Linear measure functional differential equations with infinite delay

Transcription

1 Mthemtische Nchrichten, 27 Jnry 2014 Liner mesre fnctionl differentil eqtions with infinite dely Giselle Antnes Monteiro 1, nd Antonín Slvík 2, 1 Mthemticl Institte, Acdemy of Sciences of the Czech Repblic, Žitná 25, Prh 1, Czech Repblic. 2 Chrles University in Prge, Fclty of Mthemtics nd Physics, Sokolovská 83, Prh 8, Czech Repblic Key words Mesre fnctionl differentil eqtions, generlized ordinry differentil eqtions, Krzweil-Stieltjes integrl, implsive fnctionl differentil eqtions, infinite dely, existence nd niqeness, continos dependence Sbject clssifiction 34K06, 34G10, 34K45 We se the theory of generlized liner ordinry differentil eqtions in Bnch spces to stdy liner mesre fnctionl differentil eqtions with infinite dely. We obtin new reslts concerning the existence, niqeness, nd continos dependence of soltions. Even for eqtions with finite dely, or reslts re stronger thn the existing ones. Finlly, we present n ppliction to fnctionl differentil eqtions with implses. Copyright line will be provided by the pblisher 1 Introdction In this pper, we del with liner fnctionl eqtions of the form y(t) = y() + l(y s, s) dg(s) + p(s) dg(s), t [, b], (1.1) where the fnctions y, l, nd p tke vles in R n, l is liner in the first vrible, nd both integrls re the Krzweil-Stieltjes integrls with respect to nondecresing fnction g : [, b] R. As is sl in the theory of fnctionl differentil eqtions, the symbol y s stnds for the fnction y s (θ) = y(s + θ), θ (, 0]. Eqtion (1.1) represents specil cse of the mesre fnctionl differentil eqtion y(t) = y() + f(y s, s) dg(s), t [, b] (1.2) introdced in [3] by M. Federson, J. G. Mesqit nd A. Slvík for the cse of finite dely; eqtions of this type with infinite dely were lter stdied by A. Slvík in [16]. For g(s) = s, eqtion (1.2) redces to the clssicl fnctionl differentil eqtion stdied by nmeros thors (see e.g. [10]). Moreover, it ws shown in [3] nd [4] tht implsive fnctionl differentil eqtions s well s fnctionl dynmic eqtions on time scles re specil cses of the mesre fnctionl differentil eqtion (1.2). Or min tool in the stdy of eqtion (1.1) is the theory of generlized ordinry differentil eqtions introdced by J. Krzweil in [11]. The reltion between fnctionl differentil eqtions nd generlized ordinry differentil eqtions in infinite-dimensionl Bnch spces ws first described by C. Imz, F. Oliv nd Z. Vorel in [9] nd [14]. Lter, similr correspondence ws estblished for implsive E-mil: gm@mth.cs.cz. Corresponding thor. E-mil: slvik@krlin.mff.cni.cz. Copyright line will be provided by the pblisher

2 2 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely fnctionl differentil eqtions by M. Federson nd Š. Schwbik in [5], nd for mesre fnctionl differentil eqtions with finite nd infinite dely in [3] nd [16], respectively. For eqtions with infinite dely, n importnt isse is the choice of the phse spce; this topic is discssed in Section 2. In Section 3, we smmrize the bsic fcts of the Krzweil integrtion theory needed for or prposes nd prove new convergence theorem for the Krzweil-Stieltjes integrl. Section 4 describes the correspondence between liner mesre fnctionl differentil eqtions nd generlized liner ordinry differentil eqtions. In Section 5, we prove globl existence-niqeness theorem for liner mesre fnctionl differentil eqtions. Section 6 contins the min reslts: new continos dependence theorem for generlized liner ordinry differentil eqtions (inspired by the work of G. A. Monteiro nd M. Tvrdý in [13]), nd its conterprt for fnctionl eqtions. Finlly, in Section 7, we present n ppliction of the previos reslts to implsive fnctionl differentil eqtions. Or pper confirms tht the theory of generlized ordinry differentil eqtions plys n importnt role in the stdy of fnctionl differentil eqtions. Moreover, by focsing on liner eqtions, we re ble to obtin mch stronger reslts thn in the nonliner cse (even for eqtions with finite dely). 2 Axiomtic description of the phse spce In contrst to clssicl fnctionl differentil eqtions, the soltions of mesre fnctionl differentil eqtions re no longer continos bt merely reglted fnctions. Given n intervl [, b] R nd Bnch spce X, recll tht fnction f : [, b] X is clled reglted if the limits lim f(s) = f(t ) X, t (, b] nd lim f(s) = f(t+) X, t [, b) s t s t+ exist. It is well known tht every reglted fnction f : [, b] X is bonded; the symbol f stnds for the spremm norm of f. Reglted fnctions on open or hlf-open intervls re defined in similr wy. Given n intervl I R, we se the symbol G(I, X) to denote the set of ll reglted fnctions f : I X. For eqtions with infinite dely, one of the crcil problems is the choice of sitble phse spce. In the xiomtic pproch, we do not choose fixed phse spce, bt insted del with ll spces stisfying given set of xioms. Conseqently, there is no need to prove similr reslts repetedly for different phse spces. For clssicl fnctionl differentil eqtions with infinite dely, the xiomtic pproch is well described in the pper [6] of J. K. Hle nd J. Kto, s well s in the monogrph [7] by S. Hino, S. Mrkmi, nd T. Nito. Or cndidte for the phse spce of liner mesre fnctionl differentil eqtion is spce H 0 G((, 0], R n ) eqipped with norm denoted by. We ssme tht H 0 stisfies the following conditions: (HL1) H 0 is complete. (HL2) If y H 0 nd t < 0, then y t H 0. (HL3) There exists loclly bonded fnction κ 1 : (, 0] R + sch tht if y H 0 nd t 0, then y(t) κ 1 (t) y. (HL4) There exist fnctions κ 2 : [0, ) [1, ) nd λ : [0, ) R + sch tht if t 0 nd y H 0, then y t κ 2 (t ) sp y(s) + λ(t ) y. s [,t] (HL5) There exists loclly bonded fnction κ 3 : (, 0] R + sch tht if y H 0 nd t 0, then y t κ 3 (t) y. Copyright line will be provided by the pblisher

