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The ℭ-integral of sequence-valued functions on the sequence space ℓ∞
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2017 J. Phys.: Conf. Ser. 855 012028 (http://iopscience.iop.org/1742-6596/855/1/012028) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 198.252.45.102 This content was downloaded on 08/06/2017 at 02:11 Please note that terms and conditions apply.
International Conference on Mathematics: Education, Theory and Application IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 855 (2017) 012028 doi:10.1088/1742-6596/855/1/012028
The -integral of sequence-valued functions on the sequence space Muslich1, Sutrima1, Supriyadi Wibowo1 1 Department Of Mathematics, Faculty of Mathematics and Natural Sciences, Sebelas Maret University, Surakarta, Indonesia E-mail:
[email protected],
[email protected] ,
[email protected] Abstract. In this paper the -integral of sequence-valued functions, the new concept of - integral. We report some of the integral theory, is constructed. The concept is based on the fundamental properties of the -integral including the uniqueness, linearity, the Cauchy criterion for integrability and the Henstock Lemma.
1. Introduction It is well known that the Lebesgue integral is equivalent to the McShane integral (see, e.g., [2], [3], [5], [9], [10]). In 2010, based on the McShane integral, Park et al. [6] and Park et al. [7] defined ⁄ , then the -integral, a Riemann-integral type, and constructed its properties. If a constant the -integral is called the -integral, [1]. Some extensions of the -integral have been studied for -integral and Park real-functions, such as Park et al. [6] researched the integration by parts for the [4] introduced the -integral of Banach-Valued functions concept and obtained its properties and especially, Zhao D [13] has studied the relations among Henstock, McShane and -integral. Zachriwan et al.[12] have investigated the Henstock integral of sequence-valued functions on the sequence space As an extension of the -integral, the authors are motivated by Zachriwan’s paper for studying the -integral for which the function has value in a sequence space and investigate some of their properties. 2. Basic Definitions Before introducing the -integral of sequence-valued functions concept, we start by some notations. Let be a positive function on the closed interval , i.e. , and {( } is a finite collection of interval-point pairs, where are nonoverlapping of subintervals of The following definitions can be found in [4], [7], [8], [11] and [ 13]. {( } Definition 2.1 Given (i) is a partial partition of if ⋃ (ii) is a partition of if ⋃ (iii) is a -fine McShane partition of (iv)
(v)
We say that
if
- partition of for a constant is a -fine and satisfying the condition ∑ ( | } {| is a -fine -partition of for a positive constant
of If partition ̅
(
(
(
for all
if it is a -fine McShane partition of where ( if it is a -fine McShane partition
. and satisfying the condition ∑ ( {( } is a -fine partition of and ( ( then every -fine of is a -fine partition of , and integral sum over for a function is defined as
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1
International Conference on Mathematics: Education, Theory and Application IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 855 (2017) 012028 doi:10.1088/1742-6596/855/1/012028
( ̅ where
{̅
{
}
)
(
̅(
∑
( } and ‖ ̅‖
{| |}
Definition 2.2 (Zachriwan et al. [12]) A function ̅ sequence of real valued functions ̅( ( ( { ( } { ( } ̅ ̅ Definition 2.3 Let ( ( { ( } for all ̅ ( ( ̅ if and only if ( ̅ ( ( ̅ if and only if ( ̅ ̅ on (
for each ̅
{| |}
, exists if and only if there exists a such that ̅ for all ( {| ( |}
̅( , where { ( We define for all and for all and if and only if (
(
on
(
} and
̅(
, for all
and
̅ ̅ almost everywhere on if and only if ( almost everywhere on , for all , (v) ̅ ̅ almost everywhere on if and only if almost everywhere on , for all , ( ( ( ( ̅ )( ( ( ( ̅ {( }, for all ̅ ( ( ( ( ( , . { }, for all From [1] and [8] we will define the -integral of real function and of sequence-valued functions as follow. A function is -integrable on if there exists a real number such | that for each there exists a positive function ( such that | ( for each -fine -partition of . (iv)
Definition 2.4 A function ̅ exists a sequence ̅ ( ( such that
( is -integrable on in short, ̅ if there such that for each there exists a positive function ‖ ( ̅
for each -fine The sequence ̅
)
-partition of . ̅ is called the -integral of on
̅‖ and it will be written as ̅
(
∫
̅
3. Main Results Based on the above definitions, we will provide some properties of -integral including the uniqueness, linearity, the Cauchy criterion for integrability, the Henstock Lemma and the modified Henstock Lemma. ( Theorem 3.1 If ̅ then the -integral of ̅ on ̅ ̅ Proof. Let be the -integrals of ̅ on . Given any a positive function ( such that ‖ ( ̅ ) ̅‖ for each (
-fine { (
is unique. . From the hypothesis, there exists
( -partition of Define a positive function ( }. If is a -fine -partition of , then we have ̅ ‖ ‖ ( ̅ ) ̅ ‖ ‖ ( ̅ ) ̅ ‖< + = ‖̅
2
where
International Conference on Mathematics: Education, Theory and Application IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 855 (2017) 012028 doi:10.1088/1742-6596/855/1/012028
̅ This result shows that ̅ has at most one -integral is arbitrary this implies that ̅ Hence theorem is proved. □ Let ( be the collection of all the -integrable functions ̅ on . The following result states that the set ( is a vector space over .
