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Generalized time scales.

1 The Classical Time Scales Calculus

There are several attempts to unify continuous and discrete mathematics. One of the main approaches is the time scale setting which was founded by Stefan Hilger in his Ph. D. thesis [24], see also [11]. It gives an efficient tool to unify continuous and discrete problems in one theory. Since then hundreds of papers appeared in the theory and its applications to dynamic equations, see e.g., the very interesting monographs of Bohner and Peterson [13,14]. At the beginning a time scale is defined to be an arbitrary closed subset of the real numbers , with the standard inherited topology. Examples of time scales include the real numbers R, the natural numbersN, the integers Z, the Cantor set, and any finite union of closed intervals of R. Next, we review a few basics of the time scales calculus.

For a time scale T, the forward jump operator [sigma] : T [right arrow] t is defined by [sigma](t) := inf {z [member of] T : z > t} and the backward jump operator [rho] : T [right arrow] t is defined by [rho](t) := sup{z [member of] t : z < t}. Here, we put inf [empty set] = sup t and sup [empty set] = inf T Also, the forward stepsize function, [micro] : t [right arrow] [0, [infinity]), is defined by [micro](t) := [sigma](t) - t and the backward stepsize function v : t [right arrow] [0, [infinity]), v (t) := t - [rho](t), see [2,13,14]. In [13,14,26] the stepsize functions are called graininess functions. Here, we follow the terminology of Ahlbrandt, Bohner and Ridenhour [3].

A point t [member of] with [sigma](t) = t, [sigma](t) > t, [rho](t) = t, [rho](t) < t, [rho](t) < t < [rho](t) and [sigma](t) = t = [rho](t) is called right dense, right scattered, left dense, left scattered, isolated and dense, respectively.

A real valued function f on T is called regulated on t if its right hand limits exist at all right dense points of T \ {sup t}, and left hand limits exist at all left dense points of T \ {inf t}. It is called right dense continuous, or just rd-continuous, on t if it is regulated on t and continuous at all right dense points of T.

For any time scale t, the subset [T.sup.k] is defined by [t.sup.k] := {t [member of] T : t [not equal to] sup t or t is left dense}. The delta derivative of f : T [right arrow] r at t [member of] [T.sup.k] is then defined as the number [f.sup.[DELTA]] (t) (provided it exists) with the property that given any e > 0, there is a neighborhood U of t such that

[absolute value of f([sigma](t)) - f(r) - [f.sup.[DELTA]](t)([sigma](t) - r)] [absolute value of [less than or equal to] [epsilon]] [sigma](t) - r] for all r [member of] U,

cf. [13,14,25]. If [f.sup.[DELTA]](t) exists for all t [member of] [t.sup.k], we say that f is [DELTA]-differentiable on [t.sup.k].

There are many different ways to define integration on a time scale. For example, we can use the Cauchy, Riemann or Lebesgue integrals, among other concepts. Of most importance to this work is the Cauchy integral. First, for a function f : t [right arrow] R, we say F is a [DELTA]-antiderivative of f on t, if [F.sup.[DELTA]](t) = f(t) for all t [member of] [T.sup.k]. Hilger, in [25], shows that any rd-continuous function f on T has a [DELTA]-antiderivative F on T. He then defined the [DELTA]-integral of f by

[[integral].sub.a.sup.b] f(t)[DELTA]t := F(b) - F(a),

see [13,25]. The classical time scales calculus includes three important special cases : T = R, T = [omega]Z := {k[omega] : k [member of] Z}, [omega] > 0 and T = {[q.sup.k] : k[member of] Z} [union] {0}, q > 1. One can check that in these cases we have [f.sup.[DELTA]](t) = f'(t), [f.sup.[DELTA]](t) = [[DELTA].sub.[omega]] f(t) and [f.sup.[DELTA]](t) = [D.sub.q] f(t), respectively, where [[DELTA].sub.[omega]] f(t) = f (t + [omega]) - f(t) / [omega] is the forward difference operator with stepsize [omega] [32], and [D.sub.q] f(t) = f(qt) - f(t) / t(q - 1) (for t [not equal to] 0) is the q-difference operator [10]. There are many examples and applications of time scales, see [13,14].

2 Why is Generalization Important?

In this work, we propose a generalization of the definition of time scale given by Stefan Hilger. The natural question here is that why generalization? In fact the theory outlined above does not include some important dynamical problems. The first problem concerns Jackson q-difference operator (q stands for quantum) and the associated Jackson q-integral. Jackson q-difference operator [D.sub.q] is defined to be

[D.sub.q] f(t) = f(qt) - f(t) / t(q - 1), t [member of] R {0} (2.1)

where q is a fixed number, normally taken to lie in (0,1). The derivative at zero is normally defined to be f'(0), provided that f'(0) exists, see [8,16,27,28]. Jackson also introduced q-integrals

[[integral].sup.a.sub.0] f(t)[d.sub.q]t = [infinity] [[infinity].summation over (k=0)] [aq.sup.k] (1 - q) f ([aq.sup.k]), (2.2)

and

[[integral].sub.0.sup.[infinity]] f(t)[d.sub.q]t = [[infinity].[summation over (k=1)] [aq.sup.-k] (1 - q) f ([aq.sup.-k] (2.3)

provided that the series converge [4,29]. He then defined

[[integral].sub.a.sup.b] f(t)[d.sub.q]t = [[integral].sub.0.sup.b] f(t)[d.sub.q]t - [[integral].sub.0.sup.a] f(t)[d.sub.q]t - a, b [member of] R. (2.4)

There is no unique canonical choice for the q-integral over [0, infinity) and the q-integral on R. Following Jackson, we define the q-integral over [0, [infinity]) to be

[[integral] .sub.0.sup.[infinity]] f(t)[d.sub.q]t = (1 - q) [[infinity].summation over (k=-[infinity]) [q.sup.k] f(sqk), (2.5)

[21,34]. Matsuo [37] defined the q-integral on more general sequences by

[[integral].sub.0.sup.s[infinity]] f(t)[d.sub.q]t = s(1 - q) [[infinity].summation over (k=-[infinity])] [q.sup.k] f ([sq.sup.k) (2.6)

where s is a fixed positive number, see also [33,34]. The bilateral q-integral is defined in [33] by

[[integral].sub.-s*[infinity].sup.s*[infinity]]] f(t)[d.sub.q]t = s(1 - q) [[infinity].summation over (k=-[infinity]) [[q.sup.k] f([sq.sup.k]) + [q.sup.k] f(-[sq.sup.k])], s > 0, (2.7)

provided that the sums of (2.5), (2.6) and (2.7) converge. So, the definitions in (2.6) and (2.7) are dependent on s. This calculus which is based on the Jackson q-difference operator and the associated Jackson q-integral, is the most common tongue of quantum calculus.

The second important problem which cannot be obtained from the calculus of time scales, concerns the forward difference operator

[[DELTA].sub.[omega]]f(t) = f(t + [omega]) - f(t) / [omega], t [member of] R (2.8)

where [omega] is a fixed positive number, see [12,30-32]. The associated integral of (2.8) is the well known Norlund sum

[integral] f(t)[[DELTA].sub.[omega]t = -[omega][[infinity].summation over (j=0)] f(t + j[omega]), (2.9)

see [20,30,38]. This sum is called in [31] an indefinite sum and Carmichael [17] called it just a sum.

Another problem not included in Hilger's treatment is related to the Hahn difference operator [D.sub.q,[omega]] which is defined by

[D.sub.q,[omega]]= f(tq + [omega]) - f(t) / t(q - 1) + [omega], T [member of] r {[W.sub.0]}

where q [member of] (0,1), [omega] > 0 and [w.sub.0] = [omega] / 1 - q. This operator was introduced by Hahn [22] in (1949), see also [7,19,23,35]. Some recent literature have applied these operators to construct families of orthogonal polynomials as well as to investigate some approximation problems, cf. [7,15,18,19,35]. In [5], the author introduced the q, w-integral of f from a to b,

[[integral].sub.a.sub.b] f(t)[d.sup.q,[omega]]t : = [[integral].sub.[[omega].sub.0].sup.b] f(t)[d.sub.q,[omega]]t - [[integral].sub.[[omega].sub.0].sub.a] f(t)[d.sub.q,[omega]]t (2.11)

where

[[integral].sub.[[omega].sub.0].sup.[infinity]] f(t)[d.sub.q,[omega]]t := (x(1 - q) - [omega]) [[infinity].summation over (k=0) [q.sup.k] f([xq.sup.k] + [[omega][k].sub.q]), x [member of] R, (2.12)

provided that the series converges at x = a and x = b. Also, a q, [omega]-calculus based on 2.10), (2.11) and (2.12) was introduced in [9]. The [D.sub.q[omega]des as particular cases the q-difference operator in (2.1) and the forward difference operator in (2.8). Our aim is to define a generalized time scale T and a [DELTA]-differential operator of functions defined on T such that all difference operators, mentioned above, are included. Afterwards we derive the corresponding calculus.

3 Generalized Time Scales

In the following, we generalize the classical definition of a time scale [24]. To achieve our goal we generalize and extend time scales calculus to include the three mentioned above problems together with the case of differential-difference equations. We use the terminology and notation of the classical time scales calculus as far as possible. Nonetheless, we opted to use "forward and backward" instead of "right and left".

We start with some definitions and introduce notation and terminology that will be used in the sequel.

