I-WEIGHTED LACUNARY STATISTICAL tau-CONVERGENCE IN LOCALLY SOLID RIESZ SPACES

Yıl 2019, Cilt: 2 Sayı: 1, 22 - 31, 30.01.2019

Şükran Konca , Ergin Genç

https://doi.org/10.33773/jum.492457

Öz

An ideal $I$ is a family of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. In this  paper, we introduce the notions of ideal versions of weighted lacunary statistical $\tau$-convergence, statistical $\tau$-Cauchy, weighted lacunary $\tau$-boundedness of sequences in locally solid Riesz spaces endowed with the topology $\tau$. We also prove some topological results related to these concepts in locally solid Riesz space.

Anahtar Kelimeler

Ordered topological vector space; locally solid Riesz space; $I$-weighted lacunary statistical $\tau$-convergence; I-convergence

Kaynakça

• \bibitem{1} F. Riesz, Sur la decomposition des operations fonctionnelles lineaires. In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, pp. 143-148 (1929). \bibitem{2} H. Freudenthal, Teilweise geordnete Moduln, K. Akademie van Wetenschappen, Afdeeling Natuurkunde, Proceedings of the Section of Sciences 39, 647-657 (1936). \bibitem{3} L. V. Kantorovich, Concerning the general theory of operations in partially ordered spaces, Dok. Akad. Nauk. SSSR 1, 271-274 (1936). \bibitem{4} C. D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces with applications to economics, (No. 105). American Mathematical Soc., (2003). \bibitem{5} L. V. Kantorovich, Lineare halbgeordnete Raume, Rec. Math. 2, 121-168 (1937). \bibitem{6} W. A. Luxemburg and A. C. Zaanen, Riesz spaces. Vol. I, (1971). \bibitem{7} A. C. Zannen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, 1997. \bibitem{8} H. Fast, Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951). \bibitem{9} H. Steinhaus, Sur la convergence ordinate et la convergence asymptotique, Colloq. Math., 2, 73-84 (1951). \bibitem{14} H. Albayrak and S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topology and its Applications, 159 (7) , 1887-1893 (2012). \bibitem{15} V. Karakaya and T. A. Chishti, Weighted statistical convergence, Iran. J. Sci. Technol. Trans. A Sci. 33 (1), 219-223 (2009). \bibitem{17} M. Ba\c{s}ar{\i}r and \c{S}. Konca, On some spaces of lacunary convergent sequences derived by Nörlund-type mean and weighted lacunary statistical convergence, Arab Journal of Mathematical Sciences 20 (2), 250-263 (2014). \bibitem{18} M. Ba\c{s}ar{\i}r and \c{S}. Konca, "Weighted lacunary statistical convergence in locally solid Riesz spaces." Filomat 28 (10), 2059-2067 2014. \bibitem{19} P. Kostyrko, T. Salat, and W. Wilczynski, I-convergence,Real Analysis Exchange, 26 (2), 669-686 (2000-2001). \bibitem{20} B. Hazarika, Ideal convergence in locally solid Riesz spaces. Filomat, 28 (4), 797-809 (2014). \bibitem{21} B. K. Lahiri and P. Das, $I$ and $I^*$convergence in topological space, Mathematica Bohemica, 130 (2), 153-160 (2005). \bibitem{22} B. Hazarika, On ideal convergence in topological groups. Department of Mathematics Northwest University, 7 (4), 42-48 (2011). \bibitem{23} \c{S}. Konca ,E. Gen\c{c} and S. Ekin, Ideal version of weighted lacunary statistical convergence of sequences of order, Journal of Mathematical Analysis 7 (6), (2016). \bibitem{24} S. A. Mohiuddine, B. Hazarika and M. Mursaleen, Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces, Iranian Journal of Science and Technology 38. A1 61 (2014). \bibitem{28} S. A.Mohiuddine and M. A. Alghamdi, Statistical summability through a lacunary sequence in locally solid Riesz spaces. Journal of Inequalities and Applications, 2012 (1), 225 (2012).