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1

Introduction

We consider generalized time scales (T, α) as introduced in [1], i.e., T ⊂ R is a nonempty set such that every Cauchy sequence in T converges to a point in T (with the possible exception of Cauchy sequences that converge to a finite infimum or supremum of T), and α is a function that maps T into T. A function f : T → R is called alpha differentiable at a point t ∈ T if there exists a number fα (t), the so-called alpha derivative of f at t, with the property that for every ε > 0 there exists a neighborhood U of t such that |f (α(t)) − f (s) − fα (t)(α(t) − s)| ≤ ε|α(t) − s| is true for all s ∈ U . If T is closed and α = σ, the Hilger forward jump operator, then fα = f ∆ is the usual delta derivative (see [4, 6, 7]), which contains as special cases derivatives f 0 (if T = R) and differences ∆f (if

232 E. Akin-Bohner and M. Bohner T = Z). If T is closed and α = ρ, the Hilger backward jump operator, then fα = f ∇ is the nabla derivative (see [3] and [4, Section 8.4]). In this paper we consider linear alpha dynamic equations of the form yα = p(t)y

with

1 + p(t)µα (t) 6= 0,

where µα (t) = α(t)−t is the generalized graininess. If the initial value problem yα = p(t)y,

y(t0 ) = 1

has a unique solution, we denote it by ep (t, t0 ) and call it the generalized exponential function. Note that ep also depends on α, but we choose not to indicate this dependence as it should be clear from the context. The exponential function satisfies some properties, which are presented in the next section of this paper. Similarly as in [5], the exponential function may be used to define a generalized Laplace transform, which is helpful when solving higher order linear alpha dynamic equations with constant coefficients. We illustrate this technique with an example in the last section. This example features an α which neither satisfies α(t) ≥ t for all t ∈ T nor α(t) ≤ t for all t ∈ T, and hence this example can not be accommodated in the existing literature on delta and nabla dynamic equations.

2

Alpha Derivatives, Exponentials, and Laplace Transforms

For a function f : T → R we denote by fα the alpha derivative as defined in the introductory section, and we also put f α = f ◦ α. Then the following rules (see [4, Section 8.3]) are valid: • f α = f + µα fα ; • (f g)α = f gα + fα g α (“Product Rule”); µ ¶ fα g − f gα f • = (“Quotient Rule”). g α gg α We may use these rules to find ep (α(t), t0 ) = eα p (t, t0 ) = ep (t, t0 ) + µα (t)p(t)ep (t, t0 ), i.e., • ep (α(t), t0 ) = [1 + p(t)µα (t)] ep (t, t0 ), by putting y(t) = ep (t, t0 )eq (t, t0 ), yα (t) i.e.,

= ep (t, t0 )q(t)eq (t, t0 ) + p(t)ep (t, t0 )eq (α(t), t0 ) = [p(t) + q(t) + µα (t)p(t)q(t)] y(t),

Exponential Functions and Laplace Transforms for Alpha Derivatives

• ep eq = ep⊕q ,

233

where p ⊕ q := p + q + µα pq,

and by putting y = ep (t, t0 )/eq (t, t0 ), yα (t)

= =

p(t)ep (t, t0 )eq (t, t0 ) − ep (t, t0 )q(t)eq (t, t0 ) eq (t, t0 )eq (α(t), t0 ) p(t) − q(t) y(t), 1 + µα (t)q(t)

i.e., •

ep = epªq , eq

where

p ª q :=

p−q . 1 + µα q

Note also that ªq := 0 ª q = −q/(1 + µα q) satisfies q ⊕ (ªq) = 0 and that p ª q = p ⊕ (ªq). Again, ⊕ and ª depend on α, but in order to avoid many subscripts we choose not to indicate this dependence as it should be clear from the context. We also remark that the set of alpha regressive functions Rα = {p : T → R| 1 + p(t)µα (t) 6= 0 for all t ∈ T} is an Abelian group under the addition ⊕, and ªp is the additive inverse of p ∈ Rα . Now, similarly as in [5], the Laplace transform for functions x : T → R (from now on we assume that T is unbounded above and contains 0) may be introduced as Z ∞ L{x}(z) = x(t)eα ªz (t, 0)dα t with z ∈ Rα ∩ R, 0

