Journal of Mathematical Analysis and Applications

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1 J. Mth. Anl. Appl Contents lists vilble t ScienceDirect Journl o Mthemticl Anlysis nd Applictions Dynmic equtions on time scles nd generlized ordinry dierentil equtions Antonín Slvík Chrles University, Fculty o Mthemtics nd Physics, Sokolovská 83, Prh 8, Czech Republic rticle ino bstrct Article history: Received 15 Mrch 2010 Avilbleonline30June2011 Submitted by C.E. Wyne Keywords: Vritionl stbility Stbility with respect to perturbtions Continuous dependence on prmeters Liner equtions Integrtion on time scles Kurzweil Stieltjes integrl The im o this pper is to show tht dynmic equtions on time scles cn be treted in the rmework o generlized ordinry dierentil equtions s introduced by J. Kurzweil. We lso use some known results or generlized ordinry dierentil equtions to obtin new theorems relted to stbility nd continuous dependence on prmeters or dynmic equtions on time scles Elsevier Inc. All rights reserved. 1. Introduction The nme o Jroslv Kurzweil is connected especilly with the nonbsolutely convergent Henstock Kurzweil integrl, which generlizes the integrls o Riemnn, Lebesgue, nd Newton. The roots o this integrl lie in the theory o generlized ordinry dierentil equtions, which cn be trced bck to 1957 see the pper [8]. Consider n intervl I R, set B R n, nd unction F : B I R n.aunctionx : I B is clled solution o the generlized ordinry dierentil eqution = DFx, t, dτ whenever xs 2 xs 1 = 2 s 1 DF xτ, t or ech pir o vlues s 1, s 2 I, where the integrl on the right-hnd side is the generlized Kurzweil integrl see the next section. Although generlized ordinry dierentil equtions re not s widely known s the Kurzweil integrl, they hve turned out to be quite powerul concept, which includes not only ordinry dierentil equtions, but lso dierentil equtions with impulses, mesure dierentil equtions nd other concepts. For exmple, the recent ppers [3] nd [4] demonstrte Supported by grnt KJB o the Grnt Agency o the Acdemy o Sciences o the Czech Republic, nd by grnt MSM o the Czech Ministery o Eduction. E-mil ddress: slvik@krlin.m.cuni.cz X/$ see ront mtter 2011 Elsevier Inc. All rights reserved. doi: /j.jm

2 A. Slvík / J. Mth. Anl. Appl the possibility o converting retrded unctionl dierentil equtions to generlized ordinry dierentil equtions or unctions with vlues in Bnch spce. Šten Schwbik hs suggested dierent reserch direction in his work [10], which shows tht discrete systems o the orm x k+1 = x k, k N, might be lso rewritten s generlized dierentil equtions. The present pper cn be considered s continution o his work; we concentrte on dynmic equtions on time scles o the orm x Δ t = xt, t, t T, nd show procedure which llows us to convert this dynmic eqution into generlized dierentil eqution. Finlly, to illustrte the useulness o this procedure, we obtin some new results concerning stbility nd continuous dependence on prmeters or dynmic equtions on time scles s corollries o known results or generlized dierentil equtions. Although we do not presuppose knowledge o generlized dierentil equtions, some milirity with the subject might be helpul; the monogrph [9] is recommended s good strting point. On the other hnd, we ssume bsic knowledge o dynmic equtions on time scles s presented in [1] nd [2]. 2. Integrls nd their properties We strt with short summry o the generlized Kurzweil integrl, lso known s the generlized Perron integrl. Consider unction δ :[, b] R +. A prtition D o intervl [, b] with division points = 0 1 k = b nd tgs τ i [ i 1, i ], i = 1,...,k, is clled δ-ine i [ i 1, i ] [ τ i δτ i, τ i + δτ i ], i = 1,...,k. AunctionF :[, b] [, b] R n is clled Kurzweil integrble over [, b] i there exists vector I R n such tht given n ε > 0, there is unction δ :[, b] R + such tht k F τ j, j F τ j, j 1 I < ε j=1 or every δ-ine prtition D. The vector I R n is clled the generlized Kurzweil integrl o F over [, b] nd will be denoted b by DFτ, t. b An importnt specil cse is the Kurzweil Stieltjes integrl s dgs, which is obtined rom pir o unctions :[, b] R n nd g :[, b] R by setting F τ, t = τ gt. Note lso tht the choice gt = t leds to the Henstock b Kurzweil integrl s ds mentioned in the introduction; this justiies the nme generlized Kurzweil integrl. I < b, welet DFτ b, t = b DFτ, t provided the right-hnd side exists; we lso set b DFτ, t = 0 when = b. The ollowing results will be needed lter in our development; see Chpter 1 o [9] or more inormtion bout the generlized Kurzweil integrl. Theorem 1. I :[, b] R n is regulted unction nd g :[, b] R b is nondecresing unction, then the integrl s dgs exists. Moreover, when s Coreverys [, b],then b s dgs C gb g. Proo. See Corollry 1.34 in [9]; the inequlity ollows esily rom the deinition o the integrl. Theorem 2. Let :[, b] R n nd g :[, b] R b be pir o unctions such tht g is regulted nd s dgs exists. Then the unction ht = s dgs, t [, b], is regulted nd stisies ht+ = ht + tδ + gt, t [, b, ht = ht tδ gt, t, b ], where Δ + gt = gt+ gt nd Δ gt = gt gt.

