Time Scales: From Nabla Calculus to Delta Calculus and Vice Versa via Duality

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1 Interntionl Journl of Difference Equtions ISSN , Volume 5, Number 1, pp (2010) Time Scles: From Nbl Clculus to Delt Clculus nd Vice Vers vi Dulity M. Cristin Cputo University of Texs t Austin Deprtment of Mthemtics 1 University Sttion C1200 Austin, TX , U.S.A. cputo@mth.utexs.edu Abstrct In this note we show how one cn obtin results from the nbl clculus from results on the delt clculus nd vice vers vi dulity rgument. We provide pplictions of the min results to the clculus of vritions on time scles. AMS Subject Clssifictions: 39A10, 26E70, 49K05. Keywords: Time scles, nbl clculus, delt clculus, clculus of vritions. 1 Introduction The time scle delt clculus ws introduced for the first time in 1988 by Hilger [9] to unify the theory of difference equtions nd the theory of differentil equtions. It ws extensively studied by Bohner [4] nd Hilscher nd Zeidn [10] who introduced the clculus of vritions on the time scle delt clculus (or simply delt clculus). In 2001 the time scle nbl clculus (or simply nbl clculus) ws introduced by Atici nd Guseinov [2]. Both theories of the delt nd the nbl clculus cn be pplied to ny field tht requires the study of both continuous nd discrete dt. For instnce, the nbl clculus hs been pplied to mximiztion (minimiztion) problems in economics [1, 2]. Recently severl uthors hve contributed to the development of the clculus of vritions on time scles (for instnce, see [3, 11, 12]). To the best of the uthor s knowledge there is no known technique to obtin results from the nbl clculus directly from results on the delt clculus nd vice vers. In Received October 1, 2009; Accepted Jnury 31, 2010 Communicted by Mrtin Bohner

2 26 Cristin Cputo this note we underline tht, in fct, this is possible. We show tht the two types of clculus, the nbl nd the delt on time scles, re the dul of ech other. One cn reciproclly obtin results for one type of clculus from the other nd vice vers without mking ny ssumptions on the regulrity of the time scles (s it ws done in [8]). We prove tht results for the nbl (respectively the delt) clculus cn be obtined by the dul nlogous ones which will be in the delt (respectively nbl) context. Therefore, if they hve lredy been proven for the delt cse (respectively the delt), it is not necessry to reprove them for the nbl setting (respectively nbl). This rticle is orgnized s follows: in the second section we review some bsic definitions. In third section we introduce the dul time scles. In the fourth section we derive few properties relted to dulity. In the fifth section we stte the Dulity Principle, which is the min result of the rticle, nd we pply it to few exmples. Finlly, in the lst section, we pply the Dulity Principle to the clculus of vritions on time scles. 2 Review of Bsic Definitions We first review some bsic definitions nd hence introduce both types of clculus (for complete list of definitions for the delt clculus see the pioneering book by Bohner nd Peterson [5]). A time scle T is ny closed nonempty subset T of R. The jump opertors σ, ρ : T T re defined by σ(t) = inf{s T : s > t}, nd ρ(t) = sup{s T : s < t}, with inf := sup T, sup := inf T. A point t T is clled right-dense if σ(t) = t, right-scttered if σ(t) > t, left-dense if ρ(t) = t, left-scttered if ρ(t) < t. The forwrd grininess µ : T R is defined by µ(t) = σ(t) t, nd the bckwrd grininess ν : T R is defined by ν(t) = t ρ(t). Given time scle T, we denote T κ := T \ (ρ(sup T), sup T], if sup T < nd T κ := T if sup T =. Also T κ := T \ [inf T, σ(inf T)) if inf T > nd T κ =: T if inf T =. In prticulr, if, b T with < b, we denote by [, b] the intervl [, b] T. It follows tht [, b] κ = [, ρ(b)], nd [, b] k = [σ(), b]. Of course, R itself is one trivil exmple of time scle, but one could lso tke T to be the Cntor set. For more interesting exmples of time scles we suggest reding [5]. Definition 2.1. A function f defined on T is clled rd-continuous (or right-dense continuous) (we write f C rd ) if it is continuous t the right-dense points nd its left-sided limits exist (finite) t ll left-dense points; f is ld-continuous (or left-dense continuous) if it is continuous t the left-dense points nd its right-sided limits exist (finite) t ll right-dense point.

