Exponential Dichotomies for Dynamic Equations on Measure Chains

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1 Exponenial Dichoomies for Dynamic Equaions on Measure Chains Chrisian Pözsche Insiu für Mahemaik, Universiä Augsburg, D Augsburg, Germany November 8, 2007 Absrac In his paper we inroduce he noion of an exponenial dichoomy for no necessarily inverible linear dynamic equaions in Banach spaces wihin he framework of he Calculus on Measure Chains. Paricularly his unifies he corresponding heories for difference and differenial equaions. We apply our approach o obain resuls on perurbed sysems Mahemaics Subjec Classificaion. 34D09, 34D99, 39A12 Keywords. Time Scale, Measure Chain, Linear Dynamic Equaion, Exponenial Dichoomy 1 Inroducion and Preliminaries Basically an exponenial dichoomy is a generalizaion of he concep of hyperboliciy from auonomous o nonauonomous linear equaions, where he sabiliy properies of he soluions in he nonrivial invarian ses, or precisely in he invarian vecor bundles, are uniform. Thorough inroducions ino he heory of exponenially dichoomic ordinary differenial equaions (ODEs) can be found in e.g. he books Daleckiĭ & Kreĭn [7] or Coppel [6]. For difference equaions (O Es) he lieraure is slighly sparser, bu Coffman & Schäffer [5] and Henry [9, Secion 7.6] pioneered here and meanwhile dichoomies are widely used. They are so imporan in he heory of nonauonomous dynamical sysems, since hey are a very useful ool o solve nonlinear problems as perurbaions of linear ones, like in he persisence of inegral manifolds (cf. e.g. Sakamoo [18]). Moreover he applicaions range from sabiliy heory, because a dichoomy is a ype of condiional sabiliy, o modern chaos heory (see Palmer [17], which is also a good inroducion ino discree dichoomies). In he presen paper we inroduce he noion of an exponenial dichoomy for nonregressive linear dynamic equaions in Banach spaces and prove some of is cenral properies. This allows o consider difference, differenial and equaions on inhomogeneous ime scales, i.e. closed subses of R, simulaneously. As an applicaion we also deduce some resuls abou inhomogeneously and semi-linearly perurbed sysems. Here we have o poin ou ha our exponenial growh raes are no assumed o be consan. To quoe a good reference abou dynamic equaions on measure Fax: , chrisian.poezsche@mah.uni-augsburg.de 1

2 2 2 EXPONENTIAL DICHOTOMIES chains we sill srongly recommend Hilger [11] and wih a focus on he linear heory Aulbach & Hilger [1]. Neverheless here exiss he monograph Lakshmikanham, Sivasundaram & Kaymakçalan [13]. Now suppose for he following ha (T,, µ) is an arbirary measure chain wih bounded graininess µ and X is a real or complex Banach space wih he norm. L(X 1 ; X 2 ) sands for he linear space of coninuous homomorphisms wih he norm T := sup x =1 T x for any T L(X 1 ; X 2 ), and GL(X 1 ; X 2 ) for he se of oplinear isomorphisms beween wo linear subspaces X 1, X 2 of X ; I X1 is he ideniy mapping on X 1. Addiionally we wrie L(X ) := L(X ; X ) and N (T ) := T 1 ({0}) is he nullspace and R(T ) := T X he range of T L(X ). We also shorly inroduce some noions, which are specific for he calculus on measure chains. Above all, T + and T are he T-inervals { T : } and { T : }, respecively, for any T; differing from he usual sandard, ρ + : T T is he forward jump operaor. A subse J T is said o be unbounded above (resp. below), if he se {µ(, ) R : J} is unbounded above (resp. below) for one and hence (by he properies of he