I−LACUNARY STATISTICAL CONVERGENCE OF ORDER β OF DIFFERENCE SEQUENCES OF FRACTIONAL ORDER
Abstract
In this paper, we introduce the concepts of ideal ∆α−lacunary statis- tical convergence of order β with the fractional order of α and ideal ∆α−lacunary strongly convergence of order β with the fractional order of α ( where 0 < β ≤ 1and α be a fractional order) and give some relations about these concepts.
Keywords
Full Text:
PDFReferences
bibitem{altinok}
{sc H. Alti nok, M. Et, M. {rm and} R. c{C}olak}: textit{Some remarks on
generalized sequence space of bounded variation of sequences of fuzzy numbers}.
Iran. J. Fuzzy Syst. {bf 11}(5) (2014), 39--46, 109.
bibitem{aral}
{sc N. D. Aral {rm and} M. Et}: textit{On lacunary statistical convergence of
order $beta$ of difference sequences of fractional order}. International
Conference of Mathematical Sciences, (ICMS 2019), Maltepe University,
Istanbul, Turkey.
bibitem{balarsingh1}
{sc P. Baliarsingh}: textit{Some new difference sequence spaces of
fractional order and their dual spaces}. Appl. Math. Comput. {bf 219}(18)
(2013), 9737--9742.
bibitem{balarsingh2}
{sc P. Baliarsingh, U. Kadak {rm and} M. Mursaleen}: textit{On
statistical convergence of difference sequences of fractional order and
related Korovkin type approximation theorems}. Quaest. Math. {bf41}(8)
(2018), 1117--1133.
bibitem {Mohiuddine3}
{sc N. L. Braha, H. M. Srivastava {rm and} S. A. Mohiuddine}:
textit{A Korovkin's type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallee Poussin mean}. Appl. Math. Comput. {bf228} (2014), 162--169.
bibitem {Mohiuddine1}
{sc C. Belen {rm and} S. A. Mohiuddine}:
textit{Generalized weighted statistical convergence and application}. Appl. Math. Comput. {bf219}(18) (2013), 9821--9826.
bibitem {Cakalli1}
{sc H. c{C}akalli}: textit{Lacunary statistical convergence in
topological groups}. Indian J. Pure Appl. Math. {bf26}(2) (1995), 113--119.
bibitem {Cakalli2}
{sc H. c{C}akalli, C. G. Aras {rm and} A. S"{o}nmez}: textit{Lacunary
statistical ward continuity}. AIP Conf. Proc. 1676, 020042 (2015); http://dx.doi.org/10.1063/1.4930468.
bibitem{Cakalli3}
{sc H. {c{C}akalli} {rm and} H. Kaplan}: textit{A variation on lacunary
statistical quasi Cauchy sequences}. Commun. Fac. Sci. Univ. Ank.
S'{e}r. A1 Math. Stat. {bf66}(2) (2017), 71--79.
bibitem {Caserta}
{sc A. Caserta, G. Di Maio {rm and} L. D. R. Kov{c}inac}:
textit{Statistical convergence in function spaces}. Abstr. Appl. Anal., 2011,
Art. ID 420419, 11 pp.
bibitem {Connor}
{sc J. S. Connor}: textit{The statistical and strong $p-$Cesaro
convergence of sequences}. Analysis {bf8} (1988), 47-63.
bibitem {cinar}
{sc M. c{C}i nar, M. Karakac{s} {rm and} M. Et}: textit{On pointwise and
uniform statistical convergence of order $alpha$ for sequences of functions}.
Fixed Point Theory And Applications, Article Number: 33, 2013.
bibitem {colak}
{sc R. c{C}olak}: textit{Statistical convergence of order $alpha$}.
Modern Methods in Analysis and Its Applications, New Delhi, India:
Anamaya Pub, (2010) 121--129.
bibitem {das}
{sc P. Das, E. Savac{s} {rm and} S. K. Ghosal}: textit{On
generalizations of certain summability methods using ideals}. Appl. Math.
Lett. {bf24}(9) (2011), 1509-1514.
bibitem {das2}
{sc P. Das {rm and} E. Savac{s}}: textit{On $I$-statistical and $I$-lacunary
statistical convergence of order $alpha$}. Bull. Iranian Math. Soc.
