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Time scales: from nabla calculus to delta calculus and vice versa via duality.

1 Introduction

The time scale delta calculus was introduced for the first time in 1988 by Hilger [9] to unify the theory of difference equations and the theory of differential equations. It was extensively studied by Bohner [4] and Hilscher and Zeidan [10] who introduced the calculus of variations on the time scale delta calculus (or simply delta calculus). In 2001 the time scale nabla calculus (or simply nabla calculus) was introduced by Atici and Guseinov [2].

Both theories of the delta and the nabla calculus can be applied to any field that requires the study of both continuous and discrete data. For instance, the nabla calculus has been applied to maximization (minimization) problems in economics [1, 2]. Recently several authors have contributed to the development of the calculus of variations on time scales (for instance, see [3,11,12]).

To the best of the author's knowledge there is no known technique to obtain results from the nabla calculus directly from results on the delta calculus and vice versa. In this note we underline that, in fact, this is possible. We show that the two types of calculus, the nabla and the delta on time scales, are the "dual" of each other. One can reciprocally obtain results for one type of calculus from the other and vice versa without making any assumptions on the regularity of the time scales (as it was done in [8]). We prove that results for the nabla (respectively the delta) calculus can be obtained by the dual analogous ones which will be in the delta (respectively nabla) context. Therefore, if they have already been proven for the delta case (respectively the delta), it is not necessary to reprove them for the nabla setting (respectively nabla).

This article is organized as follows: in the second section we review some basic definitions. In third section we introduce the dual time scales. In the fourth section we derive a few properties related to duality. In the fifth section we state the Duality Principle, which is the main result of the article, and we apply it to a few examples. Finally, in the last section, we apply the Duality Principle to the calculus of variations on time scales.

2 Review of Basic Definitions

We first review some basic definitions and hence introduce both types of calculus (for a complete list of definitions for the delta calculus see the pioneering book by Bohner and Peterson [5]).

A time scale T is any closed nonempty subset T of R.

The jump operators [sigma], [rho] : T [right arrow] T are defined by

[sigma](t) = inf{s [member of] T : s > t}, and [rho](t) = sup{s [member of] T : s < t},

with inf 0 := sup T, sup 0 := inf T. A point t [member of] T is called right-dense if [sigma](t) = t, right-scattered if [sigma](t) > t, left-dense if [rho](t) = t, left-scattered if [rho](t) < t.

The forward graininess [mu] : T [right arrow] R is defined by [mu](t) = [sigma](t) - t, and the backward graininess v : T [right arrow] R is defined by v(t) = t - [rho](t).

Given a time scale T, we denote [T.sup.[kappa]] := T \ ([rho](sup T), sup T], if sup T < [infinity] and [T.sup.[kappa]] := T if sup T = [infinity]. Also [T.sup.[kappa]] := T \ [inf T, [sigma](inf T)) if inf T > -[infinity] and [T.sup.[kappa]] =: T if inf T = -[infinity]. In particular, if a, b [member of] T with a < b, we denote by [a, b] the interval [a,b] [intersection] T. It follows that

[[a, b].sup.[kappa]] = [a, [rho](b)], and [[a, b].sub.k] =[[sigma](a),b].

Of course, R itself is one trivial example of time scale, but one could also take T to be the Cantor set. For more interesting examples of time scales we suggest reading [5].

Definition 2.1. A function f defined on T is called rd-continuous (or right-dense continuous) (we write f [member of] [C.sub.rd]) if it is continuous at the right-dense points and its left-sided limits exist (finite) at all left-dense points; f is ld-continuous (or left-dense continuous) if it is continuous at the left-dense points and its right-sided limits exist (finite) at all right-dense point.

2.1 Definition of Derivatives

Definition 2.2. A function f : T [right arrow] R is said to be delta differentiable at t [member of] [T.sup.[kappa]] if for all [epsilon] > 0 there exists U a neighborhood of t such that for some [alpha], the inequality

[absolute value of f ([sigma](t)) - f (s) - [alpha]([sigma](t) - s)] < [epsilon][absolute value of [sigma](t) - s],

is true for all s [member of] U. We write [f.sup.[DELTA]] (t) = [alpha].

Definition 2.3. f : T [right arrow] R is said to be delta differentiable on T if f : T [right arrow] R is delta differentiable for all t [member of] [T.sup.[kappa]].

