lebesgue (t) + L α (s) x (s) ds = f (t), t [a, b]. (V ) L Let I [a, b] be fnte. We say that f : [a, b] X s of bounded varaton on [a, b]\i whenever f s of bounded varaton on every closed nterval contaned n [a, b]. Let f 1 ([a, b], X) be contnuous or of bounded varaton on [a, b] \ I. Then t s mmedate that there exsts a sequence of sets {X n } n N such that each X n s the fnte unon of nonoverlappng closed ntervals and X n I =, X n [a, b], and for every t [a, b], L lm f (s) ds = L f (s) ds. n X n If I [a, b] s fnte and α 1 ([a, b], L (X)) s contnuous or of bounded varaton on [a, b] \ I, then there exsts a sequence {X n } n N as above such that for every t [a, b], L lm n α (s) ds = L α (s) ds. (4.1) X n Smlarly, f x G([a, b], X), then there exsts the ntegral L α x (Theorem 2.11) and we can fnd a sequence {Y n } n N such that each Y n s the fnte unon of nonoverlappng closed ntervals and Y n I =, Y n [a, b] and for every t [a, b], lm n L Y n α (s) x (s) ds = L α (s) x (s) ds. We affrm however that n fact the same sequence for α 1 ([a, b], L (X)) suts for α x 1 ([a, b], X), that s L lm α (s) x (s) ds = L α (s) x (s) ds, n X n

20 408 M. Federson and R. Bancon for every t [a, b]. Indeed. Takng approxmatng Remannan sums for the ntegrals L α (s) x (s) ds and L X n α (s) x (s) ds we have that α(ξ )x(ξ )(t t 1 ) χ Xn (ξ )α(ξ )x(ξ )(t t 1 ) x (1 χ Xn )(ξ )α(ξ )(t t 1 ) whch can be made suffcently small by (4.1) and by the Remannan defnton of Bochner-Lebesgue ntegral, (see [13] and [18]). Theorem 4.2. Let I([a, b], X) denote one ( of the two ) spaces C([a, b], X) or G([a, b], X), BV a ([a, b], X). Gven α 1 [a, b], L(X), consder the Volterra- Bochner-Lebesgue lnear ntegral equatons x (t) + L α (s) x (s) ds = f (t), t [a, b], (V ) L and x (t) + L χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Ln where x, f I([a, b], X) and the sequence {X n } n N s obtaned as n the prevous paragraph for α 1 ([a, b], L (X)). Suppose f I([a, b], X), and that χ Xn () α () 1 < 1 for every n N. Then (V ) Ln has one and only one soluton x n I([a, b], X). Consder also the followng condtons: ) {x n } n N has a convergent subsequence x nk x 0 ; ) α s bounded and T m s a contracton for some m > 1. If ) s satsfed, then x 0 I([a, b], X) s a soluton of (V ) L. If ) s satsfed, then there exsts x = lm n x n and x I([a, b], X) satsfes (V ) L. Proof. For each n N, let T n : I([a, b], X) I([a, b], X) be gven by T n x (t) = f (t) L χ Xn (s) α (s) x (s) ds, t [a, b], where x I([a, b], X). Then T n s contnuous (because χ Xn () α () 1 < ) and T n s a contracton whenever χ Xn () α () 1 < 1. The rest of the proof s analogous to the proof of Theorem 3.4 replacng the ntegral of Henstock by the ntegral of Bochner-Lebesgue. If ether I([a, b], X) = C([a, b], X) or I([a, b], X) = BV a ([a, b], X) n the theorem above, we obtan the stronger results of Theorems 4.3 and 4.5 below.

