Research ArticleNew Type Continuities via Abel Convergence

Huseyin Cakalli1 and Mehmet Albayrak2

1 Department of Mathematics Maltepe University Marmara Egıtım Koyu Maltepe 34857 Istanbul Turkey2Department of Mathematics Sakarya University Sakarya 54050 Istanbul Turkey

Correspondence should be addressed to Huseyin Cakalli huseyincakallimaltepeedutr

Received 24 January 2014 Accepted 21 March 2014 Published 27 April 2014

Academic Editor Guillermo Fernandez-Anaya

Copyright copy 2014 H Cakalli and M Albayrak This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We investigate the concept of Abel continuity A function119891 defined on a subset ofR the set of real numbers is Abel continuous if itpreserves Abel convergent sequences Some other types of continuities are also studied and interesting result is obtained It turnedout that uniform limit of a sequence of Abel continuous functions is Abel continuous and the set of Abel continuous functions is aclosed subset of continuous functions

1 Introduction

Theconcept of continuity and any concept involving continu-ity play a very important role not only in pure mathematicsbut also in other branches of sciences involving mathematicsespecially in computer sciences information theory anddynamical systems

A method of sequential convergence is a linear function119866 defined on a linear subspace of 119904 denoted by 119888119866 intoR where R and 119904 denote the set of real numbers and thespace of all sequences respectively A sequence p = (119901119899) issaid to be 119866-convergent to ℓ if p isin 119888119866 and 119866(p) = ℓ [1]A method 119866 is called regular if every convergent sequencep = (119901

119899) is 119866-convergent with 119866(p)=lim p A method 119866

is called subsequential if whenever p is 119866-convergent with119866(p) = ℓ then there is a subsequence (119901

119899119896) of p with

lim119896119901119899119896

= ℓ A function 119891 is called 119866-continuous (seealso [2 3]) if 119866(119891(p)) = 119891(119866(p)) for any 119866-convergentsequence p Any matrix summability method on a subspaceof 119904 is a method of sequential convergence Abel summabilitymethod is a regular method of sequential convergence in thismanner

The purpose of this paper is to investigate the conceptof Abel continuity for real functions and present interestingresults

2 Definitions and Notations andPreliminary Results

Wewill use boldface letters p rw for sequences p = (119901119899)

r = (119903119899) w = (119908

119899) of points in R A sequence p = (119901

119896)

of points in R is called statistically convergent [4] (see also[5ndash9]) to an element ℓ of R if

lim119899rarrinfin

1

119899

1003816100381610038161003816119896 le 119899 1003816100381610038161003816119901119896 minus ℓ

1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (1)

for every 120576 gt 0 and this is denoted by st-lim119901119896 = ℓ

A sequence (119901119896) of points in R is called lacunary statisti-

cally convergent [10] to an element ℓ of R if

lim119903rarrinfin

1

ℎ119903

1003816100381610038161003816119896 isin 119868119903 1003816100381610038161003816119901119896 minus ℓ

1003816100381610038161003816 ge 1205761003816100381610038161003816 = 0 (2)

for every 120576 gt 0 where 119868119903 = (119896119903minus1 119896119903] 1198960 = 0 ℎ119903 119896119903 minus 119896119903minus1 rarr

infin as 119903 rarr infin and 120579 = (119896119903) is an increasing sequenceof positive integers and this is denoted by 119878120579-lim119901119899 = ℓ(see also [11ndash13]) Throughout this paper we assume thatlim inf119903(119896119903119896119903minus1) gt 1 A sequence p = (119901119899) of points in R isslowly oscillating [14] (see also [15 16]) denoted by p isin SOif lim120582rarr1

+ lim119899max119899+1le119896le[120582119899]

|119909119896minus 119909119899| where [120582119899] denotes the

integer part of 120582119899

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 398379 6 pageshttpdxdoiorg1011552014398379

2 The Scientific World Journal

A sequence p = (119901119899) of real numbers is called Abel co-

nvergent (or Abel summable) to ℓ if the series Σinfin119896=0

119901119896119909119896 is

convergent for 0 le 119909 lt 1 and

lim119909rarr1

minus(1 minus 119909)

infin

sum119896=0

119901119896119909119896 = ℓ (3)

In this case we write Abel-lim119901119899

= ℓ The set of Abelconvergent sequences will be denoted by A Abel proved thatif lim

119899rarrinfin119901119899= ℓ then Abel-lim119901

119899= ℓ Abel sequential

method is regular that is every convergent sequence is Abelconvergent to the same limit ([17] see also [18 19]) As it isknown that the converse is not always true in general wesee that the sequence ((minus1)119899) is Abel convergent to 0 butconvergent in the ordinary sense

Definition 1 A subset 119864 of R is called Abel sequentiallycompact if whenever p = (119901

119899) is a sequence of points in 119864

there is an Abel convergent subsequence r = (119903119896) = (119903119899119896) of p

with lim119909rarr1

minus(1 minus 119909)suminfin

119896=0119903119896119909119896 isin 119864

Definition 2 A real number ℓ is said to be in the Abelsequentially closure of a subset 119864 of R denoted by 119864Abelif there is a sequence p = (119901119899) of points in 119864 such thatAbel-lim119901

119899= ℓ and it is called Abel sequentially closed if

119864Abel

= 119864

Note that the preceding definitions are special cases ofthe definitions of119866-sequential compactness and119866-sequentialclosure in [2]

It is clear that120601Abel

= 120601 andRAbel= R It is easily seen that

119864 sub 119864 sub 119864Abel It is not always true that (119864Abel

)Abel

= 119864Abel

for example (minus1 1Abel

)Abel

= minus1 1Abel

We note that anyAbel sequentially closed subset of Abel sequentially compactsubset of R is also Abel sequentially compact Thus inter-section of two Abel sequentially compacts Abel sequentiallyclosed subsets of R is Abel sequentially compact In generalany intersection of Abel sequentially compact and Abelsequentially closed subsets ofR is Abel sequentially compactThe condition closedness is essential here that is a subsetof an Abel sequentially compact subset need not to be Abelsequentially compact For example the interval ] minus 1 1] thatis the set of real numbers strictly greater thanminus1 and less thanor equal to 1 is a subset of Abel sequentially compact subset[minus1 1] that is the set of real numbers greater than or equalto minus1 and less than or equal to 1 but not Abel sequentiallycompact Notice that union of two Abel sequentially compactsubsets of R is Abel sequentially compact We see that anyfinite union of Abel sequentially compact subsets of R isAbel sequentially compact but any union of Abel sequen-tially compact subsets of R is not always Abel sequentiallycompact

3 Main Results

First we introduce two notions

Definition 3 A sequence (119901119896) of point in R is called Abel

quasi-Cauchy if (Δ119901119896) is Abel convergent to 0 that is

lim119909rarr1

minus(1 minus 119909)

infin

sum119896=0

Δ119901119896119909119896 = 0 (4)

where Δ119901119896= 119901119896+1

minus 119901119896for each positive integer 119896

Definition 4 A subset 119864 of R is called Abel ward compact ifwhenever p = (119901

119899) is a sequence of point in 119864 there is an

Abel quasi-Cauchy subsequence r = (119903119896) = (119903

119899119896) of p that is

lim119909rarr1minus(1 minus 119909)suminfin

119896=0Δ119903119896119909119896 = 0

We note that any Abel sequentially compact subset of Ris also Abel ward compact Intersection of two Abel wardcompact subsets of R is Abel ward compact In general anyintersection of Abel ward compact subsets of R is Abel wardcompactNotice that union of twoAbelward compact subsetsof R is Abel ward compact We see that any finite union ofAbel sequentially compact subsets ofR is Abel ward compactbut any union ofAbel ward compact subsets ofR is not alwaysAbel ward compact

Theorem 5 A subset119860 ofR is bounded if and only if it is Abelward compact

Proof It is an easy exercise to check that bounded subsets ofR are Abel ward compact To prove that Abel ward compact-ness implies boundedness suppose that119860 is unbounded If itis unbounded above then one can construct a sequence (120572

119899)

of numbers in 119860 such that 120572119899+1

gt 2119899 + 120572119899for each positive

integer 119899 Then the sequence (120572119899) does not have any Abel

quasi-Cauchy subsequence so 119860 is not Abel ward compactIf 119860 is bounded above and unbounded below then similarlywe obtain that 119860 is not Abel ward compact This completesthe proof

We now introduce a new type of continuity defined viaAbel convergent sequences

Definition 6 A function119891 is called Abel continuous denotedby f isinAC if it transforms Abel convergent sequences to Abelconvergent sequences that is (119891(119901119899)) is Abel convergent to119891(ℓ) whenever (119901119899) is Abel convergent to ℓ

We note that the sum of two Abel continuous functionsis Abel continuous and composite of two Abel continuousfunctions is Abel continuous but the product of two Abelcontinuous functions need not beAbel continuous as it can beseen by considering product of the Abel continuous function119891(119905) = 119905 with itself We see that 119866 defined by 119866(p) =Abel-lim p is a sequential method in the manner of [2] butnot subsequential so the theorems involving subsequentialityin [2] cannot be applied to Abel sequential method Inconnection with Abel convergent sequences and convergent

The Scientific World Journal 3

sequences the problem arises to investigate the followingtypes of continuity of functions on R

(119860) (119901119899) isin A 997904rArr (119891 (119901119899)) isin A

(119860119888) (119901119899) isin A 997904rArr (119891 (119901119899)) isin 119888

(119888) (119901119899) isin 119888 997904rArr (119891 (119901119899)) isin 119888

(119888119860) (119901119899) isin 119888 997904rArr (119891 (119901119899)) isin A

(5)

We see that 119860 is Abel continuity of 119891 and (119888) states theordinary continuity of 119891 We easily see that (119888) implies (119888119860)(119860) implies (119888119860) and (119860119888) implies (119860) The converses arenot always true as the identity function could be taken as acounter example for all the cases

We note that (119888) can be replaced by either statisticalcontinuity that is st-lim119891(119901119899) = 119891(ℓ) whenever p = (119901119899)is a statistically convergent sequence with st-lim119901

119899= ℓ or

lacunary statistical continuity that is 119878120579-lim119891(119901

119899) = 119891(ℓ)

whenever p = (119901119899) is a lacunary statistically convergent

sequence with 119878120579-lim119901

119899= ℓ More generally (119888) can be

replaced by 119866-sequential continuity of 119891 for any regularsubsequential method 119866

Now we give the implication that (119860) implies (119888) that isany Abel continuous function is continuous in the ordinarysense

Theorem 7 If a function 119891 is Abel continuous on a subset 119864 ofR then it is continuous on 119864 in the ordinary sense

Proof Suppose that a function 119891 is not continuous on 119864Then there exists a sequence (119901

119899) with lim

119899rarrinfin119901119899

= ℓsuch that (119891(119901

119899)) is not convergent to 119891(ℓ) If (119891(119901

119899))

exists and lim119891(119901119899) is different from 119891(ℓ) then we easily

see a contradiction Now suppose that (119891(119901119899)) has two

subsequences of 119891(119901119899) such that lim

119898rarrinfin119891(119901119896119898) = 119871

1

and lim119896rarrinfin

119891(119901119899119896) = 119871

2 Since (119901

119899119896) is subsequence of

(119901119899) by hypothesis lim119896rarrinfin119891(119901119899119896) = 119891(ℓ) and (119901119896119898) is asubsequence of (119901119899) by hypothesis lim119898rarrinfin119891(119901119896119898) = 119891(ℓ)This is a contradiction If (119891(119901119899)) is unbounded above thenwe can find an 119899

