Generalized differential equations: differentiability of solutions with respect to initial conditions and parameters

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1 Generlize ifferentil equtions: ifferentibility of solutions with respect to initil conitions n prmeters Antonín Slvík Chrles University, Fculty of Mthemtics n Physics, Sokolovská 83, Prh 8, Czech Republic Emil: Abstrct We show tht uner certin ssumptions, solutions of generlize orinry ifferentil equtions re ifferentible with respect to initil conitions n prmeters. This result unifies n extens severl existing theorems for other types of equtions, such s impulsive ifferentil equtions or ynmic equtions on time scles. Keywors: Generlize orinry ifferentil equtions, ifferentibility of solutions, Kurzweil integrl, ifferentil equtions with impulses, ynmic equtions on time scles 1 Introuction In the clssicl theory of orinry ifferentil equtions, it is well known tht uner certin ssumptions, solutions of the problem x (t) = f(x(t), t), t [, b], x(t 0 ) = x 0, re ifferentible with respect to the initil conition; tht is, if x(t, x 0 ) enotes the vlue of the solution t t [, b], then the function x 0 x(t, x 0 ) is ifferentible. The key requirement is tht the righthn sie f shoul be ifferentible with respect to x. Moreover, the erivtive s function of t is known to stisfy the soclle vritionl eqution, which might be helpful in etermining the vlue of the erivtive. Similrly, uner suitble ssumptions, solutions of ifferentil eqution whose righthn sie epens on prmeter re ifferentible with respect to tht prmeter. These two types of theorems concerning ifferentibility of solutions with respect to initil conitions n prmeters cn be foun in mny ifferentil equtions textbooks, see e.g. [3]. Theorems of similr type re lso vilble for other types of equtions, such s ifferentil equtions with impulses (see [6]) or ynmic equtions on time scles (see [2]). In 1957, Jroslv Kurzweil introuce clss of integrl equtions clle generlize orinry ifferentil equtions (see [4]). His originl motivtion ws to use them in the stuy of continuous epenence of solutions with respect to prmeters. However, it becme cler tht generlize equtions encompss vrious other types of equtions, incluing equtions with impulses (see [8]), ynmic equtions on time scles (see [11]), or mesure ifferentil equtions (see [8]). The im of this pper is to obtin ifferentibility theorems for generlize orinry ifferentil equtions. Despite the fct tht solutions of generlize equtions nee not be ifferentible or even continuous with respect to t, we show tht ifferentibility of the righthn sie with respect to x (n possibly with respect to prmeters) still gurntees tht the solutions re ifferentible with respect to initil conitions (n prmeters, respectively). Consequently, our result unifies n extens existing theorems for other types of equtions. 1
2 2 The Kurzweil integrl n generlize ifferentil equtions We strt this section by presenting short overview of the Kurzweil integrl, which is funmentl for the stuy of generlize orinry ifferentil equtions. For more informtion on Kurzweil integrtion, see [5, 8]. A function δ : [, b] R + is clle guge on [, b]. A tgge prtition of the intervl [, b] with ivision points = t 0 t 1 t k = b n tgs τ i [t i 1, t i ], i {1,..., k}, is clle δfine if [t i 1, t i ] (τ i δ(τ i ), τ i + δ(τ i )), i {1,..., k}. A mtrixvlue function U : [, b] [, b] R n m is clle Kurzweil integrble on [, b], if there is mtrix K R n m such tht for every ε > 0, there is guge δ on [, b] such tht k (U(τ i, t i ) U(τ i, t i 1 )) K < ε i=1 for every δfine tgge prtition of [, b]. In this cse, we efine D tu(τ, t) = K. Obviously, mtrixvlue function U is integrble if n only if ll its components re integrble. Note lso tht we restrict ourselves to functions with vlues in finiteimensionl spces; in this cse, the Kurzweil integrl coincies with the strong Kurzweil integrl (see Chpter 19 in [5]). An importnt specil cse is the KurzweilStieltjes integrl (lso known s the PerronStieltjes, HenstockStieltjes or HenstockKurzweilStieltjes integrl) of function f : [, b] R n with respect to function g : [, b] R, which correspons to the choice U(τ, t) = f(τ)g(t) n will be enote by f(t) g(t). We re now rey to introuce generlize orinry ifferentil equtions. Consier set B R n, n intervl [, b] R n function F : B [, b] 2 R n. A generlize orinry ifferentil eqution with the righthn sie F hs the form which is shorthn nottion for the integrl eqution x(s) = x() + t x = D tf (x, τ, t), (1) D t F (x(τ), τ, t), s [, b]. (2) In other wors, function x : [, b] B is solution of (1) if n only if (2) is stisfie. We emphsize tht (1) is symbolic nottion only n oes not men tht x hs to be ifferentible. Equtions of the form (1) hve been introuce by J. Kurzweil in [5]. In