growh calibraion µ) every T. The parial derivaive of a mapping Φ : T T X wih respec o he firs variable is denoed by 1 Φ. C rd (T κ, X ) are he rd-coninuous mappings from T κ ino X and C + rd R(Tκ, R) := {a C rd (T κ, R) : 1 + µ ()a() > 0 for T κ } is he linear space of posiively regressive funcions wih he algebraic operaions (a b)() := a() + b() + µ ()a()b(), (1 + ha()) α 1 (α a)() := lim h µ () h for T κ for a, b C + rd R(Tκ, R) and reals α R. Wih fixed T and c, d C + rd R(Tκ, R) we define he wo linear spaces { } B +,c(x ) := λ C rd (T +, X ) : sup λ() e c (, ) <, } B,d {λ (X ) := C rd (T, X ) : sup λ() e d (, ) < of so-called c + -quasibounded and d -quasibounded mappings, which are immediaely seen o be Banach spaces wih regard o he norms λ +,c := sup λ() e c (, ), λ,d := sup λ() e d (, ), respecively. 2 Exponenial Dichoomies We consider a linear dynamic equaion x = A()x, (2.1) wih coefficien operaor A C rd (T κ, L(X )) and he ransiion operaor Φ A (, ) L(X ), i.e. he soluion of he corresponding operaor-valued iniial value problem X = A()X, X() = I X in L(X ) for, T,. Since we do no assume I X + µ ()A() GL(X ; X ), in oher

3 3 words, (2.1) needs no o be regressive, Φ A (, ) in general is no inverible and exiss only for, which can be shown like in he sandard exisence and uniqueness heorem for dynamic equaions from Hilger [11, Theorem 5.7]. For all he subsequen, le J T κ be a bounded or unbounded T-inerval. A nonempy se W J X is called an invarian vecor bundle of (2.1), if he following holds: (i) W is posiively invarian wih respec o (2.1), i.e. (, ξ) W (, Φ A (, )ξ) W for,, J, (ii) for each J he fiber W() := {ξ X : (, ξ) W} is a closed subspace of X ; see also Siegmund [19]. In case of a regressive equaion (2.1) wo fibers W() and W(s) are homeomorphic by virue of he oplinear homeomorphism Φ A (, s) (s, J). Oherwise only he inclusion Φ A (, s)w(s) W() for s holds. Trivially he zero bundle J {0} and he whole exended sae space J X are invarian vecor bundles, bu also he following holds. Proposiion 2.1 (invarian projecor): If P : J L(X ) is an invarian projecor of equaion (2.1), i.e. a projecion-valued funcion (P () 2 P ()) such ha hen he wo ses P ()Φ A (, s) = Φ A (, s)p (s) for s, s, J, (2.2) S := {(, η) J X : η R(P ())}, U := {(, ξ) J X : ξ N (P ())} are invarian vecor bundles of (2.1) and i is S() U() = X for all J. Remark 2.2: (1) On a discree measure chain, i.e. if here exiss a real γ > 0 such ha γ µ () for J, a funcion of projecions P : J L(X ) is an invarian projecor of (2.1), if and only if P (ρ + ()) [I X + µ ()A()] = [I X + µ ()A()] P () for J κ holds. This is a consequence of Hilger [11, Theorem 6.2] and mahemaical inducion. (2) In case of a regressive equaion (2.1), we ge P () = Φ A (, s)p (s)φ A (s, ) for s, J and all projecions P () L(X ) are similar. In addiion o his, each P () has he same rank, if S(s) is finie-dimensional for one and consequenly every s J. Proof. The proof is a sraigh-forward verificaion of he definiion of an invarian vecor bundle. Because we do no resric our consideraions on regressive equaions, in paricular soluions in he vecor bundle U migh no exis in backward ime. To overcome his problem, we posulae inveribiliy only beween he fibers of U. Proposiion 2.3 (regulariy condiion): Le P : J L(X ) be an invarian projecor of equaion (2.1) such ha he regulariy condiion [I X + µ ()A()] U() : U() U(ρ + ()) is bijecive for righ-scaered J κ (2.3) is fulfilled. Then we have Φ A (, s) U(s) GL(U(s); U()) for any s, s, J.