3 536 A. Slvík / J. Mth. Anl. Appl Proo. The sttement is specil cse o Theorem 1.16 in [9]. Theorem 3. Let g :[, b] R be nondecresing unction. Consider sequence o unctions k :[, b] R n,k N, suchtht b kt dgt exists or every k N. Assume there exists unction m :[, b] R b such tht the integrl mt dgt exists, nd such tht k t mt, t [, b], k N. I lim k t = t or every t [, b],then b b t dgt = lim b k t dgt. t dgt exists nd Proo. The sttement ollows esily rom Corollry 1.32 in [9]. Our next gol in this section is to show tht the Riemnn Δ-integrl, which represents time scle version o the clssicl Riemnn integrl, is in ct specil cse o the Kurzweil Stieltjes integrl. We ssume tht the reder is milir with the Riemnn Δ-integrl s described in Chpter 5 o [2]. Given time scle T nd pir o numbers, b T, the symbol [, b] T will be used throughout this pper to denote compct intervl in T, i.e.[, b] T ={t T; t b}. The open nd hl-open intervls re deined in n similr wy. On the other hnd, [, b] will be used to denote intervls on the rel line, i.e. [, b]={t R; t b}. This nottionl convention should help the reder to distinguish between ordinry nd time scle intervls. Given rel number t sup T, wedeine t = in{s T; s t}. Since T is closed set, we hve t T. Further,let {, T sup T] i sup T <, =, otherwise. Given unction : T R n,wedeineunction : T R n by t = t, t T. Similrly,givensetB R n nd unction : B T R n,wedeine x, t = x, t, x B, t T. Lemm 4. I : T R n is regulted unction, then : T R n is lso regulted. I is let-continuous on T, then is letcontinuous on T. I is right-continuous on T,then is right-continuous t right-dense points o T. Proo. Let us clculte lim t t0 t, where T.I T nd it is let-dense, then lim t = lim t. t t I T nd it is let-scttered, then lim t = =. t Finlly, i / T, then lim t = t 0 =. t Now consider lim t t0 + t, where T nd < sup T.I T nd it is right-dense, then lim t = lim t. t + t + I T nd it is right-scttered, then lim t = σ. t +

4 A. Slvík / J. Mth. Anl. Appl Finlly, i / T, then lim t = t 0 =. t + Theorem 5. Let : T R n be n rd-continuous unction. Choose n rbitrry T nd deine F 1 t = F 2 t = sδs, t T, s dgs, t T, where gs = s or every s T.ThenF 2 = F 1. Proo. Note tht the unctions F 1 nd F 2 re well-deined; indeed, the Riemnn Δ-integrl in the deinition o F 1 exists becuse is rd-continuous, nd the Kurzweil Stieltjes integrl in the deinition o F 2 exists becuse is regulted use Lemm 4 nd the ct tht every rd-continuous unction is regulted nd g is nondecresing. To complete the proo, it is suicient to prove the ollowing two sttements: 1 F 1 t = F 2 t or every t T. 2 I t T nd s = sup{u T; u < t}, thenf 2 is constnt on s, t]. We strt with the second sttement, which is esy to prove: I u, v s, t] nd u < v, then v F 2 v F 2 u = u s dgs = 0, where the lst equlity ollows rom the deinition o the Kurzweil Stieltjes integrl nd the ct tht g is constnt on [u, v]. To prove the irst sttement, we note tht F 1 = F 2 = 0 nd it is thus suicient to show tht F Δ 1 t = F Δ 2 t or every t T ny two unctions with the sme Δ-derivtive dier only by constnt. It ollows rom the properties o the Riemnn Δ-integrl tht F Δ 1 t = t, nd it remins to clculte F Δ 2. When t is right-dense point, then is continuous t t nd lim s = t = t s t see Lemm 4. Thereore, given n rbitrry ε > 0, there is δ>0 such tht s t < ε whenever s t <δ. Now, consider sequence o time scle points {t k } k=1 such tht lim t k = t. Wecnindk 0 N such tht t k t <δ whenever k k 0.Thusoreveryk k 0 we hve F 2 t k F 2 t k t t k t = 1 s dgs t t k t t k 1 = s t dgs t k t ε gt k gt = ε, t k t t since gt k = t k nd gt = t. It ollows tht F 2 t k F 2 t lim = t, t k t i.e. F Δ 2 t = t. On the contrry, when t is right-scttered point, we hve F 2 σ t = F2 t+ = F 2 t + tδ + gt, where the irst equlity ollows rom the ct tht F 2 is constnt on t, σ t] nd the second equlity is consequence o Theorem 2. But Δ + gt = gt+ gt = σ t t = μt, nd it ollows tht F Δ 2 t = F 2σ t F 2 t μt = t.

