The Delta-nabla Calculus of Variations for Composition Functionals on Time Scales

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1 Interntionl Journl of Difference Equtions ISSN , Volume 8, Number, pp ) The Delt-nbl Clculus of Vritions for Composition Functionls on Time Scles Monik Dryl nd Delfim F. M. Torres Center for Reserch nd Development in Mthemtics nd Applictions Deprtment of Mthemtics, University of Aveiro Aveiro, Portugl monikdryl@u.pt, delfim@u.pt Abstrct We develop the clculus of vritions on time scles for functionl tht is the composition of certin sclr function with the delt nd nbl integrls of vector vlued field. Euler Lgrnge equtions, trnsverslity conditions, nd necessry optimlity conditions for isoperimetric problems, on n rbitrry time scle, re proved. Interesting corollries nd exmples re presented. AMS Subject Clssifictions: 6E7, 49K5. Keywords: Clculus of vritions, optimlity conditions, time scles. Introduction We study generl problem of the clculus of vritions on n rbitrry time scle T. More precisely, we consider the problem of extremizing i.e., minimizing or mximizing) delt-nbl integrl functionl Lx) = H f t, x σ t), x t)) t,..., f k t, x σ t), x t)) t, f k t, x ρ t), x t)) t,..., f kn t, x ρ t), x t)) t possibly subject to boundry conditions nd/or isoperimetric constrints. For the interest in studying such type of vritionl problems in economics, we refer the reder Received August 7, ; Accepted November 4, Communicted by Amr Debbouche

2 8 M. Dryl nd D. F. M. Torres to [] nd references therein. For review on generl pproches to the clculus of vritions on time scles, which llow to obtin both delt nd nbl vritionl clculus s prticulr cses, see [5, 9, ]. Throughout the text we ssume the reder to be fmilir with the bsic definitions nd results of time scles [3, 4, 7, 8]. The rticle is orgnized s follows. In Section we collect some necessry definitions nd theorems of the nbl nd delt clculus on time scles. The min results re presented in Section 3. We begin by proving generl Euler Lgrnge equtions Theorem 3.). Next we consider the situtions when initil or terminl boundry conditions re not specified, obtining corresponding trnsverslity conditions Theorems 3.4 nd 3.5). The results re pplied to quotient vritionl problems in Corollry 3.6. Finlly, we prove necessry optimlity conditions for generl isoperimetric problems given by the composition of delt-nbl integrls Theorem 3.9). We end with Section 4, illustrting the new results of the pper with severl exmples. Preliminries In this section we review the min results necessry in the sequel. For bsic definitions, nottions nd results of the theory of time scles, we refer the reder to the books [3,4]. The following two lemms re the extension of the Dubois Reymond fundmentl lemm of the clculus of vritions [3] to the nbl Lemm.) nd delt Lemm.) time scle clculus. We remrk tht ll intervls in this pper re time scle intervls. Lemm. See []). Let f C ld [, b], R). If ft)η t) t = for ll η C ld[, b], R) with η) = ηb) =, then ft) = c, for some constnt c, for ll t [, b] κ. Lemm. See []). Let f C rd [, b], R). If ft)η t) t = for ll η C rd[, b], R) with η) = ηb) =, then ft) = c, for some constnt c, for ll t [, b] κ. Under some ssumptions, it is possible to relte the delt nd nbl derivtives Theorem.3) s well s the delt nd nbl integrls Theorem.4). Theorem.3 See []). If f : T R is delt differentible on T κ nd f is continuous on T κ, then f is nbl differentible on T κ nd f t) = f ) ρ t) for ll t T κ..)