4 4 2 EXPONENTIAL DICHOTOMIES Remark 2.4: (1) For regressive equaions (2.1) and especially ODEs, he regulariy condiion is always fulfilled. (2) Under he regulariy condiion invarian projecors P : J L(X ) are rd-coninuously differeniable wih he derivaive P () = A()P () P (ρ + ())A() for J κ. Proof. Firs of all, he mapping (2.3) is well-defined by Proposiion 2.1. For fixed s J we use he inducion principle from Hilger [11, Theorem 1.4(c)] o prove he saemen A() : Φ A (, s) U(s) : U(s) U() is coninuous and bijecive for each J, s. Therefore we have o proceed in four seps: (I): Because of Φ A (s, s) U(s) = I U(s) obviously A(s) is fulfilled. (II): Now le J, s be righ-scaered. Since A() holds by assumpion, he mapping Φ A (ρ + (), s) U(s) = [I X + µ ()A()] U() Φ A (, s) U(s) is a composiion of wo coninuous bijecive homomorphisms, and his yields A(ρ + ()). (III): Le J, s be righ-dense and le A() be rue. Thus here exiss a T-neighborhood U T of such ha sup U µ () A() < 1 and (2.1) is regressive on U. Consequenly Φ A (r, ) is a oplinear isomorphism on X for r U, r and wih he aid of he ideniy Φ A (r, s) U(s) = Φ A (r, ) U() Φ A (, s) U(s) we obain A(r) for r U, r. (IV): Finally le J, s be lef-dense and A(r) be rue for each r J, s r. Then also A() follows similarly o sep (III). Now he proof is finished, since he coninuous isomorphism Φ A (, s) U(s) : U(s) U() has a coninuous inverse by Lang [14, p. 388, Corollary 1.4]). A his poin he Proposiion 2.3 allows us o define he exended ransiion operaor Φ A (, s) : U(s) U(), { [ΦA ] 1 (s, ) Φ A (, s) := U() if s Φ(, s) U(s) if s for any pairs (, s) J J. I is easy o show ha Φ A (, s) GL(U(s); U()) inheris cerain disincive feaures of he usual ransiion operaor, namely he cocycle propery or he wo ideniies Φ A (, ) = Φ A (, s) Φ A (s, ) for, s, J, [I X P ()] Φ A (, s) = Φ A (, s) [I X P (s)], ΦA (, s) 1 = Φ A (s, ) for s, J. (2.4) Finally one can show he differeniabiliy of Φ A (, ) : J κ L(U(); X ) wih he derivaive ( 1 ΦA )(, ) = A() Φ A (, ) for, J κ. To bring our preparaions o an end, in he definiion of an exponenial dichoomy we use he abbreviaion b a := inf T κ(b() a()) and inroduce he noaions a b : 0 < b a, a b : 0 b a, where wo posiively regressive funcions a, b C + rd R(Tκ, R) are denoed as growh raes, if sup T κ µ ()a() < and sup T κ µ ()b() <, respecively. Then we obain he limis lim e a b(, ) = 0, lim e b a(, ) = 0 (2.5) for growh raes a b and on a measure chain, which is unbounded above resp. below. The relaions (2.5) can be shown using similar argumens as in Hilger [10, Saz 9.2].

5 5 Definiion 2.5 (exponenial dichoomy): Le P : J L(X ) be an invarian projecor of (2.1) such ha he regulariy condiion (2.3) is fulfilled. Then equaion (2.1) is said o possess an exponenial dichoomy, if he esimaes Φ A (, s)p (s) K 1 e a (, s) for s, s, J, (2.6) ΦA (, s) [I X P (s)] K 2 e b (, s) for s, s, J (2.7) hold for real consans K 1, K 2 1 and growh raes a, b C + rdr(j, R), a b. Remark 2.6: (1) The growh raes a, b do no have o be consan funcions. For ODEs his goes back o Muldowney [16]. A second feaure of our definiion is ha we do no insis on a hyperboliciy condiion like a 0 b. Thus one can speak of a pseudo-hyperbolic dichoomy. Alhough his makes Definiion 2.5 more flexible, i is no a real generalizaion, because one can ransform each pseudo-hyperbolic ino a hyperbolic sysem (see Corollary 2.7). Evenually we poin ou again ha equaion (2.1) does no have o be regressive. For O Es his can be raced back o Henry [9, p. 229, Definiion 7.6.4] and wih a differen, bu equivalen definiion o Kalkbrenner [12]. A dichoomy