1such that 119891(119901

1198991) gt 21 There exists a

positive integer an 1198992 gt 1198991 such that 119891(1199011198992) gt 22 Supposethat we have chosen an 119899

119896minus1gt 119899119896minus2

such that 119891(119901119899119896minus1

) gt

2119896minus1 Then we can choose an 119899119896gt 119899119896minus1

such that 119891(119901119899119896) gt

2119896 Inductively we can construct a subsequence (119891(119901119899119896)) of

(119891(119901119899)) such that 119891(119901

119899119896) gt 2119896 Since the sequence (119901

119899119896) is a

subsequence of (119901119899) the subsequence (119901

119899119896) is convergent and

so is Abel convergent But (119891(119901119899119896)) is not Abel convergent as

we see in the line below For each positive integer 119896 we have119891(119901119899119896)119909119896 gt 2119896119909119896 The series suminfin

119896=02119896119909119896 is divergent for any 119909

satisfying 12 lt 119909 lt 1 and so is the series suminfin119896=0

119891(119901119899119896)119909119896

This is a contradiction to the Abel convergence of thesequence (119891(119901119899119896)) If (119891(119901119899)) is unbounded below similarlysuminfin

119896=0119891(119901119896)119909119896 is found to be divergent The contradiction for

all possible cases to the Abel continuity of 119891 completes theproof of the theorem

The converse is not always true for the bounded function119891(119905) = 1(1+ 1199052) defined onR as an exampleThe function 1199052is another example which is unbounded on R as well

On the other handnot all uniformly continuous functionsare Abel continuous For example the function defined by119891(119905) = 1199053 is uniformly continuous but Abel continuous

Corollary 8 If f is Abel continuous then it is statistically con-tinuous

Proof The proof follows fromTheorem 7 Corollary 4 in [6]Lemma 1 andTheorem 8 in [3]

Corollary 9 If f is Abel continuous then it is lacunarily sta-tistically sequentially continuous

Proof The proof follows fromTheorem 7 (see [20])

Now we have the following result

Corollary 10 If (119901119899) is slowly oscillating Abel convergent and

119891 is an Abel continuous function then (119891(119901119899)) is a convergent

sequence

Proof If (119901119899) is slowly oscillating and Abel convergent then

(119901119899) is convergent by the generalized Littlewood Tauberian

theorem for the Abel summability method By Theorem 7 119891is continuous so (119891(119901

119899)) is convergent This completes the

proof

Corollary 11 For any regular subsequential method 119866 anyAbel continuous function is 119866-continuous

Proof Let 119891 be an Abel continuous function and 119866 be aregular subsequential method ByTheorem 7 the function 119891is continuous Combining Lemma 1 and Corollary 9 of [3] weobtain that 119891 is 119866-continuous

For bounded functions we have the following result

Theorem 12 Any bounded Abel continuous function is Cesarocontinuous

Proof Let 119891 be a bounded Abel continuous function Nowwe are going to obtain that119891 is Cesaro continuous To do thistake any Cesaro convergent sequence (119901119899) with Cesaro limitℓ Since any Cesaro convergent sequence is Abel convergentto the same value [21] (see also [22]) (119901119899) is also Abelconvergent to ℓ By the assumption that119891 is Abel continuous(119891(119901119899)) is Abel convergent to 119891(ℓ) By the boundedness of 119891(119891(119901119899)) is bounded By Corollary to Karamatarsquos Hauptsatz onpage 108 in [18] (119891(119901

119899)) is Cesaro convergent to 119891(ℓ) Thus

119891 is Cesaro continuous at the point ℓ Hence 119891 is Cesarocontinuous at any point in the domain

Corollary 13 Any bounded Abel continuous function is uni-formly continuous

Proof Theproof follows from the preceding theorem and thetheorem on page 73 in [23]

4 The Scientific World Journal

It is well known that uniform limit of a sequence ofcontinuous functions is continuous This is also true forAbel continuity that is uniform limit of a sequence of Abelcontinuous functions is Abel continuous

Theorem 14 If (119891119899) is a sequence of Abel continuous functions

defined on a subset E of R and (119891119899) is uniformly convergent to

a function 119891 then 119891 is Abel continuous on 119864

Proof Let (119901119899) be an Abel convergent sequence of real

numbers in 119864 Write Abel-lim119901119899 = ℓ Take any 120576 gt 0Since (119891119899) is uniformly convergent to119891 there exists a positiveinteger119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

3(6)

for all 119905 isin 119864 whenever 119899 ge 119873 Hence1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (7)

As 119891119873is Abel continuous then there exist a 120575 gt 0 for 1 minus 120575 lt

119909 lt 1 such that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (119901119896) minus 119891119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (8)

Now for 1 minus 120575 lt 119909 lt 1 it follows from (6) (7) and (8) that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(9)

This completes the proof of the theorem

In the following theorem we prove that the set of Abelcontinuous functions is a closed subset of the space ofcontinuous functions

Theorem 15 The set of Abel continuous functions on a subset119864 of R is a closed subset of the set of all continuous functionson 119864 that is AC(119864) = AC(119864) where AC(119864) is the set of allAbel continuous functions on 119864 and AC(119864) denotes the set ofall cluster points of AC(119864)

Proof Let 119891 be any element in the closure of AC(119864) Thenthere exists a sequence (119891

119899) of points in AC(119864) such that

lim119899rarrinfin

119891119899= 119891 To show that 119891 is Abel continuous take any

Abel convergent sequence (119901119899) of points 119864 with Abel limit ℓ

Let 120576 gt 0 Since (119891119899) is convergent to 119891 there exists a positive

integer119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

6(10)

for all 119905 isin 119864 whenever 119899 ge 119873 Write

119872 = max 1003816100381610038161003816119891 (ℓ) minus 119891119873 (ℓ)1003816100381610038161003816

1003816100381610038161003816119891 (1199011) minus 119891119873 (1199011)1003816100381610038161003816

1003816100381610038161003816119891 (119901119873minus1) minus 119891119873 (119901119873minus1)1003816100381610038161003816

(11)

and 1205751= 1205766(119873 + 1)(119872 + 1) Then we obtain that for any 119909

satisfying 1 minus 1205751lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

119873minus1

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt (1 minus 119909)119872119873 +

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6+120576

6=120576

3

(12)

As 119891119873is Abel continuous then there exists a 120575

2gt 0 such that

for 1 minus 1205752lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (13)

Let 120575 = min1205751 1205752 Now for 1 minus 120575 lt 119909 lt 1 we have

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

119891 (119901119896) 119909119896 minus 119891 (ℓ)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896)) minus 119891

119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(14)

This completes the proof of the theorem

Corollary 16 The set of all Abel continuous functions ona subset 119864 of R is a complete subspace of the space of allcontinuous functions on 119864

The Scientific World Journal 5

Proof Theproof follows from the preceding theorem and thefact that the set of all continuous functions on 119864 is complete

Theorem 17 Abel continuous image of any Abel sequentiallycompact subset of R is Abel sequentially compact

Proof Although the proof follows from Theorem 7 in [2]we give a short proof for completeness Let 119891 be any Abelcontinuous function defined on a subset 119864 of R and let 119865 beanyAbel sequentially compact subset of119864 Take any sequencew = (119908119899) of point in119891(119865) Write119908119899 = 119891(119901119899) for each positiveinteger 119899 Since 119864 is Abel sequentially compact there existsan Abel convergent subsequence r = (119903119896) of the sequencep Write Abel-lim r = ℓ Since 119891 is Abel continuous Abel-lim119891(r) = 119891(ℓ) Thus 119891(r) = (119891(119903119896)) is Abel convergent to119891(ℓ) and a subsequence of the sequence w This completesthe proof

For 119866 = Abel-lim we have the following

Theorem 18 If a function 119891 is Abel continuous on a subset 119864of R then

119891(119860Abel

) sub (119891 (119860))Abel

(15)

for every subset 119860 of 119864

Proof The proof follows from the regularity of Abel methodandTheorem 8 on page 316 of [3]

Theorem 19 For any regular subsequential method 119866 ifa subset of R is 119866-sequentially compact then it is Abelsequentially compact

Proof The proof can be obtained by noticing the regularityand subsequentiality of 119866 (see [2] for the detail of 119866-sequential compactness)

Theorem20 A subset of R is Abel sequentially compact if andonly if it is bounded and closed

Proof It is clear that any bounded and closed subset of R isAbel sequentially compact Suppose first that119864 is unboundedso that we can construct a sequence p = (119901119899) of points in 119864such that 119901119899 gt 2119899 and 119901119899 gt 119901119899minus1 for each positive integer 119899It is easily seen that the sequence p has no Abel convergentsubsequence If it is unbounded below then similarly weconstruct a sequence of points in 119864 which has no Abelconvergent subsequence Hence 119864 is not Abel sequentiallycompact Now suppose that 119864 is not closed so that thereexists a point ℓ in 119864 minus 119864 Then there exists a sequenceof p = (119901

119899) of points in 119864 that converges to ℓ Every

subsequence of p also converges to ℓ Since Abel method isregular every subsequence of p Abel converges to ℓ Sinceℓ is not a member of 119864 119864 is not Abel sequentially compactThis contradiction completes the proof that Abel sequentiallycompactness implies boundedness and closedness

We note that an Abel sequentially compact subset ofR isslowly oscillating compact [15] an Abel sequentially compact

subset of R is ward compact [24] and an Abel sequentiallycompact subset of R is 120575-ward compact [25]

4 Conclusion

In this paper we introduce a concept of Abel continuityand a concept of Abel sequential compactness and presenttheorems related to this kind of sequential continuity thiskind of sequential compactness continuity statistical conti-nuity lacunary statistical continuity and uniform continuityOne may expect this investigation to be a useful tool in thefield of analysis in modeling various problems occurring inmany areas of science dynamical systems computer scienceinformation theory and biological science On the otherhand we suggest to introduce a concept of fuzzy Abelsequential compactness and investigate fuzzy Abel continuityfor fuzzy functions (see [26] for the definitions and relatedconcepts in fuzzy setting) However due to the change insettings the definitions andmethods of proofs will not alwaysbe the same We also suggest to investigate a theory indynamical systems by introducing the following concept twodynamical systems are called Abel-conjugate if there is a one-to-one and onto function such that ℎ and ℎminus1 are Abelcontinuous and ℎ commutes the mappings at each point Aninvestigation of Abel continuity andAbel compactness can bedone for double sequences (see [27] for basic concepts in thedouble sequences case) It seems both double Abel continuityand Abel continuity coincides but it needs proving

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthorswould like to thank the referees for a careful read-ing and several constructive comments that have improvedthe presentation of the results

References

[1] J Connor and K-G Grosse-Erdmann ldquoSequential definitionsof continuity for real functionsrdquo Rocky Mountain Journal ofMathematics vol 33 no 1 pp 93ndash121 2003

[2] H Cakallı ldquoSequential definitions of compactnessrdquo AppliedMathematics Letters vol 21 no 6 pp 594ndash598 2008

[3] H Cakallı ldquoOn G-continuityrdquo Computers amp Mathematics withApplications vol 61 no 2 pp 313ndash318 2011

[4] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[5] J A Fridy ldquoOn statistical convergencerdquoAnalysis vol 5 pp 301ndash313 1985

[6] H Cakallı ldquoA study on statistical convergencerdquo FunctionalAnalysis Approximation and Computation vol 1 no 2 pp 19ndash24 2009