It is easy to show that, if f is delta differentiable on T, then

[f.sup.[sigma]] = f + [mu][f.sup.[DELTA]],

where [f.sup.[sigma]] = f o [sigma] (the proof can be found in [5]).

Definition 2.4. A function f : T [right arrow] R is said to be nabla differentiable at t [member of] [T.sub.[kappa]] if for all [epsilon] > 0 there exists U a neighborhood of t such that for some [beta], the inequality

[absolute value of f ([rho](t)) - f (s) - [beta]([rho](t) - s)] < [epsilon][absolute value of [rho](t) - s],

is true for all s [member of] U. We write [f.sup.[nabla]] (t) = [beta].

Definition 2.5. f : T [right arrow] R is said to be nabla differentiable on T if f : T [right arrow] R is nabla differentiable for all t [member of] [T.sub.[kappa]].

It is easy to show that, if f is nabla differentiable on T, then

[f.sup.[rho]] = f - v[f.sup.[nabla]],

where [f.sup.[rho]] = f o [rho] (this formula can be seen in [1]).

Definition 2.6. f is rd-continuously delta differentiable (we write f [member of] [C.sup.1.sub.rd]) if [f.sup.[DELTA]](t) exists for all t [member of] [T.sup.k] and [f.sup.[DELTA]] [member of] [C.sub.rd], and f is ld-continuously nabla differentiable (we write f [member of] [C.sup.1.sub.ld]) if [f.sup.[nabla]](t) exists for all t [member of] [T.sub.k] and [f.sup.[nabla]] [member of] [C.sub.ld].

Remark 2.7. If T = R, then the notion of delta derivative and nabla derivative coincide and they denote the standard derivative we know from calculus, however, when T = Z, then they do not coincide (see [5]).

3 Dual Time Scales

In this section we introduce the definition of dual time scales. We will see that our main result develops merely from this basic definition. A dual time scale is just the "reverse" time scale of a given time scale. More precisely, we define it as follows:

Definition 3.1. Given a time scale T we define the dual time scale

[T.sup.*] := {s [member of] R| - s [member of] T}.

Once we have defined a dual time scale, it is natural to extend all the definitions of Section 2. We now introduce some notation regarding the correspondence between the definitions on a time scale and its dual.

Let T be a time scale. If [rho] and [sigma] denote its associated jump functions, then we denote by [??] and [??] the jump functions associated to [T.sup.*]. If [mu] and v denote, respectively, the forward graininess and backward graininess associated to T, then we denote by [??] and [??], respectively, the forward graininess and the backward graininess associated to [T.sup.*].

Next, we define another fundamental "dual" object, i.e., the "dual" function.

Definition 3.2. Given a function f : T [right arrow] R defined on time scale T we define the dual function [f.sup.*] : [T.sup.*] [right arrow] R on the time scale [T.sup.*] := {s [member of] R| - s [member of] T} by [f.sup.*](s) := f(-s) for all s [member of] [T.sup.*].

Definition 3.3. Given a time scale T we refer to the delta calculus (resp. nabla calculus) any calculation that involves delta derivatives (resp. nabla derivatives).

4 Dual Correspondences

In this section we deduce some basic lemmas which follow easily from the definitions. These lemmas concern the relationship between dual objects. We will use the following notation: given the quintuple (T, [sigma], [rho], [mu], v), where T denotes a time scale with jump functions, [sigma], [rho], and associated forward graininess f and backward graininess , its dual will be ([T.sup.*], [??],[??],[??],[??]) where [??],[??],[??], and [??] will be given as in Lemma 4.2 and 4.4 that we will prove in this section. Also, [DELTA] and [nabla] will denote the derivatives for the time scale T and [??] and [??] will denote the derivatives for the time scale [T.sup.*].

Lemma 4.1. If a, b [member of] T with a < b, then

[([a, b]).sup.*] =[-b, -a].

Proof. The proof is straightforward. In fact,

s [member of] [([a, b]).sup.*] iff - s [member of] [a, b] iff s [member of] [-b, -a].

This completes the proof.

Lemma 4.2. Given [sigma], [rho] : T [right arrow] T, the jump operators for T, then the jump operators for [T.sup.*], [??] and [??] : [T.sup.*] [right arrow] [T.sup.*], are given by the following two identities:

[??](s) = -[rho](-s)

[??](s) = -[sigma](-s)

for all s [member of] [T.sup.*].