Assume that T is a nonempty closed subset of R and E is an equivalence relation on T. Suppose also that T has the topology inherited from the standard topology on R. [bar.A], as usual, denotes the closure of A and A' the set o f all limit points of A, A [subset equal to] T. For a, b [member of] T [union] {inf T, sup T} and c [member of] T, let

* [T.sub.c] := {t [member of] T : c [member of] [bar.[t]]},

* [a, b] := {x [member of] T : a [less than or equal to] x [less than or equal to] b or b [less than or equal to] x [less than o equal to] a},

* [[a,b].sub.c] := [a, b] [intersection] [bar.[c]],

* S := {s [member of] T : [T.sub.s] [not equal to] [s]},

* [alpha] [member of] S [union] {inf T, sup T} be fix,

where [t] denotes the equivalence class that contains t [member of] T. The points of S are called special and [T.sub.t] is called the domain set of t. We define open and half-open intervals similarly. Whenever we say a neighborhood of c in a set A [subset equal to] T we mean the set A [intersection] U, where U is a neighborhood of c in R. In particular, if A = T we say a neighborhood of c without mentioning T.

One can see that [T.sub.c] = [c] [union] {t [member of] T : c [member of] [bar.[t]] \ [t]} = [union]{[t] [subset equal to] T : c [member of][bar.[t]]} for c [member of] T and s [member of] S iff there is t [member of] T such that s [member of] [bar.[t]] \ [t]. Notice also that, in this setting, if a = [infinity] or a = -[infinity], then [a, b] = (a, b] = [b, a).

Definition 3.1. A class [t] is called singular if [t] = {t}, otherwise it is called nonsingular. An interval I of T is called nonsingular if I is the union of some nonsingular classes which have the same infimum and supremum. Finally, we denote by I the family of all nonsingular intervals of T.

Note that I is a family of pairwise disjoint sets. Indeed, if I, J [member of] I and x [member of] I [intersection] J, then [x] [not equal to] {x}, [x] [subset equal to] I and [x] [subset equal to] J such that inf[x] = inf I = inf J and sup[x] = sup I = sup J which implies that I = J.

Definition 3.2. Let A, B be subsets of T. We say that:

* A is right contiguous to B, and denote it by A [much greater than] B, if A [not equal to] B and inf A = sup B.

* A is to the right of B, and denote it by A > B, if A [not equal to] B and inf A [greater than or equal to] sup B.

* A is alternating with B, and denote it by A [approximately equal to] B, if for every x [member of] A and x' [member of] B, there exist y, z [member of] B and [y.sub.1], [z.sub.1] [member of] A, such that (y, z) [intersection] (A [union] B) = {x} and (y', z') [intersection] (A [union] B) = {x'}. We write A [not approximately equal to] B, if A is not alternating with B.

We note that in the previous definition the relations > and [much greater than] are neither symmetric nor reflexive, > is transitive, [much greater than] is not transitive, while [approximately equal to] is symmetric and transitive. For A [not equal to] B and A[approximately equal to]B, there are no [a.sub.1], [a.sub.2] [member of] A such that ([a.sub.1], [a.sub.2]) [intersection] B = [empty set]. One can check that R+ [greater than or equal to] R-, [Z.sup.>0], Q[not approximately equal to]Q, N[not approximately equalN, Z[approximately equal to]Z and 2Z[approximately equal to]2Z + 1.

Definition 3.3. A family F of subsets of T is called arranged if:

(A1) For every A [member of] F with sup A [not equal to] sup[union]F (inf A = inf [union]F) there exists B [member of] F such that B [greater than or equal to] A (A [greater than or equal to] B).

(A2) For every A, B [member of] F, A [not equal to] B either A > B, B > A or A [approximately equal to] B.

Example 3.4. Let T = R, [F.sub.1] := = {[A.sub.n] := (n, n + 1) : n [member of] Z} and [F.sub.2] := {[B.sub.m,n] : (m + n - 1 / n, m + n / n + 1) : m, n [member of] N}. Then, we can see that [A.sub.n+1] [much greater than] [A.sub.n], n [member of] Z and [A.sub.n] > [A.sub.m], n, m [member of] Z, n > m. That is [F.sub.1] is arranged, but [F.sub.2] is not. This is because for any m [member of] N, there is no B [member of] [F.sub.2] such that [B.sub.m,1] [much greater than] B.

Definition 3.5. An equivalence relation E on t is called arranged if the family of all equivalence classes is arranged.

The universal equivalence relation E = T x T is arranged while the identity relation E = {(x,x) : x [member of] t} is not whenever t contains more than one element. In [5], the author gave necessary and sufficient conditions for an equivalence relation E to be arranged.

We are now in a position to give a definition that widens the scope of time scales and preserves the classical ones as special cases.

Definition 3.6. A triple (t, E, [alpha]) is called a time scale if t is a nonempty closed subset of r and E is an arranged equivalence relation on t.

Throughout this work, one can see that all concepts (jump operators, derivatives, Cauchy integrals, etc.) with respect to the triple (t, E, [alpha]) where T [subset equal to] R is a nonempty closed subset, E is the universal equivalence relation and [alpha] = sup t coincide with the corresponding ones with respect to the classical time scale, defined by Stefan Hilger [13,14,24-26].

Remark 3.7. On any nonempty closed subset t of r, we can obtain as many time scales as the number of arranged equivalence relations defined on t and the number of values of [alpha] [member of] S [union] {inf T, sup t}. [alpha] represents a point where the "time" changes its direction.

For more generality, we can replace the condition of closedness of T by the following condition: T is a nonempty subset of R such that every Cauchy sequence in T converges to a point of T with the possible exception of Cauchy sequences which converge to a finite infimum or finite supremum of t. See [3].

Example 3.8. Fix q [member of] R, 0 < q < 1. Let T = R and E be defined by

xEy [??] [there exists]k [member of] Z such that x = [q.sup.k] y, x, y [member of] T.

It is easily seen that E is an equivalence relation. For t [not equal to] 0, we have [t] = {[tq.sup.k] : k [member of] Z}, [bar.[t]] = [t] [union] {0}, [t]' = {0}, and t [not member of] [bar.[c]] for any c [member of] t \ {t} which implies [T.sub.t] = [t]. While [0] = {0}, and 0 = inf [t] for any t [member of] (0, [infinity]), 0 = sup[t] for any t [member of] (-[infinty], 0) which implies 0 [member of] [t] for any t [member of] T, and then [T.sub.0] = [U.sub.t[member] of]T [bar.[t]] = r. Thus, s = {0} and all classes are nonsingular except the class of zero. For [t] [subset] t, we have either [t] = {0}, [t] [subset] (0, [infinity]) or [t] [subset] (-[infinity], 0). If [t] [subset] [0, [infinity]), then [t] [greater than or equal to] [r] for any r [member of] (-[infinity], 0] and if [t] [subset] (-[infinity], 0], then [r] [much greater than] [t] for any r [member of] [0, [infinity]). Then, E satisfies property (A1). For r, t [member of] (0, [infinity]) with [r] [not equal to] [t] we claim that [r] [approximately equal to] [t]. Note that [r] [intersection] (0,1] [not equal to] [empty set] and [t] [intersection] (0,1] [not equal to] [empty set]. Let us put [r.sub.0] := max{[r] [intersection] (0,1]} and [t.sub.0] := max{[t] [intersection] (0,1]}, then [r.sub.0] [not equal to] [t.sub.0]. Assume that [r.sub.0] < [t.sub.0], which implies [t.sub.0] [less than or equal to] 1 < [q.sup.-1] [r.sub.0]. Since the function f(t) = qt is strictly increasing on R, we have

... < [q.sup.2][r.sub.0] < [q.sup.2][t.sub.0] < [qr.sub.0] < [qt.sub.0] < [r.sub.0] < [t.sub.0] < [q.sup.-1][r.sub.0] < [q.sup.-1][t.sub.0] < [q.sup.-2][r.sub.0] < [q.sup.-2][t.sub.0] < ....

Thus, for x [member of] [r] and x' [member of] [t] there exists [k.sub.1], [k.sub.2] [member of] Z such that x = [q.sup.[k.sub.1]][r.sub.0], x' = [q.sup.[k.sub.2]][t.sub.0], and then ([q.sup.[k.sub.1] + 1] [t.sub.0], [q.sup.[k.sub.1]] [t.sub.0]) [intersection] ([r] [union] [t]) = {x} and ([q.sup.[k.sub.2]][r.sub.0], [q.sup.[k.sub.2] - 1][r.sub.0]) [intersection] ([r] [union] [t]) = {x'}. Thus, [r] [approximately equal to] [t]. In a similar way, we can show that [r] [approximately equal to] [t] for any r, t [member of] (-[infinity], 0). While [r] [much greater than] [0] for any r [member of] (0, [infinity]) and [0] [much greater than] [r] for any r [member of] (-[infinity], 0). Therefore, E satisfies properties (A2), and so (T, E, 0) is a time scale. From now on, we call this time scale the Jackson difference time scale.