noion for O Es wihou any regulariy condiion is conained in Aulbach & Kalkbrenner [2]. (2) The equaion (2.1) is said o have an ordinary dichoomy if he esimaes (2.6) and (2.7) hold wih a = b = 0. Having he curren concep available, one can generalize resuls from Bohner & Luz [3] o nonregressive equaions in finie-dimensional Banach spaces, like i has been done in Elaydi, Papaschinopoulos & Schinas [8] for O Es. (3) Seing s =, he inequaliies (2.6) and (2.7) imply he boundedness of he invarian projecors P and I X P, respecively. The following resul can be seen as a firs sep o inroduce a specral noion for linear dynamic equaions (see Siegmund [19]). Corollary 2.7 (shifed sysem): If equaion (2.1) possesses an exponenial dichoomy wih a, b, K 1, K 2 and P on a T-inerval J, hen for arbirary growh raes c C + rdr(j, R) also he shifed sysem x = (A ci X )()x (2.8) has an exponenial dichoomy wih a c, b c, K 1, K 2 and P on J. Remark 2.8: Specifically for c := 1 2 (a b) he shifed sysem (2.8) has a hyperbolic exponenial dichoomy wih he growh raes 1 2 (a b), 1 2 (b a). Proof. For define he mapping Φ(, ) := e c (, )Φ A (, ). Then he produc rule (cf. Hilger [11, Theorem 2.6(ii)]) leads o ( 1 Φ)(, ) = e c (, ρ + ())A()Φ A (, ) c()e c (, ρ + ())Φ A (, ) = = e c (, ρ + ()) [A() c()i X ] e c (, )Φ A (, ) = = [1 + µ ()c()] 1 [A() c()i X ] Φ(, ) for, J κ and consequenly Φ(, ) is he ransiion operaor of he shifed sysem (2.8) since we already have Φ(, ) = I X. Now i is easy o see ha P is also an invarian projecor of (2.8) saisfying he regulariy condiion (2.3), and ha he dichoomy esimaes (2.6) and (2.7) for Φ(, ) hold rue.

6 6 2 EXPONENTIAL DICHOTOMIES Evenually, for sysems possessing an exponenial dichoomy, we can characerize he invarian vecor bundles S and U dynamically. Theorem 2.9 (dynamical characerizaion of S and U): Le equaion (2.1) possess an exponenial dichoomy wih a, b and P on J. For a fixed 0 T κ he following holds: (a) If J = T + 0 is unbounded above and c C + rdr(j, R) wih a c b, hen S = { (, η) J X : Φ A (, )η B +,c(x ) }, (b) if J = T 0 is unbounded below and d C + rdr(j, R) wih a d b, hen U = { (, ξ) J X : here exiss a soluion ν : T } X of (2.1) wih ν() = ξ and ν B,d (X ). Hence we call S and U he pseudo-sable and he pseudo-unsable vecor bundle of (2.1), respecively. Remark 2.10: In paricular he ses S and U are independen of he growh raes c and d, respecively, since hey are defined in Proposiion 2.1 only using invarian projecors. Proof. We have o show wo inclusions each. (a)( ) For (, η) S, i.e. η = P ()η i follows Φ A (, )η e c (, ) (2.6) K 1 e a c (, ) η K 1 η for and we obain Φ A (, )η B +,c(x ) by passing over o he leas upper bound over T + in he las esimae. ( ) Because of Φ A (, )η B +,c(x ) here exiss a real C 0 such ha Φ A (, )η C e c (, ) η for and he ideniy [I X P ()] η = Φ A (, ) [I X P ()] Φ A (, )η yields [I X P ()] η ΦA (, ) [I X P ()] Φ A (, )η (2.7) C K 2 e c b (, ) η for. By aking he limi in his esimae and considering (2.5), i follows η = P ()η and consequenly (, η) S. (b)( ) This inclusion resuls similarly o he firs inclusion in (a), if one defines he funcion ν() := Φ A (, )ξ for. ( ) Vice versa le ν B,d (X ) be a soluion of (2.1) wih ν() = ξ. Then we have Φ A(, )ν() = ξ for and one ges P ()ξ (2.2) = Φ A (, )P ()ν() (2.6) K 1 e d a (, ) ν,d for. Now passing over o he limi in his esimae and by considering (2.5) we obain P ()ξ = 0 and (, ξ) U. Wihou proof we sae an immediae consequence of Theorem 2.9 abou he possible choices of he invarian projecors. Is parial converse can be found in Siegmund [19, Lemma 1.1].

