On Zweier Ideal Convergence of Double Sequences In Probabilistic Normed Spaces

Abstract

Background and Objectives: In 2007, M Sengönül defined the notion of zweier sequence spacesand  consisting of the sequences that are convergent and null convergent respectively. Lately motivated by the above two concepts, Khan and Nazneen proposed the zweier convergent sequence spaces [1]. In 2010, M. Mursaleen et al [2] introduced the idea of probabilistic normed linear space (PNS) and the ideal convergence in PNS. In this paper, we introduce a new type of sequence spaces denoted asand which are zweier convergent in PNS. We studied the algebraic and topological properties of these spaces and certain inclusion relations. Methodology: This stimulus to study some new sequence spaces i.e., zweier ideal convergence in probabilistic normed spaces. Moreover, some fundamental properties and inclusion relations between these said spaces and related to these notions. Results: In this section by combining zweier ideal convergence and probabilistic normed linear space, we define some new notions related to the notion of zweier ideal convergence in probabilistic norm linear spaces. Furthermore, we prove some important results  and inclusion relations on said spaces. Conclusion: In the present article, we have introduced new kind of sequence spaces using the concept of zweier sequence spaces in probabilistic normed space. We investigated the elementary properties of these spaces such as linearity, inclusion relations etc. These results will provide new tool to deal with the convergence problems in the field of science and engineering.