Example 3.9. Fix [omega] [member of] R, [omega] > 0. Let T = R, E be defined by

xEy [??] [there exists]k [member of] Z such that x = y + k[omega],

and [alpha] = [infinity]. E is an equivalence relation and for t [member of] T, we have [t] = {t + k[omega] : k [member of] Z}, inf [t] = -[infinity], sup[t] = [infinity] and [bar.[t]] = [t] which implies [T.sub.t] = [t], i.e., S = [empty set]. Here, all classes of T are nonsingular and I = {T}. We claim that the triple (T, E, [infinity]) is a time scale. We need to prove that E is arranged. First, E satisfies (A1), since inf [t] = -[infinity] and sup[t] = [infinity] for any t [member of] T. Now, for any r, t [member of] T we have [r] [approximately equal to] [t]. Indeed, let [t.sub.0] := min{[t] [intersection] [0, [infinity])} and [r.sub.0] := min{[r] [infinity] [0, [infinity])}. We may assume that [r.sub.0] < [t.sub.0]. This implies that

... < [r.sub.0]-2[omega] < [t.sub.0]-2[omega] < [r.sub.0]-[omega] < [t.sub.0]-[omega] < [r.sub.0] < [t.sub.0] < [r.sub.0]+[omega] < [t.sub.0]+[omega] < [r.sub.0]+2[omega] < .... (3.1)

For x [member of] [r] and x' [member of] [t], there are k, k' [member of] Z such that x = [r.sub.0] + k[omega] and x' = [t.sub.0] + k'[omega]. By (3.1), ([t.sub.0] + (k - 1)[omega], [t.sub.0] + k[omega]) [intersection] ([r][intersection][t]) = {x} and ([r.sub.0] + k'[omega], [r.sub.0] + (k' + 1)[omega]) [intersection] ([r][intersetion][t]) = {x'}, i.e., [r] [approximately equal to] [t]. Thus, E satisfies (A2), and so E is arranged. Therefore, (T, E, [infinity]) is a time scale. This time scale will be called the Norlund difference time scale.

Example 3.10. Fix q [member of] R; 0 < q < 1 and [omega] > 0. Assume that T = R, E is defined by

xEy [??] [there exists]k [member of] Z such that x = [q.sup.k]y + [[omega][k].sub.q], x, y [member of] T,

and [alpha] = [[omega].sub.0] := [omega] / 1 - q where [[k].sub.q] = 1 - [q.sup.k] / 1 - q for any k [member of] {0, 1, 2, -[infinity]}. Easily, one can check that E is an equivalence relation. Now, we show that E is arranged. Let h denote the function h(t) = qt + [omega], t [member of] R. One can see that h(t) < t for t > [[omega].sub.0], h(t) > t for t < [[omega].sub.0], and h([[omega].sub.0]) = [[omega].sub.0]. The function h has the inverse [h.sup.-1](t) = [q.sup.-1](t - [omega]), t [member of] R. It is not hard to prove the following results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

Furthermore, [{[h.sup.k](t)}.sup.[infinity].sub.k=1] is a decreasing (an increasing) sequence in k when t > [[omega].sub.0] (t < [[omega].sub.0]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

see Figure 3.1 [9]. The sequence [{[h.sup.-k](t)}.sup.[infinity].sub.k=1] is increasing (decreasing), t > [[omega].sub.0] (t < [[omega].sub.0]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.5)

Now, if t [not equal to] [[omega].sub.0], then [t] = {[tq.sup.k] + [[omega]k.sub.q] : k [member of] Z}, [t] = [t] [union] {[[omega].sub.0]}, and t [not member of] [c]' for any c [member of] T which implies [T.sub.t] = [t]. Whilst [[[omega].sub.0]] = {[[omega].sub.0]}, [[omega].sub.0] = inf [t] for any t [member of] [[[omega].sub.0], [infinity]), [[omega].sub.0] = sup[t] for any t [member of] (-[i0nfinity], [[omega].sub.0]], and {[[omega].sub.0]} = [t]' for any t [member of] T \ {[[omega].sub.0]}, then [T.sub.[omega]0] = [[union].sub.t[member of]T] [bar.[t]] = R. Thus, S = {[[omega].sub.0]} and all classes are nonsingular except the class of [[omega].sub.0]. For t [member of] [[[omega].sub.0], [infinity]) (t [member of] (-[infinity], [[omega].sub.0]]), we have [t] [much greater than] [r] ([r] [much greater than] [t]) for all r [member of] (-[infinity], [[omega].sub.0]] (r [member of] [[[omega].sub.0], [infinity])), i.e., E satisfies (A1) property. Now, for [r] [not equal to] [t], with either r, t [member of] ([[omega].sub.0], [infinity]) or r, t [member of] (-[infinity],[[omega].sub.0]), we claim that [r] [approximately equal to] [t]. By (3.4), [r] [intersection] ([[omega].sub.0], 1 + [[omega].sub.0]] [not equal to] [empty set] and [t] [instersection] ([[omega].sub.0], 1 + [[omega].sub.0]] [not equal to] [empty set]. Let [r.sub.0] := max{[r] [intersection] ([[omega].sub.0], 1 + [[omega].sub.0]]} and [t.sub.0] := max{[t] n ([[omega].sub.0], 1 + [[omega].sub.0]]}. The values [r.sub.0], [t.sub.0] exist, since [r]' = [t]' = {[[omega].sub.0]}. Here, [r.sub.0] = [t.sub.0]. We may assume that [r.sub.0] < [t.sub.0], and then [t.sub.0] < 1 + [[omega].sub.0] < [h.sup.-1] ([r.sub.0]), and since the function h is strictly increasing on R, then ... < [h.sup.2]([r.sub.0]) < [h.sup.2]([t.sub.0]) < h([r.sub.0]) < h([t.sub.0]) < [r.sub.0] < [t.sub.0] < [h.sup.-1] ([r.sub.0]) < [h.sup.-1] ([t.sub.0]) < [h.sup.-2]([r.sub.0]) < [h.sup.-2]([t.sub.0]) < .... This implies for x [member of] [r] and x' [member of] [t] there exist k, k' [member of] Z such that x = [h.sup.k]([r.sub.0]), x' = [h.sup.k'] ([t.sub.0]), then ([h.sup.k + 1]([t.sub.0]), [h.sup.k]([t.sub.0])) [intersection] ([r] [union] [t]) = {x} and ([h.sup.k' ([r.sub.0]), [h.sup.k' - 1] [r.sub.0]) [intersection] ([r] [union] [t]) = {x'}. That is [r] [approximately equal to] [t]. Similarly, we can see that [r] [approximately equal to] [t] for any r, t [member of] (-[infinity], [[omega].sub.0]). Also, we have [r] > [t] for any r [member of] ([[omega].sub.0], [infinity]) and t [member of] (-[infinity], [[omega].sub.0]]. So, (A2) property is true which implies that E is arranged. Therefore the triple (T, E, [[omega].sub.0]) is a time scale. We call this time scale the Hahn difference time scale.

[FIGURE 3.1 OMITTED]

Example 3.11. Consider (T, E, [alpha]), where T = Z + - 1 / N := (z + 1 / n : z [member of] Z and n [member of] N), E is defined by xEy [??] [there exists]z [member of] Z such that x, y [member of] (z, z + 1] and [alpha] [member of] Z [union] {-[infinity], [infinity]}. One can see that T is closed and E is an equivalence relation. For z [member of] Z, we have [z] = {z - : n - 1 / n : n [member of] N} = (z - 1, z], [T.sub.z] = [z] [union] [z + 1] = (z - 1, z + 1], and [T.sub.t] = [t] for all t [member of] T \ Z which imply S = Z and z = sup[z] = inf [z + 1]. Consequently, for z [member of] Z we have T = [union]t[member of]Z [t], and [z + 1] [much greater than] [z]. For [z.sub.1], [z.sub.2] [member of] Z, [z.sub.1] > [z.sub.2] we note that [[z.sub.1] ] > [[z.sub.2]]. Hence E is arranged. Therefore (T, E, [alpha]) is a time scale for any a [member of] z U {-[infinity], [infinity]}.

Example 3.12. Let t be defined as in Example 3.11 and E be the equivalence relation on t defined by xEy [??] [there exists]z [member of] z such that x, y [member of] [z, z + 1) and a [member of] {-[infinity], [infinity]}. For z [member of] Z we have [z] = {z, z + 1/ 2, z + 1 / 3, -[infinity]} = [z, z + 1 / 2]. Here [T.sub.t] = [t] for all t [member of] T, that is S = [empty set]. For [z.sub.1], [z.sub.2] [member of] z, [z.sub.1] > [z.sub.2], we have [[z.sub.1]] > [[z.sub.2]], that is E satisfies property (A2). Since z + 1 / 2 = sup[z] < inf [z + 1] = z + 1, z [member of] z, then there is no z' [member of] Z such that [z'] [much greater than] [z]. Thus, (A1) is not true and hence (T, E, [alpha]) is not a time scale.

One of the main properties of a time scale is the following.

Theorem 3.13. Assume that (T, E, [alpha]) is a time scale and x, y [member of] T, x > y. Then, there is a sequence of nonsingular intervals {[I.sub.i]} [{[I.sup.i]}.sup.n.sub.i=1] such that

[I.sub.1] > [I.sub.2] > -[infinity] > [I.sub.n], x [member of] [bar.[I.sub.1]] and y [member of] [bar.[I.sub.n]]. (3.6)

In addition, either [x, y] [intersection] S = [empty set], or [x, y] [intersection] S = {[s.sub.i]} [{[s.sup.i]}.sup.m.sub.i=1], m [member of] N such that

[x, [s.sub.1]] [much greater than] [[s.sub.1], [s.sub.2]] [much greater than] ... [much greater than] [[s.sub.m] - 1, [s.sub.m]] [much greater than] [[s.sub.m], y]. (3.7)

We postpone the proof of this theorem to Section 4. The following lemma will be needed in order to define jump operators. Also, it plays an essential role in this work.

