TAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS ON TIME SCALES
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1 TAYLOR POLYNOMIALS FOR NABLA DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS R. ANDERSON Abtract. We are concerned with the repreentation of polynomial for nabla dynamic equation on time cale. Once etablihed, thee polynomial will be ued to derive Taylor Formula for dynamic function. Several example are given for pecial time cale uch a Z, R, and qz for q > 1. Thee polynomial will be related to thoe given for delta dynamic equation. 1. Preliminarie on Time Scale The following definition can be found in Agarwal and Bohner [1], Atici and Gueinov [2], and Bohner and Peteron [3]; thee are baed on the notion firt introduced by Hilger in hi Ph.D. thei [4]. A time cale T i any nonempty cloed ubet of R. It follow that the jump operator σ, ρ : T T σ(t) inf{ T : > t} and ρ(t) up{ T : < t} (upplemented by inf : up T and up : inf T) are well defined. The point t T i left-dene, left-cattered, right-dene, right-cattered if ρ(t) t, ρ(t) < t, σ(t) t, σ(t) > t, repectively. If T ha a right-cattered minimum m, define T κ : T {m}; otherwie, et T κ T. If T ha a left-cattered maximum M, define T κ : T {M}; otherwie, et T κ T. The forward grainine i µ(t) : σ(t) t. Similarly, the backward grainine i ν(t) : t ρ(t). For f : T R and t T κ, the delta derivative [3] of f at t, denoted f (t), i the number (provided it exit) with the property that given any ε > 0, there i a neighborhood U of t uch that f(σ(t)) f() f (t)[σ(t) ] ε σ(t) for all U. For T R, we have f f, the uual derivative, and for T Z we have the forward difference operator, f (t) f(t + 1) f(t). For f : T R and t T κ, the nabla derivative [2] of f at t, denoted f (t), i the number (provided it exit) with the property that given any ε > 0, there i a 1991 Mathematic Subject Claification. 34B10, 39A10. Key word and phrae. Cauchy Function, Variation of Contant, Taylor Formula.
2 2 D. ANDERSON neighborhood U of t uch that f(ρ(t)) f() f (t)[ρ(t) ] ε ρ(t) for all U. For T R, we have f f, the uual derivative, and for T Z we have the backward difference operator, f (t) f(t) f(t 1). A function f : T R i left-dene continuou or ld-continuou provided it i continuou at left-dene point in T and it right-ided limit exit (finite) at rightdene point in T. If T R, then f i ld-continuou if and only if f i continuou. If T Z, then any function i ld-continuou. It i known [3] that if f i ld-continuou, then there i a function F (t) uch that F (t) f(t). In thi cae, we define b a f(t) t F (b) F (a). Remark 1. The following theorem delineate everal propertie of the delta and nabla derivative; they are found in [2] and [3, p ]. Theorem 1. Aume f : T R i a function and let t T κ. Then we have the following: (i) If f i delta differentiable at t, then f i continuou at t. (ii) If f i continuou at t and t i right-cattered, then f i delta differentiable at t with f f(σ(t)) f(t) (t). µ(t) (iii) If t i right-dene, then f i delta differentiable at t iff the limit f(t) f() lim t t exit a a finite number. In thi cae f f(t) f() (t) lim. t t (iv) If f i delta differentiable at t, then where f σ f σ. f σ (t) f(t) + µ(t)f (t), Theorem 2. Aume f : T R i a function and let t T κ. Then we have the following: (i) If f i nabla differentiable at t, then f i continuou at t. (ii) If f i continuou at t and t i left-cattered, then f i nabla differentiable at t with f f(t) f(ρ(t)) (t). ν(t)
3 NABLA TAYLOR POLYNOMIALS 3 (iii) If t i left-dene, then f i nabla differentiable at t iff the limit f(t) f() lim t t exit a a finite number. In thi cae f f(t) f() (t) lim. t t (iv) If f i nabla differentiable at t, then where f ρ f ρ. f ρ (t) f(t) ν(t)f (t), Theorem 3. Aume f, g : T R are nabla differentiable at t T κ. Then: (i) The um f + g : T R i nabla differentiable at t with (f + g) (t) f (t) + g (t). (ii) The product fg : T R i nabla differentiable at t, and we get the product rule (fg) (t) f (t)g(t) + f ρ (t)g (t) f(t)g (t) + f (t)g ρ (t). (iii) If g(t)g ρ (t) 0, then f g rule i nabla differentiable at t, and we get the quotient ( ) f (t) f (t)g(t) f(t)g (t). g g(t)g ρ (t) (iv) If f and f are continuou, then ( f(t, ) ) f(ρ(t), t) + a 2. Monomial a f (t, ). The generalized monomial, that will alo occur in Taylor formula, are the function ĥk : T 2 R, k N 0, defined recurively a follow: The function ĥ0 i (1) ĥ 0 (t, ) 1 for all, t T, and, given ĥk for k N 0, the function ĥk+1 i (2) ĥ k+1 (t, ) ĥ k (τ, ) τ for all, t T. Note that the function ĥk are all well defined, ince each i ld-continuou. If we let ĥ k (t, ) denote for each fixed the derivative of ĥk(t, ) with repect to t, then (3) ĥ k (t, ) ĥk 1(t, ) for k N, t T κ.