21 Lnear Integral Equatons of Volterra 409 Theorem 4.3. Gven α 1 ([a, b], L (X)), and I [a, b] fnte, suppose that α s contnuous on [a, b] \ I and consder the Volterra-Bochner-Lebesgue lnear ntegral equatons and x (t) + L x (t) + L α (s) x (s) ds = f (t), t [a, b], (V ) L χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Ln where x, f C([a, b], X) and the sequence {X n } n N satsfes the condtons of the paragraph before Theorem 4.2. Then gven f C([a, b], X) and n N, (V ) Ln has a soluton x n C([a, b], X). Besdes, there exsts x = lm x n and n x C([a, b], X) satsfes (V ) L. Proof. For each n N, (V ) Ln has a soluton x n C([a, b], X). Indeed. For every n N, let X n = kn [a n, b n]. Then α C ( [a n, b n], L (X) ), for =1 every n N and = 1,..., k n and by a well-known result from the Theory of Integral Equatons (see, for nstance, [15], p. 74), there exsts a soluton x n C ( [a n, b n], X ) of x n (t) + L [a n,t] α [a n,b n ] (s) x (s) ds = f (t), t [a n, b n], (V ) Ln, such that x n (t) = f [a n,b n ] (t) L [a n,t] ρ n (t, s) x n (s) ds, t [a n, b n], wth contnuous kernel ρ n:[a n, b n] [a n, b n] L (X) determned by the Neumann seres method. For every n N and every {1,..., k n }, let yn : [a, b] X be gven by yn = x n on [a n, b n] and yn = 0 otherwse, and let φ n : [a, b] [a, b] L (X) be defned by φ n = ρ n on [a n, b n] [a n, b n] and φ n = 0 otherwse. Then x n = kn yn s a soluton of (V ) Ln and x n (t) = f (t) L =1 X n ρ n (t, s) x n (s) ds, t [a, b],

22 410 M. Federson and R. Bancon where ρ n (t, s) = kn φ n (t, s), for each n N. =1 Moreover x n C([a, b], X), snce f and the ndefnte ntegral are contnuous (Theorem 2.2). Hence, for every n N, the mappng T n : C([a, b], X) C([a, b], X) gven by (T n x) (t) = f (t) L χ Xn (s) α (s) x (s) ds, t [a, b], has a fxed pont. The rest of the demonstraton follows the steps of Theorem 3.4 and the observaton n the proof of Theorem 4.2. Let L 1 ([a, b], X) A denote the space of all equvalence classes of functons of 1 ([a, b], X) endowed wth the Alexewcz norm. When we wrte f L 1 ([a, b], X) A we mean that we have chosen a functon f = f Φ Φ, where Φ L 1 ([a, b], X) A. The next result s a consequence of [11], Th. 3.5 and the Remark that follows t. Theorem 4.4. Gven α SV ([a, b], L (X)) wth α (b) = 0, consder the Volterra-Bochner-Lebesgue lnear ntegral equaton x (t) + L α (s) x (s) ds = f (t), t [a, b], (V ) L where x, f L 1 ([a, b], X) A. Then for every f L 1 ([a, b], X) A there exsts one and only one soluton x L 1 ([a, b], X) A wth x (t) = f (t) L ρ (t, s) x (s) ds, t [a, b], where the kernel ρ:[a, b] [a, b] L (X) s bounded and can be gven by the Neumann seres whch converges n L (L 1 ([a, b], X) A ). Theorem 4.5. Gven α 1 ([a, b], L (X)) and I [a, b] fnte, suppose that α s of bounded varaton on [a, b] \ I and consder the Volterra-Bochner- Lebesgue lnear ntegral equatons x (t) + L α (s) x (s) ds = f (t), t [a, b], (V ) L and x (t) + L χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Ln where x, f BV a ([a, b], X) and the sequence {X n } n N satsfes the condtons of the paragraph before Theorem 4.2. Then gven f BV a ([a, b], X) and n N, (V ) Ln has a unque soluton x n BV a ([a, b], X). Moreover, there exsts x = lm x n and x BV a ([a, b], X) satsfes (V ) L. n