mny situtions, it is sufficient to consier the less generl type of eqution t x = D tf (x, t), (3) where the righthn sie oes not epen on τ (in fct, most existing sources evote to generlize equtions focus on this less generl type; see e.g. the pioneering pper [4] by J. Kurzweil, or the monogrph [8] by Š. Schwbik). The corresponing integrl eqution hs the form x(s) = x() + D t F (x(τ), t), s [, b]. Uner certin conitions, n eqution of the form (1) cn be trnsforme to n eqution of the form (3) (see Chpter 27 in [5]). However, s will be cler in Section 4, the more generl type (1) is quite useful for our purposes. The rest of this section summrizes some bsic fcts concerning the Kurzweil integrl tht will be neee lter. The following existence theorem cn be foun in [8, Corollry 1.34] or [5, Chpter 20]. 2
3 Theorem 2.1. If f : [, b] R n is regulte function n g : [, b] R is nonecresing function, then the integrl f(s) g(s) exists. The next estimte follows irectly from the efinition of the Kurzweil integrl. Lemm 2.2. Let U : [, b] 2 R n n be Kurzweil integrble function. Assume there exists pir of functions f : [, b] R n g : [, b] R such tht f is regulte, g is nonecresing, n U(τ, t) U(τ, s) f(τ) g(t) g(s), τ, t, s [, b]. Then D t U(τ, t) f(τ) g(τ). For the first prt of the following sttement, see [5, Corollry 14.18] or [8, Theorem 1.16]; the secon prt is irect consequence of Lemm 2.2. Theorem 2.3. Assume tht U : [, b] 2 R n m is Kurzweil integrble n u : [, b] R n m is its primitive, i.e., u(s) = u() + D t U(τ, t), s [, b]. If U is regulte in the secon vrible, then u is regulte n stisfies u(τ+) = u(τ) + U(τ, τ+) U(τ, τ), τ [, b) u(τ ) = u(τ) + U(τ, τ ) U(τ, τ), τ (, b]. Moreover, if there exists nonecresing function h : [, b] R such tht U(τ, t) U(τ, s) h(t) h(s), τ, t, s [, b], then u(t) u(s) h(t) h(s), t, s [, b]. The following theorem represents n nlogue of Gronwll s inequlity for the KurzweilStieltjes integrl; the proof cn be foun in [8, Corollry 1.43]. Theorem 2.4. Let h : [, b] [0, ) be nonecresing leftcontinuous function, k > 0, l 0. Assume tht ψ : [, b] [0, ) is boune n stisfies ψ(ξ) k + l Then ψ(ξ) ke l(h(ξ) h()) for every ξ [, b]. 3 Liner equtions ξ ψ(τ) h(τ), ξ [, b]. In this section, we consier the homogeneous liner generlize orinry ifferentil eqution t z = D t[a(τ, t)z], (4) where A : [, b] 2 R n n is given mtrixvlue function n the solution z tkes vlues in R n. The integrl form of this eqution is z(s) = z() + D t [A(τ, t)z(τ)], s [, b]. The cse when A oes not epen on τ hs been stuie in numerous works (see e.g. [8, 9]). However, to prove the min result of this pper, we nee some bsic fcts concerning the more generl type (4). For convenience, let us introuce the following conition: 3
4 (A) There exists nonecresing function h : [, b] R such tht A(τ, t) A(τ, s) h(t) h(s), τ, t, s [, b]. Note tht if (A) is stisfie, then for every fixe τ [, b], the function t A(τ, t) is regulte (this follows from the Cuchy conition for the existence of onesie limits). Also, if h is leftcontinuous, then t A(τ, t) is leftcontinuous s well. Lemm 3.1. Assume tht A : [, b] 2 R n n stisfies (A). Let y, z : [, b] R n be pir of functions such tht Then z is regulte on [, b]. z(s) = z() + Proof. Let U(τ, t) = A(τ, t)y(τ); note tht D t [A(τ, t)y(τ)], s [, b]. z(s) = z() + D t U(τ, t), s [, b]. By conition (A), U is regulte in the secon vrible. By Theorem 2.3, z is regulte. A simple consequence of the previous lemm is tht every solution of the liner generlize ifferentil eqution (4) is regulte. Lemm 3.2. Assume tht A : [, b] 2 R n n stisfies (A) with leftcontinuous function h. Then for every z 0 R n, the initil vlue problem hs t most one solution. t z = D t[a(τ, t)z], z() = z 0 Proof. Consier pir of functions z 1, z 2 : [, b] R n such tht Then t z i = D t [A(τ, t)z i ], for i {1, 2}. z 1 (s) z 2 (s) z 1 () z 2 () + z 1 () z 2 () + D t [A(τ, t)(z 1 (τ) z 2 (τ))] z 1 (τ) z 2 (τ) h(τ), s [, b] (the lst inequlity follows from Lemm 2.2). By Gronwll s inequlity from Theorem 2.4, we obtin z 1 (s) z 2 (s) z 1 () z 2 () e h(b) h(), s [, b]. Thus, if z 1 () = z 2 (), then z 1, z 2 coincie on [, b]. The proof of the following lemm is lmost ienticl to the proof of Theorem 3.14 in [8]; however, since our ssumptions re ifferent, we repet the proof here for reer s convenience. Lemm 3.3. Let z k : [, b] R n, k N, be uniformly boune sequence of functions, which is pointwise convergent to function z : [, b] R n. Assume tht A : [, b] 2 R n n stisfies (A) n the integrl D t[a(τ, t)z k (τ)] exists for every k N. Then D t[a(τ, t)z(τ)] exists s well n equls lim k D t[a(τ, t)z k (τ)]. 4
