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Time scales inequalities.

1 Preliminaries

Here mainly we follow [6]. We are also inspired by [4,5].

Definition 1.1. A time scale is an arbitrary nonempty closed subset of the real numbers, e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.2. If T is a time scale, then we define the forward jump operator [sigma] : T [??] T by [sigma] (t) = inf {s [member of] T|s > t}, [for all]t [member of] T; the backward jump operator [rho] : T [??] T by [rho] (t) = sup{s [member of] T|s < t), [for all]t [member of] T; and the graininess function [mu] : T [right arrow] [R.sub.+] = [0, [infinity]), by [mu] (t) = [sigma] (t) - t, [for all]t [member of] T. Furthermore for a function f : T [right arrow] R, we define [f.sup.[sigma]] (t) = f ([sigma] (t)), [for all]t [member of] T; and [f.sup.[rho]] (t) = f ([rho] (t)), [for all]t [member of] T. In this definition we use inf 0 = sup T (i.e., [sigma] (t) = t if t is the maximum of T) and sup 0 = inf T (i.e., [rho] (t) = t if t is the minimum of T).

We call T [member of] T right-scattered if t < [sigma] (t), T [member of] T right-dense if t = [sigma] (t), T [member of] T left-scattered if [rho] (r) < t, T [member of] T left-dense if [rho] (t) = t, T [member of] T isolated if [rho] (t) < t < [sigma] (t), T [member of] T dense if [rho] (t) = t = [sigma] (t). We notice that [rho] is an increasing function, so is [[rho].sup.2] (t) = p ([rho] (t)),..., so that [[rho].sup.n] (t) = [rho] ([[rho].sup.n-1] (t)) is increasing in t for n [member of] N. Since T is closed subset of R we have that [sigma] (t), [rho] (t) [member of] T, for T [member of] T.

Definition 1.3 (see [6]). A function f : T [right arrow] R is called rd-continuous (denoted by [C.sub.rd]) if it is continuous at right-dense points of T and its left-sided limits are finite at left-dense points of T. If T = R, then f : R [right arrow] R is rd-continuous iff f is continuous. Also, if T = Z, then any function defined on Z is rd-continuous (see [7]).

Definition 1.4 (see [6]). If sup T < [infinity] and sup T is left-scattered, we let [T.sup.k] := T -{sup T}, otherwise we let [T.sup.k] := T the time scale. In summary,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.5 (see [6]). Assume f : T [right arrow] R is a function and let T [member of] [T.sup.k]. Then we define [f.sup.[DELTA]] (t) to be the number (provided it exists) with the property that given any [epsilon] > 0, there is a neighborhood U of t such that

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

We call [f.sup.[DELTA]] (t) the delta (or Hilger [8]) derivative of f at t. If T = R, then [f.sup.[DELTA]] = f', whereas if T = Z, then [f.sup.[DELTA]] (t) = [DELTA]f (t) = f (t + 1) - f (t), the usual forward difference operator.

Theorem 1.6 (Existence of Antiderivatives [6]). Let f be rd-continuous. Then f has an antiderivative F satisfying [F.sup.[DELTA]] = f.

Definition 1.7 (see [6]). If f is rd-continuous and [t.sub.0] [member of] T, then we define the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore for f [member of] [C.sub.rd] (T) we have by definition

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

where [F.sup.[DELTA]] = f. If T = R, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the integral on the right-hand side is the Riemann integral [7].

If every point in T is isolated and a < b are in T, then [7]

[[integral].sup.b.sub.a] f (t) [DELTA]t = [[rho](b).summation over (t=a)] f (t) [mu] (t).

Theorem 1.8 (see [6]). Let f, g be rd-continuous on T, a,b,c [member of] T and [alpha], [beta] [member of] R. Then

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

(6) [[integral].sup.b.sub.a] 1[DELTA]t = b - a.

Theorem 1.9 (Holder's Inequality [1]). Let a, b [member of] T, a [less than or equal to] b, and f, g : T [right arrow] R be rd-continuous. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where p,q > 1 satisfy [1/p] + [1/q] = 1.

Theorem 1.10 (see [6]). Let f,g [member of] [C.sub.rd] (T), a,b [member of] T, a [less than or equal to] b. Then

1) if [absolute value of f (t)] [less than or equal to] g (t) on [a,b) [intersection] T, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

2) if f (t) [greater than or equal to] 0, for all a [less than or equal to] t < b and T [member of] T, then [[integral].sup.b.sub.a] f (t) [DELTA]t [greater than or equal to] 0.