3 mn heder will be provided by the pblisher 3 Or conditions (HL1) (HL5) re lmost identicl to conditions (H1) (H5) in [16], except tht (HL4) is stronger thn (H4). Indeed, ssme tht σ > 0 nd y H 0 is fnction whose spport is contined in [ σ, 0]. Using (HL4) with t = 0 nd = σ, we obtin y κ 2 (σ) sp t [ σ,0] y(t), which is precisely condition (H4). On the other hnd, there is n dditionl condition (H6) in [16], which is however not strictly necessry (see [16, Remrk 3.10]) nd we omit it here. Remrk 2.1. One cn replce (HL3) by the following condition, which is known from the xiomtic theory of clssicl fnctionl differentil eqtions with infinite dely (see [7]): There exists constnt β > 0 sch tht if y H 0 nd t 0, then y(t) β y t. Indeed, combining this ssmption with (HL5), we obtin y(t) β y t βκ 3 (t) y, i.e., (HL3) is stisfied with κ 1 = βκ 3. The following exmple of phse spce is simple modifiction of the spce C ϕ ((, 0], R n ), which is well known from the clssicl theory of fnctionl differentil eqtions with infinite dely (see [7]). Exmple 2.2. Consider the spce G ϕ ((, 0], R n ) = y G((, 0], R n ); y/ϕ is bonded}, where ϕ : (, 0] R is fixed continos positive fnction. The norm of fnction y G ϕ ((, 0], R n ) is defined s y ϕ = Assme tht y(t) sp t (,0] ϕ(t). ϕ(s + t) γ 1 (t) = sp <, t 0, (2.1) s (, t] ϕ(s) ϕ(s + t) γ 2 (t) = sp <, t 0, (2.2) s (,0] ϕ(s) nd tht γ 2 is loclly bonded fnction. (In the typicl cse when ϕ is nonincresing, the first condition is stisfied tomticlly.) The following clcltions show tht nder these hypotheses, the spce G ϕ ((, 0], R n ) stisfies conditions (HL1) (HL5). The mpping y y/ϕ is n isometric isomorphism between G ϕ ((, 0], R n ) nd the spce BG((, 0], R n ) of ll bonded reglted fnctions on (, 0], which is endowed with the spremm norm. The ltter spce is complete, nd ths G ϕ ((, 0], R n ) is complete, too. For every t < 0 nd y G ϕ ((, 0], R n ), we hve y t (s) sp s (,0] ϕ(s) y(t + s) sp s (,0] ϕ(t + s) ϕ(t + s) sp y ϕ γ 2 (t), s (,0] ϕ(s) which shows tht (HL2) is tre nd (HL5) is stisfied with κ 3 (t) = γ 2 (t). Since y(t) ϕ(t) y(t) ϕ(t) ϕ(t) sp s (,0] y(s) ϕ(s), we see tht (HL3) is stisfied with κ 1 (t) = ϕ(t). Copyright line will be provided by the pblisher

4 4 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely For n rbitrry t 0, we hve y t ϕ = y(t + s) sp = sp s (,0] ϕ(s) s (,t] We estimte the first term s follows: sp s (,] y(s) ϕ(s t) sp s (,] y(s) ϕ(s ) y(s) ϕ(s t) sp s (,] ϕ(s ) sp s (,] ϕ(s t) ϕ(s + t ) = y ϕ sp y ϕ γ 1 (t ) s (, t] ϕ(s) For the second term, we hve the following estimte: Ths, sp s [,t] y(s) ϕ(s t) sp s [,t] y(s) inf s [,t] ϕ(s t) = sp s [,t] y(s) inf s [ t,0] ϕ(s). y t ϕ y ϕ γ 1 (t ) + sp s [,t] y(s) inf s [ t,0] ϕ(s), y(s) ϕ(s t) + sp s [,t] y(s) ϕ(s t). which shows tht (HL4) is stisfied with κ 2 (σ) = 1/(inf s [ σ,0] ϕ(s)) nd λ(σ) = γ 1 (σ), for σ [0, ). For exmple, when ϕ(t) = 1 for every t (, 0], then G ϕ ((, 0], R n ) coincides with the spce BG((, 0], R n ) of ll bonded reglted fnctions on (, 0] nd endowed with the spremm norm (see [16, Exmple 2.2]). Another importnt specil cse, which is phse spce commonly sed for deling with nbonded fnctions, is obtined by tking n rbitrry γ 0 nd letting ϕ(t) = e γt (see [16, Exmple 2.5]). Besides the phse spce H 0, we lso need sitble spces H of reglted fnctions defined on (, ], where R. We obtin these spces by shifting the fnctions from H 0. More precisely, for every R, denote H = y G((, ], R n ); y H 0 }. Finlly, define norm on H by letting y = y for every y H. Exmple 2.3. Let H 0 be one of the phse spces G ϕ described in Exmple 2.2. Then H consists of ll reglted fnctions y : (, ] R n sch tht sp t (,] y(t) ϕ(t ) <. In this cse, the vle of the spremm eqls y. The following lemm is strightforwrd conseqence of (HL1) (HL5). Lemm 2.4. If H 0 G((, 0], R n ) is spce stisfying conditions (HL1) (HL5), then the following sttements re tre for every R: 1. H is complete. 2. If y H nd t, then y t H If t nd y H, then y(t) κ 1 (t ) y. 4. If y H nd t, then y t κ 2 (t ) sp y(s) + λ(t ) y. s [,t] Copyright line will be provided by the pblisher

5 mn heder will be provided by the pblisher 5 5. If y H nd t, then y t κ 3 (t ) y. The next lemm nlyzes the prticlr cse when fnction y H b is defined s the prolongtion of fnction in H 0. Lemm 2.5. Let φ H 0,, b R, with < b, be given nd consider fnction x H b of the form φ(ϑ ), ϑ (, ], x(ϑ) = φ(0), ϑ [, b]. Then, x (κ 2 (b )κ 1 (0) + λ(b )) φ. P r o o f. Using the definition of the norm in H b, Lemm 2.4 nd (HL3), we obtin x = x b κ 2 (b ) sp x(s) + λ(b ) x s [,b] = κ 2 (b ) φ(0) + λ(b ) φ (κ 2 (b )κ 1 (0) + λ(b )) φ. 3 Krzweil integrtion The integrls which occr in this pper represent specil cses of the integrl introdced by J. Krzweil in [11] nder the nme generlized Perron integrl. For the reder s convenience, let s recll its definition. Consider Bnch spce X nd let X denote its norm. As sl, the symbol L(X) denotes the spce of ll bonded liner opertors on X. A pir (D, ξ) is clled prtition of the intervl [, b], if D : = s 0 < s 1 < < s m = b is division of [, b], nd τ j [s j 1, s j ] for every j 1, 2,..., m}. Given fnction δ : [, b] R + (clled gge on [, b]), prtition is sid to be δ-fine, if [s j 1, s j ] (τ j δ(τ j ), τ j + δ(τ j )), j 1, 2,..., m}. A fnction U : [, b] [, b] X is Krzweil integrble on [, b], if there exists vector I X sch tht for every ε > 0, there is gge δ : [, b] R + sch tht m [U(τ j, s j ) U(τ j, s j 1 )] I < ε X j=1 for every δ-fine prtition of [, b]. In this cse, we define the Krzweil integrl s b DU(τ, s) = I. Bsic properties of the Krzweil integrl, sch s linerity, dditivity with respect to djcent intervls, s well s vrios convergence theorems, cn be fond in [17], [12]. When U(τ, s) = f(τ)s, where f : [, b] X is given fnction, the definition bove redces to the definition of the well-known Henstock-Krzweil integrl, which generlizes the Lebesge integrl. In this pper, we re prticlrly interested in Stieltjes-type integrls. The Krzweil-Stieltjes integrl b f dg of fnction f : [, b] Rn with respect to fnction g : [, b] R is obtined by letting U(τ, s) = f(τ) g(s). This is the integrl which ppers in the definition of mesre fnctionl differentil eqtion. Secondly, we need the bstrct Krzweil-Stieltjes integrl b d[a] g, where A : [, b] L(X) nd g : [, b] X (see [18]). This integrl corresponds to the choice U(τ, s) = A(s) g(τ), nd it will pper in the definition of generlized liner ordinry differentil eqtion. Let s recll tht fnction f : [, b] X hs bonded vrition on [, b], if vr [,b] f = sp m f(s j ) f(s j 1 ) X <, j=1 Copyright line will be provided by the pblisher

