Henstock Kurzweil delta and nabla integrals

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1 Henstock Kurzweil delt nd nbl integrls Alln Peterson nd Bevn Thompson Deprtment of Mthemtics nd Sttistics, University of Nebrsk-Lincoln Lincoln, NE Mthemtics, SPS, The University of Queenslnd Brisbne Qld 4072, Austrli Abstrct We will study the Henstock Kurzweil delt nd nbl integrls, which generlize the Henstock Kurzweil integrl. Mny properties of these integrls will be obtined. These results will enble time scle reserchers to study more generl dynmic equtions. The Hensock Kurzweil delt (nbl) integrl contins the Riemnn delt (nbl) nd Lebesque delt (nbl) integrls s specil cses. Key words: Henstock Kurzweil integrl, time scles AMS Subject Clssifiction: 34B10, 39A10. 1 Introduction In this pper we shll study the so-clled Henstock Kurzweil delt nd nbl integrls, which generlize the so-clled delt nd nbl integrls respectively, which hve been widely used in the study of dynmic equtions on time scle. This will inble one to solve more generl dynmic equtions nd hence will be of gret interest to reserchers in this re. First we introduce the following concepts relted to the notion of time scles. A time scle T is just ny closed nonempty subset of the rel numbers R. Definition 1.1 Let T be time scle nd define the forwrd jump opertor σ(t) t t, for t T, by σ(t) := inf{τ > t : τ T}, nd the bckwrd jump opertor ρ(t) t t, for t T, by ρ(t) := sup{τ < t : τ T} where inf := sup T nd sup := inf T, where denotes the empty set. 1

2 We ssume thoughout tht T hs the topology tht it inherits from the stndrd topology on the rel numbers R. If σ(t) > t, we sy t is right-scttered, while if ρ(t) < t we sy t is left-scttered. If t < sup T nd σ(t) = t we sy t is right-dense, while if t > inf T nd ρ(t) = t we sy t is left-dense. A function f : T R is sid to be right-dense continuous provided f is continuous t right-dense points in T nd t left-dense points in T, left hnd limits exist nd re finite. We shll lso use the nottion µ(t) := σ(t) t, where µ is clled the grininess function nd ν(t) := t ρ(t), where ν is clled the left-grininess function. We denote the nturl numbers by N nd the nonnegtive integers by N 0. Definition 1.2 Thoughout this pper we mke the blnket ssumption tht, b T nd we define the time scle intervl in T by [, b] T := {t T such tht t b}. The notion of time scle ws introducted by S. Hilger [5]. Relted work on the clculus of time scles my be found in Agrwl nd Bohner [1], nd Erbe nd Hilger [4]. See lso the introductory books on time scles [2], [3], nd [6]. Definition 1.3 Assume x : T R nd fix t T, then we define x (t) to be the number (provided it exists) with the property tht given ny ɛ > 0, there is neighborhood U of t such tht [x(σ(t)) x(s)] x (t)[σ(t) s] ɛ σ(t) s, for ll s U. We cll x (t) the delt derivtive of x t t. It cn be shown tht [2, Theorem 1.16] if x : T R is continuous t t T nd t is rightscttered, then x x(σ(t)) x(t) (t) =. σ(t) t Note tht if T=N 0, then x (t) = x(t) := x(t + 1) x(t), here is the forwrd difference opertor [7]. If t is right-dense, then it cn be shown tht [2, Theorem 1.16] x x(t) x(s) (t) = lim. s t t s In prticulr, if T is the rel intervl [, ), then x (t) = x (t). Definition 1.4 We sy δ = (δ L, δ R ) is -guge for [, b] T provided δ L (t) > 0 on (, b] T, δ R (t) > 0 on [, b) T, δ L () 0, δ R (b) 0, nd δ R (t) µ(t) for ll t [, b) T. Similrly we sy γ = (γ L, γ R ) is -guge for [, b] T provided γ L (t) > 0 on (, b] T, γ R (t) > 0 on [, b) T, γ L () 0, γ R (b) 0, nd γ L (t) ν(t) for ll t (, b] T. 2