whenever this Cauchy alpha integral is well defined. As an example, we calculate L{ec (·, 0)}, where c ∈ Rα is a constant such that limt→∞ ecªz (t, 0) = 0. Then Z ∞ L{ec (·, 0)}(z) = ec (t, 0)eα ªz (t, 0)dα t 0 Z ∞ = [1 + µα (t)(ªz)(t)] ec (t, 0)eªz (t, 0)dα t ¸ Z0 ∞ · µα (t)z = 1− ecªz (t, 0)dα t 1 + µα (t)z Z0 ∞ 1 = ecªz (t, 0)dα t 1 + µ α (t)z 0 Z ∞ 1 = (c ª z)(t)ecªz (t, 0)dα t c−z 0 Z ∞ 1 = (ecªz (·, 0))α dα t c−z 0 1 . = z−c Under appropriate assumptions we can also show

234 E. Akin-Bohner and M. Bohner • L{xα }(z) = zL{x}(z) − x(0); • L{xαα }(z) = z 2 L{x}(z) − zx(0) − xα (0); Z t 1 x(τ )dα τ . • L{X}(z) = L{x}(z), where X(t) = z 0 Further results can be derived as in [4, Section 3.10].

3

An Example

To illustrate the use of our Laplace transform, we consider the initial value problem yαα − 5yα + 6y = 0, y(0) = 1, yα (0) = 5. By formally taking Laplace transforms, we find 0 =

z 2 L{y}(z) − zy(0) − yα (0) − 5 [zL{y}(z) − y(0)] + 6L{y}(z)

= (z 2 − 5z + 6)L{y}(z) − z = (z − 2)(z − 3)L{y}(z) − z so that L{y}(z) =

3 2 z = − = L{3e3 (·, 0) − 2e2 (·, 0)}(z). (z − 2)(z − 3) z−3 z−2

Hence, if e2 (·, 0) and e3 (·, 0) exist, we let y = 3e3 (·, 0) − 2e2 (·, 0), and then yα = 9e3 (·, 0) − 4e2 (·, 0)

and yαα = 27e3 (·, 0) − 8e2 (·, 0)

so that indeed y(0) = 3 − 2 = 1, yα (0) = 9 − 4 = 5, and yαα − 5yα + 6y = 0. Let us now consider several special cases of this example. (a) T = R and α(t) = t for all t ∈ T. Then ec (t, 0) = ect for any constant c ∈ R, and the solution is given by y(t) = 3e3t − 2e2t . (b) T = N0 and α(t) = 2t + 1 for all t ∈ T. Note that ec (·, 0) is only defined on {tm = 2m − 1| m ∈ N0 } ⊂ T. Since µα (t) = t + 1, we find that ec satisfies ec (tk+1 , 0) = (1 + cµα (tk ))ec (tk , 0) = (1 + c2k )ec (tk , 0),

Exponential Functions and Laplace Transforms for Alpha Derivatives

235

Table 1: y = 3e3 (·, 0) − 2e2 (·, 0) for (b) t 0 1 3 7 15 31

e2 (t, 0) 1 3 15 135 2295 75735

e3 (t, 0) 1 4 28 364 9100 445900

y(t) 1 6 54 822 22710 1186230

yα (t) 5 24 192 2736 72720

yαα (t) 19 84 636 8748

Table 2: y = 3e3 (·, 0) − 2e2 (·, 0) for (c) t -6 -4 -2 0 2 4

e2 (t, 0) -27 9 -3 1 -0.¯3 0.¯1

e3 (t, 0) -125 25 -5 1 -0.2 0.04

y(t) -321 57 -9 1 0.0¯6 -0.10¯2

yα (t)

yαα (t)

189 -33 5 -0.4¯6 -0.08¯4

-111 19 -2.7¯3 0.19¯1

and hence we obtain for constant c ∈ Rα ec (tm , 0) =

m−1 Y

(1 + c2k ).