Lemma 3.14. In a time scale (T, E, [alpha]), either

[bar.[x]] [subset equal to] [[alpha], sup T] or [bar.[x]] [subset equal to] [inf T, [alpha]], x [member of] T. (3.8)

Proof. First, we can see that [alpha] [member of] {inf [[alpha]], sup[[alpha]]}. Indeed, assume the contrary: i.e., [alpha] [member of] (inf [[alpha]], sup[[alpha]]). Then [alpha] [not member of] {inf T, sup T}, that is [alpha] [member of] S \ {inf T, sup T}. Thus, there is c [member of] [T.sub.[alpha]] \ [[alpha]], i.e., a [member of] [bar.[c]] [c]. Since [alpha] [member of] [c]', then (inf[[alpha]], sup[[alpha]]) [intersection] [c] [not equal to] [empty set]. Hence, neither [[alpha]] > [c] nor [c] > [[alpha]]. By property (A2), [[alpha]][approximately equal to][c]. Then there are y, z [member of][c] such that (y, z) [infinity] ([[alpha]] [union] [c]) = {[alpha]}, which contradicts [alpha] [member of] [c]'. Therefore [alpha] [member of] {inf[[alpha]], sup[[alpha]]}. Let x [member of] T. We distinguish between the following three cases: First case: x = [alpha]. Statement (3.8) is true, since a {inf[[alpha]], sup[[alpha]]}. Second case: x > [alpha]. If x [member of] [[alpha]], then [x] = [[alpha]]. Hence (3.8) is true, by the first case. If inf [x] > [alpha], thus [bar.[x]] [subset equal to] [[alpha], sup T]. When inf[x] < [alpha], neither [[alpha]] > [x] nor [x] > [[alpha]]. So, [x][approximately equal to][[alpha]], by property (A2). Recall that [alpha] [member of] {inf [[alpha]], sup[[alpha]]}. For [alpha] = inf [[alpha]] ([alpha] = sup[[alpha]]), we have t [member of] [[inf[x], [alpha]).sub.x] (t [member of] ([alpha], [sup[x]].sub.x]). Then there are no y, z [member of] [[alpha]] such that (y, z) [intersection] ([[alpha]] [union] [x]) = {t}. This leads to a contradiction with [[alpha]][approximately equal to][x]. Therefore [bar.[x]] [subset equal to] [[alpha], sup T]. Third case: x < [alpha]. In a similar way, we can show that [bar.[x]] [subset equal to] [inf T, [alpha]].

Definition 3.15. Let (T, E, [alpha]) be a time scale. We define the jump operators [sigma], [rho]: T [right arrow] T as follows:

The forward jump operator

[sigma](t) = {sup{z [member of] [t] : z < t}, if [bar.[t]] [subset equal to] [[alpha],sup T], t [not equal to] inf[t], [sigma](t) = {inf{z [member of] [t] : z > t}, if [bar.[t]] [subset equal to] [inf T, [alpha]], t [not equal to] sup[t], [sigma](t) = t, otherwise.

The backward jump operator

[rho](t) = {inf{z [member of] [t] : z > t}, if [bar.[t]] [subset equal to] [[alpha], sup T], t [not equal to] sup[t], [rho](t) = {sup{z [member of] [t] : z < t}, if [bar.[t]] [subset equal to] [inf T, [alpha]], t [not equal to] inf[t], [rho](t) = t, otherwise.

We also define the following functions: the forward stepsize function [micro] : T [right arrow] R; [micro](t) = [sigma](t) - t, and the backward stepsize function v : T [right arrow] R; v (t) = t - [rho](t).

We can see that for t [member of] T \ {[alpha]}:

[sigma](t) = r iff r [member of] [[t, [alpha]].sub.t] and [absolute value of t - r] = inf{[absolute value of t - z] : z [member of] [(t, [alpha]).sub.t]},

[rho](t) = r iff r [member of] [t] \ (t, [alpha]) and [absolute value of t - r] = inf{[absolute value of t - z] : z [member of] [t] [t, [alpha]]}.

For a function f : t [right arrow] R, we denote by [f.sup.[infinity]] (t) := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [f.sup.n](t) whenever the limit exists. If [f.sup.-1] exists, then f-[infinity](t) := ([f.sup.-1])[infinity](t). Here, [f.sup.n] denotes the n-th iteration of f for n [member of] N and [f.sup.0] is the identity function.

Definition 3.16. For t [member of] T, we say that t is forward scattered, forward dense, backward scattered, backward dense, isolated and dense if [sigma](t) [not equal to] t, [sigma](t) = t, [rho](t) [not equal to] t, [rho](t) = t, [rho](t) [not equal to] t [not equal to] [sigma] (t), and [rho](t) = t = [sigma](t), respectively.

Note that [alpha] = inf[[alpha]] ([alpha] = sup[[alpha]]) when [alpha] [member of] T and [bar.[[alpha]]] [subset equal to] [[alpha], sup t] ([bar.[[alpha]]] [subset equal to] [inf T, [alpha]]). Thus, we have always [sigma]([alpha]) = [alpha] while it is not true in general for [rho]([alpha]).

Example 3.17. On the following time scales, we find a and p:

(i) t is a closed subset of R and E is the universal equivalence relation. In this case, we have only one equivalence class T, and hence S = 0. Also, t is the only nonsingular interval when t has more than one point. For [alpha] = sup T, we have [sigma](t) = inf{z [member of] T : z > t} and [rho](t) = sup{z [member of] T : z < t} which are the forward and backward jump operators in the classical time scale calculus, respectively. For [alpha] = inf T, we have [sigma](t) = sup{z [member of] T : z < t} and [rho](t) = inf {z [member of] T : z > t}. That is to say that [alpha] and [rho] interchange their roles when a changes between inf T and sup t.

(ii) <T, E, 0> is the q-difference time scale. We note that for t [member of] (0, [infinity]),

0 < ... < [q.sup.2]t < qt < t < [q.sup.-1]t < [q.sup.-2]t < ...

which implies that [sigma](t) = sup {[q.sup.k]t : k [member of] N} = qt, [rho](t) = inf{[q.sup.-k]t : k [member of] n} = [q.sup.-1]t, [micro](t) = t(q - 1), v (t) = t(1 - [q.sup.-1]), [[sigma].sup.[infinity]] (t) = 0, and [rho][infinity](t) = [infinity]. If t [member of] (-[infinity], 0), then

... < [q.sup.-2]t < [q.sup.-1]t < t < qt < [q.sup.2]t ... < 0,

[sigma](t) = qt, [rho](t) = [q.sup.-1]t, [micro](t) = t(q - 1), v(t) = t(1 - [q.sup.-1]), [[sigma].sup.[infinity]] (t) = 0, and [[rho].sup.[infinity]](t) = -[infinity].

(iii) (T, E, [infinity]) is the Norlund difference time scale. For t [member of] T, we have

... < t - 2[omega] < t - [omega] < t < t + [omega] < t + 2[omega] < ... ,

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] t + k[omega] = [infinity] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] t - k[omega] = -[infinity]. Thus, [sigma](t) = t + [omega], [rho](t) = t - [omega], [micro](t) = v(t) = [omega], [[sigma].sup.[infinity]] (t) = [infinity], and [[rho].sup.[infinity]](t) = -[infinity]. This implies that all points of T are isolated.

(iv) (T, E, [[omega].sub.0]) is the Hahn difference time scale. First, for t [member of] ([[omega].sub.0], [infinity]), we have [[omega].sub.0] < ... < [h.sup.2](t) < h(t) < t < [h.sup.-1] (t) < [h.sup.-2] (t) < ... and for t [member of] (-[infinity], [[omega].sub.0]), ... < [h.sup.-2](t) < [h.sup.-1](t) < t < h(t) < [h.sup.2](t) ... < [[omega].sub.0]. Then [sigma](t) = h(t) = qt + [omega], [rho](t) = [h.sup.-1](t) = [q.sup.-1](t - [omega]), [micro](t) = t(q - 1) + [omega], v (t) = t(1 - [q.sup.-1]) + [q.sup.-1][omega],

[[sigma].sup.[infinity]](t) = [[omega].sub.0] and [[rho].sup.[infinity]](t) = {[infinity], if t > [[omega].sub.0],

[[sigma].sup.[infinity]](t) = [[omega].sub.0] and [[rho].sup.[infinity]](t) = {[[omega].sub.0] , if t = [[omega].sub.0],

[[sigma].sup.[infinity]](t) = [[omega].sub.0] and [[rho].sup.[infinity]](t) = {-[infinity], if t < [[omega].sub.0].

