$(A)_ {\Delta}$ - double Sequence Spaces of fuzzy numbers via Orlicz Function

Document Type : Research Paper

Author

Istanbul Ticaret University, Department of Mathematics, Uskudar-Istanbul, Turkey

Abstract

The aim of this paper is to introduce and study a new concept of
strong double $(A)_ {\Delta}$-convergent sequence of
fuzzy numbers with respect to an Orlicz function and also some
properties of the resulting sequence spaces of fuzzy   numbers are
examined. In addition, we define the double
$(A,\Delta)$-statistical convergence of fuzzy  numbers and
establish some connections between the spaces of strong double
$(A)_ {\Delta}$-convergent sequence and  double $(A
,\Delta)$-statistical convergent sequence.

Keywords


\bibitem{DK}
P. Diomand  and P. Kloeden, \emph{Metric spaces of fuzzy sets},
Fuzzy Sets and Systems, \textbf{33} (1989), 123-126.
\bibitem{Fa}  H. Fast, \emph{Sur la convergence statistique}, Collog. Math., \textbf{2} (1951), 241-244.
\bibitem{FS} A. R. Freedman and J. J. Sember, \emph{Densities and summability}, Pacific J. Math., \textbf{95} (1981), 293-305.
\bibitem{Ham1} H. J. Hamilton, \emph{Transformations of multiple sequences}, Duke Math. J., \textbf{2} (1936), 29-60.
\bibitem{KR} M. A. Krasnoselskii and Y. B. Rutisky, \emph{Convex
function and Orlicz spaces}, Groningen, Netherlands, 1961.
\bibitem{mu1} M. Mursaleen and M. Basarir, \emph{$A-$statistical convergence of a sequence of fuzzy numbers}, Indian J. Pure appl. Math.,
\textbf{34} (2003), 1351-1357.
\bibitem{Na}
S. Nanda, \emph{On sequence of fuzzy numbers}, Fuzzy Sets and Systems,
 \textbf{33} (1989), 123-126.
\bibitem{NE}
F. Nuray and E. Savas, \emph{Statistical convergence of fuzzy numbers},
Math. Slovaca, \textbf{45} (1995), 269-273.
\bibitem{Nu}
F. Nuray, \emph{Lacunary statistical convergence of sequences of
fuzzy numbers}, Fuzzy Sets and Systems, \textbf{99} (1998), 353-355.
\bibitem{PC}
S. D. Parashar and B. Choudhary, \emph{Sequence spaces defined
by Orlicz functions}, Indian J. Pure appl. Math., \textbf{25} (1994), 419-428.
\bibitem{pringsheim} A. Pringsheim, \emph{Zur theorie der zweifach
unendlichen Zahlenfolgen}, Math. Ann., \textbf{53} (1900), 289-321.
\bibitem{robison} G. M. Robison, \emph{Divergent double sequences and series}, Amer. Math. Soc. Trans., \textbf{28} (1926), 50-73.
\bibitem{Es96}
 E. Sava\c{s}, \emph{A note on double sequence of fuzzy numbers}, Turk J. Math., \textbf{20} (1996), 175-178.
\bibitem{Es00}
 E. Sava\c{s}, \emph{A note on sequence of fuzzy numbers}, Information Sciences, \textbf{124} (2000), 297-300.
\bibitem{E00}
 E. Sava\c{s}, \emph{On strongly $\lambda-$summable sequences  of fuzzy numbers},
Information Sciences, \textbf{125} (2000), 181-186.
 \bibitem{Es01}
 E. Sava\c{s}, \emph{On statistically convergent sequence of fuzzy numbers}, Information Sciences, \textbf{137} (2001),
 272-282.
\bibitem{Es06}
 E. Sava\c{s}, \emph{Difference sequence spaces of fuzzy numbers}, J. Fuzzy
Math., \textbf{14} (2006), 967-975.
 \bibitem{Es08}
 E. Sava\c{s}, \emph{On lacunary statistical convergent double sequences of fuzzy numbers},
 Appl. Math. Lett., \textbf{21} (2008), 134-141.
 \bibitem{EM}
 E. Sava\c{s} and M. Mursaleen, \emph{On statistically convergent double sequence of fuzzy numbers}, Information Sciences, \textbf{162} (2004),
 183-192.
  \bibitem{EP}
 E. Sava\c{s} and R. F. Patterson, \emph{$(A_{\sigma})_{\Delta}$-double sequence spaces via Orlicz functions
 and double statistical convergence}, Iran. J. Sci. Technol. Trans. A, \textbf{31} (2007), 357-367.
  \bibitem{T007}
 B. C. Tripathy and A. J. Dutta, \emph{On fuzzy real-valued double sequence space}, Math. Comput. Modelling,
 \textbf{46} (2007), 1294-1299.
\bibitem{007}
  B. C. Tripathy and A. J. Dutta, \emph{Statistically convergent and Ces\`{a}ro summable double sequences of fuzzy real numbers}, Soochow J. Math.,
 \textbf{33} (2007), 835-848.
 \bibitem{009}
 W. Yao, \emph{On L-fuzzifying convergence spaces}, Iranian Journal of Fuzzy Systems, \textbf{6(1)} (2009), 63-80.
  \bibitem{65}
  A. Zadeh, \emph{Fuzzy Sets}, Information and Control, \textbf{8} (1965), 338-353.