6 6 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely where the spremm is tken over ll divisions D : = s 0 < s 1 <... < s m = b of the intervl [, b]. Let BV ([, b], X) denote the set of ll fnctions f : [, b] X with bonded vrition. It is worth mentioning tht BV ([, b], X) G([, b], X). The next theorem (see [18, Proposition 15]) provides simple criterion for the existence of the bstrct Krzweil-Stieltjes integrl. Theorem 3.1. If A BV ([, b], L(X)) nd g : [, b] X is reglted fnction, then the integrl b d[a]g exists nd we hve b d[a]g (vr [,b] A) g. X The following property of the indefinite Krzweil-Stieltjes integrl (see [17, Theorem 1.16]) implies tht soltions of mesre fnctionl differentil eqtions re reglted fnctions. Theorem 3.2. Let f : [, b] R n nd g : [, b] R be sch tht g is reglted nd the integrl b f dg exists. Then the fnction (t) = f dg, t [, b], is reglted. Moreover, if g is left-continos, then so is the fnction. The next convergence theorem for the Krzweil-Stieltjes integrl is inspired by similr reslt from [13, Theorem 2.2]. Insted of reqiring the niform convergence of the seqence A k } k=1 to A 0, we show tht weker ssmption is sfficient. Theorem 3.3. Let A k BV ([, b], L(X)), g k G([, b], X) for k N 0. Assme tht the following conditions re stisfied: lim k g k g 0 = 0. There exists constnt γ > 0 sch tht vr [,b] A k γ for every k N. lim k sp t [,b] [A k (t) A 0 (t)]x X = 0 for every x X. Then lim sp k t [,b] d[a k ]g k d[a 0 ]g 0 X = 0. P r o o f. Let ε > 0 be given. Since g 0 is reglted, there exists step fnction g : [, b] X sch tht g 0 g < ε (see [8, Theorem I.3.1]). Also, there exist division = t 0 < t 1 < < t m = b of [, b] nd elements c 1,..., c m X sch tht g(t) = c j for every t (t j 1, t j ). Let k 0 N be sch tht g k g 0 < ε, (A k A 0 )(τ)g(t j ) X < ε/m, j 0,..., m}, (A k A 0 )(τ)c j X < ε/m, j 1,..., m} for every k k 0 nd τ [, b]. For n rbitrry t [, b], we hve d[a k ]g k X d[a 0 ]g 0 d[a k ](g k g) + X d[a k A 0 ]g + X d[a 0 ](g g 0 ). X Copyright line will be provided by the pblisher

7 mn heder will be provided by the pblisher 7 For k k 0, the first nd third term on the right-hnd side cn be estimted sing Theorem 3.1: d[a k ](g k g) γ g k g γ( g k g 0 + g 0 g ) < 2γε, X d[a 0 ](g g 0 ) (vr [,b] A 0 ) g g 0 < (vr [,b] A 0 )ε. X Since g is step fnction, we cn clclte the integrl d[a k A 0 ]g in the second term (see [18, Proposition 14]), nd obtin the following estimte: Conseqently, sp t [,b] m d[a k A 0 ]g [A k A 0 ](t j 1 +)g(t j 1 ) [A k A 0 ](t j 1 )g(t j 1 ) X X d[a k ]g k + + j=1 m [A k A 0 ](t j )c j [A k A 0 ](t j 1 +)c j X j=1 m [A k A 0 ](t j )g(t j ) [A k A 0 ](t j )g(t j ) X 6ε. j=1 for every k k 0, which completes the proof. d[a 0 ]g 0 X < ε(2γ + vr [,b] A 0 + 6) Remrk 3.4. For the so-clled interior integrl, reslt similr to Theorem 3.3 ws proved by C. Hönig in [8, Theorem I.5.8]. 4 Liner mesre fnctionl differentil eqtions nd generlized liner ordinry differentil eqtions A generlized liner ordinry differentil eqtion is n integrl eqtion of the form x(t) = x + d[a] x + h(t) h(), t [, b], (4.1) where A : [, b] L(X), h G([, b], X), nd x X. We sy tht fnction x : [, b] X is soltion of (4.1) on the intervl [, b], if the integrl b d[a] x exists nd eqlity (4.1) holds for ll t [, b]. Eqtions of the form (4.1), which represent specil cse of generlized ordinry differentil eqtions introdced by J. Krzweil in [11], hve been stdied by nmeros thors. The sittion when X is generl Bnch spce ws for the first time investigted by Š. Schwbik in [19], [20]. In this section, we clrify the reltion between liner mesre fnctionl differentil eqtions nd generlized liner ordinry differentil eqtions. Consider the fnctionl eqtion y(t) = y() + l(y s, s) dg(s) + p(s) dg(s), t [, b], where g : [, b] R is nondecresing, l : H 0 [, b] R n is liner in the first vrible nd p : [, b] R n. We will show tht, nder certin ssmptions, this fnctionl eqtion is eqivlent to the generlized eqtion (4.1), where X = H b nd the fnctions A, h re defined s follows: Copyright line will be provided by the pblisher

