INSTITUTE of MATHEMATICS. ACADEMY of SCIENCES of the CZECH REPUBLIC

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1 INSTITUTE of MATHEMATICS Acdemy of Sciences Czech Republic INSTITUTE of MATHEMATICS ACADEMY of SCIENCES of the CZECH REPUBLIC Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter Giselle A. Monteiro Miln Tvrdý Preprint No (Old Series No. 239) PRAHA 2011

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3 Preprint, Institute of Mthemtics, AS CR, Prgue INSTITUTE of MATHEMATICS Acdemy of Sciences Czech Republic Generlized liner differentil equtions in Bnch spce: Continuous dependence on prmeter Giselle A. Monteiro nd Miln Tvrdý December 16, 2010 Abstrct The pper dels with integrl equtions in Bnch spce X of the form x(t) = x + d[a] x + f(t) f(), t [, b ], (0.1) where < < b <, x X, f : [, b ] X is regulted on [, b ], nd A(t) is for ech t [, b ] liner bounded opertor on X, while the mpping A: [, b ] L(X) hs bounded vrition on [, b ]. Such equtions re clled generlized liner differentil equtions. Our im is to present new results on the continuous dependence of solutions of such equtions on prmeter. In prticulr, in Sections 3 nd 4 we give sufficient conditions ensuring tht the sequence {x n } of the solutions of generlized liner differentil equtions x n (t) = x n + d[a n ] x n + f n (t) f n (), t [, b ], n N, tends to the solution x of (0.1). Crucil ssumptions of Section 3 re the uniform boundedness of the vritions vr b A n of A n nd uniform convergence of A n to A. In Section 4, we present the extension of the clssicl result by Opil to the cse X R n, i.e. we do not require the uniform boundedness of vr b A n while the uniform convergence is replced by properly stronger concept. Finlly in Section 5 we present prtil result for the cse when the uniform convergence of A n to A is violted Mthemtics Subject Clssifiction: 34A37, 45A05, 34A30. Key words. Kurzweil-Stieltjes integrl, generlized differentil equtions in Bnch spce, continuous dependence. Universidde de São Pulo, Instituto de Ciêncis Mtemátics e Computção, ICMC-USP, São Crlos, SP, Brsil, gm@icmc.usp.br. Supported by CAPES BEX 5320/09-7 Supported by the Institutionl Reserch Pln No. AV0Z

4 2 G. Monteiro & M. Tvrdý 1 Introduction The theory of generlized differentil equtions enbles the investigtion of continuous nd discrete systems, including the equtions on time scles, from the common stndpoint. This fct cn be observed in severl ppers relted to specil kinds of equtions, such s e.g. those by Imz nd Vorel [11], Oliv nd Vorel [22], Federson nd Schwbik [4], Schwbik [24] or Slvík [30]. This pper is devoted to generlized liner differentil equtions of the form ((0.1)) in Bnch spce X. A complete theory in cse of X = R n cn be found, for instnce, in the monogrphs by Schwbik [24] or Schwbik, Tvrdý nd Vejvod [29]. See lso the pioneering pper by Hildebrndt [9]. As concerns integrl equtions in generl Bnch spce, it is worth to highlight the monogrph by Hönig [10] hving s bckground the interior (Dushnik) integrl. On the other hnd, deling with the Kurzweil-Stieltjes integrl, the contributions by Schwbik in [26] nd [27] represent the bse of this pper. In the cse X = R n, for ordinry differentil equtions, fundmentl results on the continuous dependence of solutions on prmeter bsed on the verging principle hve been delivered by Krsnoselskii nd Krejn [13], Kurzweil nd Vorel [15], Kurzweil[16], Opil [23] nd Kigurdze [12]. In prticulr, the problem of continuous dependence gve n inspirtion to Kurzweil to introduce the notion of generlized differentil eqution in the ppers [16] nd [17]. For liner ordinry differentil equtions, the most generl result seems to be tht given by Opil. An interesting observtion is contined in the fundmentl pper by Artstein [1]. A different pproch cn be found in the ppers [18] [20] by Meng Gng nd Zhng Meirong deling lso with mesure differentil nlogues of Sturm-Liouville equtions nd, in prticulr, describing the wek nd wek*continuous dependence of relted Dirichlet or Neumnn eigenvlues on potentil. After Kurzweil, problem of the continuous dependence for generlized differentil equtions hs been treted by severl uthors, see e.g. Schwbik [24], Ashordi [2], Frňková [5], Tvrdý [33], Hls [6], Hls nd Tvrdý [7]. Up to now, to our knowledge, only Federson nd Schwbik [4] delt with the cse of generl Bnch spce X. Our im is to prove new results vlid lso for X R n nd such tht, on the contrry to ll the bove mentioned ppers, they cover lso the Opil s result. 2 Preliminries Throughout these notes X is Bnch spce nd L(X) is the Bnch spce of bounded liner opertors on X. By X we denote the norm in X. Similrly, L(X) denotes the usul opertor norm in L(X). Assume tht < < b < + nd [, b ] denotes the corresponding closed intervl.