3 From Nbl Clculus to Delt Clculus nd Vice Vers Definition of Derivtives Definition 2.2. A function f : T R is sid to be delt differentible t t T κ if for ll ɛ > 0 there exists U neighborhood of t such tht for some α, the inequlity f(σ(t)) f(s) α(σ(t) s) < ɛ σ(t) s, is true for ll s U. We write f (t) = α. Definition 2.3. f : T R is sid to be delt differentible on T if f : T R is delt differentible for ll t T κ. It is esy to show tht, if f is delt differentible on T, then f σ = f + µf, where f σ = f σ (the proof cn be found in [5]). Definition 2.4. A function f : T R is sid to be nbl differentible t t T κ if for ll ɛ > 0 there exists U neighborhood of t such tht for some β, the inequlity f(ρ(t)) f(s) β(ρ(t) s) < ɛ ρ(t) s, is true for ll s U. We write f (t) = β. Definition 2.5. f : T R is sid to be nbl differentible on T if f : T R is nbl differentible for ll t T κ. It is esy to show tht, if f is nbl differentible on T, then f ρ = f νf, where f ρ = f ρ (this formul cn be seen in [1]). Definition 2.6. f is rd-continuously delt differentible (we write f C 1 rd) if f (t) exists for ll t T k nd f C rd, nd f is ld-continuously nbl differentible (we write f C 1 ld) if f (t) exists for ll t T k nd f C ld. Remrk 2.7. If T = R, then the notion of delt derivtive nd nbl derivtive coincide nd they denote the stndrd derivtive we know from clculus, however, when T = Z, then they do not coincide (see [5]).

4 28 Cristin Cputo 3 Dul Time Scles In this section we introduce the definition of dul time scles. We will see tht our min result develops merely from this bsic definition. A dul time scle is just the reverse time scle of given time scle. More precisely, we define it s follows: Definition 3.1. Given time scle T we define the dul time scle T := {s R s T}. Once we hve defined dul time scle, it is nturl to extend ll the definitions of Section 2. We now introduce some nottion regrding the correspondence between the definitions on time scle nd its dul. Let T be time scle. If ρ nd σ denote its ssocited jump functions, then we denote by ˆρ nd ˆσ the jump functions ssocited to T. If µ nd ν denote, respectively, the forwrd grininess nd bckwrd grininess ssocited to T, then we denote by ˆµ nd ˆν, respectively, the forwrd grininess nd the bckwrd grininess ssocited to T. Next, we define nother fundmentl dul object, i.e., the dul function. Definition 3.2. Given function f : T R defined on time scle T we define the dul function f : T R on the time scle T := {s R s T} by f (s) := f( s) for ll s T. Definition 3.3. Given time scle T we refer to the delt clculus (resp. nbl clculus) ny clcultion tht involves delt derivtives (resp. nbl derivtives). 4 Dul Correspondences In this section we deduce some bsic lemms which follow esily from the definitions. These lemms concern the reltionship between dul objects. We will use the following nottion: given the quintuple (T, σ, ρ, µ, ν), where T denotes time scle with jump functions, σ, ρ, nd ssocited forwrd grininess µ nd bckwrd grininess ν, its dul will be (T, ˆσ, ˆρ, ˆµ, ˆν) where ˆσ, ˆρ, ˆµ, nd ˆν will be given s in Lemm 4.2 nd 4.4 tht we will prove in this section. Also, nd will denote the derivtives for the time scle T nd ˆ nd ˆ will denote the derivtives for the time scle T. Lemm 4.1. If, b T with < b, then ([, b]) = [, ]. Proof. The proof is strightforwrd. In fct, This completes the proof. s ([, b]) iff s [, b] iff s [, ].