8 8 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely For every t [, b], A(t) : H b G((, b], R n ) is the opertor given by 0, < ϑ, (A(t)y)(ϑ) = l(y s, s) dg(s), ϑ t b, l(y s, s) dg(s), t ϑ b, (4.2) nd h(t) G((, b], R n ) is the fnction given by 0, < ϑ, h(t)(ϑ) = p(s) dg(s), ϑ t b, (4.3) p(s) dg(s), t ϑ b. We introdce the following system of conditions, which will be sefl lter (in prticlr, conditions (A) nd (E) grntee tht the integrls in (4.2) nd (4.3) exist): (A) The integrl b l(y t, t) dg(t) exists for every y H b. (B) There exists fnction M : [, b] R +, which is Krzweil-Stieltjes integrble with respect to g, sch tht (l(y t, t) l(z t, t)) dg(t) M(t) y t z t dg(t) whenever y, z H b nd [, v] [, b]. (C) For every y H b, A(b)y is n element of H b. (D) H b hs the prolongtion property for t, i.e., for every y H b nd t [, b), the fnction ȳ : (, b] R n given by y(s), s (, t], ȳ(s) = y(t), s [t, b] is n element of H b. (E) The integrl b p(t) dg(t) exists. (F) There exists fnction N : [, b] R +, which is Krzweil-Stieltjes integrble with respect to g, sch tht p(t) dg(t) N(t) dg(t) whenever [, v] [, b]. (G) h(b) is n element of H b. We strt with the following xiliry sttements. Lemm 4.1. Assme tht l : H 0 [, b] R n is liner in the first vrible nd conditions (A), (B) re stisfied. Let α = sp t [,b] κ 3 (t b). Then l(y t, t) dg(t) y whenever y H b nd [, v] [, b]. α M(t) dg(t) Copyright line will be provided by the pblisher

9 mn heder will be provided by the pblisher 9 P r o o f. For every y H b nd t b, we hve y t κ 3 (t b) y α y (the first ineqlity follows from Lemm 2.4). z 0. To finish the proof, it is enogh to pply (B) with Corollry 4.2. Assme tht l : H 0 [, b] R n is liner in the first vrible nd conditions (A), (B) re stisfied. For every bonded O H b, there exists fnction K : [, b] R +, which is Krzweil-Stieltjes integrble with respect to g, sch tht l(y t, t) dg(t) K(t) dg(t) whenever y O nd [, v] [, b]. The properties of the fnction A defined in (4.2), sch s the fct tht it hs bonded vrition on [, b], re described in the next lemm. Lemm 4.3. Assme tht l : H 0 [, b] R n is liner in the first vrible nd conditions (A) (D) re stisfied. For every t [, b], let A(t) : H b G((, b], R n ) be given by (4.2). Then, the fnction A tkes vles in L(H b ) nd hs bonded vrition on [, b]. Moreover, vr [,b] A κ 2 (b ) where α = sp t [,b] κ 3 (t b). b α M(s) dg(s), P r o o f. It is cler tht for every t [, b], A(t) is liner opertor defined on H b. Using (C), (D), nd the definition of A, we see tht A(t)y H b for every y H b nd t [, b]. Note tht (A(t)y) 0 for every y H b. Ths, by Lemm 2.4, we hve A(t)y = (A(t)y) b κ 2 (b ) sp (A(t)y)(ϑ) = κ 2 (b ) sp ϑ [,b] By Lemm 4.1, A(t) is bonded liner opertor on H b nd A(t) L(Hb ) κ 2 (b ) α M(s) dg(s). ϑ [,t] l(y s, s) dg(s). To show tht A : [, b] L(H b ) hs bonded vrition, consider < v b nd y H b. By Lemms 2.4 nd 4.1, Hence, [A(v) A()]y κ 2 (b ) sp ([A(v) A()]y)(ϑ) ϑ [,b] = κ 2 (b ) sp l(y s, s) dg(s) ϑ [,v] κ 2 (b ) y α M(s) dg(s). A(v) A() L(Hb ) κ 2 (b ) which concldes the proof. α M(s) dg(s), Copyright line will be provided by the pblisher

10 10 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely The following two theorems describe the reltion between liner mesre fnctionl differentil eqtions nd generlized liner ordinry differentil eqtions. Similr reslts for nonliner eqtions were lredy obtined in [16]; therefore, it is sfficient to verify tht the ssmptions from [16] re stisfied. Theorem 4.4. Assme tht g : [, b] R is nondecresing fnction, l : H 0 [, b] R n is liner in the first vrible, φ H 0, nd conditions (A) (G) re stisfied. If y H b is soltion of the mesre fnctionl differentil eqtion y(t) = y() + y = φ, l(y s, s) dg(s) + p(s) dg(s), t [, b], then the fnction x : [, b] H b given by y(ϑ), ϑ (, t], x(t)(ϑ) = y(t), ϑ [t, b], is soltion of the generlized ordinry differentil eqtion x(t) = x() + where A, h re given by (4.2), (4.3). d[a]x + h(t) h(), t [, b], (4.4) P r o o f. Consider the set O = x(t); t [, b]} H b. Clerly, O hs the prolongtion property for t. Observe tht for every t [, b], the spport of x(t) x() is contined in [, b]. Ths, by Lemm 2.4, we hve x(t) x(t) x() + x() κ 2 (b ) sp x(t)(τ) x()(τ) + x() τ [,b] κ 2 (b ) sp x(b)(τ) x()(τ) + x(). τ [,b] The right-hnd side does not depend on t, which mens tht the set O is bonded. Let f(y, t) = l(y, t) + p(t) for every t [, b] nd y H 0. For every y O nd [, v] [, b], Corollry 4.2 nd condition (F) led to the estimte v f(y s, s) dg(s) v l(y s, s) dg(s) + v p(s) dg(s) (K(s) + N(s)) dg(s). The fnction F given by F (y, t) = A(t)y + h(t) for every t [, b], y H b is well defined thnks to (A) nd (E), nd hs vles in H b by (C), (D), nd (G). This shows tht ll ssmptions of Theorem 3.6 from [16] re stisfied. Conseqently, x is soltion of the generlized ordinry differentil eqtion whose right-hnd side is F ; however, this is eqtion coincides with (4.4). Theorem 4.5. Assme tht g : [, b] R is nondecresing fnction, l : H 0 [, b] R n is liner in the first vrible, φ H 0, nd conditions (A) (G) re stisfied. Let A, h be given by (4.2), (4.3). If x : [, b] H b is soltion of the generlized ordinry differentil eqtion x(t) = x() + with the initil condition φ(ϑ ), x()(ϑ) = φ(0), d[a]x + h(t) h(), ϑ (, ], ϑ [, b], t [, b], Copyright line will be provided by the pblisher