23 Lnear Integral Equatons of Volterra 411 Proof. It follows from Theorem 4.4, each (V ) Ln has a unque soluton x n L 1 ([a, b], X) A and x (t) = f (t) L ρ (t, s) x (s) ds, t [a, b], where the kernel ρ:[a, b] [a, b] L (X) s bounded and can be gven by the Neumann seres. As a matter of fact, x n BV a ([a, b], X) for every n N, snce f BV a ([a, b], X) and the ndefnte ntegral of a Bochner-Lebesgue ntegrable functon s of bounded varaton (Theorem 2.10). Ths means that each T n has a fxed pont, where T n : BV a ([a, b], X) BV a ([a, b], X) s gven by (T n x) (t) = f (t) L χ Xn (s) α (s) x (s) ds, t [a, b]. The rest of the demonstraton follows the steps of Theorem 3.4 and the observaton n the proof of Theorem Applcatons 5.1 Applcatons to Ordnary Dfferental Equatons Consder the equaton ẋ = f (t, x) (5.1) where f : [a, b] B R s a functon and B R s an open set. Let J [a, b] be a closed nterval. We say that x : J R s a Carathéodory soluton (respectvely a Henstock soluton) of equaton (4.1) f and only f the followng condtons are satsfed: ) x (t) B m-almost everywhere on J; ) x (t) = x (c) + f (s, x (s)) ds, for every t, c J; [c,t] where m denotes Lebesgue measure and = L (resp. = K ), provded the ntegral exsts. From the Fundamental Theorem of Calculus t s mmedate that f x satsfes ) for = L (resp. = K ), then x s absolutely contnuous and dfferentable (resp. x satsfes the Strong Lusn Condton and s dfferentable m-almost everywhere) and ẋ = f (t, x) m-almost everywhere. Example 5.1. Let F : [0, 1] R be gven by F (t) = t sn 1 t f t 0, F (0) = 0, and f = F, that s f (t) = sn 1 t 1 t cos 1 f t 0 and f (0) = 0. Then t f H ([0, 1], R) \ 1 ([0, 1], R) and F = f (from the Fundamental Theorem of Calculus). Consder the ntal value problem

24 412 M. Federson and R. Bancon ẋ + αx = f, x (a) = 0 (5.2) where x G ([0, 1], R) and α 1 ([0, 1], R). Integratng (5.2) we obtan the Volterra-Lebesgue lnear ntegral equaton x (t) + K α (s) x (s) ds = f (t), t [a, b], (V ) L By Theorem 4.1, f α 1 < 1, then there exsts a unque soluton x G ([0, 1], R) of equaton (V ) H (note that the ndefnte ntegral F s contnuous - Theorem 2.2). In other words, f α 1 < 1, then there exsts a unque Carathéodory soluton x (whch s n fact absolutely contnuous by the Fundamental Theorem of Calculus) of equaton (5.1). Example 5.2. Consder the ntal value problem ẋ + αx = βu, x (a) = 0, (5.3) where x C ([a, b], R), α 1 ([a, b], R) and ether β H ([a, b], R) and u BV ([a, b], R) or β BV ([a, b], R) and u H ([a, b], R). In any case there exsts F (t) = K β (s) u (s) ds, for every t [a, b] (see [2] or [5]). Integratng (5.3) we obtan the lnear ntegral equaton of Volterra-Lebesgue x (t) + K α (s) x (s) ds = f (t), t [a, b], (V ) L If α 1 < 1, then there s a unque Carathéodory soluton x C([a, b], X) of equaton (5.3) (Theorem 4.1). Now we change the hypothess. Let x BV a ([a, b], R), α 1 ([a, b], R) and ether β 1 ([a, b], R) and u BV ([a, b], R) or β BV ([a, b], R) and u 1 ([a, b], R), then there exsts F (t) = L β (s) u (s) ds, for every t [a, b] (see [2] or [5]) and F BV a ([a, b], R) (Theorem 2.10). By Theorem 4.1, there s a unque Carathéodory soluton x BV a ([a, b], X) of equaton (5.3) whenever α 1 < 1. Example 5.3. The contnuous-tme lnear dynamc system ẋ (t) = A (t) x (t) + B (t) u (t) ẏ (t) = C (t) x (t) + D (t) u (t) (5.4) where A, B, C and D are contnuous matrx 1 1, has soluton gven by R x (t) = e A(s)ds x (a) + R e [a,τ] A(s)ds B (τ) u (τ) dτ.