(v) Fix a positive integer n > 1. Assume that the triple (T, E, [alpha]) with T = R+, xEy if there exists k [member of] Z such that x = [y.sup.[n.sup.k]] for x, y [member of] R+, and [alpha] [member of] {0, 1, [infinity]}. Easily, one check that E is an equivalence relation. Note that [t] = {[t.sup.[n.sup.k]] : k [member of] Z} and [T.sub.t] = [t] for all t [member of] T \ {0,1}, and [0] = {0}, [1] = {1}, T) = [0,1) and [T.sub.1] = (0, [infinity]). Furthermore, 0 and 1 are the special points. Also, the intervals (0,1) and (1, [infinity]) are the nonsingular intervals of T. We can show easily that E is arranged. If [alpha] = [infinity], then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, that [[sigma].sup.[infinity]](t) = {0, if t = 0, [[sigma].sup.[infinity]](t) = {1, if 0 < t [less than or equal to] 1, [[rho].sup.[infinity]](t) [[sigma].sup.[infinity]](t) = {[infinity], if t > 1,

[[rho].sup.[infinity]](t) = {0, if 0 [less than or equal to] t < 1, [[rho].sup.[infinity]](t) = {1, if t [greater than or equal to] 1.

If [alpha] = 1, then [sigma](t) = [nth root of (t)], [rho](t) = [t.sup.n], for all t [member of] T. This implies that

[[sigma].sup.[infinity]](t) = {t, if t = 0 [[sigma].sup.[infinity]](t) = {1, otherwise

[[rho].sup.[infinity]](t) = {0, if 0 [less than or equal to] t < 1, [[rho].sup.[infinity]](t) = {1, if t = 1, [[rho].sup.[infinity]](t) = {[infinity], if t > 1.

Finally, when [alpha] = 0, we get

[sigma](t) = {[t.sup.n], if 0 [less than or equal to] t [less than or equal to] 1, [sigma](t) = {[nth root of (t)], if t [greater than or equal to] 1,

[rho](t) = {[nth root of (t)], if 0 [less than or equal to] t [less than or equal to] 1, [rho](t) = {[t.sup.n], if t [greater than or equal to] 1.

and hence [[sigma].sup.[infinity]](t) = {0, if 0 [less than or equal to] t < 1, [[sigma].sup.[infinity]](t) = {1, if t [greater than or equal to] 1,

[[rho].sup.[infinity]](t) = {0, if t = 0, [[rho].sup.[infinity]](t) = {1, if 0 < t [less than or equal to] 1, [[rho].sup.[infinity]](t) = {[infinity], if t > 1.

4 Preliminary Results

In this section, we study some properties of time scales that will be needed in later sections. In the beginning, we state some preliminary lemmas, for the proofs see [5].

The following lemma gives us equivalent statements for a class [t] to be alternating with itself.

Lemma 4.1. Let (t, E, [alpha]) be a time scale and t [member of] T. Then the following statements are equivalent:

(i) [t] [approximately equal to] [t].

(ii) [t]' = {inf[t], sup[t]} [intersection] T and both inf[t], sup[t] [not member of] [t].

(iii) All points of[t] are isolated.

(iv) [t] = {... , [[rho].sup.2](t), [rho](t), t, [sigma](t), [[sigma].sup.2](t), ...} whose elements are distinct.

(v) [t] [approximately equal to] [d] for some d [member of] t.

Lemmas 4.2, 4.3, 4.4, 4.5 describe some properties of alternating classes, general classes, special points and nonsingular intervals of a time scale, respectively.

Lemma 4.2. Let (T, E, [alpha]) be a time scale and t [member of] T be such that [t][approximately equal to][t]. The following statements are true.

(i) If [t] [approximately equal to] [d] for d [member of] t, then [t]' = [d]'.

(ii) t [not member of] [c]' for every c [member of] T.

(iii) [T.sub.t] = [t].

(iv) [[sigma].sup.[infinity]](t) [member of] S [union] {[alpha]} and [[rho].sup.[infinity]] (t) [member of] S [union] {infT, supT}.

(v) If c [member of] (inf[t], sup[t]), then [c] [approximately equal to] [t].

The following lemma indicates possible forms and some properties of the classes of E.

Lemma 4.3. Let (T, E, [alpha]) be a time scale and t [member of] T. Then we have:

(i) [t] is of the following types: [t] is an interval of T, [t] = {t}, and

[t] = {..., [[rho].sup.2](t), [rho](t), t, [sigma](t), [[sigma].sup.2] (t), ...}.

(ii) [t] = [[rho](t)] = [[sigma](t)].

(iii) If [t] is nonsingular, then there exists a unique nonsingular interval I such that [t] [subset equal to] I.

(iv) If sup[t] [not equal to] sup T (inf[t] = inf T), then sup[t] [member of] S (inf[t] [member of] S).

(v) If [t] is nonsingular, then [bar.[t]] = ([t] [union] {inf[t], sup[t]}) [intersection] t and [t]' = I' [subset equal to] I [union] {inf I, sup I} where I is the nonsingular interval which contains t.

Lemma 4.4. Let (T, E, [alpha]) be a time scale. Assume that S = [empty set]. Then:

(i) E is not the universal equivalence relation.

(ii) For s [member of] S, [T.sub.s] = [union]{I [member of] I : s [member of] [bar.I] \ I} [union] [s].

(iii) For s [member of] S, either s = inf I = sup[s] or s = sup J = inf[s] for some I, J [member of] I.

(iv) If inf S [not equal to] inf T, then either [inf T, inf S) or [inf T, inf S] is a nonsingular interval.

(v) If sup S [not equal to] sup T, then either (sup S, sup T] or [sup S, sup T] is a nonsingular interval.

(vi) If [t] = {t}, then t [member of] S \ [union] I.

In the following lemma, a detailed description of the nonsingular intervals is given.

Lemma 4.5. Assume that (T, E, [alpha]) is a time scale and I [member of] I. Then,

(i) If all points of/ are isolated, then I is the union of some alternating classes, i.e., classes which are alternating with each other.

(ii) If s [member of] S [intersection] I, then I = [s], s [member of] {inf I, sup I} and s is not isolated.

(iii) If inf I [member of] I (sup I [member of] I), then I = [inf I] (I = [sup I).

(iv) If inf I [not equal to] inf T (sup I = sup T), then inf I [member of] S and I [subset equal to] [T.sub.inf I] (sup I [member of] S and I [subset equal to] [T.sub.sup I]).

(v) S [union] (inf I, sup I) = [empty set].

(vi) If T is not singleton, then I is an arranged countable family of pairwise disjoint intervals and T = [bar.[union]I] = [union]I [union]S.

The domain sets [T.sub.t], t [member of] T play an important role in the calculus of time scales suggested in this paper. Recall that [T.sub.t] = [t] iff t [not member of] S. The following lemma combines the most important properties for domain sets.

Lemma 4.6. For a time scale (T, E, a), t [member of] T we have the following:

(i) ([rho](t), t) [intersection] [T.sub.t] = ([sigma](t), t) [intersection] [T.sub.t] = [empty set].

(ii) If t [not equal to] [alpha] and [bar.[t]] [subset equal to] [[alpha], sup T] ([bar.[t]] [subset equal to] [inf T, [alpha]]), then [T.sub.t] [subset equal to] [[alpha], sup T] ([T.sub.t] [[subset].bar.] [inf T, [alpha]]).

(iii) If c [member of] [T.sub.t] and V is a neighborhood of t in [T.sub.t], then V [intersection] [bar.[c]] is a neighborhood of t in [bar.[c]].

We are now in a position to prove Theorem 3.13.

Proof of Theorem 3.13. Assume that x, y [member of] T, x > y. By Lemma 4.5 (vi), there are I, J [member of] I such that x [member of] [bar.I], y [member of] [bar.J]. There are two cases: I = J or I > J. If I = J, then (3.6) is true and n = 1. If not, then I > J. We put [I.sub.1] := I. By Lemma 4.5 (vi), I is arranged, then by property (A1) there is [I.sub.2] [member of] I with [I.sub.1] [much greater than] [I.sub.2]. Now, given {[I.sub.1], ..., [I.sub.n]}, n [member of] N, there exists [I.sub.n + 1] such that [I.sub.n] [much greater than] [I.sub.n]+1. So, we construct a sequence {[I.sub.n]} [[subset].bar.] I such that [I.sub.1] [much greater than] [I.sub.2] [much greater than] .... We have the following two possibilities: 1) [I.sub.n] > J for all n [member of] n. 2) [I.sub.n] = J for some n. Assume, towards a contradiction, that [I.sub.n] > J for all n. The sequence [{inf [I.sub.n]}.sub.n[member of]N] is decreasing and bounded below by sup J, then it converges to a point of t say z. Now, there are two cases. Case 1: sup[z] [not equal to] z. Then [z] [not equal to] {z}. By using Lemma 4.3 (iii), there is [I.sub.z] [member of] I containing [z] and hence sup [I.sub.z] [not equal to] z. Now, we have z [member of] (inf [I.sub.z], sup [I.sub.z]) and (inf [I.sub.z], sup [I.sub.z]) [intersection] [({inf [I.sub.n]}.sup.n[member of]N] \ {z}) = [empty set] which contradicts our definition of z. Case 2: sup[z] = z. Now, by property (A1), there is [t] [not equal to] {t}, [t] [much greater than] [z]. Let [I.sub.t] be the nonsingular interval which contains t, in view of Lemma 4.3 (iii). Then inf [I.sub.t] = z, this contradicts the definition of z. Therefore (3.6) is true. Now, if [x, y] [intersection] S [not equal to] [empty set], then by using (3.6) and Lemma 4.5 (iv), (v), we have (3.7) is true.

