Volume 27, No. 2, May 2014
http://dx.doi.org/10.14403/jcms.2014.27.2.327
THE RIEMANN DELTA INTEGRAL ON TIME SCALES
Jae Myung Park*, Deok Ho Lee**, Ju Han Yoon***, Young Kuk Kim****, and Jong Tae Lim*****
Abstract. In this paper, we define the extension f∗: [a, b] → R of a function f : [a, b]T → R for a time scale T and show that f is Riemann delta integrable on [a, b]T if and only if f∗ is Riemann integrable on [a, b].
1. Introduction and preliminaries
Let T be a time scale, a < b be points in T, and [a, b]T be the closed interval in T. A partition P = {(ξi, [ti−1, ti])}ni=1 of [a, b]T is a collection of tagged intervals such that
a = t0 < t1 < · · · < tn= b, ti∈ T for each i = 1, 2, · · · , n, and ξi is an arbitrary point on [ti−1, ti)T.
Let f be a real-valued bounded function on [a, b]T. Let Mi = sup{f (t) : t ∈ [ti−1, ti)T} and mi = inf{f (t) : t ∈ [ti−1, ti)T}. The upper ∆−sum SP(f ) and the lower ∆−sum SP(f ) of f with respect to P are defined by
SP(f ) = Xn i=1
Mi(ti− ti−1), SP(f ) = Xn i=1
mi(ti− ti−1).
Let {(ak, bk)}∞k=1 be the sequence of intervals contiguous to [a, b]T in [a, b].
For a function f : [a, b]T → R, define the extension f∗ : [a, b] → R of f by
f∗(t) = (
f (ak) if t ∈ (ak, bk) for some k f (t) if t ∈ [a, b]T.
Received April 07, 2014; Accepted April 16, 2014.
2010 Mathematics Subject Classification: Primary 26A39; Secondary 26E70.
Key words and phrases: time scales, Riemann delta integral, δ-partition.
Correspondence should be addressed to Young Kuk Kim, ykkim@dragon.seo- won.ac.kr.
It is well-known [7] that f : [a, b]T → R is McShane delta integrable on [a, b]T if and only if f∗: [a, b] → R is McShane integrable on [a, b].
In this paper, we consider the Riemann delta integral and show that a function f : [a, b]T → R is Riemann delta integrable on [a, b]T if and only if f∗ : [a, b] → R is Riemann integrable on [a, b].
2. The Riemann delta integral
Definition 2.1. For given δ > 0, a partition P = {(ξi, [ti−1, ti])}ni=1 is a δ-partition of [a, b]T if for each i ∈ {1, 2, · · · , n} either ti− ti−1≤ δ or ti− ti−1> δ and σ(ti−1) = ti, where σ(t) = inf{s ∈ T : s > t}.
Definition 2.2. A bounded function f : [a, b]T → R is Riemann delta integrable (or R∆−integrable) on [a, b]T if there exists a number A such that for each ² > 0 there exists δ > 0 such that
¯¯
¯ Xn
i=1
f (ξi)(ti− ti−1) − A
¯¯
¯ < ²
for every δ−partition P = {(ξi, [ti−1, ti])}ni=1of [a, b]T. The number A is called the Riemann delta integral of f on [a, b]T and we write
A = (R∆) Z b
a
f.
The following theorem gives a Cauchy criterion for R∆−integrability.
Theorem 2.3. [3] A bounded function f : [a, b]T→ R is R∆−integrable on [a, b]Tif and only if for each ² > 0 there exists a partition P of [a, b]T such that SP(f ) − SP(f ) < ².
Let P = {(ξi, [ti−1, ti])}ni=1 and Q = {(ζj, [xj−1, xj])}mj=1 be two par- titions of [a, b](or [a, b]T). If {t0, t1, · · · , tn} ⊂ {x0, x1, · · · , xm}, then we say that Q is a refinement of P and we denote Q ≥ P.
Recall that f : [a, b] → R is Riemann integrable on [a, b] with value A if for each ² > 0 there exists a partition P = {(ξi, [ti−1, ti])} of [a, b]
such that ¯¯
¯X
j
f (ζj)(xj− xj−1) − A
¯¯
¯ < ² for every refinement Q = {(ζi, [xj−1, xj])} of P.