3 Since for -guge, δ, we lwys ssume δ L () 0 nd δ R (b) 0, we will sometimes not even point this out. Similrly for -guge, γ, we will not lwys mke the point tht γ L () 0 nd γ R (b) 0. Definition 1.5 A prtition P for [, b] T is division of [, b] T denoted by P = { = t 0 ξ 1 t 1 t n 1 ξ n t n = b} with t i > for 1 i n nd t i, ξ i T. We cll the points ξ i tg points nd the points t i end points. As in Peng Yee [8], we sometimes denote such prtition by P = {[u, v]; ξ}, where [u, v] denotes typicl intervl in P nd ξ is the ssocited tg point in [u, v]. Definition 1.6 If δ is -guge for [, b] T, then we sy prtition P is δ-fine if ξ i δ L (ξ i ) < t i ξ i + δ R (ξ i ) for 1 i n. Similrly, if γ is -guge for [, b] T, then we sy the prtition P is γ-fine if for 1 i n. ξ i γ L (ξ i ) < t i ξ i + γ R (ξ i ) Now we cn define the Henstock Kurzweil delt nd nbl integrl. Definition 1.7 We sy tht f : [, b] T R is Henstock Kurzweil delt integrble on [, b] T with vlue I = f(t) t, provided given ny ɛ > 0 there exists -guge, δ, for [, b] T such tht I f(ξ i )(t i ) < ɛ for ll δ-fine prtitions P of [, b] T. Similrly we sy tht f : [, b] T R is Henstock Kurzweil nbl integrble on [, b] T with vlue I = f(t) t, provided given ny ɛ > 0 there exists -guge, γ, for [, b] T such tht I f(ξ i )(t i ) < ɛ for ll γ-fine prtitions P of [, b] T. Remrk 1.8 If f is Riemnn delt integrble on [, b] T ccording to the Riemnn sums definition given for bounded function f(t) on [, b] T in Definition 5.10 in [3], then it is -delt integrble on [, b] T nd f(t) t = f(t) t. 3

4 The proof of this remrk follows from the fct tht if we define δ(t) = (δ L (t), δ R (t)), by δ L (t) = δ L > 0 constnt for t [, b] T nd δ R (t) = δ L for ll right-dense points in [, b] T nd δ R (t) = µ(t) for ll right-scttered points in [, b] T, then δ is -guge of [, b] T. As in Lee Peng Yee [8] we sometimes in proofs use the bbrevition f(ξ)(u v) := f(ξ i )(t i ). For the Henstock Kurzweil delt nd nbl integrls to mke ny sense we prove the following lemm. In this lemm nd lter we use the nottion, if s [, b], then β L (s) := sup{t [, b] T : t s}, β R (s) := inf{t [, b] T : t s}. Lemm 1.9 If δ is -guge for [, b] T, then there is δ-fine prtition P for [, b] T. Similrly if γ is -guge for [, b] T, then there is γ-fine prtition P for [, b] T. Proof: We just prove the first sttement. Assume there is no such δ-fine prtition P for [, b] T. Let c = 1 2 (b ) nd let d = β L (c), e = β R (c). Then either [, d] T or [e, b] T does not hve δ-fine prtition. Assume [ 1, b 1 ] T is one of these two intervls such tht [ 1, b 1 ] T does not hve δ-fine prtition. Repeting this rgument we get nested sequence of intervls [ i, b i ] T with b i i b 2 i, ech of which does not hve δ-fine prtition. Let Then for i sufficiently lrge ξ 0 = lim i i = lim i b i. ξ 0 δ L (ξ 0 ) i < b i ξ 0 + δ R (ξ 0 ). But then { i ξ 0 b i } is δ-fine prtition of [ i, b i ] T, which is contrdiction. We next give n interesting introductory exmple. In this exmple we use the fct [2, Theorem 1.79] tht if [c, d] T = {t 0 = c, t 1, t 2,, t n = d}, then d c n 1 n 1 f(t) t = f(t i )(t i+1 t i ) = f(t i )µ(t i ). i=0 i=0 Exmple 1.10 Let T = { 1 n : n N} {0}. Let f : T R be defined by f(t) = { 1, t 0 t+ σ(t) 0, t = 0, 4