5 Limits and Continuity

In this section, we generalize the concepts of limits and continuity in the classical real analysis by introducing dense limit (d-limit) and dense continuity (d-continuity). Furthermore, we give analogous concepts of right dense and left dense continuity in the classical time scale calculus. Throughout the remainder of this work, we suppose that (T, E, [alpha]) is a time scale. Often, if there is no confusion concerning E and a, we denote (T, E, [alpha]) simply by T When we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we mean t, r [member of] T.

Definition 5.1. Let A be a subset of T. A point t [member of] T is called:

(i) a forward dense limit point, or shortly a fd-limit point, of A if T [member of] T \ {[alpha]} and it is a limit point of A [intersection] [T.sub.t] [intersection] ([alpha], t),

(ii) a backward dense limit point, or shortly a bd-limit point, of A if t [member of] t \ {[alpha], inf T, sup T} and it is a limit point of A [intersection] [T.sub.t] \ [[alpha], t], and

(iii) a dense limit point, or shortly d-limit point, of A if it is a limit point of A [intersection] [T.sub.t] .

The sets fd-L(A), bd-L(A) and d-L(A) denote all fd-limit, bd-limit, and d-limit points of A, respectively.

Note that d-L(A) = fd-L(A) [union] bd-L(A), A [subset equal to] T. In the Jackson, Norlund, Hahn difference time scales, d-LT = {0}, d-LT = [empty set], d-L(t) = {[[omega].sub.0]}, respectively. The following lemma investigates some properties of the dense limit points.

Lemma 5.2. Let t [[contains].bar] {t}. Then the following statements hold:

(i) If t [member of] d-LT, then [T.sub.t] is a neighborhood of t.

(ii) t [member of] d-LT iff one of the following cases holds: t is dense, t [member of] {inf T, sup T} is forward dense, t [member of] {inf T, sup T} is backward dense.

(iii) Ift = a, then t is a forward dense point ifft fd-LT.

(iv) If t [member of] {a, inf T, sup T}, then t is a backward dense point iff t [member of] bd-LT.

Throughout the rest of the paper, X denotes a Banach space.

Definition 5.3. Let f : T [right arrow] X. Assume that t [member of] fd-LT (t [member of] bd-LT). We say that the forward (backward) dense limit; or just fd-limit (bd-limit); of f exists at t, if there is an element l [member of] X such that for every [epsilon] > 0 there is a neighborhood U of t in [T.sub.t] such that:

[paralle]f (r) - l[paralle] < [epsilon] for all r [member of] [intersection] ([alpha], t) (r [member of] U \ [[alpha], t]).

In this case, we write fd-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) = l (bd-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(r) = l). For t [member of] d-LT, we say that the dense limit; or just d-limit; of f exists at t, if there is l [member of] X such that for every [epsilon] > 0 there is a neighborhood U of t in [T.sub.t] such that:

[parallel]f (r) - l[parallel] < [epsilon] for all r [epsilon] U \ {t}.

Here, we write d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) = l. When t [not member of] d-LT, then we put d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) := f (t), and when t [not member of] fd-LT (t [not member of] bd-LT) and t [not equal to] [alpha], then we put fd-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) := f (t) (bd-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r):= f (t)).

For more details about d-limits, see [5,6].

Example 5.4. Consider the time scale (T, E, a) which is defined in Example 3.17(v). Recall that [T.sub.1] = (0, [infinity]), [T.sub.0] = [0,1), and [T.sub.t] = [t] for any t [member of] T \ {0,1}. Note that t [not member of] T for t [member of] T \ {0,1}, 0 [member of] [T'.sub.0] and 1 [member of] [T'.sub.1], i.e., d-LT = S = {0,1}. Assume that f : T [right arrow] R, then

d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r), d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r),

when these limits exist. Now, if we define f by

f(t) = {t, if t [member of] T [intersection] Q, f(t) = {-t, otherwise,

then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) does not exist for any t [member of] T, but d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(r) = f(t) for t [member of] T \ {0,1}.

Definition 5.5. Let f : T [right arrow] X. Then f is called:

(i) Regulated on A [subset] T if the limits fd-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) for t [member of] fd-L(A) and bd-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) for t [member of] bd-L(A) exist, and if [alpha] [member of] d-L(A) then d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f (r) exist.

(ii) Dense continuous; or just d-continuous; at a point t [member of] T if for any [epsilon] > 0, there is a neighborhood U of t in [T.sub.t] such that [parallel]f (r) - f (t)[parallel] < [epsilon] whenever r [member of] U and f is said to be d-continuous on A [subset equal to] T if it is d-continuous at all points of A.

(iii) Forward dense continuous; or just fd-continuous; on A [subset equal to] T provided it is regulated on A and d-continuous at all forward dense points in A.

(iv) Backward dense continuous; or just bd-continuous; on A [subset] T provided it is regulated on A and d-continuous at all backward dense points in A.

Remark 5.6. For any function f : T [right arrow] X, we note the following:

(i) f is d-continuous on A [subset equal to] T iff it is continuous at all points of d-L(A), in view of Lemma 5.2 (i).

(ii) f is d-continuous on A [subset equal to] T iff it is d-continuous at all forward dense points in A and at all backward dense points in A, i.e., f is d-continuous on A iff f is fd-continuous and bd-continuous on A.

(iii) In general, fd-continuity (bd-continuity) does not imply d-continuity.

(iv) Continuity implies d-continuity, the converse is not necessarily true. In Example 5.4, f is d-continuous at points of T {0,1} and discontinuous at all points of T.

(v) f is fd-continuous on A [subset equal to] T iff f is d-continuous at all forward dense points in A and all its bd-limits exist at all backward dense points in A \ {inf T, sup T}.

We can show the following lemma, see [5].

Lemma 5.7. The following statements are true:

(i) [sigma] and [rho] are nondecreasing.

(ii) [sigma] and [rho] are fd-continuous and bd-continuous, respectively.

(iii) If f : T [right arrow] R is a regulated (fd-continuous) function, then so is f [omicron] [sigma].

(iv) Let [{[f.sub.n]}.sub.n[member of]N] be a sequence of functions which is locally uniformly convergent to f. If [f.sub.n] is a regulated, fd-continuous or d-continuous for all n [member of] N, then so is f.

6 Differentiation

In this section, we define the [DELTA]-derivative on a time scale (T, E, a) and we study their main properties. In this setting, the [DELTA]-derivative generalizes both the A and [nabla]-derivatives of the classical time scales calculus. Throughout this section, we suppose that (T, E, [alpha]) is a time scale and X is a Banach space. We denote by

[T.sup.k] = {T \ {[alpha]}, if [alpha] [member of] {inf T, sup T} is backward scattered, [T.sup.k] = {T, otherwise.

For a subset A [subset equal to] T, we denote the set A [intersection] [T.sup.k] by [A.sup.k].

Definition 6.1 (The delta derivative). A function f : T [right arrow] X is called delta differentiate ([DELTA]-differentiable) at t [member of] [T.sup.k] if f is d-continuous at t and

[f.sup.[DELTA]](t) := d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f([sigma](t)) - f(r) / [sigma](t) - r

exists. We say that [f.sup.[DELTA]](t) is the delta derivative ([DELTA]-derivative) of f at t. For A [subset equal to] T, if f is [DELTA]-differentiable at every t [member of] [A.sup.k], then f is said to be [DELTA]-differentiable on [A.sup.k].

Now, we state without proofs some relationships concerning the [DELTA]-derivative, see [5].

Theorem 6.2. Assume that f : T [right arrow] X and t [member of] [T.sub.k]. The following statements are true:

(i) Iff is [DELTA]-differentiable at t, then f is d-continuous at t.

(ii) Iff is d-continuous at a forward scattered point t, then it is [DELTA]-differentiable at t and

[f.sup.[DELTA]](t) = f([sigma](t)) - f(t) / [sigma](t) - t.

(iii) Iff is [DELTA]-differentiable at a forward dense point t, then

[f.sup.[DELTA]](t) = d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(t) - f(r) / t - r.

(iv) Iff is [DELTA]-differentiable at t, then

f([sigma](t)) = f(t)+ [micro](t) [f.sup.[DELTA]](t). (6.1)

Remark 6.3. In Example 5.4, the function

f(t) = {t, if t [member of] T [member of] Q,

f(t) = {-t, otherwise,

is [DELTA]-differentiable on T \ {0, 1}, but it is not continuous, in the ordinary sense, at any point of t in stark contrast to the classical time scales calculus, where f is [DELTA]-differentiable at t [member of] T implies that it is continuous at that point.

Example 6.4. (i) In the Jackson difference time scale, we have [T.sub.0]' = R and [T.sub.t]' = [t]' = {0} which implies that d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(r) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(r) and d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(r) = f(t) for any t [member of] r \ {0} when these limits exist. Note also that [T.sup.k] = t, since [alpha] = 0, and then

[f.sup.[DELTA]](t) = {fq(t)- f(t) / t(q - 1), if t [not equal to] 0,

[f.sup.[DELTA]](t) = {f' (0), t = 0.

Therefore for t [member of] R, we have [f.sup.[DELTA]](t) = [D.sub.q]f(t), where [D.sub.q] is the Jackson q-difference operator which is defined in (2.1).