7 7 Corollary 2.11: Le equaion (2.1) possess an exponenial dichoomy wih a, b and he invarian projecors P and Q on J. For fixed 0 T κ and c, d C + rdr(j, R) he following holds: (a) If J = T + 0 is unbounded above, J, hen R(P ()) = R(Q()) and for c b he equaion (2.1) has no nonrivial c + -quasibounded soluion in U on T +, (b) if J = T 0 is unbounded below, J, hen N (P ()) = N (Q()) and for a d he equaion (2.1) has no nonrivial d -quasibounded soluion in S on T, (c) if J = T is unbounded above and below, hen P = Q and for a c b, a d b he equaion (2.1) has no nonrivial c + - and d -quasibounded soluion on T. 3 Perurbaion Resuls To prepare he proofs of our perurbaion heorems, we have o derive an elemenary lemma. Ye is imporance should no be underesimaed, since i is he key o consider nonconsan growh raes, oo. Lemma 3.1: For,, 1, 2 T κ, 1 2 and a, b C + rd R(Tκ, R) we obain 2 1 e a (, ρ + (s))e b (s, ) s { ea(,) b a [e b a( 2, ) e b a ( 1, )] if a b e a(,) a b [e b a( 1, ) e b a ( 2, )] if b a. (3.1) Proof. The desired esimaes follow from he ideniy 2 1 e a (, ρ + (s))e b (s, ) s = e a (, ) 2 ( 1 e b a )(s, ) 1 b(s) a(s) by he properies of he Cauchy-Inegral (cf. Hilger [11, Theorem 4.3]). s, Or primary aim is o show he exisence of some quasibounded soluions of perurbed dynamic equaions. Here we follow closely o Palmer [17, Lemma 2.7, Proposiion 2.8], who considered finie-dimensional inverible O Es. One can also apply he Theorems 3.2 and 3.4 in he siuaion of a rivial dichoomy, where P () I X or P () 0. In his case hey sae ha cerain exponenial growh properies of he soluions of (2.1) are preserved under linear-inhomogeneous and semi-linear perurbaions. On he oher hand, boh subsequen resuls can be seen as a generalizaion of Corollary 2.11, while he following heorem is also relaed o he problem of admissibiliy (cf. Massera & Schäffer [15, pp. 165ff]), because i gives sufficien condiions, under which he pairs (B +,c(x ), B +,c(x )) and (B,d (X ), B,d (X )) are admissible for he linear dynamic equaion (2.1). Theorem 3.2 (inhomogeneous perurbaions): Le equaion (2.1) possess an exponenial dichoomy wih a, b, K 1, K 2 and P on J. For he linear-inhomogeneous equaion x = A()x + r() (3.2) and fixed 0 T κ, c, d C + rdr(j, R), a c b, a d b he following holds:

8 8 3 PERTURBATION RESULTS (a) If J = T + 0 is unbounded above, hen for each J, x 0 X and r B +,c(x ) here exiss exacly one soluion λ B +,c(x ) of (3.2) wih (, λ () x 0 ) U; addiionally i is λ +,c K 1 x 0 + C(c) r +,c, (3.3) (b) if J = T 0 is unbounded below, hen for each J, x 0 X and r B,d (X ) here exiss exacly one soluion λ B,d (X ) of (3.2) wih (, λ () x 0 ) S; addiionally i is λ,d K 2 x 0 + C(d) r,d, where we have used he abbreviaion C(c) := K 1 c a + K 2 b c. Remark 3.3: The inclusion (, λ () x 0 ) U (resp. (, λ () x 0 ) S) means ha he iniial value λ () and he poin x 0 have he same projecion ono he fiber S() (resp. U()). Proof. (a) Keep he poins J, x 0 X and he inhomogeneiy r B +,c(x ) fixed. Above all he funcion λ : T + X, λ () := Φ A (, )P ()x 0 + Φ A (, ρ + (s))p (ρ + (s))r(s) s Φ A (, ρ + (s)) [ I X P (ρ + (s)) ] r(s) s (3.4) is well-defined, since he inegrands are rd-coninuous in he variable s (hence hey have an ani-derivaive by Hilger [11, Theorem 4.4]) and using Lemma 3.1 we obain he esimae (3.4) λ () K 1 e a (, ) x 0 + K 1 e a (, ρ + (s)) r(s) s + K 2 e b (, ρ + (s)) r(s) s K 1 e a (, ) x [K 1 e a (, ρ + (s))e c (s, ) s + K 2 (3.1) K 1 e a (, ) x 0 + ] e b (, ρ + (s))e c (s, ) s r +,c [ K1 c a (e c(, ) e a (, )) + K 2 b c e c(, ) ] r +,c for, which on he oher side implies he c + -quasiboundedness of λ as well as he inequaliy (3.3). Now he Lemma 4.1 applies o he firs inegral in (3.4) and Lemma 4.2 can be applied o he indefinie inegral in (3.4), which leads o λ () (3.4) = A()Φ A (, )P ()x 0 + P (ρ + ())r() + + [ I X P (ρ + ()) ] r() (3.4) = A()λ () + r() for ; A()Φ A (, ρ + (s))p (ρ + (s))r(s) s + A() Φ A (, ρ + (s)) [ I X P (ρ + (s)) ] r(s) s = whence λ is a soluion of (3.2). I remains o show (, λ () x 0 ) U, which however follows from P ()λ () (3.4) = P ()x 0 P () Φ A (, ρ + (s)) [ I X P (ρ + (s)) ] r(s) s (2.4) = P ()x 0.

