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Wijsman asymptotic lacunary \(\mathcal {I}_2\)-invariant equivalence for double set sequences

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Abstract

In this study, for double set sequences, we present the notions of Wijsman asymptotic lacunary invariant equivalence, Wijsman asymptotic lacunary \(\mathcal {I}_2\)-invariant equivalence and Wijsman asymptotic lacunary \(\mathcal {I}_2^{*}\)-invariant equivalence. Also, we examine the relations between these notions and Wijsman asymptotic lacunary invariant statistical equivalence studied in this field before.

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References

  • Baronti M, Papini P (1986) Convergence of sequences of sets. In: Methods of Functional Analysis in Approximation Theory (pp.133–155). Birkhäuser, Basel

  • Beer G (1985) On convergence of closed sets in a metric space and distance functions. Bull. Aust. Math. Soc. 31(3):421–432

    Article  MathSciNet  MATH  Google Scholar 

  • Beer G (1994) Wijsman convergence: a survey. Set-Valued Anal. 2(1–2):77–94

    Article  MathSciNet  MATH  Google Scholar 

  • Çakan C, Altay B, Mursaleen M (2006) The -convergence and -core of double sequences. Appl. Math. Lett. 19:1122–1128

    Article  MathSciNet  MATH  Google Scholar 

  • Das P, Kostyrko P, Wilczyński W, Malik P (2008) \(\cal{I}\) and \(\cal{I}^*\)-convergence of double sequences. Math. Slovaca 58(5):605–620

    Article  MathSciNet  MATH  Google Scholar 

  • Dündar E, Altay B (2014) \(\cal{I}_2\)-convergence and \(\cal{I}_2\)-Cauchy of double sequences. Acta Math. Sci. 34B(2):343–353

    Article  MATH  Google Scholar 

  • Dündar E, Akın, Pancaroǧlu, Wijsman N (2020) lacunary ideal invariant convergence of double sequences of sets. Honam Math. J., 42(2), 345–358

  • Esi A (2009) On asymptotically double lacunary statistically equivalent sequences. Appl. Math. Lett. 22:1781–1785

    Article  MathSciNet  MATH  Google Scholar 

  • Et M, Şengül H (2016) On \((\Delta ^m,\cal{I})\)-lacunary statistical convergence of order \(\alpha \). J. Math. Anal. 7(5):78–84

    MathSciNet  MATH  Google Scholar 

  • Fridy JA, Orhan C (1993) Lacunary statistical convergence. Pacific J. Math. 160(1):43–51

    Article  MathSciNet  MATH  Google Scholar 

  • Hazarika B, Esi A, Braha NL (2013) On asympotically Wijsman lacunary -statistical convergence of set sequences. J. Math. Anal. 4(3):33–46

    MathSciNet  MATH  Google Scholar 

  • Hazarika B, Kumar V (2013) On asymptotically double lacunary statistical equivalent sequences in ideal context. J. Ineq. Appl., 2013(543), 15 pages

  • Hazarika B (2015) On asymptotically ideal equivalent sequences. J. Egyptian Math. Soc. 23(1):67–72

    Article  MathSciNet  MATH  Google Scholar 

  • Kara EE, Datan M, İlkhan M (2017) On lacunary ideal convergence of some sequences. New Trends Math. Sci. 5(1):234–242

    Article  Google Scholar 

  • Kişi, Ö, Nuray F (2013) New convergence definitions for sequences of sets. Abstr. Appl. Anal., 2013(Article ID 852796), 6 pages

  • Kişi, Ö, Nuray F (2013) \(S^L_{\lambda }({\cal{I}})\)-asymptotically statistical equivalence of sequence of sets. ISRN Math. Anal., 2013(Article ID 602963), 6 pages

  • Kostyrko P, Wilczyński W, Šalát T (2000) \(\cal{I}\)-convergence. Real Anal. Exchange 26(2):669–686

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar V (2007) On \(\cal{I}\) and \(\cal{I}^{*}\)-convergence of double sequences. Math. Commun. 12:171–181

    MathSciNet  Google Scholar 

  • Marouf M (1993) Asymptotic equivalence and summability. Int. J. Math. Math. Sci. 16(4):755–762