k=0

See Table 1 for some numeric values. Note that yαα − 5yα + 6y = 0 in each row. (c) T = Z and α(t) = t − 2 for all t ∈ T. Note that ec (t, 0) is only defined for all even integers. Since µα (t) ≡ −2, we find that ec satisfies ec (α(t), 0) = (1 + cµα (t))ec (t, 0) = (1 − 2c)ec (t, 0), and hence we obtain for constant c 6= 1/2 ec (t, 0) = (1 − 2c)−t/2 . See Table 2 for some numeric values. (d) T = Z and α(t) = t + 1 + 2(−1)t for all t ∈ T. Note that while in the previous two examples α(t) ≥ t for all t ∈ T and α(t) ≤ t for all t ∈ T, respectively, none of these properties hold in the current example. This time α : T → T is additionally a bijection and hence ec (t, 0) is defined on the entire set T when c 6∈ {−1/3, 1}. We have n 3 if t is even µα (t) = 1 + 2(−1)t = −1 if t is odd.

236 E. Akin-Bohner and M. Bohner

Table 3: y = 3e3 (·, 0) − 2e2 (·, 0) for (d) t 0 1 2 3 4 5 6 7 8 9

e2 (t, 0) 1 -1 -7 7 49 -49 -343 343 2401 -2401

e3 (t, 0) 1 -0.5 -20 10 400 -200 -8000 4000 160000 -80000

y(t) 1 0.5 -46 16 1102 -502 -23314 11314 475198 -235198

yα (t) 5 -0.5 -152 62 3404 -1604 -70628 34628

yαα (t) 19 -5.5 -484 214 10408 -5008

As an example we calculate e2 (t, 0) for some values of t. Since e2 (0, 0) = 1 and α(0) = 3, we find e2 (3, 0) = e2 (α(0), 0) = (1 + 2µα (0))e2 (0, 0) = 7. Next, e2 (2, 0) = e2 (α(3), 0) = (1 + 2µα (3))e2 (3, 0) = −7, and similarly, e2 (5, 0) = −72 ,

e2 (4, 0) = 72 ,

e2 (7, 0) = 73 ,

and so on. In general, we find ( {(1 − c)(1 + 3c)}t/2 ec (t, 0) = (1 − c)(t−3)/2 (1 + 3c)(t−1)/2

e2 (6, 0) = −73

if t is even if t is odd,

which can be written in closed formula as ec (t, 0) =

{(1 − c)(1 + 3c)}bt/2c {(1 − c)(1 + 3c)}bt/2c = , (1 − c)χ(t) (1 − c)dt/2−bt/2ce

where χ = χ2Z+1 is the characteristic function for the odd integers. Again we refer to Table 3 for some numeric values.

References [1] C. D. Ahlbrandt, M. Bohner, and J. Ridenhour. Hamiltonian systems on time scales. J. Math. Anal. Appl., 250:561–578, 2000.

Exponential Functions and Laplace Transforms for Alpha Derivatives

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[2] E. Akın, L. Erbe, B. Kaymak¸calan, and A. Peterson. Oscillation results for a dynamic equation on a time scale. J. Differ. Equations Appl., 7:793–810, 2001. [3] F. M. Atıcı and G. Sh. Guseinov. On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math., 2002. Special Issue on “Dynamic Equations on Time Scales”, edited by R. P. Agarwal, M. Bohner, and D. O’Regan. To appear. [4] M. Bohner and A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkh¨auser, Boston, 2001. [5] M. Bohner and A. Peterson. Laplace transform and Z-transform: Unification and extension. Methods Appl. Anal., 2002. To appear. [6] S. Hilger. Analysis on measure chains — a unified approach to continuous and discrete calculus. Results Math., 18:18–56, 1990. [7] B. Kaymak¸calan, V. Lakshmikantham, and S. Sivasundaram. Dynamic Systems on Measure Chains, volume 370 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1996. [8] W. G. Kelley and A. C. Peterson. Difference Equations: An Introduction with Applications. Academic Press, San Diego, second edition, 2001.

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