Corollary 1.11. Let f [member of] [C.sub.rd] (T) ; a,b,c [member of] T, with c [member of] [a, b]; f (t) > 0, [for all] t [member of] [a, b]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.12 (see [6]). For a function f : T [right arrow] R we consider the second derivative [f.sup.[DELTA][DELTA]] provided [f.sup.[DELTA]] is differentiable on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with derivative [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly we define higher order derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly we define [[sigma].sup.2] (t) = [sigma] ([sigma] (t)), ..., [[sigma].sup.n] (t) = [sigma] ([[sigma].sup.n-l] (t)), n [member of] N. For convenience we put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notice [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 1.13 (Taylor's Formula [2]). Let f be n-times differentiable on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], t [member of] T, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 1.14 (see [2]). Let f be n-times differentiable on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and m [member of] N with m < n. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote by [C.sup.n.sub.rd] (T) the space of all functions f [member of] [C.sub.rd] (T) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 1,...,n [member of] N. In this last case [T.sup.k] = T.

We need the following result.

Theorem 1.15 (Taylor's Formula [3,7]). Assume [T.sup.k] = T and f [member of] [C.sup.n.sub.rd] (T), n [member of] N and s, t [member of] T. Here [h.sub.0] (t, s) = 1, [for all] s, t [member of] T; k [member of] [N.sub.0], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 1.16 (to Theorem 1.15). By [7], we have [h.sub.1] (t,s) = t - s, [for all] s, t [member of] T. So if t [greater than or equal to] s, then [h.sub.1] (t, s) [greater than or equal to] 0, [h.sub.2] (t,s) [greater than or equal to] 0,..., [h.sub.n-1] (t, s) [greater than or equal to] 0. However for n odd, [h.sub.n-1] (t, [sigma] ([tau])) [greater than or equal to] 0 for all s [less than or equal to] [tau] [less than or equal to] t (see proof of Theorem 2.5). Also it holds [2]

[h.sub.k] (t,s) [less than or equal to] [[(t - s).sup.k]/k!], [for all] t [greater than or equal to] s, k [member of] [N.sub.0].

Corollary 1.17 (to Theorem 1.15). Assume f [member of] [C.sup.n.sub.rd] (T) and s, t [member of] T. Let m [member of] N with m < n. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Use Theorem 1.15 with n and f substituted by n - m and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively.

Corollary 1.18. Let f [member of] [C.sub.rd] (T); a,b [member of] T, such that f (t) > 0, [for all] t [member of] [a, b] [intersection] T, then [[integral].sup.b.sub.a] f (t) [DELTA]t> 0.

Proof. Since f (t) > 0, [for all] t [member of] [a, b] [intersection] T by Theorem 1.10 (2) we get [[integral].sup.b.sub.a] f (t) [DELTA]t [greater than or equal to] 0.

Assume that [[integral].sup.b.sub.a] f (t) [DELTA]t = 0. Then F (t) = [[integral].sup.t.sub.a] f (t) [DELTA]t = 0, [for all] t [member of] [a, b] [intersection] T. Thus by [6] we get [F.sup.[DELTA]] (t) = f (t) = 0, [for all] t [member of] [a,b] [intersection] T, a contradiction.

We need the following result.

Lemma 1.19. Let the time scale T be such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [h.sub.k] (t, s) is continuous in s [member of] T, k [member of] [N.sub.0], for each fixed t [member of] T; and continuous in t [member of] T for each fixed s [member of] T. Also it holds that [h.sub.k] (t, [sigma] (s)) is rd- continuous in s [member of] T for each fixed t [member of] T; for all k [member of] [N.sub.0].

Proof. Consider also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have that [h.sup.[DELTA].sub.k] (t, s) = [h.sub.k-1] (t,s), k [member of] N, [for all] t [member of] T,

for each fixed s [member of] T. Also we have

[g.sup.[DELTA].sub.k] (t,s) = [g.sub.k-1] ([sigma] (t), s), k [member of] N, [for all] t [member of] T,

for each fixed s [member of] T. Clearly [g.sub.1] (t, s) = [h.sub.1] (t,s) = t - s, [for all] s, t [member of] T. By [6, Theorem 1.112] we obtain that

[h.sub.k] (t,s) = [(-1).sup.k] [g.sub.k] (s,t), [for all] t, s [member of] T, for all k [member of] [N.sub.0].

