IDEAL CONVERGENCE OF DOUBLE SEQUENCES OF CLOSED SETS

Ozer Talo, Yurdal Sever

DOI Number
https://doi.org/10.22190/FUMI210121045T
First page
605
Last page
617

Abstract


In the present paper, we introduce the concepts of ideal inner and ideal outer limits which always exist even if empty sets for double sequences of closed sets in Pringsheim's sense. Next, we give some formulas for finding ideal inner and outer limits in a metric space. After then, we define Kuratowski ideal convergence of double sequences of closed sets by means of the ideal inner and ideal outer limits of a double sequence of closed sets. Additionally, we give some examples that our result is more general than the results obtained before.

Keywords

Double sequence of sets, ideal convergence, Kuratowski convergence.

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References


bibitem{Aubin}

{sc J. P. Aubin {rm and} H. Frankowska}: textit{Set-Valued Analysis}. Birkhauser, Boston, 1990.

bibitem{Beer}

{sc G. Beer}: textit{Topologies on closed and closed convex sets}. Kluwer Academic, Dordrecht, 1993.

bibitem{das 1}

{sc P. Das, P. Kostyrko, W. Wilczy'{n}ski {rm and} P. Malik}:

textit{$mathcal{I}$ and $mathcal{I}^{*}$-convergence of double sequences}. Math. Slovaca, textbf{58} (5) (2008), 605--620.

bibitem{das 2}

{sc P. Das {rm and} P. Malik}: textit{On extremal $mathcal{I}$-limit points of double sequences}. Tatra Mt. Math. Publ., textbf{40} (2008), 91--102.

bibitem{Dontchev}

{sc A. L. Dontchev {rm and} R. T. Rockafellar}:

textit{Implicit functions and solution mappings: A view from variational analysis}. Springer, 2009.

bibitem{dutta}

{sc H. L. Dutta {rm and} E. R. Billy}:

textit{Current topics in summability theory and applications}. Puchong: Springer Singapore, 2016.

bibitem{fast}

{sc H. Fast}: textit{Sur la convergence statistique}, Colloq. Math., textbf{2} (1951), 241--244.

bibitem{gurdal}

{sc M. G"{u}rdal {rm and} A. c{S}ahiner}:

textit{Extremal $mathcal{I}$-limit points of double sequences}. Applied Mathematics E-Notes, textbf{8} (2008), 131--137.

bibitem{hausdorff}

{sc F. Hausdorff}:

textit{ Mengenlehre}. Walter de Gruyter and Co., Berlin (1927).

bibitem{hazarika}

{sc B. Hazarika {rm and} A. Esi}:

textit{On asympotically Wijsman lacunary statistical convergence of set sequences in ideal context}. Filomat, textbf{31}(9) (2017), 2691--2703.

bibitem{hazarika2}

{sc B. Hazarika {rm and} A. Esi}:

textit{Lacunary ideal summability and its applications to approximation theorem}. The Journal of Analysis, textbf{27}(4) (2019), 997--1006.

bibitem{kosa}

{sc P. Kostyrko, T. u{S}al'{a}t {rm and} W. Wilczy'{n}ski}:

textit{$mathcal{I}$-convergence}. Real Analysis Exchange textbf{26}(2) (2000), 669--686.

bibitem{kos2}

{sc P. Kostyrko, M. Mav{c}aj T. u{S}al'{a}t {rm and} M. Sleziak}:

textit{$mathcal{I}$-convergence and extremal $mathcal{I}$-limit points}. Math. Slovaca textbf{55} (2005), 443--464.

bibitem{kumar}

{sc V. Kumar}: textit{On $mathcal{I}$ and $mathcal{I}^{*}$- convergence of double sequences}.

Mathematical communications textbf{12}(2) (2007), 171--181.

bibitem{Kuratowski}

{sc C. Kuratowski}: textit{Topologie, vol.I}. PWN, Warszawa, 1958.

bibitem{Lohne}

{sc A. L"ohne {rm and} C. Zalinescu}:

textit{On convergence of closed convex sets}. J. Math. Anal. Appl., textbf{319} (2006), 617--634.

bibitem{moricz}

{sc F. M'{o}ricz}: textit{Statistical convergence of multiple sequences}. Archiv der Mathematik, textbf{81}(1) (2003), 82--89.

bibitem{me}

{sc M. Mursaleen {rm and} O. H. H. Edely}:

textit{Statistical convergence of double sequences}. J. Math. Anal. Appl., textbf{288} (2003), 223--231.

bibitem{Nuray}

{sc F. Nuray {rm and} B. E. Rhoades}:

textit{Statistical convergence of sequences of sets}. Fasc. Math., textbf{49} (2012), 87--99.

bibitem{prn}

{sc A. Pringsheim}: textit{Elementare Theorie der unendliche Doppelreihen}. Sitsungs berichte der Math. Akad. der Wissenschafftenzu M"{u}nch. Ber.,

textbf{7} (1898), 101--153.

bibitem{Rockafellar}

{sc R. T. Rockafellar {rm and} R. J.-B. Wets}: textit{Variational Analysis}. Springer, Berlin, 1998.

bibitem{Salinetti}

{sc G. Salinetti {rm and} R. J.-B. Wets}:

textit{On the convergence of sequences of convex sets in finite dimensions}. SIAM Rev., textbf{21} (1979), 18--33.

bibitem{scho}

{sc I. J. Schoenberg}: textit{The integrability of certain functions and related summability methods}. Amer. Math. Monthly, textbf{66} (1959), 361--375.

bibitem{sever}

{sc Y. Sever, "O. Talo {rm and} B. Altay}:

textit{On convergence of double sequences of closed sets}. Contemp. Anal. Appl. Math., textbf{3}(1) (2015), 30--49.

bibitem{sever1}

{sc Y. Sever {rm and} "O. Talo}:

textit{On Statistical Convergence of Double Sequences of Closed Sets} Filomat, textbf{30}(3) (2016), 533--539.

bibitem{talo1}

{sc"O. Talo, Y. Sever {rm and} F. Bac sar}:

textit{On statistically convergent sequences of closed set}. Filomat, textbf{30}(6) (2016), 1497--1509.

bibitem{talo2}

{sc"O. Talo, {rm and} Y. Sever}:

textit{On Kuratowski $mathcal{I}$-convergence of sequences of closed sets}. Filomat, textbf{31}(4) (2017), 899--912.

bibitem{tripathy3}

{sc B. C. Tripathy}: textit{Statistically convergent double sequences}. Tamkang Jour. Math., textbf{34}(3) (2003), 231--237.

bibitem{tripathy}

{sc B. Tripathy {rm and} B. C. Tripathy}:

textit{On $mathcal{I}$-convergent double sequences}. Soochow Journal of Mathematics, textbf{31}(4) (2005), 549--560.

bibitem{tripathy2}

{sc B. C. Tripathy {rm and} B. Sarma}:

textit{Vector valued paranormed statistically convergent double sequence spaces}. Math. Slovaca, textbf{57}(2)(2007), 179--188.




DOI: https://doi.org/10.22190/FUMI210121045T

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