Since on
̅ ( ( Theorem 3.2 If ̅ ̅ then ̅ for every ̅ ̅ ( ∫ ( ̅) ( ∫ ( ∫ ̅ Proof. Without lose of the generality, we assume and are nonzero. Given any ̅ ̅ ̅ ̅ ̅ ( ∫ and ( ∫ ̅ , where There exists two positive functions ̅ ̅ for each -fine -partition of such that ‖ ( ) ‖
, and Let and
| |
̅ ‖
‖ ( ̅ ( have
||
for each
-fine { (
where (
̅ ‖ ( ̅ ) ( ̅ ̅ Definition 2.4 shows that ̅ (
∫ (
-partition
̅
. Define a positive function
( }, then for each
̅ ‖ ( ̅)
of
| |‖ ( ̅ and
)
̅
(
̅
̅ ‖
-fine -partition
(
∫
, we
̅ ‖< .
| |‖ ( ̅ ̅
∫
of
̅
Hence the theorem is proved. □ ( . A function ̅ if and Theorem 3.3 (Cauchy’s Criterion Theorem) Let ̅ only if for every there exsits a positive function ( such that )‖ ‖ ( ̅ ) ( ̅ for every two -fine -partitions and of ̅ ( Proof. ( ) Let be given. Since then there exists a positive ̅ ̅ function ( such that ‖ ( for every -fine -partition of ) ( ∫ ‖ . Therefore, for two arbitrary -fine -partitions and of we have ) ( ∫ ̅‖ ) ( ∫ ̅‖ ‖ ( ̅ )‖ ‖ ( ̅ ) ( ̅ ‖ ( ̅ . ⁄ , ( ) We choose . There is a decreasing sequence { } and a positive function and of we have ( such that for every -fine -partitions ̅ ̅ , )‖ ) ( ‖ ( ∑ ( ( where ( ̅ } . Therefore, we have )= {( ‖{(
∑
(
(
}
{(
∑
(
(
}
‖
This implies that ∑ ∑ ( ( ( ( ( , Now, we show that |( | ⁄ , and so we have two is -integrable on for Given consequences. First, there is a decreasing sequence { ( } of positive functions on , where { ( ( ( ( ( } Second, there is a Cauchy sequence of real number ∑ ( ( is a -fine - partition of . By the Cauchy {( } , where sequence properties, we have
is
-integrable on
Therefore
̅
{ }
( This completes the proof of the Theorem. □ See Park et al. [7] Theorem 2.2 for details. ( ( Theorem 3.4 Let ̅ . If ̅ then ̅ for every . 3
International Conference on Mathematics: Education, Theory and Application IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 855 (2017) 012028 doi:10.1088/1742-6596/855/1/012028
Proof. Assume that ̅ such that
(
there exists a positive function (
. Given
‖ ( ̅ ) ( ̅ )‖ for every -fine -partitions and of We define -fine -partitions and of of . Let = and = then the unions of and form the -fine -partitions of where and are always a partition point of , So we have ‖ ( ̅ ) ( ̅ )‖ ‖ ( ̅ ) ( ̅ )‖ ̅ ̅ ‖ ( ) ( )‖ ̅ ( By Theorem 3.3, then Hence the theorem is proved. □ Theorem (
̅ 3.5 Let and satisfying
If (
̅
∫
(
̅
(
∫
̅
̅ (
̅
̅
̅
(
̅
̅
∫
and on of
, and and
We define a positive function ( ( and { ( ( } if
of { ( } if , { ( ( } if Therefore for each -fine -partition of we have ̅ ‖ ‖ ( ̅ ) ̅ ‖ ‖ ( ̅ ) (̅ ( Definition 2.4 shows that ̅ and ∫
then
̅ be the -integrals of ̅ on two positive functions ( for each -fine -partition
Proof. Given any . Let the sequence ̅ and respectively. From the hypothesis, there exists ( such that ‖ ( ̅ ) ̅ ‖ for each -fine -partition ‖ ( ̅ ) ̅ ‖
(
(
∫
‖ ( ̅
)
̅
∫
(
̅ ‖