    Article  MathSciNet  MATH  Google Scholar 

  • Mursaleen M (1979) Invariant means and some matrix transformations. Tamkang J. Math. 10(2):183–188

    MathSciNet  MATH  Google Scholar 

  • Mursaleen M (1983) Matrix transformation between some new sequence spaces. Houston J. Math. 9(4):505–509

    MathSciNet  MATH  Google Scholar 

  • Mursaleen M, Edely OHH (2009) On the invariant mean and statistical convergence. Appl. Math. Lett. 22(11):1700–1704

    Article  MathSciNet  MATH  Google Scholar 

  • Nuray F, Rhoades BE (2012) Statistical convergence of sequences of sets. Fasc. Math. 49(2):87–99

    MathSciNet  MATH  Google Scholar 

  • Nuray F, Dündar E, Ulusu U (2014) Wijsman \(\cal{I}_2\)-convergence of double sequences of closed sets. Pure Appl. Math. Lett. 2:35–39

    Google Scholar 

  • Nuray F, Ulusu U, Dündar E (2016) Lacunary statistical convergence of double sequences of sets. Soft Comput. 20:2883–2888

    Article  MATH  Google Scholar 

  • Nuray F, Patterson RF, Dündar E (2016) Asymptotically lacunary statistical equivalence of double sequences of sets. Demonstratio Math. 49(2):183–196

    MathSciNet  MATH  Google Scholar 

  • Pancaroǧlu N, Nuray F (2013) On invariant statistically convergence and lacunary invariant statistically convergence of sequences of sets. Progress Appl. Math. 5(2):23–29

    Google Scholar 

  • Pancaroǧlu N, Nuray F (2013) Statistical lacunary invariant summability. Theoretical Math. Appl. 3(2):71–78

    MATH  Google Scholar 

  • Pancaroǧlu N, Nuray F, Savaş E (2013) On asymptotically lacunary invariant statistical equivalent set sequences. AIP Conf. Proc. 1558(1):780–781

    Article  Google Scholar 

  • Pancaroǧlu Akın N, Dündar E, Ulusu U, lacunary, Wijsman (2020) \(\cal{I}\)-invariant convergence of sequences of sets. Proc. Nat. Acad. Sci. India Sect. A. https://doi.org/10.1007/s40010-020-00694-w

  • Patterson RF (2002) Rates of convergence for double sequences. Southeast Asian Bull. Math. 26(3):469–478

    Article  MathSciNet  MATH  Google Scholar 

  • Patterson RF, Savaş E (2005) Lacunary statistical convergence of double sequences. Math. Commun. 10:55–61

    MathSciNet  MATH  Google Scholar 

  • Patterson RF, Savaş E (2006) On asymptotically lacunary statistically equivalent sequences. Thai J. Math. 4(2):267–272

    MathSciNet  MATH  Google Scholar 

  • Pringsheim A (1900) Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53:289–321

    Article  MathSciNet  MATH  Google Scholar 

  • Raj K, Anand R (2018) Double difference spaces of almost null and almost convergent sequences for Orlicz function. J. Comput. Anal. Appl. 24(4):773–783

    MathSciNet  Google Scholar 

  • Raj K, Choudhary A, Sharma C (2018) Almost strongly Orlicz double sequence spaces of regular matrices and their applications to statistical convergence. Asian-Eur. J. Math. 11(5):1850073

    Article  MathSciNet  MATH  Google Scholar 

  • Raj K, Jamwal S (2019) Difference sequence spaces for K-functions and its extension to double by a new matrix theoretic approach. Ann. Acad. Rom. Sci. Ser. Math. Appl. 11:5–23

    MathSciNet  MATH  Google Scholar 

  • Savaş E (1989) Some sequence spaces involving invariant means. Indian J. Math. 31:1–8

    MathSciNet  MATH  Google Scholar 

  • Savaş E (1990) On lacunary strong -convergence. Indian J. Pure Appl. Math. 21(4):359–365

    MathSciNet  MATH  Google Scholar 

  • Savaş E, Nuray F (1993) On -statistically convergence and lacunary -statistically convergence. Math. Slovaca. 43(3):309–315