By [6, Theorem 1.16(i)], we have that since [g.sub.k] is differentiable for any t [member of] T (the first variable), then it is continuous for any T [member of] T; for all k [member of] [N.sub.0]. Thus, by the last equation just above, we obtain that [h.sub.k] (t, s) is continuous in s [member of] T; and of course [h.sub.k] is also continuous in t [member of] T; for all k [member of] [N.sub.0]. By [6, Theorem 1.60(iii)], we have that the jump operator [sigma] is rd-continuous, and by the same [6, Theorem 1.60(v)], we get that [h.sub.k] (t, a (s)) is rd-continuous, for all k [member of] [N.sub.0]. The lemma now is proved.

2 Main Results

In this article we assume [T.sup.k] = T. We present first a time scales Poincare type inequality.

Theorem 2.1. Let f [member of] [C.sup.n.sub.rd] (T), n is an odd number, a, b [member of] T; a [less than or equal to] b; p, q > 1 satisfy [1/p] + [1/q] = 1. Assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here [sigma] is continuous and [h.sub.n-1] (t,s) jointly continuous. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by Theorem 1.15 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[for all] t [member of] [a,b] [intersection] T, where a, b [member of] T. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

for all a [less than or equal to] t [less than or equal to] b. Next by integrating (2.1) we are proving the claim.

Next we give a time scales Sobolev type inequality.

Theorem 2.2. Here all terms and assumptions are as in Theorem 2.1. Let r [greater than or equal to] 1. Denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. As in the proof of Theorem 2.1 we have (a [less than or equal to] t [less than or equal to] b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next raise both sides of the last inequality to the power 1/r, thus proving the claim.

We present a time scales Opial type inequality.

Theorem 2.3. Let f [member of] [C.sup.n.sub.rd] (T), n is an odd number, a, b [member of] T; a [less than or equal to] b; p, q > 1 satisfy [1/p] + [1/q] = 1. Assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is increasing on [a, b] [intersection] T. Here [sigma] is continuous and [h.sub.n-1] (t, s) jointly continuous. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. It holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[for all] t [member of] [a, b] [intersection] T, where a, b [member of] T. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all a [less than or equal to] t [less than or equal to] b. Consequently we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

proving the claim.

We make the following remark.

Remark 2.4 (to Theorems 2.1-2.3 and their proofs). As we know [6], we have that

[h.sup.[DELTA].sub.n-1] (t, [sigma] ([tau])) = [h.sub.n-2] (t, [sigma] ([tau])), [for all] t [member of] [a, b] [intersection] T.

Also [([h.sub.n-1] (t, [sigma] (t))).sup.p] is continuous at (t,t), t > a; p > 1. By the chain rule, [6, Theorem 1.90], we get that [([h.sub.n-1] [(t, [sigma] ([tau])).sup.p]).sup.[DELTA]] exists in T [member of] T, where [tau] is fixed in T; p > 1 , and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here by assumption [sigma] is continuous and [h.sub.n-1] (t,s) is jointly continuous. So that [([h.sub.n-1] (t, [sigma] ([tau]))).sup.p] is jointly continuous in (t, [tau]), that is rd-continuous in t and [tau]; p [greater than or equal to] 1. Here [T.sup.k] = T, and by Lemma 1.19 we get that [h.sub.n-2] (t, [sigma] ([tau])) is continuous in t and [tau]. By bounded convergence theorem, using the last formula above, we get that [([([h.sub.n-1] (t, [sigma] ([tau]))).sup.p]).sup.[DELTA]] is continuous in t and [tau]; p > 1, and thus rd-continuous in t and [tau].

Consider now the function

u (t) = [[integral].sup.t.sub.a] [h.sub.n-1] [(t, [sigma] ([tau])).sup.p] [DELTA][tau], [for all] t [member of] [a, b] [intersection] T.

Clearly u (a) = 0. Furthermore, by [6, Theorem 1.117], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, u (t) is differentiable, hence continuous and therefore rd-continuous on [a, b] [intersection] T.

We proceed with a time scales Ostrowski type inequality.

Theorem 2.5. Let f [member of] [C.sup.n.sub.rd] (T), n is odd, a, b, c [member of] T with a [less than or equal to] c [less than or equal to] b. Assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By assumptions and Theorem 1.15, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

However we observe that (c [less than or equal to] t [less than or equal to] b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is, [h.sub.2] (t, s) [greater than or equal to] 0, for any t, s [member of] T. We continue with (t [less than or equal to] s)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently by induction, we obtain (t [less than or equal to] s)

[absolute value of [h.sub.k] (t,s)] = [(-1).sup.k] [h.sub.k] (t,s), k [member of] [N.sub.0].

