## Abstract and Applied Analysis

### Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces

#### Abstract

We introduce some vector-valued sequence spaces defined by a Musielak-Orlicz function and the concepts of lacunary convergence and strong ($A$)-convergence, where $A=({a}_{ik})$ is an infinite matrix of complex numbers. We also make an effort to study some topological properties and some inclusion relations between these spaces.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 858504, 8 pages.

Dates
First available in Project Euclid: 7 October 2014

https://projecteuclid.org/euclid.aaa/1412687039

Digital Object Identifier
doi:10.1155/2014/858504

Mathematical Reviews number (MathSciNet)
MR3246362

Zentralblatt MATH identifier
07023208

#### Citation

Mursaleen, M.; Alotaibi, A.; Sharma, Sunil K. Some New Lacunary Strong Convergent Vector-Valued Sequence Spaces. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 858504, 8 pages. doi:10.1155/2014/858504. https://projecteuclid.org/euclid.aaa/1412687039

#### References

• J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,” Israel Journal of Mathematics, vol. 10, pp. 379–390, 1971.
• L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Polish Academy of Science, 1989.
• J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.
• A. Wilansky, Summability through Functional Analysis, vol. 85, North-Holland, Amsterdam, Netherlands, 1984.
• M. Mursaleen and A. K. Noman, “On some new sequence spaces of non absolute type related to the spaces l$_{p}$ and ${l}_{\infty }$ II,” Mathematical Communications, vol. 16, pp. 383–398, 2011.
• S. D. Parashar and B. Choudhary, “Sequence spaces defined by Orlicz functions,” Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 419–428, 1994.
• K. Raj and S. K. Sharma, “Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function,” Acta Universitatis Sapientiae: Mathematica, vol. 3, no. 1, pp. 97–109, 2011.
• K. Raj and S. K. Sharma, “Some generalized difference double sequence spaces defined by a sequence of Orlicz-functions,” Cubo, vol. 14, no. 3, pp. 167–189, 2012.
• K. Raj and S. K. Sharma, “Some multiplier sequence spaces defined by a Musielak-Orlicz function in $n$-normed spaces,” New Zealand Journal of Mathematics, vol. 42, pp. 45–56, 2012.
• A. Gökhan, M. Et, and M. Mursaleen, “Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 548–555, 2009.
• M. Gungor and M. Et, “${\Delta }^{T}$-strongly almost summable sequences defined by Orlicz functions,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 8, pp. 1141–1151, 2003.
• A. R. Freedman, J. J. Sember, and M. Raphael, “Some Cesàro-type summability spaces,” Proceedings of the London Mathematical Society, vol. 37, no. 3, pp. 508–520, 1978.
• T. Bilgin, “Lacunary strong $A$-convergence with respect to a modulus,” Studia Universitatis Babeş-Bolyai, vol. 46, no. 4, pp. 39–46, 2001.
• T. Bilgin, “Lacunary strong \emphA-convergence with respect to a sequence of modulus functions,” Applied Mathematics and Computation, vol. 151, no. 3, pp. 595–600, 2004.
• M. Mursaleen and A. K. Noman, “On some new sequence spaces of non-absolute type related to the spaces ${l}_{p}$ and ${l}_{\infty }$ I,” Filomat, vol. 25, no. 2, pp. 33–51, 2011.
• H. Fast, “Sur la convergence statistique,” Colloquium Mathematicae, vol. 2, pp. 241–244, 1951.
• I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, pp. 361–375, 1959.
• J. A. Fridy, “On statistical convergence,” Analysis, vol. 5, no. 4, pp. 301–313, 1985.
• J. S. Connor, “A topological and functional analytic approach to statistical convergence,” in Applied and Numerical Harmonic Analysis, vol. 8 of Analysis of Divergence, pp. 403–413, 1999.
• T. Šalát, “On statistically convergent sequences of real numbers,” Mathematica Slovaca, vol. 30, no. 2, pp. 139–150, 1980.
• M. Mursaleen and O. H. H. Edely, “Generalized statistical convergence,” Information Sciences, vol. 162, no. 3-4, pp. 287–294, 2004.
• M. Isk, “On statistical convergence of generalized difference sequences,” Soochow Journal of Mathematics, vol. 30, no. 2, pp. 197–205, 2004.
• S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,” Journal of Inequalities and Applications, vol. 2012, article 225, 2012.
• B. Hazarika, S. A. Mohiuddine, and M. Mursaleen, “Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces,” Iranian Journal of Science Technology, vol. 38, no. A1, pp. 61–68, 2014.
• E. Kolk, “The statistical convergence in Banach spaces,” Acta et Commentationes Universitatis Tartuensis, vol. 928, pp. 41–52, 1991.
• I. J. Maddox, “Statistical convergence in a locally convex space,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104, no. 1, pp. 141–145, 1988.
• A. Alotaibi and M. Mursaleen, “Statistical convergence in random paranormed space,” Journal of Computational Analysis and Applications, vol. 17, no. 2, pp. 297–304, 2014.
• S. A. Mohiuddine, K. Raj, and A. Alotaibi, “Some paranormed double difference sequence spaces for Orlicz functions and bounded-regular matrices,” Abstract and Applied Analysis, vol. 2014, Article ID 419064, 10 pages, 2014.
• S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 581–585, 2012. \endinput