DSpace at VNU: Some mean value theorems for integrals on time scales

Applied Mathematics and Computation 213 (2009) 322–328

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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc

Some mean value theorems for integrals on time scales
^c-Anh Ngô *
Quo
Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viêt Nam
Department of Mathematics, National Univesity of Singapore, 2 Science Drive 2, Singapore 117543, Singapore

a r t i c l e

i n f o

Keywords:
Inequality
Time scales
Integral
Mean value theorem

a b s t r a c t
In this short paper, we present time scales version of mean value theorems for integrals in
the single variable case.
Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries
The following two mean value theorems for time scales are due to M. Bohner and G. Guseinov.
Theorem A (See [1], Theorem 4.1). Suppose that f is continuous on ½a; bŠ and has a delta derivative at each point of ½a; bÞ. If

f ðaÞ ¼ f ðbÞ, then there exist points n; g 2 ½a; bÞ such that

f D ðnÞ 5 0 5 f D ðgÞ:
Theorem B (See [1], Theorem 4.2). Suppose that f is continuous on ½a; bŠ and has a delta derivative at each point of ½a; bÞ. If
f ðaÞ ¼ f ðbÞ, then there exist points n; g 2 ½a; bÞ such that

f D ðnÞðb À aÞ 5 f ðbÞ À f ðaÞ 5 f D ðgÞðb À aÞ:
Motivated by Theorem A, the main aim of this paper is to present time scale version of mean value results for integrals in
the single variable case. We first introduce some preliminaries on time scales (see [2,3,5] for details).
Definition 1. A time scale T is an arbitrary nonempty closed subset of real numbers.
The calculus of time scales was initiated by Stefan Hilger in his PhD thesis [4] in order to create a theory that can unify
discrete and continuous analysis. Let T be a time scale. T has the topology that it inherits from the real numbers with the
standard topology.
Definition 2. Let rðtÞ and qðtÞ be the forward and backward jump operators in T, respectively. For t 2 T, we define the
forward jump operator r : T ! T by

rðtÞ ¼ inf fs 2 T : s > tg;
while the backward jump operator q : T ! T is defined by

qðtÞ ¼ sup fs 2 T : s < tg:
If

rðtÞ > t, then we say that t is right-scattered, while if qðtÞ < t then we say that t is left-scattered.

* Address: Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viêt Nam.
E-mail address:
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2009.03.025

^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328
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323

In this definition we put inf ; ¼ sup T (i.e., rðtÞ ¼ t if T has a maximum t) and sup ; ¼ inf T (i.e., qðtÞ ¼ t if T has a minimum t), where ; denotes the empty set.
Points that are right-scattered and left-scattered at the same time are called isolated. If rðtÞ ¼ t and t– sup T, then t is
called right-dense, and if qðtÞ ¼ t and t – inf T, then t is called left-dense. Points that are right-dense and left-dense at the
same time are called dense.
Definition 3. Let t 2 T, then two mappings

l; m : T ! ½0; þ1Þ satisfying

lðtÞ :¼ rðtÞ À t; mðtÞ :¼ t À qðtÞ
are called the graininess functions.
We now introduce the set Tj which is derived from the time scales T as follows. If T has a left-scattered maximum t, then
T :¼ T À ftg, otherwise Tj :¼ T.
j

Definition 4. Let f : T ! R be a function on time scales. Then for t 2 Tj , we define f D ðtÞ to be the number, if one exists
(finite), such that for all e > 0 there is a neighborhood U of t such that for all s 2 U






f ðrðtÞÞ À f ðsÞ À f D ðtÞðrðtÞ À sÞ
5 ejrðtÞ À sj:
We say that f is D-differentiable on Tj provided f D ðtÞ exists for all t 2 Tj .

Assume that f : T ! R is a function and let t 2 Tj (t– min T). Then we have the following
(i) If f is D-differentiable at t, then f is continuous at t.
(ii) If f is left continuous at t and t is right-scattered, then f is D-differentiable at t with

f D ðtÞ ¼

f ðrðtÞÞ À f ðtÞ
:
lðtÞ

(iii) If t is right-dense, then f is D-differentiable at t if and only if

lims!t

f ð t Þ À f ðsÞ
;
tÀs

exists a finite number. In this case

f D ðtÞ ¼ lims!t

f ðtÞ À f ðsÞ
:
tÀs

(iv) If f is D-differentiable at t, then

f ðrðtÞÞ ¼ f ðtÞ þ lðtÞf D ðtÞ:
Proposition 1 (See [2], Theorem 1.20). Let f ; g : T ! R be differentiable at t 2 Tj . Then

