Studies on some fuzzy real valued sequence spaces and density of subsets of natural numbers

Abstract

Studies on sequence spaces are very old and these were further extended through summability theory. The summability theory originated from the attempts made by the mathematicians to give limits to the divergent sequences, on taking its transformations. O. Toeplitz was the first person to study the summability methods as a class of transformations of complex sequences by complex infinite matrices. It was followed by the works due to I. Schur, S. Mazur, W. Orlicz, K. Knopp, G. M. Petersen, S. Banach, B. Kuttner, G. Kothe and I.J. Maddox are few to be named. We the people are using mathematics directly or indirectly in our daily world. It is observed that all our concept (idea, expression) used in our natural language is not possible to express by classical set theory only. With a view to describe this concept more precisely the notion of fuzzy set was introduced in the mid nineteen hundred sixties by Zadeh [105]. The notion was applied to introduce the notion of fuzzy real numbers which helped in constructing different fuzzy real-valued sequence spaces. The fuzzy real-valued sequences were studied at the initial stage by Nuray and Savas [65], Tripathy and Nanda [100], Choudhury and Tripathy [8], Das; Das and Choudhury [16], Et; Altin and Altinok [22], Esi [19], Matloka [54] and many others. The studies on sequence spaces were extended by introducing the notion of statistical convergence of sequences. The notion of statistical convergence of sequences was introduced by Fast [26] and Schoenbeg [80] independently. Later on it was studied from sequence space point of view by Fridy [31], Salat [69], Connor [10], Kolk [47], Tripathy ([87], [95]) Rath and Tripathy ([68], [69]) and many others. The idea of statistical convergence of sequences depends on certain density of subsets of N, the set of natural numbers. Throughout wF, lF, cF, cF0, c-F, mF, c-F0 and mFo denote the class of all, bounded, convergent, null, statistically convergent, bounded statistically convergent, statistically null and bounded statistically null...