3 The Delt-nbl Clculus of Vritions for Composition Functionls 9 If f : T R is nbl differentible on T κ nd f is continuous on T κ, then f is delt differentible on T κ nd f t) = f ) σ t) for ll t T κ..) Theorem.4 See [6]). Let, b T with < b. If function f : T R is continuous, then ft) t = ft) t = f ρ t) t,.3) f σ t) t..4) 3 Min Results By C k,n[, b], R) we denote the clss of functions x : [, b] R such tht: if n =, then x is continuous on [, b] κ ; if k =, then x is continuous on [, b] κ ; if k nd n, then x is continuous on [, b] κ κ nd x is continuous on [, b] κ κ, where [, b] κ κ := [, b] κ [, b] κ. We consider the following problem of clculus of vritions: Lx) = H f t, x σ t), x t)) t,..., f k t, x ρ t), x t)) t,..., f k t, x σ t), x t)) t, f kn t, x ρ t), x t)) t extr, 3.) x) = x ), xb) = x b ), 3.) where extr mens minimize or mximize. The prentheses in 3.), round the end-point conditions, mens tht those conditions my or my not occur it is possible tht both x) nd xb) re free). A function x C k,n is sid to be dmissible provided it stisfies the boundry conditions 3.) if ny is given). For k = problem 3.) reduces to nbl problem no delt integrl nd delt derivtive is present); for n = problem 3.) reduces to delt problem no nbl integrl nd nbl derivtive is present). We ssume tht:. the function H : R nk R hs continuous prtil derivtives with respect to its rguments, which we denote by, i =,..., n k;. functions t, y, v) f i t, y, v) from [, b] R to R, i =,..., n k, hve prtil continuous derivtives with respect to y nd v for ll t [, b], which we denote by f iy nd f iv ;

4 3 M. Dryl nd D. F. M. Torres 3. f i, f iy, f iv re continuous on [, b] κ, i =,..., k, nd continuous on [, b] κ, i = k,..., k n, for ll x C k,n. The following norm in C k,n is considered: where x := sup xt). x, := x σ x x ρ x, Definition 3.. We sy tht n dmissible function ˆx is wek locl minimizer respectively wek locl mximizer) to problem 3.) 3.) if there exists δ > such tht Lˆx) Lx) respectively Lˆx) Lx)) for ll dmissible functions x C k,n stisfying the inequlity x ˆx, < δ. For simplicity, we introduce the opertors [ ] nd [ ] by [x] t) = t, x σ t), x t)) nd [x] t) = t, x ρ t), x t)). Along the text, c denotes constnts tht re generic nd my chnge t ech occurrence. 3. Euler Lgrnge Equtions Depending on the given boundry conditions, we cn distinguish four different problems. The first is problem P b ), where the two boundry conditions re specified. To solve this problem we need type of Euler Lgrnge necessry optimlity condition. This is given by Theorem 3. below. Next two problems denoted by P ) nd P b ) occur when x) is given nd xb) is free problem P )) nd when x) is free nd xb) is specified problem P b )). To solve both of them we need to use n Euler Lgrnge eqution nd one trnsverslity condition. The lst problem denoted by P ) occurs when both boundry conditions re not specified. To find solution for such problem we need to use n Euler Lgrnge eqution nd two trnsverslity conditions one t ech time nd b). Trnsverslity conditions re the subject of Section 3.. Theorem 3. Euler Lgrnge equtions in integrl form). If ˆx is wek locl solution to problem 3.) 3.), then the Euler Lgrnge equtions f iv [ˆx] ρt)) kn i=k ρt) f iy [ˆx] τ) τ f iv [ˆx] t) f iy [ˆx] τ) τ = c, t T κ, 3.3) For brevity, we re omitting the rguments of, i.e., := F ˆx),..., F kn ˆx)), where F i ˆx) = f i [ˆx] t) t, i =,..., k, nd F i ˆx) = f i [ˆx] t) t, i = k,..., k n.

