Integral Tranform and Special Function, 2013 Vol. 24, No. 4, 289 301, http://d.doi.org/10.1080/10652469.2012.689300 Further propertie of the Laplace tranform on time cale with arbitrary grainine Martin Bohner a *, Guein Sh. Gueinov b and Başak Karpuz c a Department of Mathematic and Statitic, Miouri S&T, Rolla, MO 65409-0020, USA; b Department of Mathematic, Atilim Univerity, 06836 Incek, Ankara, Turkey; c Department of Mathematic, Afyon Kocatepe Univerity, 03200 Afyonkarahiar, Turkey (Received 7 March 2012; final verion received 25 April 2012) In thi work, we generalize everal propertie of the uual Laplace tranform to the Laplace tranform on arbitrary time cale. Among them are tranlation theorem, tranform of periodic function, integration of tranform, tranform of derivative and integral, and aymptotic value. Keyword: time cale; Laplace tranform; time cale eponential function; tranlation theorem AMS Subject Claification: 44A10; 34N05; 26E70; 39A10; 39A12; 39A13 1. Introduction If f : [0, ) C i a given function and if the integral F(z) L{f }(z) 0 f (τ) e zτ dτ eit, then the function F of a comple variable i called the Laplace tranform of the original function f. The operation which yield F from a given f i alo called the Laplace tranform (or Laplace tranformation). A large cla of function poee a Laplace tranform. One ay that a function f ha eponential order α if there eit real contant K > 0 and α uch that for ome t 0 0, f (t) K e αt, t t 0. A function f i piecewie continuou on the interval [0, ) if f i continuou on every finite interval [0, b) ecept poibly at a finite number of point in [0, b) at which f ha a jump dicontinuity. If f i piecewie continuou on [0, ) and of eponential order α, then the Laplace tranform L{f }(z) eit for Rez >α. Net, we will aume that f (t) 0 for t < 0. *Correponding author. Email: bohner@mt.edu 2013 Taylor & Franci
290 M. Bohner et al. For convenience of the reader, let u lit here (following [14]) everal propertie of the Laplace tranform which we will generalize in thi paper to the Laplace tranform on arbitrary time cale. 1. There are two imple and ueful tranlation theorem for the Laplace tranform. The firt pertain to a tranlation in the z-domain and the econd to a tranlation in the t-domain. The firt tranlation theorem tate that if F(z) L{f (t)}(z) for Rez > 0, then for any real number a, F(z a) L{e at f (t)}(z) for Rez > a. The econd tranlation theorem tate that if F(z) L{f (t)}(z), then L{u a (t)f (t a)}(z) e az F(z), a 0, where u a i the o-called unit tep function defined by { 1 ift > a, u a (t) 0 if t < a, for a 0. The function u a i alo known a the Heaviide function. 2. If F(z) L{f }(z) and f i a periodic function of period T > 0, then 1 T F(z) f (τ) e zτ dτ. 1 e zt 0 3. If f i piecewie continuou on [0, ) and of eponential order α, with F(z) L{f }(z) and uch that lim t 0 + f (t)/t eit, then { } f (t) F(y) dy L () for >α. t 4. In order to olve differential equation, it i neceary to know the Laplace tranform of derivative of a function f.iff, f,..., f (n 1) are continuou on [0, ) and of eponential order, while f (n) i piecewie continuou on [0, ), then n 1 L{f (n) }(z) z n L{f }(z) z n 1 k f (k) (0). 5. If f i piecewie continuou on [0, ) and of eponential order α 0, then { } L f (τ) dτ (z) 1 L{f (t)}(z) for Rez α. z 0 6. If f i continuou on [0, ) and of eponential order α, and f i piecewie continuou on [0, ), then f (0) lim {zf(z)}, z,z R where F(z) L{f }(z). 7. If f i continuou on [0, ) and of eponential order α and if furthermore lim t f (t) eit, then lim f (t) lim {zf(z)}, t z 0,z R where F(z) L{f }(z). k0