5 then we clim tht f is Henstock Kurzweil delt integrble with 1 0 f(t) t = 1 even though the delt integrl 1 f(t) t does not exist (lthough it does exist s n improper 0 delt integrl). To see tht f is Henstock Kurzweil delt integrble on [0, 1] T, let 0 < < 1 be given. Assume tht δ is -guge on [0, 1] T stisfying δ L (t) = 1ν(t) for t (0, 1] 2 T nd δ R (t) = µ(t), for t (0, 1) T, with δ R (0) = 2. Let P be δ-fine prtition of [0, 1] T. Since δ L (t) = 1ν(t) on (0, 1] 2 T, we hve tht our first tg point is ξ 1 = 0. Also for 1 i n 1 we hve δ R (t i ) = µ(t i ) nd δ L (σ(t i )) = 1ν(σ(t 2 i)) which implies tht Now consider ξ i+1 = t i, nd t i+1 = σ(t i ), 1 i n 1. 1 f(ξ i )[t i ] = = = = 1 f(ξ i )[t i ] i=2 1 f( )µ( )] i=2 1 1 f(t) t t 1 1 [ t] 1 t 1 = t 1 δ R (0) =, where we used the fct tht F (t) = t is delt ntiderivtive of f(t) on (0, 1) T. Exmple 1.11 Assume [, b) T contins countble infinite subset {r i } with σ(r i ) = r i. Define f : [, b] T R by { 1, t = r i f(t) = 0, t r i. Let > 0 be given. Then define -guge, δ, on [, b] T by δ L (r i ) = δ R (r i ) =, i 1, 2 i+2 δ R (t) = mx{1, µ(t)} nd δ L (t) = 1 for t [, b] T \ {r i }. Let P be δ-fine prtition of [, b] T, then f(ξ i )(t i ) (δ L (r i ) + δ R (r i )) = <. 2i+1 Hence, even though in mny cses f is not delt integrble on [, b] T, we get f is Henstock Kurzweil delt integrble on [, b] T nd f(t) t = 0. 5

6 Similrly, one cn show tht if bove {r i } is countble infinite subset of left dense points in (, b] T, then f s given bove is Henstock Kurtzweil nbl integrble on [, b] T with 2 Min Results f(t) t = 0. The results in the following theorem in the specil cse of delt nd nbl integrls re used ll the time in the study of dynmic equtions on time scles. Theorem 2.1 Assume f : [, b] T R. If f is -delt integrble on [, b] T, then the vlue of the integrl f(t) t does not depend on f(b). On the other hnd if c [, b) T nd c is right scttered, then f(t) t does depend on the vlue f(c)µ(c). Also, if f is -nbl f(t) t does not depend on f() nd f(t) t does depend on the vlue f(c)ν(c). integrble on [, b] T, then the vlue of the integrl if c (, b] T nd c is left scttered, then Proof: We will just prove the sttements concerning the -delt integrls. Assume tht f is -delt integrble on [, b] T. We consider the two cses ρ(b) < b nd ρ(b) = b. If ρ(b) < b, we choose δ L (b) < ν(b). In this cse b is not tg point for ny δ-fine prtition nd hence f(t) t does not depend on the vlue f(b). Next consider the cse ρ(b) = b. In this cse given ny > 0 we cn choose so if b = ξ n is tg point, then δ L (b) < ɛ f(b) + 1 f(ξ n )(t n t n 1 ) = f(b)(b t n 1 ) f(b) δ L (b) < f(b) f(b) + 1 < nd the result follows. Next ssume tht c [, b) T nd c is right scttered. From Theorem 2.12 we get tht = = c c f(t) t f(t) t + σ(c) f(t) t + f(c)µ(c) + c f(t) t + σ(c) σ(c) f(t) t. f(t) t Since the first nd lst terms do not depend on f(c)µ(c) we get the desired result. Remrk 2.2 From the proof of the first sttement in Theorem 2.1 we see tht in the definition of the Henstock Kurzweil delt integrl we cn without loss of generlity ssume tht the lst tg point stisfies ξ n b. 6