(ii) Consider the Norlund difference time scale (T, E, [infinity]). Here, for any t [member of] T, we have [T.sub.t]' = [t]' = [empty set] which implies fd-L(A) = bd-L(A) = d-L(A) = [empty set] for A [subset equal to] r. For a function f : T [right arrow] X and t [member of] T, we have d-[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] f(r) = f(t). Easily, we can see that [T.sup.k] = T. Recall that [sigma](t) = t + [omega] for any t [member of] T, then by Definition 6.1 we have

[f.sup.[DELTA]](t) = f (t + [omega]) - f(t) / [omega] = [[DELTA].sub.[omega]] f(t),

where [[DELTA].sub.[omega]] is the usual forward difference operator with step size [omega], see [12,30, 31,36].

(iii) In the Hahn difference time scale (T, E, [[omega].sub.0]), [T.sub.[omega]0] = {t [member of] T : [[omega].sub.0] [member of] [bar.[t]]} = R, [T.sub.t] = [t] for any t [not equal to] [[omega].sub.0], and [T.sup.k] = T. Here, fd-LT = bd-LT = d-LT = {[[omega].sub.0]}. Then by Definition 6.1,

[f.sup.[DELTA]](t) = {(qt + [omega]) - f (t) / t(q - 1) + [omega], if t [not equal to] [[omega].sub.0], [f.sup.[DELTA]](t) = {f' ([[omega].sub.0]), if t = [[omega].sub.0].

Therefore for t [member of] R, we have [D.sub.q,[omega]]f(t) = [f.sub.[DELTA]](t), where [d.sub.q,[omega]] is the Hahn difference operator which is defined in (2.10).

Lemma 6.5. Let f : T [right arrow] X and t [member of] [T.sub.k]. Then f is [DELTA]-differentiable at t iff there is l [member of] X with the property that given [epsilon] > 0, there is a neighborhood U of t in [T.sub.t] such that:

[parallel]f([sigma](t)) - f(r) - l([sigma](t) - r)[parallel] < [epsilon] [absolute value of [sigma](t) - r] for all r [member of] U. (6.2)

In this case l = [f.sup.[DELTA]](t).

In the next theorem, we calculate the [DELTA]-derivative of sums, products and quotients of [DELTA]-differentiable functions.

Theorem 6.6. Assume that f, g : T [right arrow] X are [DELTA]-differentiable at t [member of] [T.sub.k] then:

(i) The sum f + g is [DELTA]-differentiable at t and

(f + g)[sigma](t) = f [sigma](t) + g[sigma](t).

(ii) For any constant c, cf : T [right arrow] X is [DELTA]-differentiable at t and

[(cf).sup.[DELTA]](t) = [cf.sup.[DELTA]](t).

(iii) If X = R, the product fg : T [right arrow] R is [DELTA]-differentiable at t and

[(fg).sup.[DELTA]](t) = [f.sup.[DELTA]](t)g(t) + f([sigma](t))[g.sup.[DELTA]](t) = f(t)[g.sup.[DELTA]](t) + [f.sup.[DELTA]](t)g([sigma](t)).

(iv) If X = R and g(t)g([sigma](t)) [not equal to] 0, then f/g is [DELTA]-differentiable at t and

[(f / g).sup.[DELTA]](t) = [f.sup.[DELTA]](t)g(t) - f(t)[g.sup.[DELTA]](t) / g(t)g([sigma](t)).

Example 6.7. Let (T, E, [alpha]) be a time scale. For n [member of] [N.sub.0], by induction one can show easily that

[([(at + b).sup.n]).sup.[DELTA]] = [alpha] [n-1.summation over (k=0)] [(a[sigma](t) + b).sup.-n+k] [(at + b).sup.n-k-1]. (6.3)

By using Theorem 6.6 (iv) and (6.3), we obtain that

[([(at + b).sup.-n]).sup.[DELTA]] = - [alpha] [n-1.summation over (k=0)] [(a[sigma](t) + b).sup.-n+k] [(at + b).sup.-k-1], (6.4)

where a, b [member of] R, provided that (a[sigma](t) + b) (at + b) [not equal to] 0.

Definition 6.8. Let A be a closed subset of T. A function F : T [right arrow] X is called pre [DELTA]-differentiable on A with region D if:

(D1) F is d-continuous on A,

(D2) F is [DELTA]-differentiable on D, and

(D3) D [subset equal to] [A.sub.k] and [A.sub.k] \ D is a countable subset, that contains no forward scattered points from A.

A pre [DELTA]-differentiable function F on A with region D is called pre [DELTA]-antiderivative of f : T [right arrow] X on A if:

[F.sup.[DELTA]](t) = f(t) for all t [member of] D.

Finally, we say that a function F : A [right arrow] X is a [DELTA]-antiderivative of f on A if [F.sup.[DELTA]](t) = f(t) for all t [member of] [A.sub.k].

Remark 6.9. In [5], the author defined [NABLA]-derivative and established analogues of the previous theorems and relationships of this section. Also, he gave analogues of Leibniz' formula, mean value theorem and chain rules of the classical time scale.

7 Integration

This section introduces the theory of [DELTA]-integration on a time scale <T, E, [alpha]>. This theory extends the theory of integration derived by Hilger [24, 25] to include some new examples. To simplify the discussion, we provide the following notation:

[[alpha].sub.c] = (inf [bar.[c]] if [bar.[c]] [subset equal to] [[alpha], sup T] and [[gamma].sub.c] = (sup [bar.[c]] if [bar.[c]] [subset equal to] [[alpha], sup T],

[[alpha].sub.c] = (sup [bar.[c]] if [bar.[c]] [subset equal to] [inf T, [alpha]], and [[gamma].sub.c] = (inf [bar.[c]] if [bar.[c]] [subset equal to] [inf T, [alpha]]. (7.1)

That is, [[alpaha].sub.c] and [[gamma].sub.c] are the nearest and the farthest points respectively in [bar.[c]] from a. Assume that (T, E, [alpha]) is a time scale throughout this section.

Remark 7.1. For any class [c] of T, we denote by [E.sub.c], the universal equivalence relation on [bar[c]]. The triple <[c], [E.sub.c], [[alpha].sub.c]] is a time scale. For any function f defined on T we have [f.sup.[DELTA]](t) = [f.sup.[DELTA]c](t) for any t [member of] [c] \ S where [DELTA] and [DELTA]c are the delta derivatives in the time scales <T, E, [alpha]> and <[bar.[c]], [E.sub.c], [[alpha].sub.c]>, respectively. Here S is the set of special points in the time scale <T, E, [alpha]>.

Now, we give the main existence theorems for [DELTA]-antiderivatives, which is the precursor to defining [DELTA]-integration, see [5]. Throughout this section, X is a Banach space.

Theorem 7.2. Let f : T [right arrow] X be a regulated function and c [member of] T. For t [member of] [c] and x [member of] X there exists a unique pre [[DELTA].sub.c]-differentiable function [F.sub.c] : [c] [right arrow] X with region [D.sub.c] [subset equal to] [bar.[c]] such that

[F.sup.[DELTA]c.sub.c](t) = f(t) for all t [member of] [D.sub.c] and [F.sub.c]T = x.

Theorem 7.3 (Existence of [DELTA]-antiderivative). Assume that f : T [right arrow] X is fd-continuous and c [member of] T. Then f has a [[DELTA].sub.c]-antiderivative function [F.sub.c] on [bar.[c]].

The previous theorems are analogues for the results in classical time scales calculus, see [13, 25].

Assume that RG(T, X) is the set of all regulated functions from T to X. Let f [member of] RG(T, X) and [F.sub.c] be a pre Ac-antiderivative of f on [c], c [member of] T which exists by Theorem 7.2. In view of Remark 7.1, [F.sub.c] is a pre [DELTA]-antiderivative of f on [bar.[c]] \ S.

Our approach for integration depends on the choice of some points [c.sub.I], I [member of] I. This is similar to (2.6) and (2.7). From now on, for any I [member of] I, we fix a point [c.sub.I] [member of] I. For a, b [member of] T, by Theorem 3.13 and Lemma 4.5 (iv), there are m special points in the interval [a, b], m [member of] [N.sub.0]. If m [greater than or equal to] 1, then we denote those special points by [{[s.sub.i]}.sup.m.sub.i=1] such that either

[a, [s.sub.1]] [much greater than] [[s.sub.1], [s.sub.2]] [much greater than] ... [much greater than] [[s.sub.m-1], [s.sub.m]] [much greater than] [[s.sub.m], b] (7.2)

or

[[s.sub.b] [much greater than] [[s.sub.m-1], [s.sub.m]] [much greater than] ... [much greater than] [[s.sub.1], [s.sub.2]] [much greater than] [a, [s.sub.1]]. (7.3)

For m [greater than or equal to] 1 , put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.4)

where I [member of] [bar.[I.sub.i]] I = [[s.sub.i], [s.sub.i+1]] for i = 1, ... , m. For m = 0, we use the convention

[s.sub.0] = [c.sub.0] = [alpha] and [s.sub.1] = [c.sub.0]' = b. (7.5)

Define the Cauchy [DELTA]-integral of f from a to b by

[[integral].sub.a.sup.b] f(t)[DELTA]t := [m.summation over (i=0)] {[F.sub.[c.sub.i]'] ([s.sub.i+1]) - [F.sub.[c.sub.i]]([s.sub.i])}. (7.6)

Remark 7.4. Notice that this definition yields the Cauchy [DELTA]-integral (resp. [NABLA]-integral) in the classical time scales calculus when E is the universal equivalence relation and [alpha] = sup T (resp. [alpha] = inf T). Indeed, we have no special points, i.e., m = 0. Also, there is only one class [c] = T, c [member of] T. Consequently, if F is a pre [DELTA]-antiderivative of f on T, then F is also a pre [DELTA]-antiderivative of f on [bar.[t]] for any t [member of] T.