5 The Delt-nbl Clculus of Vritions for Composition Functionls 3 nd f iv [ˆx] t) hold. kn i=k f iy [ˆx] τ) τ f iv [ˆx] σt)) σt) f iy [ˆx] τ) τ = c, t T κ, 3.4) Proof. Suppose tht L x) hs wek locl extremum t ˆx. Consider vrition h C k,n of ˆx for which we define the function φ : R R by φε) = L ˆx εh). A necessry condition for ˆx to be n extremizer for L x) is given by φ ε) = for ε =. Using the chin rule, we obtin tht = φ ) = fiy [ˆx] t)h σ t) f iv [ˆx] t)h t) ) t kn i=k fiy [ˆx] t)h ρ t) f iv [ˆx] t)h t) ) t. Integrtion by prts of the first terms of both integrls gives f iy [ˆx] t)h σ t) t = f iy [ˆx] t)h ρ t) t = f iy [ˆx] τ) τht) f iy [ˆx] τ) τht) Thus, the necessry condition φ ) = cn be written s f iy [ˆx] τ) τht) b b b f iy [ˆx] τ) τ h t) t f iy [ˆx] τ) τ h t) t, f iy [ˆx] τ) τ h t) t. f iv [ˆx] t)h t) t

6 3 M. Dryl nd D. F. M. Torres kn i=k f iy [ˆx] τ) τht) b f iy [ˆx] τ) τ h t) t f iv [ˆx] t)h t) t =. 3.5) In prticulr, condition 3.5) holds for ll vritions tht re zero t both ends: h) = nd hb) =. Then, we obtin: H ih t) f iv [ˆx] t) f iy [ˆx] τ) τ t Introducing ξ nd χ by nd ξt) := χt) := kn i=k kn i=k H ih t) f iv [ˆx] t) f iv [ˆx] t) we then obtin the following reltion: h t)ξt) t f iv [ˆx] t) f iy [ˆx] τ) τ t =. f iy [ˆx] τ) τ 3.6) f iy [ˆx] τ) τ, 3.7) h t)χt) t =. 3.8) We consider two cses. i) Firstly, we chnge the first integrl of 3.8) nd we obtin two nbl-integrls nd, subsequently, the eqution 3.3). ii) In the second cse, we chnge the second integrl of 3.8) to obtin two delt-integrls, which leds us to 3.4). i) Using reltion.3) of Theorem.4, we obtin: h t) ) ρ ξ ρ t) t Using.) of Theorem.3 we hve h t)χt) t =. h t) ξ ρ t) χt)) t =.

7 The Delt-nbl Clculus of Vritions for Composition Functionls 33 By the Dubois Reymond Lemm. nd we obtin 3.3). ii) From 3.8), nd using reltion.4) of Theorem.4, h t)ξt) t ξ ρ t) χt) = const 3.9) h t)) σ χ σ t) t =. Using.) of Theorem.3, we get: h t)ξt) χ σ t)) t =. From the Dubois Reymond Lemm., it follows tht ξt) χ σ t) = const. Hence, we obtin the Euler Lgrnge eqution 3.4). A time scle T is sid to be regulr if the following two conditions re stisfied simultneously for ll t T: σρt)) = t nd ρσt)) = t. For regulr time scles, the Euler Lgrnge equtions 3.3) nd 3.4) coincide; on generl time scle, they re different. Such difference is illustrted in Exmple 3.3. Exmple 3.3. Let us consider the irregulr time scle T = P, = [k, k ]. We show tht for this time scle there is difference between the Euler Lgrnge equtions 3.3) nd 3.4). The forwrd nd bckwrd jump opertors re given by t, t k, k ], t, t [k, k ), k= k= σt) = ρt) = t, t {k}, t, t {k }, k= k=, t =. For t = nd t k= k, k ), equtions 3.3) nd 3.4) coincide. We cn distinguish between them for t {k } nd t k= nottions 3.6) nd 3.7). If t k= {k}. In wht follows we use the k= {k }, then we obtin from 3.3) nd 3.4) the k=