5 Preliminries 3 A set D = {α 0, α 1,..., α m } [, b ] is sid to be division of [, b ] if = α 0 < α 1 <... < α m = b. The set of ll divisions of [, b ] is denoted by D[, b ]. A function f : [, b ] X is clled finite step function on [, b ] if there exists division D = {α 0, α 1,..., α m } of [, b ] such tht f is constnt on every open intervl (α j 1, α j ), j = 1, 2,..., m. For n rbitrry function f : [, b ] X we set nd vr b f = f = sup f(t) X t [,b ] sup D D[,b ] m f(α j ) f(α j 1 ) X j=1 is the vrition of f over [, b ]. If vr b f < we sy tht f is function of bounded vrition on [, b ]. BV ([, b ], X) denotes the Bnch spce of functions f : [, b ] X of bounded vrition on [, b ] equipped with the norm f BV = f() X + vr b f. Given f : [, b ] X, the function f is clled regulted on [, b ] if, for ech t [, b) there is f(t+) X such tht lim f(s) f(t+) X = 0, s t+ nd for ech t (, b ] there is f(t ) X such tht lim f(s) f(t ) X = 0. s t By G([, b ], X) we denote the set of ll regulted functions f : [, b ] X. For t [, b), s (, b ] we put + f(t)=f(t+) f(t) nd f(s)=f(s) f(s ). Recll tht BV ([, b ], X) G([, b ], X) cf. e.g. [26, 1.5]. Moreover, it is known tht regulted function re uniform limits of finite step functions (see [10, Theorem I.3.1 ]). In wht follows, by n integrl we men the Kurzweil-Stieltjes integrl. recll its definition. Let us As usul, prtition of [, b ] is tgged system, i.e., couple P = (D, ξ) where D D[, b ], D = {α 0, α 1,..., α m }, nd ξ = (ξ 1,..., ξ m ) [, b ] m with α j 1 ξ j α j, j = 1, 2,..., m.

6 4 G. Monteiro & M. Tvrdý The set of ll prtitions of [, b ] is denoted by P[, b ]. Furthermore, ny function δ : [, b ] (0, ) is clled guge on [, b ]. Given guge δ, the prtition P is clled δ-fine [α j 1, α j ] (ξ j δ(ξ j ), ξ j + δ(ξ j )) We remrk tht for n rbitrry guge δ on [, b ] there lwys exists δ-fine prtition of [, b ]. It is stted by the Cousin lemm (see [24, Lemm 1.4]). For given functions F : [, b ] L(X) nd g : [, b ] X nd prtition P = (D, ξ) of [, b ], where D = {α 0, α 1,..., α m }, ξ = (ξ 1,..., ξ m ), we define S(dF, g, P ) = m [F (α j ) F (α j 1 )] g(ξ j ). j=1 We sy tht I X is the Kurzweil-Stieltjes integrl (or shortly KS-integrl) of g with respect to F on [, b ] nd denote I = d[f ] g if for every ε > 0 there exists guge δ on [, b ] such tht S(dF, g, P ) I < ε for ll δ fine prtitions P of [, b ]. X Anlogously, we define the integrl S(F, dg, P ) = F d[g] using sums of the form m F (ξ j ) [g(α j ) g(α j 1 )] j=1 For the reder s convenience some of the further results needed lter re summrized in the following ssertions: 2.1. Proposition. Let F : [, b ] L(X) nd g : [, b ] X. (i) [25, Proposition 10] Let F BV ([, b ], L(X)) nd g: [, b ] X be such tht d[f ] g (vr b X F ) g. d[f ] g exists. Then

7 Preliminries 5 (ii) [21, Lemm 2.2] Let F G([, b ], L(X)) nd g BV ([, b ], X) be such tht d[f ] g 2 F g BV. X d[f ] g exists. Then (iii) [28, Corollry 14] If F BV ([, b ], L(X)) nd g BV ([, b ], X) then both the integrls nd d[f ] g exist, the sum + F (τ) + g(τ) F (τ) g(τ) τ<b <τ b F d[g] converges in X nd the equlity is true. F d[g] + d[f ] g = F (b) g(b) F () g() t<b + F (t) + g(t)+ <t b F (t) g(t) (iv) [21, Theorem 2.11] Let F BV ([, b ], L(X)) nd let g: [, b ] X be bounded nd such tht the integrl d[f ] g exists. Then both the integrls exist nd the equlity H(s) d s [ s holds for ech H G([, b ], L(X)). ] d[f ] g nd [ ] H(s) d d[f ] g = In ddition, we need the following convergence result. H d[f ] g H d[f ] g