[7] A Caserta andD R Lj Kocinac ldquoOn statistical exhaustivenessrdquoApplied Mathematics Letters vol 25 no 10 pp 1447ndash1451 2012

6 The Scientific World Journal

[8] H Cakallı andMK Khan ldquoSummability in topological spacesrdquoApplied Mathematics Letters vol 24 pp 348ndash352 2011

[9] A Caserta G Di Maio and D R Lj Kocinac ldquoStatisticalconvergence in function spacesrdquo Applied Mathematics Lettersvol 2011 Article ID 420419 11 pages 2011

[10] J A Fridy and C Orhan ldquoLacunary statistical convergencerdquoPacific Journal of Mathematics vol 160 no 1 pp 43ndash51 1993

[11] H Cakallı ldquoLacunary statistical convergence in topologicalgroupsrdquo Indian Journal of Pure and Applied Mathematics vol26 no 2 pp 113ndash119 1995

[12] E Savas ldquoA study on absolute summability factors for atriangular matrixrdquo Mathematical Inequalities and Applicationsvol 12 no 1 pp 141ndash146 2009

[13] E Savas and F Nuray ldquoOn 120590-statistically convergence andlacunary 120590-statistically convergencerdquoMathematica Slovaca vol43 no 3 pp 309ndash315 1993

[14] F Dik M Dik and I Canak ldquoApplications of subsequentialTauberian theory to classical Tauberian theoryrdquo Applied Math-ematics Letters vol 20 no 8 pp 946ndash950 2007

[15] H Cakallı ldquoSlowly oscillating continuityrdquo Abstract and AppliedAnalysis vol 2008 Article ID 485706 5 pages 2008

[16] M Dik and I Canak ldquoNew types of continuitiesrdquo Abstract andApplied Analysis vol 2010 Article ID 258980 6 pages 2010

[17] N H Abel ldquoRecherches sur la srie 1 + (1198981) 119909 + (119898 (119898 minus 1) 12) 1199092+sdot sdot sdot rdquo Journal fur die Reine undAngewandteMathematikvol 1 pp 311ndash339 1826

[18] C V Stanojevic and V B Stanojevic ldquoTauberian retrievaltheoryrdquo Publications de lrsquoInstitut Mathematique Nouvelle Serievol 71 no 85 pp 105ndash111 2002

[19] J A Fridy and M K Khan ldquoStatistical extensions of someclassical tauberian theoremsrdquo Proceedings of the AmericanMathematical Society vol 128 no 8 pp 2347ndash2355 2000

[20] H Cakalli A Sonmez andG C Aras ldquo120582-statistically wardcon-tinuityrdquo Analele Stiintifice ale Universitatii Al I Cuza din IasiSerie Noua Matematica In press

[21] C T Rajagopal ldquoOn Tauberian theorems for Abel-Cesarosummabilityrdquo Proceedings of the Glasgow Mathematical Associ-ation vol 3 pp 176ndash181 1958

[22] A J Badiozzaman and B Thorpe ldquoSome best possible Taube-rian results for Abel and Cesaro summabilityrdquo Bulletin of theLondon Mathematical Society vol 28 no 3 pp 283ndash290 1996

[23] E C Posner ldquoSummability preserving functionsrdquo Proceedingsof the American Mathematical Society vol 12 pp 73ndash76 1961

[24] H Cakallı ldquoForward continuityrdquo Journal of ComputationalAnalysis and Applications vol 13 no 2 pp 225ndash230 2011

[25] H Cakallı ldquo120575-quasi-Cauchy sequencesrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 397ndash401 2011

[26] H Cakallı and P Das ldquoFuzzy compactness via summabilityrdquoAppliedMathematics Letters vol 22 no 11 pp 1665ndash1669 2009

[27] H Cakallı and E Savas ldquoStatistical convergence of doublesequences in topological groupsrdquo Journal of ComputationalAnalysis and Applications vol 12 no 2 pp 421ndash426 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

A sequence p = (119901119899) of real numbers is called Abel co-

nvergent (or Abel summable) to ℓ if the series Σinfin119896=0

119901119896119909119896 is

convergent for 0 le 119909 lt 1 and

lim119909rarr1

minus(1 minus 119909)

infin

sum119896=0

119901119896119909119896 = ℓ (3)

In this case we write Abel-lim119901119899

= ℓ The set of Abelconvergent sequences will be denoted by A Abel proved thatif lim

119899rarrinfin119901119899= ℓ then Abel-lim119901

119899= ℓ Abel sequential

method is regular that is every convergent sequence is Abelconvergent to the same limit ([17] see also [18 19]) As it isknown that the converse is not always true in general wesee that the sequence ((minus1)119899) is Abel convergent to 0 butconvergent in the ordinary sense

Definition 1 A subset 119864 of R is called Abel sequentiallycompact if whenever p = (119901

119899) is a sequence of points in 119864

there is an Abel convergent subsequence r = (119903119896) = (119903119899119896) of p

with lim119909rarr1

minus(1 minus 119909)suminfin

119896=0119903119896119909119896 isin 119864

Definition 2 A real number ℓ is said to be in the Abelsequentially closure of a subset 119864 of R denoted by 119864Abelif there is a sequence p = (119901119899) of points in 119864 such thatAbel-lim119901

119899= ℓ and it is called Abel sequentially closed if

119864Abel

= 119864

Note that the preceding definitions are special cases ofthe definitions of119866-sequential compactness and119866-sequentialclosure in [2]

It is clear that120601Abel

= 120601 andRAbel= R It is easily seen that

119864 sub 119864 sub 119864Abel It is not always true that (119864Abel

)Abel

= 119864Abel

for example (minus1 1Abel

)Abel

= minus1 1Abel

We note that anyAbel sequentially closed subset of Abel sequentially compactsubset of R is also Abel sequentially compact Thus inter-section of two Abel sequentially compacts Abel sequentiallyclosed subsets of R is Abel sequentially compact In generalany intersection of Abel sequentially compact and Abelsequentially closed subsets ofR is Abel sequentially compactThe condition closedness is essential here that is a subsetof an Abel sequentially compact subset need not to be Abelsequentially compact For example the interval ] minus 1 1] thatis the set of real numbers strictly greater thanminus1 and less thanor equal to 1 is a subset of Abel sequentially compact subset[minus1 1] that is the set of real numbers greater than or equalto minus1 and less than or equal to 1 but not Abel sequentiallycompact Notice that union of two Abel sequentially compactsubsets of R is Abel sequentially compact We see that anyfinite union of Abel sequentially compact subsets of R isAbel sequentially compact but any union of Abel sequen-tially compact subsets of R is not always Abel sequentiallycompact

3 Main Results

First we introduce two notions

Definition 3 A sequence (119901119896) of point in R is called Abel

quasi-Cauchy if (Δ119901119896) is Abel convergent to 0 that is

lim119909rarr1

minus(1 minus 119909)

infin

sum119896=0

Δ119901119896119909119896 = 0 (4)

where Δ119901119896= 119901119896+1

minus 119901119896for each positive integer 119896

Definition 4 A subset 119864 of R is called Abel ward compact ifwhenever p = (119901

119899) is a sequence of point in 119864 there is an

Abel quasi-Cauchy subsequence r = (119903119896) = (119903

119899119896) of p that is

lim119909rarr1minus(1 minus 119909)suminfin

119896=0Δ119903119896119909119896 = 0

We note that any Abel sequentially compact subset of Ris also Abel ward compact Intersection of two Abel wardcompact subsets of R is Abel ward compact In general anyintersection of Abel ward compact subsets of R is Abel wardcompactNotice that union of twoAbelward compact subsetsof R is Abel ward compact We see that any finite union ofAbel sequentially compact subsets ofR is Abel ward compactbut any union ofAbel ward compact subsets ofR is not alwaysAbel ward compact

Theorem 5 A subset119860 ofR is bounded if and only if it is Abelward compact

Proof It is an easy exercise to check that bounded subsets ofR are Abel ward compact To prove that Abel ward compact-ness implies boundedness suppose that119860 is unbounded If itis unbounded above then one can construct a sequence (120572

119899)

of numbers in 119860 such that 120572119899+1

gt 2119899 + 120572119899for each positive

integer 119899 Then the sequence (120572119899) does not have any Abel

quasi-Cauchy subsequence so 119860 is not Abel ward compactIf 119860 is bounded above and unbounded below then similarlywe obtain that 119860 is not Abel ward compact This completesthe proof

We now introduce a new type of continuity defined viaAbel convergent sequences

Definition 6 A function119891 is called Abel continuous denotedby f isinAC if it transforms Abel convergent sequences to Abelconvergent sequences that is (119891(119901119899)) is Abel convergent to119891(ℓ) whenever (119901119899) is Abel convergent to ℓ

We note that the sum of two Abel continuous functionsis Abel continuous and composite of two Abel continuousfunctions is Abel continuous but the product of two Abelcontinuous functions need not beAbel continuous as it can beseen by considering product of the Abel continuous function119891(119905) = 119905 with itself We see that 119866 defined by 119866(p) =Abel-lim p is a sequential method in the manner of [2] butnot subsequential so the theorems involving subsequentialityin [2] cannot be applied to Abel sequential method Inconnection with Abel convergent sequences and convergent

The Scientific World Journal 3

sequences the problem arises to investigate the followingtypes of continuity of functions on R

(119860) (119901119899) isin A 997904rArr (119891 (119901119899)) isin A

(119860119888) (119901119899) isin A 997904rArr (119891 (119901119899)) isin 119888

(119888) (119901119899) isin 119888 997904rArr (119891 (119901119899)) isin 119888

(119888119860) (119901119899) isin 119888 997904rArr (119891 (119901119899)) isin A

(5)

We see that 119860 is Abel continuity of 119891 and (119888) states theordinary continuity of 119891 We easily see that (119888) implies (119888119860)(119860) implies (119888119860) and (119860119888) implies (119860) The converses arenot always true as the identity function could be taken as acounter example for all the cases

We note that (119888) can be replaced by either statisticalcontinuity that is st-lim119891(119901119899) = 119891(ℓ) whenever p = (119901119899)is a statistically convergent sequence with st-lim119901

119899= ℓ or

lacunary statistical continuity that is 119878120579-lim119891(119901

119899) = 119891(ℓ)

whenever p = (119901119899) is a lacunary statistically convergent

sequence with 119878120579-lim119901

119899= ℓ More generally (119888) can be

replaced by 119866-sequential continuity of 119891 for any regularsubsequential method 119866

Now we give the implication that (119860) implies (119888) that isany Abel continuous function is continuous in the ordinarysense

Theorem 7 If a function 119891 is Abel continuous on a subset 119864 ofR then it is continuous on 119864 in the ordinary sense

Proof Suppose that a function 119891 is not continuous on 119864Then there exists a sequence (119901

119899) with lim

119899rarrinfin119901119899

= ℓsuch that (119891(119901

119899)) is not convergent to 119891(ℓ) If (119891(119901

119899))

exists and lim119891(119901119899) is different from 119891(ℓ) then we easily

see a contradiction Now suppose that (119891(119901119899)) has two

subsequences of 119891(119901119899) such that lim

119898rarrinfin119891(119901119896119898) = 119871

1

and lim119896rarrinfin

119891(119901119899119896) = 119871

2 Since (119901

119899119896) is subsequence of

(119901119899) by hypothesis lim119896rarrinfin119891(119901119899119896) = 119891(ℓ) and (119901119896119898) is asubsequence of (119901119899) by hypothesis lim119898rarrinfin119891(119901119896119898) = 119891(ℓ)This is a contradiction If (119891(119901119899)) is unbounded above thenwe can find an 119899