11 mn heder will be provided by the pblisher 11 then the fnction y H b defined by x()(ϑ), ϑ (, ], y(ϑ) = x(ϑ)(ϑ), ϑ [, b] is soltion of the mesre fnctionl differentil eqtion y(t) = y() + y = φ. l(y s, s) dg(s) + p(s) dg(s), t [, b], P r o o f. From the definition of A, it follows tht the fnctions x(t), where t [, b], coincide on (, ]. As in the proof of the previos theorem, the set O = x(t), t [, b]} is bonded, hs the prolongtion property for t (this follows from [16, Lemm 3.5]), nd the fnctions f, F given by f(y, t) = l(y, t) + p(t) nd F (y, t) = A(t)y + h(t) stisfy ll ssmptions of Theorem 3.7 from [16]; conseqently, y is soltion of the given mesre fnctionl differentil eqtion. 5 Existence nd niqeness of soltions A locl existence nd niqeness theorem for nonliner mesre fnctionl differentil eqtions with finite dely ws obtined in [3, Theorem 5.3]. A generlized version for eqtions with infinite dely ws proved in [16, Theorem 3.12]. For liner eqtions, it is possible to prove mch stronger globl existence nd niqeness theorem; this is the content of the present section. The following theorem grntees the existence nd niqeness of soltion of the generlized liner ordinry differentil eqtion, nd corresponds to specil cse of Proposition 2.8 from [19]. Theorem 5.1. Consider Bnch spce X nd let A BV ([, b], L(X)) be left-continos fnction. Then, for every x X nd every h G([, b], X), the eqtion x(t) = x + d[a] x + h(t) h(), t [, b] hs niqe soltion x on [, b]. Moreover, x is reglted fnction. With this in mind nd sing the reltion estblished in the previos section, we derive the following reslt. Theorem 5.2. Assme tht g : [, b] R is nondecresing left-continos fnction, l : H 0 [, b] R n is liner in the first vrible, φ H 0, conditions (A) (G) re stisfied, nd the fnction x 0 : (, b] R n given by φ(ϑ ), ϑ (, ], x 0 (ϑ) = φ(0), ϑ [, b], is n element of H b. Then, the mesre fnctionl differentil eqtion y(t) = y() + y = φ, hs niqe soltion on [, b]. l(y s, s) dg(s) + p(s) dg(s), t [, b], P r o o f. Let A, h be given by (4.2), (4.3). It follows from Theorem 3.2 tht A, h re reglted leftcontinos fnctions. Moreover, by Lemm 4.3, A hs bonded vrition on [, b]. Ths, Theorem 5.1 ensres the existence of soltion of the generlized liner ordinry differentil eqtion x(t) = x 0 + d[a]x + h(t) h(), t [, b], Copyright line will be provided by the pblisher

12 12 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely where x tkes vles in the Bnch spce X = H b. By Theorem 4.5, there exists corresponding soltion of the given mesre fnctionl differentil eqtion. If this eqtion hd two different soltions, then, by Theorem 4.4, the corresponding generlized ordinry differentil eqtion wold hve two different soltions, which is contrdiction. Ths, the soltion hs to be niqe. 6 Continos dependence theorems In this section, we se the theory of generlized liner ordinry differentil eqtions (especilly the reslts from [13]) to prove new continos dependence reslt for liner mesre fnctionl differentil eqtions with infinite dely. We remrk tht continos dependence theorem for nonliner mesre fnctionl differentil eqtions with finite dely is vilble in [3, Theorem 6.3]. Althogh this theorem is lso pplicble to liner eqtions, or reslt is mch stronger. The following continos dependence theorem (inclding its proof) is lmost identicl to Theorem 3.4 from [13]; however, or ssmptions re weker since we do not reqire tht A k A 0 0. (On the other hnd, [13, Theorem 3.4] does not reqire tht A k, k N 0, re left-continos fnctions.) Also, the reslt from the next theorem is more generl thn [1, Proposition A.3] when restricted to the liner cse. Theorem 6.1. Let X be Bnch spce. Consider A k BV ([, b], L(X)), h k G([, b], X) nd x k X, for k N 0. Assme tht the following conditions re stisfied: For every k N 0, A k is left-continos fnction. lim k x k x 0 X = 0. lim k sp t [,b] [A k (t) A 0 (t)]x X = 0 for every x X. lim k h k h 0 = 0. There exists constnt γ > 0 sch tht vr [,b] A k γ for every k N. Then, for every k N 0, the eqtion x k (t) = x k + d[a k ] x k + h k (t) h k (), t [, b] hs niqe soltion x k on [, b], nd lim k x k x 0 = 0. P r o o f. Existence nd niqeness of soltions follow from Theorem 5.1. Next, observe tht where Note tht lim x k x 0 lim h k h 0 + lim x k h k x 0 + h 0 = lim w k, k k k k w k = x k h k x 0 + h 0, k N. w k () = x k h k () x 0 + h 0 (), k N, nd therefore lim k w k () X = 0. A simple clcltion revels tht w k (t) = x k + = w k () + = w k () + = w k () + d[a k ]x k h k () x 0 d[a k ]x k d[a k ]w k + d[a 0 ]x 0 d[a k ]h k d[a k ]w k + z k (t) z k (), d[a 0 ]x 0 + h 0 () d[a 0 ]h 0 + d[a k A 0 ](x 0 h 0 ) Copyright line will be provided by the pblisher

13 mn heder will be provided by the pblisher 13 where z k (t) = By Theorem 3.3, lim sp k t [,b] lim sp k t [,b] d[a k ]h k d[a 0 ]h 0 + d[a k ]h k d[a 0 ]h 0 X = 0, d[a k A 0 ](x 0 h 0 ) = 0, X d[a k A 0 ](x 0 h 0 ). nd conseqently lim k z k = 0. According to [13, Lemm 3.2], we hve w k (t) X ( w k () X + z k )e γ, t [, b], which implies tht lim k w k = 0. For every k N 0, we consider the liner mesre fnctionl differentil eqtion y k (t) = y k () + (y k ) = φ k, l k ((y k ) s, s) dg k (s) + p k (s) dg k (s), t [, b], where g k : [, b] R is nondecresing, l k : H 0 [, b] R n is liner in the first vrible, p k : [, b] R n nd φ k H 0. We define the corresponding opertors A k (t) : H b G((, b], R n ) by 0, < ϑ, (A k (t)y)(ϑ) = l k(y s, s) dg k (s), ϑ t b, l k(y s, s) dg k (s), t ϑ b, (6.1) nd fnctions h k (t) G((, b], R n ) by 0, < ϑ, (h k (t))(ϑ) = p k(s) dg k (s), ϑ t b, p k(s) dg k (s), t ϑ b, (6.2) for every k N 0 nd t [, b]. To reflect the fct tht we re deling with seqences of fnctions, we modify conditions (A) (G) from Section 4 s follows: (A) The integrl b l k(y t, t) dg k (t) exists for every k N 0 nd y H b. (B) For every k N 0, there exists fnction M k : [, b] R +, which is Krzweil-Stieltjes integrble with respect to g k, sch tht (l k (y t, t) l k (z t, t)) dg k (t) M k (t) y t z t dg k (t) whenever y, z H b nd [, v] [, b]. (C) For every k N 0 nd y H b, the fnction A k (b)y is n element of H b. (D) H b hs the prolongtion property for t. Copyright line will be provided by the pblisher