5 Proof. Let A i enote the ith row of A. Clerly, it is enough to prove tht for every i {1,..., n}, the integrl D t[a i b (τ, t)z(τ)] exists n equls lim k D t[a i (τ, t)z k (τ)]. Let i {1,..., n} be fixe, U(τ, t) = A i (τ, t)z(τ) n U k (τ, t) = A i (τ, t)z k (τ) for every k N (note tht U n U k re sclr functions). Consier n rbitrry fixe ε > 0. For every τ [, b], there is number p(τ) N such tht z k (τ) z(τ) < ε h(b) h() + 1, k p(τ). Let M > 0 be such tht z k (τ) M for every k N, τ [, b]. The function µ(t) = εh(t) h(b) h() + 1, t [, b], is nonecresing n µ(b) µ() < ε. If τ, t 1, t 2 [, b], t 1 t 2, n k p(τ), we hve n U k (τ, t 2 ) U k (τ, t 1 ) U(τ, t 2 ) + U(τ, t 1 ) A i (τ, t 2 ) A i (τ, t 1 ) z k (τ) z(τ) A(τ, t 2 ) A(τ, t 1 ) z k (τ) z(τ) ε(h(t 2) h(t 1 )) h(b) h() + 1 = µ(t 2) µ(t 1 ) M( h(t 2 ) + h(t 1 )) U k (τ, t 2 ) U k (τ, t 1 ) M(h(t 2 ) h(t 1 )). The conclusion now follows from the ominte convergence theorem for the Kurzweil integrl ([8, Corollry 1.31]). Lemm 3.4. Assume tht A : [, b] 2 R n n is Kurzweil integrble n stisfies (A). Then for every regulte function y : [, b] R n, the integrl D t[a(τ, t)y(τ)] exists. Proof. Every regulte function is uniform limit of step functions. Thus, in view of Lemm 3.3, it is sufficient to prove tht the sttement is true for every step function y : [, b] R n. Let = t 0 < t 1 < < t k = b be prtition of [, b] such tht y is constnt on ech intervl (t i 1, t i ). For t i 1 < u < σ < v < t i, the integrbility of A implies tht the integrls σ u D t[a(τ, t)y(τ)] n v σ D t[a(τ, t)y(τ)] exist n re regulte functions of u, v (this follows from (A) n Lemm 3.1). Accoring to Hke s theorems for the Kurzweil integrl (see Theorems n in [5], or Theorem 1.14 n Remrk 1.15 in [8]), the integrls σ t i 1 D t [A(τ, t)y(τ)] n i σ D t[a(τ, t)y(τ)] exist s well. Thus, i t i 1 D t [A(τ, t)y(τ)] exists for every i {1,..., k}, which proves the sttement. Up to smll etil, the proof of the following theorem is the sme s the proof of Theorem 23.4 in [5]. Therefore, we provie only sketch of the proof n leve the etils to the reer. Theorem 3.5. Assume tht A : [, b] 2 R n n is Kurzweil integrble n stisfies (A) with leftcontinuous function h. Then for every z 0 R n, the initil vlue problem hs unique solution z : [, b] R n. t z = D t[a(τ, t)z], z() = z 0 (5) Proof. Uniqueness of solutions follows immeitely from Lemm 3.2. To prove the existence of solution of (5) on [, b], choose prtition = s 0 < s 1 < < s k = b of [, b] such tht h(s i ) h(s i 1 +) 1 2 for every i {1,..., k}. Now, it is sufficient to prove tht for every w 0 R n n i {1,..., k}, the initil vlue problem t z = D t[a(τ, t)z], z(s i 1 ) = w 0 5
6 hs solution on [s i 1, s i ]. This solution cn be obtine by the metho of successive pproximtions: Let v 0 (s) = w 0, s [s i 1, s i ], v j (s) = w 0 + s i 1 D t [A(τ, t)v j 1 (τ)], s [s i 1, s i ], j N. The existence of the integrl on the righthn sie is gurntee by Lemm 3.4 (this is the only ifference ginst the proof of Theorem 23.4 in [5]). By Theorem 2.3, we hve v j (s) = ŵ 0 + D t [A(τ, t)v j 1 (τ)], s [s i 1, s i ], j N, s i 1+ where ŵ 0 = w 0 + A(s i 1, s i 1 +)w 0 A(s i 1, s i 1 )w 0. Using Lemm 2.2, it is not ifficult to see tht v j+1 (s) v j (s) v 1 (s) v 0 (s) w 0 (h(s i ) h(s i 1 )), s [s i 1, s i ], sup v j (τ) v j 1 (τ) (h(s i ) h(s i 1 +)), s (s i 1, s i ], j N. τ [s i 1,s i] Since h(s i ) h(s i 1 +) 1 2, it follows by inuction tht v j+1 (s) v j (s) w 0 ( ) j 1 (h(s i ) h(s i 1)), s (s i 1, s i ], j N. 2 This implies tht {v j } j=1 is uniformly convergent to function v : [s i 1, s i ] R n, which stisfies v(s) = lim j(s) = w 0 + lim j j D t [A(τ, t)v j 1 (τ)] = w 0 + s i 1 D t [A(τ, t)v(τ)] s i 1 for every s [s i 1, s i ] (we hve use Lemm 3.3). Throughout this section, we focuse our ttention on equtions of the form t z = D t[a(τ, t)z], z() = z 0, where z tkes vlues in R n. More generlly, it is possible to consier equtions where the unknown function hs vlues in R n n. For exmple, if Z 0 R n n is n rbitrry mtrix, then is shorthn nottion for the integrl eqution t Z = D t[a(τ, t)z], Z() = Z 0 Z(s) = Z 0 + D t [A(τ, t)z(τ)], s [, b]. (6) For n rbitrry mtrix X R n n, let X i be the ith column of X. Then it is obvious tht (6) hols if n only if Z i (s) = Z i 0 + D t [A(τ, t)z i (τ)], s [, b], i {1,..., n}. We will encounter equtions with mtrixvlue solutions in the following section. 6