9 9 Evenually λ is uniquely defined, because if ν would be anoher soluion of (3.2) wih (, ν () x 0 ) U, hen he difference λ ν B +,c(x ) would be a soluion of he homogeneous equaion (2.1) and P () [λ () ν ()] = P ()x 0 P ()x 0 = 0. Consequenly λ ν is a c + -quasibounded soluion of (2.1) in U which has o vanish idenically on T + by Corollary 2.11(a). (b) Compleely analogously o (a), he funcion λ : T X, λ () := Φ A (, ) [I X P ()] x 0 + Φ A (, ρ + (s))p (ρ + (s))r(s) s Φ A (, ρ + (s)) [ I X P (ρ + (s)) ] r(s) s is a d -quasibounded soluion of equaion (3.2) wih [I X P ()] λ () = [I X P ()] x 0, i.e. (, λ () x 0 ) S. The uniqueness in he presen case follows from Corollary 2.11(b). Theorem 3.4 (semi-linear perurbaions): Le equaion (2.1) possess an exponenial dichoomy wih a, b, K 1, K 2 and P on J. For he semi-linear equaion x = A()x + g(, x) + h(), (3.5) fixed 0 T κ, c, d C + rdr(j, R), a c b, a d b and under he assumpions (i) g : J X X is rd-coninuous and fulfills for a real consan L 0, (ii) L max {C(c), C(d)} < 1, he following holds: g(, x) g(, x) L x x for J, x, x X, (3.6) (a) If J = T + 0 is unbounded above and g(, 0) B +,c(x ), hen for each J, x 0 X and h B +,c(x ) here exiss exacly one soluion λ B +,c(x ) of (3.5) wih (, λ () x 0 ) U; addiionally i is λ +,c 1 [ ] K 1 x 0 + C(c) g(, 0) + h +,c, (3.7) 1 LC(c) (b) if J = T 0 is unbounded below and g(, 0) B,d (X ), hen for each J, x 0 X and h B,d (X ) here exiss exacly one soluion λ B,d (X ) of (3.5) wih (, λ () x 0 ) S; addiionally i is λ,d 1 [ ] K 2 x 0 + C(d) g(, 0) + h,d, 1 LC(d) where he consan C(c) > 0 is given in Theorem 3.2. Remark 3.5: Under he addiional assumpions g(, 0) 0, h() 0 on J, he semilinear equaion (3.5) has no nonrivial c + -quasibounded (resp. d -quasibounded) soluion wih (, λ ()) U on T + (resp. (, λ ()) S on T ).