Integral Tranform and Special Function 291 The concept of Laplace tranform on time cale wa introduced in [11] and further developed in [1,4 6,8]. More pecific reult are etablihed for the Laplace tranform on iolated time cale [2,3]. See alo [13] for a reult related to uniquene of the Laplace tranform. Thi paper i a continuation of the author work [4] and generalize the above formulated propertie of the uual Laplace tranform to the Laplace tranform on arbitrary time cale. 2. Summary of the reult and notation from [4] For the reader convenience, we ummarize in thi ection the main reult and notation from [4]. For the tandard theory of time cale calculu, we refer to [5,7,10]. A function f C rd (T, C) i called regreive if 1 + μf 0onT, and poitively regreive if it i real valued and 1 + μf > 0onT. The et of regreive function and the et of poitively regreive function are denoted by R(T, C) and R + (T, R), repectively, and R (T, R) i defined imilarly. For implicity, we denote by R c (T, C) the et of comple regreive contant and, imilarly, we define the et R + c (T, R) and R c (T, R). Let f R(T, C). Then the eponential function e f (, ) i defined to be the unique olution of the initial value problem f, () 1 for ome fied T. Forh > 0, et { C h : z C : z 1 } {, Z h : z C : π h h < Im(z) π }, h and C 0 : Z 0 : C.Forh > 0, the Hilger real part and imaginary part of a comple number are given by Re h (z) : 1 h ( 1 + hz 1) and Im h(z) : 1 Arg(1 + hz), h repectively, where Arg denote the principle argument function. If we define m λ (t, ) : then the following reult hold (ee [4, Lemma 3.1]). 1 1 + μ(τ)λ τ for, t T and λ R c(t, R), Lemma 2.1 Aume up T. Let T and λ R + c ([, ) T, R). Then we have: (i) m 1 λ (t, ) >0 for all t (, ) T, where 1 denote the delta differentiation with repect to t. (ii) lim t m λ (t, ). For h > 0, let R h : C h R { λ R : λ 1 }, h and let R 0 R. Define the minimal grainine function μ : T R + 0 by and for h 0, we define μ () : The net reult i given in [4, Theorem 3.4]. inf μ(τ) for T, τ [, ) T C h (λ) : {z C h :Re h (z) >λ}.
292 M. Bohner et al. Theorem 2.2 Aume up T. Let T and λ R + c ([, ) T, R). Then for any z C μ ()(λ), we have the following propertie: (i) e λ z (t, ) e λ Reμ () (z)(t, ) for all t [, ) T. (ii) lim t e λ Reμ () (z)(t, ) 0. (iii) lim t e λ z (t, ) 0. Throughout thi paper, we aume that up T. The net definition and reult were given in [4, Definition 4.1 and Lemma 4.4]. Definition 2.3 Let T. A function f C rd (T, C) ha eponential order α on [, ) T if (i) α R + c ([, ) T, R), (ii) there eit K > 0 uch that f (t) Ke α (t, ) for all t [, ) T. Lemma 2.4 Let T and f C rd ([, ) T, C) be a function of eponential order α. Then lim f (t)e z(t, ) 0, (1) t where z C μ ()(α). Now we give the definition of the Laplace tranform (ee [4, Definition 4.5]). Definition 2.5 Let f C rd (T, C) be a function. Then the Laplace tranform about the point T of the function f i defined by F(z; ) : L{f }(z; ) : f (τ)e z (σ (τ), )τ for z D, where D C conit of comple number z R c (T, C) for which the improper integral converge. The function f i referred to a the determining function, and F( ; ) i called the generating function about the point T. The following reult about the Laplace tranform were proved in [4, Theorem 5.1, Corollary 5.2, Theorem 5.3, and Theorem 5.4]. Theorem 2.6 Let f C rd ([, ) T, C) be of eponential order α. Then the Laplace tranform L{f }( ; ) eit on C μ ()(α) and converge abolutely. Corollary 2.7 Let f C rd ([, ) T, C) be of ome eponential order. Then lim Re(z) L{f } (z; ) 0 ince Re(z) implie Re μ ()(z). If μ () >0, then z implie Re μ ()(z). Therefore, lim z L{f }(z; ) 0 in the cae μ () >0. Theorem 2.8 Let f C rd ([, ) T, C) be of eponential order α. Then the Laplace tranform L{f } converge uniformly in the half-plane C μ ()(β), where β>α. Theorem 2.9 Let f 1, f 2 C rd ([, ) T, C) be of eponential order α 1, α 2, repectively. Then for any c 1, c 2 R, we have L{c 1 f 1 + c 2 f 2 }c 1 L{f 1 }+c 2 L{f 2 } on C μ ()(ma{α 1, α 2 }).