7 In the proof of the next theorem we use the nottion nd note tht N µ is countble set. N µ := {z j [, b) T : µ(z j ) > 0} Theorem 2.3 Assume F : [, b] T R is continuous, f : [, b] T R, nd there is set D with N µ D [, b] κ T such tht F (t) = f(t) for t D nd [, b] T \D is countble, then f is delt integrble on [, b] T with f(t) t = F (b) F (). Proof: By hypothesis there is set D with N µ D [, b] κ T such tht F (t) = f(t) for t D nd [, b] T \D is countble. Let Y := [, b] T \D = {y 1, y 2, }, which is countble (could be finite). Let > 0 be given. We now define -guge, δ, for [, b] T. First let t = z i N µ, then we define δ R (z i ) = µ(z i ). Since F is delt differentible t z i, there is δ 1 L (z i) > 0 such tht F (σ(z i )) F (s) F (z i )[σ(z i ) s] 4(b ) [σ(z i) s] (2.1) for ll s [z i δ 1 L (z i), z i ] T. Also, since F is continuous t z i, we get there is δ 2 L (z i) > 0 such tht F (z i ) F (s) F (z i )[z i s] 2 i+2 (2.2) for ll s [z i δ 2 L (z i), z i ] T. Then we define δ L (z i ) = min{δ 1 L(z i ), δ 2 L(z i )} so tht (2.1) nd (2.2) both hold for s [z i δ L (z i ), z i ] T. Next ssume t Y, then t = y j for some j. In this cse, since F is continuous t y j, there is n η(y j ) > 0 such tht F (r) F (s) f(y j )(r s) 2 j+2 (2.3) for ll r, s [y j η(y j ), y j + η(y j )] T. In this cse we define δ R (y j ) = δ L (y j ) = η(y j ). Finlly, consider the cse t D\N µ. Since F is differentible t t we get tht there is n α(t) > 0 such tht F (t) F (s) F (t)(t s) 7 t s (2.4) 4(b )

8 for s [t α(t), t + α(t)] T. In this cse we define δ L (t) = δ R (t) = α(t). Hence δ is -guge on [, b] T. Now ssume P is δ-fine prtition of [, b] T. Consider F (b) F () f(ξ i )[t i ] = {[F (t i ) F ( )] + f(ξ i )[t i ]} [F (t i ) F ( )] + f(ξ i )[t i ]. (2.5) We now look t the terms in this lst sum for the cses ξ i N µ, ξ i Y, nd ξ i D\N µ respectively. First ssume ξ i N µ, then ξ i = z j for some j nd σ(z j ) > z j. Then, since δ R (ξ i ) = δ R (z j ) = µ(z j ), we hve tht either t i = z j or t i = σ(z j ). If t i = z j, then using (2.2) we hve tht F (t i ) F ( ) f(ξ i )[t i ] = F (z j ) F ( ) F (z j )[z j ], (2.6) 2j+4 where s = [z j δ L (z j ), z j ] T. On the other hnd if t i = σ(z j ), then using (2.1) with s = we get tht F (t i ) F ( ) f(ξ i )[t i ] = F (σ(z j )) F ( ) F (z j )[σ(z j ) ] 4(b ) [σ(z j) ] 4(b ) [t i ]. (2.7) Next ssume ξ i Y, so ξ i = y j for some j. Then by (2.3) with r = t i nd s = F (t i ) F ( ) f(ξ i )[t i ] = F (t i ) F ( ) f(y j )[t i ]. (2.8) 2j+2 Finlly, ssume tht ξ i D\N µ. Then by the tringle inequlity nd (2.4) F (t i ) F ( ) f(ξ i )[t i ] = F (t i ) F ( ) F (ξ i )[t i ] F (t i ) F (ξ i ) F (ξ i )[t i ξ i ] + F (ξ i ) F ( ) F (ξ i )[ξ i ] 4(b ) [t i ]. (2.9) 8