D

ðfg Þ ðtÞ ¼ f D ðtÞg ðtÞ þ f ðrðt ÞÞg D ðt Þ ¼ f ðt Þg D ðt Þ þ f D ðt Þg ðrðt ÞÞ:
Definition 5. A mapping f : T ! R is called rd-continuous provided if it satisfies
(1) f is continuous at each right-dense point.
(2) The left-sided limit lims!tÀ f ðsÞ ¼ f ðtÀÞ exists at each left-dense point t of T.
Remark 1. It follows from Theorem 1.74 of Bohner and Peterson [2] that every rd-continuous function has an antiderivative.
Definition 6. A function F : T ! R is called a D-antiderivative of f : T ! R provided f D ðtÞ ¼ f ðtÞ holds for all t 2 Tj . Then the
D-integral of f is defined by

Z

b

f ðtÞDt ¼ F ðbÞ À F ðaÞ:

a

Proposition 2 (See [2], Theorem 1.77). Let f ; g be rd-continuous, a; b; c 2 T and a; b 2 R. Then
Rb
Rb
ðaf ðtÞ þ bgðtÞÞDt ¼ a a f ðtÞDt þ b a gðtÞDt;
Ra
(2) a f ðtÞDt ¼ À b f ðtÞDt;
Rc
Rb
Rb
(3) a f ðtÞDt ¼ a f ðtÞDt þ c f ðtÞDt;
Ra
(4) a f ðtÞDt ¼ 0:

(1)

Rb
a

Rb


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Definition 7. We say that a function p : T ! R is regressive provided

8t 2 T j

1 þ lðt Þpðt Þ – 0;
holds.

Definition 8. If a function p is regressive, then we define the exponential function by

ep ðt; sÞ ¼ exp

Z


nlðsÞ ðpðsÞÞDs ;

t

8s; t 2 T

s

where nh ðzÞ is the cylinder transformation which is defined by

(
n h ð sÞ ¼

1
Logð1
h

s;

þ shÞ; if h > 0;
if h ¼ 0;

where Log is the principal logarithm function.
Remark 2. It is obviously to see that e1 ðt; sÞ is well-defined and e1 ðt; sÞ > 0 for all t; s 2 T.
We now list here two properties of ep ðt; sÞ which we will use in the rest of this paper.
Theorem C (See [2], Theorem 2.33). If p is regressive, then for each t 0 2 T fixed, ep ðt; sÞ is a solution of the initial value problem

yD ¼ pðt Þy;

yðt0 Þ ¼ 1

on T.

Theorem D (See [2], Theorem 2.36). If p is regressive, then
1
.
(1) ep ðt; sÞ ¼ ep ðs;tÞ
Dt
1
(2) ðep ðt;sÞ Þ ¼ ep ðrÀpðtÞ;sÞ.

Throughout this paper, we suppose that T is a time scale, a; b 2 T with a < b and an interval means the intersection of real
interval with the given time scale.
2. Main results
Theorem 1. Let f be a continuous function on ½a; bŠ such that

Z

b

f ðxÞDx ¼ 0:

a

Then there exist n; g 2 ½a; bÞ so that

f ðn Þ 5

Z

Z

n

f ðxÞDx;

g

f ðxÞDx 5 f ðgÞ:

a

a

Proof of Theorem 1. Let

hðxÞ ¼ e1 ða; xÞ

Z

x

f ðtÞDt;

x 2 ½a; bÞ:

a

Then


Z
D

h ðxÞ ¼ e1 ða; xÞ

f ðt ÞDt

D

¼ ðe1 ða; xÞÞD

a


¼

x

1
e1 ðx; aÞ

x

Z
f ðt ÞDt þ e1 ða; rðxÞÞ

a

D Z

¼ Àe1 ða; rðxÞÞ

Z

x

f ðtÞDt þ e1 ða; rðxÞÞ

a

Z

Z

f ðt ÞDt

a
x

f ðt ÞDt þ e1 ða; rðxÞÞf ðxÞ:

a

Since hðaÞ ¼ hðbÞ then there exists n; g 2 ½a; bÞ such that
D

D

h ðnÞ 5 0 5 h ðgÞ:

a

D

x

x

¼

À1
e1 ðrðxÞ; xÞ

f ð t Þ Dt
Z
a

D

x

f ðtÞDt þ e1 ða; rðxÞÞf ðxÞ


^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328
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Hence

Àe1 ða; rðnÞÞ

Z

n

f ðtÞDt þ e1 ða; rðnÞÞf ðnÞ 5 0 5 À e1 ða; rðgÞÞ

a

Z

g

f ðt ÞDt þ e1 ða; rðgÞÞf ðgÞ;

a

which implies that

f ðnÞ 5

Z

Z

n

f ðxÞDx;

g

f ðxÞDx 5 f ðgÞ:

a

a

The proof is complete.