7 4 The min results Consier generlize orinry ifferentil eqution of the form t x = D tf (x, τ, t), x() = x 0 (λ), (7) where the solution x tkes vlues in R n, n x 0 : R l R n is function which escribes the epenence of the initil conition on prmeter λ R l. Let x(s, λ) be the vlue of the solution t s [, b]. Our gol is to show tht uner certin conitions, x(s, λ) is ifferentible with respect to λ. Using the efinition of the Kurzweil integrl, we see tht the vlue of x(s, λ) cn be pproximte by x 0 (λ) + k (F (x(τ j, λ), τ j, t j ) F (x(τ j, λ), τ j, t j 1 )), j=1 where = t 0 < t 1 < < t k = s is sufficiently fine prtition of [, s] with tgs τ j [t j 1, t j ], j {1,..., k}. Assuming tht ll expressions re ifferentible with respect to λ t λ 0 R l, we see tht the erivtive x λ (s, λ 0 ), i.e., the mtrix { xi λ j (s, λ 0 )} i,j R n l, shoul be pproximtely equl to x 0 λ(λ 0 ) + k (F x (x(τ j, λ 0 ), τ j, t j )x λ (τ j, λ 0 ) F x (x(τ j, λ), τ j, t j 1 )x λ (τ j, λ 0 )). j=1 Now, the righthn sie is n pproximtion to x 0 λ(λ 0 ) + D t [F x (x(τ, λ 0 ), τ, t)x λ (τ, λ 0 )]. Thus, it seems resonble to expect tht the erivtive Z(t) = x λ (t, λ 0 ), t [, b], is solution of the liner eqution t z = D tg(z, τ, t), z() = x 0 λ(λ 0 ), (8) where G(z, τ, t) = F x (x(τ, λ 0 ), τ, t)z. This provies motivtion for the following theorem. Note tht even in the cse when the righthn sie of Eq. (7) oes not epen on τ n hs the form F (x, t), the righthn sie of Eq. (8) hs the form G(z, τ, t) = F x (x(τ, λ 0 ), t)z, i.e., still epens on τ. Tht is why we h to stuy the more generl type of equtions in the previous section. The following proof is bse on elementry estimtes n Gronwll s inequlity; it is inspire by proof of Theorem 3.1 in the pper [2], which is concerne with ynmic equtions on time scles. Theorem 4.1. Let B R n be n open set, λ 0 R l, ρ > 0, Λ = {λ R l ; λ λ 0 < ρ}, x 0 : Λ B, F : B [, b] 2 R n. Assume tht F is regulte n leftcontinuous in the thir vrible, n tht for every λ Λ, the initil vlue problem t x = D tf (x, τ, t), x() = x 0 (λ) (9) hs solution efine on [, b]; let x(t, λ) be the vlue of tht solution t t [, b]. Moreover, let the following conitions be stisfie: 1. For every fixe pir (t, τ) [, b] 2, the function x F (x, τ, t) is continuously ifferentible on B. 2. The function x 0 is ifferentible t λ There exists leftcontinuous nonecresing function h : [, b] R such tht F x (x, τ, t) F x (x, τ, s) h(t) h(s), s, t, τ [, b], x B. 7
8 4. There exists continuous incresing function ω : [0, ) [0, ) such tht ω(0) = 0 n F x (x, τ, t) F x (x, τ, s) F x (y, τ, t)+f x (y, τ, s) ω( x y ) h(t) h(s), s, t, τ [, b], x, y B. 5. There exists leftcontinuous nonecresing function k : [, b] R such tht F (x, τ, t) F (x, τ, s) F (y, τ, t) + F (y, τ, s) x y k(t) k(s), s, t, τ [, b], x, y B. 6. There exists number η > 0 such tht if x R n stisfies x x(t, λ 0 ) < η for some t [, b], then x B (i.e., the ηneighborhoo of the solution t x(t, λ 0 ) is contine in B). Then the function λ x(t, λ) is ifferentible t λ 0, uniformly for ll t [, b]. Moreover, its erivtive Z(t) = x λ (t, λ 0 ), t [, b], is the unique solution of the generlize ifferentil eqution Z(s) = x 0 λ(λ 0 ) + Proof. Accoring to the ssumptions, we hve D t [F x (x(τ, λ 0 ), τ, t)z(τ)], s [, b]. (10) x(s, λ) = x 0 (λ) + D t F (x(τ, λ), τ, t), λ Λ, s [, b]. By Theorem 2.3, every solution x is regulte n leftcontinuous function on [, b]. If λ R l is such tht < ρ, then x(s, λ 0 + λ) x(s, λ 0 ) x 0 (λ 0 + λ) x 0 (λ 0 ) + D t V (τ, t), where V (τ, t) = F (x(τ, λ 0 + λ), τ, t) F (x(τ, λ 0 ), τ, t). By ssumption 5, we obtin V (τ, t 1 ) V (τ, t 2 ) x(τ, λ 0 + λ) x(τ, λ 0 ) k(t 1 ) k(t 2 ), n consequently (using Lemm 2.2) x(s, λ 0 + λ) x(s, λ 0 ) x 0 (λ 0 + λ) x 0 (λ 0 ) + for every s [, b]. Gronwll s inequlity from Theorem 2.4 implies x(τ, λ 0 + λ) x(τ, λ 0 ) k(τ). x(s, λ 0 + λ) x(s, λ 0 ) x 0 (λ 0 + λ) x 0 (λ 0 ) e k(b) k(), s [, b]. Thus we see tht for λ 0, x(s, λ 0 + λ) pproches x(s, λ 0 ) uniformly for ll s [, b]. By ssumption 3, the function A(τ, t) = F x (x(τ, λ 0 ), τ, t) stisfies conition (A). By Theorem 3.5, Eq. (10) hs unique solution Z : [, b] R n n. By Lemm 3.1, the solution is regulte. Consequently, there exists constnt M > 0 such tht Z(t) M for every t [, b]. For every λ R l such tht < ρ, let ξ(r, λ) = x(r, λ 0 + λ) x(r, λ 0 ) Z(r) λ, r [, b]. Our gol is to prove tht if λ 0, then ξ(r, λ) 0 uniformly for r [, b]. Let ε > 0 be given. There exists δ > 0 such tht if λ R l n < δ, then x(t, λ 0 + λ) x(t, λ 0 ) < min(ε, η), t [, b], n x 0 (λ 0 + λ) x 0 (λ 0 ) x 0 λ (λ 0) λ < ε. 8
9 Observe tht where ξ(, λ) = x0 (λ 0 + λ) x 0 (λ 0 ) x 0 λ (λ 0) λ, ξ(r, λ) ξ(, λ) = x(r, λ 0 + λ) x(, λ 0 + λ) = r D t U(τ, t), x(r, λ 0) x(, λ 0 ) (Z(r) Z()) λ U(τ, t) = F (x(τ, λ 0 + λ), τ, t) F (x(τ, λ 0 ), τ, t) F x (x(τ, λ 0 ), τ, t)z(τ) λ. For every τ [, b] n u [0, 1], we hve ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ) x(τ, λ 0 ) x(τ, λ 0 + λ) x(τ, λ 0 ) < η. By ssumption 6, the point ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ) is n element of B. In other wors, the segment connecting x(τ, λ 0 + λ) n x(τ, λ 0 ) is contine in B. Thus we cn use the menvlue theorem for vectorvlue functions (see e.g. [3, Lemm 8.11]) to exmine the following ifference: U(τ, t) U(τ, s) = F (x(τ, λ 0 + λ), τ, t) F (x(τ, λ 0 ), τ, t) F (x(τ, λ 0 + λ), τ, s) + F (x(τ, λ 0 ), τ, s) (F x(x(τ, λ 0 ), τ, t) F x (x(τ, λ 0 ), τ, s))(x(τ, λ 0 + λ) x(τ, λ 0 )) + (F x(x(τ, λ 0 ), τ, t) F x (x(τ, λ 0 ), τ, s))(x(τ, λ 0 + λ) x(τ, λ 0 ) Z(τ) λ) ( = 1 1 (F x (ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ), τ, t) F x (ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ), τ, s)) u 0 ) 1 0 (F x (x(τ, λ 0 ), τ, t) F x (x(τ, λ 0 ), τ, s)) u +(F x (x(τ, λ 0 ), τ, t) F x (x(τ, λ 0 ), τ, s))ξ(τ, λ) (x(τ, λ 0 + λ) x(τ, λ 0 )) (In the secon integrl bove, we re simply integrting constnt function.) If < δ, then (by ssumption 4) F x (ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ), τ, t) F x (ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ), τ, s) F x (x(τ, λ 0 ), τ, t) +F x (x(τ, λ 0 ), τ, s) ω( ux(τ, λ 0 + λ) + (1 u)x(τ, λ 0 ) x(τ, λ 0 ) ) h(t) h(s) n thus (using ssumption 3) = ω( u(x(τ, λ 0 + λ) x(τ, λ 0 )) ) h(t) h(s) ω(ε) h(t) h(s), U(τ, t) U(τ, s) ω(ε) h(t) h(s) x(τ, λ 0 + λ) x(τ, λ 0 ) + h(t) h(s) ξ(τ, λ) ( h(t) h(s) ω(ε) x(τ, λ ) 0 + λ) x(τ, λ 0 ) Z(τ) λ + Z(τ) λ + ξ(τ, λ) Consequently, by Lemm 2.2, ξ(r, λ) ξ(, λ) = h(t) h(s) (ω(ε)( ξ(τ, λ) + M) + ξ(τ, λ) ). r r D t U(τ, t) (ω(ε)( ξ(τ, λ) + M) + ξ(τ, λ) ) h(τ) 9
10 It follows tht = ω(ε)m(h(r) h()) + (1 + ω(ε)) ω(ε)m(h(b) h()) + (1 + ω(ε)) r r ξ(τ, λ) h(τ) ξ(τ, λ) h(τ). ξ(r, λ) ξ(r, λ) ξ(, λ) + ξ(, λ) ε + ω(ε)m(h(b) h()) + (1 + ω(ε)) Finlly, Gronwll s inequlity les to the estimte r ξ(τ, λ) h(τ). ξ(r, λ) (ε + ω(ε)m(h(b) h()))e (1+ω(ε))(h(r) h()) (ε + ω(ε)m(h(b) h()))e (1+ω(ε))(h(b) h()). Since lim ε 0+ ω(ε) = 0, we see tht if λ 0, then ξ(r, λ) 0 uniformly for r [, b]. In the simplest cse when l = n, Λ B n x 0 (λ) = λ for every λ Λ, the previous theorem sys tht solutions of t x = D tf (x, τ, t), x() = λ re ifferentible with respect to λ, n the erivtive Z(t) = x λ (t, λ 0 ), t [, b], is the unique solution of the generlize ifferentil eqution Z(s) = I + D t [F x (x(τ, λ 0 ), τ, t)z(τ)], s [, b]. Remrk 4.2. In Theorem 4.1, we re ssuming the existence of ρ > 0 such tht for every λ R l stisfying λ λ 0 < ρ, the initil vlue problem (9) hs solution t x(t, λ) efine on [, b] n tking vlues in B. For equtions whose righthn sie F oes not epen on τ, this ssumption cn be replce by the following simple conition: F (x, t) F (x, s) k(t) k(s), s, t [, b], x B. (11) Let us explin why this conition is sufficient. (We re still ssuming tht conitions 1 6 from Theorem 4.1 re stisfie. In prticulr, we re ssuming tht the initil vlue problem (9) hs solution corresponing to λ = λ 0.) Observe tht if c [, b) n y x(c+, λ 0 ) < η/2, then y B. (Choose δ > 0 such tht x(c+, λ 0 ) x(c + δ, λ 0 ) < η/2; then y x(c + δ, λ 0 ) y x(c+, λ 0 ) + x(c+, λ 0 ) x(c + δ, λ 0 ) < η, n thus y B by ssumption 6.) Following the first prt of proof of Theorem 4.1, we observe tht there is δ > 0 such tht if λ λ 0 < δ n if the solution t x(t, λ) exists on [, c] [, b], then x(t, λ) x(t, λ 0 ) < η ) (1, 4 min 1, t [, c]. k(b) k() + 1 If λ R l stisfies λ λ 0 < δ n the solution t x(t, λ) exists on [, c] [, b] with c [, b), then (x(c, λ) + F (x(c, λ), c+) F (x(c, λ), c)) (x(c, λ 0 ) + F (x(c, λ 0 ), c+) F (x(c, λ 0 ), c)) x(c, λ) x(c, λ 0 ) + F (x(c, λ), c+) F (x(c, λ), c) F (x(c, λ 0 ), c+) + F (x(c, λ 0 ), c) < η/4 + x(c, λ) x(c, λ 0 ) (k(c+) k(c)) < η/4 + x(c, λ) x(c, λ 0 ) (k(b) k()) < η/2 (we hve use ssumption 5). By Theorem 2.3, x(c+, λ 0 ) = x(c, λ 0 ) + F (x(c, λ 0 ), c+) F (x(c, λ 0 ), c). Thus the previous inequlity n ssumption 6 imply x(c, λ) + F (x(c, λ), c+) F (x(c, λ), c) B. All ssumptions of the locl existence theorem for generlize ifferentil equtions (see [8, Theorem 4.2]) re stisfie (we nee (11) t this moment), n thus the solution t x(t, λ) cn be extene to lrger intervl [, ], (c, b]. Consequently, for λ λ 0 < δ, the solution must exist on the whole intervl [, b]. 10
11 Remrk 4.3. Assume tht the set B from Theorem 4.1 is convex. Then it is esy to see tht ssumption 5 in this theorem is reunnt. Inee, the menvlue theorem for vectorvlue functions n ssumption 3 le to the estimte ( 1 0 F (x, τ, t) F (x, τ, s) F (y, τ, t) + F (y, τ, s) ) F x (ux + (1 u)y, τ, t) F x (ux + (1 u)y, τ, s) u x y h(t) h(s) x y for ll s, t, τ [, b], x, y B, i.e., ssumption 5 is stisfie with k = h. With the help of Theorem 4.1, it is esy to obtin n even more generl theorem for equtions where not only the initil conition, but lso the righthn sie of the eqution epens on the prmeter λ. The proof is inspire by similr proof of Theorem 8.49 in [3]. Theorem 4.4. Let B R n be n open set, λ 0 R l, ρ > 0, Λ = {λ R l ; λ λ 0 < ρ}, x 0 : Λ B, F : B [, b] 2 Λ R n. Assume tht F is regulte n leftcontinuous in the thir vrible, n tht for every λ Λ, the initil vlue problem t x = D tf (x, τ, t, λ), x() = x 0 (λ) hs solution in B; let x(t, λ) be the vlue of tht solution t t [, b]. Moreover, let the following conitions be stisfie: 