8 6 G. Monteiro & M. Tvrdý 2.2. Theorem. Let g, g n G([, b ], X), F, F n BV ([, b ], L(X)) for n N. Assume tht lim g n g = 0, lim F n F = 0 nd Then, ϕ := sup{vr b F n ; n N} <. ( { lim sup d[f n ] g n }) d[f ] g ; t [, b ] = 0. (2.1) X Proof. Let ε > 0 be given. By [10, Theorem I.3.1 ], we cn choose finite step function g : [, b ] X such tht g g < ε. Furthermore, let n 0 N be such tht g n g < ε nd F n F < ε for n n 0. For fixed t [, b ], by Proposition 2.1 (i) nd (ii), we obtin for n n 0 d[f n ] g n d[f ] g X d[f n ] ( g n g ) X + d[f n F ] g + X d[f ] ( g g ) X (vr t F n ) g n g + 2 F n F g BV + (vr t F ) g g ϕ ( g n g + g g ) + 2 g BV ε + (vr b F ) ε ( 2 ϕ + 2 g BV + vr b F ) ε = K ε, where K = ( 2 ϕ + 2 g BV + vr b F ) (0, ) does not depend on n. This proves (2.1) Remrk. In the cse tht X is Hilbert spce, Theorem 2.2 hs been lredy given by Krejčí nd Lurençot [14, Proposition 3.1] or Brokte nd Krejčí [3, Proposition 1.10]. 3 Continuous dependence on prmeter in the cse of uniformly bounded vritions Given A BV ([, b ], L(X)), f G([, b ], X) nd x X, consider the integrl eqution x(t) = x + d[a] x + f(t) f(), t [, b ]. (3.1)

9 3. Cse of uniformly bounded vrition 7 A function x : [, b ] X is clled solution of (3.1) on [, b ] if the integrl nd x stisfies the equlity (3.1) for ech t [, b ]. d[a] x exists For our purposes the following property is crucil [ I A(t) ] 1 L(X) for ll t (, b ]. (3.2) In prticulr, tking into ccount the closing remrk in [26] we cn see tht the following result is prticulr cse of [26, Proposition 2.10] Proposition. Let A BV ([, b ], L(X)) stisfy (3.2) Then, for every x X nd every f G([, b ], X), the eqution (3.1) possesses unique solution x on [, b ] nd x G([, b ], X). Moreover, if A nd f re left-continuous on (, b ], then x is lso left-continuous on (, b ]. In ddition, the following two importnt uxiliry ssertions re true: 3.2. Lemm. Let A BV ([, b ], L(X)) stisfy (3.2), f G([, b ], X) nd x X nd let x be the corresponding solution of (3.1) on [, b ]. Then vr b (x f) (vr b A) x < (3.3) nd Proof. c A := sup{ [I A(t)] 1 L(X) ; t (, b ]} (0, ), (3.4) x(t) X c A ( x X + f() X + f ) exp (c A vr t A) for t [, b ]. (3.5) i) Let D = {α 0, α 1,..., α m } be n rbitrry division of [, b ]. Then m x(α j ) f(α j ) x(α j 1 ) + f(α j 1 ) j=1 = m j=1 αj d[a] x α j 1 X m j=1 X [ (vr α j α j 1 A) x ] = (vr b A) x <, i.e. (3.3) is true. ii) For t (, b ] such tht A(t) L(X) < 1 2 [I A(t)] 1 L(X) we hve 1 1 A(t) L(X) < 2 (cf. e.g. [31, Lemm 4.1-C]). Therefore, 0 c A < due to the fct tht the set {t [, b ]; A(t) L(X) 1 2 } hs t most finitely mny elements. As the cse c A = 0 is impossible, this proves (3.4).

10 8 G. Monteiro & M. Tvrdý iii) Now, let x be solution of (3.1). Put B() = A() nd B(t) = A(t ) for t (, b ]. Then, by [26, Corollry 2.6] nd [26, Proposition 2.7], we get nd Consequently A(t) B(t) = A(t), [I A(t)] x(t) = x + nd (cf. Proposition 2.1 (i)) where A B BV ([, b ], L(X)), vr b B vr b A d[a B] x = A(t) x(t) for t (, b ]. d[b] x + f(t) f() for t (, b ] x(t) X K 1 + K 2 d[h] x X for t [, b ], K 1 = c A ( x X + f() X + f ), K 2 = c A nd h(t) = vr t B. The function h is nondecresing nd, since B is left-continuous on (, b ], h is lso leftcontinuous on (, b ]. Therefore we cn use the generlized Gronwll inequlity (see e.g. [29, Lemm I.4.30] or [24, Corollry 1.43]) to get the estimte (3.5) Lemm. Let A, A n BV ([, b ], L(X)), n N, be such tht (3.2) nd re stisfied. Then lim A n A = 0 (3.6) [ I A n (t) ] 1 L(X) (3.7) for ll t (, b ] nd ll n N sufficiently lrge. Moreover, there is µ (0, ) such tht for ll n N sufficiently lrge. c An := sup{ [I A n (t)] 1 L(X) ; t (, b ]} µ (3.8) Proof. First, notice tht, since A BV ([, b ], L(X)), the set hs t most finite number of elements. D := {t (, b ]; A(t) L(X) 1 4 } Let c A be defined s in (3.4). Then, s by (3.6) lim A n A = 0, there is n 0 N such tht Thus, A n (t) A(t) L(X) < 1 4 min{1, 1 c A } for t [, b ] nd n n 0. (3.9) A n (t) L(X) A(t) L(X) + A n (t) A(t) L(X) < 1 2 for t [, b ] \ D, n n 0.