Integral Tranform and Special Function 293 3. Tranlation theorem We tart thi ection with the following reult, which will play a major role in the proof of the firt tranlation theorem for time cale with contant grainine. However, thi reult itelf cannot be called a tranlation theorem ince a uch it hould in general contain ome tranlation in the z-domain. Lemma 3.1 Let f C rd ([, ) T, C) be a function of eponential order α. Then L{f }(z; ) L{fe σ w z (, )}(w; ) for all z C μ ()(α), where w R c ([, ) T, C). Proof Let z C μ ()(α) and w R c ([, ) T, C). Then we have L{f }(z; ) f (τ)e z (σ (τ), )τ f (τ)e (w z) w (σ (τ), )τ f (τ)e w z (σ (τ), )e w (σ (τ), )τ L{fe σ w z (, )}(w; ). Thu, the proof i completed. Note that if the grainine i a contant function, i.e. μ(t) h 0 for all t T, we ee that for z, w R c (T, C), z w and z w are independent of the time cale variable t. Hence, we have the following theorem. Theorem 3.2 (Firt tranlation theorem) Suppoe that the time cale T ha contant grainine, i.e. μ(t) h 0 for all t T. Let f C rd ([, ) T, C) be a function of eponential order α. Then L{f }(z w; ) L{fe σ w (, )}(z; ) for all z C h(α w ), where w R c ([, ) T, C). Proof It follow from the proof of Lemma 3.1 that L{f }(z w; ) L{fe σ w (, )}(z; ) for all z w C h(α). On the other hand, uing z w C h (α),wehave α<re h (z w) Re h (z) Re h (w) Re h(z) Re h (w) 1 + hre h (w) Re h(z) w 1 + h w ince 1 + hre h (w) 0 and Re h (w) lim r Re r (w) w (note that for any fied z C, the Hilger real part Re h (z) i a nondecreaing function of h). Thi implie α w < Re h (z), i.e. z C μ ()(α w ). The proof i complete.
294 M. Bohner et al. Theorem 3.3 (Second tranlation theorem) Let f C rd ([, ) T, C) be a function of eponential order α and ϕ C 1 rd ([, ) T, T) be a trictly increaing function uch that ϕ(t) t for all t [, ) T and ϕ([, ) T ) [ϕ(), ) T. Define by (z; t) : e z(σ (ϕ 1 (t)), σ(t)) ϕ (ϕ 1 (t)) for z C μ ()(α) and t [ϕ(), ) T. (2) Then L{(χf ) ϕ}(z; ) L{(z; )f }(z; ) for all z C μ ()(α), where χ i the characteritic function of the interval [, ) T. Proof Clearly, (χf ) ϕ 0on(, ϕ 1 ()) T and (χf ) ϕ f ϕ on [ϕ 1 (), ) T.Now, we can compute L{(χf ) ϕ}(z; ) ϕ 1 () ϕ 1 () ϕ 1 () (χf )(ϕ(τ))e z (σ (τ), )τ f (ϕ(τ))e z (σ (τ), )τ ϕ (τ) e z(σ (τ), σ (ϕ(τ))) f (ϕ(τ))e ϕ z (σ (ϕ(τ)), )τ (τ) ϕ (τ)(z; ϕ(τ))f (ϕ(τ))e z (σ (ϕ(τ)), )τ (3) for all z C μ ()(α). Keeping in mind that ϕ hift the half-ray [, ) T onto [ϕ(), ) T,an application of [5, Theorem 1.98] to (3) yield L{(χf ) ϕ}(z; ) for all z C μ ()(α). The proof i completed. (z; ζ)f (ζ )e σ z (ζ, )ζ L{(z; )f }(z; ) Note that Theorem 3.3 i particularly ueful when the function (z; t) defined by (2) doe not depend on t [, ) T. Denoting it in thi cae by (z; ) or imply by (z), we obtain L{χ(ϕ(t))f (ϕ(t))}(z; ) (z)l{f (t)}(z; ) for z C μ ()(α). Eample 3.4 Let T R and a be a poitive real number. Letting ϕ(t) t a,wehaveϕ 1 (t) t + a, ϕ (t) 1, and we ee that ( +a ) (z; t) e z (t + a, t) ep z dτ e az. Therefore, Theorem 3.3 give L{χ(t a)f (t a)} e az L{f (t)}(z; ). Eample 3.5 Let T hz with h > 0. Let n be a fied poitive integer and let ϕ(t) t nh. Then ϕ 1 (t) t + nh, ϕ (t) 1, and taking into account that for T hz we have (ee [2]) e z (t, t 0 ) (1 + hz) (t t 0)/h, t we ee that (z; t) e z (t + (n + 1)h, t + h) (1 + hz) n.