9 The result now follows from (2.5) (2.9). In most ppers concerning dynmic equtions on time scles the uthor(s) define the Cuchy Newton delt integrl s follows: If F : [, b] T R is delt ntiderivtive of f(t) on [, b] T, then we sy f is CN-delt integrble on [, b] T nd we define CN f(t) t := F (b) F (). It follows from Theorem 2.3 tht every CN-delt integrble function f(t) on [, b] T is -delt integrble on [, b] T nd f(t) t = CN f(t) t. Hence the clss of -delt integrble functions on [, b] T contins the clss of Riemnn delt integrble functions on [, b] T nd the clss of CN-delt integrble functions on [, b] T. If T := [0, 1] [2, 3], nd f : T R is defined by f(t) := 0, t 1 nd f(1) := 1, then f is simple exmple of function which is Riemnn delt integrble on [0, 2] T (with f(t) t = 1), but is not CN-delt integrble on [0, 2] T. For the next exmple let T := {t 2n = 1 : n N} {t n 2n+1 = 1 1 : n = 2, 3, 4, } {0} nd let F : T R be defined by F (t n n 3 2n ) := t 2 2n, n N F (t 2n+1 ) = 0 = F (0), n 2 nd note tht F (0) = 0. Let f(t) := F (t), t [0, 1) T, f(1) = 1, then f(t) is CN-delt integrble on [0, 1] T with CN 1 f(t) t = F (1) F (0) = 1, but since, 0 for n 3, f(t 2n+1 ) = F (t 2n+1 ) = F (t 2n) F (t 2n+1 ) t 2n t 2n+1 = n we hve f(t) is not bounded on [0, 1] T, so f(t) is not Riemnn delt integrble on [0, 1] T. Similr to the proof of Theorem 2.3 one cn prove the following theorem. Theorem 2.4 Assume f : [, b] T R nd F : [, b] T R is continuous nd there is set D [, b] T such tht F (t) = f(t) for t D nd [, b] T \D is countble nd contins no left-scttered, right-dense points, then f is nbl integrble on [, b] T with f(t) t = F (b) F (). For ech of the remining exmples nd theorems concerning Henstock Kurzweil delt integrtion there re the corresponding results for Henstock Kurzweil nbl integrtion which we won t bother to stte. We next give n exmple of function which is not delt integrble on [0, 1] T, but is delt integrble on [0, 1] T, but is not bsolutely -delt integrble on [0, 1] T. Exmple 2.5 Let T = {t = 1 n : n N} {0} nd define f : T R by f(t) := { ( 1) n n, t = 1 n L, t = 0, 9

10 where L is ny constnt. Note tht f is not delt integrble on [0, 1] T. Then it cn be shown tht if F : T R is defined by 0, t = 1 F (t) := n ( 1) k+1 k=2, t = 1 k 1 n ln 2, t = 0, then F (t) = f(t) for t (0, 1) T nd F is continuous [0, 1] T ( note the vlue of F (0) is determined so tht F is continuous t 0). It follows by Theorem 2.3 with D := (0, 1) T, tht f is -delt integrble on [0, 1] T nd 1 0 f(t) t = F (1) F (0) = ln 2. However it cn be shown tht 1 f(t) t does not exist (f is not bsolutely -delt 0 integrble on [0, 1] T ). Remrk 2.6 Lebesgue integrtion is n bsolute integrtion. By this we men f is Lesbesgue integrble on [, b] iff f is Lebesgue integrble on [, b]. Exmple 2.5 shows tht Henstock- Kurzweil delt integrtion does not hve this wekness nd is nonbsolute integrtion. Indeed the Henstock-Kurtzweil delt integrl is designed to integrte highly oscilltory functions. Corollry 2.7 If f : T R is regulted nd, b T, then f is -delt integrble on [, b] T nd f(t) t = f(t) t. Moreover, if G(t) := t f(s) s, then G (t) = f(t) except for countble set. Proof: By [2, Theorem 1.70] we hve, since f is regulted, tht there is function F : [, b] T R which is continuous on [, b] T nd there is set D with D N µ [, b] κ T such tht F (t) = f(t) for t D nd [, b] T \D is countble. Then using Theorem 2.3, f is -delt integrble on [, b] T with f(t) t = F (b) F (). (2.10) Since f is regulted we hve by [3, Theorem 5.21] tht f is delt integrble on [, b] T, nd then by [3, Theorem 5.39] Hence from (2.10) nd (2.11) we hve the desired result f(t) t = F (b) F (). (2.11) f(t) t = f(t) t. Mny of the simple properties of the -delt integrl go through like the clssicl Henstock Kurzweil integrl where the following remrks re useful. 10