5 538 A. Slvík / J. Mth. Anl. Appl Note tht the integrl s dgs in the deinition o F 2 lso exists s the Lebesgue Stieltjes integrl. We hve chosen the Kurzweil Stieltjes integrl simply becuse it seems to be more nturl in the context o generlized dierentil equtions. On the other hnd, the integrl need not exist s the Riemnn Stieltjes integrl, becuse there might be points where both nd g re discontinuous this is in ct typicl behvior t right-scttered points. 3. Min result This section describes the correspondence between dynmic equtions on time scles nd generlized ordinry dierentil equtions. To obtin resonble theory, we restrict ourselves to dierentil nd dynmic equtions whose right-hnd sides re unctions stisying the conditions given below. Assume tht G = B I, where I R is n intervl nd B R n.givenunctionf : G R n, we introduce the ollowing conditions, which ply n importnt role in the theory o generlized ordinry dierentil equtions: F1 There exists nondecresing unction h : I R such tht F x, t2 F x, t 1 ht2 ht 1 or every x B nd t 1, t 2 I. F2 There exists continuous incresing unction ω :[0, R with ω0 = 0 such tht F x, t2 F x, t 1 F y, t 2 + F y, t 1 ω x y ht2 ht 1 or every x, y B nd t 1, t 2 I. Now, consider set B R n nd unction : B T R n. Let us introduce the ollowing three conditions: C1 is rd-continuous, i.e. the unction t xt, t is rd-continuous whenever x : T B is continuous unction. C2 There exists regulted unction m : T R such tht x, t mt or every x B nd t T. C3 There exists continuous incresing unction ω :[0, R with ω0 = 0 nd regulted unction l : T R such tht x, t y, t ltω x y or every x, y B nd t T. The ollowing lemm describes the reltion between the two sets o conditions. Lemm 6. Consider set B R n. Assume tht : B T R n stisies conditions C1 C3.Deinegs = s or every s T.Then or rbitrry T, the unction F x, t = x, s dgs, x B, t T, stisies conditions F1 F2 on the set G = B T with ht = l s + m s dgs. Proo. For ixed x B, condition C1 implies tht the unction t x, t is rd-continuous on T, nd thereore t x, t is regulted on T. The unction g is nondecresing, nd thus the Kurzweil Stieltjes integrl exists nd F is well deined. Similrly, the unctions l nd m re regulted, nd thus the integrl in the deinition o h exists. When t 1 t 2,wehve F x, t2 F x, t 1 2 = x, s dgs t 1 2 x, s t 2 dgs m s dgs ht 2 ht 1 t 1 t 1

6 A. Slvík / J. Mth. Anl. Appl nd F x, t2 F x, t 1 F y, t 2 + F y, t 1 = 2 t x, s dgs t 1 y, s dgs = x, s y, s t 2 dgs ω x y l s dgs 2 t 1 x, s y, s dgs t 1 ω x y ht 2 ht 1. t 1 The cse t 1 > t 2 is similr nd is let to the reder. Beore proceeding to the min result, we need the ollowing uxiliry lemms. Lemm 7. Let G = B [,β],whereb R n. Consider unction F : G R n such tht t F x, t is regulted on I or every x B. I x :[,β] B is step unction, i.e. i there exists prtition = s 0 < s 1 < < s k = β nd vectors c 1,...,c k R n such tht xs = c i or every s s i 1, s i, then DF xτ, k t = F c j, s j F c j, s j 1 + j=1 + F xs j 1, s j 1 + F xs j 1, s j 1 + F xs j, s j F xs j, s j. Proo. See the proo o Corollry 3.15 in [9]. Lemm 8. Consider set B R n nd unction : B T R n such tht t x, t is regulted on T or every x B. Deine gt = t or every t T, choose n rbitrry T nd let F x, t = x, s dgs, x B, t T. I [,β] T nd x :[,β] B is step unction, then DF xτ, t = xt, t dgt. Proo. By Lemm 4, the unction t x, t is regulted on T or every x B, nd thus the integrl in the deinition o F exists. Given step unction x :[,β] B, there exists prtition = s 0 < s 1 < < s k = β nd vectors c 1,...,c k R n such tht xs = c i or every s s i 1, s i.