11 3. Cse of uniformly bounded vrition 9 By [31, Lemm 4.1-C], this implies tht [I A n (t)] is invertible nd [I A n (t)] 1 L(X) < 2 for t [, b ] \ D nd n n 0. Notice tht, due to (3.2), the reltion I A n (t) = [I A(t)] [ I [I A(t)] 1 ( A n (t) A(t)) ] (3.10) holds for ll t [, b ] nd n N. Denote T n (t) := [I A(t)] 1 ( A n (t) A(t)) for n N nd t [, b ]. Then (3.10) mens tht, I A n (t) is invertible if nd only if I T n (t) is invertible. Now, let t D nd n n 0 be given. Then, due to (3.4) nd (3.9), we hve T n (t) L(X) < 1 4. Consequently, by [31, Lemm 4.1-C], I T n (t) nd therefore lso [I A n (t)] re invertible. Moreover, tking into ccount (3.4) nd (3.10), we cn see tht is true. [I A n (t)] 1 L(X) 4 3 c A < 2 c A To summrize, there exists n 0 N such tht [I A n (t)] is invertible nd [I A n (t)] 1 L(X) µ = 2 mx{1, c A } for ll t (, b ] nd n n 0. This completes the proof. The min result of this section is the following Theorem, which generlizes in liner cse the recent results by Federson nd Schwbik [4]) nd covers the results for generlized liner differentil equtions known for the cse X = R n. Unlike [2], to prove it we do not utilize the vrition-of-constnts formul. Therefore it is not necessry to ssume the dditionl condition [I + A(t)] 1 L(X), t [, b ] Theorem. Let A, A n BV ([, b ], L(X)), f, f n G([, b ], X), x, x n X for n N. Furthermore, let A stisfy (3.2), (3.6), nd α := sup{vr b A n ; n N} < (3.11) lim x n x X = 0 nd lim f n f = 0. (3.12) Then eqution (3.1) hs unique solution x on [, b ]. Furthermore, for ech n N lrge enough there is unique solution x n on [, b ] to the eqution nd lim x n x = 0. x n (t) = x n + d[a n ] x n + f n (t) f n (), t [, b ] (3.13)

12 10 G. Monteiro & M. Tvrdý Proof. Due to (3.2) eqution (3.1) hs unique solution x on [, b ]. Furthermore, by Lemm 3.2, there is n 0 N such tht (3.7) is true for n n 0. Hence, for ech n n 0, eqution (3.13) possesses unique solution x n on [, b ]. Set w n = (x n f n ) (x f) (3.14) Then w n (t) = w n + d[a n ] w n + h n (t) h n () for n N nd t [, b ], where w n = ( x n f n ()) ( x f()) nd h n (t) = First, notice tht ccording to (3.12) we hve ( d[a n A] (x f) + d[a n ] f n Furthermore, in view of Theorem 2.2, we hve lim ) d[a] f. lim w n X = 0. (3.15) d[a n ] f n d[a] f = 0. X Moreover, since (x f) BV ([, b ], X) by (3.3), we get by Proposition 2.1 (ii) d[a n A] (x f) 2 A n A x f BV for ll t [, b ]. X Hving in mind (3.6), we cn see tht the reltion holds. To summrize, lim By (3.11) nd by Lemms 3.2 nd 3.3 we hve d[a n A] (x f) = 0 X lim h n = 0. (3.16) w n (t) X µ ( w n X + h n ) exp (µ vr b A n ) for ll t [, b ]. Consequently, using (3.15) nd (3.16) we deduce tht lim w n X = 0. Now, by (3.12) nd (3.14) we conclude finlly tht lim x n x = 0. We will close this section by comprison of Theorem 3.4 with two similr results presented for dim X < by Schwbik in [24]. First, when restricted to the liner cse, Theorem 8.2 from [24] modifies to

13 3. Cse of uniformly bounded vrition Theorem. Let A, A n BV ([, b ], L(X) nd f n (t) f n ()=f(t) f()=0 for n N nd t [, b ]. Further, let nondecresing function h : [, b ] R be given such tht lim A n(t) = A(t) on [, b ], (3.17) { An (t 2 ) A n (t 1 ) L(X) h(t 2 ) h(t 1 ), A(t 2 ) A(t 1 ) L(X) h(t 2 ) h(t 1 ) (3.18) for t 1, t 2 [, b ] nd n N. Let x n, n N, be solutions of (3.13) nd let lim x n(t) x(t) X for t [, b ]. Then x BV ([, b ], X) is solution of (3.1) on [, b ] Proposition. Under the ssumptions of Theorem 3.5 the reltions (3.6) nd (3.11) re stisfied. Proof. i) The reltion (3.11) follows immeditely from (3.18). ii) Notice tht (3.17) nd (3.18) imply tht { An (t ) A n (s) L(X) h(t ) h(s), A(t ) A(s) L(X) h(t ) h(s) for t (, b ], s [, b ], n N, (3.19) nd { An (t+) A n (s) L(X) h(t+) h(s), A(t+) A(s) L(X) h(t+) h(s) for t [, b), s [, b ], n N. (3.20) iii) Let ε > 0 nd t (, b ] be given nd let us choose s 0 (, t) nd n 0 N so tht h(t ) h(s 0 ) < ε 3 Then, by (3.19) nd (3.21), nd A n (s 0 ) A(s 0 ) L(X) < ε 3 for n n 0. (3.21) This mens tht A n (t ) A(t ) L(X) A n (t ) A n (s 0 ) L(X) + A n (s 0 ) A(s 0 ) L(X) Similrly, using (3.20) we get + A(s 0 ) A(t ) L(X) < h(t ) h(s 0 ) + ε 3 + h(t ) h(s 0) < ε. lim A n(t ) = A(t ) holds for t (, b ]. (3.22) lim A n(t+) = A(t+) holds for t [, b). (3.23)

