All Title Author
Keywords Abstract


On a New I-Convergent Double-Sequence Space

DOI: 10.1155/2013/126163

Full-Text   Cite this paper   Add to My Lib

Abstract:

The sequence space was introduced and studied by Mursaleen (1983). In this article we introduce the sequence space 2 and study some of its properties and inclusion relations. 1. Introduction and Preliminaries Let , , and be the sets of all natural, real, and complex numbers, respectively. We write showing the space of all real or complex sequences. Definition 1. A double sequence of complex numbers is defined as a function . We denote a double sequence as where the two subscripts run through the sequence of natural numbers independent of each other [1]. A number is called a double limit of a double sequence if for every there exists some such that (see??[2]). Let and denote the Banach spaces of bounded and convergent sequences, respectively, with norm . Let denote the space of sequences of bounded variation; that is, where is a Banach space normed by (see??[3]). Definition 2. Let be a mapping of the set of the positive integers into itself having no finite orbits. A continuous linear functional on is said to be an invariant mean or -mean if and only if(i) when the sequence has for all ;(ii) , where ;(iii) for all . In case is the translation mapping , a -mean is often called a Banach limit (see [4]), and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [5]). If , then . Then it can be shown that where , . Consider where denote the th iterate of at . The special case of (5) in which was given by Lorentz [5, Theorem 1], and that the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on . Theorem 3. A -mean extends the limit functional on in the sense that for all if and only if has no finite orbits; that is to say, if and only if, for all , , (see [3]) Put assuming that . A straight forward calculation shows (see [6]) that For any sequence , , and scalar , we have Definition 4. A sequence is of -bounded variation if and only if (i) converges uniformly in ;(ii) , which must exist, should take the same value for all . We denote by , the space of all sequences of -bounded variation (see [7]): Theorem 5. is a Banach space normed by (see [8]). Subsequently, invariant means have been studied by Ahmad and Mursaleen [9], Mursaleen et al. [3, 6, 8, 10–14], Raimi [15], Schaefer [16], Savas and Rhoades [17], Vakeel et al. [18–20], and many others [21–23]. For the first time, I-convergence was studied by Kostyrko et al. [24]. Later on, it was studied by ?alát et al. [25, 26], Tripathy and Hazarika [27], Ebadullah et al. [18–20, 28], and