Integral Tranform and Special Function 295 Therefore, Theorem 3.3 give L{χ(t nh)f (t nh)}(z; ) (1 + hz) n L{f (t)}(z; ). Eample 3.6 Let T be an ω-periodic time cale, i.e. T + ω T, and T. Some eample of uch a time cale are R, Z, and hz for h > 0. Then letting ϕ(t) t nω for ome n N, one may ee that (z; ) (z; ) on [ϕ(), ) T, and thu L{(χf ) ϕ}(z; ) (z; )L{f }( ; ) hold on C m (α), where m : inf t [,+ω)t μ(t). Eample 3.7 Let T P a,b : k Z [k(a + b), k(a + b) + a] R, where a, b > 0 and ϕ(t) t n(a + b) for ome n N. Fork 1, 2,..., n, denote by r k (t) the right-cattered point in [t, t + n(a + b)] T uch that r k+1 (t) >r k (t) for k 1, 2,..., n 1. Let t T. Ifμ(t) 0, then (z; t) e z ((t + n(a + b), t) { r1 (t) n 1 ep ( z) dτ + and if μ(t) >0, then n k1 Hence, in either cae, we have (z; t) t 1 1 + μ(r k (t))z, k1 k1 σ(r k (t)) rk+1 (t) σ(r k (t)) ( z) dτ + k1 +n(a+b) r n (t) (z; t) e z (r n+1 (t), r 1 (t)) { n } rk+1 (t) n 1 ep ( z) dτ 1 + μ(r k (t))z. ( z) dτ e zna (1 + bz) n (z;0) for all z R c(p a,b, C) C b and t [0, ) Pa,b, where we have ued the convention that r n+1 (t) : t + n(a + b) if μ(t) >0. Note that μ(t) μ(t + n(a + b)), σ(r k (t)) r k (t) + b and r k+1 (t) σ(r k (t)) a for all t [0, ) Pa,b and k 1, 2,..., n. Hence, Theorem 3.3 and Eample 3.6 imply that L{(χf ) ϕ}(z; ) provided that f i of eponential order α. e zna (1 + bz) n L{f }(z; ) for all z C μ ()(α) } 4. Tranform of periodic function Definition 4.1 Let T be fied and f C rd ([, ) T, C) be a function and ϕ C 1 rd ([, ) T, T) be a trictly increaing function uch that ϕ(t) t for all t [, ) T and ϕ([, ) T ) [ϕ(), ) T.Iff f ϕ hold on [ϕ 1 (), ) T, then f i aid to be generalized ϕ-periodic on [, ) T. The following reult concern the Laplace tranform of a function atifying the generalized periodicity property.