8 34 M. Dryl nd D. F. M. Torres Euler Lgrnge equtions ξt) χt) = c nd ξt) χt ) = c, respectively. If t {k}, then the Euler Lgrnge eqution 3.3) hs the form ξt ) χt) = c k= while 3.4) tkes the form ξt) χt) = c. 3. Nturl Boundry Conditions In this section we consider the sitution when we wnt to minimize or mximize the vritionl functionl 3.), but boundry conditions x) nd/or xb) re free. Theorem 3.4 Trnsverslity condition t the initil time t = ). Let T be time scle for which ρσ)) =. If ˆx is wek locl solution to 3.) with x) not specified, then f iv [ˆx] ) kn i=k f iv [ˆx] σ)) σ) holds together with the Euler Lgrnge equtions 3.3) nd 3.4). f iy [ˆx] t) t = 3.) Proof. From 3.5) nd 3.9), we hve f iy [ˆx] τ) τht) b kn i=k f iy [ˆx] τ) τht) b h t) c t =. Therefore, f iy [ˆx] τ) τht) b kn i=k f iy [ˆx] τ) τht) b ht) c b =. Next, we deduce tht hb) h) f iy [ˆx] τ) τ kn i=k f iy [ˆx] τ) τ kn i=k f iy [ˆx] τ) τ c f iy [ˆx] τ) τ c =, 3.) where c = ξρt)) χt). 3.)

9 The Delt-nbl Clculus of Vritions for Composition Functionls 35 The Euler Lgrnge eqution 3.3) of Theorem 3. or 3.)) is given t t = σ) s ρσ)) f iv [ˆx] ρσ))) f iy [ˆx] τ) τ We conclude tht f iv [ˆx] ) kn i=k kn i=k f iv [ˆx] σ)) f iv [ˆx] σ)) σ) σ) f iy [ˆx] τ) τ = c. f iy [ˆx] τ) τ = c. Restricting the vritions h to those such tht hb) =, it follows from 3.) tht h) c =. From the rbitrriness of h, we conclude tht c =. Hence, we obtin 3.). Theorem 3.5 Trnsverslity condition t the terminl time t = b). Let T be time scle for which σρb)) = b. If ˆx is wek locl solution to 3.) with xb) not specified, then f iv [ˆx] ρb)) f iy [ˆx] kn t) t f iv [ˆx] b) = 3.3) ρb) i=k holds together with the Euler Lgrnge equtions 3.3) nd 3.4). Proof. The clcultions in the proof of Theorem 3.4 give us 3.). When h) =, the Euler Lgrnge eqution 3.4) of Theorem 3. hs the following form t t = ρb): ρb) f iv [ˆx] ρb)) f iy [ˆx] τ) τ Then, kn i=k f iv [ˆx] ρb)) ρb) f iv [ˆx] σρb))) f iy [ˆx] τ) τ kn i=k We obtin 3.3) from 3.) nd 3.4). f iv [ˆx] b) σρb)) f iy [ˆx] t) τ = c. f iy [ˆx] t) τ = c. 3.4)

10 36 M. Dryl nd D. F. M. Torres Severl new interesting results cn be immeditely obtined from Theorems 3., 3.4 nd 3.5. An exmple of such results is given by Corollry 3.6. Corollry 3.6. If ˆx is solution to the problem Lx) = then the Euler Lgrnge equtions F f v [ˆx] ρt)) ρt) f t, x σ t), x t)) t f t, x ρ t), x t)) t x) = x ), xb) = x b ), f y [ˆx] τ) τ F F extr, f v [ˆx] t) f y [ˆx] τ) τ = c nd F f v [ˆx] t) f y [ˆx] τ) τ F F f v [ˆx] σt)) σt) f y [ˆx] τ) τ = c hold for ll t [, b] κ κ, where F := f t, ˆx σ t), ˆx t)) t nd F := f t, ˆx ρ t), ˆx t)) t. Moreover, if x) is free nd ρσ)) =, then f v [ˆx] ) F f F F v [ˆx] σ)) σ) f y [ˆx] t) t = ; if xb) is free nd σρb)) = b, then F f v [ˆx] ρb)) ρb) f y [ˆx] t) t F f v [ˆx] b) =. F

