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THE χγ2F − SUMMABLE SEQUENCES OF STRONGLY FUZZY NUMBERS

Year 2011, Volume 01, Issue 1, 98 - 108, 01.06.2011

Abstract

We introduce the classes of χ 2F λ A, p − summable sequences of strongly fuzzy numbers and give some relations between these classes. We also give a natural relationship between χ 2F λ − summable sequences of strongly fuzzy numbers and strongly χ 2F λ A − statistical convergence of sequences of fuzzy numbers.

References

  • Savas, E., (2000),A note on sequences of fuzzy numbers, Information Sciences, 124, 297-300.
  • Savas, E., (1996), A note on double sequences of fuzzy numbers, Turkish Journal of Mathematics, 20(2), 175-178.
  • Savas, E., (2008), On λ− statistically convergent double sequences of fuzzy numbers, Journal of
  • Inequalities and Applications, Article ID 147827, 6 pages, doi:10.1155/2008/147827.
  • Savas, E., (2000), On strongly λ− summable sequences of fuzzy numbers, Information Science, 125, 181-186.
  • Esi, A., (2011), On Some Double λ (∆, F ) − Statistical Convergence of Fuzzy numbers, Acta Univer
  • sittis Apulensis, 25, 99-104.
  • Nanda, S., (1989), On sequences of fuzzy numbers, Fuzzy Sets System, 33, 123-126.
  • Apostol, T., (1978), Mathematical Analysis, Addison-wesley, London.
  • Bromwich, T. J. I’A., (1965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York.
  • Basarir, M. and Solancan, O., (1999), On some double sequence spaces, J. Indian Acad. Math., 21(2), 193-200.
  • Bektas, C. and Altin, Y., (2003), The sequence space `M(p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4), 529-534.
  • Hardy, G. H., (1917), On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19, 86-95.
  • Krasnoselskii, M. A. and Rutickii, Y. B., (1961), Convex functions and Orlicz spaces, Gorningen, Netherlands.
  • Lindenstrauss, J. and Tzafriri, L., (1971), On Orlicz sequence spaces, Israel J. Math., 10, 379-390.
  • Maddox, I. J., (1986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1), 161-166.
  • Moricz, F., (1991), Extentions of the spaces c and c0from single to double sequences, Acta. Math. Hung., 57(1-2), 129-136.
  • Moricz, F. and Rhoades, B.E., (1988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, 283-294.
  • Mursaleen, M., (1999),Khan, M. A. and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, 145-150.
  • Nakano, H., (1953), Concave modulars, J. Math. Soc. Japan, 5, 29-49.
  • Orlicz, W., (1936), ¨Uber RaumeLM, Bull. Int. Acad. Polon. Sci. A, 93-107.
  • Parashar, S. D. and Choudhary, B., (1994), Sequence spaces defined by Orlicz functions, Indian J. ` ´ Pure Appl. Math., 25(4), 419-428.
  • Chandrasekhara Rao, K. and Subramanian, N., (2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, 3755-3764.
  • Ruckle, W. H., (1973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25, 973-978.
  • Tripathy, B. C., (2003), On statistically convergent double sequences, Tamkang J. Math., 34(3), 231- 237.
  • Tripathy, B. C., Et, M. and Altin, Y., (2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 1(3), 175-192.
  • Turkmenoglu, A., (1999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1), 23-31.
  • Kamthan, P. K. and Gupta, M., (1981), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
  • G¨okhan, A. and C¸ olak, R., (2004), The double sequence spaces cP(p) and cP B(p), Appl. Math. 22 Comput., 157(2), 491-501.
  • G¨okhan, A. and C¸ olak, R., (2005), Double sequence spaces `∞, ibid., 160(1), 147-153. 2
  • Zeltser, M., (2001), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu.
  • Mursaleen, M. and Edely, O. H. H., (2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), 223-231.
  • Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 523-531.
  • Mursaleen, M. and Edely, O. H. H., (2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 532-540.
  • Altay, B. and Basar, F., (2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), 70- 90.
  • Basar, F. and Sever, Y., (2009), The space Lpof double sequences, Math. J. Okayama Univ, 51, 149- 157.
  • Subramanian, N. and Misra, U. K., (2010), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46.
  • Kizmaz, H., (1981), On certain sequence spaces, Cand. Math. Bull., 24(2), 169-176.
  • Kuttner, B., (1946), Note on strong summability, J. London Math. Soc., 21, 118-122.
  • Maddox, I. J., (1979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 85(2), 345- 350.
  • Cannor, J., (1989), On strong matrix summability with respect to a modulus and statistical conver- gence, Canad. Math. Bull., 32(2), 194-198.
  • Pringsheim, A., (1900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, 289-321.
  • Hamilton, H. J., (1936), Transformations of multiple sequences, Duke Math. J., 2, 29-60.
  • ———-, (1938), A Generalization of multiple sequences transformation, Duke Math. J., 4, 343-358.
  • ———-, (1938), Change of Dimension in sequence transformation , Duke Math. J., 4, 341-342.
  • ———-, (1939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, 293-297.
  • Robison, G. M., (1926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, 50-73.
  • Silverman, L. L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series.
  • Toeplitz, O., (1911), ¨Uber allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne (war- saw), 22.
  • Basar, F. and Altay, B., (2003), On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J., 55(1), 136-147.
  • Altay, B. and Basar, F., (2007), The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p, (0 < p < 1), Commun. Math. Anal., 2(2), 1-11.
  • C¸ olak, R., Et, M. and Malkowsky, E., (2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 2004, pp. 1-63, Firat Univ. Press, ISBN: 975-394-0386-6.