296 M. Bohner et al. Theorem 4.2 Let ϕ and be a in Theorem 3.3. Let f C rd ([, ) T, C) be a generalized ϕ-periodic function of eponential order α. Then, we have L{(1 (z; ))f }(z; ) ϕ 1 () f (τ)e z (σ (τ), )τ for all z C μ ()(α). (4) Proof We can eaily calculate by following tep imilar to thoe in the proof of Theorem 3.3 that and ϕ 1 () L{f }(z; ) ϕ 1 () f (τ)e z (σ (τ), )τ f (τ)e z (σ (τ), )τ f (τ)e z (σ (τ), )τ + ϕ 1 () ϕ 1 () ϕ 1 () f (ϕ(τ))e z (σ (τ), )τ f (τ)e z (σ (τ), )τ ϕ (τ)(z; ϕ(τ))f (ϕ(τ))e z (σ (ϕ(τ)), )τ (z; ζ)f (ζ )e σ z (ζ, )ζ L{(z; )f }(z; ) hold for all z C μ ()(α), from which (4) follow. Eample 4.3 Let T be a ω-periodic time cale and T. Then letting ϕ(t) t nω for ome n N, one may ee that (z; ) (z; ) on [ϕ(), ) T.If( ; ) 1onC m (α), where m : inf t [,+ω)t μ(t), then L{f }(z; ) 1 1 (z; ) ϕ 1 () f (τ)e z (σ (τ), )τ for all z C m (α). 5. Integration of tranform Now we hall prove the formula for the integral of the tranform of a function. To thi end, we introduce the notation v(; t, ) : Theorem 5.1 (Integral of the tranform) order α and let the integral e y (t, ) dy for R + c ([, ) T, R) and t [, ) T. eit (finite) for each fied R μ ()(α). Then we have Let f C rd ([, ) T, R) be a function of eponential e α (τ, )v(; σ(τ), )e (σ (τ), )τ (5) L{f }(y; ) dy L{v(; σ( ), )f }(; ) for all R μ ()(α). (6)
Proof We have Integral Tranform and Special Function 297 r r lim L{f }(y; ) dy lim r r f (τ)e y (σ (τ), )τ dy. A L{f }(; ) converge uniformly for all R μ ()(β) R μ ()(α), we are allowed to revere the order of the integration. The dominated convergence theorem for -integral (ee [9]) and (5) imply that lim r r L{f }(y; ) dy lim r r f (τ) f (τ) lim r ( r e y (σ (τ), ) dyτ ) e y (σ (τ), ) dy e (σ (τ), )τ f (τ)v(; σ(τ), )e (σ (τ), )τ for all R μ ()(α) ince we may pick β ufficiently cloer to α. Thi prove (6) and complete the proof. By induction, we have the following reult. Corollary 5.2 Let f C rd ([, ) T, R) be a function of eponential order α and let the integral e α (τ, )v k (; σ(τ), )e (σ (τ), )τ for all k [1, n] N eit (finite) for each fied R μ ()(α) and ome n N. Then we have where (y ) n L{f }(y; ) dy n!l{v n (; σ( ), )f }(; ) for all R μ ()(α), (7) for R c (T, R) and, t T. 1, n 0 v n (; t, ) : v n 1 (y; t, )e y (t, ) dy, n N (8) 6. Tranform of derivative and integral Firt, following [5, Section 1.6], let u recall the generalized monomial h k : T T T, k N 0, defined recurively by It follow that where h 1 k h 0 (t, ) 1, h k (t, ) h k 1 (τ, )τ for all k N and, t T. (9) h 1 k (t, ) h k 1(t, ) for all k N, (10) denote the -derivative of h k with repect to t. The definition (9) obviouly implie h 1 (t, ) t and h 2 (t, ) and hence, finding h k eplicitly for k 2 i not eay in general. (τ )τ for all, t T,
298 M. Bohner et al. For all k N 0 and, t T with t,wehave 0 h k (t, ) and for any comple contant α and t T with t, e α (t, ) (t )k, k! α k h k (t, ), (11) k0 where the erie converge uniformly with repect to α and t in bounded region (ee [1]). Therefore, e α (t, ) e α (t, ) e α (t ). Note that if a given function f : T C i of ome eponential order, it derivative f need not be of ome eponential order (for eample, T R, f (t) in(t 2 )). However, if f i of ome eponential order, then f itelf i of ome eponential order too. Namely, the following lemma hold. Lemma 6.1 If f i of eponential order α, then f i of eponential order α if α 0, and of eponential order ε for any ε>0if α 0. Proof Suppoe that α R c ([, ) T, R) and f (t) Ke α (t, ) for all t [, ) T. By [4, Lemmma 3.2], we have e α (t, ) >0 for all t [, ) T and it follow from (11) that e α (t, ) e α (t, ), t [, ) T. Therefore, in the cae α 0, auming K i choen uch that f () K/ α, wehave f (t) f (τ)τ + f () f (τ) τ + f () K K α e α (τ, )τ + f () K e α (τ, )τ + f () e K τ α (τ, )τ + f () α (e α (t, ) 1) + f () K α (e α (t, ) 1) + K α K α e α (t, ). If α 0, then taking into account that e 0 (t, ) 1 and e ε (t, ) h 0 (t, ) + εh 1 (t, ) 1 + ε(t ) for t [, ) T,wehave f (t) K C ε e ε (t, ) e 0 (τ, )τ + f () K(t ) + f () for ome contant C ε. In order to olve dynamic equation, it i ueful to know the Laplace tranform of the derivative function f in term of the Laplace tranform of the function f.