11 The Delt-nbl Clculus of Vritions for Composition Functionls Isoperimetric Problems Let us consider the generl composition isoperimetric problem on time scles subject to given boundry conditions. The problem consists of minimizing or mximizing Lx) = H f t, x σ t), x t)) t,..., f k t, x σ t), x t)) t, f k t, x ρ t), x t)) t,..., f kn t, x ρ t), x t)) t 3.5) in the clss of functions x C k,n stisfying the boundry conditions nd the generlized isoperimetric constrint Kx) = P x) = x, xb) = x b, 3.6) g t, x σ t), x t)) t,..., g m t, x ρ t), x t)) t,..., g m t, x σ t), x t)) t, g mp t, x ρ t), x t)) t = d, 3.7) where x, x b, d R. We ssume tht:. the functions H : R nk R nd P : R mp R hve continuous prtil derivtives with respect to ll their rguments, which we denote by H i, i =,..., nk, nd P i, i =,..., m p;. functions t, y, v) f i t, y, v), i =,..., n k, nd t, y, v) g j t, y, v), j =,..., m p, from [, b] R to R, hve prtil continuous derivtives with respect to y nd v for ll t [, b], which we denote by f iy, f iv, nd g jy, g jv ; 3. for ll x C km,np, f i, f iy, f iv nd g j, g jy, g jv re continuous in t [, b] κ, i =,..., k, j =,..., m, nd continuous in t [, b] κ, i = k,..., k n, j = m,..., m p. Definition 3.7. We sy tht n dmissible function ˆx is wek locl minimizer respectively wek locl mximizer) to the isoperimetric problem 3.5) 3.7), if there exists δ > such tht Lˆx) Lx) respectively Lˆx) Lx)) for ll dmissible functions x C km,np stisfying the boundry conditions 3.6), the isoperimetric constrint 3.7), nd inequlity x ˆx, < δ.

12 38 M. Dryl nd D. F. M. Torres nd Let us define u nd w by ut) := wt) := m mp i=m P i P i g iv [ˆx] t) g iv [ˆx] t) g iy [ˆx] τ) τ 3.8) g iy [ˆx] τ) τ, 3.9) where we omit, for brevity, the rgument of P i : P i := P i G ˆx),..., G mp ˆx)) with G i ˆx) = i = m,..., m p. g i t, ˆx σ t), ˆx t)) t, i =,..., m, nd G i ˆx) = g i t, ˆx ρ t), ˆx t)) t, Definition 3.8. An dmissible function ˆx is sid to be n extreml for K if ut) wσt)) = const nd uρt)) wt) = const for ll t [, b] κ κ. An extremizer i.e., wek locl minimizer or wek locl mximizer) to problem 3.5) 3.7) tht is not n extreml for K is sid to be norml extremizer; otherwise i.e., if it is n extreml for K), the extremizer is sid to be bnorml. Theorem 3.9 Optimlity condition to the isoperimetric problem 3.5) 3.7)). Let ξ nd χ be given s in 3.6) nd 3.7), nd u nd w be given s in 3.8) nd 3.9). If ˆx is norml extremizer to the isoperimetric problem 3.5) 3.7), then there exists rel number λ such tht. ξ ρ t) χt) λ u ρ t) wt)) = const;. ξt) χ σ t) λ u ρ t) wt)) = const; 3. ξ ρ t) χt) λ ut) w σ t)) = const; 4. ξt) χ σ t) λ ut) w σ t)) = const; for ll t [, b] κ κ. Proof. We prove the first item of Theorem 3.9. The other items re proved in similr wy. Consider vrition of ˆx such tht x = ˆx ε h ε h, where h i Ckm,np nd h i ) = h i b) =, i =,, nd prmeters ε nd ε re such tht x ˆx, < δ for some δ >. Function h is rbitrry nd h will be chosen lter. Define Kε, ε ) = Kx) = P g t, x σ t), x t)) t,..., g m t, x ρ t), x t)) t,..., g m t, x σ t), x t)) t, g mp t, x ρ t), x t)) t d.