7 540 A. Slvík / J. Mth. Anl. Appl The unction t F x, t is regulted by Theorem 2 nd we my use Lemm 7 to obtin DF xτ, t = lim k F c j, s j ε F c j, s j 1 + ε 1 j=1 + lim + lim k F xs j 1, s j 1 + ε F xs j 1, s j 1 j=1 k F xs j, s j F xs j, s j ε. 3 j=1 Now, since x is step unction, it is not diicult to see tht t xt, t is regulted, nd thus the integrl xt, t dgt exists. In this cse, we obtin 2 xt, t dgt = k j j=1 s j 1 = lim k + lim xs, s dgs s j 1 +ε j=1 s j 1 k s j ε j=1 s j 1 +ε xs, s dgs 4 xs, s dgs 5 + lim k j j=1 s j ε xs, s dgs. 6 Obviously, or every i {1,...,k} we hve F c j, s j ε F c j, s j 1 + ε = s j ε s j 1 +ε xt, t dgt, nd thus 1 equls 5. Theorem 2 gives lim F xs j 1, s j 1 + ε F xs j 1, s j 1 = lim s j 1 +ε s j 1 xs j 1, s dgs = xs j 1, s j 1 Δ + gs j 1 nd lim s j 1 +ε s j 1 xs, s dgs = xs j 1, s j 1 Δ + gs j 1 nd thus 2 equls 4. Finlly, nd lim F xs j, s j F xs j, s j ε j = lim xs j, s dgs = xs j, s j Δ gs j s j ε

8 A. Slvík / J. Mth. Anl. Appl j lim xs, s dgs = xs j, s j Δ gs j s j ε nd thus 3 equls 6. Lemm 9. Let G = B [,β], whereb R n. Assume tht F : G R n stisies conditions F1 F2 or some h nd ω. I x :[,β] B is pointwise limit o step unctions x k :[,β] B, then DF xτ, t = lim DF x k τ, t. Proo. See Corollry 3.15 in [9]. It is known ct tht given regulted unction x :[,β] R n nd number ε > 0, there is step unction ϕ :[,β] R n such tht xt ϕt < ε or every t [,β]. In other words, every regulted unction is uniorm limit o step unctions. Now suppose there is set B R n such tht xt B or every t [,β]; we wish to show tht x cn be uniormly pproximted by step unctions with vlues in B. Assume tht the bove mentioned step unction ϕ is constnt on intervls s i 1, s i, where = s 0 < s 1 < < s k = β is prtition o [,β]. Now, choose t i s i 1, s i or every i {1,...,k} nd construct unction ψ :[,β] B s ollows: { xsi or s = s i, ψs = xt i or s s i 1, s i. It is cler tht ψ is step unction. Moreover, when s s i 1, s i,then xs ψs xs ϕs + ϕs ψs = xs ϕs + ϕt i xt i < 2ε. It ollows tht xt ψt < 2ε or every t [,β]. This mens tht x cn be uniormly pproximted by step unctions with vlues in B. Lemm 10. Let B R n nd ssume tht : B T R n stisies conditions C1 C3. Deinegt = t or every t T, choose n rbitrry T nd let F x, t = x, s dgs, x B, t T. I [,β] T nd x :[,β] B is regulted unction, then DF xτ, t = xt, t dgt. Proo. Given regulted unction x :[,β] B, there is sequence o step unctions x k :[,β] B which converge uniormly to x on [,β]. Condition C3 implies x k t, t xt, t l t ω xk t xt, k N, t [,β], nd thus lim x k t, t = xt, t or every t [,β]. Using irst Lemm 9 the ssumptions re stisied by Lemm 6 nd then Lemm 8, we obtin DF xτ, t = lim DF x k τ, t = lim x k t, t dgt = xt, t dgt, where the lst equlity ollows rom Theorem 3 note tht xt, t m t, m is regulted, nd thus the ssumptions re stisied.

9 542 A. Slvík / J. Mth. Anl. Appl Lemm 11. I x :[,β] R n is solution o the generlized ordinry dierentil eqution then = DFx, t, dτ lim xu F xt, u + F xt, t = xt u t or every t [,β]. Proo. See Proposition 3.6 in [9]. Now we hve ll prerequisites necessry or the proo o the min result. Theorem 12. Let X R n nd ssume tht : X T R n is such tht conditions C1 C3 re stisied on every set G = B [,β] T, where B X is bounded. I x : T X is solution o x Δ t = xt, t, t T, 7 then x : T X is solution o where dτ = DFx, t, t T, F x, t = x, s dgs, x X, t T, 8 T,ndgs = s or every s T. Moreover, every solution y : T Xo8 hs the orm y = x,wherex: T Xissolution o 7. Proo. Choose n rbitrry T. Ix : T R n is solution o 7, then xs = x + It ollows tht x s = x + xt, t Δt, s T. xt, t Δt, s T. Using Theorem 5, we rewrite the lst eqution s x s = x + x t, t dgt, s T. 9 Let I be compct intervl in T contining both nd s.sincex is continuous, it is bounded on I. Thereore it is possible to ind bounded set B X such tht xt B or every t I. The unction stisies conditions C1 C3 on B I nd we my use Lemm 10 to replce the lst equlity by x s = x + DF x τ, t, s T, which mens tht x is solution o the generlized eqution 8. To prove the second ssertion, let y : T X be solution o 8. Then ys = y + DF yτ, t, s T.