11 Remrk 2.8 Let δ 1, δ be -guges for [, b] T such tht 0 < δ 1 L (t) δ L(t) for t (, b] T nd 0 < δ 1 R (t) δ R(t) for t [, b) T (write δ 1 δ nd we sy δ 1 is finer thn δ). If P 1 is δ 1 -fine prtition of [, b] T, then P 1 is δ-fine prtition of [, b]. Remrk 2.9 If c [, b] T nd P is δ-fine prtition, then there is δ-fine prtition with c s tg point. Remrk 2.10 If c [, b] T nd c t i is tg point in δ-fine prtition P, then P := {t 0 = ξ 1 c c c t i t n = b} where c is n end point nd tg point for the two intervls [, c] nd [c, t i ] is δ-fine prtition nd the Riemnn sum corresponding to these two prtitions is the sme. This follows from the simple fct tht f(c)[t i ] = f(c)[t i c] + f(c)[c ]. Remrk 2.11 Let < c < b be points in T, then we my choose our guge δ so tht δ R (t), δ L (t) t c for ll t T\{c}. Then, if P is δ-fine prtition of [, b] T, then ξ i0 = c for some i 0. If x i0 1 < ξ i0, then we my dd y i0 to prtition so tht so tht nd { = t 0 ξ 1 t 1 t i0 1 ξ i0 = y i0 b} { = t 0 ξ 1 t 1 t i0 1 ξ i0 = y i0 = c} {c = y i0 ξ 0 t n = b} re δ-fine prtitions of [, c] T nd [c, b] T respectively. Using these remrks nd simple djustments of the results in Lee Peng Yee [8] one cn prove the following theorem. Theorem 2.12 Let f : [, b] T R. Then f is -delt integrble on [, b] T iff f is -delt integrble on [, c] T nd [c, b] T. Moreover, in this cse f(t) t = c f(t) t + c f(t) t. (2.12) Also if f, g : [, b] T R re -delt integrble on [, b] T, then αf +βg is -delt integrble on [, b] T nd (αf(t) + βg(t)) t = α ( ) ( f(t) t + β ) f(t) t. 11

12 Proof: We will just show tht if f is -delt integrble on [, c] T nd [c, b] T, then f is -delt integrble on [, b] T nd (2.12) holds. Let A := c f(t) t, B := c f(t) t. Let > 0 be given. Then there is -guge, δ 1, of [, c] T nd -guge, δ 2, of [c, b] T such tht for ll δ 1 -fine prtitions P of [, c] T nd ll δ 2 -fine prtitions P of [c, b] T we hve tht A f(ξ i )(t i ) < ɛ m 2, B f(ξ i)(t i t i 1) < ɛ 2. (2.13) We define -guge, δ = (δ L, δ R ), on [, b] T by first defining δ L s follows: δ L (t) = δl 1 (t), t [, c) T, { δ 1 δ L (c) = L (c), ν(c) = 0 min{δl 1 ν(c) (c), }, ν(c) > 0, 2 δ L (t) = min{δl 2 t c (t), }, t (c, b] 2 T, nd then defining δ R s follows: δ R (t) = δr 2 (t), t [c, b] T, δ R (t) = min{δr(t), 1 mx{µ(t), c t 2 }}, t [, c) T. Now let P be δ-fine prtition of [, b] T. Becuse of the wy we defined δ, there re two cses: either c is tg point for P, sy c = ξ k, nd t k > c or ρ(c) < c, ρ(c) is tg point for P, sy ρ(c) = ξ k, nd t k = c. In the first cse we hve using tht (A + B) p f(c)(t k t k 1) = f(c)(t k c) + f(c)(c t k 1) f(ξ i )(t i t i ) k 1 A f(ξ )(t i t i) f(c)(c t k 1) + B < =, p i=k+1 f(ξ )(t i t i ) f(c)(t k c) by (2.12), where we hve used tht the prtition corresponding to the first term is finer thn the prtition P of [, c] T nd the prtition corresponding to the second term is finer thn the prtition P of [c, b] T. The other cse is esy nd hence will be omited. The proofs of the following two results re very similr to the proofs of Theorem 3.6 nd Theorem 3.7 in [8] respectively nd hence the proofs re omited. Theorem 2.13 If f nd g re -delt integrble on [, b] T nd then f(t) g(t) f(t) t.e. on [, b) T 12 g(t) t.