{bf40}(2) (2014), 459-472.
bibitem {et1}
{sc M. Et {rm and} R. c{C}olak}:textit{On some generalized difference
sequence spaces}. Soochow J. Math. {bf21}(4) (1995), 377-386.
bibitem {et2}
{sc M. Et {rm and} F. Nuray}: textit{$Delta^{m}-$Statistical convergence}.
Indian J. Pure appl. Math. {bf32}(6) (2001), 961 - 969.
bibitem {etsengul}
{sc M. Et {rm and} H. c{S}eng"{u}l}: textit{On $(Delta^{m},I)$-Lacunary
Statistical Convergence of order $alpha$}. J. Math. Analysis {bf7}(5) (2016), 78-84.
bibitem {EtSengul}
{sc M. Et {rm and} H. c{S}eng"{u}l}: textit{Some Cesaro-Type Summability
Spaces of Order alpha and Lacunary Statistical Convergence of Order $alpha$}.
Filomat {bf28}(8),1593-1602, 2014.
bibitem {et6}
{sc M. Et, R. c{C}olak {rm and} Y. Altin}: textit{Strongly almost summable
sequences of order $alpha$}. Kuwait J. Sci. {bf41}(2) (2014), 35--47.
bibitem {et3}
{sc M. Et, M. Karakac{s} {rm and} V. Karakaya}: textit{Some geometric
properties of a new difference sequence space defined by de la
Vall'{e}e-Poussin mean}. Appl. Math. Comput. {bf234} (2014), 237--244.
bibitem {et4}
{sc M. Et, M. Mursaleen {rm and} M. Ic{s}i k}: textit{On a class of fuzzy
sets defined by Orlicz functions}. Filomat {bf27}(5) (2013), 789--796.
bibitem {et5}
{sc M. Et, B. C. Tripathy {rm and} A. J. Dutta}: textit{On pointwise
statistical convergence of order $alpha$ of sequences of fuzzy mappings}.
Kuwait J. Sci. {bf41}(3) (2014), 17--30.
bibitem {Fast}
{sc H. Fast}: textit{Sur la convergence statistique}. Colloq.
Math. {bf2} (1951), 241-244.
bibitem {freedman}
{sc A. R. Freedman, J. J. Sember {rm and} M. Raphael}: textit{Some
Cesaro-type summability spaces}, Proc. Lond. Math. Soc. {bf37}(3) (1978), 508--520.
bibitem {Fridy}
{sc J. Fridy}: textit{On statistical convergence}. Analysis {bf5}
(1985), 301--313.
bibitem {fridyandorhan}
{sc J. Fridy {rm and} C. Orhan}: textit{Lacunary statistical
convergence}. Pacific J. Math. {bf160} (1993), 43--51.
bibitem {fridyorhan2}
{sc J. A. Fridy {rm and} C. Orhan}: textit{Lacunary statistical
summability}. J. Math. Anal. Appl. {bf173}(2) (1993), 497--504.
bibitem {isik1}
{sc M. Ic{s}i k {rm and} K. E. Akbac{s}}: textit{On $lambda-$statistical
convergence of order $alpha$ in probability}. J. Inequal. Spec. Funct. {bf8}(4)
(2017), 57--64.
bibitem {isik2}
{sc M. Ic{s}i k {rm and} K. E. Et}: textit{On lacunary statistical
convergence of order $alpha$ in probability}. AIP Conference Proceedings 1676,
(2015); doi: http://dx.doi.org/10.1063/1.4930471.
bibitem {isik3}
{sc M. Ic{s}i k {rm and} K. E. Akbac{s}}: textit{On Asymptotically Lacunary
Statistical Equivalent Sequences of Order $alpha$ in Probability}. ITM Web of
Conferences 13, 01024 (2017). DOI: 10.1051/itmconf/20171301024.
bibitem {kadak}
{sc U. Kadak}: textit{Generalized lacunary statistical difference
sequence spaces of fractional order}. Int. J. Math. Math. Sci. 2015,
Art. ID 984283, 6 pp.
bibitem {Mohiuddine2}
{sc U. Kadak {rm and} S. A. Mohiuddine}:textit
{Generalized statistically almost convergence based on the difference operator which includes the $(p, q)$-Gamma function and related approximation theorems}. Results Math. {bf73}(1) (2018) Article 9.
bibitem {karakas et al}
{sc M. Karakac{s}, M. Et {rm and} V. Karakaya}: textit{Some
geometric properties of a new difference sequence space involving lacunary
sequences}. Acta Math. Sci. Ser. B (Engl. Ed.) {bf33}(6) (2013), 1711--1720.
bibitem {kizmaz}
{sc H. Ki zmaz}: textit{On certain sequence spaces}. Canad.