4 4 D. ANDERSON The above definition implie ĥ 1 (t, ) t for all, t T. Finding the ĥk for k > 1 i not eay in general. But for a particular given time cale it might be eay to find thee function. We will conider everal example firt before we preent Taylor formula for arbitrary time cale. Example 1. For the cae T R and T Z it i eay to find the function ĥk: Firt, conider T R. Then ρ(t) t for t R, o that ĥ k (t, ) (t )k for, t R, k N 0. We note that, for an n time differentiable function f : R R, the following well-known Taylor formula hold: Let R be arbitrary. Then, for all t R, the repreentation (4) f(t) (t ) k f (k) () + 1 (t τ) n f (n+1) (τ)dτ n! ĥ k (t, )f (k) () + ĥ n (t, ρ(τ))f (n+1) (τ)dτ are valid, where f (k) denote a uual the kth derivative of f. Next, conider T Z. Then ρ(t) t 1, f(t) f(t) f(t 1) for t Z, and [3, p333] b b f(t) t f(t). a ta+1 For a nonnegative integer k and t Z, define the dicrete factorial function t to the k riing by t 0 : 1 for k 0, and for k 1; then For, t Z and k N 0 we have t k : t(t + 1) (t + k 1) t k kt k 1. (5) ĥ k (t, ) (t )k for all, t Z.
5 NABLA TAYLOR POLYNOMIALS 5 Therefore the dicrete nabla verion of Taylor formula read a follow: Let f : Z R be a function, and let Z. Then, for all t Z with t > +n, the repreentation (t ) k f(t) k f() + 1 t (t τ + 1) n n+1 f(τ) n! (6) ĥ k (t, ) k f() + t τ+1 τ+1 ĥ n (t, ρ(τ)) n+1 f(τ) hold, where k i the k-time iterated backward difference operator. Example 2. We conider the time cale T q Z {0, 1, q, q 1, q 2, q 2,...} for ome q > 1. Here ν(t) t ρ(t) t t/q (q 1)t/q. From [3, Example 1.104] and Theorem 9 later in thi paper we have that (7) ĥ k (t, ) hold for all k N 0. k 1 r0 q r t r j0 qj for all, t T 3. Taylor Theorem We will conider Taylor Theorem in the context of higher-order dynamic equation. In thi etting, the theorem will be proved uing the variation of contant formula. With thi in mind, we begin thi ection with a general higher-order equation. For n N 0 and ld-continuou function p i : T R, 1 i n, with regreivity condition 1 + (ν(t)) i p i (t) 0 t T κ, i1 we conider the nth order linear dynamic equation (8) Ly 0, where Ly y n + p i y n i. A function y : T R i a olution of equation (8) on T provided y i n time nabla differentiable on T κ n and atifie Ly(t) 0 for all t T κ n. It follow that y n i an ld-continuou function on T κ n. Now let f : T R be ld-continuou and conider the nonhomogeneou equation (9) y n (t) + p i (t)y n i (t) f(t). i1 i1