Proof. We show the first identity. Using the definition and some simple algebra,

[??](s) = inf{-w [member of] T : -w < -s} = - sup{v [member of] T : v < -s} = -[rho](-s).

The second identity follows similarly.

Lemma 4.3. If T is a time scale, then

[([T.sup.[kappa]]).sup.*] = [([T.sup.*]).sub.[kappa]], and [([T.sub.[kappa]]).sup.*] = [([T.sup.*]).sup.[kappa]].

Proof. We first observe that sup T = - inf [T.sup.*]. If sup T = [infinity], then

[([T.sup.[kappa]]).sup.*] = [(T).sup.*] = [([T.sup.*]).sub.[kappa]].

If sup T < [infinity], then

[([T.sup.[kappa]]).sup.*] = (T \ [([rho](sup T), sup T]).sup.*] = [T.sup.*] \ [([rho](sup T), sup T]).sup.*] = [([T.sup.*]).sub.[kappa]].

Similarly, [([T.sub.[kappa]]).sup.*] = [([T.sup.*]).sup.[kappa]].

Lemma 4.4. Given [mu] : T [right arrow] R, the forward graininess of T, then the backward graininess of [T.sup.*], [??] : [T.sup.*] [right arrow] R, is given by the identity

[??](s) = [[mu].sup.*](s) for all s [member of] [T.sup.*].

Also, given v : T [right arrow] R, the backward graininess of T, then the forward graininess of [T.sup.*], [??] : [T.sup.*] [right arrow] R, is given by the identity

[??](s) = [v.sup.*(s)] for all s [member of] [T.sup.*].

Proof. We prove the first identity. Let s [member of] [T.sup.*], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The second identity follows analogously.

Lemma 4.5. Given f : T [right arrow] R, f is rd continuous (resp. ld continuous) if and only if its dual [f.sup.*] : [T.sup.*] [right arrow] R is ld continuous (resp. rd continuous).

Proof. We will only show the statement for rd continuous functions as the proof for ld continuous functions is analogous. We first observe that t [member of] T isa right-dense point iff -t [member of] [T.sup.*] isa left-dense point. Also, f : T [right arrow] R is continuous at t iff [f.sup.*] : [T.sup.*] [right arrow] R is continuous at -t. Let f : T [right arrow] R be a function, then, the following is true: f : T [right arrow] R is rd continuous iff f is continuous at the right-dense points and its left-sided limits exist (finite) at all left-dense points iff [f.sup.*] is continuous at the left-dense points and its right-sided limits exist (finite) at all right-dense points iff [f.sup.*] : [T.sup.*] [right arrow] R is ld continuous.

The next lemma links delta derivatives to nabla derivatives, showing that the two fundamental concepts of the two types of calculus are, in a certain sense, the dual of each other. In fact, this is the key lemma for our main results.

Lemma 4.6. Let f : T [right arrow] R be delta (resp. nabla) differentiable at [t.sub.0] [member of] [T.sup.[kappa]] (resp. at [t.sub.0] [member of] [T.sub.[kappa]]), then [f.sup.*] : [T.sup.*] [right arrow] R is nabla (resp. delta) differentiable at -[t.sub.0] [member of] [([T.sup.*]).sub.[kappa]] (resp. at -[t.sub.0] [member of] [([T.sup.*]).sup.[kappa]]), and the following identities hold true

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [DELTA], [nabla] denote the derivatives for the time scale T and [??] , [??] denote the derivatives for the time scale [T.sup.*].

Proof. The proof is trivial but for the sake of completeness we will write all the details. We will prove that if f : T [right arrow] R is delta differentiable at [t.sub.0] [member of] [T.sup.[kappa]], then [f.sup.*] is nabla differentiable at -[t.sub.0] [member of] [([T.sup.*]).sub.[kappa]]. Let f : T [right arrow] R be delta differentiable at [t.sub.0] [member of] [T.sup.[kappa]]. Then for all [epsilon] > 0 there exists U a neighborhood of [t.sub.0] such that the inequality

[absolute value of f ([sigma]([t.sub.0])) - f (s) - [f.sup.[DELTA]] ([t.sub.0])([sigma]([t.sub.0]) - s)] < [epsilon][absolute value of [sigma]([t.sub.0]) - s],

is true for all s [member of] U. Next, using Lemma 4.2, as well as the definition of dual function [f.sup.*], we rewrite the above inequality as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for all s [member of] [U.sup.*]. Let [U.sup.*] be the dual of U. Let t [member of] [U.sup.*], then -t [member of] U. Hence, by replacing s by -t, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By definition, this implies that the function [f.sup.*] is nabla differentiable at -[t.sub.0], and

[([f.sup.*]).sup.[??]] (-[t.sub.0]) = -[f.sup.[DELTA]] ([t.sub.0]).