10 A. Slvík / J. Mth. Anl. Appl Fix n rbitrry s T nd let [,β] T be time scle intervl such tht, s [,β]. Foreveryτ [,β, Lemm 11 implies tht yτ = lim yu F yτ, u + F yτ, τ u τ + u yu yτ, s dgs = lim u τ + τ = lim yu yτ, τ Δ + gτ, u τ + nd thereore lim u τ + yu exists. Similrly, or every τ,β], wehve yτ = lim yu F yτ, u + F yτ, τ u τ u = lim yu yτ, s dgs u τ τ = lim u τ yu + yτ, τ Δ gτ = lim u τ yu, becuse g is let-continuous unction. Since y is regulted nd thereore bounded on [,β], it is possible to ind bounded set B X such tht yt B or every t [,β]. The unction stisies conditions C1 C3 on B [,β] T nd Lemm 6 gurntees tht the unction F stisies conditions F1 F2 on B [,β]. Using Lemm 10 gin, we obtin ys = y + yt, t dgt, s T. But the right-hnd side is constnt on every intervl s, t], where t T nd s = sup{u T; u < t} see the rgument in the proo o Theorem 5. Thus y = x, where x : T B is the restriction o y to T. This implies 9, nd consequently lso 7 note tht, ccording to Theorem 2, x is rd-continuous unction. From now on, the letter g will lwys denote the unction gs = s. Let us puse or moment to discuss conditions C1 C3. Condition C1 is irly common in the theory o dynmic equtions; its purpose is to ensure tht the integrl eqution xs = x + xt, t Δt cn be dierentited to obtin x Δ t = xt, t. In more generl setting, we could ocus our interest on the integrl eqution only; in this cse, it would be suicient to ssume tht t xt, t is regulted whenever x : T X is regulted unction. Conditions C2 C3 were used to prove tht the unction F x, t = x, s dgs, x B, t T, stisies conditions F1 F2. Condition C3 represents generliztion o Lipschitz-continuity with respect to x, which corresponds to the specil cse ωr = r nd lt = L. Agin, this is irly stndrd condition. In mny cses, the unction is deined on R n T nd hs continuous prtil derivtives with respect to x 1,...,x n. Since we require the conditions to be stisied only or sets o the orm G = B [,β] T with B X bounded, it is esy to see tht both C2 nd C3 re stisied. Moreover, i B ={x R n ; x r}, condition C3 cn be wekened; in this cse, it is suicient to ssume tht x x, t is continuous or every t T see Chpter 5 o [9], which describes the cse T = R, but the sme resoning cn be used or generl time scle. 4. Liner equtions To illustrte Theorem 12 on simple exmple, consider the liner dynmic eqution x Δ t = txt + ht, t T, 10

