On ideal convergence in probabilistic normed spaces
An interesting generalization of statistical convergence is I -convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence , Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I -convergence, I *-convergence, I...
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| Vydáno v: | Mathematica Slovaca Ročník 62; číslo 1; s. 49 - 62 |
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01.02.2012
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| Abstract | An interesting generalization of statistical convergence is
I
-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.:
I-Convergence
, Real Anal. Exchange
26
(2000–2001), 669–686]. In this paper, we define and study the concept of
I
-convergence,
I
*-convergence,
I
-limit points and
I
-cluster points in probabilistic normed space. We discuss the relationship between
I
-convergence and
I
*-convergence, i.e. we show that
I
*-convergence implies the
I
-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general,
I
-convergence does not imply
I
*-convergence in probabilistic normed space. |
|---|---|
| AbstractList | An interesting generalization of statistical convergence is
I
-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.:
I-Convergence
, Real Anal. Exchange
26
(2000–2001), 669–686]. In this paper, we define and study the concept of
I
-convergence,
I
*-convergence,
I
-limit points and
I
-cluster points in probabilistic normed space. We discuss the relationship between
I
-convergence and
I
*-convergence, i.e. we show that
I
*-convergence implies the
I
-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general,
I
-convergence does not imply
I
*-convergence in probabilistic normed space. An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space. |
| Author | Mohiuddine, S. A. Mursaleen, M. |
| Author_xml | – sequence: 1 givenname: M. surname: Mursaleen fullname: Mursaleen, M. email: mursaleenm@gmail.com organization: Department of Mathematics, Aligarh Muslim University – sequence: 2 givenname: S. A. surname: Mohiuddine fullname: Mohiuddine, S. A. organization: Department of Mathematics Faculty of Science, King Abdulaziz University |
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| References | Balcerzak, Dems, Komisarski (CR2) 2007; 328 Alsina, Schweizer, Sklar (CR1) 1997; 208 Komisarski (CR13) 2008; 340 Fridy (CR7) 1985; 5 Dems (CR3) 2004; 30 Mursaleen, Edely (CR16) 2003; 288 Schweizer, Sklar (CR20) 1960; 10 Freedman, Sember, Raphael (CR6) 1978; 37 Mursaleen, Mohiuddine (CR17) 2009; 233 Mursaleen, Çakan, Mohiuddine, Savasş (CR15) 2010; 26 Fast (CR4) 1951; 2 Steinhaus (CR22) 1951; 2 Mursaleen, Mohiuddine, Edely (CR19) 2010; 59 Guillén, Lallena, Sempi (CR9) 1999; 232 Schweizer, Sklar (CR21) 1983 Karakus (CR11) 2007; 12 Frank (CR5) 1971; 34 Mursaleen, Mohiuddine (CR18) 2009; 41 Kostyrko, Šalát, Wilczyński (CR12) 2000; 26 Guillén, Sempi (CR10) 2003; 280 Fridy (CR8) 1993; 118 Menger (CR14) 1942; 28 |
| References_xml | – volume: 59 start-page: 603 year: 2010 end-page: 611 ident: CR19 article-title: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces publication-title: Comput.Math. Appl. doi: 10.1016/j.camwa.2009.11.002 – volume: 340 start-page: 770 year: 2008 end-page: 779 ident: CR13 article-title: Pointwise I-convergence and I*-convergence in measure of sequences of functions publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2007.09.016 – volume: 328 start-page: 715 year: 2007 end-page: 729 ident: CR2 article-title: Statistical convergence and ideal convergence for sequences of functions publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2006.05.040 – volume: 10 start-page: 313 year: 1960 end-page: 334 ident: CR20 article-title: Statistical metric spaces publication-title: Pacific J. Math. – volume: 232 start-page: 183 year: 1999 end-page: 196 ident: CR9 article-title: A study of boundedness in probabilistic normed spaces publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1998.6261 – volume: 2 start-page: 73 year: 1951 end-page: 74 ident: CR22 article-title: Surla convergence ordinaire et la convergence asymptotique publication-title: Colloq. Math. – volume: 208 start-page: 446 year: 1997 end-page: 452 ident: CR1 article-title: Continuity properties of probabilistic norms publication-title: J. Math. Anal. Appl. doi: 10.1006/jmaa.1997.5333 – volume: 34 start-page: 67 year: 1971 end-page: 81 ident: CR5 article-title: Probabilistic topological spaces publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(71)90158-2 – volume: 12 start-page: 11 year: 2007 end-page: 23 ident: CR11 article-title: Statistical convergence on probabilistic normed spaces publication-title: Math. Commun. – volume: 28 start-page: 535 year: 1942 end-page: 537 ident: CR14 article-title: Statistical metrics publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.28.12.535 – volume: 280 start-page: 9 year: 2003 end-page: 16 ident: CR10 article-title: Probabilistic norms and convergence of random variables publication-title: J. Math. Anal. Appl. doi: 10.1016/S0022-247X(02)00577-2 – volume: 2 start-page: 241 year: 1951 end-page: 244 ident: CR4 article-title: Surla convergence statistique publication-title: Colloq. Math. – year: 1983 ident: CR21 publication-title: Probabilistic Metric Spaces – volume: 233 start-page: 142 year: 2009 end-page: 149 ident: CR17 article-title: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2009.07.005 – volume: 118 start-page: 1187 year: 1993 end-page: 1192 ident: CR8 article-title: Statistical limit points publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1993-1181163-6 – volume: 26 start-page: 669 year: 2000 end-page: 686 ident: CR12 article-title: I-Convergence publication-title: Real Anal. Exchange – volume: 288 start-page: 223 year: 2003 end-page: 231 ident: CR16 article-title: Statistical convergence of double sequences publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2003.08.004 – volume: 5 start-page: 301 year: 1985 end-page: 313 ident: CR7 article-title: On statistical convergence publication-title: Analysis – volume: 26 start-page: 1 year: 2010 end-page: 14 ident: CR15 article-title: Generalized statistical convergence and core of double sequences publication-title: Acta Math. Sin. (Engl. Ser.) doi: 10.1007/s10114-010-9050-2 – volume: 41 start-page: 2414 year: 2009 end-page: 2421 ident: CR18 article-title: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces publication-title: Chaos Solitons Fractals doi: 10.1016/j.chaos.2008.09.018 – volume: 37 start-page: 508 year: 1978 end-page: 520 ident: CR6 article-title: Some Cesàro type summability spaces publication-title: Proc. London Math. Soc. (3) doi: 10.1112/plms/s3-37.3.508 – volume: 30 start-page: 123 year: 2004 end-page: 128 ident: CR3 article-title: On I-Cauchy sequences publication-title: Real Anal. Exchange |
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I
-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.:... An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.:... |
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| SubjectTerms | Algebra I-cluster points I-convergence I-limit points Mathematics Mathematics and Statistics Primary 40A05 probabilistic normed space Secondary 46A70 t-norm |
| Title | On ideal convergence in probabilistic normed spaces |
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