25 Lnear Integral Equatons of Volterra 413 If x C ([a, b], R), A 1 ([a, b], R) and ether B H ([a, b], R) and u BV ([a, b], R) or B BV ([a, b], R) and u H ([a, b], R), then x (t) = e L R A(s)ds x (a) + K e L R [a,τ] A(s)ds B (τ) u (τ) dτ (5.5) s a Henstock soluton of the system. If moreover we have that A 1 < 1, then (5.5) s the unque (contnuous) soluton of (5.4) and, accordng to the frst part of Example 5.2, t s n fact a Carathéodory soluton. Example 5.4. Consder the ntal value problem ẋ + αx = f, x (a) = 0, (5.6) where x BV 0 ([0, 1], R), f 1 ([0, 1], R) and α : [0, 1] R s gven by α (t) = sn t f t 0 and α (0) = 0. Hence α H ([0, 1], R) \ 1 ([0, 1], R), t α BV ( [ 1 n, 1], R) for every n N, and F = f BV 0 ([0, 1], R) (by Theorem 2.10). Let us consder the Volterra-Henstock lnear ntegral equaton obtaned from (5.6) x(t) + K α(s)xt(s)ds = f(t), t [a, b], (V ) H as well as the lnear ntegral equatons of Volterra-Henstock x (t) + K χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Hn By Theorem 3.6, for every n N, there exsts a unque soluton x n H ([0, 1], R) A of (V ) Hn whose resolvent s gven by the Neumann seres: x n (t) = F (t) + K ( 1) (j 1) sn(s)( S(t) + S(s)) (j 1) F (s) ds [0,t] [ 1 n,1] (j 1)! s j=1 = F (t) + K sn(s)e (S(t) S(s)) F (s) ds, for t [0, 1] [0,t] [ 1 n,1] s where S (t) = sn s ds. However, snce F BV ([0, 1], R) and the kernel [0,t] s sn (s) e(s(t) S(s)) ρ n (t, s) = s bounded (Theorem 3.6), then t follows that s x n s of bounded varaton. From Theorem 3.7, we have that x (t) = lm x n (t) = F (t) + K sn (s) e (S(t) S(s)) F (s) ds, t [0, 1], n [0,t] s (5.7)

26 414 M. Federson and R. Bancon s of bounded varaton and satsfes (V ) H. Thus x gven by (5.7) s a Henstock soluton of (5.6). 5.2 Applcatons to Sngular Integral Equatons We call sngular ntegral equatons those ntegral equatons whose kernel has a sngularty. Such equatons play an mportant role n the Theory of Integral Equatons wth applcatons n varous areas. Example 5.5. Let x, f BV ([0, 1], R) and α : [0, 1] R be as n Example 4.4. Then accordng to Example 4.4, gven f BV 0 ([0, 1], R), the sngular ntegral equaton x(t) + K α(s)xt(s)ds = f(t), t [a, b], (V ) H admts a soluton of bounded varaton whch s gven by x (t) = f (t) + K sn (s) e (S(t) S(s)) f (s) ds, t [0, 1]. s [0,t] Example 5.6. Consder the unbounded functon α (t) = 1 t for t [0, 1], and α (0) = 0, and the sngular ntegral equaton x (t) + K α (s) x (s) ds = f (t), t [a, b], (V ) L where x, f ( C ([0, 1], R). Snce α s Lebesgue ntegrable on the nterval [0, 1], then α 1 [ 1 n, 1], R) for each n N, and we can consder the lnear ntegral Volterra equatons x (t) + L χ Xn (s) α (s) x (s) ds = f (t), t [a, b], and n N, (V ) Ln where x, f C ([0, 1], R) and X n = [ 1 n, 1] for each n N. Gven f C ([0, 1], R), snce α C ( [ 1 n, 1], R) for every n N, then there exsts a contnuous soluton x n of (V ) Ln whose resolvent s gven by the Neumann seres: ( ) t s x n (t) = f (t) + L f (s) ds [0,t] [ 1 n,1] =1 s ( ) = f (t) + L 1 ( ) f (s) ds, t [0, 1]. s t s 1 [0,t] [ 1 n,1]