1. For every fixe pir (t, τ) [, b] 2, the function (x, λ) F (x, τ, t, λ) is continuously ifferentible on B Λ. 2. The function x 0 is ifferentible t λ There exists leftcontinuous nonecresing function h : [, b] R such tht for ll s, t, τ [, b], x B, λ Λ. F x (x, τ, t, λ) F x (x, τ, s, λ) h(t) h(s), F λ (x, τ, t, λ) F λ (x, τ, s, λ) h(t) h(s) 4. There exists continuous incresing function ω : [0, ) [0, ) such tht ω(0) = 0 n F x (x, τ, t, λ 1 ) F x (x, τ, s, λ 1 ) F x (y, τ, t, λ 2 )+F x (y, τ, s, λ 2 ) ω( x y + λ 1 λ 2 ) h(t) h(s), F λ (x, τ, t, λ 1 ) F λ (x, τ, s, λ 1 ) F λ (y, τ, t, λ 2 )+F λ (y, τ, s, λ 2 ) ω( x y + λ 1 λ 2 ) h(t) h(s) for ll s, t, τ [, b], x, y B, λ 1, λ 2 Λ. 5. There exists leftcontinuous nonecresing function k : [, b] R such tht F (x, τ, t, λ 1 ) F (x, τ, s, λ 1 ) F (y, τ, t, λ 2 ) + F (y, τ, s, λ 1 ) ( x y + λ 1 λ 2 ) k(t) k(s) for ll s, t, τ [, b], x, y B, λ 1, λ 2 Λ. 6. There exists number η > 0 such tht if x R n stisfies x x(t, λ 0 ) < η for some t [, b], then x B (i.e., the ηneighborhoo of the solution t x(t, λ 0 ) is contine in B). Then the function λ x(t, λ) is ifferentible t λ 0, uniformly for ll t [, b]. Moreover, its erivtive Z(t) = x λ (t, λ 0 ), t [, b], is the unique solution of the generlize ifferentil eqution Z(s) = x 0 λ(λ 0 ) + D t [F x (x(τ, λ 0 ), τ, t, λ 0 )Z(τ) + F λ (x(τ, λ 0 ), τ, t, λ 0 )], s [, b]. 11
12 Proof. Let B = B Λ. Without loss of generlity, ssume tht ll finiteimensionl spces we re working with re equippe with the L 1 norm. In prticulr, when (x, λ) B, then (x, λ) = Define F : B [, b] 2 R n+l by n l x i + λ j = x + λ. i=1 j=1 F ((x, λ), τ, t) = (F (x, τ, t, λ), 0,..., 0) R n+l, x B, λ Λ, t, τ [, b], n y 0 : Λ B by y 0 (λ) = (x 0 (λ), λ), λ Λ. From these efinitions, it is cler tht for every λ Λ, the function y(t, λ) = (x(t, λ), λ), t [, b], is solution of the initil vlue problem t y = D t F (y, τ, t), y() = y 0 (λ) (note tht by the efinition of F, the lst l components of every solution re constnt on [, b]). Without loss of generlity, ssume tht η < ρ. If (x, λ) B is such tht (x, λ) (x(t, λ 0 ), λ 0 ) < η for some t [, b], then x x(t, λ 0 ) < η n λ λ 0 < η, i.e., x B n λ Λ. In other wors, the ηneighborhoo of the solution t y(t, λ 0 ) is contine in B. The erivtive of F with respect to y is the (n + l) (n + l) mtrix F 1 y 1 (y, τ, t) F 1 y n+l (y, τ, t) F y (y, τ, t) =..... F n+l y 1 (y, τ, t) F n+l y n+l (y, τ, t) F 1 F x 1 (x, τ, t, λ) 1 F x n (x, τ, t, λ) 1 F λ 1 (x, τ, t, λ) 1 λ l (x, τ, t, λ) F n F = x 1 (x, τ, t, λ) n F x n (x, τ, t, λ) n F λ 1 (x, τ, t, λ) n λ l (x, τ, t, λ) , where y = (x, λ) B Λ n t, τ [, b]. Similrly, the erivtive of y 0 with respect to λ t λ 0 is the (n + l) l mtrix x 0 1 x λ 1 (λ 0 ) 0 1 λ l (λ 0 )..... yλ(λ 0 x 0 n x 0 ) = λ 1 (λ 0 ) 0 n λ l (λ 0 ) Using ssumptions 3, 4 n 5, it is not ifficult to see tht F n F y stisfy ssumptions 3, 4 n 5 of Theorem 4.1. For exmple, let s, t, τ [, b], y 1, y 2 B, where y 1 = (x 1, λ 1 ) n y 2 = (x 2, λ 2 ). Then F y (y 1, τ, t) F y (y 1, τ, s) F y (y 2, τ, t) + F y (y 2, τ, s) 12
13 = F x (x 1, τ, t, λ 1 ) F x (x 1, τ, s, λ 1 ) F x (x 2, τ, t, λ 2 ) + F x (x 2, τ, s, λ 2 ) + F λ (x 1, τ, t, λ 1 ) F λ (x 1, τ, s, λ 1 ) F λ (x 2, τ, t, λ 2 ) + F λ (x 2, τ, s, λ 2 ) 2ω( x 1 x 2 + λ 1 λ 2 ) h(t) h(s) = ω( y 1 y 2 ) 2h(t) 2h(s), which verifies ssumption 4 of Theorem 4.1. Now, ccoring to Theorem 4.1, the function λ y(t, λ) is ifferentible t λ 0, uniformly for ll t [, b], n its erivtive Z(t) = y λ (t, λ 0 ), t [, b], is the unique solution of the generlize ifferentil eqution Z(s) = y 0 λ(λ 0 ) + D t [ F y (y(τ, λ 0 ), τ, t) Z(τ)], s [, b]. Let Z(t) = x λ (t, λ 0 ), t [, b]; observe tht Z is the submtrix of Z corresponing to the first n rows. Also, note tht the lst l rows of Z form the ientity mtrix. Thus it follows tht Z(s) = x 0 λ(λ 0 ) + D t [F x (x(τ, λ 0 ), τ, t, λ 0 )Z(τ) + F λ (x(τ, λ 0 ), τ, t, λ 0 )], 5 Reltion to other types of equtions s [, b]. In this section, we show tht for impulsive ifferentil equtions n for ynmic equtions on time scles, ifferentibility of solutions with respect to initil conitions follows from our Theorem 4.1. (Similrly, it cn be shown tht ifferentibility with respect to prmeters follows from Theorem 4.4). The reson is tht both types of equtions cn be rewritten s generlize equtions, whose righthn sies o not epen on τ. Assume tht r > 0 is fixe number. We restrict ourselves to the cse B = {x R n ; x < r}; we use B to enote the closure of B. Lemm 5.1. Let µ be the LebesgueStieltjes mesure generte by leftcontinuous nonecresing function g : [, b] R (i.e., µ([c, )) = g() g(c) for every intervl [c, ) [, b]). Assume tht f : B [, b] R m n stisfies the following conitions: For every x B, the function s f(x, s) is mesurble on [, b] with respect to the mesure µ. There exists µmesurble function M : [, b] R such tht M(s) µ < + n f(x, s) M(s), x B, s [, b]. For every s [, b], the function x f(x, s) is continuous in B. Consier the function F given by F (x, t) = f(x, s) µ = [,t) f(x, s) g(s), x B, t [, b]. Then the following sttements re true: 1. There exists nonecresing leftcontinuous function h : [, b] R n continuous incresing function ω : [0, ) [0, ) such tht ω(0) = 0 n F (x, t) F (x, s) h(t) h(s), s, t [, b], x B, F (x, t) F (x, s) F (y, t) + F (y, s) ω( x y ) h(t) h(s), s, t [, b], x, y B. 2. If x : [, b] B, Z : [, b] R n l re regulte functions, then D t [F (x(τ), t)z(τ)] = f(x(τ), τ)z(τ) g(τ). (12) 13