Analogously, it follows that, if f : T [right arrow] R is nabla differentiable at [t.sub.0] [member of] [T.sub.[kappa]], then [f.sup.*] : [T.sup.*] [right arrow] R is delta differentiable at -[t.sub.0] [member of] [([T.sup.*]).sup.[kappa]], and

[([f.sup.*]).sup.[??]] (-[t.sub.0]) = -[f.sup.[nabla]]([t.sub.0]).

The proof is complete.

The next two lemmas link the notions of [C.sup.1.sub.rd] and [C.sup.1.sub.ld] functions.

Lemma 4.7. Given a function f : T [right arrow] R, f belongs to [C.sup.1.sub.rd] (resp. [C.sup.1.sub.ld]) if and only if its dual [f.sup.*] : [T.sup.*] [right arrow] R belongs to [C.sup.1.sub.ld] (resp. [C.sup.1.sub.rd]) .

Lemma 4.8. Given a function f : T [right arrow] R, f belongs to [C.sup.1.sub.prd] (resp. [C.sup.1.sub.pld]) if and only if its dual [f.sup.*] : [T.sup.*] [right arrow] R belongs to [C.sup.1.sub.pld] (resp. [C.sup.1.sub.prd]).

In the following example we derive a well-known formula for derivatives. We will deduce the formula for the nabla derivative using the one for the delta derivative.

Example 4.9 (Formula for Derivatives). It is well known (see [4]) that if f is delta differentiable on T, with j the associated forward graininess, then

[f.sup.[sigma]](t) = f (t) + [mu](t)[f.sup.[DELTA]](t) for all t [member of] [T.sup.[kappa]], (4.1)

where [f.sup.[sigma]] = fo[sigma]. We will use it to derive the analogous formula for the nabla derivative. Suppose that h is nabla differentiable on T, with v its associated backward graininess, then its dual function [h.sup.*] is delta differentiable on [T.sup.*]. Hence, we apply (4.1) to [h.sup.*]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

We observe that [??] = [v.sup.*], while [([h.sup.*]).sup.[??]] = [h.sup.[rho]] by Lemma 4.2, and Lemma 4.4, with [h.sup.[rho]] = h o [rho], and [([h.sup.*]).sup.[??]] = -[h.sup.[nabla]] by Lemma 4.6. So,

[h.sup.[rho]](t) = h(t) - v(t)[h.sup.[nabla]](t) for all t [member of] [T.sub.[kappa]]. (4.3)

We recall that this formula (4.3) has appeared in the nabla context in [1].

Next, using Lemma 4.5 and Lemma 4.6, we show in the following proposition how to compare nabla and delta integrals.

Proposition 4.10. (i) If f : [a, b] [right arrow] R is rd continuous, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(ii) If f : [a, b] [right arrow] R is ld continuous, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Proof of (i). By definition of the integral,

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

is an antiderivative of f, i.e.,

[F.sup.[DELTA]](t) = f(t).

We have seen in Lemma 4.6 that [f.sup.*](s) = [([F.sup.[DELTA]]).sup.*](s) = -[([F.sup.*]).sup.[??]](s). Also, again by definition,

[[integral].sup.-a.sub.-b] [f.sup.*](s)[??]s = G(-a) - G(-b), where G

is an antiderivative of [f.sup.*], i.e.,

[G.sup.[nabla]](s) = [f.sup.*](s).

It follows that G = -[F.sup.*] + c, where c [member of] R, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of (ii). We apply (i) to [f.sup.*],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [([f.sup.*]).sup.*] = f, (ii) follows immediately.

5 Main Result

The main result of this article will be the following Duality Principle which asserts that given certain results in the nabla (resp. delta) calculus under certain hypotheses, one can obtain the dual results by considering the corresponding dual hypotheses and the dual conclusions in the delta (resp. nabla) setting.

Given a statement in the delta calculus (resp. nabla calculus), the corresponding dual statement is obtained by replacing any object in the given statement by the corresponding dual one.

Duality Principle For any statement true in the nabla (resp. delta) calculus in the time scale T there is an equivalent dual statement in the delta (resp.nabla) calculus for the dual time scale [T.sup.*].