References

[1]  A. K. Vakeel and S. Tabassum, “On some new double sequence spaces of invariant means defined by Orlicz functions,” Communications de la Faculté des Sciences de l'Université d'Ankara Séries A, vol. 60, no. 2, pp. 11–21, 2011.
[2]  E. D. Habil, “Double sequences and double series,” The Islamic University Journal, Series of Natural Studies and Engineering, vol. 14, pp. 1–32, 2006.
[3]  M. Mursaleen, “On some new invariant matrix methods of summability,” The Quarterly Journal of Mathematics, vol. 34, no. 133, pp. 77–86, 1983.
[4]  S. Banach, Theorie des Operations Lineaires, Warszawa, Poland, 1932.
[5]  G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167–190, 1948.
[6]  M. Mursaleen, “Matrix transformations between some new sequence spaces,” Houston Journal of Mathematics, vol. 9, no. 4, pp. 505–509, 1983.
[7]  V. A. Khan, “On a new sequence space defined by Orlicz functions,” Communications de la Faculté des Sciences de l'Université d'Ankara Séries A, vol. 57, no. 2, pp. 25–33, 2008.
[8]  M. Mursaleen and S. A. Mohiuddine, “Some new double sequence spaces of invariant means,” Glasnik Matemati?ki, vol. 45, no. 1, pp. 139–153, 2010.
[9]  Z. U. Ahmad and Mursaleen, “An application of Banach limits,” Proceedings of the American Mathematical Society, vol. 103, no. 1, pp. 244–246, 1988.
[10]  M. Mursaleen and A. Alotaibi, “On -convergence in random 2-normed spaces,” Mathematica Slovaca, vol. 61, no. 6, pp. 933–940, 2011.
[11]  M. Mursaleen, S. A. Mohiuddine, and O. H. H. Edely, “On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 603–611, 2010.
[12]  M. Mursaleen and S. A. Mohiuddine, “On ideal convergence of double sequences in probabilistic normed spaces,” Mathematical Reports, vol. 12, no. 4, pp. 359–371, 2010.
[13]  M. Mursaleen and S. A. Mohiuddine, “On ideal convergence in probabilistic normed spaces,” Mathematica Slovaca, vol. 62, no. 1, pp. 49–62, 2012.
[14]  M. Mursaleen, A. Alotaibi, and M. A. Alghamdi, “ -summability and -approximation through invariant mean,” Journal of Computational Analysis and Applications, vol. 14, no. 6, pp. 1049–1058, 2012.
[15]  R. A. Raimi, “Invariant means and invariant matrix methods of summability,” Duke Mathematical Journal, vol. 30, pp. 81–94, 1963.
[16]  P. Schaefer, “Infinite matrices and invariant means,” Proceedings of the American Mathematical Society, vol. 36, pp. 104–110, 1972.
[17]  E. Savas and B. E. Rhoades, “On some new sequence spaces of invariant means defined by Orlicz functions,” Mathematical Inequalities & Applications, vol. 5, no. 2, pp. 271–281, 2002.
[18]  A. K. Vakeel, K. Ebadullah, and S. Suantai, “On a new I-convergent sequence space,” Analysis, vol. 32, no. 3, pp. 199–208, 2012.
[19]  A. K. Vakeel and K. Ebadullah, “On some I-Convergent sequence spaces defined by a modullus function,” Theory and Applications of Mathematics and Computer Science, vol. 1, no. 2, pp. 22–30, 2011.
[20]  A. K. Vakeel and K. Ebadullah, “I-convergent difference sequence spaces defined by a sequence of moduli,” Journal of Mathematical and Computational Science, vol. 2, no. 2, pp. 265–273, 2012.
[21]  A. Komisarski, “Pointwise -convergence and -convergence in measure of sequences of functions,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 770–779, 2008.
[22]  V. Kumar, “On I and -convergence of double sequences,” Mathematical Communications, vol. 12, no. 2, pp. 171–181, 2007.
[23]  A. ?ahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,” Taiwanese Journal of Mathematics, vol. 11, no. 5, pp. 1477–1484, 2007.
[24]  P. Kostyrko, T. ?alát, and W. Wilczyński, “ -convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–685, 2000.
[25]  T. ?alát, B. C. Tripathy, and M. Ziman, “On some properties of -convergence,” Tatra Mountains Mathematical Publications, vol. 28, pp. 279–286, 2004.
[26]  T. ?alát, B. C. Tripathy, and M. Ziman, “On -convergence field,” Italian Journal of Pure and Applied Mathematics, no. 17, pp. 45–54, 2005.
[27]  B. C. Tripathy and B. Hazarika, “Paranorm -convergent sequence spaces,” Mathematica Slovaca, vol. 59, no. 4, pp. 485–494, 2009.
[28]  V. A. Khan and K. Ebadullah, “On a new difference sequence space of invariant means defined by Orlicz functions,” Bulletin of the Allahabad Mathematical Society, vol. 26, no. 2, pp. 259–272, 2011.
[29]  A. K. Vakeel and T. Sabiha, “On ideal convergent difference double sequence spaces in 2-normed spaces defined by Orlicz function,” JMI International Journal of Mathematical Sciences, vol. 1, no. 2, pp. 1–9, 2010.
[30]  P. Das, P. Kostyrko, W. Wilczyński, and P. Malik, “I and -convergence of double sequences,” Mathematica Slovaca, vol. 58, no. 5, pp. 605–620, 2008.
[31]  M. Gurdal and M. B. Huban, “On I-convergence of double sequences in the topology induced by random 2-norms,” Matemati?ki Vesnik, vol. 65, no. 3, pp. 1–13, 2013.
[32]  M. Gürdal and A. ?ahiner, “Extremal -limit points of double sequences,” Applied Mathematics E-Notes, vol. 8, pp. 131–137, 2008.
[33]  S. Roy, “Some new type of fuzzy I-convergent double difference sequence spaces,” International Journal of Soft Computing and Engineering, vol. 1, pp. 429–431, 2012.
[34]  I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 1970.
[35]  H. Nakano, “Concave modulars,” Journal of the Mathematical Society of Japan, vol. 5, pp. 29–49, 1953.

Full-Text

comments powered by Disqus

Contact Us

service@oalib.com

QQ:3279437679

微信:OALib Journal