10 10 4 APPENDIX: PARAMETER INTEGRALS Proof. (a) Firs of all, he soluions of equaion (3.5) are unique and hey exis on T +, which follows from he proof of Hilger [11, Theorem 5.7]. Keep J, x 0 X fixed and consider any funcions λ, λ B +,c(x ) for he momen. Then he mapping r λ : J X, r λ () := g(, λ())+h() has he propery r λ () g(, λ()) g(, 0) + g(, 0) + h() (3.6) L λ() + g(, 0) + h() for and accordingly i is c + -quasibounded wih r λ +,c L λ +,c + g(, 0) + h +,c. (3.8) Therefore Theorem 3.2(a) implies ha x = A()x + r λ () (3.9) has a unique c + -quasibounded soluion T x0 λ : T + given by he expression (3.4), namely X wih (, (T x0 λ)() x 0 ) U, which is (T x0 λ)() := Φ A (, )P ()x 0 + Φ A (, ρ + (s))p (ρ + (s))r λ (s) s Φ A (, ρ + (s)) [ I X P (ρ + (s)) ] r λ (s) s. In paricular he operaor T x0 : B +,c(x ) B +,c(x ) is well-defined and he difference T x0 λ T x0 λ B +,c (X ) solves he equaion x = A()x + g(, λ()) g(, λ()) (3.10) wih (, (T x0 λ)() (T x0 λ)()) U, where he inhomogeneiy fulfills g(, λ( )) g(, λ( )) +,c L λ λ + by (3.6). Now Theorem 3.2(a) applied o he equaion (3.10) has he consequence,c (3.3) T x0 λ T x0 λ + C(c) g(, λ( )) g(, λ( )) +,c,c LC(c) λ λ +,c ; hence T x0 is a conracion on he Banach space B +,c(x ) by assumpion (ii). Using he conracion principle (see e.g. Lang [14, p. 360, Lemma 1.1]), T x0 has a unique fixed poin λ B +,c(x ) wih (, λ () x 0 ) U. Moreover λ is a soluion of (3.9) (for λ = λ ) and hus also a soluion of (3.5). Finally we ge he esimae λ +,c which implies (3.7) (3.3) K 1 x 0 + C(c) r λ + (3.8) [ ],c K 1 x 0 + C(c) L λ +,c + g(, 0) + h +,c, (b) One has o proceed analogously o sep (a) and use Theorem 3.2(b) hereby. Consequenly he proof is complee. 4 Appendix: Parameer Inegrals Here we sae wo resuls abou he differeniabiliy of inegrals, which depend on a parameer. So far hey canno be quoed from anoher reference concerning he calculus on measure chains.

11 REFERENCES 11 Lemma 4.1 (parameer inegrals): Assume, T κ, and r C rd (T κ, X ). If Φ : {(, ) T T κ : } L(X ) is a coninuous mapping, Φ(, 0 ) possesses a coninuous exension o a neighborhood of each righ-dense 0 T κ and if Φ(, s) is rd-coninuously differeniable such ha he limi ( 1 Φ)( 0, s) = lim 0 Φ(, s) Φ( 0, s) µ(, 0 ) exiss uniformly in s on compac subses in each righ-dense 0 T κ. Then he funcion F : T X, F () := Φ(, s)r(s) s is differeniable on Tκ wih derivaive F () = Φ(ρ + (), )r() + ( 1 Φ)(, s)r(s) s. Proof. The proof can be done similarly o Bohner & Peerson [4, Theorem 1.73]. Lemma 4.2 (improper parameer inegrals): Assume ha T is unbounded o he righ, Φ : T T L(X ) is a coninuous mapping, r C rd (T, X ) and (i) for arbirary T i is Φ(, s)r(s) s <, (ii) Φ(, s) is rd-coninuously differeniable, where he limi ( 1 Φ)( 0, s) = lim 0 Φ(, s) Φ( 0, s) µ(, 0 ) exiss uniformly in s on compac subses of T in each righ-dense 0 T, (iii) here exiss a locally bounded funcion c : T R + 0 and a rd-coninuous funcion m : T R + 0 wih T m(s) s < such ha ( 1Φ)(, s)r(s) c()m(s) for s, T. Then he mapping F : T X, F () := Φ(, s)r(s) s is differeniable wih derivaive F () = Φ(ρ + (), )r() + ( 1 Φ)(, s)r(s) s. Proof. The proof can be done using argumens from o he well-known case of he ime scale T = R, which can be found in sandard calculus exbooks on he Riemann inegral. References [1] B. Aulbach and S. Hilger, Linear dynamic processes wih inhomogeneous ime scale, in Nonlinear Dynamics and Quanum Dynamical Sysems, G. A. Leonov, V. Reimann, W. Timmermann, eds., Mahemaical Research 59, Akademie-Verlag, Berlin, 1990, pp [2] B. Aulbach and J. Kalkbrenner, Exponenial forward spliing for noninverible difference equaions, Compuers & Mahemaics wih Applicaions (o appear). [3] M. Bohner and D. A. Luz, Asympoic behavior of dynamic equaions on ime scales, Journal of Difference Equaions and Applicaions, 7(1), 2001, pp

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