5 From Nbl Clculus to Delt Clculus nd Vice Vers 29 Lemm 4.2. Given σ, ρ : T T, the jump opertors for T, then the jump opertors for T, ˆσ nd ˆρ : T T, re given by the following two identities: ˆσ(s) = ρ( s), for ll s T. ˆρ(s) = σ( s), Proof. We show the first identity. Using the definition nd some simple lgebr, ˆσ(s) = inf{ w T : w < s} = sup{v T : v < s} = ρ( s). The second identity follows similrly. Lemm 4.3. If T is time scle, then (T κ ) = (T ) κ, nd (T κ ) = (T ) κ. Proof. We first observe tht sup T = inf T. If sup T =, then If sup T <, then (T κ ) = (T) = (T ) κ. (T κ ) = (T \ (ρ(sup T), sup T]) = T \ (ρ(sup T), sup T]) = (T ) κ. Similrly, (T κ ) = (T ) κ. Lemm 4.4. Given µ : T R, the forwrd grininess of T, then the bckwrd grininess of T, ˆν : T R, is given by the identity ˆν(s) = µ (s) for ll s T. Also, given ν : T R, the bckwrd grininess of T, then the forwrd grininess of T, ˆµ : T R, is given by the identity ˆµ(s) = ν (s) for ll s T. Proof. We prove the first identity. Let s T, then The second identity follows nlogously. ˆν(s) = s ˆρ(s) = s + σ (s) = µ (s). Lemm 4.5. Given f : T R, f is rd continuous (resp. ld continuous) if nd only if its dul f : T R is ld continuous (resp. rd continuous).

6 30 Cristin Cputo Proof. We will only show the sttement for rd continuous functions s the proof for ld continuous functions is nlogous. We first observe tht t T is right-dense point iff t T is left-dense point. Also, f : T R is continuous t t iff f : T R is continuous t t. Let f : T R be function, then, the following is true: f : T R is rd continuous iff f is continuous t the right-dense points nd its left-sided limits exist (finite) t ll left-dense points iff f is continuous t the left-dense points nd its rightsided limits exist (finite) t ll right-dense points iff f : T R is ld continuous. The next lemm links delt derivtives to nbl derivtives, showing tht the two fundmentl concepts of the two types of clculus re, in certin sense, the dul of ech other. In fct, this is the key lemm for our min results. Lemm 4.6. Let f : T R be delt (resp. nbl) differentible t t 0 T κ (resp. t t 0 T κ ), then f : T R is nbl (resp. delt) differentible t t 0 (T ) κ (resp. t t 0 (T ) κ ), nd the following identities hold true or, or, f (t 0 ) = (f ) ˆ ( t 0 ) (resp. f (t 0 ) = (f ) ˆ ( t 0 )), f (t 0 ) = ((f ) ˆ ) (t 0 ) (resp. f (t 0 ) = ((f ) ˆ ) (t 0 )), (f ) ( t 0 ) = ((f ) ˆ )( t 0 ) (resp. (f ) ( t 0 ) = (f ) ˆ ( t 0 )), where, denote the derivtives for the time scle T nd ˆ, ˆ denote the derivtives for the time scle T. Proof. The proof is trivil but for the ske of completeness we will write ll the detils. We will prove tht if f : T R is delt differentible t t 0 T κ, then f is nbl differentible t t 0 (T ) κ. Let f : T R be delt differentible t t 0 T κ. Then for ll ɛ > 0 there exists U neighborhood of t 0 such tht the inequlity f(σ(t 0 )) f(s) f (t 0 )(σ(t 0 ) s) < ɛ σ(t 0 ) s, is true for ll s U. Next, using Lemm 4.2, s well s the definition of dul function f, we rewrite the bove inequlity s f( ˆρ( t 0 )) f ( s) f (t 0 )( ˆρ( t 0 ) s) < ɛ ˆρ( t 0 ) s, for ll s U. Let U be the dul of U. Let t U, then t U. Hence, by replcing s by t, we obtin (f (ˆρ( t 0 )) f (t) f (t 0 )( ˆρ( t 0 ) + t) < ɛ ˆρ( t 0 ) + t, f (ˆρ( t 0 )) f (t) ( f (t 0 ))(ˆρ( t 0 ) t) < ɛ ˆρ( t 0 ) t.