14 14 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely (E) For every k N 0, the integrl b p k(t) dg k (t) exists. (F) For every k N 0, there exists fnction N k : [, b] R +, which is Krzweil-Stieltjes integrble with respect to g k, sch tht p k (t) dg k (t) N k (t) dg k (t) whenever [, v] [, b]. (G) For every k N 0, the fnction h k (b) is n element of H b. Now, sing the correspondence estblished in Theorems 4.4 nd 4.5, we derive the following reslt. Theorem 6.2. For every k N 0, sppose tht g k : [, b] R is nondecresing left-continos fnction, nd l k : H 0 [, b] R n is fnction liner with respect to the first vrible. Assme tht conditions (A) (G) s well s the following conditions re stisfied: For every y H b, lim k sp t [,b] l k(y s, s)dg k (s) l 0(y s, s) dg 0 (s) = 0. lim k sp t [,b] p k(s)dg k (s) p 0(s) dg 0 (s) = 0. There exists constnt γ > 0 sch tht b M k(s)dg k (s) γ for ll k N. Frther, consider seqence of fnctions φ k H 0, k N 0, sch tht lim φ k φ 0 = 0, k nd sch tht for every k N 0, the fnction φ k (ϑ ), ϑ (, ], x k (ϑ) = φ k (0), ϑ [, b], is n element of H b. Then, for ech k N 0, there exists soltion y k : (, b] R n of the mesre fnctionl differentil eqtion y k (t) = y k () + l k((y k ) s, s) dg k (s) + p k(s) dg k (s), (y k ) = φ k, nd the seqence y k } k=1 converges niformly to y 0 on [, b]. t [, b], P r o o f. Consider A k, h k, for k N 0, given by (6.1), (6.2). By Theorem 3.2, h k nd A k re reglted left-continos fnctions. In ddition, by Lemm 4.3, A k BV ([, b], L(H b )) with vr [,b] A k κ 2 (b ) b } (6.3) α M k (s) dg k (s) κ 2 (b ) α γ, k N 0, (6.4) where α = sp t [,b] κ 3 (t ). For every k N 0, Theorem 5.1 ensres the existence of niqe soltion x k : [, b] H b of the generlized ordinry differentil eqtion x k (t) = x k + d[a k ] x k + h k (t) h k (), t [, b]. Copyright line will be provided by the pblisher

15 mn heder will be provided by the pblisher 15 Given y H b, the definition of A k, k N 0, together with Lemm 2.4, implies A k (t)y A 0 (t)y κ 2 (b ) sp [A k (t)y A 0 (t)y](ϑ) ϑ [,b] = κ 2 (b ) sp l k (y s, s)dg k (s) for every t [, b]. Conseqently, nd we conclde tht ϑ [,b] sp [A k (t) A 0 (t)] y κ 2 (b ) sp t [,b] lim sp k t [,b] Anlogosly, we hve ϑ [,b] [A k (t) A 0 (t)] y = 0, y H b. h k (t) h 0 (t) κ 2 (b ) sp ϑ [,b] p k (s)dg k (s) l k (y s, s)dg k (s) l 0 (y s, s) dg 0 (s) l 0 (y s, s) dg 0 (s), p 0 (s) dg 0 (s), t [, b], nd ths lim k h k h 0 = 0 holds. To show tht lim k x k x 0 = 0, it is enogh to notice tht, by Lemm 2.5, we hve x k x 0 (κ 2 (b )κ 1 (0) + λ(b )) φ k φ 0. In smmry, ll hypotheses of Theorem 6.1 re stisfied, which proves tht lim k x k x 0 = 0. By Theorems 4.4 nd 4.5, for ech k N 0, the fnction x k ()(ϑ), ϑ (, ], y k (ϑ) = x k (ϑ)(ϑ), ϑ [, b] is the niqe soltion of Eq. (6.3). For t [, b], we cn se Lemm 2.4 to see tht y k (t) y 0 (t) = x k (t)(t) x 0 (t)(t) κ 1 (t b) x k (t) x 0 (t) β sp x k (τ) x 0 (τ), τ [,b] where β = sp σ [,b] κ 1 (σ ). Ths, the seqence y k } k=1 is niformly convergent to y 0. Remrk 6.3. By the forth prt of Lemm 2.4, y k y 0 = (y k y 0 ) b κ 2 (b ) sp y k (s) y 0 (s) + λ(b ) φ k φ 0, s [,b] i.e., the seqence of soltions y k } k=1 from Theorem 6.2 converges to y 0 lso in the norm. 7 Fnctionl differentil eqtions with implses As n exmple, we show how or reslts pply to fnctionl differentil eqtions with implses. For simplicity, we restrict orselves to the cse when the phse spce H 0 coincides with one of the spces G ϕ from Exmple 2.2. Recll tht G ϕ ((, 0], R n ) = y G((, 0], R n ); y/ϕ is bonded}, nd for H 0 = G ϕ, the norm of fnction y H is given by y = y(t + ) y(t) sp = sp t (,0] ϕ(t) t (,] ϕ(t ). Copyright line will be provided by the pblisher

16 16 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely We consider liner implsive fnctionl differentil eqtions of the form y (t) = l(y t, t) + p(t),.e. in [, b], + y(t i ) = A i y(t i ) + b i, i 1,..., k}, (7.1) where l : H 0 [, b] R n is liner in the first vrible, p : [, b] R n, t 1 < < t k < b, A 1,..., A k R n n, b 1,..., b k R n, nd, s sl, + y(s) = y(s+) y(s), s [, b). In ddition, ssme the following conditions re stisfied: (1) The Lebesge integrl b l(y t, t) dt exists for every y H b. (2) There exists Lebesge integrble fnction M : [, b] R + sch tht (l(y t, t) l(z t, t)) dt M(t) y t z t dt whenever y, z H b nd [, v] [, b]. (3) The Lebesge integrl b p(t) dt exists. (4) There exists Lebesge integrble fnction N : [, b] R + sch tht p(t) dt N(t) dt whenever [, v] [, b]. The soltions of Eq. (7.1) re ssmed to be left-continos on [, b], nd bsoltely continos on [, t 1 ], (t 1, t 2 ],..., (t k, b]. The eqivlent integrl form of Eq. (7.1) is y(t) = y() + (l(y s, s) + p(s)) ds + (A i y(t i ) + b i ). (7.2) i; t i<t We need the following sttement, which is conseqence of [4, Lemm 2.4]. Lemm 7.1. Let k N, t 1 < t 2 < < t k < b, nd g(s) = s + k χ (ti, )(s), s [, b] i=1 (the symbol χ A denotes the chrcteristic fnction of set A R). Consider n rbitrry fnction f : [, b] R nd let f : [, b] R be sch tht f(s) = f(s) for every s [, b]\t 1,..., t k }. Then the integrl b f(s) dg(s) exists if nd only if the integrl b f(s) ds exists; in tht cse, we hve f(s) dg(s) = f(s) ds + i 1,...,k}, t i<t f(t i ), Using the previos lemm, Eq. (7.2) cn be rewritten s t [, b]. where y(t) = y() + l(y s, s) dg(s) + p(s) dg(s), (7.3) l(z, t) = l(z, t) if t [, b]\t 1,..., t k }, A i z(0) if t = t i for some i 1,..., k} Copyright line will be provided by the pblisher