13 The Delt-nbl Clculus of Vritions for Composition Functionls 39 A direct clcultion gives K ε =,) m P i giy [ˆx] t)h σ t) g iv [ˆx] t)h t) ) t mp i=m P i giy [ˆx] t)h ρ t) g iv [ˆx] t)h t) ) t. Integrtion by prts of the first terms of both integrls gives: m b P b i g iy [ˆx] τ) τh t) g iy [ˆx] τ) τ h t) t mp i=m P i g iy [ˆx] τ) τh t) Since h ) = h b) =, we hve m P i h t) g iv [ˆx] t) Therefore, K ε =,) mp i=m g iv [ˆx] t)h t) t b g iy [ˆx] τ) τ t P g iy [ˆx] τ) τ h t) t i h t) g iv [ˆx] t) h t)ut) t Using reltion.) of Theorem.3, we obtin tht h ) ρ t)u ρ t) t h t)wt) t = h t)wt) t. g iv [ˆx] t)h t) t. g iy [ˆx] τ) τ t. h t) u ρ t) wt)) t.

14 4 M. Dryl nd D. F. M. Torres By the Dubois Reymond Lemm., there exists function h such tht K ε.,) Since K, ) =, there exists function ε, defined in the neighborhood of zero, such tht Kε, ε ε )) =, i.e., we my choose subset of vritions ˆx stisfying the isoperimetric constrint. Let us consider the rel function Lε, ε ) = Lx) = H f t, x σ t), x t)) t,..., f k t, x σ t), x t)) t, f k t, x ρ t), x t)) t,..., f kn t, x ρ t), x t)) t. The point, ) is n extreml of L subject to the constrint K = nd K, ). By the Lgrnge multiplier rule, there exists λ R such tht L, ) λk, ) ) =. Becuse h ) = h b) =, we hve L ε =,) fiy [ˆx] t)h σ t) f iv [ˆx] t)h t) ) t kn i=k Integrting by prts, nd using h ) = h b) =, gives L ε =,) h t)ξt) t fiy [ˆx] t)h ρ t) f iv [ˆx] t)h t) ) t. h t)χt) t. Using.3) of Theorem.4 nd.) of Theorem.3, we obtin tht nd L ε =,) K ε =,) m h ) ρ t)ξ ρ t) t P i h t)χt) t = giy [ˆx] t)h σ t) g iv [ˆx] t)h t) ) t mp i=m h t) ξ ρ t) χt)) t P i giy [ˆx] t)h ρ t) g iv [ˆx] t)h t) ) t.

15 The Delt-nbl Clculus of Vritions for Composition Functionls 4 Integrting by prts, nd reclling tht h ) = h b) =, K ε =,) h t)ut) t h t)wt) t. Using reltion.3) of Theorem.4 nd reltion.) of Theorem.3, we obtin tht K ε =,) ) b h ρ t)u ρ t) t Since L ε λ K,) ε =, we hve,) h t)wt) t = h t) u ρ t) wt)) t. h t) [ξ ρ t) χt) λ u ρ t) wt))] t = for ny h C km,np. Therefore, by the Dubois Reymond Lemm., one hs ξ ρ t) χt) λ u ρ t) wt)) = c, where c R. Remrk 3.. One cn esily cover both norml nd bnorml extremizers with Theorem 3.9, if in the proof we use the bnorml Lgrnge multiplier rule [3]. 4 Illustrtive Exmples We begin with nonutonomous problem. Exmple 4.. Consider the problem Lx) = tx t) t min, x t)) t x) =, x) =. 4.) If x is locl minimizer to problem 4.), then the Euler Lgrnge equtions of Corollry 3.6 must hold, i.e., ρt) F x t) = c F F nd t F x σt)) = c, F F

16 4 M. Dryl nd D. F. M. Torres where F := F x) = tx t) t nd F := F x) = the second eqution. Using.) of Theorem.3, it cn be written s x t)) t. Let us consider t F x t) = c. 4.) F Solving eqution 4.) nd using the boundry conditions x) = nd x) =, xt) = Q F τ τ t Q τ τ, 4.3) where Q := F. Therefore, the solution depends on the time scle. Let us consider two F exmples: T = R nd T = {, },. With T = R, from 4.3) we obtin xt) = 4Q t 4Q 4Q t, x t) = x t) = x t) = Q t 4Q 4Q. 4.4) Substituting 4.4) into F nd F gives F = Q 4Q nd F = 48Q, tht is, 48Q Q = Solving eqution 4.5) we get Q 3 } 3, 3 3. Becuse 4.) is mini- mizing problem, we select Q = 3 3 If T = { QQ ) 48Q. 4.5) nd we get the extreml xt) = 3 3)t 4 3)t. 4.6) {, },, then from 4.3) we obtin xt) = t k 8Q t, tht is, 8Q 8Q k=, if t =, 8Q xt) = 6Q, if t =,, if t =.