7 From Nbl Clculus to Delt Clculus nd Vice Vers 31 By definition, this implies tht the function f is nbl differentible t t 0, nd (f ) ˆ ( t 0 ) = f (t 0 ). Anlogously, it follows tht, if f : T R is nbl differentible t t 0 f : T R is delt differentible t t 0 (T ) κ, nd T κ, then (f ) ˆ ( t 0 ) = f (t 0 ). The proof is complete. The next two lemms link the notions of C 1 rd nd C 1 ld functions. Lemm 4.7. Given function f : T R, f belongs to C 1 rd (resp. C 1 ld) if nd only if its dul f : T R belongs to C 1 ld (resp. C 1 rd). Lemm 4.8. Given function f : T R, f belongs to C 1 prd (resp. C 1 pld) if nd only if its dul f : T R belongs to C 1 pld (resp. C 1 prd). In the following exmple we derive well-known formul for derivtives. We will deduce the formul for the nbl derivtive using the one for the delt derivtive. Exmple 4.9 (Formul for Derivtives). It is well known (see [4]) tht if f is delt differentible on T, with µ the ssocited forwrd grininess, then f σ (t) = f(t) + µ(t)f (t) for ll t T κ, (4.1) where f σ = f σ. We will use it to derive the nlogous formul for the nbl derivtive. Suppose tht h is nbl differentible on T, with ν its ssocited bckwrd grininess, then its dul function h is delt differentible on T. Hence, we pply (4.1) to h : (h )ˆσ (s) = h (s) + ˆµ(s)(h ) ˆ (s) for ll s (T ) κ. (4.2) We observe tht ˆµ = ν, while (h )ˆσ = h ρ by Lemm 4.2, nd Lemm 4.4, with h ρ = h ρ, nd (h ) ˆ = h by Lemm 4.6. So, h ρ (t) = h(t) ν(t)h (t) for ll t T κ. (4.3) We recll tht this formul (4.3) hs ppered in the nbl context in [1]. Next, using Lemm 4.5 nd Lemm 4.6, we show in the following proposition how to compre nbl nd delt integrls. Proposition (i) If f : [, b] R is rd continuous, then f(t) t = f (s) ˆ s;

8 32 Cristin Cputo (ii) If f : [, b] R is ld continuous, then f(t) t = Proof. Proof of (i). By definition of the integrl, is n ntiderivtive of f, i.e., f(t) t = F (b) F (), F (t) = f(t). f (s) ˆ s. where F We hve seen in Lemm 4.6 tht f (s) = (F ) (s) = (F ) ˆ (s). Also, gin by definition, is n ntiderivtive of f, i.e., f (s) ˆ s = G( ) G(), G (s) = f (s). It follows tht G = F + c, where c R, nd where G f (s) ˆ s = F ( ) + F () = F () + F (b) = Proof of (ii). We pply (i) to f, f (s) ˆ s = Since (f ) = f, (ii) follows immeditely. (f ) (t) t. f(t) t. 5 Min Result The min result of this rticle will be the following Dulity Principle which sserts tht given certin results in the nbl (resp. delt) clculus under certin hypotheses, one cn obtin the dul results by considering the corresponding dul hypotheses nd the dul conclusions in the delt (resp. nbl) setting. Given sttement in the delt clculus (resp. nbl clculus), the corresponding dul sttement is obtined by replcing ny object in the given sttement by the corresponding dul one. Dulity Principle For ny sttement true in the nbl (resp. delt) clculus in the time scle T there is n equivlent dul sttement in the delt (resp.nbl) clculus for the dul time scle T. In the next exmple we further illustrte how the Dulity Principle pplies.