6 6 D. ANDERSON Definition 1. We define the Cauchy function y : T T κ n R for the linear dynamic equation (8) to be for each fixed T κ n the olution of the initial value problem Ly 0, y i (ρ(), ) 0, 0 i n 2, y n 1 (ρ(), ) 1. Remark 2. Note that i the Cauchy function for y n 0. y(t, ) : ĥn 1(t, ρ()) Theorem 4 (Variation of Contant). Let T κ n and t T. If f C ld, then the olution of the initial value problem i given by Ly f(t), y i () 0, 0 i n 1 y(t) where y(t, τ) i the Cauchy function for (8). y(t, τ)f(τ) τ, Proof. With y defined a above and by the propertie of the Cauchy function we have y i (t) for 0 i n 1 and y n (t) y i (t, τ)f(τ) τ + y i 1 (ρ(t), t)f(t) y n (t, τ)f(τ) τ + y n 1 (ρ(t), t)f(t) It follow from thee equation that and Ly(t) and the proof i complete. y i () 0, 0 i n 1 Ly(t, τ)f(τ) τ + f(t) f(t), y i (t, τ)f(τ) τ y n (t, τ)f(τ) τ + f(t). Theorem 5 (Taylor Formula). Let n N. Suppoe f i n + 1 time nabla differentiable on T κ n+1. Let T κ n, t T, and define the function ĥk by (1) and (2), i.e., Then we have ĥ 0 (r, ) 1 and ĥ k+1 (r, ) f(t) ĥ k (t, )f k () + r ĥ k (τ, ) τ for k N 0. ĥ n (t, ρ(τ))f n+1 (τ) τ.
7 NABLA TAYLOR POLYNOMIALS 7 Proof. Let g(t) : f n+1 (t). Then f olve the initial value problem x n+1 g, x k () f k (), 0 k n. Note that the Cauchy function for y n+1 0 i y(t, ) ĥn(t, ρ()). By the variation of contant formula, f(t) u(t) + where u olve the initial value problem ĥ n (t, ρ(τ))g(τ) τ, (10) u n+1 0, u m () f m (), 0 m n. To validate the claim that u(t) n ĥk(t, )f k (), et w(t) : ĥ k (t, )f k (). By the propertie of the ĥk given in (1) and (3), w n+1 0. We have moreover that w m (t) ĥ k m (t, )f k (), o that w m () km ĥ k m (, )f k () f m () km for 0 m n. We conequently have that w alo olve (10), whence u w by uniquene. Remark 3. The reader may compare Example 1 (i.e., the cae T R and T Z) to the above preented theory. Taylor formula correpond to formula (4) and (6). Corollary 6. Let, β T be fixed. For any t T and any poitive integer n, ĥ n (t, β) ĥ k (t, )ĥn k(, β). Proof. Take f ĥn(, β) in Taylor formula. Example 3. For T Z, conider f(t) ( 1 2 )t for t Z. If we expand f about 0, then Taylor formula (6) for f i given by ( ) t 1 P n (t) + E n (t), 2
8 8 D. ANDERSON where the Taylor polynomial P n i defined by ( 1) k t k P n (t) : and the error term E n i given by { ( 1) n+1 t n! τ1 E n (t) (t + 1 ( τ)n 1 τ 2) if t > 0 0 if t 0 For t 1, P n (1) 1 ( 1)n+1 and E 2 n (1) ( 1)n+1, o that the Taylor polynomial P 2 n will not converge to f at 1 a n. If t 0, however, then ( ) t 1 ( 1) k t k t ( 1) k t k t ( ) t P t (t) 2 k converge eaily ince t i nonpoitive. 4. Further Propertie In thi final ection we relate the function ĥk a introduced in (1) and (2) (which we repeat below) to the function h k and g k in the delta cae [1, 3], and the function ĝ k. Definition 2. For t, T define the function and given h n, g n, ĥn, ĝ n for n N 0, h 0 (t, ) g 0 (t, ) ĥ0(t, ) ĝ 0 (t, ) 1, h n+1 (t, ) g n+1 (t, ) ĥ n+1 (t, ) ĝ n+1 (t, ) We will need the following theorem. h n (τ, ) τ g n (σ(τ), ) τ ĥ n (τ, ) τ ĝ n (ρ(τ), ) τ. Theorem 7. [3, Theorem 1.112] The function g k and h k defined above atify for all t T and all T κn. h n (t, ) ( 1) n g n (, t)