13 Theorem 2.14 Assume f is -delt integrble on [, b] T. Then given ny > 0 there is -guge, δ, of [, b] T such tht for ll δ-fine prtitions P of [, b] T. f(t) t f(ξ i )(t i ) < Definition 2.15 We sy subset S of time scle T hs delt mesure zero provided S contins no right-scttered points nd S hs Lebesgue mesure zero. We sy property A holds delt lmost everywhere (delt.e.) on T provided there is subset S of T such tht the property A holds for ll t S nd S hs delt mesure zero. Theorem 2.16 (Monotone Convergence Theorem) Let f k, f : [, b] T tht T nd ssume (i) f k is -delt integrble, k N; (ii) f k f delt e in [, b) T ; (iii) f k f n+1 delt e on [, b) T, k N; (iv) lim k f k(t) t = I. Then f is -delt integrble on [, b] T nd I = f(t) t. Proof: Considering f k f 1 if necessry we cn ssume without loss of generlity tht f k (t) 0 delt e on [, b) T. Also to keep things esier we will just do the proof in the cse where we replce (ii) by nd in plce of (iii) we ssume f k f for ech t [, b) T (2.14) f k (t) f k+1 (t), t [, b) T. (2.15) Let > 0 be given. Since f k(t) t is monotone nondecresing with limit I, we cn pick positive integer k 0 so tht 0 I f k (t) t < 3 (2.16) 13

14 for ll k k 0. From (2.14) we hve for ech t [, b) T there is positive integer m(, t) k 0 such tht f m(,t) (t) f(t) < 3(b ). (2.17) Since ech f k is -delt integrble on [, b] T we hve from Theorem 2.14 there is -guge, δ k, of [, b] T such tht v f k (t) t f k (ξ)(v u) < u for ech δ k -fine prtition of [, b] T. Now define -guge, δ, on [, b] T by δ(t) := δ m(,t) (t). Let P be δ-fine prtition, then using nd (2.17) nd (2.18) + + < f(ξ i )(t i ) I f(ξ i ) f m(,ξi )(ξ i ) (t i ) f m(,ξi )(ξ i )(t i ) 3(b ) (b ) + 3 = f m(,ξi )(t) t I i f m(,ξi )(t) t I f m(,ξi )(t) t ɛ 3 2 k (2.18) f m(,ξi )(t) t I To complete the proof we need to show tht the lst term bove is less thn. To see this let 3 p := min{m(t, ξ i ) : 1 i n} k 0. Then since f k (t) t 14

15 is monotone nondecresing with respect to k we hve f p (t) t = f p (t) t f m(,ξi )(t) t = lim k lim f k (t) t k f(t) t = I It follows from this tht nd the proof is complete. I f m(,ξi )(t) t 3 The proofs of the remining results in this section re now very similr to the proofs for the nlogous results given in Lee Peng Yee [8] nd hence the proofs re omitted. Theorem 2.17 (Dominted Convergence Theorem) Assume (i) f k f e on [, b] T ; (ii) g f k h e on [, b] T ; (iii) f k, g, h re -delt integrble on [, b]. Then f is -delt integrble on [, b] T nd lim f k (t) t = k f(t) t. Definition 2.18 We sy f : [, b] T R is bsolutely -delt integrble on [, b] T provided f nd f re delt integrble on [, b] T. Theorem 2.19 The function f is Lebesgue delt integrble on [, b] T iff f is bsolutely delt integrble on [, b] T. Theorem 2.20 The function F is bsolutely continuous on [, b] T iff F (t) = f(t) delt e on [, b] T is bsolutely -delt integrble function f on [, b] T. Moreover F (t) t = F (t) dt + F σ (t i ) F (t i ). [,b] T t i N µ Theorem 2.21 If f(t) = 0 delt.e. in [, b] T, then f is -delt integrble on [, b] T nd f(t) t = 0. 15