9 From Nbl Clculus to Delt Clculus nd Vice Vers 33 Exmple 5.1 (Integrtion by Prts). We show how the Dulity Principle cn be pplied to prove the integrtion by prts formul. In delt settings the integrtion by prts formul is given by the following identity: f(t)g (t) t = f(b)g(b) f()g() f (t)g σ (t) t, (5.1) for ll functions f, g : [, b] R, with f, g Crd. 1 Now, let h, j : [, b] R, with h, j Cld, 1 then, the dul functions h, j : [, ] R re in Crd. 1 Next, we will pply the identity (5.1) to h nd j : h (t)(j ) ˆ (t) ˆ t = h ( )j ( ) h ()j () The LHS of the lst identity cn be written s h (t)(j ) ˆ (t) ˆ t = (h j ) (t) ˆ t = (h ) ˆ (t)(j ) σ (t) ˆ t. h(s) j (s) s, (5.2) becuse (h j ) (t) = h (t)(j ) (t) = h (t)(j ) ˆ (t). The second term in the RHS cn be written s (h ) ˆ (t)(j )ˆσ (t) t = ((h ) ˆ (j )ˆσ ) (s) s = h (s)j ρ (s) s, (5.3) becuse of the identity ((j )ˆσ ) (s) = j ρ (s). To obtin the desired formul we substitute the RHS of (5.3) in the integrtion by prts formul (5.1): h(s) j (s) (s) = h()j() + h(b)j(b) h (s)j ρ (s) s. (5.4) It follows tht the identity (5.4) is the integrtion by prts formul for the nbl setting. 6 Appliction of the Dulity Principle to the Clculus of Vritions on Time Scles 6.1 Euler Lgrnge Eqution We consider the Euler Lgrnge eqution using the identity of Proposition We will use Bohner s results in [4] in the delt settings to prove similr results in the nbl settings s done in [1] (one could lso do the vice vers). We review few definitions. Definition 6.1. A function f : [, b] R belongs to the spce Crd 1 if the following norm is finite: f C 1 rd = f 0,r + mx f (t), where f 0,r = mx f σ (t) ; lso, t [,b] κ t [,b] κ function f : [, b] R belongs to the spce Cld 1 if the following norm is finite: f C 1 ld = f 0,l + mx f (t), where f 0,l = mx f ρ (t). t [,b] κ t [,b] κ

10 34 Cristin Cputo Definition 6.2. A function f is delt regulted if the right-hnd limit f(t+) exists (finite) t ll right-dense points t T nd the left-hnd limit f(t ) exists t ll left-dense points t T; f is regulted if the left-hnd limit f(t+) exists (finite) t ll left-dense points t T nd the right-hnd limit f(t ) exists t ll right-dense points t T. Definition 6.3. A function f is delt piecewise rd-continuous (we write f C prd ) if it is regulted nd if it is rd continuous t ll, except possibly t finitely mny, right-dense points t T; f is nbl piecewise ld-continuous (we write f C pld ) if it is nbl regulted nd if it is ld continuous t ll, except possibly t nitely mny, left-dense points t T. Definition 6.4. f is delt piecewise rd-continuously differentible (we write f C 1 prd ) if f is rd continuous nd f C prd ; f is delt piecewise ld-continuously differentible (we write f C 1 pld ) if f is ld continuous nd f C pld. Definition 6.5. Assume the function L : T R R R is of clss C 2 in the second nd third vrible, nd rd continuous in the first vrible. Then, y 0 is sid to be wek (resp. strong) locl minimum of the problem L(y) = L(t, y σ (t), y (t)) t y() = α, y(b) = β, (6.1) where, b T, with < b; α, β R, nd L : T R R R, if y 0 () = α, y 0 (b) = β, nd L(y 0 ) L(y) for ll y C 1 rd with y() = α, y(b) = β nd y y 0 C 1 rd δ (resp. y y 0 0,r δ) for some δ > 0. We refer to the function L s to the Lgrngin for the bove problem. Moreover, if L = L(t, x, v), then L v, L x represent, respectively, the prtil derivtives of L with respect to v, nd x. Definition 6.6. Assume the function L : T R R R is of clss C 2 in the second nd third vrible, nd rd continuous in the first vrible. Then, y 0 is sid to be wek (strong) locl minimum of the problem L(h) = d c L(s, h ρ (s), h (s)) s h(c) = A, h(d) = B, (6.2) where c, d T, with c < d; A, B R, nd L : T R R R, if y 0 (c) = A, y 0 (d) = B, nd L(y 0 ) L(y) for ll y C 1 ld with y(c) = A, y(d) = B nd y y 0 C 1 ld δ (resp. y y 0 0,l δ) for some δ > 0. Definition 6.7. Given Lgrngin L : T R R R, we define the dul (corresponding) Lgrngin L : T R R R by L (s, x, v) = L( s, x, v) for ll (s, x, v) T R R.