1such that 119891(119901

1198991) gt 21 There exists a

positive integer an 1198992 gt 1198991 such that 119891(1199011198992) gt 22 Supposethat we have chosen an 119899

119896minus1gt 119899119896minus2

such that 119891(119901119899119896minus1

) gt

2119896minus1 Then we can choose an 119899119896gt 119899119896minus1

such that 119891(119901119899119896) gt

2119896 Inductively we can construct a subsequence (119891(119901119899119896)) of

(119891(119901119899)) such that 119891(119901

119899119896) gt 2119896 Since the sequence (119901

119899119896) is a

subsequence of (119901119899) the subsequence (119901

119899119896) is convergent and

so is Abel convergent But (119891(119901119899119896)) is not Abel convergent as

we see in the line below For each positive integer 119896 we have119891(119901119899119896)119909119896 gt 2119896119909119896 The series suminfin

119896=02119896119909119896 is divergent for any 119909

satisfying 12 lt 119909 lt 1 and so is the series suminfin119896=0

119891(119901119899119896)119909119896

This is a contradiction to the Abel convergence of thesequence (119891(119901119899119896)) If (119891(119901119899)) is unbounded below similarlysuminfin

119896=0119891(119901119896)119909119896 is found to be divergent The contradiction for

all possible cases to the Abel continuity of 119891 completes theproof of the theorem

The converse is not always true for the bounded function119891(119905) = 1(1+ 1199052) defined onR as an exampleThe function 1199052is another example which is unbounded on R as well

On the other handnot all uniformly continuous functionsare Abel continuous For example the function defined by119891(119905) = 1199053 is uniformly continuous but Abel continuous

Corollary 8 If f is Abel continuous then it is statistically con-tinuous

Proof The proof follows fromTheorem 7 Corollary 4 in [6]Lemma 1 andTheorem 8 in [3]

Corollary 9 If f is Abel continuous then it is lacunarily sta-tistically sequentially continuous

Proof The proof follows fromTheorem 7 (see [20])

Now we have the following result

Corollary 10 If (119901119899) is slowly oscillating Abel convergent and

119891 is an Abel continuous function then (119891(119901119899)) is a convergent

sequence

Proof If (119901119899) is slowly oscillating and Abel convergent then

(119901119899) is convergent by the generalized Littlewood Tauberian

theorem for the Abel summability method By Theorem 7 119891is continuous so (119891(119901

119899)) is convergent This completes the

proof

Corollary 11 For any regular subsequential method 119866 anyAbel continuous function is 119866-continuous

Proof Let 119891 be an Abel continuous function and 119866 be aregular subsequential method ByTheorem 7 the function 119891is continuous Combining Lemma 1 and Corollary 9 of [3] weobtain that 119891 is 119866-continuous

For bounded functions we have the following result

Theorem 12 Any bounded Abel continuous function is Cesarocontinuous

Proof Let 119891 be a bounded Abel continuous function Nowwe are going to obtain that119891 is Cesaro continuous To do thistake any Cesaro convergent sequence (119901119899) with Cesaro limitℓ Since any Cesaro convergent sequence is Abel convergentto the same value [21] (see also [22]) (119901119899) is also Abelconvergent to ℓ By the assumption that119891 is Abel continuous(119891(119901119899)) is Abel convergent to 119891(ℓ) By the boundedness of 119891(119891(119901119899)) is bounded By Corollary to Karamatarsquos Hauptsatz onpage 108 in [18] (119891(119901

119899)) is Cesaro convergent to 119891(ℓ) Thus

119891 is Cesaro continuous at the point ℓ Hence 119891 is Cesarocontinuous at any point in the domain

Corollary 13 Any bounded Abel continuous function is uni-formly continuous

Proof Theproof follows from the preceding theorem and thetheorem on page 73 in [23]

4 The Scientific World Journal

It is well known that uniform limit of a sequence ofcontinuous functions is continuous This is also true forAbel continuity that is uniform limit of a sequence of Abelcontinuous functions is Abel continuous

Theorem 14 If (119891119899) is a sequence of Abel continuous functions

defined on a subset E of R and (119891119899) is uniformly convergent to

a function 119891 then 119891 is Abel continuous on 119864

Proof Let (119901119899) be an Abel convergent sequence of real

numbers in 119864 Write Abel-lim119901119899 = ℓ Take any 120576 gt 0Since (119891119899) is uniformly convergent to119891 there exists a positiveinteger119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

3(6)

for all 119905 isin 119864 whenever 119899 ge 119873 Hence1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (7)

As 119891119873is Abel continuous then there exist a 120575 gt 0 for 1 minus 120575 lt

119909 lt 1 such that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (119901119896) minus 119891119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (8)

Now for 1 minus 120575 lt 119909 lt 1 it follows from (6) (7) and (8) that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(9)

This completes the proof of the theorem

In the following theorem we prove that the set of Abelcontinuous functions is a closed subset of the space ofcontinuous functions

Theorem 15 The set of Abel continuous functions on a subset119864 of R is a closed subset of the set of all continuous functionson 119864 that is AC(119864) = AC(119864) where AC(119864) is the set of allAbel continuous functions on 119864 and AC(119864) denotes the set ofall cluster points of AC(119864)

Proof Let 119891 be any element in the closure of AC(119864) Thenthere exists a sequence (119891

119899) of points in AC(119864) such that

lim119899rarrinfin

119891119899= 119891 To show that 119891 is Abel continuous take any

Abel convergent sequence (119901119899) of points 119864 with Abel limit ℓ

Let 120576 gt 0 Since (119891119899) is convergent to 119891 there exists a positive

integer119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

6(10)

for all 119905 isin 119864 whenever 119899 ge 119873 Write

119872 = max 1003816100381610038161003816119891 (ℓ) minus 119891119873 (ℓ)1003816100381610038161003816

1003816100381610038161003816119891 (1199011) minus 119891119873 (1199011)1003816100381610038161003816

1003816100381610038161003816119891 (119901119873minus1) minus 119891119873 (119901119873minus1)1003816100381610038161003816

(11)

and 1205751= 1205766(119873 + 1)(119872 + 1) Then we obtain that for any 119909

satisfying 1 minus 1205751lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

119873minus1

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt (1 minus 119909)119872119873 +

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6+120576

6=120576

3

(12)

As 119891119873is Abel continuous then there exists a 120575

2gt 0 such that

for 1 minus 1205752lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (13)

Let 120575 = min1205751 1205752 Now for 1 minus 120575 lt 119909 lt 1 we have

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

119891 (119901119896) 119909119896 minus 119891 (ℓ)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896)) minus 119891

119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(14)

This completes the proof of the theorem

Corollary 16 The set of all Abel continuous functions ona subset 119864 of R is a complete subspace of the space of allcontinuous functions on 119864

The Scientific World Journal 5

Proof Theproof follows from the preceding theorem and thefact that the set of all continuous functions on 119864 is complete

Theorem 17 Abel continuous image of any Abel sequentiallycompact subset of R is Abel sequentially compact

Proof Although the proof follows from Theorem 7 in [2]we give a short proof for completeness Let 119891 be any Abelcontinuous function defined on a subset 119864 of R and let 119865 beanyAbel sequentially compact subset of119864 Take any sequencew = (119908119899) of point in119891(119865) Write119908119899 = 119891(119901119899) for each positiveinteger 119899 Since 119864 is Abel sequentially compact there existsan Abel convergent subsequence r = (119903119896) of the sequencep Write Abel-lim r = ℓ Since 119891 is Abel continuous Abel-lim119891(r) = 119891(ℓ) Thus 119891(r) = (119891(119903119896)) is Abel convergent to119891(ℓ) and a subsequence of the sequence w This completesthe proof

For 119866 = Abel-lim we have the following

Theorem 18 If a function 119891 is Abel continuous on a subset 119864of R then

119891(119860Abel

) sub (119891 (119860))Abel

(15)

for every subset 119860 of 119864

Proof The proof follows from the regularity of Abel methodandTheorem 8 on page 316 of [3]

Theorem 19 For any regular subsequential method 119866 ifa subset of R is 119866-sequentially compact then it is Abelsequentially compact

Proof The proof can be obtained by noticing the regularityand subsequentiality of 119866 (see [2] for the detail of 119866-sequential compactness)

Theorem20 A subset of R is Abel sequentially compact if andonly if it is bounded and closed

Proof It is clear that any bounded and closed subset of R isAbel sequentially compact Suppose first that119864 is unboundedso that we can construct a sequence p = (119901119899) of points in 119864such that 119901119899 gt 2119899 and 119901119899 gt 119901119899minus1 for each positive integer 119899It is easily seen that the sequence p has no Abel convergentsubsequence If it is unbounded below then similarly weconstruct a sequence of points in 119864 which has no Abelconvergent subsequence Hence 119864 is not Abel sequentiallycompact Now suppose that 119864 is not closed so that thereexists a point ℓ in 119864 minus 119864 Then there exists a sequenceof p = (119901

119899) of points in 119864 that converges to ℓ Every

subsequence of p also converges to ℓ Since Abel method isregular every subsequence of p Abel converges to ℓ Sinceℓ is not a member of 119864 119864 is not Abel sequentially compactThis contradiction completes the proof that Abel sequentiallycompactness implies boundedness and closedness

We note that an Abel sequentially compact subset ofR isslowly oscillating compact [15] an Abel sequentially compact

subset of R is ward compact [24] and an Abel sequentiallycompact subset of R is 120575-ward compact [25]

4 Conclusion

In this paper we introduce a concept of Abel continuityand a concept of Abel sequential compactness and presenttheorems related to this kind of sequential continuity thiskind of sequential compactness continuity statistical conti-nuity lacunary statistical continuity and uniform continuityOne may expect this investigation to be a useful tool in thefield of analysis in modeling various problems occurring inmany areas of science dynamical systems computer scienceinformation theory and biological science On the otherhand we suggest to introduce a concept of fuzzy Abelsequential compactness and investigate fuzzy Abel continuityfor fuzzy functions (see [26] for the definitions and relatedconcepts in fuzzy setting) However due to the change insettings the definitions andmethods of proofs will not alwaysbe the same We also suggest to investigate a theory indynamical systems by introducing the following concept twodynamical systems are called Abel-conjugate if there is a one-to-one and onto function such that ℎ and ℎminus1 are Abelcontinuous and ℎ commutes the mappings at each point Aninvestigation of Abel continuity andAbel compactness can bedone for double sequences (see [27] for basic concepts in thedouble sequences case) It seems both double Abel continuityand Abel continuity coincides but it needs proving

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthorswould like to thank the referees for a careful read-ing and several constructive comments that have improvedthe presentation of the results

References

[1] J Connor and K-G Grosse-Erdmann ldquoSequential definitionsof continuity for real functionsrdquo Rocky Mountain Journal ofMathematics vol 33 no 1 pp 93ndash121 2003

[2] H Cakallı ldquoSequential definitions of compactnessrdquo AppliedMathematics Letters vol 21 no 6 pp 594ndash598 2008

[3] H Cakallı ldquoOn G-continuityrdquo Computers amp Mathematics withApplications vol 61 no 2 pp 313ndash318 2011

[4] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[5] J A Fridy ldquoOn statistical convergencerdquoAnalysis vol 5 pp 301ndash313 1985

[6] H Cakallı ldquoA study on statistical convergencerdquo FunctionalAnalysis Approximation and Computation vol 1 no 2 pp 19ndash24 2009

[7] A Caserta andD R Lj Kocinac ldquoOn statistical exhaustivenessrdquoApplied Mathematics Letters vol 25 no 10 pp 1447ndash1451 2012