17 mn heder will be provided by the pblisher 17 p(t) = p(t) if t [, b]\t 1,..., t k }, b i if t = t i for some i 1,..., k} for every t [, b] nd z H 0. Ths, the originl liner implsive fnctionl differentil eqtion (7.1) is eqivlent to the liner mesre fnctionl differentil eqtion (7.3). It follows from Lemm 7.1 tht ssmptions (A), (E) from Section 4, where l, p re replced by l, p, re stisfied. For [, v] [, b], we obtin p(t) dg(t) = v p(t) dt + b i N(t) dt + b i = Ñ(t) dg(t), i; t i [,v) i; t i [,v) where Ñ(t) = N(t) if t [, b]\t 1,..., t k }, b i if t = t i for some i 1,..., k}. This verifies ssmption (F). Similrly, for every y, z H b, we get ( l(y t, t) l(z t, t)) dg(t) = v (l(y t, t) l(z t, t)) dt + where M(t) = M(t) y t z t dt + i; t i [,v) A i κ 1 (0) y ti z ti = M(t) if t [, b]\t 1,..., t k }, A i κ 1 (0) if t = t i for some i 1,..., k}. i; t i [,v) A i (y(t i ) z(t i )) M(t) y t z t dg(t), Hence, ssmption (B) from Section 4 is stisfied. The remining ssmptions (C), (D), nd (G) re flfilled, too: (C) nd (G) follow from the fct tht A(b)y nd h(b) re reglted fnctions with compct spport contined in [, b], nd hence elements of H b. Also, it is cler tht or spce H b hs the prolongtion property for t. Finlly, let φ H 0. We clim tht the fnction y : (, b] R n given by φ(ϑ ), ϑ (, ], y(ϑ) = φ(0), ϑ [, b] is n element of H b. (Recll tht ssmptions of this type pper both in the existence-niqeness theorem nd in the continos dependence theorem.) The clim follows from the estimte sp t (,b] y(t) ϕ(t b) sp t (,] since the second spremm is finite, nd ( y(t) ϕ(t b) sp t (,] sp t (,] y(t) ϕ(t b) + sp t [,b] ) ( y(t) ϕ(t ) y(t) ϕ(t b), sp t (,] ) ( ϕ(t ) = φ ϕ(t b) sp t (,] ) ϕ(t ) ϕ(t b) is finite, too (we need (2.1) here). Or clcltions led to the following existence-niqeness theorem, which is n immedite conseqence of Theorem 5.2. (For locl existence-niqeness theorem for nonliner implsive fnctionl differentil eqtions with finite dely, see [5, Theorem 5.1].) Copyright line will be provided by the pblisher

18 18 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely Theorem 7.2. Assme tht H 0 = G ϕ nd conditions (1) (4) re stisfied. Then for every φ G ϕ, the implsive fnctionl differentil eqtion y (t) = l(y t, t) + p(t),.e. in [, b], + y(t i ) = A i y(t i ) + b i, i 1,..., k}, y = φ hs niqe soltion on [, b]. We now proceed to continos dependence nd consider seqence of implsive fnctionl differentil eqtions of the form y j (t) = l j((y j ) t, t) + p j (t),.e. in [, b], + y j (t i ) = A i j y j(t i ) + b i j, i 1,..., k}, where l j : H 0 [, b] R n is liner in the first vrible, p j : [, b] R n, A 1 j,..., Ak j Rn n, nd b 1 j,..., bk j Rn for every j N 0. For liner eqtions, the following reslt is mch stronger thn the continos dependence reslt stted in [5, Theorem 4.1]. Theorem 7.3. Assme tht H 0 = G ϕ nd the following conditions re stisfied: The Lebesge integrl b l j(y t, t) dt exists for every j N 0 nd y H b. For every j N 0, there exists Lebesge integrble fnction M j : [, b] R + sch tht (l j (y t, t) l j (z t, t)) dt M j (t) y t z t dt whenever y, z H b nd [, v] [, b]. For every j N 0, the Lebesge integrl b p j(t) dt exists. For every j N 0, there exists Lebesge integrble fnction N j : [, b] R + sch tht p j (t) dt N j (t) dt whenever [, v] [, b]. For every y H b, lim j sp t [,b] l j(y s, s) ds l 0(y s, s) ds = 0. lim j sp t [,b] p j(s) ds p 0(s) ds = 0. There exists constnt γ > 0 sch tht b M j(s) ds γ for ll j N. For every i 1,..., k}, lim j A i j = Ai 0 nd lim j b i j = bi 0. Frther, consider seqence of fnctions φ j H 0, j N 0, sch tht lim j φ j φ 0 = 0. Then, for ech j N 0, there exists soltion y j : (, b] R n of the implsive fnctionl differentil eqtion y j (t) = l j((y j ) t, t) + p j (t),.e. in [, b], + y j (t i ) = A i j y j(t i ) + b i j, i 1,..., k}, (y j ) = φ j, nd the seqence y j } j=1 converges niformly to y 0 on [, b]. Copyright line will be provided by the pblisher