14 12 G. Monteiro & M. Tvrdý iv) Now, suppose tht (3.6) is not vlid. Then there is ε > 0 such tht for ny l N there exist m l l nd t l [, b ] such tht We my ssume tht m l+1 > m l for ny l N nd A ml (t l ) A(t l ) L(X) ε. (3.24) lim t l = t 0 [, b ]. (3.25) l Let t 0 (, b ] nd ssume tht the set of those l N for which t l (, t 0 ) hs infinitely mny elements, i.e. there is sequence {l k } N such tht t lk (, t 0 ) for ll k N nd lim k t lk = t 0. Denote s k = t lk nd B k = A mlk for k N. Then, in view of (3.24), we hve nd s k (, t 0 ) for k N, lim k s k = t 0 (3.26) B k (s k ) A(s k ) L(X) ε for k N. (3.27) By (3.19), we hve nd A(t 0 ) A(s k ) L(X) h(t 0 ) h(k n ) B k (t 0 ) B k (s k ) L(X) h(t 0 ) h(k n ) for k N. Therefore, by (3.22) nd since lim k (h(t 0 ) h(s k )) = 0 due to (3.26), we cn choose k 0 N so tht B k0 (t 0 ) A(t 0 ) L(X) < ε 3 nd A(t 0 ) A(s k0 ) L(X) h(t 0 ) h(s k0 ) < ε 3 B k0 (t 0 ) B k0 (s k0 ) L(X) < ε 3. As consequence, we get finlly by (3.27) ε B k0 (s k0 ) A(s k0 ) L(X) B k0 (s k0 ) B k0 (t 0 ) L(X) + B k0 (t 0 ) A(t 0 ) L(X) + A(t 0 ) A(s k0 ) L(X) < ε, contrdiction. If t 0 [, b) nd the set of those l N for which t l (, t 0 ) hs only finitely mny elements, then there is sequence {l k } N such tht t lk (t 0, b) for ll k N nd lim k t lk = t 0. As before, let s k = t lk nd B k = A mlk for k N nd notice tht s k (t 0, b) for k N, lim k s k = t 0

15 3. Cse of uniformly bounded vrition 13 nd (3.27) re true. Arguing similrly s before we get tht there is k 0 N such tht ε B k0 (s k0 ) A(s k0 ) L(X) B k0 (s k0 ) B k0 (t 0 +) L(X) + B k0 (t 0 +) A(t 0 +) L(X) + A(t 0 +) A(s k0 ) L(X) < ε, contrdiction. Similrly, when restricted to the liner cse, Theorem 8.8 from [24] modifies to 3.7. Theorem. Let A, A n BV ([, b ], X), f n (t) f n ()=f(t) f()=0 for n N nd t [, b ]. Furthermore, let (3.2) hold nd let x be the corresponding solution of (3.1). Finlly, let sclr nondecresing nd left-continuous on (, b ] functions h n, n N, nd h be given such tht h is continuous on [, b ] nd lim A n(t) = A(t) on [, b ], (3.28) { An (t 2 ) A n (t 1 ) L(X) h n (t 2 ) h n (t 1 ), A(t 2 ) A(t 1 ) L(X) h(t 2 ) h(t 1 ) (3.29) for ll t 1, t 2 [, b ] nd n N, { [ lim sup hn (t 2 ) h n (t 1 ) ] h(t 2 ) h(t 1 ) whenever t 1 t 2 b. (3.30) Then, for ny n N sufficiently lrge, eqution (3.13) hs unique solution x n on [, b ] nd lim x n(t) = x(t) uniformly on [, b ] Proposition. Under the ssumptions of Theorem 3.7 the reltions (3.6) nd (3.11) re stisfied. Proof (tken from [33]). i) By (3.30) there is n 0 N such tht Hence for ny n N we hve h n (b) h n () h(b) h() + 1 for ll n n 0. vra n α 0 = mx ( {vran ; n n 0 } { h(b) h() + 1 } ) <. Thus we conclude tht (3.11) is true. ii) Suppose tht (3.6) does not hold. Then there is ε > 0 such tht for ny l N there exist m l l nd t l [, b ] such tht We my ssume tht m l+1 > m l for ny l N nd A ml (t l ) A(t l ) L(X) ε. (3.31) lim t l = t 0 [, b ]. (3.32) l