Year 2011, Volume 01, Issue 1, 98 - 108, 01.06.2011

Abstract

References

  • Savas, E., (2000),A note on sequences of fuzzy numbers, Information Sciences, 124, 297-300.
  • Savas, E., (1996), A note on double sequences of fuzzy numbers, Turkish Journal of Mathematics, 20(2), 175-178.
  • Savas, E., (2008), On λ− statistically convergent double sequences of fuzzy numbers, Journal of
  • Inequalities and Applications, Article ID 147827, 6 pages, doi:10.1155/2008/147827.
  • Savas, E., (2000), On strongly λ− summable sequences of fuzzy numbers, Information Science, 125, 181-186.
  • Esi, A., (2011), On Some Double λ (∆, F ) − Statistical Convergence of Fuzzy numbers, Acta Univer
  • sittis Apulensis, 25, 99-104.
  • Nanda, S., (1989), On sequences of fuzzy numbers, Fuzzy Sets System, 33, 123-126.
  • Apostol, T., (1978), Mathematical Analysis, Addison-wesley, London.
  • Bromwich, T. J. I’A., (1965), An introduction to the theory of infinite series, Macmillan and Co.Ltd., New York.
  • Basarir, M. and Solancan, O., (1999), On some double sequence spaces, J. Indian Acad. Math., 21(2), 193-200.
  • Bektas, C. and Altin, Y., (2003), The sequence space `M(p, q, s) on seminormed spaces, Indian J. Pure Appl. Math., 34(4), 529-534.
  • Hardy, G. H., (1917), On the convergence of certain multiple series, Proc. Camb. Phil. Soc., 19, 86-95.
  • Krasnoselskii, M. A. and Rutickii, Y. B., (1961), Convex functions and Orlicz spaces, Gorningen, Netherlands.
  • Lindenstrauss, J. and Tzafriri, L., (1971), On Orlicz sequence spaces, Israel J. Math., 10, 379-390.
  • Maddox, I. J., (1986), Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc, 100(1), 161-166.
  • Moricz, F., (1991), Extentions of the spaces c and c0from single to double sequences, Acta. Math. Hung., 57(1-2), 129-136.
  • Moricz, F. and Rhoades, B.E., (1988), Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc., 104, 283-294.
  • Mursaleen, M., (1999),Khan, M. A. and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Math., Vol. XXXII, 145-150.
  • Nakano, H., (1953), Concave modulars, J. Math. Soc. Japan, 5, 29-49.
  • Orlicz, W., (1936), ¨Uber RaumeLM, Bull. Int. Acad. Polon. Sci. A, 93-107.
  • Parashar, S. D. and Choudhary, B., (1994), Sequence spaces defined by Orlicz functions, Indian J. ` ´ Pure Appl. Math., 25(4), 419-428.
  • Chandrasekhara Rao, K. and Subramanian, N., (2004), The Orlicz space of entire sequences, Int. J. Math. Math. Sci., 68, 3755-3764.
  • Ruckle, W. H., (1973), FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25, 973-978.
  • Tripathy, B. C., (2003), On statistically convergent double sequences, Tamkang J. Math., 34(3), 231- 237.
  • Tripathy, B. C., Et, M. and Altin, Y., (2003), Generalized difference sequence spaces defined by Orlicz function in a locally convex space, J. Anal. Appl., 1(3), 175-192.
  • Turkmenoglu, A., (1999), Matrix transformation between some classes of double sequences, J. Inst. Math. Comp. Sci. Math. Ser., 12(1), 23-31.
  • Kamthan, P. K. and Gupta, M., (1981), Sequence spaces and series, Lecture notes, Pure and Applied Mathematics, 65 Marcel Dekker, In c., New York.
  • G¨okhan, A. and C¸ olak, R., (2004), The double sequence spaces cP(p) and cP B(p), Appl. Math. 22 Comput., 157(2), 491-501.