Theorem 2.4. A bounded function f : [a, b]T→ R is R∆−integrable on [a, b]Tif and only if f∗ : [a, b] → R is Riemann integrable on [a, b]. In that case, (R)Rb
af∗= (R∆)Rb
a f.
Proof. Let f : [a, b]T → R be R∆−integrable on [a, b]T and let ² > 0.
Then there exists a partition P0 = {(ξj0, [t0j−1, t0j])}mj=1of [a, b]Tsuch that
(2.1)
¯¯
¯ Xn
i=1
f (ξi)(ti− ti−1) − (R∆) Z b
a
f
¯¯
¯ < ²
for every partition P = {(ξi, [ti−1, ti])}ni=1≥ P0 of [a, b]T.
Assume that P0 = {(ξ0i, [t0i−1, t0i])}ni=1is a partition of [a, b] with P0 ≥ P0, where we regard P0 as a partition of [a, b].
If i ≤ n, then there is a unique j ≤ m such that [t0i−1, t0i] ⊆ [t0j−1, t0j] and there is a ξi00 ∈ [t0j−1, t0j]T with f∗(ξi0) = f (ξi00). For each j ≤ m, there are i1j, i2j ≤ n such that [t0i1j−1, t0i1j], [t0i2j−1, t0i2j] ⊆ [t0j−1, t0j] and
f (ξi001j) = min
[t0i−1,t0i]⊆[t0j−1,t0j]
f (ξi00), f (ξ00i2j) =max
[t0i−1,t0i]⊆[t0j−1,t0j]
f (ξi00).
By (2.1), we have Xn i=1
f∗(ξi0)(t0i− t0i−1)
= Xm j=1
X
[t0i−1,t0i]⊆[t0j−1,t0j]
f (ξi00)(t0i− t0i−1)
= Xm j=1
³ X
[t0i−1,t0i]⊆[t0j−1,t0j]
f (ξi00)t0i− t0i−1 t0j− t0j−1
´
(t0j − t0j−1)
≤ Xm j=1
f (ξi002j)(t0j − t0j−1)
<
Xm j=1
f (ξj0)(t0j − t0j−1) + 2².
(2.2)
Similarly, we have (2.3)
Xn i=1
f∗(ξ0i)(t0i− t0i−1) >
Xm j=1
f (ξj0)(t0j − t0j−1) − 2².
From (2.1), (2.2), (2.3) we have
¯¯
¯ Xn
i=1
f∗(ξi0)(t0i− t0i−1) − (R∆) Z b
a
f
¯¯
¯
≤
¯¯
¯ Xn
i=1
f∗(ξi0)(t0i− t0i−1) − Xm j=1
f (ξj0)(t0j − t0j−1)
¯¯
¯
+
¯¯
¯ Xm j=1
f (ξj0)(t0j − t0j−1) − (R∆) Z b
a
f
¯¯
¯
< 2² + ² = 3².
Thus f∗ is Riemann integrable on [a, b] and Rb
af∗ = (R∆)Rb
af.
Conversely, suppose that f∗ : [a, b] → R is Riemann integrable on [a, b]. Let ² > 0. Then there exists a partition P = {[xi, yi]}ni=1 of [a, b]
such that
SP(f∗) − SP(f∗) < ².
Let {(ak, bk)} be the sequence of intervals contiguous to [a, b]T in [a, b]. Put
A = {i|[xi, yi] ⊂ [ak, bk] for some k ∈ N, i = 1, 2, · · · , n}, B = {1, 2, · · · , n} − A.
We see that [xi, yi]T6= ∅ for each i ∈ B. Put
si = inf[xi, yi]T, ti= sup[xi, yi]T for each i ∈ B.
Put B1 = {i ∈ B | xi < si}, B2 = {i ∈ B | ti < yi} B3 = {i ∈ B | si < ti}.
Let K = {k ∈ N | [xi, yi] ⊂ [ak, bk] for some i ∈ A}
∪ {k ∈ N | [xi, si] ⊂ [ak, bk] for some i ∈ B1}
∪ {k ∈ N | [ti, yi] ⊂ [ak, bk] for some i ∈ B2}.