16 14 G. Monteiro & M. Tvrdý Let t 0 (, b) nd let n rbitrry ε > 0 be given. Since h is continuous, we my choose η > 0 in such wy tht t 0 η, t 0 + η [, b ] nd Furthermore, by (3.28) there is l 1 N such tht h(t 0 + η) h(t 0 η) < ε. (3.33) A ml (t 0 ) A(t 0 ) L(X) < ε for ll l l 1 (3.34) nd by (3.29), (3.30) nd (3.33) there is l 2 N, l 2 l 1, such tht A ml (τ 2 ) A ml (τ 1 ) L(X) h(t 0 + η) h(t 0 η) + ε < 2 ε (3.35) whenever τ 1, τ 2 (t 0 η, t 0 + η) nd l l 2. The reltions (3.28) nd (3.35) imply immeditely tht { A(τ2 ) A(τ 1 ) L(X) = lim l A ml (τ 2 ) A ml (τ 1 ) L(X) 2ε whenever τ 1, τ 2 (t 0 η, t 0 + η). (3.36) Finlly, let l 3 N be such tht l 3 l 2 nd then in virtue of the reltions (3.32) (3.37) we hve A ml (t l ) A(t l ) L(X) t l t 0 < η for ll l l 3, (3.37) A ml (t l ) A ml (t 0 ) L(X) + A ml (t 0 ) A(t 0 ) L(X) + A(t 0 ) A(t l ) L(X) 5 ε. Hence, choosing ε < 1 5 ε, we obtin by (3.31) tht ε > A ml (t l ) A(t l ) L(X) ε. This being impossible, the reltion (3.6) hs to be true. The modifiction of the proof in the cses t 0 = or t 0 = b is obvious. 4 Continuous dependence on prmeter in the cse of vritions bounded with weight In this section we restrict ourselves to homogeneous generlized liner differentil equtions x(t) = x + d[a] x, t [, b ], (4.1) where, s before, A BV ([, b ], L(X)) nd x X. As in the previous section we will ssume tht the fundmentl existence ssumption (3.2) is stisfied. The min result of this section extends tht obtined by Z. Opil for the cse dim X < in [23]. To this im, we recll n estimte presented in [21].

17 4. Weighted convergence Lemm. If F G([, b ], L(X)) nd G BV ([, b ], L(X)) then + F (t) + G(t) L(X) + F (t) G(t) L(X) 2 F vr b G. (4.2) t [,b) t (,b ] 4.2. Theorem. Let A, A n BV ([, b ], L(X)) nd x, x n X for n N. Assume (3.2) nd nd lim A ( ) n A 1 + vr b A n = 0 (4.3) lim x n x X = 0. (4.4) Then (4.1) hs unique solution x on [, b ]. Moreover, for ech n N sufficiently lrge, the eqution x n (t) = x n + d[a n ] x n, t [, b ] (4.5) hs unique solution x n on [, b ] nd lim x n x = 0. Proof. First, notice tht, since A n A A n A ( 1 + vr b A n ) for ll n N, (4.3) implies (3.6). Therefore, by Lemm 3.3, there is n 0 N such tht (3.7) holds for ech t (, b ] nd ech n n 0. Assume n n 0. Let x nd x n be the solutions on [, b ] of (4.1) nd (4.5), respectively. Then (x n (t) x(t)) = ( x n x) + d[a] (x n x) + h n (t) for t [, b ], (4.6) where By Lemm 3.2 we hve h n (t) = d[a n A] x n for t [, b ]. (4.7) x n x c A ( x n x X + h n ) exp (c A vr b A). (4.8) (Notice tht h n () = 0 for ll k.) Thus, in view of the ssumption (4.4), to prove the ssertion of the theorem, we hve to show tht lim h n = 0. To this im, we integrte by prts (cf. Proposition 2.1 (iii)) in the right-hnd side of (4.7) nd use Substitution Formul (cf. Proposition 2.1 (iv)). Then we get h n (t) = [A n (t) A(t)] x n (t) [A n () A()] x n for t [, b ], where t (A n A, x n ) = s<t (A n A) d[a n ] x n t (A n A, x n ) (4.9) [ + (A n (s) A(s)) + x n (s)] [ (A n (s) A(s)) x n (s)]. (4.10) <s t

18 16 G. Monteiro & M. Tvrdý Inserting the reltions (cf. [26, Proposition 2.3]) + x n (t) = + A n (t) x n (t) for t [, b) nd x n (t) = A n (t) x n (t) for t (, b ] into the right-hnd side of (4.10) nd using Lemm 4.1, we obtin the estimtes t (A n A, x n ) X 2 A n A (vr t A n ) x n for t [, b ]. Hence h n (t) X A n A ( (vr t A n ) ) x n, tht is, h n α n x n, (4.11) ( where α n = A n A 2+3 vr b A n ). Note tht, due to (4.3), we hve lim α n = 0. (4.12) We cn see tht to show tht lim h n = 0, it is sufficient to prove tht the sequence { x n } is bounded. By (4.8) nd (4.11) we hve Hence ( ) x n x n x + x c A xn x X + α n x n exp (c A vr b A) + x. ( 1 ca α n exp (c A vr b A) ) x n c A x n x X exp (c A vr b A) + x for n n 0. By (4.4) nd (4.12), there is n 1 n 0 such tht x n x X < 1 nd c A α n exp (c A vr b A) < 1 2 for n n 1. In prticulr, x n < 2 ( c A exp (c A vr b A) + x ) for n n 1, i.e. the sequence { x n } is bounded nd this completes the proof Remrk. In comprison with Theorem 3.4, the uniform boundedness of vrition (3.11) ws not needed in Theorem 4.2. On the other hnd, if (3.11) is ssumed, Theorem 4.2 reduces to Theorem 3.4. Let us note tht, on the contrry to the finite dimensionl cse, in the cse of generl Bnch spce X it is not possible to extend esily the convergence result Theorem 4.2 to the the nonhomogeneous equtions.