14 Proof. For the first sttement, see Proposition 5.9 in [8] n the references given there. Let us prove the secon sttement. Accoring to Proposition 5.12 in [8], we hve D t F (x(τ), t) = f(x(τ), τ) g(τ) (13) for every regulte function x : [, b] B. Let [α, β] [, b] n ssume tht Z : [α, β] R n l is constnt on (α, β). Then, by (13) n Theorem 2.3, = lim ε 0+ = lim ε 0+ ( β ε α+ε β ( β ε D t [F (x(τ), t)z(τ)] + α+ε f(x(τ), τ)z(τ) g(τ) ) D t [F (x(τ), t)z(τ)] α α+ε α ) β D t [F (x(τ), t)z(τ)] + D t [F (x(τ), t)z(τ)] β ε +(F (x(α), α+) F (x(α), α))z(α)+(f (x(β), β) F (x(β), β ))Z(β) ( β ε ) = lim ε 0+ f(x(τ), τ)z(τ) g(τ) + f(x(α), α)z(α)(g(α+) g(α)) + f(x(β), β)z(β)(g(β) g(β )) α+ε = β α f(x(τ), τ)z(τ) g(τ). This shows tht (12) is stisfie for ll step functions Z : [, b] R n l. For generl regulte function Z, let {Z k } k=1 be sequence of step functions tht is uniformly convergent to Z. Then = lim k D t [F (x(τ), t)z(τ)] = lim k f(x(τ), τ)z k (τ) g(τ) = D t [F (x(τ), t)z k (τ)] f(x(τ), τ)z(τ) g(τ), where the first equlity follows from Lemm 3.3 n the lst equlity from [8, Corollry 1.32]. Let us strt by consiering ifferentil equtions with impulses. Assume tht C is n open neighborhoo of B, f : C [, b] R n is continuous function whose erivtive f x exists n is continuous on C [, b], n I 1,..., I k : C R n re continuously ifferentible functions. Then it is known (see [8, Chpter 5]) tht the impulsive ifferentil eqution x (t) = f(x(t), t), t [, b]\{t 1,..., t k }, x(t i +) x(t i ) = I i (x(t i )), i {1,..., k}, x() = x 0 (λ), whose solutions re ssume to be leftcontinuous, is equivlent to the generlize ifferentil eqution where F (x, t) = F 1 (x, t) + F 2 (x, t) n F 1 (x, t) = t x = D tf (x, t), t [, b], x() = x 0 (λ), f(x, s) s, F 2 (x, t) = k I i (x)χ (ti, )(t) (the symbol χ A enotes the chrcteristic function of set A R). More precisely, x : [, b] B is solution of the impulsive eqution (14) if n only if it is solution of the generlize eqution (see [8, Theorem 5.20]). Now, F 1 n F 2 re ifferentible with respect to x n F 1 x (x, t) = f x (x, s) s, F 2 x (x, t) = 14 i=1 k Ix(x)χ i (ti, )(t). i=1 (14)
15 By the first prt of Lemm 5.1 (where we tke g(s) = s n µ is the Lebesgue mesure), we obtin the existence of functions h 1 : [, b] R n ω 1 : [0, ) [0, ) such tht F 1 x (x, t) F 1 x (x, s) h 1 (t) h 1 (s), s, t [, b], x B, F 1 x (x, t) F 1 x (x, s) F 1 x (y, t) + F 1 x (y, s) ω 1 ( x y ) h 1 (t) h 1 (s), s, t [, b], x, y B. By continuity, there exists K 1 such tht Ix(x) i K for ll x B, i {1,..., k}. Let h 2 (t) = K k i=1 χ (t i, )(t) for t [, b], n let ω 2 be the common moulus of continuity of the mppings I 1,..., I k on B. Then simple clcultion revels tht F 2 x (x, t) F 2 x (x, s) h 2 (t) h 2 (s), s, t [, b], x B, F 2 x (x, t) F 2 x (x, s) F 2 x (y, t) + F 2 x (y, s) ω 2 ( x y ) h 2 (t) h 2 (s), s, t [, b], x, y B. Consequently, the function F x = F 1 x + F 2 x stisfies ssumptions 3, 4 of Theorem 4.1 with h = h 1 + h 2 n ω = ω 1 + ω 2. By Remrk 4.3, ssumption 5 is stisfie with k = h. It follows tht solutions of the impulsive eqution (14) re ifferentible with respect to λ, uniformly on [, b]. (Actully, the whole proceure still works uner weker hypotheses on f; the crucil thing is to ensure tht f n f x stisfy the ssumptions of Lemm 5.1.) To obtin n eqution for the erivtive Z(s) = x λ (s, λ 0 ), we mke use of the fct tht F x (x, t) = f x (x, s) g(s), t [, b], where g(s) = s + k i=1 χ (t i, )(s), n { f x (x, t) if t [, b]\{t 1,..., t k }, f x (x, t) = Ix(x) i if t = t i for some i {1,..., k} (see e.g. [1, Remrk 3.12]). By Theorem 4.1 n the secon prt of Lemm 5.1, Z(s) = x 0 λ(λ 0 ) + D t [F x (x(τ, λ 0 ), t)z(τ)] = x 0 λ(λ 0 ) + f x (x(τ, λ 0 ), τ)z(τ) g(τ), s [, b]. This integrl eqution cn be rewritten bck (see gin [1, Remrk 3.12]) s the impulsive eqution Z (t) = f x (x(t, λ 0 ), t)z(t), t [, b]\{t 1,..., t k }, Z(t i +) Z(t i ) = Ix(x(t i i, λ 0 ))Z(t i ), i {1,..., k}, x() = x 0 λ (λ 0), which grees with the result from [6]. Next, let us turn our ttention to ynmic equtions on time scles. Let T be time scle,, b T, < b. We use the nottion [, b] T = [, b] T. For every t [, b], let t = inf{s [, b] T ; s t}. Given n rbitrry function f : [, b] T R n, we efine function f : [, b] R n by f (t) = f(t ), t [, b]. Similrly, for every function f : C [, b] T R n, where C R n, let f : C [, b] R n be efine by f (x, t) = f(x, t ), t [, b], x C. Assume tht C is n open neighborhoo of B n f : C [, b] T R n stisfies the following conitions: For every t [, b] T, the function x f(x, t) is continuously ifferentible on C. 15