In the next example we further illustrate how the Duality Principle applies.

Example 5.1 (Integration by Parts). We show how the Duality Principle can be applied to prove the integration by parts formula. In delta settings the integration by parts formula is given by the following identity:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.1)

for all functions f, g : [a, b] [right arrow] R, with f, g [member of] [C.sup.1.sub.rd]. Now, let h, j : [a, b] [right arrow] R, with h,j [member of] [C.sup.1.sub.ld], then, the dual functions [h.sup.*], [j.sup.*] : [-b, -a] [right arrow] R are in [C.sup.1.sub.rd]. Next, we will apply the identity (5.1) to [h.sup.*] and [j.sup.*]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The LHS of the last identity can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

because [(h[j.sup.[nabla]]).sup.*](t) = [h.sup.*](t)[([j.sup.[nabla]]).sup.*](t) = - [h.sup.*](t)[([j.sup.*]).sup.[??]](t). The second term in the RHS can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

because of the identity ([([j.sup.*]).sup.[??]])[sup.*](s) = [j.sup.[rho]](s). To obtain the desired formula we substitute the RHS of (5.3) in the integration by parts formula (5.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.4)

It follows that the identity (5.4) is the integration by parts formula for the nabla setting.

6 Application of the Duality Principle to the Calculus of Variations on Time Scales

6.1 Euler-Lagrange Equation

We consider the Euler-Lagrange equation using the identity of Proposition 4.10. We will use Bohner's results in [4] in the delta settings to prove similar results in the nabla settings as done in [1] (one could also do the vice versa). We review a few definitions.

Definition 6.1. A function f : [a, b] [right arrow] R belongs to the space [C.sup.1.sub.rd] if the following norm is finite: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; also, a function f : [a, b] [right arrow] R belongs to the space [C.sup.1.sub.ld] if the following norm is finite: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 6.2. A function f is delta regulated if the right-hand limit f (t+) exists (finite) at all right-dense points t [member of] T and the left-hand limit f (t-) exists at all left-dense points t [member of] T; f is regulated if the left-hand limit f (t+) exists (finite) at all left-dense points t [member of] T and the right-hand limit f (t-) exists at all right-dense points t [member of] T.

Definition 6.3. A function f is delta piecewise rd-continuous (we write f [member of] [C.sub.prd]) if it is regulated and if it is rd continuous at all, except possibly at finitely many, right-dense points t [member of] T; f is nabla piecewise ld-continuous (we write f [member of] [C.sub.pld]) if it is nabla regulated and if it is ld continuous at all, except possibly at nitely many, left-dense points t [member of] T.

Definition 6.4. f is delta piecewise rd-continuously differentiable (we write f [member of] [C.sup.1.sub.prd] ) if f is rd continuous and [f.sup.[DELTA]] [member of] [C.sub.prd]; f is delta piecewise ld-continuously differentiable (we write f [member of] [C.sup.1.sub.pld] ) if f is ld continuous and [f.sup.[nabla]] [member of] [C.sub.pld].

Definition 6.5. Assume the function L : T x R x R [right arrow] R is of class [C.sup.2] in the second and third variable, and rd continuous in the first variable. Then, [y.sub.0] is said to be a weak (resp. strong) local minimum of the problem

L(y) = [[integral].sup.b.sub.a] L(t, [y.sup.[sigma]](t), [y.sup.[DELTA]](t))[DELTA]t y(a) = [alpha], y(b) = [beta], (6.1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We refer to the function L as to the Lagrangian for the above problem. Moreover, if L = L(t,x,v), then [L.sub.v], [L.sub.x] represent, respectively, the partial derivatives of L with respect to v, and x.

Definition 6.6. Assume the function [bar.L]: T x R x R [right arrow] R is of class [C.sup.2] in the second and third variable, and rd continuous in the first variable. Then, [y.sub.0] is said to be a weak (strong) local minimum of the problem

[bar.L](h) = [[integral].sup.d.sub.c] [bar.L](s, [h.sup.[rho]] (s), [h.sup.[nabla]](s))[nabla]s h(c) = A, h(d) = B, (6.2)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 6.7. Given a Lagrangian L : T x R x R [right arrow] R, we define the dual (corresponding) Lagrangian [L.sup.*] : [T.sup.*] x R x R [right arrow] R by [L.sup.*](s,x,v) = L(-s,x, -v) for all (s,x,v) [member of] [T.sup.*] x R x R.