19 4. Weighted convergence 17 5 Emphtic convergence In this section we del with the cse tht the uniform convergence is violted. The ssumptions of Theorems 5.1 nd 5.2 re relted to the notion of emphtic convergence introduced by Kurzweil in [17]. More precisely, together with the loclly uniform convergence we infer some control condition for points sufficiently close to the end points, b of the intervl [, b ]. These results extend the work of Hls nd Tvrdý deling with X = R n (c.f. [6], [8] nd [34]). If {f n } is sequence of X-vlued functions defined on [, b ], we sy tht it tends to f loclly uniformly on J [, b ] if lim (sup{ f n(t) f(t) X ; t I}) = 0 for ll closed subintervls I J. In such cse we write f n f loclly on J. Of course, f n f loclly on J implies lim f n(t) f(t) X = 0 for ll interior points t of J Theorem. Let A, A n BV ([, b ], L(X)), f, f n G([, b ], X), x, x n X for n N. Assume (3.2), (4.4), A n A loclly on (, b ] nd f n f loclly on (, b ], (5.1) nd tht there is N N such tht (3.7) is true for ll t (, b ] nd ll n N such tht n N. Then, for n N sufficiently lrge, there exist unique solutions x nd x n on [, b ] to (3.1) nd (3.13), respectively. In ddition, let (3.11) nd hold. Then nd x n x loclly on (, b ]. ε > 0 δ > 0 such tht t (, + δ) n 0 N such tht ( ) ( ) + A() x + + X f() x n (t) x n < ε for n n 0 lim x n(t) x(t) X = 0 for ny t [, b ] (5.2) Proof. Without ny loss of generlity we my ssume A n () = A() = 0 nd f n () = f() = 0 for n N. Due to ssumptions (3.2) nd (3.7), the existence nd uniqueness of solutions to (3.1) nd (3.13) re gurnteed by Proposition 3.1. Denote by x nd x n the corresponding solutions. Let ε > 0 be given. Then, s x is regulted, there is δ 0 > 0 such tht x(s) x(+) X < ε for ll s (, + δ 0 ).

20 18 G. Monteiro & M. Tvrdý Furthermore, by (4.4) there is n 1 N such tht x n x X < ε for n n 1. By (5.2) there is δ (0, δ 0 ) such tht for ech t (, + δ) we cn find n 0 n 1 so tht ( + A() x + + f() ) (x n (t) x n ) X < ε for n n 0. To summrize, for ny t (, + δ 0 ) nd n n 0, we hve x(t) x n (t) X x(t) x(+) X + x(+) x + x n x n (t) X + x x n X = x(t) x(+) X + + A() x + + f() + x n x n (t) X + x x n X < 3 ε. This implies lso tht lim x n (t) x(t) X = 0 for ll t [, + δ). Now, let n rbitrry c (, + δ) be given. Then lim x n(c) = x(c). Therefore, by Theorem 3.4 nd due to the uniqueness of solutions to nd x n (t) = x n (c) + d[a n ] x n + f n (t) f n (c), t [c, b] c x(t) = x(c) + d[a] x + f(t) f(c), c t [c, b], x n tend to x uniformly on [c, b ] s n. More precisely, lim (sup{ x n(t) x(t) X ; t [c, b]}) = 0. Since c ws rbitrry, this mens tht x n (t) x(t) for ech t [, b ] nd x n x loclly on (, b ]. The result symmetricl to the previous theorem slightly differs. However, its proof is very similr Theorem. Let A, A n BV ([, b ], L(X)), f, f n G([, b ], X), x, x n X for n N. Assume (3.2), (4.4) nd A n A loclly on [, b) nd f n f loclly on [, b) (5.3) nd tht there is N N such tht (3.7) is true for ll t (, b ] nd ll n N such tht n N. Then for n N sufficiently lrge there exist unique solutions x nd x n to (3.1) nd (3.13), respectively.