6 The Scientific World Journal

[8] H Cakallı andMK Khan ldquoSummability in topological spacesrdquoApplied Mathematics Letters vol 24 pp 348ndash352 2011

[9] A Caserta G Di Maio and D R Lj Kocinac ldquoStatisticalconvergence in function spacesrdquo Applied Mathematics Lettersvol 2011 Article ID 420419 11 pages 2011

[10] J A Fridy and C Orhan ldquoLacunary statistical convergencerdquoPacific Journal of Mathematics vol 160 no 1 pp 43ndash51 1993

[11] H Cakallı ldquoLacunary statistical convergence in topologicalgroupsrdquo Indian Journal of Pure and Applied Mathematics vol26 no 2 pp 113ndash119 1995

[12] E Savas ldquoA study on absolute summability factors for atriangular matrixrdquo Mathematical Inequalities and Applicationsvol 12 no 1 pp 141ndash146 2009

[13] E Savas and F Nuray ldquoOn 120590-statistically convergence andlacunary 120590-statistically convergencerdquoMathematica Slovaca vol43 no 3 pp 309ndash315 1993

[14] F Dik M Dik and I Canak ldquoApplications of subsequentialTauberian theory to classical Tauberian theoryrdquo Applied Math-ematics Letters vol 20 no 8 pp 946ndash950 2007

[15] H Cakallı ldquoSlowly oscillating continuityrdquo Abstract and AppliedAnalysis vol 2008 Article ID 485706 5 pages 2008

[16] M Dik and I Canak ldquoNew types of continuitiesrdquo Abstract andApplied Analysis vol 2010 Article ID 258980 6 pages 2010

[17] N H Abel ldquoRecherches sur la srie 1 + (1198981) 119909 + (119898 (119898 minus 1) 12) 1199092+sdot sdot sdot rdquo Journal fur die Reine undAngewandteMathematikvol 1 pp 311ndash339 1826

[18] C V Stanojevic and V B Stanojevic ldquoTauberian retrievaltheoryrdquo Publications de lrsquoInstitut Mathematique Nouvelle Serievol 71 no 85 pp 105ndash111 2002

[19] J A Fridy and M K Khan ldquoStatistical extensions of someclassical tauberian theoremsrdquo Proceedings of the AmericanMathematical Society vol 128 no 8 pp 2347ndash2355 2000

[20] H Cakalli A Sonmez andG C Aras ldquo120582-statistically wardcon-tinuityrdquo Analele Stiintifice ale Universitatii Al I Cuza din IasiSerie Noua Matematica In press

[21] C T Rajagopal ldquoOn Tauberian theorems for Abel-Cesarosummabilityrdquo Proceedings of the Glasgow Mathematical Associ-ation vol 3 pp 176ndash181 1958

[22] A J Badiozzaman and B Thorpe ldquoSome best possible Taube-rian results for Abel and Cesaro summabilityrdquo Bulletin of theLondon Mathematical Society vol 28 no 3 pp 283ndash290 1996

[23] E C Posner ldquoSummability preserving functionsrdquo Proceedingsof the American Mathematical Society vol 12 pp 73ndash76 1961

[24] H Cakallı ldquoForward continuityrdquo Journal of ComputationalAnalysis and Applications vol 13 no 2 pp 225ndash230 2011

[25] H Cakallı ldquo120575-quasi-Cauchy sequencesrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 397ndash401 2011

[26] H Cakallı and P Das ldquoFuzzy compactness via summabilityrdquoAppliedMathematics Letters vol 22 no 11 pp 1665ndash1669 2009

[27] H Cakallı and E Savas ldquoStatistical convergence of doublesequences in topological groupsrdquo Journal of ComputationalAnalysis and Applications vol 12 no 2 pp 421ndash426 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

sequences the problem arises to investigate the followingtypes of continuity of functions on R

(119860) (119901119899) isin A 997904rArr (119891 (119901119899)) isin A

(119860119888) (119901119899) isin A 997904rArr (119891 (119901119899)) isin 119888

(119888) (119901119899) isin 119888 997904rArr (119891 (119901119899)) isin 119888

(119888119860) (119901119899) isin 119888 997904rArr (119891 (119901119899)) isin A

(5)

We see that 119860 is Abel continuity of 119891 and (119888) states theordinary continuity of 119891 We easily see that (119888) implies (119888119860)(119860) implies (119888119860) and (119860119888) implies (119860) The converses arenot always true as the identity function could be taken as acounter example for all the cases

We note that (119888) can be replaced by either statisticalcontinuity that is st-lim119891(119901119899) = 119891(ℓ) whenever p = (119901119899)is a statistically convergent sequence with st-lim119901

119899= ℓ or

lacunary statistical continuity that is 119878120579-lim119891(119901

119899) = 119891(ℓ)

whenever p = (119901119899) is a lacunary statistically convergent

sequence with 119878120579-lim119901

119899= ℓ More generally (119888) can be

replaced by 119866-sequential continuity of 119891 for any regularsubsequential method 119866

Now we give the implication that (119860) implies (119888) that isany Abel continuous function is continuous in the ordinarysense

Theorem 7 If a function 119891 is Abel continuous on a subset 119864 ofR then it is continuous on 119864 in the ordinary sense

Proof Suppose that a function 119891 is not continuous on 119864Then there exists a sequence (119901

119899) with lim

119899rarrinfin119901119899

= ℓsuch that (119891(119901

119899)) is not convergent to 119891(ℓ) If (119891(119901

119899))

exists and lim119891(119901119899) is different from 119891(ℓ) then we easily

see a contradiction Now suppose that (119891(119901119899)) has two

subsequences of 119891(119901119899) such that lim

119898rarrinfin119891(119901119896119898) = 119871

1

and lim119896rarrinfin

119891(119901119899119896) = 119871

2 Since (119901

119899119896) is subsequence of

(119901119899) by hypothesis lim119896rarrinfin119891(119901119899119896) = 119891(ℓ) and (119901119896119898) is asubsequence of (119901119899) by hypothesis lim119898rarrinfin119891(119901119896119898) = 119891(ℓ)This is a contradiction If (119891(119901119899)) is unbounded above thenwe can find an 119899

1such that 119891(119901

1198991) gt 21 There exists a

positive integer an 1198992 gt 1198991 such that 119891(1199011198992) gt 22 Supposethat we have chosen an 119899

119896minus1gt 119899119896minus2

such that 119891(119901119899119896minus1

) gt

2119896minus1 Then we can choose an 119899119896gt 119899119896minus1

such that 119891(119901119899119896) gt

2119896 Inductively we can construct a subsequence (119891(119901119899119896)) of

(119891(119901119899)) such that 119891(119901

119899119896) gt 2119896 Since the sequence (119901

119899119896) is a

subsequence of (119901119899) the subsequence (119901

119899119896) is convergent and

so is Abel convergent But (119891(119901119899119896)) is not Abel convergent as

we see in the line below For each positive integer 119896 we have119891(119901119899119896)119909119896 gt 2119896119909119896 The series suminfin

119896=02119896119909119896 is divergent for any 119909

satisfying 12 lt 119909 lt 1 and so is the series suminfin119896=0

119891(119901119899119896)119909119896

This is a contradiction to the Abel convergence of thesequence (119891(119901119899119896)) If (119891(119901119899)) is unbounded below similarlysuminfin

119896=0119891(119901119896)119909119896 is found to be divergent The contradiction for

all possible cases to the Abel continuity of 119891 completes theproof of the theorem

The converse is not always true for the bounded function119891(119905) = 1(1+ 1199052) defined onR as an exampleThe function 1199052is another example which is unbounded on R as well

On the other handnot all uniformly continuous functionsare Abel continuous For example the function defined by119891(119905) = 1199053 is uniformly continuous but Abel continuous

Corollary 8 If f is Abel continuous then it is statistically con-tinuous

Proof The proof follows fromTheorem 7 Corollary 4 in [6]Lemma 1 andTheorem 8 in [3]

Corollary 9 If f is Abel continuous then it is lacunarily sta-tistically sequentially continuous

Proof The proof follows fromTheorem 7 (see [20])

Now we have the following result

Corollary 10 If (119901119899) is slowly oscillating Abel convergent and

119891 is an Abel continuous function then (119891(119901119899)) is a convergent

sequence

Proof If (119901119899) is slowly oscillating and Abel convergent then

(119901119899) is convergent by the generalized Littlewood Tauberian

theorem for the Abel summability method By Theorem 7 119891is continuous so (119891(119901

119899)) is convergent This completes the

proof

Corollary 11 For any regular subsequential method 119866 anyAbel continuous function is 119866-continuous

Proof Let 119891 be an Abel continuous function and 119866 be aregular subsequential method ByTheorem 7 the function 119891is continuous Combining Lemma 1 and Corollary 9 of [3] weobtain that 119891 is 119866-continuous

For bounded functions we have the following result

Theorem 12 Any bounded Abel continuous function is Cesarocontinuous

Proof Let 119891 be a bounded Abel continuous function Nowwe are going to obtain that119891 is Cesaro continuous To do thistake any Cesaro convergent sequence (119901119899) with Cesaro limitℓ Since any Cesaro convergent sequence is Abel convergentto the same value [21] (see also [22]) (119901119899) is also Abelconvergent to ℓ By the assumption that119891 is Abel continuous(119891(119901119899)) is Abel convergent to 119891(ℓ) By the boundedness of 119891(119891(119901119899)) is bounded By Corollary to Karamatarsquos Hauptsatz onpage 108 in [18] (119891(119901

119899)) is Cesaro convergent to 119891(ℓ) Thus

119891 is Cesaro continuous at the point ℓ Hence 119891 is Cesarocontinuous at any point in the domain

Corollary 13 Any bounded Abel continuous function is uni-formly continuous

Proof Theproof follows from the preceding theorem and thetheorem on page 73 in [23]

4 The Scientific World Journal

It is well known that uniform limit of a sequence ofcontinuous functions is continuous This is also true forAbel continuity that is uniform limit of a sequence of Abelcontinuous functions is Abel continuous

Theorem 14 If (119891119899) is a sequence of Abel continuous functions

defined on a subset E of R and (119891119899) is uniformly convergent to

a function 119891 then 119891 is Abel continuous on 119864

Proof Let (119901119899) be an Abel convergent sequence of real

numbers in 119864 Write Abel-lim119901119899 = ℓ Take any 120576 gt 0Since (119891119899) is uniformly convergent to119891 there exists a positiveinteger119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

3(6)

for all 119905 isin 119864 whenever 119899 ge 119873 Hence1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (7)

As 119891119873is Abel continuous then there exist a 120575 gt 0 for 1 minus 120575 lt

119909 lt 1 such that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (119901119896) minus 119891119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (8)

Now for 1 minus 120575 lt 119909 lt 1 it follows from (6) (7) and (8) that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(9)

This completes the proof of the theorem

In the following theorem we prove that the set of Abelcontinuous functions is a closed subset of the space ofcontinuous functions

Theorem 15 The set of Abel continuous functions on a subset119864 of R is a closed subset of the set of all continuous functionson 119864 that is AC(119864) = AC(119864) where AC(119864) is the set of allAbel continuous functions on 119864 and AC(119864) denotes the set ofall cluster points of AC(119864)

Proof Let 119891 be any element in the closure of AC(119864) Thenthere exists a sequence (119891