As a consequence of the definition of the dual Lagrangian and Proposition 4.10 we have the following useful lemma.

Lemma 6.8. Given a Lagrangian L : [a, b] x R x R [right arrow] R, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all functions y [member of] [C.sup.1.sub.rd]([a, b]).

The next theorem is a result by Bohner [4] in one dimension (the results we will present can be obtained without this restriction, but we prefer one dimension to have an immediate comparison with the results in [1]).

Theorem 6.9 (Euler-Lagrange Necessary Condition in Delta Setting). If [y.sub.0] is a (weak) local minimum of the variational problem (6.1), then the Euler-Lagrange equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds.

Now, we will use Bohner's theorem to prove the Euler-Lagrange equation in the nabla context. We recall that the Euler-Lagrange equation in the nabla context was shown in [1]. Here we will reprove it using our technique. (Also, see Remark 6.11.)

Theorem 6.10 (Euler-Lagrange Necessary Condition in Nabla Setting). If [[bar.y].sub.0] is a local (weak) minimum for the variational problem (6.2), then the Euler-Lagrange equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

holds.

Proof. This theorem is essentially a corollary of Theorem 6.9. Since [[bar.y].sub.0] is a local minimum for (6.2), it follows from Lemma 6.8 that [[bar.y].sup.*.sub.0] is local minimum for the variational problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.3)

where g [member of] [C.sup.1.sub.rd]. The variational problem (6.3) is the same as (6.1) for the Lagrangian [[bar.L].sup.*] (with a = -d, b = -c, [alpha] = B and [beta] = A). Hence, we can apply Theorem 6.9. The Euler-Lagrange equation for the Lagrangian [[bar.L].sup.*] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.4)

Our goal is now to rewrite (6.4) for the Lagrangian [bar.L]. It is easy to check that

[[bar.L].sup.*.sub.v](t,x,v) = -[[bar.L].sub.w](-t, x, -v), and [[bar.L].sup.*.sub.x](t, x, v) = [[bar.L].sub.x](-t, x, -v),

where [[bar.L].sub.w] is the partial derivative of L with respect to the third variable. Let us substitute x by [([[bar.y].sup.*.sub.0]).sup.[??]](t), and v by [([[bar.y].sup.*.sub.0]).sup.[??]](t), in the previous identities. We get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Lemma 4.6, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, let s [member of] [[c, d].sub.[kappa]] and set -t = s. Then by (6.4),

[p.sup.[nabla]](s) = [[bar.L].sub.x](s, [([[bar.y].sub.0]).sup.[rho]](s), [([[bar.y].sub.0]).sup.[nabla]](s)), (6.5)

and, finally, revealing the definition of p, from (6.5) we obtain the Euler-Lagrange equation in the nabla setting:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof is complete.

Remark 6.11. Theorem 6.10 states the same result as the main theorem proven in [1]. The only difference is the interval of points for which the Euler-Lagrange equation holds. In fact, since in [1] the interval of integration for the Lagrangian is [[[rho].sup.2](a)), [rho](b)], it follows from our results that the Euler-Lagrange equation has to hold in the interval [[[[rho].sup.2](a)), [rho](b)].sub.[kappa]] and not [[rho](a)),b] as in [1]. This claim can be also justified by noticing that, in order of applying [1, Lemma 2.1], the test functions have to vanish at the limit points of integration. Another observation about such interval was pointed out in [6].

Remark 6.12. Theorem 6.10 can be easily generalized to the higher-order results of [12] by applying our Duality Principle to the results in [7].

6.2 Weierstrass Necessary Condition on Time Scales

We first review a few definitions. Let L be a Lagrangian. Let E : [[a, b].sup.[kappa]] x [R.sup.3] [right arrow] R be the function defined as

E(t, x, r, q) = L{t, x, q) - L(t, x, r) - (q - r)[L.sub.r](t, x, r).

This function E is called the Weierstrass excess function of L.

The Weierstrass necessary optimality condition on time scales was proven in the delta setting in [11]. This theorem is stated as follows.