21 4. Weighted convergence 19 Let, in ddition, (3.11) nd ε > 0 δ > 0 such tht t (b δ, b) n 0 N such tht ( ) ( A(b)[I A(b)] 1 x(b )+[I A(b)] 1 X f(b) x n (b) x n (t)) < ε for n n 0 hold. Then x n x loclly on [, b). (5.4) Proof. Similrly to the previous theorem, the existence nd uniqueness of solutions to (3.1) nd (3.13) re gurnteed by Proposition 3.1. Denote by x nd x n the corresponding solutions. Then, due to Theorem 3.4 nd due to the uniqueness of solutions to nd x n (t) = x n () + d[a n ] x n + f n (t) f n (), t [, c ] x(t) = x() + d[a] x + f(t) f(), t [, c ], the sequence {x n } tends for ech c [, b) to x uniformly on [, c ] s n. In prticulr, lim x n(t) = x(t) for ll t [, b). It remins to show tht lim x n (b) x(b) X = 0. Let ε > 0 be given. By (5.4), there is δ > 0 such tht for ech t (b δ, b) we cn find n 1 N such tht n 1 N nd ( ) ( A(b)[I A(b)] 1 x(b )+[I A(b)] 1 X f(b) x n (b) x n (t)) < ε holds for ll n n 0. As x is regulted on [, b ], we cn lso ssume tht x(s) x(b ) X < ε for ll s (b δ, b). Now, choose nd rbitrry τ (b δ, b). Then x n (τ) x(τ). Hence there is n 0 n 1 such tht x n (τ) x(τ) X < ε for n n 0. To summrize, for n n 0 we hve x(b) x n (b) X x(b) ( A(b) x(b) + f(b)) x(τ) X + x(τ) x n (τ) X + x n (τ) + A(b) x(b) + f(b) x n (b) X = x(b ) x(τ) X + x(τ) x n (τ) X + (x n (τ) x n (b)) A(b) [I A(b)] 1 (x(b ) + f(b)) X

22 20 G. Monteiro & M. Tvrdý = x(b ) x(τ) X + + x(τ) x n (τ) X + (x n (τ) x n (b)) A(b) [I A(b)] 1 x(b ) [I A(b)] 1 f(b)) X < 3 ε, where we mde use of the following well-known reltions: nd x(b) = A(b) x(b) + f(b), x(b) = [I A(b)] 1 (x(b ) + f(b)) I + A(b)[I A(b)] 1 = [I A(b)] 1. Therefore lim x n (b) x(b) X = 0 nd this completes the proof Remrk. Let us notice tht, due to Lemm 3.3, we cn, insted of: there is N N such tht (3.7) is true for ll t (, b ] nd ll n N such tht n N ssume only there re n N N nd < 0 such tht (3.7) is true for ll t (, + ] nd ll n N such tht n N Remrk. It is esy to combine Theorems 5.1 nd 5.2 to formulte corresponding result for the cse tht the uniform convergence is violted t finitely mny points in [, b ]. We leve it to the reder. References [1] Artstein Z.: Continuous dependence on prmeters: On the best possible results. J. Differentil Equtions 19 (1975), [2] Ashordi, M.: On the correctness of liner boundry vlue problems for systems of generlized ordinry differentil equtions. Proc.Georgin Acd.Sci.Mth. 1 (1993), No. 4, [3] Brokte, M. nd Krejčí, P.: Dulity in the spce of regulted functions nd the ply opertor. Mth. Z. 245 (2003), [4] Federson, M nd Schwbik, Š: Generlized ordinry differentil equtions pproch to impulsive retrded functionl differentil equtions. Differentil nd Integrl Equtions 19 (2006), [5] Frňková, D.: Continuous dependence on prmeter of solutions of generlized differentil equtions. Čsopis pěst. mt.114 (1989), [6] Hls, Z.: Continuous dependence of solutions of generlized liner ordinry differentil equtions on prmeter. Mthemtic Bohemic 132 (2007),

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24 Powered by TCPDF ( 22 G. Monteiro & M. Tvrdý [23] Opil, Z.: Continuous Prmeter Dependence in Liner Systems of Differentil Equtions. J. Differentil Equtions 3 (1967), [24] Schwbik, Š.: Generlized Ordinry Differentil Equtions. World Scientific. Singpore, [25] Schwbik, Š.: Abstrct Perron-Stieltjes integrl. Mth. Bohem. 121(1996), [26] Schwbik, Š.: Liner Stieltjes integrl equtions in Bnch spces. Mth. Bohem. 124 (1999), [27] Schwbik, Š.: Liner Stieltjes integrl equtions in Bnch spces II; Opertor vlued solutions. Mth. Bohem , [28] Schwbik, Š.: A note on integrtion by prts for bstrct Perron-Stieltjes integrls. Mth. Bohem , [29] Schwbik, Š., Tvrdý, M. nd Vejvod, O.: Differentil nd Integrl Equtions: Boundry Vlue Problems nd Adjoint. Acdemi nd Reidel. Prh nd Dordrecht, 1979 [30] Slvík, A.: Dynmic equtions on time scles nd generlized ordinry differentil equtions. [31] Tylor, A.: Introduction to Functionl Anlysis. Wiley, [32] Tvrdý, M.: Regulted functions nd the Perron-Stieltjes integrl. Čs.pěst.mt. 114 (1989), [33] Tvrdý, M: On the continuous dependence on prmeter of solutions of initil vlue problems for liner generlized differentil equtions. Funct.Differ.Equ. 5 (1999), [34] Tvrdý, M.: Differentil nd Integrl equtions in the Spce of Regulted Functions. Mem. Differentil Equtions Mth. Phys. 25 (2002),