16 3 Applictions In this section we indicte tht our results led to more generl Tylor s theorem with reminder nd generl men vlue theorem. First we need the following lemm. Lemm 3.1 (Integrtion by Prts) Assume f, g : [, b] T R re differentible nd f nd g re -delt integrble on [, b] T, then Proof: This follows directly from the product rule f(t)g (t) t = [f(t)g(t)] b f (t)g(t) t. (f(t)g(t)) = f(t)g (t) + f (t)g(σ(t)) nd properties of -delt integrls. Before we stte our next result we define s in Section 1.6 in [2] the so-clled Tylor monomils {h k (t, s)} k=0 s follows: h 0 (t, s) = 1, h n+1 (t, s) = t s h n (t, σ(s)) s, t, s T. Note if T = R, then h n (t, s) = (t s)n nd if T = Z, then h n! n (t, s) = (t s)n, where (t s) n := n! (t s)(t s 1) (t s n + 1). Now we cn prove very generl Tylor s theorem with reminder (we do not need to ssume f n+1 (t) is rd-continuous!). Theorem 3.2 (Tylor s Theorem) Assume f : [, b] T R, is such tht f n+1 (t) exists for t T κn+1 nd f n+1 (t) is -delt integrble on [, b] T. Then t f(t) = f k (t 0 )h k (t, t 0 ) + (s)h n+1 (t, σ(s)) s where t 0, t T. k=0 Proof: Integrting by prts we get tht t t 0 f n+1 t 0 f n+1 (s)h n+1 (t, σ(s)) s = [ f n (t)h n+1 (t, s) ] s=t s=t 0 = f n (t 0 )h n+1 (t, t 0 ) t t t 0 f n t 0 f n (s)h s n+1(t, s) s (s)h s n+1(t, s) s. Simplifying nd repeted integrtion by prts s in the proof of Theorem in [2] leds to the desired result. In the next theorem we use the essentil sup nd the essentil inf which we define s follows essinf [,b) f(t) := inf N inf t/ N {f(t), N [, b) T hs delt mesure zero} 16

17 nd esssup [,b] f(t) := sup N sup{f(t), N [, b) T t/ N hs delt mesure zero}. Theorem 3.3 (Men Vlue Theorem) Assume f : [, b] T R nd f is -integrble on [, b] T, then essinf [,b) f f(b) f() (t) (t σ(s)) s esssup [,b)f (t). Proof: By Tylor s theorem (Theorem 3.2) with n = 0, f(b) = f() + = f() + Hence f(b) f() = The result follows esily from this lst equlity. f (s)h 1 (t, σ(s)) s f (s)(t σ(s)) s. f (s)(t σ(s)) s. A specil cse of this men vlue theorem is Corollry 1.68 in Bohner nd Peterson [2]. References [1] R. Agrwl nd M. Bohner, Bsic clculus on time scles nd some of its pplictions, Results Mth., 35, (1999), [2] Mrtin Bohner nd Alln Peterson, Dynmic Equtions On Time Scles: An Introduction With Appliction, Birkhäuser, Boston, [3] Mrtin Bohner nd Alln Peterson, Advnces in Dynmic Equtions On Time Scles, Editors, Birkhäuser, Boston, [4] L. Erbe nd S. Hilger, Sturmin Theory on Mesure Chins, Differentil Equtions nd Dynmicl Systems, 1 (1993), [5] S. Hilger, Anlysis on mesure chins- unified pproch to continuous nd discrete clculus, Results in Mthemtics, 18 (1990), [6] B. Kymkçln, V. Lksmiknthm, nd S. Sivsundrm, Dynmicl Systems on Mesure Chins, Kluwer Acdemic Publishers, Boston, [7] W. Kelley nd A. Peterson, Difference Equtions: An Introduction with Applictions, Acdemic Press, Second Edition, [8] Lee Peng Yee, Lnzhou Lectures on Henstock Integrtion, Series in Rel Anlysis Volume 2, World Scientific, Singpore,