17 The Delt-nbl Clculus of Vritions for Composition Functionls 43 Direct clcultions show tht x ) = x ) x) ) = 8Q 8Q, x ) x = x ) x) Substituting 4.7) into the integrls F nd F gives = x) x ) = 8Q 8Q, x ) = x) x ) F = 8Q 3Q, F = 64Q 64Q, Q = F F = = 8Q 8Q, = 8Q 8Q. Q8Q ) 64Q. 4.7) Thus, { we obtin the eqution 64Q 6Q =. The solutions to this eqution re: Q, }. We re interested in the minimum vlue Q, so we select 8 8 Q = 8 to get the extreml, if t =, xt) =, if t =,, if t =. 4.8) Note tht the extremls 4.6) nd 4.8) re different: for 4.6) one hs x/) = We now present problem where, in contrst with Exmple 4., the extreml does not depend on the time scle T. Exmple 4.. Consider the utonomous problem Lx) = x t) ) t [ x t) x t)) ] t min, 4.9) x) =, x) = 4. If x is locl minimizer to 4.9), then the Euler Lgrnge equtions must hold, i.e., x t) F x t) ) = c nd F F where F := F x) = x t) ) t nd F := F x) = x t) F x t) ) = c, 4.) F F [ x t) x t) ) ] t. Choosing one of the equtions of 4.), for exmple the first one, we get x t) = c F ) F. 4.) F F F

18 44 M. Dryl nd D. F. M. Torres Using 4.) with boundry conditions x) = nd x) = 4, we obtin, for ny given time scle T, the extreml xt) = t. In the previous two exmples, the vritionl functionl is given by the rtio of delt nd nbl integrl. We now discuss vritionl problem where the composition is expressed by the product of three time-scle integrls. Exmple 4.3. Consider the problem Lx) = tx t) t x t) t) t x t) ) t min, x) =, x) =. 4.) If x is locl minimizer to problem 4.), then the Euler Lgrnge equtions must hold, nd we cn write tht where c is constnt, F := F x) = nd F 3 := F 3 x) = F F 3 F F 3 ) t F F 3 F F x σt)) = c, 4.3) tx t) t, F := F x) = x t) t) t, x t) ) t. Using reltion.), we cn write 4.3) s F F 3 F F 3 ) t F F 3 F F x t) = c. 4.4) Using the boundry conditions x) = nd x) =, we get from 4.4) tht xt) = Q τ τ t Q τ τ, 4.5) where Q = F F 3 F F 3. Therefore, the solution depends on the time scle. Let us F F consider T = R nd T = {, },. With T = R, expression 4.5) gives ) Q xt) = t Q t, x t) = x t) = x t) = Q Qt. 4.6) Substituting 4.6) into F, F nd F 3 gives: F = 6 Q, F = 8 Q, F 3 = Q.

19 The Delt-nbl Clculus of Vritions for Composition Functionls 45 One cn proceed ) by solving the eqution Q 3 8Q 6Q 7 =, to find the extreml Q xt) = t Q t with Q = ) 8 6. Let us consider now the time scle T = {, },. From 4.5), we obtin ) 4 Q xt) = t Q 4 4 Substituting 4.7) into F, F nd F 3, we obtin, if t =, t 4 Q k =, if t = k= 8,, if t =. F = 4 Q 6, F = Q, F 3 = Q ) nd the eqution Q 3 8Q 48Q 96 =. Solving this eqution, we find the extreml, if t =, xt) = , if t = 4,, if t =. Finlly, we pply the results of Section 3.3 to n isoperimetric vritionl problem. Exmple 4.4. Let us consider the problem of extremizing Lx) = x t)) t tx t) t subject to the boundry conditions x) = nd x) =, nd the constrint Kt) = tx t) t =. Applying Theorem 3.9, we get the nbl differentil eqution x t) λ F ) t = c. 4.8) F F ) Solving this eqution, we obtin xt) = Q τ τ t Q τ τ, 4.9)