19 mn heder will be provided by the pblisher 19 P r o o f. The theorem is conseqence of Theorem 6.2. Indeed, or seqence of implsive eqtions is eqivlent to the seqence of mesre fnctionl differentil eqtions y j (t) = y j () + l j ((y j ) s, s) dg(s) + p j(s) dg(s), (y j ) = φ j, t [, b], where the fnctions l j, p j, nd g re defined s in the beginning of the present section. Or previos clcltions show tht ssmptions (A) (G) from Section 5 (with l j, p j replced by l j, p j ) re stisfied. In prticlr, we hve the estimte ( l j (y t, t) l v j (z t, t)) dg(t) M j (t) y t z t dg(t), where M j (t) = M j (t) if t [, b]\t 1,..., t k }, nd M j (t) = A i j κ 1(0) if t = t i for some i 1,..., k}. Hence, b M j (s) dg(s) = b M j (s) ds + k A i j κ 1 (0) γ + κ 1 (0) i=1 k i=1 sp j N A i j for every j N. By Lemm 7.1, we hve t ( p j (s) p 0 (s)) dg(s) (p j (s) p 0 (s)) ds + (b j i b0 i ), nd it follows tht lim sp j t [,b] p j (s) dg(s) p 0 (s) dg(s) = 0. i; t i<t Similrly, for every y H b, we hve ( l j (y s, s) l t 0 (y s, s)) dg(s) (l j (y s, s) l 0 (y s, s)) ds + (A j i A0 i )y(t i ). Since the lst term cn be mjorized by k i=1 Aj i A0 i y(t i), we obtin lim sp t j l j (y s, s) dg(s) l 0 (y s, s) dg(s) = 0. t [,b] Ths, we hve verified tht ll ssmptions of Theorem 6.2 re stisfied. 8 Conclsion i; t i<t Finlly, let s mention tht the reslts of this pper re lso pplicble to liner fnctionl dynmic eqtions on time scles. As shown in [3], fnctionl dynmic eqtions represent nother specil cse of mesre fnctionl differentil eqtions. Let T be time scle, i.e., nonempty closed sbset of R. Given, b T, < b, let [, b] T = [, b] T. Using the nottion from [15], let t = infs T; s t} for every t (, b]. Recll tht in the time scle clcls, the sl derivtive f is replced by the -derivtive f (see [2] for the bsic definitions). Therefore, we consider the liner fnctionl dynmic eqtion y (t) = l(y t, t) + q(t), t [, b] T, (8.1) Copyright line will be provided by the pblisher

20 20 G. A. Monteiro, A. Slvík: Liner mesre fnctionl differentil eqtions with infinite dely where l : H 0 [, b] R n is liner in the first vrible nd q : [, b] T R n. Here, the symbol y t shold be interpreted s (y ) t ; s explined in [3], the reson for working with y t insted of jst y t is tht the first rgment of l hs to be fnction defined on the whole intervl (, 0]. For exmple, liner dely dynmic eqtion of the form y (t) = k p i (t) y(τ i (t)) + q(t), i=1 where τ i : T T, p i : [, b] T R n nd τ i (t) t for every t [, b] T, i 1,..., k}, is specil cse of Eq. (8.1) corresponding to the choice l(y, t) = k p i (t) y(τ i (t) t). i=1 Now, nder certin ssmptions on l nd q, the liner fnctionl dynmic eqtion (8.1) is eqivlent to the mesre fnctionl differentil eqtion z(t) = z() + l(z s, s ) dg(s) + q(s ) dg(s), t [, b], where g(s) = s, s [, b]. More precisely, fnction z : [, b] R n is soltion of the lst eqtion if nd only if z(t) = y(t ), t [, b], where y : [, b] T R n is soltion of (8.1). Therefore, the existence-niqeness theorem s well s the continos dependence theorem for liner fnctionl dynmic eqtions of the form (8.1) re simple conseqences of or reslts for mesre fnctionl differentil eqtions. Moreover, the sme pproch lso works for fnctionl dynmic eqtions with implses (see [4]). Since the whole procedre wold be very similr to the one described in Section 7, we omit the detils. References [1] S. M. Afonso, E. M. Bonotto, M. Federson, Š. Schwbik, Discontinos locl semiflows for Krzweil eqtions leding to LSlle s invrince principle for differentil systems with implses t vrible times, J. Differ. Eqtions 250 (2011), [2] M. Bohner, A. Peterson, Dynmic Eqtions on Time Scles: An Introdction with Applictions, Birkhäser, Boston, [3] M. Federson, J. G. Mesqit, A. Slvík, Mesre fnctionl differentil nd fnctionl dynmic eqtions on time scles, J. Differ. Eqtions 252 (2012), [4] M. Federson, J. G. Mesqit, A. Slvík, Bsic reslts for fnctionl differentil nd dynmic eqtions involving implses, Mth. Nchr. 286 (2013), no. 2 3, [5] M. Federson nd Š. Schwbik, Generlized ODE pproch to implsive retrded fnctionl differentil eqtions, Differ. Integrl Eq. 19, no. 11, (2006). [6] J. K. Hle, J. Kto, Phse spce for retrded eqtions with infinite dely, Fnkcil. Ekvc. 21 (1978), [7] Y. Hino, S. Mrkmi, T. Nito, Fnctionl Differentil Eqtions with Infinite Dely, Springer-Verlg, [8] C. S. Hönig, Volterr Stieltjes-Integrl Eqtions, North Hollnd nd Americn Elsevier, Mthemtics Stdies 16. Amsterdm nd New York, [9] C. Imz, Z. Vorel,Generlized ordinry differentil eqtions in Bnch spces nd pplictions to fnctionl eqtions, Bol. Soc. Mt. Mexicn 11 (1966), [10] J. K. Hle, S. M. V. Lnel, Introdction to Fnctionl Differentil Eqtions, Springer-Verlg, New York, [11] J. Krzweil, Generlized ordinry differentil eqtion nd continos dependence on prmeter, Czech. Mth. J. 7 (82) (1957), [12] J. Krzweil, Generlized Ordinry Differentil Eqtions. Not Absoltely Continos Soltions, World Scientific, Copyright line will be provided by the pblisher

21 mn heder will be provided by the pblisher 21 [13] G. Monteiro, M. Tvrdý, Generlized liner differentil eqtions in Bnch spce: Continos dependence on prmeter, Discrete Contin. Dyn. Syst. 33 (2013), no. 1, [14] F. Oliv, Z. Vorel, Fnctionl eqtions nd generlized ordinry differentil eqtions, Bol. Soc. Mt. Mexicn 11 (1966), [15] A. Slvík, Dynmic eqtions on time scles nd generlized ordinry differentil eqtions, J. Mth. Anl. Appl. 385 (2012), [16] A. Slvík, Mesre fnctionl differentil eqtions with infinite dely, Nonliner Anl. 79 (2013), [17] Š. Schwbik, Generlized Ordinry Differentil Eqtions, World Scientific, Singpore, [18] Š. Schwbik, Abstrct Perron-Stieltjes integrl, Mth. Bohem. 121 (1996), no. 4, [19] Š. Schwbik, Liner Stieltjes integrl eqtions in Bnch spces, Mth. Bohem. 124 (1999), no. 4, [20] Š. Schwbik, Liner Stieltjes integrl eqtions in Bnch spces II; Opertor vled soltions, Mth. Bohem. 125 (2000), Copyright line will be provided by the pblisher