119899) of points in AC(119864) such that

lim119899rarrinfin

119891119899= 119891 To show that 119891 is Abel continuous take any

Abel convergent sequence (119901119899) of points 119864 with Abel limit ℓ

Let 120576 gt 0 Since (119891119899) is convergent to 119891 there exists a positive

integer119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

6(10)

for all 119905 isin 119864 whenever 119899 ge 119873 Write

119872 = max 1003816100381610038161003816119891 (ℓ) minus 119891119873 (ℓ)1003816100381610038161003816

1003816100381610038161003816119891 (1199011) minus 119891119873 (1199011)1003816100381610038161003816

1003816100381610038161003816119891 (119901119873minus1) minus 119891119873 (119901119873minus1)1003816100381610038161003816

(11)

and 1205751= 1205766(119873 + 1)(119872 + 1) Then we obtain that for any 119909

satisfying 1 minus 1205751lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

119873minus1

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt (1 minus 119909)119872119873 +

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6+120576

6=120576

3

(12)

As 119891119873is Abel continuous then there exists a 120575

2gt 0 such that

for 1 minus 1205752lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (13)

Let 120575 = min1205751 1205752 Now for 1 minus 120575 lt 119909 lt 1 we have

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

119891 (119901119896) 119909119896 minus 119891 (ℓ)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896)) minus 119891

119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(14)

This completes the proof of the theorem

Corollary 16 The set of all Abel continuous functions ona subset 119864 of R is a complete subspace of the space of allcontinuous functions on 119864

The Scientific World Journal 5

Proof Theproof follows from the preceding theorem and thefact that the set of all continuous functions on 119864 is complete

Theorem 17 Abel continuous image of any Abel sequentiallycompact subset of R is Abel sequentially compact

Proof Although the proof follows from Theorem 7 in [2]we give a short proof for completeness Let 119891 be any Abelcontinuous function defined on a subset 119864 of R and let 119865 beanyAbel sequentially compact subset of119864 Take any sequencew = (119908119899) of point in119891(119865) Write119908119899 = 119891(119901119899) for each positiveinteger 119899 Since 119864 is Abel sequentially compact there existsan Abel convergent subsequence r = (119903119896) of the sequencep Write Abel-lim r = ℓ Since 119891 is Abel continuous Abel-lim119891(r) = 119891(ℓ) Thus 119891(r) = (119891(119903119896)) is Abel convergent to119891(ℓ) and a subsequence of the sequence w This completesthe proof

For 119866 = Abel-lim we have the following

Theorem 18 If a function 119891 is Abel continuous on a subset 119864of R then

119891(119860Abel

) sub (119891 (119860))Abel

(15)

for every subset 119860 of 119864

Proof The proof follows from the regularity of Abel methodandTheorem 8 on page 316 of [3]

Theorem 19 For any regular subsequential method 119866 ifa subset of R is 119866-sequentially compact then it is Abelsequentially compact

Proof The proof can be obtained by noticing the regularityand subsequentiality of 119866 (see [2] for the detail of 119866-sequential compactness)

Theorem20 A subset of R is Abel sequentially compact if andonly if it is bounded and closed

Proof It is clear that any bounded and closed subset of R isAbel sequentially compact Suppose first that119864 is unboundedso that we can construct a sequence p = (119901119899) of points in 119864such that 119901119899 gt 2119899 and 119901119899 gt 119901119899minus1 for each positive integer 119899It is easily seen that the sequence p has no Abel convergentsubsequence If it is unbounded below then similarly weconstruct a sequence of points in 119864 which has no Abelconvergent subsequence Hence 119864 is not Abel sequentiallycompact Now suppose that 119864 is not closed so that thereexists a point ℓ in 119864 minus 119864 Then there exists a sequenceof p = (119901

119899) of points in 119864 that converges to ℓ Every

subsequence of p also converges to ℓ Since Abel method isregular every subsequence of p Abel converges to ℓ Sinceℓ is not a member of 119864 119864 is not Abel sequentially compactThis contradiction completes the proof that Abel sequentiallycompactness implies boundedness and closedness

We note that an Abel sequentially compact subset ofR isslowly oscillating compact [15] an Abel sequentially compact

subset of R is ward compact [24] and an Abel sequentiallycompact subset of R is 120575-ward compact [25]

4 Conclusion

In this paper we introduce a concept of Abel continuityand a concept of Abel sequential compactness and presenttheorems related to this kind of sequential continuity thiskind of sequential compactness continuity statistical conti-nuity lacunary statistical continuity and uniform continuityOne may expect this investigation to be a useful tool in thefield of analysis in modeling various problems occurring inmany areas of science dynamical systems computer scienceinformation theory and biological science On the otherhand we suggest to introduce a concept of fuzzy Abelsequential compactness and investigate fuzzy Abel continuityfor fuzzy functions (see [26] for the definitions and relatedconcepts in fuzzy setting) However due to the change insettings the definitions andmethods of proofs will not alwaysbe the same We also suggest to investigate a theory indynamical systems by introducing the following concept twodynamical systems are called Abel-conjugate if there is a one-to-one and onto function such that ℎ and ℎminus1 are Abelcontinuous and ℎ commutes the mappings at each point Aninvestigation of Abel continuity andAbel compactness can bedone for double sequences (see [27] for basic concepts in thedouble sequences case) It seems both double Abel continuityand Abel continuity coincides but it needs proving

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthorswould like to thank the referees for a careful read-ing and several constructive comments that have improvedthe presentation of the results

References

[1] J Connor and K-G Grosse-Erdmann ldquoSequential definitionsof continuity for real functionsrdquo Rocky Mountain Journal ofMathematics vol 33 no 1 pp 93ndash121 2003

[2] H Cakallı ldquoSequential definitions of compactnessrdquo AppliedMathematics Letters vol 21 no 6 pp 594ndash598 2008

[3] H Cakallı ldquoOn G-continuityrdquo Computers amp Mathematics withApplications vol 61 no 2 pp 313ndash318 2011

[4] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[5] J A Fridy ldquoOn statistical convergencerdquoAnalysis vol 5 pp 301ndash313 1985

[6] H Cakallı ldquoA study on statistical convergencerdquo FunctionalAnalysis Approximation and Computation vol 1 no 2 pp 19ndash24 2009

[7] A Caserta andD R Lj Kocinac ldquoOn statistical exhaustivenessrdquoApplied Mathematics Letters vol 25 no 10 pp 1447ndash1451 2012

6 The Scientific World Journal

[8] H Cakallı andMK Khan ldquoSummability in topological spacesrdquoApplied Mathematics Letters vol 24 pp 348ndash352 2011

[9] A Caserta G Di Maio and D R Lj Kocinac ldquoStatisticalconvergence in function spacesrdquo Applied Mathematics Lettersvol 2011 Article ID 420419 11 pages 2011

[10] J A Fridy and C Orhan ldquoLacunary statistical convergencerdquoPacific Journal of Mathematics vol 160 no 1 pp 43ndash51 1993

[11] H Cakallı ldquoLacunary statistical convergence in topologicalgroupsrdquo Indian Journal of Pure and Applied Mathematics vol26 no 2 pp 113ndash119 1995

[12] E Savas ldquoA study on absolute summability factors for atriangular matrixrdquo Mathematical Inequalities and Applicationsvol 12 no 1 pp 141ndash146 2009

[13] E Savas and F Nuray ldquoOn 120590-statistically convergence andlacunary 120590-statistically convergencerdquoMathematica Slovaca vol43 no 3 pp 309ndash315 1993

[14] F Dik M Dik and I Canak ldquoApplications of subsequentialTauberian theory to classical Tauberian theoryrdquo Applied Math-ematics Letters vol 20 no 8 pp 946ndash950 2007

[15] H Cakallı ldquoSlowly oscillating continuityrdquo Abstract and AppliedAnalysis vol 2008 Article ID 485706 5 pages 2008

[16] M Dik and I Canak ldquoNew types of continuitiesrdquo Abstract andApplied Analysis vol 2010 Article ID 258980 6 pages 2010

[17] N H Abel ldquoRecherches sur la srie 1 + (1198981) 119909 + (119898 (119898 minus 1) 12) 1199092+sdot sdot sdot rdquo Journal fur die Reine undAngewandteMathematikvol 1 pp 311ndash339 1826

[18] C V Stanojevic and V B Stanojevic ldquoTauberian retrievaltheoryrdquo Publications de lrsquoInstitut Mathematique Nouvelle Serievol 71 no 85 pp 105ndash111 2002

[19] J A Fridy and M K Khan ldquoStatistical extensions of someclassical tauberian theoremsrdquo Proceedings of the AmericanMathematical Society vol 128 no 8 pp 2347ndash2355 2000

[20] H Cakalli A Sonmez andG C Aras ldquo120582-statistically wardcon-tinuityrdquo Analele Stiintifice ale Universitatii Al I Cuza din IasiSerie Noua Matematica In press

[21] C T Rajagopal ldquoOn Tauberian theorems for Abel-Cesarosummabilityrdquo Proceedings of the Glasgow Mathematical Associ-ation vol 3 pp 176ndash181 1958

[22] A J Badiozzaman and B Thorpe ldquoSome best possible Taube-rian results for Abel and Cesaro summabilityrdquo Bulletin of theLondon Mathematical Society vol 28 no 3 pp 283ndash290 1996

[23] E C Posner ldquoSummability preserving functionsrdquo Proceedingsof the American Mathematical Society vol 12 pp 73ndash76 1961

[24] H Cakallı ldquoForward continuityrdquo Journal of ComputationalAnalysis and Applications vol 13 no 2 pp 225ndash230 2011

[25] H Cakallı ldquo120575-quasi-Cauchy sequencesrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 397ndash401 2011

[26] H Cakallı and P Das ldquoFuzzy compactness via summabilityrdquoAppliedMathematics Letters vol 22 no 11 pp 1665ndash1669 2009

[27] H Cakallı and E Savas ldquoStatistical convergence of doublesequences in topological groupsrdquo Journal of ComputationalAnalysis and Applications vol 12 no 2 pp 421ndash426 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

It is well known that uniform limit of a sequence ofcontinuous functions is continuous This is also true forAbel continuity that is uniform limit of a sequence of Abelcontinuous functions is Abel continuous

Theorem 14 If (119891119899) is a sequence of Abel continuous functions

defined on a subset E of R and (119891119899) is uniformly convergent to

a function 119891 then 119891 is Abel continuous on 119864

Proof Let (119901119899) be an Abel convergent sequence of real

numbers in 119864 Write Abel-lim119901119899 = ℓ Take any 120576 gt 0Since (119891119899) is uniformly convergent to119891 there exists a positiveinteger119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

3(6)

for all 119905 isin 119864 whenever 119899 ge 119873 Hence1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (7)

As 119891119873is Abel continuous then there exist a 120575 gt 0 for 1 minus 120575 lt

119909 lt 1 such that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (119901119896) minus 119891119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (8)

Now for 1 minus 120575 lt 119909 lt 1 it follows from (6) (7) and (8) that1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(9)

This completes the proof of the theorem

In the following theorem we prove that the set of Abelcontinuous functions is a closed subset of the space ofcontinuous functions

Theorem 15 The set of Abel continuous functions on a subset119864 of R is a closed subset of the set of all continuous functionson 119864 that is AC(119864) = AC(119864) where AC(119864) is the set of allAbel continuous functions on 119864 and AC(119864) denotes the set ofall cluster points of AC(119864)

Proof Let 119891 be any element in the closure of AC(119864) Thenthere exists a sequence (119891