27 Lnear Integral Equatons of Volterra 415 By Theorem 4.3 and the Monotone Convergence Theorem, ( ) x (t) = lm x n (t) = f (t)+ L 1 ( ) f (s) ds, t [0, 1], n s t s 1 [0,t] s a contnuous soluton for the sngular ntegral equaton (V ) L. Let us suppose now that x, f BV 0 ([0, 1], R) nstead of x, f C ([0, 1], R). From the fact that α BV ( [ 1 n, 1], R) for every n N, and gven that f BV 0 ([0, 1], R) H ([0, 1], R), we can calculate the soluton x n of each (V ) Ln by the Neumann seres method (see Theorem 3.6). We have that x n (t) = f (t) + K = f (t) + K [0,t] [ 1 n,t] =1 [0,t] [ 1 n,t] ( ) t s f (s) ds s ( ) 1 ( ) f (s), ds, t [0, 1]. s t s 1 Then, by Theorem 4.4 and the Monotone Convergence Theorem, ( ) x (t) = lm x n (t) = f (t)+ K 1 ( ) f (s) ds, t [0, 1], n s t s 1 [0,t] s a soluton of bounded varaton of the sngular ntegral equaton (V ) L. References The bass for ths paper s [4]. [1] G. Brkhoff, Integraton of functons on Banach spaces, Trans. Am. Math. Socety, 38(1935), [2] M. Federson, Fórmulas de Substtucão para ntegras de Gauge, MS Dssertaton, Insttute of Mathematcs and Statstcs, Unversty of São Paulo, [3] M. Federson, The Monotone Convergence Theorem for mult-dmensonal Kurzwel vector ntegrals, Semnáro Braslero de Análse, 45(1997), [4] M. Federson, Sobre a exstênca de soluções para Equações Integras Lneares com respeto a Integras de Gauge, Doctoral Thess, Insttute of Mathematcs and Statstcs, Unversty of São Paulo, Brazl, 1998.

28 416 M. Federson and R. Bancon [5] M. Federson, The Fundamental Theorem of Calculus for multdmensonal Banach space-valued Henstock vector ntegrals, Real Analyss Exchange, 1999, to appear. [6] M. Federson, Substtuton Formulas for Kurzwel and Henstock vector ntegrals, pre-prnt, [7] L. Gengan, On necessary condtons for Henstock ntegrablty, Real Analyss Exchange, 18( ), [8] C. S. Höng, The abstract Remann-Steltjes ntegral and ts Applcatons to lnear dfferental equatons wth generalzed boundary condtons, Notes of the Insttute of Mathematcs and Statstcs, Unversty of São Paulo, Mathematcs Seres #1, [9] C. S. Höng, Volterra-Steltjes ntegral equatons, Math. Studes, 16, North-Holland Publ. Comp., Amsterdam, [10] C. S. Höng, Equatons ntégrales generalsées et applcatons, Publ. Math. d Orsay, 83-01, exposé 5, [11] C. S. Höng, On lnear Kurzwel-Henstock ntegral equatons, Semnáro Braslero de Análse, 32(1990), [12] C. S. Höng, On a remarkable dfferental characterzaton of the functons that are Kurzwel-Henstock ntegrals, Semnáro Braslero de Análse, 33(1991), [13] C. S. Höng, A Remannan characterzaton of the Bochner-Lebesgue ntegral, Semnáro Braslero de Análse, 35(1992), [14] V. I. Istratescu, Fxed Pont Theory - An ntroducton, D. Redel Pub. Comp., [15] A. J. Jerr, Introducton to ntegral equatons wth applcatons, Marcel Dekker, Inc., New York and Basel, [16] J. Kurzwel and J. Jarník, Equntegrablty and controlled convergence of Perron-type ntegrable functons, Real Analyss Exchange 17( ), [17] R. M. Mc Leod, The generalzed Remann ntegral, Carus Math. Monog., 20, The Math. Ass. of Amerca, [18] E. J. McShane, A unfed theory of ntegraton, Am. Math. Monthly, 80(1973),

29 Lnear Integral Equatons of Volterra 417 [19] S. Schwabk, Abstract Perron-Steltjes ntegral, Math. Bohem., 121(1996), 4,

30 418 M. Federson and R. Bancon