  • G¨okhan, A. and C¸ olak, R., (2005), Double sequence spaces `∞, ibid., 160(1), 147-153. 2
  • Zeltser, M., (2001), Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu.
  • Mursaleen, M. and Edely, O. H. H., (2003), Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1), 223-231.
  • Mursaleen, M., (2004), Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 523-531.
  • Mursaleen, M. and Edely, O. H. H., (2004), Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl., 293(2), 532-540.
  • Altay, B. and Basar, F., (2005), Some new spaces of double sequences, J. Math. Anal. Appl., 309(1), 70- 90.
  • Basar, F. and Sever, Y., (2009), The space Lpof double sequences, Math. J. Okayama Univ, 51, 149- 157.
  • Subramanian, N. and Misra, U. K., (2010), The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46.
  • Kizmaz, H., (1981), On certain sequence spaces, Cand. Math. Bull., 24(2), 169-176.
  • Kuttner, B., (1946), Note on strong summability, J. London Math. Soc., 21, 118-122.
  • Maddox, I. J., (1979), On strong almost convergence, Math. Proc. Cambridge Philos. Soc., 85(2), 345- 350.
  • Cannor, J., (1989), On strong matrix summability with respect to a modulus and statistical conver- gence, Canad. Math. Bull., 32(2), 194-198.
  • Pringsheim, A., (1900), Zurtheorie derzweifach unendlichen zahlenfolgen, Math. Ann., 53, 289-321.
  • Hamilton, H. J., (1936), Transformations of multiple sequences, Duke Math. J., 2, 29-60.
  • ———-, (1938), A Generalization of multiple sequences transformation, Duke Math. J., 4, 343-358.
  • ———-, (1938), Change of Dimension in sequence transformation , Duke Math. J., 4, 341-342.
  • ———-, (1939), Preservation of partial Limits in Multiple sequence transformations, Duke Math. J., 4, 293-297.
  • Robison, G. M., (1926), Divergent double sequences and series, Amer. Math. Soc. Trans., 28, 50-73.
  • Silverman, L. L., On the definition of the sum of a divergent series, unpublished thesis, University of Missouri studies, Mathematics series.
  • Toeplitz, O., (1911), ¨Uber allgenmeine linear mittel bridungen, Prace Matemalyczno Fizyczne (war- saw), 22.
  • Basar, F. and Altay, B., (2003), On the space of sequences of p− bounded variation and related matrix mappings, Ukrainian Math. J., 55(1), 136-147.
  • Altay, B. and Basar, F., (2007), The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p, (0 < p < 1), Commun. Math. Anal., 2(2), 1-11.
  • C¸ olak, R., Et, M. and Malkowsky, E., (2004), Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Firat Univ. Elazig, Turkey, 2004, pp. 1-63, Firat Univ. Press, ISBN: 975-394-0386-6.

Details

Primary Language English
Journal Section Research Article
Authors

N. SUBRAMANİAN This is me
Department of Mathematics, SASTRA University, Thanjavur-613 401, India


U. K. MİSRA This is me
Department of Mathematics, Berhampur University, Berhampur-760 007,Odissa, India

Publication Date June 1, 2011
Published in Issue Year 2011, Volume 01, Issue 1

Cite

Bibtex @ { twmsjaem761815, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2011}, volume = {01}, number = {1}, pages = {98 - 108}, title = {THE χγ2F − SUMMABLE SEQUENCES OF STRONGLY FUZZY NUMBERS}, key = {cite}, author = {Subramanian, N. and Misra, U. K.} }