119899) of points in AC(119864) such that

lim119899rarrinfin

119891119899= 119891 To show that 119891 is Abel continuous take any

Abel convergent sequence (119901119899) of points 119864 with Abel limit ℓ

Let 120576 gt 0 Since (119891119899) is convergent to 119891 there exists a positive

integer119873 such that

1003816100381610038161003816119891119899 (119905) minus 119891 (119905)1003816100381610038161003816 lt

120576

6(10)

for all 119905 isin 119864 whenever 119899 ge 119873 Write

119872 = max 1003816100381610038161003816119891 (ℓ) minus 119891119873 (ℓ)1003816100381610038161003816

1003816100381610038161003816119891 (1199011) minus 119891119873 (1199011)1003816100381610038161003816

1003816100381610038161003816119891 (119901119873minus1) minus 119891119873 (119901119873minus1)1003816100381610038161003816

(11)

and 1205751= 1205766(119873 + 1)(119872 + 1) Then we obtain that for any 119909

satisfying 1 minus 1205751lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

119873minus1

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt (1 minus 119909)119872119873 +

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=119873

(119891 (119901119896) minus 119891119873 (119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

6+120576

6=120576

3

(12)

As 119891119873is Abel continuous then there exists a 120575

2gt 0 such that

for 1 minus 1205752lt 119909 lt 1

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896) minus 119891119873 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816lt120576

3 (13)

Let 120575 = min1205751 1205752 Now for 1 minus 120575 lt 119909 lt 1 we have

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

119891 (119901119896) 119909119896 minus 119891 (ℓ)

1003816100381610038161003816100381610038161003816100381610038161003816

le

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891 (119901119896) minus 119891119873(119901119896)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873(119901119896)) minus 119891

119873 (ℓ)) 119909119896

1003816100381610038161003816100381610038161003816100381610038161003816

+

1003816100381610038161003816100381610038161003816100381610038161003816(1 minus 119909)

infin

sum119896=0

(119891119873 (ℓ) minus 119891 (ℓ)) 119909

119896

1003816100381610038161003816100381610038161003816100381610038161003816

lt120576

3+120576

3+120576

3= 120576

(14)

This completes the proof of the theorem

Corollary 16 The set of all Abel continuous functions ona subset 119864 of R is a complete subspace of the space of allcontinuous functions on 119864

The Scientific World Journal 5

Proof Theproof follows from the preceding theorem and thefact that the set of all continuous functions on 119864 is complete

Theorem 17 Abel continuous image of any Abel sequentiallycompact subset of R is Abel sequentially compact

Proof Although the proof follows from Theorem 7 in [2]we give a short proof for completeness Let 119891 be any Abelcontinuous function defined on a subset 119864 of R and let 119865 beanyAbel sequentially compact subset of119864 Take any sequencew = (119908119899) of point in119891(119865) Write119908119899 = 119891(119901119899) for each positiveinteger 119899 Since 119864 is Abel sequentially compact there existsan Abel convergent subsequence r = (119903119896) of the sequencep Write Abel-lim r = ℓ Since 119891 is Abel continuous Abel-lim119891(r) = 119891(ℓ) Thus 119891(r) = (119891(119903119896)) is Abel convergent to119891(ℓ) and a subsequence of the sequence w This completesthe proof

For 119866 = Abel-lim we have the following

Theorem 18 If a function 119891 is Abel continuous on a subset 119864of R then

119891(119860Abel

) sub (119891 (119860))Abel

(15)

for every subset 119860 of 119864

Proof The proof follows from the regularity of Abel methodandTheorem 8 on page 316 of [3]

Theorem 19 For any regular subsequential method 119866 ifa subset of R is 119866-sequentially compact then it is Abelsequentially compact

Proof The proof can be obtained by noticing the regularityand subsequentiality of 119866 (see [2] for the detail of 119866-sequential compactness)

Theorem20 A subset of R is Abel sequentially compact if andonly if it is bounded and closed

Proof It is clear that any bounded and closed subset of R isAbel sequentially compact Suppose first that119864 is unboundedso that we can construct a sequence p = (119901119899) of points in 119864such that 119901119899 gt 2119899 and 119901119899 gt 119901119899minus1 for each positive integer 119899It is easily seen that the sequence p has no Abel convergentsubsequence If it is unbounded below then similarly weconstruct a sequence of points in 119864 which has no Abelconvergent subsequence Hence 119864 is not Abel sequentiallycompact Now suppose that 119864 is not closed so that thereexists a point ℓ in 119864 minus 119864 Then there exists a sequenceof p = (119901

119899) of points in 119864 that converges to ℓ Every

subsequence of p also converges to ℓ Since Abel method isregular every subsequence of p Abel converges to ℓ Sinceℓ is not a member of 119864 119864 is not Abel sequentially compactThis contradiction completes the proof that Abel sequentiallycompactness implies boundedness and closedness

We note that an Abel sequentially compact subset ofR isslowly oscillating compact [15] an Abel sequentially compact

subset of R is ward compact [24] and an Abel sequentiallycompact subset of R is 120575-ward compact [25]

4 Conclusion

In this paper we introduce a concept of Abel continuityand a concept of Abel sequential compactness and presenttheorems related to this kind of sequential continuity thiskind of sequential compactness continuity statistical conti-nuity lacunary statistical continuity and uniform continuityOne may expect this investigation to be a useful tool in thefield of analysis in modeling various problems occurring inmany areas of science dynamical systems computer scienceinformation theory and biological science On the otherhand we suggest to introduce a concept of fuzzy Abelsequential compactness and investigate fuzzy Abel continuityfor fuzzy functions (see [26] for the definitions and relatedconcepts in fuzzy setting) However due to the change insettings the definitions andmethods of proofs will not alwaysbe the same We also suggest to investigate a theory indynamical systems by introducing the following concept twodynamical systems are called Abel-conjugate if there is a one-to-one and onto function such that ℎ and ℎminus1 are Abelcontinuous and ℎ commutes the mappings at each point Aninvestigation of Abel continuity andAbel compactness can bedone for double sequences (see [27] for basic concepts in thedouble sequences case) It seems both double Abel continuityand Abel continuity coincides but it needs proving

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthorswould like to thank the referees for a careful read-ing and several constructive comments that have improvedthe presentation of the results

References

[1] J Connor and K-G Grosse-Erdmann ldquoSequential definitionsof continuity for real functionsrdquo Rocky Mountain Journal ofMathematics vol 33 no 1 pp 93ndash121 2003

[2] H Cakallı ldquoSequential definitions of compactnessrdquo AppliedMathematics Letters vol 21 no 6 pp 594ndash598 2008

[3] H Cakallı ldquoOn G-continuityrdquo Computers amp Mathematics withApplications vol 61 no 2 pp 313ndash318 2011

[4] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[5] J A Fridy ldquoOn statistical convergencerdquoAnalysis vol 5 pp 301ndash313 1985

[6] H Cakallı ldquoA study on statistical convergencerdquo FunctionalAnalysis Approximation and Computation vol 1 no 2 pp 19ndash24 2009

[7] A Caserta andD R Lj Kocinac ldquoOn statistical exhaustivenessrdquoApplied Mathematics Letters vol 25 no 10 pp 1447ndash1451 2012

6 The Scientific World Journal

[8] H Cakallı andMK Khan ldquoSummability in topological spacesrdquoApplied Mathematics Letters vol 24 pp 348ndash352 2011

[9] A Caserta G Di Maio and D R Lj Kocinac ldquoStatisticalconvergence in function spacesrdquo Applied Mathematics Lettersvol 2011 Article ID 420419 11 pages 2011

[10] J A Fridy and C Orhan ldquoLacunary statistical convergencerdquoPacific Journal of Mathematics vol 160 no 1 pp 43ndash51 1993

[11] H Cakallı ldquoLacunary statistical convergence in topologicalgroupsrdquo Indian Journal of Pure and Applied Mathematics vol26 no 2 pp 113ndash119 1995

[12] E Savas ldquoA study on absolute summability factors for atriangular matrixrdquo Mathematical Inequalities and Applicationsvol 12 no 1 pp 141ndash146 2009

[13] E Savas and F Nuray ldquoOn 120590-statistically convergence andlacunary 120590-statistically convergencerdquoMathematica Slovaca vol43 no 3 pp 309ndash315 1993

[14] F Dik M Dik and I Canak ldquoApplications of subsequentialTauberian theory to classical Tauberian theoryrdquo Applied Math-ematics Letters vol 20 no 8 pp 946ndash950 2007

[15] H Cakallı ldquoSlowly oscillating continuityrdquo Abstract and AppliedAnalysis vol 2008 Article ID 485706 5 pages 2008

[16] M Dik and I Canak ldquoNew types of continuitiesrdquo Abstract andApplied Analysis vol 2010 Article ID 258980 6 pages 2010

[17] N H Abel ldquoRecherches sur la srie 1 + (1198981) 119909 + (119898 (119898 minus 1) 12) 1199092+sdot sdot sdot rdquo Journal fur die Reine undAngewandteMathematikvol 1 pp 311ndash339 1826

[18] C V Stanojevic and V B Stanojevic ldquoTauberian retrievaltheoryrdquo Publications de lrsquoInstitut Mathematique Nouvelle Serievol 71 no 85 pp 105ndash111 2002

[19] J A Fridy and M K Khan ldquoStatistical extensions of someclassical tauberian theoremsrdquo Proceedings of the AmericanMathematical Society vol 128 no 8 pp 2347ndash2355 2000

[20] H Cakalli A Sonmez andG C Aras ldquo120582-statistically wardcon-tinuityrdquo Analele Stiintifice ale Universitatii Al I Cuza din IasiSerie Noua Matematica In press

[21] C T Rajagopal ldquoOn Tauberian theorems for Abel-Cesarosummabilityrdquo Proceedings of the Glasgow Mathematical Associ-ation vol 3 pp 176ndash181 1958

[22] A J Badiozzaman and B Thorpe ldquoSome best possible Taube-rian results for Abel and Cesaro summabilityrdquo Bulletin of theLondon Mathematical Society vol 28 no 3 pp 283ndash290 1996

[23] E C Posner ldquoSummability preserving functionsrdquo Proceedingsof the American Mathematical Society vol 12 pp 73ndash76 1961

[24] H Cakallı ldquoForward continuityrdquo Journal of ComputationalAnalysis and Applications vol 13 no 2 pp 225ndash230 2011

[25] H Cakallı ldquo120575-quasi-Cauchy sequencesrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 397ndash401 2011

[26] H Cakallı and P Das ldquoFuzzy compactness via summabilityrdquoAppliedMathematics Letters vol 22 no 11 pp 1665ndash1669 2009

[27] H Cakallı and E Savas ldquoStatistical convergence of doublesequences in topological groupsrdquo Journal of ComputationalAnalysis and Applications vol 12 no 2 pp 421ndash426 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Proof Theproof follows from the preceding theorem and thefact that the set of all continuous functions on 119864 is complete

Theorem 17 Abel continuous image of any Abel sequentiallycompact subset of R is Abel sequentially compact

Proof Although the proof follows from Theorem 7 in [2]we give a short proof for completeness Let 119891 be any Abelcontinuous function defined on a subset 119864 of R and let 119865 beanyAbel sequentially compact subset of119864 Take any sequencew = (119908119899) of point in119891(119865) Write119908119899 = 119891(119901119899) for each positiveinteger 119899 Since 119864 is Abel sequentially compact there existsan Abel convergent subsequence r = (119903119896) of the sequencep Write Abel-lim r = ℓ Since 119891 is Abel continuous Abel-lim119891(r) = 119891(ℓ) Thus 119891(r) = (119891(119903119896)) is Abel convergent to119891(ℓ) and a subsequence of the sequence w This completesthe proof