11 544 A. Slvík / J. Mth. Anl. Appl where : T R n n nd h : T R n re rd-continuous unctions we use the symbol R n n to denote the set o ll n n mtrices. It is esy to see tht the unction x, t = tx + ht stisies conditions C1 C3 on every set G = B [,β] T, where B X is bounded. To obtin the corresponding generlized dierentil eqution, we choose n rbitrry τ 0 T nd let F x, t = τ 0 sx + h s dgs = Atx + Ht, where At = τ 0 s dgs nd Ht = τ h 0 s dgs. Now, Theorem 12 sys tht i x : T X is solution o 10, then the unction x is solution o the liner generlized dierentil eqution dτ = D Atx + Ht, t T. Conversely, every solution o this generlized eqution hs the orm x, where x : T R n is solution o the dynmic eqution 10. The monogrph [9] contins irly complete theory o liner generlized equtions. For exmple, Eq. 11 is known to hve unique solution stisying x = x 0, whenever I At At nd I + At+ At re regulr or every t. 12 Let us rephrse this condition in the lnguge o Eq. 10; Theorem 2 gives At+ = At + tδ + gt, At = At tδ gt. First, i t T \T, thenδ + gt = Δ gt = 0 nd 12 is stisied. Next, ssume t T. Sinceg is let-continuous unction, we lwys hve Δ gt = 0 nd thereore I At At = I is regulr. Finlly, i t is right-dense point, then Δ + gt = 0 nd I + At+ At = I is regulr; i t is right-scttered, then Δ + gt = μt nd I + At+ At is regulr i nd only i I + tμt is regulr. The lst condition is clled regressivity nd is well known in the theory o liner dynmic equtions. Let us mention one more result: Under ssumption 12, there exists unction U : T T R n n such tht the unction xt = Ut, x 0 represents the unique solution o the homogeneous eqution dτ = D Atx, t T, x = x 0. The unction U hs the ollowing properties: i Ut, t = I or every t T, ii Ut, s = Ut, rur, s or every r, s, t T, iii Ut+, s = I + At+ AtUt, s or every s, t T, iv Ut, s is lwys regulr mtrix nd Ut, s 1 = Us, t. We lredy know tht or t T, the third condition might be written s Ut+, s = I + tμtut, s. It is esy to recognize tht the restriction o U to T T is the mtrix exponentil unction, which is denoted by e t, in the book [1]. In his pper [11], Š. Schwbik presents the ollowing interesting construction o the unction U : He deines the Perron product integrl b I + das s mtrix P Rn n such tht or every ε > 0, there is unction δ :[, b] R + which stisies 1 I + A j A j 1 P < ε j=k or every δ-ine prtition with division points = 0 1 k = b nd tgs τ i [ i 1, i ], i = 1,...,k. Now,the unction U is obtined by considering the product integrl s unction o its upper bound, i.e. 11 Ut, = t I + das. A similr result or liner systems on time scles is given in the pper [12], which shows tht the mtrix exponentil unction e t, corresponding to n rd-continuous unction : T R n n cn be written in the orm

12 A. Slvík / J. Mth. Anl. Appl e t, = t I + sδs, where the symbol on the right-hnd side stnds or the product Δ-integrl. Thus our considertions imply tht t I + sδs = t I + das, t0, t T or every rd-continuous unction : T R n n nd At = τ 0 s dgs. 5. Continuous dependence on prmeter In this section, we use two known results concerning continuous dependence o generlized equtions on prmeters to obtin new theorems bout dynmic equtions on time scles. The symbol B r will be used to denote the open bll {x R n ; x < r} nd B r stnds or the corresponding closed bll. Theorem 13. Let c > 0, G= B c [,β], nd consider sequence o unctions F k : G R n,k N 0,suchtht lim F kx, t = F 0 x, t, x B c, t [,β]. Assume there exist unctions h nd ω such tht F k stisies conditions F1 F2 or every k N 0. Finlly, suppose there exist unction x :[,β] B c nd sequence o unctions x k :[,β] B c such tht Then k dτ = DF kx k, t, t [,β], k N, lim ks = xs, s [,β]. dτ = DF 0x, t, t [,β]. Proo. See Theorem 8.2 in [9]. Theorem 14. Consider sequence o unctions k : B c [,β] T R n,k N 0. Assume there exist unctions l, m nd ω such tht ech unction k,k N 0, stisies conditions C1 C3. Suppose tht lim k x, sδs = 0 x, sδs or every x B c nd t [,β] T. Finlly, suppose there exist unction x :[,β] T B c nd sequence o unctions x k :[,β] T B c, k N,suchtht 13 Then x Δ k t = k xk t, t, t [,β] T, lim x ks = xs, s [,β] T. x Δ t = 0 xt, t, t [,β]t. Proo. Let G = B c [,β] nd F k x, t = Eq. 13 together with Theorem 5 imply lim F kx, t = F 0 x, t, k x, s dgs, x B c, t [,β], k N 0. x B c, t [,β]. It ollows rom Lemm 6 tht F k stisies conditions F1 F2 or every k N 0. It is cler tht