16 For every continuous function x : [, b] T B, the functions t f(x(t), t) n t f x (x(t), t) re rcontinuous. f x is boune in B [, b] T. A consequence of these conitions is tht f is boune in B [, b] T (use the estimte f(x, t) f(x, t) f(0, t) + f(0, t) n pply the menvlue theorem in the first norm). Uner these ssumptions, it is known (see [11, Theorem 12]) tht the ynmic eqution x (t) = f(x(t), t), t [, b] T, x() = x 0 (λ) (15) is equivlent to the generlize orinry ifferentil eqution where t x = D tf (x, t), t [, b], x() = x 0 (λ), (16) F (x, t) = f (x, s) g(s), n g(s) = s for every s [, b]. More precisely, if x : [, b] T B is solution of (15), then the function x : [, b] B is solution of (16). Conversely, every solution y : [, b] B of (16) hs the form y = x, where x : [, b] T B is solution of (15). We hve F x (x, t) = f x(x, s) g(s). By the first prt of Lemm 5.1, there exist functions h : [, b] R n ω : [0, ) [0, ) such tht F x (x, t) F x (x, s) h(t) h(s), s, t [, b], x B, F x (x, t) F x (x, s) F x (y, t) + F x (y, s) ω( x y ) h(t) h(s), s, t [, b], x, y B. Thus, F x stisfies ssumptions 3, 4 of Theorem 4.1. By Remrk 4.3, ssumption 5 is stisfie with k = h. This mens tht solutions of the ynmic eqution (15) re ifferentible with respect to λ, uniformly on [, b] T. Let Z(s) = x λ (s, λ 0 ) be the corresponing erivtive t λ 0. By Theorem 4.1 n the secon prt of Lemm 5.1, Z(s) = x 0 λ(λ 0 ) + D t [F x (x(τ, λ 0 ), t)z(τ)] = x 0 λ(λ 0 ) + f x(x(τ, λ 0 ), τ)z(τ) g(τ), s [, b]. Consequently (see [11, Theorem 5]), n therefore Z(s) = x 0 λ(λ 0 ) + which grees with the result obtine in [2]. 6 Conclusion f x (x(τ, λ 0 ), τ)z(τ) τ, s [, b] T, Z (t) = f x (x(t, λ 0 ), t)z(t), t [, b] T, Z() = x 0 λ(λ 0 ), Besies impulsive ifferentil equtions n ynmic equtions on time scles, our ifferentibility results re lso pplicble to the soclle mesure ifferentil equtions of the form x(t) = x() + f(x(s), s) s + g(x(s), s) u(s), t [, b], 16
17 where u is leftcontinuous function with boune vrition. It ws shown in [8, Chpter 5] tht uner certin ssumptions, this eqution is equivlent to the generlize orinry ifferentil eqution whose righthn sie is F (x, t) = f(x, s) s + g(x, s) u(s). As in the previous section, simple ppliction of Lemm 5.1 shows tht uner suitble ssumptions on f n g, the hypotheses of Theorem 4.1 re stisfie, i.e., solutions of mesure ifferentil equtions re ifferentible with respect to initil conitions. An interesting open question is whether the results in this pper cn be extene to generlize equtions whose solutions tke vlues in infiniteimensionl Bnch spces. Numerous uthors hve lrey investigte equtions of this type (see e.g. [5, 7]). For exmple, it is known tht uner certin ssumptions, mesure functionl ifferentil equtions re equivlent to generlize orinry ifferentil equtions with vectorvlue solutions (see e.g. [1, 10] n the references there). Therefore, ifferentibility results for the ltter type of equtions woul be irectly pplicble in the stuy of functionl ifferentil equtions. References [1] M. Feerson, J. G. Mesquit, A. Slvík, Mesure functionl ifferentil equtions n functionl ynmic equtions on time scles, J. Differentil Equtions 252 (2012), [2] R. Hilscher, V. Zein, n W. Krtz, Differentition of Solutions of Dynmic Equtions on Time Scles with Respect to Prmeters, Av. Dyn. Syst. Appl. 4 (2009), no. 1, [3] W. G. Kelley n A. C. Peterson, The Theory of Differentil Equtions (Secon Eition), Springer, [4] J. Kurzweil, Generlize orinry ifferentil equtions n continuous epenence on prmeter, Czechoslovk Mth. J. 7 (82), 1957, [5] J. Kurzweil, Generlize Orinry Differentil Equtions. Not Absolutely Continuous Solutions, Worl Scientific, [6] V. Lkshmiknthm, D. D. Binov, P. S. Simeonov, Theory of Impulsive Differentil Equtions, Worl Scientific, [7] G. Monteiro, M. Tvrý, Generlize liner ifferentil equtions in Bnch spce: Continuous epenence on prmeter, Discrete Contin. Dyn. Syst. 33 (2013), [8] Š. Schwbik, Generlize Orinry Differentil Equtions, Worl Scientific, [9] Š. Schwbik, M. Tvrý, n O. Vejvo, Differentil n Integrl Equtions: Bounry Vlue Problems n Ajoints, Acemi, Prh, [10] A. Slvík, Mesure functionl ifferentil equtions with infinite ely, Nonliner Anl. 79 (2013), [11] A. Slvík, Dynmic equtions on time scles n generlize orinry ifferentil equtions, J. Mth. Anl. Appl. 385 (2012),
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