For 119866 = Abel-lim we have the following

Theorem 18 If a function 119891 is Abel continuous on a subset 119864of R then

119891(119860Abel

) sub (119891 (119860))Abel

(15)

for every subset 119860 of 119864

Proof The proof follows from the regularity of Abel methodandTheorem 8 on page 316 of [3]

Theorem 19 For any regular subsequential method 119866 ifa subset of R is 119866-sequentially compact then it is Abelsequentially compact

Proof The proof can be obtained by noticing the regularityand subsequentiality of 119866 (see [2] for the detail of 119866-sequential compactness)

Theorem20 A subset of R is Abel sequentially compact if andonly if it is bounded and closed

Proof It is clear that any bounded and closed subset of R isAbel sequentially compact Suppose first that119864 is unboundedso that we can construct a sequence p = (119901119899) of points in 119864such that 119901119899 gt 2119899 and 119901119899 gt 119901119899minus1 for each positive integer 119899It is easily seen that the sequence p has no Abel convergentsubsequence If it is unbounded below then similarly weconstruct a sequence of points in 119864 which has no Abelconvergent subsequence Hence 119864 is not Abel sequentiallycompact Now suppose that 119864 is not closed so that thereexists a point ℓ in 119864 minus 119864 Then there exists a sequenceof p = (119901

119899) of points in 119864 that converges to ℓ Every

subsequence of p also converges to ℓ Since Abel method isregular every subsequence of p Abel converges to ℓ Sinceℓ is not a member of 119864 119864 is not Abel sequentially compactThis contradiction completes the proof that Abel sequentiallycompactness implies boundedness and closedness

We note that an Abel sequentially compact subset ofR isslowly oscillating compact [15] an Abel sequentially compact

subset of R is ward compact [24] and an Abel sequentiallycompact subset of R is 120575-ward compact [25]

4 Conclusion

In this paper we introduce a concept of Abel continuityand a concept of Abel sequential compactness and presenttheorems related to this kind of sequential continuity thiskind of sequential compactness continuity statistical conti-nuity lacunary statistical continuity and uniform continuityOne may expect this investigation to be a useful tool in thefield of analysis in modeling various problems occurring inmany areas of science dynamical systems computer scienceinformation theory and biological science On the otherhand we suggest to introduce a concept of fuzzy Abelsequential compactness and investigate fuzzy Abel continuityfor fuzzy functions (see [26] for the definitions and relatedconcepts in fuzzy setting) However due to the change insettings the definitions andmethods of proofs will not alwaysbe the same We also suggest to investigate a theory indynamical systems by introducing the following concept twodynamical systems are called Abel-conjugate if there is a one-to-one and onto function such that ℎ and ℎminus1 are Abelcontinuous and ℎ commutes the mappings at each point Aninvestigation of Abel continuity andAbel compactness can bedone for double sequences (see [27] for basic concepts in thedouble sequences case) It seems both double Abel continuityand Abel continuity coincides but it needs proving

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Theauthorswould like to thank the referees for a careful read-ing and several constructive comments that have improvedthe presentation of the results

References

[1] J Connor and K-G Grosse-Erdmann ldquoSequential definitionsof continuity for real functionsrdquo Rocky Mountain Journal ofMathematics vol 33 no 1 pp 93ndash121 2003

[2] H Cakallı ldquoSequential definitions of compactnessrdquo AppliedMathematics Letters vol 21 no 6 pp 594ndash598 2008

[3] H Cakallı ldquoOn G-continuityrdquo Computers amp Mathematics withApplications vol 61 no 2 pp 313ndash318 2011

[4] H Fast ldquoSur la convergence statistiquerdquo Colloquium Mathe-maticum vol 2 pp 241ndash244 1951

[5] J A Fridy ldquoOn statistical convergencerdquoAnalysis vol 5 pp 301ndash313 1985

[6] H Cakallı ldquoA study on statistical convergencerdquo FunctionalAnalysis Approximation and Computation vol 1 no 2 pp 19ndash24 2009

[7] A Caserta andD R Lj Kocinac ldquoOn statistical exhaustivenessrdquoApplied Mathematics Letters vol 25 no 10 pp 1447ndash1451 2012

6 The Scientific World Journal

[8] H Cakallı andMK Khan ldquoSummability in topological spacesrdquoApplied Mathematics Letters vol 24 pp 348ndash352 2011

[9] A Caserta G Di Maio and D R Lj Kocinac ldquoStatisticalconvergence in function spacesrdquo Applied Mathematics Lettersvol 2011 Article ID 420419 11 pages 2011

[10] J A Fridy and C Orhan ldquoLacunary statistical convergencerdquoPacific Journal of Mathematics vol 160 no 1 pp 43ndash51 1993

[11] H Cakallı ldquoLacunary statistical convergence in topologicalgroupsrdquo Indian Journal of Pure and Applied Mathematics vol26 no 2 pp 113ndash119 1995

[12] E Savas ldquoA study on absolute summability factors for atriangular matrixrdquo Mathematical Inequalities and Applicationsvol 12 no 1 pp 141ndash146 2009

[13] E Savas and F Nuray ldquoOn 120590-statistically convergence andlacunary 120590-statistically convergencerdquoMathematica Slovaca vol43 no 3 pp 309ndash315 1993

[14] F Dik M Dik and I Canak ldquoApplications of subsequentialTauberian theory to classical Tauberian theoryrdquo Applied Math-ematics Letters vol 20 no 8 pp 946ndash950 2007

[15] H Cakallı ldquoSlowly oscillating continuityrdquo Abstract and AppliedAnalysis vol 2008 Article ID 485706 5 pages 2008

[16] M Dik and I Canak ldquoNew types of continuitiesrdquo Abstract andApplied Analysis vol 2010 Article ID 258980 6 pages 2010

[17] N H Abel ldquoRecherches sur la srie 1 + (1198981) 119909 + (119898 (119898 minus 1) 12) 1199092+sdot sdot sdot rdquo Journal fur die Reine undAngewandteMathematikvol 1 pp 311ndash339 1826

[18] C V Stanojevic and V B Stanojevic ldquoTauberian retrievaltheoryrdquo Publications de lrsquoInstitut Mathematique Nouvelle Serievol 71 no 85 pp 105ndash111 2002

[19] J A Fridy and M K Khan ldquoStatistical extensions of someclassical tauberian theoremsrdquo Proceedings of the AmericanMathematical Society vol 128 no 8 pp 2347ndash2355 2000

[20] H Cakalli A Sonmez andG C Aras ldquo120582-statistically wardcon-tinuityrdquo Analele Stiintifice ale Universitatii Al I Cuza din IasiSerie Noua Matematica In press

[21] C T Rajagopal ldquoOn Tauberian theorems for Abel-Cesarosummabilityrdquo Proceedings of the Glasgow Mathematical Associ-ation vol 3 pp 176ndash181 1958

[22] A J Badiozzaman and B Thorpe ldquoSome best possible Taube-rian results for Abel and Cesaro summabilityrdquo Bulletin of theLondon Mathematical Society vol 28 no 3 pp 283ndash290 1996

[23] E C Posner ldquoSummability preserving functionsrdquo Proceedingsof the American Mathematical Society vol 12 pp 73ndash76 1961

[24] H Cakallı ldquoForward continuityrdquo Journal of ComputationalAnalysis and Applications vol 13 no 2 pp 225ndash230 2011

[25] H Cakallı ldquo120575-quasi-Cauchy sequencesrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 397ndash401 2011

[26] H Cakallı and P Das ldquoFuzzy compactness via summabilityrdquoAppliedMathematics Letters vol 22 no 11 pp 1665ndash1669 2009

[27] H Cakallı and E Savas ldquoStatistical convergence of doublesequences in topological groupsrdquo Journal of ComputationalAnalysis and Applications vol 12 no 2 pp 421ndash426 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

[8] H Cakallı andMK Khan ldquoSummability in topological spacesrdquoApplied Mathematics Letters vol 24 pp 348ndash352 2011

[9] A Caserta G Di Maio and D R Lj Kocinac ldquoStatisticalconvergence in function spacesrdquo Applied Mathematics Lettersvol 2011 Article ID 420419 11 pages 2011

[10] J A Fridy and C Orhan ldquoLacunary statistical convergencerdquoPacific Journal of Mathematics vol 160 no 1 pp 43ndash51 1993

[11] H Cakallı ldquoLacunary statistical convergence in topologicalgroupsrdquo Indian Journal of Pure and Applied Mathematics vol26 no 2 pp 113ndash119 1995

[12] E Savas ldquoA study on absolute summability factors for atriangular matrixrdquo Mathematical Inequalities and Applicationsvol 12 no 1 pp 141ndash146 2009

[13] E Savas and F Nuray ldquoOn 120590-statistically convergence andlacunary 120590-statistically convergencerdquoMathematica Slovaca vol43 no 3 pp 309ndash315 1993

[14] F Dik M Dik and I Canak ldquoApplications of subsequentialTauberian theory to classical Tauberian theoryrdquo Applied Math-ematics Letters vol 20 no 8 pp 946ndash950 2007

[15] H Cakallı ldquoSlowly oscillating continuityrdquo Abstract and AppliedAnalysis vol 2008 Article ID 485706 5 pages 2008

[16] M Dik and I Canak ldquoNew types of continuitiesrdquo Abstract andApplied Analysis vol 2010 Article ID 258980 6 pages 2010

[17] N H Abel ldquoRecherches sur la srie 1 + (1198981) 119909 + (119898 (119898 minus 1) 12) 1199092+sdot sdot sdot rdquo Journal fur die Reine undAngewandteMathematikvol 1 pp 311ndash339 1826

[18] C V Stanojevic and V B Stanojevic ldquoTauberian retrievaltheoryrdquo Publications de lrsquoInstitut Mathematique Nouvelle Serievol 71 no 85 pp 105ndash111 2002

[19] J A Fridy and M K Khan ldquoStatistical extensions of someclassical tauberian theoremsrdquo Proceedings of the AmericanMathematical Society vol 128 no 8 pp 2347ndash2355 2000

[20] H Cakalli A Sonmez andG C Aras ldquo120582-statistically wardcon-tinuityrdquo Analele Stiintifice ale Universitatii Al I Cuza din IasiSerie Noua Matematica In press

[21] C T Rajagopal ldquoOn Tauberian theorems for Abel-Cesarosummabilityrdquo Proceedings of the Glasgow Mathematical Associ-ation vol 3 pp 176ndash181 1958

[22] A J Badiozzaman and B Thorpe ldquoSome best possible Taube-rian results for Abel and Cesaro summabilityrdquo Bulletin of theLondon Mathematical Society vol 28 no 3 pp 283ndash290 1996

[23] E C Posner ldquoSummability preserving functionsrdquo Proceedingsof the American Mathematical Society vol 12 pp 73ndash76 1961

[24] H Cakallı ldquoForward continuityrdquo Journal of ComputationalAnalysis and Applications vol 13 no 2 pp 225ndash230 2011

[25] H Cakallı ldquo120575-quasi-Cauchy sequencesrdquo Mathematical andComputer Modelling vol 53 no 1-2 pp 397ndash401 2011

[26] H Cakallı and P Das ldquoFuzzy compactness via summabilityrdquoAppliedMathematics Letters vol 22 no 11 pp 1665ndash1669 2009

[27] H Cakallı and E Savas ldquoStatistical convergence of doublesequences in topological groupsrdquo Journal of ComputationalAnalysis and Applications vol 12 no 2 pp 421ndash426 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of