13 546 A. Slvík / J. Mth. Anl. Appl lim x k s = x s, Theorem 12 implies s [,β]. k dτ = DF k x k, t, t [,β], k N. Thus the ssumptions o Theorem 13 re stisied nd dτ = DF 0 x, t, t [,β]. The unction x is bounded nd it ollows rom Theorem 12 tht x Δ t = 0 xt, t, t [,β]t. Given unction F : B I R n nd n intervl [,β] I, solution x :[,β] B o the generlized dierentil eqution = DFx, t dτ is sid to be unique i every other solution y :[, γ ] B o 14 such tht x = y stisies xt = yt or every t [, γ ] [,β]. Theorem 15. Let c > 0, G= B c [,β], nd consider sequence o unctions F k : G R n,k N 0,suchtht lim F kx, t = F 0 x, t, x B c, t [,β]. Assume there exist let-continuous unction h nd unction ω such tht F k stisies conditions F1 F2 or every k N 0.Let x :[,β] B c be unique solution o dτ = DF 0x, t. Finlly, ssume there exists ρ > 0 such tht y xs < ρ implies y B c whenever s [,β] i.e., ρ-neighborhood o x is contined in B c. Then, given n rbitrry sequence o n-dimensionl vectors {y k } k=1 such tht lim y k = x, thereisk 0 N nd sequence o unctions x k :[,β] B c,k k 0, which stisy k dτ = DF kx k, t, t [,β], x k = y k, lim ks = xs, s [,β]. 14 Proo. See Theorem 8.6 in [9]. In nlogy with the previous cse, we sy tht solution x :[,β] T B o the dynmic eqution x Δ t = xt, t is unique i every other solution y :[, γ ] B such tht x = y stisies xt = yt or every t [, γ ] [,β]. Theorem 16. Consider sequence o unctions k : B c [,β] T R n,k N 0. Assume there exist unctions l, m, ω such tht ech unction k,k N 0, stisies conditions C1 C3. Suppose tht lim k x, sδs = 0 x, sδs or every x B c nd t [,β] T.Letx:[,β] T B c be unique solution o 15 x Δ t = 0 xt, t. Finlly, ssume there exists ρ > 0 such tht y xs < ρ implies y B c whenever s [,β] T. Then, given n rbitrry sequence o n-dimensionl vectors {y k } k=1 such tht lim y k = x, thereisk 0 N nd sequence o unctions x k :[,β] T B c, k k 0, which stisy x Δ k t = k xk t, t, t [,β] T, x k = y k, lim x ks = xs, s [,β] T.

14 A. Slvík / J. Mth. Anl. Appl Proo. Using the sme resoning s in the proo o Theorem 14, we construct sequence o unctions {F k } deined on k=0 G = B c [,β]. All these unctions stisy conditions F1 F2 with ht = t l 0 s + m s dgs; note tht gt = t is let-continuous unction, nd thus h is let-continuous ccording to Theorem 2. It ollows rom Theorem 12 tht x is unique solution o dτ = DF 0 x, t, t [,β]. The proo is inished by pplying Theorem 15. Let us note tht Theorem 4.11 in [9] sttes tht i the unction F stisies conditions F1 F2 with ω such tht lim v 0+ u v dr ωr = or every u > 0, then every solution x :[,β] B c o the generlized eqution = DFx, t dτ isunique.now,itisotenthecsethttheunction is Lipschitz-continuous with respect to x on B c [,β], nd thus stisies conditions C1 C3 with lt = L nd ωr = r. In this cse, we see tht 16 is true nd thereore every solution is unique. 6. Stbility The dynmic eqution x Δ t = xt, t hs the trivil solution x 0 i nd only i 0, t = 0oreveryt T. The present section is devoted to the investigtion o stbility o this trivil solution. The problem o stbility hs lredy been considered in number o ppers, see e.g. [5 7]. However, the theorem which will be obtined in this section describes two slightly dierent types o stbility. Consider set I R nd unction : I R n. Given inite set o points D ={, t 1,...,t k } I such tht t 1 t k,let v, D = k ti t i 1. i=1 The vrition o over I is deined s vr t I t = sup v, D, D where the supremum rnges over ll inite subsets D o I. Note tht when I is n intervl on the rel line, then we obtin the usul vrition o unction over n intervl, but our slightly more generl deinition permits us to clculte the vrition o unction deined on time scle intervl [, b] T. 16 Lemm 17. Given n rbitrry unction :[, b] T R n,wehve vr t = vr t. t [,b] T t [,b] Proo. The sttement ollows rom the ct tht i D ={, t 1,...,t k } [, b], then D ={t 0, t 1,...,t k } [, b] T v, D = v, D. nd We strt with Lypunov-type theorem or the generlized eqution = DFx, t. dτ Note tht this eqution hs the trivil solution x 0onnintervlI R i nd only i F 0, t 1 = F 0, t 2 or ech pir t 1, t 2 I.