11 From Nbl Clculus to Delt Clculus nd Vice Vers 35 As consequence of the definition of the dul Lgrngin nd Proposition 4.10 we hve the following useful lemm. Lemm 6.8. Given Lgrngin L : [, b] R R R, then L(t, y σ (t), y (t)) t = for ll functions y C 1 rd([, b]). L (s, (y )ˆρ (s), (y ) ˆ (s)) ˆ s, The next theorem is result by Bohner [4] in one dimension (the results we will present cn be obtined without this restriction, but we prefer one dimension to hve n immedite comprison with the results in [1]). Theorem 6.9 (Euler Lgrnge Necessry Condition in Delt Setting). If y 0 is (wek) locl minimum of the vritionl problem (6.1), then the Euler Lgrnge eqution L v (t, y σ 0 (t), y 0 (t)) = L x (t, y σ 0 (t), y 0 (t)), for ll t [, b] κ, holds. Now, we will use Bohner s theorem to prove the Euler Lgrnge eqution in the nbl context. We recll tht the Euler Lgrnge eqution in the nbl context ws shown in [1]. Here we will reprove it using our technique. (Also, see Remrk 6.11.) Theorem 6.10 (Euler Lgrnge Necessry Condition in Nbl Setting). If ȳ 0 is locl (wek) minimum for the vritionl problem (6.2), then the Euler Lgrnge eqution L x (s, (ȳ 0 ) ρ (s), (ȳ 0 ) (s)) = ( L w ) (s, (ȳ 0 ) ρ (s), (ȳ 0 ) (s)) for ll s [c, d] κ, holds. Proof. This theorem is essentilly corollry of Theorem 6.9. Since ȳ 0 is locl minimum for (6.2), it follows from Lemm 6.8 tht ȳ 0 is locl minimum for the vritionl problem ( L) (g) = c d L (t, gˆσ (t), g ˆ (t)) ˆ t, g( c) = A, g( d) = B, (6.3) where g C 1 rd. The vritionl problem (6.3) is the sme s (6.1) for the Lgrngin L (with = d, b = c, α = B nd β = A). Hence, we cn pply Theorem 6.9. The Euler Lgrnge eqution for the Lgrngin L is given by ( L v) ˆ (t, (ȳ 0)ˆσ (t), (ȳ 0) ˆ (t)) = L x(t, (ȳ 0)ˆσ (t), (ȳ 0) ˆ (t)), for ll t [ d, c] κ. (6.4)

12 36 Cristin Cputo Our gol is now to rewrite (6.4) for the Lgrngin L. It is esy to check tht L v(t, x, v) = L w ( t, x, v), nd L x(t, x, v) = L x ( t, x, v), where L w is the prtil derivtive of L with respect to the third vrible. Let us substitute x by (ȳ 0)ˆσ (t), nd v by (ȳ 0) ˆ (t), in the previous identities. We get L v(t, (ȳ 0)ˆσ (t), (ȳ 0) ˆ (t)) = L w ( t, (ȳ 0 ) ρ ( t), (ȳ 0 ) ( t)), nd L x(t, (ȳ 0)ˆσ (t), (ȳ 0) ˆ (t)) = L x ( t, (ȳ 0 ) ρ ( t), (ȳ 0 ) ( t)). From Lemm 4.6, it follows tht where g ˆ (t) = p ( t) for ll t [ d, c] κ, g(t) = L v(t, (ȳ 0)ˆσ (t), (ȳ 0) ˆ (t)) nd p( t) = L w ( t, (ȳ 0 ) ρ ( t), (ȳ 0 ) ( t)). Next, let s [c, d] κ nd set t = s. Then by (6.4), p (s) = L x (s, (ȳ 0 ) ρ (s), (ȳ 0 ) (s)), (6.5) nd, finlly, reveling the definition of p, from (6.5) we obtin the Euler Lgrnge eqution in the nbl setting: L x (s, (ȳ 0 ) ρ (s), (ȳ 0 ) (s)) = ( L w ) (s, (ȳ 0 ) ρ (s), (ȳ 0 ) (s)) for ll s [c, d] κ. The proof is complete. Remrk Theorem 6.10 sttes the sme result s the min theorem proven in [1]. The only difference is the intervl of points for which the Euler Lgrnge eqution holds. In fct, since in [1] the intervl of integrtion for the Lgrngin is [ρ 2 ()), ρ(b)], it follows from our results tht the Euler Lgrnge eqution hs to hold in the intervl [ρ 2 ()), ρ(b)] κ nd not [ρ()), b] s in [1]. This clim cn be lso justified by noticing tht, in order of pplying [1, Lemm 2.1], the test functions hve to vnish t the limit points of integrtion. Another observtion bout such intervl ws pointed out in [6]. Remrk Theorem 6.10 cn be esily generlized to the higher-order results of [12] by pplying our Dulity Principle to the results in [7].