Then the partition
P0 = {[xi, yi] | i ∈ A} ∪ {[xi, si] | i ∈ B1} ∪ {[ti, yi] | i ∈ B2}
∪ {[si, ti] | i ∈ B3} is a refinement of P. Hence, SP0(f∗) − SP0(f∗) < ².
Put P00 = {[si, ti] | i ∈ B3}, Q = {[ak, bk] | k ∈ K} ∪ P00. Then Q is a partition of [a, b]T and
SQ(f ) − SQ(f ) = SP00(f ) − SP00(f )
= SP0(f∗) − SP0(f∗) < ².
By Theorem 2.3, f is R∆−integrable on [a, b]T.
Theorem 2.5. Let f be a bounded R∆−integrable function on [a, b]T. Then f is R∆−integrable on every subinterval [c, d]T of [a, b]T.
Proof. Let f be a bounded R∆-integrable function on [a, b]T. By The- orem 2.4, f∗ : [a, b] → R is Riemann integrable on [a, b]. By the property of the Riemann integral, f∗ is Riemann integrable on every subinterval [c, d] ⊂ [a, b]. By Theorem 2.4, f is R∆−integrable on every subinterval [c, d]T ⊂ [a, b]T.
Theorem 2.6. Let f and g be R∆−integrable on [a, b]T and α, β be real numbers. Then αf + βg is R∆−integrable on [a, b]T and
(R∆) Z b
a
(αf + βg) = α(R∆) Z b
a
f + β(R∆) Z b
a
g.
Proof. Let f and g be R∆−integrable on [a, b]T. By Theorem 2.4, αf∗+ βg∗ is Riemann integrable on [a, b] and
(R) Z b
a
(αf∗+ βg∗) = α (R) Z b
a
f∗+ β (R) Z b
a
g∗. Hence, αf + βg is R∆−integrable on [a, b]T and
(R∆) Z b
a
(αf + βg) = α (R∆) Z b
a
f + β (R∆) Z b
a
g.
Theorem 2.7. Let f be a bounded function on [a, b]T and let c ∈ T with a < c < b. If f is R∆−integrable on each of intervals [a, c]T and [c, b]T, then f is R∆−integrable on [a, b]T and
(R∆) Z b
a
f = (R∆) Z c
a
f + (R∆) Z b
c
f.
Proof. If f is R∆−integrable on [a, c]Tand [c, b]T, then f∗is Riemann integrable on [a, c] and [c, b]. By the property of the Riemann integral, f∗ is Riemann integrable on [a, b] and
(R) Z b
a
f∗ = (R) Z c
a
f∗+ (R) Z b
c
f∗. By Theorem 2.4, f is R∆−integrable on [a, b]T and
(R∆) Z b
a
f = (R∆) Z c
a
f + (R∆) Z b
c
f.
Theorem 2.8. Let {fn} be a sequence of R∆−integrable functions on [a, b]T such that fn → f uniformly on [a, b]T. Then f is R∆−integrable on [a, b]T and
(R∆) Z b
a
f = lim
n→∞(R∆) Z b
a
fn.
Proof. Let {fn} be a sequence of R∆−integrable functions on [a, b]T such that fn → f uniformly on [a, b]T. By Theorem 2.4, {fn∗} is a sequence of Riemann integrable functions on [a, b] such that fn∗ → f∗ uniformly on [a, b].
By the property of Riemann integral, f∗ is Riemann integrable on [a, b] and
(R) Z b
a
f∗= lim
n→∞(R) Z b
a
fn∗. By Theorem 2.4, f is R∆−integrable on [a, b]T and
(R∆) Z b
a
f = lim
n→∞(R∆) Z b
a
fn.
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*
Department of Mathematics Chungnam National University Daejeon 305-764, Republic of Korea E-mail: parkjm@cnu.ac.kr
**
Department of Mathematics Education KongJu National University
Kongju 314-701, Republic of Korea E-mail: dhlee@kongju.ac.kr
***
Department of Mathematics Education Chungbuk National University
Chungju 360-763, Republic of Korea E-mail: yoonjh@cbnu.ac.kr
****
Department of Mathematics Education Seowon University
Chungju 361-742, Republic of Korea E-mail: ykkim@dragon.seowon.ac.kr
*****
Department of Mathematics Chungnam National University Daejeon 305-764, Republic of Korea E-mail: shiniljt@gmail.com