15 548 A. Slvík / J. Mth. Anl. Appl Theorem 18. Let c > 0, R nd G = B c [,. Consider unction F : G R n which stisies conditions F1 F2 nd F 0, t 1 = F 0, t 2 or every t 1, t 2. Assume there exists number 0, c nd unction V :[, B R with the ollowing properties: V1 t V t, x is let-continuous or every x B. V2 There exists continuous incresing unction b :[0, R such tht bρ = 0 i nd only i ρ = 0 nd V t, x b x or every t [, nd x B. V3 V t, 0 = 0 or every t [,. V4 There exists constnt K > 0 such tht V t, x V t, y K x y or every t [, nd x, y B. V5 t V t, xt is nonincresing long every solution x o the generlized eqution = DFx, t. dτ Then the ollowing sttements re true: 1 For every ε > 0, thereisδ>0 such tht i nd y :[,β] B c is let-continuous unction with bounded vrition which stisies y <δnd vr s [,β] ys DF yτ, t <δ, then yt < ε or every t [,β]. 2 For every ε > 0,thereisδ>0 such tht i P :[,β] B c is let-continuous unction with vr Ps<δ, s [,β] then n rbitrry unction y :[,β] B c which is solution o dτ = D F x, t + Pt nd y <δstisies yt < ε or every t [,β]. Proo. See Theorem 10.8 nd Theorem in [9]. The sttement 1 is clled vritionl stbility; it sys tht unctions which re initilly smll, nd which re lmost solutions o the given generlized eqution, re close to zero in the whole intervl. The sttement 2 is clled stbility with respect to perturbtions; it sys tht unctions which re initilly smll, nd which re solutions o generlized eqution with smll perturbtion term, re gin close to zero in the whole intervl. We now proceed to similr theorem concerning dynmic equtions on time scles. Theorem 19. Let c > 0 nd T. Consider unction : B c [, T R n which stisies conditions C1 C3 nd 0, t = 0 or every t [, T. Assume there exists number 0, c nd unction V :[, T B R with the ollowing properties: V1 t V t, x is let-continuous or every x B. V2 There exists continuous incresing unction b :[0, R such tht bρ = 0 i nd only i ρ = 0 nd V t, x b x or every t [, T nd x B. V3 V t, 0 = 0 or every t [, T. V4 There exists constnt K > 0 such tht V t, x V t, y K x y or every t [, T nd x, y B. V5 t V t, xt is nonincresing long every solution x o the dynmic eqution x Δ t = xt, t. Then the ollowing sttements re true: 1 For every ε > 0, thereisδ>0 such tht i nd y :[,β] T B c is let-continuous unction with bounded vrition which stisies y <δnd vr s [,β] T ys yt, t Δt then yt < ε or every t [,β] T. <δ,

16 A. Slvík / J. Mth. Anl. Appl For every ε > 0,thereisδ>0 such tht i p :[,β] T B c is n rd-continuous unction nd pt Δt <δ, then every unction y :[,β] T B c such tht y <δnd y Δ t = yt, t + pt, stisies yt < ε or every t [,β] T. t [,β] T Proo. It is suicient to pply Theorem 18 to the unctions F x, t = V t, x = V t, x. To prove 1, note tht vr s [,β] T x, s dgs, ys yt, t Δt <δ implies vr y s DF y τ, t <δ s [,β] this ollows rom Lemm 17, Theorem 5, nd Lemm 10. To prove 2, note tht implies y Δ t = yt, t + pt, dτ = D F x, t + Pt with Pt = p s dgs, nd tht pt Δt <δ t [,β] T implies s vr ptδt <δ s [,β] T this is esy to see rom the deinition o vrition nd consequently s vr Ps = vr p t dgs <δ s [,β] s [,β] this ollows rom Lemm 17 nd Theorem Conclusion We hve outlined method which enbles us to trnslte existing results concerning generlized ordinry dierentil equtions into the lnguge o dynmic equtions on time scles. The reders re invited to exmine existing sources on generlized dierentil equtions to ind theorems which might be interesting in the time scle setting; the mount o literture devoted to generlized equtions is still growing.

17 550 A. Slvík / J. Mth. Anl. Appl Acknowledgments The uthor thnks the nonymous reeree whose suggestions helped to improve this pper. Reerences [1] M. Bohner, A. Peterson, Dynmic Equtions on Time Scles: An Introduction with Applictions, Birkhäuser, Boston, [2] M. Bohner, A. Peterson, Advnces in Dynmic Equtions on Time Scles, Birkhäuser, Boston, [3] M. Federson, Š. Schwbik, Generlized ODE pproch to impulsive retrded unctionl dierentil equtions, Dierentil Integrl Equtions [4] M. Federson, Š. Schwbik, Stbility or retrded unctionl dierentil equtions, Ukr. Mth. J [5] J. Hocker, C.C. Tisdell, Stbility nd instbility or dynmic equtions on time scles, Comput. Mth. Appl [6] B. Kymkçln, Lypunov stbility theory or dynmic systems on time scles, J. Appl. Mth. Stoch. Anl [7] B. Kymkçln, L. Rngrjn, Vrition o Lypunov s method or dynmic systems on time scles, J. Mth. Anl. Appl [8] J. Kurzweil, Generlized ordinry dierentil equtions nd continuous dependence on prmeter, Czechoslovk Mth. J [9] Š. Schwbik, Generlized Ordinry Dierentil Equtions, World Scientiic, Singpore, [10] Š. Schwbik, Generlized ordinry dierentil equtions nd discrete systems, Arch. Mth. Brno [11] Š. Schwbik, The Perron product integrl nd generlized liner dierentil equtions, Čsopis Pěst. Mt [12] A. Slvík, Product integrtion on time scles, Dynm. Systems Appl