By applying (7.6), one can obtain directly the following lemma.

Lemma 7.5. For a, b [member of] T, the Cauchy [DELTA]-integral satisfies the following properties:

(i) If [a, b] [intersection] S = [empty set], then [[integral].sub.a.sup.b] f(t)[DELTA]t = - [F.sub.b](b) - [F.sub.a](a).

(ii) If [a, b] [intersection] S = {a, b}, then [[integral].sub.a.sup.b] f(t)[DELTA]t = - [[integral].sub.a.sup.b] (t)[DELTA]t = [F.sub.[c.sub.I]](b) - [F.sub.[c.sub.I]](a), where I [member of] I with [bar.I] = [a, b].

(iii) If [a, b] [intersection] S = {b}, then [[integral].sub.a.sup.b] f(t)[DELTA]t = - [[integral].sub.a.sup.b] (t)[DELTA]t = [F.sub.a](b) - [F.sub.a](a).

(iv) If [a, b] [intersection] S = [{[s.sub.i]}.sup.m.sub.i=1], m > 1, then

[[integral].sub.a.sub.b] f(t)[DELTA]t = [F.sub.b](b) - [F.sub.b]([s.sub.m]) + [m-1.summation over (i=1)] {[F.sub.[c.sub.I]]([s.sub.i+1]) - [F.sub.[c.sub.I]]([s.sub.i])} + [F.sub.a] ([s.sub.1]) - [F.sub.a](a).

Using these relations, (7.6) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.7)

when m [greater than or equal to] 1.

In the following, we define the improper [DELTA]-integrals.

Definition 7.6. Let a [member of] T. If sup T = [integral], we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] otherwise, (7.8)

whenever the limits exist. Similarly, when inf t = -[infinity], we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] otherwise, (7.9)

whenever the limits exist.

In the next theorems, we give some properties of the Cauchy [DELTA]-integral, for the proofs see [5]. The first theorem is an analogue of [13, Theorem 1.77],

Theorem 7.7. Let f, g : T [right arrow] X be fd-continuous functions. For k [member of] R and a, b [member of] T, we have the following:

(i) [[integral].sub.a.sup.a] f(t)[DELTA]t = 0.

(ii) [[integral].sub.a.sup.b] kf(t)[DELTA]t = k [[integral].sub.a.sup.b] f(t)[DELTA]t.

(iii) [[integral].sub.a.sup.b] f(t)[DELTA]t = [[integral].sub.b.sup.a] f(t)[DELTA]t.

(iv) [[integral].sub.a.sup.b] f(t)[DELTA]t = [[integral].sub.a.sup.c] f(t)[DELTA]t + [[integral].sub.c.sup.b] f(t)[DELTA]t when [a, c] [intersection] S = [empty set], [b, c] [intersection] S = [empty set] or c [member of] S [union] {inf T, sup T}.

(v) [[integral].sub.a.sup.b] (f(t) + g (t)) [DELTA]t = [[integral].sub.a.sup.b] f(t)[DELTA]t + [[integral].sub.a.sup.b] g(t)[DELTA]t.

(vi) If X = R, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (integration by parts formulas).

(vii) If X = R, a, b [member of] [bar.[c]] and g(t) [greater than or equal to] [absolute value of f(t)] on [[a, b].sub.c], then [[integral].sub.a.sup.b] g(t) [DELTA]t [greater than or equal to] [absolute value of [[integral].sub.a.sup.b] f(t)[DELTA]t].

Theorem 7.8. Let f and g be fd-continuous functions defined on a time scale T. Then, the following statements are true:

(i) If t [member of] T is isolated, then

[[integral].sub.t.sup.[sigma](t)] f(s)[DELTA]s = [micro](t)f(t) and [[integral].sub.[rho](t).sup.t] f(s)[DELTA]s = [micro]([rho](t))f([rho](t)).

(ii) If a, b [member of] [c] such that b [member of] (a, [alpha]) and all points of [[a, b].sub.c] are isolated, then

[[integral].sub.a.sup.b] f(t)[DELTA]t = [summation over t[member of][[a, b).sub.c]] [micro](t)f(t).

(iii) If [c][approximately equal to][c], then for any a [member of] [c] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.10)

provided that these sums are convergent.

(iv) If I is an isolated nonsingular interval, [bar.I] = [r, s] and r [member of] (s, [alpha]], then

[[integral].sub.r.sup.s] f(t)[DELTA]t = - [[infinity].summation over (k=-[infinity])] [micro]([[sigma].sup.k](([c.sub.I]))f([[sigma].sup.k]([c.sub.I])), (7.11)

provided that this sum is convergent (here [[sigma].sup.-1] := [rho]).

Example 7.9. In q-difference time scale, if we choose [c.sub.(0,[infinity])] := c and [c.sub.(-[infinity],0)] := -c

for fixed c [member of] (0, [infinity]), then by Theorem 7.8, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.16)

Formulas (7.12), (7.13), (7.14) and (7.16) are the Jackson q-integrals, setting c =1, provided that the sums are convergent [1,4,29,34]. Therefore, if we fix [c.sub.(0,[infinity])] = [-c.sub.(-[infinity],0)] = 1, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, one can check that for any fd-continuous f, the function F(t) := [[infinity].summation over (k=0)] t[q.sup.k] (1 - q)f(t[q.sup.k]) is a [DELTA]-antiderivative of f on T.

Example 7.10. Assume that T is the Norlund difference time scale. For any t [member of] T, [[sigma].sup.[infinity]](t) = [infinity], then by Theorem 7.8 (iii), the [DELTA]-antiderivative of any function f : T [right arrow] X is

F(t) = -[omega] [[infinity].summation over (k=0)] f(t + k[omega]),

provided this sum is convergent. This function is the well known Norlund sum, see [20,30,38]. If [omega] = 1, then obtain the so called indefinite sum in [31]. Therefore, for a, b T, we have

[[integral].sub.a.sup.b] f(t)[DELTA]t = [[infinity].summation over (j=0)] [omega][f (a + j[omega]) - f(b + j[omega])],

and if we choose [c.sub.(-[infinity],[infinity])] = 0, then

[[integral].sub.a.sup.b] f(t)[DELTA]t = w [[infinity].summation over (k=-[infinity])] f(k[omega]).

Example 7.11. Let T be the Hahn difference time scale. For any x [member of] T, [[alpha].sub.x] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [h.sup.n](x) = [[omega].sub.0], and [x] [approximately equal to] [x], then by Theorem 7.8 (iii),

[[integral].sub.a.sup.b] f(t)[DELTA]t = (x(1 - q) - [omega]) [[infinity].summation over (k=0)] [q.sup.k]f([xq.sup.k] + [omega][[k].sub.q]),

and by Theorem 7.7 (iv), we have

[[integral].sub.a.sup.b] f(t)[DELTA]t = [[integral].sub.[[omega].sub.0].sup.b] f(t)[DELTA]t - [[integral].sub.[[omega].sub.0].sup.a] f(t)[DELTA]t,

provided that the series converges at x = a and x = b. Therefore,

[[integral].sub.a.sup.b] f(t)[DELTA]t = [[integral].sub.a.sup.b] f(t)[d.sub.q,[omega]]t, a,b [member of] R (7.17)

where q, [omega]-integral is defined in (2.11) and (2.12).

Example 7.12. In Example 3.17 (v), recall that there are two nonsingular intervals (0,1), (1, [infinity]) and two special points 0, 1. For simplicity, we put [c.sub.(0,1)] := [c.sub.1] and [c.sub.(1,[infinity])] := [c.sub.2]. We have two cases. First case: [alpha] = [infinity]. By (7.11) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.18)

If x [member of] (0,1) (x [member of] (1, [infinity])), then [[alpha].sub.x] = 0 ([[alpha].sub.x] = [infinity]) and [[gamma].sub.x] = 0 ([[gamma].sub.x] = 1). By (7.10), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.21)

Second case [alpha] = 1. In this case [[alpha].sub.x] = [alpha] for all x [member of] (0, [infinity]). By Theorem 7.8 (iii), we have the following integrals:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.24)

Finally, by (7.11), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7.26)

In the two cases, [alpha] = [infinity] and [alpha] = 1, using (7.7) and Theorem 7.7 (iii), we have

[[integral].sub.a.sup.b] f(t)[DELTA]t = [[integral].sub.a.sup.1] f(t)[DELTA]t - [[integral].sub.a.sup.b] f(t)[DELTA]t, a, b [member of] (0, [infinity]). (7.27)

In the above, we assumed that all sums are convergent.

For the definition and theorems of [NABLA]-integral of this setting, see [5].

Acknowledgements

The first author acknowledges support from the Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia, where he is currently visiting while being on leave from Hajjah University, Hajjah, Yemen.

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K. A. Aldwoah

Hajjah University Department of Mathematics Hajjah, Yemen aldwoah@yahoo.com

A. E. Hamza

Cairo University Department of Mathematics Giza, Egypt hamzaaeg2003@yahoo.com

Received October 12, 2010; Accepted February 18, 2011 Communicated by Peter Kloeden
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Author:Aldwoah, K.A.; Hamza, A.E.
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