Integral Tranform and Special Function 299 Theorem 6.2 (Tranform of the derivative function) eponential order α. Then L{f }(z; ) zl{f }(z; ) f () Let f C 1 rd ([, ) T, C) be a function of for all z C μ ()(α). Proof Performing an integration by part and uing Lemma 2.4 yield that L{f }(z; ) f (τ)e z (σ (τ), )τ [f (t)e z (t, )] t t f () + z f () + zl{f }(z; ) f (τ)e z (σ (τ), )τ f (τ)( z)(τ)e z (τ, )τ hold for all z C μ ()(α). Thi complete the proof. By induction, one can prove the following. Corollary 6.3 For ome n N, let f C n rd ([, ) T, C) and f, f,..., f n 1 be function of eponential order α. Then n 1 L{f n }(z; ) z n L{f }(z; ) z n 1 k f k () k0 for all z C μ ()(α). An important feature of the derivative theorem i that we obtain L{f } without requiring that f itelf be of eponential order. Theorem 6.4 (Tranform of the integral function) eponential order α. Then L{F(, )}(z; ) 1 L{f }(z; ) z for Let f C rd ([, ) T, C) be a function of all z C μ ()( α ), where F(t, ) : f (τ)τ for, t T. Proof Since f i of eponential order α and F 1 (, ) f, it follow from Lemma 6.1 that F(, ) i of eponential order α +ε for any ε>0. Applying Theorem 6.2 to the function F(, ) complete the proof. By induction, we can obtain the following corollary. Corollary 6.5 Let f C rd ([, ) T, C) be a function of eponential order α. Then L{F n (, )}(z; ) 1 L{f }(z; ) z for n all z C μ ()( α ), where F n (t, ) : h n 1 (t, σ(τ))f (τ)τ for, t T and n N.
300 M. Bohner et al. Proof The proof make ue of mathematical induction and the formula τn τ2 f (τ 1 )τ 1 τ 2 τ n h n 1 (t, σ(τ))f (τ)τ (12) for, t T, which hold for the n-fold integration. Note that the formula (12) appear in [12, Lemma 1]. However, the proof of (12) given in [12] require the continuity of the function h n (t, σ(τ))f (τ) with repect to (t, τ) a the proof ue [5, Theorem 1.117]. On the other hand, f and the forward jump function σ are in general only rd-continuou. Let u preent here a proof of (12) which ue only the rd-continuity of f : Applying the integration by part formula [5, Theorem 1.77(v)], we can write { τ h n 1 (t, σ(τ))f (τ)τ h n 1 (t, σ(τ)) τ h n 1 (t, τ) f (ζ )ζ where we have ued the equation Thu, we have obtained the formula h n 1 (t, σ(τ))f (τ)τ h τ n 1 (t, τ) { τ { τ h n 2 (t, σ(τ)) f (ζ )ζ } τ τt τ f (ζ )ζ } τ } f (ζ )ζ τ, h τ n 1 (t, τ) h n 2(t, σ(τ)). h τ n 1 (t, τ) { τ } f (ζ )ζ τ { τ } h n 2 (t, σ(τ)) f (ζ )ζ τ (13) for all n 2. Iterating formula (13) and taking into account that h 0 (t, σ(τ)) 1, we obtain (12). 7. Aymptotic value Theorem 7.1 (Initial value) eponential order. Then Let f C 1 rd ([, ) T, C) be a function uch that f i of ome lim zl{f }(z; ) f (). Re(z) Proof Due to the general property given in Corollary 2.7 that Laplace tranform tend to zero aymptotically, i.e. lim Re(z) L{f }(z; ) 0, we obtain by Theorem 6.2 that 0 lim L{f }(z; ) lim (zl{f }(z; ) f ()). Re(z) Re(z) Therefore, the proof i completed.