Thus [h.sub.k](t, s) [greater than or equal to] 0, for any t, s [member of] T, when k is even. Therefore when a [less than or equal to] t [less than or equal to] c, we derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By [6, (1.7), (1.8), (1.9)] and [6, Theorem 1.112], we notice that (c [less than or equal to] t [less than or equal to] b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also it holds (a [less than or equal to] t [less than or equal to] c)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So we got that (c [less than or equal to] t [less than or equal to] b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

proving the claim.

It follows a time scales Hilbert-Pachpatte type inequality.

Theorem 2.6. Let [epsilon] > 0, i = 1, 2; [f.sub.i] [member of] [C.sup.n.sub.rd] ([T.sub.i]), n is odd, with ffk (a,) = 0, k = 0,1,... ,n - 1; [a.sub.i] [less than or equal to] [b.sub.i]; [a.sub.i], [b.sub.i] [member of] [T.sub.i], time scale. Let also p, q > 1 such that [1/p] + [1/q] = 1.

Call

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all [t.sub.2] [member of] [[a.sub.2], [b.sub.2]] [intersection] [T.sub.2] (where [h.sup.(i).sub.n-1], [[sigma].sup.(i)] the corresponding [h.sup.n-1], [sigma] to [T.sub.i], i = 1 , 2). Here [[sigma].sub.i] is continuous and [h.sup.(i).sub.n-1] ([t.sub.i], [s.sub.i]) jointly continuous in [t.sub.i], [s.sub.i] [member of] [T.sub.i]. We further assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an rd-continuous function on [T.sub.1] . Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(above double time scales integration is considered in the natural iterative way). Proof. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by Theorem 1.15 we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Young's inequality for a, b [greater than or equal to] 0 says that

[a.sup.1/p] [b.sup.1/q] [less than or equal to] [a/p] + [b/q].

Therefore we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The last gives ([epsilon] > 0)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [t.sub.i] [member of] [[a.sub.i], [b.sub.i]] [intersection] [T.sub.i], i = 1, 2. Next we observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

proving the claim.

Based on Corollary 1.17 we get the following results. First we present a generalized time scales Poincare type inequality.

Proposition 2.7. Let f [member of] [C.sup.n.sub.rd] (T), m,n [member of] N, m < n, n - m is odd, a,b [member of] T; a [less than or equal to] b; p,q > 1 satisfy [1/p] + [1/q] = 1. Assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here [sigma] is continuous and [h.sub.n-m-1] (t,s) jointly continuous. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. As in Theorem 2.1.

It follows a generalized time scales Sobolev type inequality.

Proposition 2.8. Here all terms and assumptions are as in Proposition 2.7. Let r [greater than or equal to] 1.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. As in Theorem 2.2.

Next comes a generalized time scales Opial type inequality.

Proposition 2.9. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is increasing on [a, b] [intersection] T. Here [sigma] is continuous and [h.sub.n-m-1] (t,s) jointly continuous. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. As in Theorem 2.3.

We continue with a generalized Ostrowski type inequality over time scales.

Proposition 2.10. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. As in Theorem 2.5.

We finish with the generalized Hilbert-Pachpatte type inequality on time scales.

Proposition 2.11. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], k = 0,1,... ,n - m - 1; [a.sub.i] [less than or equal to] [b.sub.i]; [a.sub.i], [b.sub.i] [member of] [T.sub.i], time scale. Let also p,q> 1 satisfy [1/p] + [1/q] = 1. Call

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all [t.sub.2] [member of] [[a.sub.2], [b.sub.2]] [intersection] [T.sub.2] (where [h.sup.(i).sub.n-m-1], [[sigma].sup.(i)] the corresponding [h.sub.n-m-1], [sigma] to [T.sub.i], i = 1, 2). Here [[sigma].sub.i] is continuous and [h.sup.(i).sub.n-m-1] ([t.sub.i], [s.sub.i]) jointly continuous in [t.sub.i], [s.sub.i] [member of] [T.sub.i]. We further assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an rd-continuous function on [T.sub.1]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. As in Theorem 2.6.

3 Applications

We need the following remark.

Remark 3.1 (see [6]). i) When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; rd-continuous corresponds to f continuous.

ii) When [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

rd-continuous f corresponds to any f.

Next we present a Poincare inequality.

Corollary 3.2. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Proof. Based on Theorem 2.1 and Remark 3.1 (i).

A discrete Poincare inequality follows.

Corollary 3.3. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Based on Theorem 2.1 and Remark 3.1 (ii).

A Sobolev inequality is presented next.

Corollary 3.4. All as in Corollary 3.2. Let r [greater than or equal to] 1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Based on Theorem 2.2 and Remark 3.1 (i).

A discrete Sobolev inequality follows.

Corollary 3.5. All as in Corollary 3.3 and let r [greater than or equal to] 1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Based on Theorem 2.2 and Remark 3.1 (ii).