Math. Bull. {bf24}(2) (1981), 169-176.
bibitem {kostyrko}
{sc P. Kostyrko, T. v{S}al'at {rm and} W. Wilczy'{n}ski}: textit{
$I$-convergence}. Real Anal. Exchange 26 (2000/2001), 669-686.
bibitem {kostyrko2}
{sc P. Kostyrko, M. Mav{c}aj, T. v{S}al'at {rm and} M. Sleziak}: textit
{$I$-convergence and extremal $I$-limit points}. Math. Slovaca
{bf55}(4) (2005), 443-464.
bibitem {Mohiuddine4}
{sc S. A. Mohiuddine {rm and} B. A. S. Alamri}: textit
{Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems}. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, {bf113}(3) (2019), 1955--1973.
bibitem {Mohiuddine5}
{sc S. A. Mohiuddine, A. Asiri {rm and} B. Hazarika}: textit
{Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems}. Int. J. Gen. Syst. {bf48}(5) (2019), 492--506.
bibitem {Mursaleen}
{sc M. Mursaleen}: textit{ $lambda$- statistical convergence}.
Math. Slovaca {bf50}(1) (2012), 111--115.
bibitem {Mohiuddine6}
{sc M. Mursaleen {rm and} S. A. Mohiuddine}: textit{
On ideal convergence in probabilistic normed spaces}. Math. Slovaca {bf62} (2012), 49--62.
bibitem {nayak}
{sc L. Nayak, M. Et {rm and} P. Baliarsingh}: textit{On certain generalized
weighted mean fractional difference sequence spaces}. Proc. Nat. Acad.
Sci. India Sect. A {bf89}(1) (2019), 163--170.
bibitem {salat}
{sc T. v{S}al'at, B. C. Tripathy {rm and} M. Ziman}: textit{On
$I$-convergence field}. Italian J. Pure Appl. Math. {bf17} (2005), 45--54.
bibitem {savas}
{sc E. Savac{s} {rm and} M. Et}: textit{On $(Delta_{lambda}^{m}%
,I)-$statistical convergence of order $alpha$}. Period. Math. Hungar.
{bf71}(2) (2015), 135--145.
bibitem {Schoenberg}
{sc I. J. Schoenberg}: textit{The integrability of certain functions
and related summability methods}. Amer. Math. Monthly {bf66} (1959), 361--375.
bibitem {srivastava}
{sc H. M. Srivastava {rm and} M. Et}: textit{Lacunary statistical
convergence and strongly lacunary summable functions of order $alpha$}.
Filomat {bf31}(6) (2017), 1573--1582.
bibitem {Salat}
{sc T. Salat}: textit{On statistically convergent sequences of real
numbers}. Math. Slovaca {bf30} (1980), 139--150.
bibitem {Steinhaus}
{sc H. Steinhaus}: textit{Sur la convergence ordinaire et la
convergence asymptotique}. Colloq. Math. {bf2} (1951), 73--74.
bibitem {sengulet}
{sc H. c{S}eng"{u}l {rm and} M. Et}: textit{On lacunary statistical
convergence of order $alpha$}. Acta Mathematica Scientia {bf34}(2) (2014), 473-482.
bibitem {sengulet2}
{sc H. c{S}eng"{u}l {rm and} M. Et}: textit{On $I-$lacunary statistical
convergence of order $alpha$ of sequences of sets}. Filomat {bf31}(8)
(2017), 2403--2412.
bibitem {tripathy}
{sc B. C. Tripathy {rm and} M. Et}:textit{On generalized difference
lacunary statistical convergence}. Studia Univ. Babec{s}-Bolyai Math.
{bf50}(1) (2005), 119--130.
bibitem {tripathy2}
{sc B. C. Tripathy, B. Hazarika {rm and} B. Choudhary}: textit {Lacunary
$I$-Convergent Sequences}. Kyungpook Math. J. {bf 52} (2012), 473--482.
bibitem {zygmund}
{sc A. Zygmund}: textit{Trigonometric Series}. Cambridge University
Press, Cambridge, UK, 1979.
DOI: https://doi.org/10.22190/FUMI200117004A
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)