13 From Nbl Clculus to Delt Clculus nd Vice Vers Weierstrss Necessry Condition on Time Scles We first review few definitions. Let L be Lgrngin. Let E : [, b] κ R 3 R be the function defined s E(t, x, r, q) = L(t, x, q) L(t, x, r) (q r)l r (t, x, r). This function E is clled the Weierstrss excess function of L. The Weierstrss necessry optimlity condition on time scles ws proven in the delt setting in [11]. This theorem is stted s follows. Theorem 6.13 (Weierstrss Necessry Optimlity Condition with Delt Setting). Let T be time scle, nd b T, < b. Assume tht the function L(t, x, r) in (6.1) stisfies the following condition: µ(t)l(t, x, γr 1 + (1 γ)r 2 ) µ(t)γl(t, x, r 1 ) + µ(t)(1 γ)l(t, x, r 2 ), (6.6) for ech (t, x) [, b] κ R nd ll r 1, r 2 R, γ [0, 1]. Let x be piecewise continuous function. If x is strong locl minimum for (6.1), then E[t, x σ (t), x (t), q] 0 for ll t [, b] κ nd q R, where we replce x (t) by x (t ) nd x (t+) t finitely mny points t where x (t) does not exist. Let E be the Weierstrss excess function of L. Theorem 6.14 (Weierstrss Necessry Optimlity Condition with Nbl Setting). Let T be time scle, nd b T, < b. Assume tht the function L(t, x, r) in (6.2) stisfies the following condition: ν(t) L(t, x, γr 1 + (1 γ)r 2 ) ν(t)γ L(t, x, r 1 ) + ν(t)(1 γ) L(t, x, r 2 ), (6.7) for ech (t, x) [, b] κ R nd ll r 1, r 2 R, γ [0, 1]. Let x be piecewise continuous function. If x is strong locl minimum for (6.2), then E[t, x ρ (t), x (t), q] 0 for ll t [, b] κ nd q R where we replce x (t) by x (t ) nd x (t+) t finitely mny points t where x (t) does not exist. Proof. Let L be the dul Lgrngin of L. It is esy to prove (similrly s we did in Theorem 6.10, lthough here x is strong minimum), tht x is strong locl minimum for (6.1). Then, (6.7) cn be written on the dul time scle T s µ(s) L (s, x, γr 1 (1 γ)r 2 ) µ(s)γ L (s, x, r 1 ) + µ(s)(1 γ) L (s, x, r 2 ),