An Opial inequality follows.

Corollary 3.6. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Based on Theorem 2.3 and Remark 3.1 (i).

A discrete Opial inequality follows.

Corollary 3.7. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By Theorem 2.3 and Remark 3.1 (ii).

An Ostrowski inequality follows.

Corollary 3.8. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Based on Theorem 2.5 and Remark 3.1 (i).

A discrete Ostrowski inequality follows.

Corollary 3.9. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Theorem 2.5 and Remark 3.1 (ii).

A Hilbert-Pachpatte inequality is next.

Corollary 3.10. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. Based on Theorem 2.6 and Remark 3.1 (i). Notice here that [lambda] ([t.sub.1]) is a continuous function on [[a.sub.1], [b.sub.1]] by bounded convergence theorem.

It follows a discrete Hilbert-Pachpatte inequality.

Corollary 3.11. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By Theorem 2.6 and Remark 3.1 (ii).

Another generalized Poincare inequality is next.

Corollary 3.12. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.2, n [??] n - m, f [??] [f.sup.(m)], [f.sup.(k)] [??] [f.sup.(k+m)] into (3.1).

A generalized discrete Poincare inequality follows.

Corollary 3.13. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By Corollary 3.3.

A generalized Sobolev inequality is next.

Corollary 3.14. All as in Corollary 3.12, r [greater than or equal to] 1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By Corollary 3.4.

A generalized discrete Sobolev inequality follows.

Corollary 3.15. All as in Corollary 3.13, r [greater than or equal to] 1. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.5.

A generalized Opial inequality follows.

Corollary 3.16. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.6.

A generalized discrete Opial inequality follows.

Corollary 3.17. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.7.

A generalized Ostrowski inequality follows.

Corollary 3.18. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.8.

A generalized discrete Ostrowski inequality follows.

Corollary 3.19. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.9.

A generalized Hilbert-Pachpatte is next.

Corollary 3.20. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.10.

It follows a generalized discrete Hilbert-Pachpatte inequality.

Corollary 3.21. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. By Corollary 3.11.

Remark 3.22 (see [2, 6]). Consider q > 1, [q.sup.Z] = {[q.sup.k] : k [member of] Z}, and the time scale T = [q.sup.[bar.Z]] = [q.sup.Z] [union] {0}, which is very important in q-difference equations. It holds [2,6] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We give a related q-Ostrowski type inequality.

Corollary 3.23. Let f [member of] [C.sup.n.sub.rd] ([q.sup.[bar.Z]]), n is odd, a, b, c [member of] [q.sup.[bar.Z]] with a [less than or equal to] c [less than or equal to] b. Assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. By Theorem 2.5.

We finish with a generalized q-Ostrowski type inequality.

Corollary 3.24. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One can give many similar applications for other time scales.

Acknowledgment

The author wants to thank Professors M. Bohner and A. Peterson for helping him understand time scales.

References

[1] R. Agarwal, M. Bohner, and A. Peterson. Inequalities on time scales: A survey. Math. Inequal. Appl., 4(4):535-557, 2001.

[2] R. P. Agarwal and M. Bohner. Basic calculus on time scales and some of its applications. ResultsMath., 35(1-2):3-22, 1999.

[3] M. Bohner and G. Sh. Guseinov. The convolution on time scales. Abstr. Appl. Anal., 2007:24, Art. ID 54989, 2007.

[4] M. Bohner and B. Kaymakcalan. Opial inequalities on time scales. Ann. Polon. Math., 77(1):11-20, 2001.

[5] M. Bohner and T. Matthews. Ostrowski inequalities on time scales. JIPAM. J. Inequal. Pure Appl. Math., 9(1):8, 2008. Article 6.

[6] M. Bohner and A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston, 2001.

[7] R. Higgins and A. Peterson. Cauchy functions and Taylor's formula for time scales T. In B. Aulbach, S. Elaydi, and G. Ladas, editors, Proceedings of the Sixth International Conference on Difference Equations, pages 299-308, Boca Raton, FL, 2004. CRC.

[8] S. Hilger. Ein Mafikettenkalkiil mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis, Universitat Wurzburg, 1988.

George A. Anastassiou

University of Memphis

Department of Mathematical Sciences

Memphis, TN 38152, U.S.A.

ganastss@memphis.edu

Received December 15, 2009; Accepted February 5, 2010 Communicated by Martin Bohner
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Author:Anastassiou, George A.
Publication:International Journal of Difference Equations
Article Type:Report
Geographic Code:1USA
Date:Jun 1, 2010
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