Integral Tranform and Special Function 301 Theorem 7.2 (Terminal value) Aume that f C 1 rd ([, ) T, C) i a function uch that l : lim t f (t) eit and i finite. Then lim zl{f }(z; ) l. z 0 Proof Firt, note that f i bounded by the hypothei and i, therefore, of eponential order 0. By the boundedne of f together with it finite limit at infinity, we infer that lim t f (t) 0 hold, and thu f i alo bounded and of eponential order 0, too. Then, from Theorem 6.2, we have and thi complete the proof. lim zl{f }(z; ) lim L{f }(z; ) + f () z 0 z 0 lim z 0 f (τ)e z (σ (τ), )τ + f () f (τ)τ + f () l, Reference [1] M. Bohner and G.Sh. Gueinov, The convolution on time cale, Abtr. Appl. Anal. 2007 (2007), pp. 1 24, Art. ID 54989. [2] M. Bohner and G.Sh. Gueinov, The h-laplace and q-laplace tranform, J. Math. Anal. Appl. 365(1) (2010), pp. 75 92. [3] M. Bohner and G.Sh. Gueinov, The Laplace tranform on iolated time cale, Comput. Math. Appl. 60(6) (2010), pp. 1536 1547. [4] M. Bohner, G.Sh. Gueinov, and B. Karpuz, Propertie of the Laplace tranform on time cale with arbitrary grainine, Integ. Tran. Spec. Funct. 22(11) (2011), pp. 785 800. [5] M. Bohner and A. Peteron, Dynamic Equation on Time Scale: An Introduction with Application, Birkhäuer, Boton, MA, 2001. [6] M. Bohner and A. Peteron, Laplace tranform and Z-tranform: Unification and etenion, Meth. Appl. Anal. 9(1) (2002), pp. 151 157. Preprint in Ulmer Seminare 6. [7] M. Bohner and A. Peteron, Advance in Dynamic Equation on Time Scale, Birkhäuer, Boton, MA, 2003. [8] J.M. Davi, I.A. Gravagne, B.J. Jackon, R.J. Mark II, and A.A. Ramo, The Laplace tranform on time cale reviited, J. Math. Anal. Appl. 332(2) (2007), pp. 1291 1307. [9] G.Sh. Gueinov, Integration on time cale, J. Math. Anal. Appl. 285 (2003), pp. 107 127. [10] S. Hilger, Analyi on meaure chain a unified approach to continuou and dicrete calculu, Reult Math. 18 (1990), pp. 18 56. [11] S. Hilger, Special function, Laplace and Fourier tranform on meaure chain, Dynam. Syt.Appl. 8(3 4) (1999), pp. 471 488. Special Iue on Dicrete and continuou hamiltonian ytem, edited by R. P. Agarwal and M. Bohner. [12] B. Karpuz, Unbounded ocillation of higher-order nonlinear delay dynamic equation of neutral type with ocillating coefficient, Electron. J. Qual. Theory Differ. Equ. 2009(34), pp. 1 14. [13] B. Karpuz, On uniquene of the Laplace tranform on time cale, Panamer. Math. J. 21(2) (2011), pp. 101 110. [14] J.L. Schiff, The Laplace Tranform: Theory and Application, Undergraduate Tet in Mathematic, Springer-Verlag, New York, 1999.