14 38 Cristin Cputo for ech (s, x) [, ] κ R nd ll r 1, r 2 R, γ [0, 1]. We recognize tht the lst inequlity is the sme s (6.6) in Theorem 6.13 for the Lgrngin L. Hence, we pply Theorem 6.13, E [s, ( x )ˆσ (s), ( x ) ˆ (s), q] 0 for ll s [, ] κ nd q R, where E is the Weierstrss excess function of L. Also, we notice tht E [s, ( x )ˆσ (s), ( x ) ˆ (s), q] = E[ s, ( x )ˆσ (s), ( x ) ˆ (s), q], where E is the Weierstrss excess function of L. Finlly, E[t, x ρ (t), x (t), q] 0 for ll t [, b] κ nd ll q R, becuse ( x )ˆσ (s) = x ρ ( s), nd ( x ) ˆ (s) = x ( s). We observe tht, the fct tht we cn replce x (t) by where x (t ) nd x (t+) t finitely mny points t, x (t) does not exist, follows s well from Theorem Acknowledgments The uthor would like to thnk Professors D. Torres nd A. Mlinowsk for hving brought this problem to her ttention while they were visiting the University of Texs, t Austin, in the Fll Also, she thnks them for ptiently reding some drfts of this rticle nd mking very helpful suggestions. Appendix: Tble of Dul Objects Bsed on the bove definitions, remrks nd lemms we summrize in Tble 1 for ech object its dul one. Nturlly, Tble 1 my be extended to more objects.

15 From Nbl Clculus to Delt Clculus nd Vice Vers 39 Tble 1: Tble of Dul Objects Object T f : T R f : T R Corresponding dul object T f : T R f : T R t 0 right-dense (left-dense) t 0 left-dense (right-dense ) t 0 right-scttered (left-scttered) t 0 left-scttered (right-scttered) µ, ν ˆν(= µ ), ˆµ(= ν ) σ, ρ ˆρ(= σ ), ˆσ(= ρ ) f (t 0 ) (f ) ˆ ( t 0 ) f (t 0 ) (f ) ( t 0 ) f (t 0 ) ((f ) ) (t 0 ) (f ) ( t 0 ) ((f ) ˆ )( t 0 )) f C rd ( f C ld ) f C ld (f C rd ) f C 1 rd ( f C 1 ld) f C 1 ld (f C 1 rd) f C prd ( f C pld ) f C pld ( f C prd ) f C 1 prd (f C 1 pld) f(t) t L : T R 2 R, L(t, x, v) f C 1 pld( f C 1 prd) f (s) ˆ s L : T R 2 R, L (s, x, w)(= L( s, x, w))

16 40 Cristin Cputo References [1] F. M. Atıcı, D. C. Biles, nd A. Lebedinsky. An ppliction of time scles to economics. Mth. Comput. Modelling, 43(7-8): , [2] F. M. Atıcı nd G. Sh. Guseinov. On Green s functions nd positive solutions for boundry vlue problems on time scles. J. Comput. Appl. Mth., 141(1-2):75 99, Specil Issue on Dynmic Equtions on Time Scles, edited by R. P. Agrwl, M. Bohner, nd D. O Regn. [3] F. M. Atici nd C. S. McMhn. A comprison in the theory of clculus of vritions on time scles with n ppliction to the Rmsey model. Nonliner Dyn. Syst. Theory, 9(1):1 10, [4] M. Bohner. Clculus of vritions on time scles. Dynm. Systems Appl., 13: , [5] M. Bohner nd A. Peterson. Dynmic Equtions on Time Scles: An Introduction with Applictions. Birkhäuser, Boston, [6] R. A. C. Ferreir nd D. F. M. Torres. Remrks on the clculus of vritions on time scles. Int. J. Ecol. Econ. Stt., 9(F07):65 73, [7] R. A. C. Ferreir nd D. F. M. Torres. Higher-order clculus of vritions on time scles. In Mthemticl control theory nd finnce, pges Springer, Berlin, [8] M. Gürses, G. Sh. Guseinov, nd B. Silindir. Integrble equtions on time scles. J. Mth. Phys., 46(11):113510, 22, [9] S. Hilger. Ein Mßkettenklkül mit Anwendung uf Zentrumsmnnigfltigkeiten. PhD thesis, Universität Würzburg, [10] R. Hilscher nd V. Zeidn. Clculus of vritions on time scles: wek locl piecewise C 1 rd solutions with vrible endpoints. J. Mth. Anl. Appl., 289: , [11] A. B. Mlinowsk nd D. F. M. Torres. Strong minimizers of the clculus of vritions on time scles nd the Weierstrss condition. Proc. Est. Acd. Sci., 58(4): , [12] N. Mrtins nd D. F. M. Torres. Clculus of vritions on time scles with nbl derivtives. Nonliner Anl., 71(12):e , 2009.