Theorem 6.13 (Weierstrass Necessary Optimality Condition with Delta Setting). Let T be a time scale, a and b [member of] T, a < b. Assume that the function L(t, x, r) in (6.1) satisfies the following condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6.6)

for each (t, x) [member of] [[a, b].sup.[kappa]] x R and all [r.sub.1], [r.sub.2] [member of] R, [gamma] [member of] [0,1]. Let [bar.x] be a piecewise continuous function. If [bar.x] is a strong local minimum for (6.1), then

E[t, [[bar.x].sup.[sigma]](t), [[bar.x].sup.[DELTA]](t),q] [greater than or equal to] 0 for all t [member of] [[a, b].sup.[kappa]] and q [member of] R,

where we replace [[bar.x].sup.[DELTA]](t) by [[bar.x].sup.[DELTA]](t-) and [[bar.x].sup.[DELTA]](t+) at finitely many points t where [[bar.x].sup.[DELTA]](t) does not exist.

Let E be the Weierstrass excess function of [bar.L].

Theorem 6.14 (Weierstrass Necessary Optimality Condition with Nabla Setting). Let T be a time scale, a and b [member of] T, a < b. Assume that the function [bar.L](t, x, r) in (6.2) satisfies the following condition:

v(t)[bar.L](t, x, [gamma][r.sub.1] + (1 - [gamma])[r.sub.2]) [less than or equal to] v(t)[gamma][bar.L](t, x, [r.sub.1]) + v(t)(1 - [gamma])[bar.L](t, x, [r.sub.2]), (6.7)

for each (t, x) [member of] [[a, b].sup.[kappa]] x R and all [r.sub.1], [r.sub.2] [member of] R, [gamma] [member of] [0,1]. Let [bar.x] be a piecewise continuous function. If [bar.x] is a strong local minimum for (6.2), then

E[t, [[bar.x].sup.[rho]](t), [[bar.x].sup.[nabla]](t),q] [greater than or equal to] 0 for all t [member of] [[a, b].sup.[kappa]] and q [member of] R

where we replace [[bar.x].sup.[nabla]](t) by [[bar.x].sup.[nabla]](t-) and [[bar.x].sup.[nabla]](t+) at finitely many points t where [[bar.x].sup.[nabla]](t) does not exist.

Proof. Let [[bar.L].sup.*] be the dual Lagrangian of [bar.L]. It is easy to prove (similarly as we did in Theorem 6.10, although here [bar.x] is a strong minimum), that [[bar.x].sup.*] is a strong local minimum for (6.1). Then, (6.7) can be written on the dual time scale [T.sup.*] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for each (s, x) [member of] [[-b, -a].sup.[kappa]] x R and all [r.sub.1], [r.sub.2] [member of] R, y [member of] [0,1]. We recognize that the last inequality is the same as (6.6) in Theorem 6.13 for the Lagrangian [L.sup.*]. Hence, we apply Theorem 6.13,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [E.sup.*] is the Weierstrass excess function of [[bar.L].sup.*]. Also, we notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where E is the Weierstrass excess function of [bar.L]. Finally,

E [t, [[bar.x].sup.[rho]](t), [[bar.x].sup.[nabla]](t), -q] [greater than or equal to] 0 for all t [member of] [[a, b].sup.[kappa]] and all q [member of] R,

because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We observe that, the fact that we can replace

[[bar.x].sup.[nabla]](t)

by

[[bar.x].sup.[nabla]](t-) and [[bar.x].sup.[nabla]](t+) at finitely many points t,

where

[[bar.x].sup.[nabla]](t)

does not exist, follows as well from Theorem 6.13.

Acknowledgments

The author would like to thank Professors D. Torres and A. Malinowska for having brought this problem to her attention while they were visiting the University of Texas, at Austin, in the Fall 2009. Also, she thanks them for patiently reading some drafts of this article and making very helpful suggestions.

Appendix: Table of Dual Objects

Based on the above definitions, remarks and lemmas we summarize in Table 1 for each "object" its dual one. Naturally, Table 1 may be extended to more objects.

References

[1] F. M. Atici, D. C. Biles, and A. Lebedinsky. An application of time scales to economics. Math. Comput. Modelling, 43(7-8):718-726, 2006.

[2] F. M. Atici and G. Sh. Guseinov. On Green's functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math., 141(1-2):75-99, 2002. Special Issue on "Dynamic Equations on Time Scales", edited by R. P. Agarwal, M. Bohner, and D. O'Regan.

[3] F. M. Atici and C. S. McMahan. A comparison in the theory of calculus of variations on time scales with an application to the Ramsey model. Nonlinear Dyn. Syst. Theory, 9(1):1-10, 2009.