9 NABLA TAYLOR POLYNOMIALS 9 Theorem 8. [2, Theorem 2.5, 2.6] If f : T R i delta differentiable on T κ and if f i continuou on T κ, then f i nabla differentiable on T κ and f (t) f (ρ(t)) for all t T κ. If f : T R i nabla differentiable on T κ and if f i continuou on T κ, then f i delta differentiable on T κ and for all t T κ. f (t) f (σ(t)) Theorem 9. Let t T κ T κ and T κn. Then ĥn(t, ) ( 1) n h n (, t) for all n 0. In other word, Proof. For n 0, Aume ĥ n (t, ) g n (t, ), h n (t, ) ĝ n (t, ). h 0 (t, ) 1 ( 1) 0 ĥ 0 (, t). h n 1 (t, ) ( 1) n 1 ĥ n 1 (, t), n 0. Note that ince h n 1 (, ) i nabla differentiable, it i continuou for each fixed T. Then, uing Theorem 7 and 8 Therefore ĥ n (t, ) ĥ σ n (t, ) ĥn 1(σ(t), ) ( 1) n 1 h n 1 (, σ(t)) g n 1 (σ(t), ) g n (t, ). ĥ n (t, ) g n (t, ) + c() for all t T κ T κ and T κn. But ĥn(, ) 0 g n (, ), o that c() 0. It follow that ĥ n (t, ) g n (t, ) ( 1) n h(, t) and the induction i complete. The proof relating h n and ĝ n i imilar. Remark 4. Recall that o that for t T κ and T. f ρ f νf, ĥ n (ρ(t), ) ĥn(t, ) ν(t)ĥn 1(t, )
10 10 D. ANDERSON Lemma 10. Let n N 0. For t T and T κ, ĥ n (t, ρ()) ν k ()ĥn k(t, ). Proof. For n 0, Aume Then 0 ĥ 0 (t, ρ()) 1 ν k ()ĥ0 k(t, ). ĥ n (t, ρ()) ν k ()ĥn k(t, ). ĥ n+1 (t, ρ()) ρ() ĥ n (τ, ρ()) τ ( ) + ν k ()ĥn k(τ, ) τ ρ() ν k () ĥ n k (τ, ) τ +ν() ν k ()ĥn k(, ) ν k ()ĥn+1 k(t, ) + ν n+1 () n+1 ν k ()ĥn+1 k(t, ), which complete the induction. Corollary 11. Fix c T κn and n N 0. Then ĥ n+1(c, t) ν k (t)ĥn k(c, t) for t T κ T κ.
11 NABLA TAYLOR POLYNOMIALS 11 Proof. Let n N 0. Then ĥ n+1(c, t) ( 1) n+1 h n+1(t, c) where we have ued Theorem 8 and 9. ( 1) n+1 h ρ n+1(t, c) ( 1) n+1 h n (ρ(t), c) ( 1) n+1 ( 1) n ĥ n (c, ρ(t)) ν k (t)ĥn k(c, t), Reference [1] R. P. Agarwal and M. Bohner. Baic calculu on time cale and ome of it application. Reult Math., 35(1-2):3 22, [2] F. M. Atici and G. Sh. Gueinov. On Green function and poitive olution for boundary value problem on time cale. J. Comput. Appl. Math., 141 (2002) Special Iue on Dynamic Equation on Time Scale, edited by R. P. Agarwal, M. Bohner, and D. O Regan. [3] M. Bohner and A. Peteron. Dynamic Equation on Time Scale: An Introduction with Application. Birkhauer, New York, [4] S. Hilger. Ein Maßkettenkalkül mit Anwendung auf Zentrummannigfaltigkeiten. PhD thei, Univerität Würzburg, Concordia College, Department of Mathematic and Computer Science, Moorhead, MN USA addre: anderod@cord.edu
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