20 46 M. Dryl nd D. F. M. Torres where Q = F ) F F ) λ. Therefore, the solution of eqution 4.8) depends on {, },. the time scle. As before, let us consider T = R nd T = For T = R, we obtin from 4.9) tht xt) = Q t Q t. Substituting this expression for x into the integrls F nd F, gives F = Q nd F = Q 6. Using the given isoperimetric constrint, we obtin Q = 6, λ = 8, nd xt) = 3t t. Let us consider now the time scle T = {, },. From 4.9), we hve xt) = 4 3Q 4 Simple clcultions show tht F = k= F = nd Kt) = Q 6 t t Q k= k 4 =, if t =, 4 Q, if t = 8,, if t =. )) k x = x ) ) )) x = Q 6, 6 k= 4 kx Acknowledgements ) k = ) 4 x x ) = Q 6 =. Therefore, Q = 4, λ = 6, nd we hve the extreml {, if t, }, xt) =, if t =. This work ws supported by FEDER funds through COMPETE Opertionl Progrmme Fctors of Competitiveness Progrm Opercionl Fctores de Competitividde ) nd by Portuguese funds through the Center for Reserch nd Development in Mthemtics nd Applictions University of Aveiro) nd the Portuguese Foundtion for Science nd Technology FCT Fundção pr Ciênci e Tecnologi ), within project PEst-C/MAT/UI46/ with COMPETE number FCOMP--4- FEDER-69. Dryl ws lso supported by FCT through the Ph.D. fellowship SFRH/ BD/563/. The uthors would like to thnk n nonymous referee for creful reding of the submitted mnuscript nd for suggesting severl useful chnges.

21 The Delt-nbl Clculus of Vritions for Composition Functionls 47 References [] F. M. Atici nd G. Sh. Guseinov, On Green s functions nd positive solutions for boundry vlue problems on time scles, J. Comput. Appl. Mth. 4 ), no. -, [] M. Bohner, Clculus of vritions on time scles, Dynm. Systems Appl. 3 4), no. 3-4, [3] M. Bohner nd A. Peterson, Dynmic equtions on time scles, Birkhäuser Boston, Boston, MA,. [4] M. Bohner nd A. Peterson, Advnces in dynmic equtions on time scles, Birkhäuser Boston, Boston, MA, 3. [5] E. Girejko, A. B. Mlinowsk nd D. F. M. Torres, Delt-nbl optiml control problems, J. Vib. Control 7 ), no., [6] M. Gürses, G. Sh. Guseinov nd B. Silindir, Integrble equtions on time scles, J. Mth. Phys. 46 5), no., 35, pp. [7] S. Hilger, Anlysis on mesure chins unified pproch to continuous nd discrete clculus, Results Mth. 8 99), no. -, [8] S. Hilger, Differentil nd difference clculus unified!, Nonliner Anl ), no. 5, [9] A. B. Mlinowsk nd D. F. M. Torres, The delt-nbl clculus of vritions, Fsc. Mth. No. 44 ), [] A. B. Mlinowsk nd D. F. M. Torres, Euler-Lgrnge equtions for composition functionls in clculus of vritions on time scles, Discrete Contin. Dyn. Syst. 9 ), no., [] N. Mrtins nd D. F. M. Torres, Clculus of vritions on time scles with nbl derivtives, Nonliner Anl. 7 9), no., e763 e773. [] D. F. M. Torres, The vritionl clculus on time scles, Int. J. Simul. Multidisci. Des. Optim. 4 ), no., 5. [3] B. vn Brunt, The clculus of vritions, Universitext, Springer, New York, 4.

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