[4] M. Bohner. Calculus of variations on time scales. Dynam. Systems Appl., 13:339-349, 2004.

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

[6] R. A. C. Ferreira and D. F. M. Torres. Remarks on the calculus of variations on time scales. Int. J. Ecol. Econ. Stat., 9(F07):65-73, 2007.

[7] R. A. C. Ferreira and D. F. M. Torres. Higher-order calculus of variations on time scales. In Mathematical control theory and finance, pages 149-159. Springer, Berlin, 2008.

[8] M. Gurses, G. Sh. Guseinov, and B. Silindir. Integrable equations on time scales. J.Math. Phys., 46(11):113510, 22, 2005.

[9] S. Hilger. Ein Ma[beta]kettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis, Universitat Wurzburg, 1988.

[10] R. Hilscher and V. Zeidan. Calculus of variations on time scales: weak local piecewise [C.sup.1.sub.rd]solutions with variable endpoints. J. Math. Anal. Appl., 289:143-166, 2004.

[11] A. B. Malinowska and D. F. M. Torres. Strong minimizers of the calculus of variations on time scales and the Weierstrass condition. Proc. Est. Acad. Sci., 58(4):205-212, 2009.

[12] N. Martins and D. F. M. Torres. Calculus of variations on time scales with nabla derivatives. Nonlinear Anal., 71(12):e763-4773, 2009.

M. Cristina Caputo

University of Texas at Austin

Department of Mathematics

1 University Station C1200

Austin, TX 78712-0257, U.S.A.

caputo@math.utexas.edu

Received October 1, 2009; Accepted January 31, 2010 Communicated by Martin Bohner
Table 1: Table of Dual Objects

Object                                 Corresponding dual object

T                                              [T.sup.*]
f: T [right arrow] R             [f.sup.*]: [T.sup.*] [right arrow] R
[f.sup.*]: [T.sup.*]                     f: T  [right arrow] R
  [right arrow] R
[t.sub.0] right-dense             -[t.sub.0] left-dense (right-dense)
  (left-dense)
[t.sub.0] right-scattered             -[t.sub.0] left-scattered
  (left-scattered)                         (right-scattered)
[mu], v                                          [??]
[sigma], [rho]                                   [??]
[f.sup.[DELTA]] ([t.sub.0])       -[([f.sup.*]).sup.[??]](-[t.sub.0])
[f.sup.[nabla]] ([t.sub.0])        -[([f.sup.*]).sup.[bar.[DELTA]]]
                                             (-[t.sub.0])
[f.sup.[DELTA]]([t.sub.0])             -[([(f.sup.*]).sup.[bar.
                                     [nabla]]]).sup.*]([t.sub.0])
[([f.sup.[DELTA]]).sup.*]             -([([f.sup.*]).sup.[??]])
  (-[t.sub.0])                               (-[t.sub.0]))
f [member of] [C.sub.rd]           [f.sup.*] [member of] [C.sub.ld]
  (f [member of] [C.sub.ld])      ([f.sup.*] [member of] [C.sub.rd])
f [member of]                           [f.sup.*] [member of]
  [C.sup.1.sub.rd]                   [C.sup.1.sub.ld] ([f.sup.*]
  (f [member of]                     [member of] [C.sup.1.sub.rd])
  [C.sup.1.sub.ld])
f [member of] [C.sub.prd]               [f.sup.*] [member of]
  (f [member of] [C.sub.pld])        [C.sup.1.sub.pld] ([f.sup.*]
                                       [member of] [C.sub.prd])
f [member of]                            [f.sup.*] [member of]
  [C.sup.1.sub.prd]                  [C.sup.1.dub.pld] ([f.sup.*]
  (f [member of]                    [member of] [C.sup.1.sub.prd])
  [C.sup.1.sub.pld])
[[integral].sup.b.sub.a]              [[integral].sup.-a.sub.-b]
  f(t)[DELTA]t                         [f.sup.*](s)[[??].sub.s]
L: T x [R.sup.2]                   [L.sup.*]: [T.sup.*] x [R.sup.2]
  [right arrow] R, L(t, x, v)         [right arrow] R, [L.sup.*]
                                      (s, x, w) (= L(-s, x, -w))
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Author:Caputo, M. Cristina
Publication:International Journal of Difference Equations
Article Type:Report
Geographic Code:1USA
Date:Jun 1, 2010
Words:6009
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