h

Theorem 2. Let f be a continuous function on ½a; bŠ such that

Z

b

f ðxÞDx ¼ 0:

a

Then there exist n; g 2 ½a; bÞ so that

e1 ða; nÞ
f ðnÞ 5
e1 ða; rðnÞÞ

Z rðnÞ

f ðtÞDt;

a

and

Z rðgÞ

e1 ða; gÞ
f ðgÞ:
e1 ða; rðgÞÞ

f ðt ÞDt 5

a

Proof of Theorem 2. Let

hðxÞ ¼ e1 ða; xÞ

Z

x

f ðtÞDt:

a

Then
D

h ðxÞ ¼ e1 ða; xÞf ðxÞ À e1 ða; rðxÞÞ

Z rðxÞ

f ðt ÞDt:

a

Since hðaÞ ¼ hðbÞ then there exists n; g 2 ½a; bÞ such that

g D ðnÞ 5 0 5 g D ðgÞ:
Hence

e1 ða; nÞf ðnÞ À e1 ða; rðnÞÞ

Z rðnÞ

f ðtÞDt 5 0 5 e1 ða; gÞf ðgÞ À e1 ða; rðgÞÞ

a

Z rðgÞ

f ðt ÞDt;

a

which implies that

Z rðnÞ
e1 ða; nÞ

f ðt ÞDt;
f ðnÞ 5
e1 ða; rðnÞÞ
a
Z rðgÞ
e1 ða; gÞ
f ðt ÞDt 5
f ðgÞ:
e1 ða; rðgÞÞ
a
The proof is complete.

h

Corollary 1. Let T ¼ R, from Theorems 1 and 2 together with the continuity of f we deduce that the existence of c 2 ½a; bŠ such that

f ðcÞ ¼

Z

c

f ðxÞdx

a

provided

Z

b

f ðxÞdx ¼ 0:

a

Theorem 3. Let f be a continuous function on ½a; bŠ such that

Z

b

f ðxÞDx ¼ 0:

a

Then for each T 3 c < a, there exist n; g 2 ½a; bÞ so that


^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328
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326

f ðnÞðn À cÞ 5

Z

Z

n

f ðt ÞDt;
a

g

f ðt ÞDt 5 f ðgÞðg À cÞ:

a

Proof of Theorem 3. Let

1
xÀc

hðxÞ ¼

Z

x

f ðt ÞDt;

x 2 ½a; bÞ; T 3 c < a:

a

Therefore

À1
ðrðxÞ À cÞðx À cÞ

D

h ðxÞ ¼

Z

x

f ðt ÞDt þ

a

1

rðxÞ À c

f ðxÞ:

Since hðaÞ ¼ hðbÞ then there exists n; g 2 ½a; bÞ such that
D

D

h ðnÞ 5 0 5 h ðgÞ:
Hence

Z

À1
ðrðnÞ À cÞðn À cÞ

n

f ðt ÞDt þ

a

1
À1
f ðnÞ 5 0 5
rðnÞ À c
ðrðgÞ À cÞðg À cÞ

Z
a

g

f ð t Þ Dt þ

1

rðgÞ À c

f ðgÞ;

which implies

Rn
f ð t Þ Dt
f ðnÞ
a
5
;
rðnÞ À c ðrðnÞ À cÞðn À cÞ
Rg
f ðt ÞDt
f ðgÞ
a
5
:
ðrðgÞ À cÞðg À cÞ
rðgÞ À c
Thus

f ðnÞðn À cÞ 5

Z

Z

n

f ðt ÞDt;

g

f ðt ÞDt 5 f ðgÞðg À cÞ:

a

a

The proof is complete.

h

Corollary 2. Let T ¼ R, from Theorem 3 together with the continuity of f we deduce the existence of c 2 ½a; bŠ such that

f ðcÞðn À cÞ ¼

Z

c

f ðxÞdx;

a

for each c < a provided

Z

b

f ðxÞDx ¼ 0:

a

Theorem 4. Let f ; g be a continuous function on ½a; bŠ. Then there exist n; g 2 ½a; bÞ so that

f ðn Þ

Z

!

b

g ðt ÞDt

Z rðnÞ
5


f ðt ÞDt g ðnÞ;

a

n

and

f ðgÞ

Z

!

b

g ðt ÞDt

Z rðgÞ
=

g


f ðtÞDt g ðgÞ:

a

Proof of Theorem 4

Z
hðxÞ ¼ À

x

f ðt ÞDt

a

Z

x


g ðt ÞDt ;

x 2 ½a; bÞ:

b

Then

Z
D
h ðxÞ ¼ Àf ðxÞ

x
b

 Z rðxÞ

g ð t Þ Dt À
f ðtÞDt gðxÞ:
a

Since hðaÞ ¼ hðbÞ then there exist n; g 2 ½a; bÞ such that
D

D

h ðnÞ 5 0 5 h ðgÞ:

^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328
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327

Hence

Àf ðnÞ

Z

n

 Z rðnÞ

Z
g ðt ÞDt À
f ðtÞDt g ðnÞ 5 0 5 À f ðgÞ

b

a

g

 Z rðgÞ

g ð t Þ Dt À
f ðt ÞDt g ðgÞ;

a

b

which implies

Z
0 5 f ðnÞ

 Z rðnÞ

g ðt ÞDt þ
f ðtÞDt g ðnÞ;

n

b

Z
0 = f ðgÞ

a



g

Z rðgÞ

g ðt ÞDt þ


f ðtÞDt g ðgÞ;

a

b

or equivalently

f ðnÞ

Z

!

b

g ðt ÞDt

n

f ðgÞ

Z

Z rðnÞ


f ðt ÞDt g ðnÞ;

5
a

!

b

Z rðgÞ

g ð t Þ Dt

=

g


f ðt ÞDt g ðgÞ:

a

The proof is complete. h
Corollary 3. Let T ¼ R, from Theorem 4 together with the continuity of f and g we deduce the existence of c 2 ½a; bŠ such that

Z

f ðcÞ

!

b

gðxÞdx

¼

Z

c

c


f ðxÞdx g ðcÞ:

a

Theorem 5. Let f ; g be continuous functions on ½a; bŠ. Then there exist n; g 2 ½a; bÞ so that

Z

n

f ðt ÞDt

Z

a

n

g ðt ÞDt



Z
5 f ðn Þ

b

n

 Z rðnÞ

g ð t Þ Dt þ
f ðtÞDt g ðnÞ

b

a

and

Z

g

f ðt ÞDt

Z

a

g


Z
g ðt ÞDt = f ðgÞ

b

g

 Z rðgÞ

g ðt ÞDt þ
f ðt ÞDt g ðgÞ:

b

a

Proof of Theorem 5. Let

hðxÞ ¼ Àe1 ða; xÞ

Z

x

f ð t Þ Dt

Z

a

x


g ðt ÞDt :

b

Then
D

h ðxÞ ¼ e1 ða; rðxÞÞ

Z

x

f ð t Þ Dt

a

¼ e1 ða; rðxÞÞ

Z

Z

x

b

x

f ð t Þ Dt

a

Z

x


Z
g ðt ÞDt À e1 ða; rðxÞÞ

x

f ð t Þ Dt

Z

a



g ðt ÞDt

b


Z
g ðt ÞDt À e1 ða; rðxÞÞ f ðxÞ

b

x



x

g ðt ÞDt þ
b

D

Z rðxÞ



f ðtÞDt gðxÞ :

a

Since hðaÞ ¼ hðbÞ then there exists n; g 2 ½a; bÞ such that

D

D

h ðnÞ 5 0 5 h ðgÞ:
Hence

e1 ða; rðnÞÞ

Z

n

f ðt ÞDt

Z

a

n



Z
g ðt ÞDt À e1 ða; rðnÞÞ f ðnÞ

b

n

 Z rðnÞ


g ðt ÞDt þ
f ðt ÞDt g ðnÞ 5 0;

b

a

and

Z
e1 ða; rðgÞÞ

g

f ðt ÞDt

Z

a

g



Z
g ðt ÞDt À e1 ða; rðgÞÞ f ðgÞ

b

g

 Z rðgÞ


g ðt ÞDt þ
f ðt ÞDt g ðgÞ = 0:

b

Thus

Z
a

n

f ðt ÞDt

Z
b

n

g ðt ÞDt



Z
5 f ðn Þ

n
b

 Z rðnÞ

g ð t Þ Dt þ
f ðtÞDt g ðnÞ
a

a


^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328
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328

and

Z

g

f ðt ÞDt

Z

a



g

g ðt ÞDt
b

= f ðgÞ

Z

g

a

b

The proof is complete.

 Z rðgÞ

g ð t Þ Dt þ
f ðt ÞDt g ðgÞ:

h

Corollary 4. Let T ¼ R, from Theorem 4 together with the continuity of f and g we deduce the existence of c 2 ½a; bŠ such that

Z

c

f ðxÞdx

Z

a

b

c


Z
gðxÞdx ¼ f ðcÞ

c
b

 Z
gðxÞdx þ

c


f ðxÞdx g ðcÞ:

a

Acknowledgements
The author wishes to express gratitude to the anonymous referee(s) for a number of valuable comments and suggestions
which helped to improve the presentation of the present paper from line to line.
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[2]
[3]
[4]
[5]

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