    MathSciNet  MATH  Google Scholar 

  • Savaş E, Patterson RF (2006) -asymptotically lacunary statistical equivalent sequences. Cent. Eur. J. Math. 4(4):648–655

    Article  MathSciNet  MATH  Google Scholar 

  • Savaş E, Patterson RF (2009) Double -convergence lacunary statistical sequences. J. Comput. Anal. Appl. 11(4):610–615

    MathSciNet  MATH  Google Scholar 

  • Sever Y, Ulusu U, Dündar E (2014) On strongly \(\cal{I}\) and \(\cal{I}^*\)-lacunary convergence of sequences of sets. AIP Conf. Proc. 1611(1):357–362

    Article  Google Scholar 

  • Şengül H (2018) On Wijsman \(\cal{I}\)-lacunary statistical equivalence of order \((\eta,\mu )\). J. Ineq. Spec. Funct. 9(2):92–101

    MathSciNet  Google Scholar 

  • Şengül H, Et M, Işık M (2019) On \(\cal{I}\)-deferred statistical convergence of order \(\alpha \). Filomat 33(9):2833–2840

    Article  MathSciNet  Google Scholar 

  • Şengül, H., Çakallı, H. and Et, M. A variation on strongly ideal lacunary ward continuity. Bol. Soc. Parana. Mat. (3), 38(7) (2020), 99–108

  • Tortop Ş, Dündar E (2018) Wijsman \(\cal{I}_2\)-invariant convergence of double sequences of sets. J. Ineq. Spec. Funct. 9(4):90–100

    Google Scholar 

  • Ulusu U, Nuray F (2012) Lacunary statistical convergence of sequences of sets. Progress Appl. Math. 4(2):99–109

    MATH  Google Scholar 

  • Ulusu U, Nuray F (2013) On asymptotically lacunary statistical equivalent set sequences. Journal of Math., 2013(Article ID 310438), 5 pages

  • Ulusu U, Nuray F (2013) On strongly lacunary summability of sequences of sets. J. Appl. Math. Bio. 3(3):75–88

    MATH  Google Scholar 

  • Ulusu U, Savaş E (2014) An extension of asymptotically lacunary statistical equivalence set sequences. J. Ineq. Appl., 2014(134) , 8 pages

  • Ulusu U, Dündar E (2016) Asymptotically \(\cal{I}_2\)-lacunary statistical equivalence of double sequences of sets. J. Ineq. Spec. Funct. 7(2):44–56

    Google Scholar 

  • Ulusu U, Dündar E, Nuray F (2018) Lacunary \(\cal{I}_2\)-invariant convergence and some properties. Int. J. Anal. Appl. 16(3):317–327

    MATH  Google Scholar 

  • Ulusu U, Dündar E (2019) Asymptotically lacunary \(\cal{I}_2\)-invariant equivalence. J. Intell. Fuzzy Systems 36(1):467–472

    Article  Google Scholar 

  • Ulusu U, Dündar E, Pancaroǧlu Akın N. Lacunary invariant statistical equivalence for double set sequences. (in review)

  • Wijsman RA (1964) Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc. 70(1):186–188

    Article  MathSciNet  MATH  Google Scholar 

  • Wijsman RA (1966) Convergence of sequences of convex sets, cones and functions II. Trans. Amer. Math. Soc. 123(1):32–45

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Sivas Cumhuriyet University, 58140, Sivas, Turkey

    Uǧur Ulusu

  2. Afyon Kocatepe University, 03200, Afyonkarahisar, Turkey

    Erdinç Dündar & Nimet Pancaroǧlu Akın

Authors
  1. Uǧur Ulusu
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  2. Erdinç Dündar
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  3. Nimet Pancaroǧlu Akın
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Correspondence to Uǧur Ulusu.

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Ulusu, U., Dündar, E. & Akın, N.P. Wijsman asymptotic lacunary \(\mathcal {I}_2\)-invariant equivalence for double set sequences. Soft Comput 25, 13805–13811 (2021). https://doi.org/10.1007/s00500-021-06195-1

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