A Study on Rough \(\mathcal {I}\)-Deferred Statistical Convergence in Gradual Normed Linear Spaces

Abstract

In the present paper, we set forth with the new notion of rough \(\mathcal {I}\)-deferred statistical convergence of order \(\alpha (0<\alpha \le 1)\) in gradual normed linear spaces (GNLS). We prove some fundamental features and implication relations of this convergence method. Also, we put forward the notion of gradual rough \(\mathcal {I}\)-deferred statistical limit set of order \(\alpha \) and prove some of its properties such as closedness and convexity. We prove that the gradual rough \(\mathcal {I}\)-deferred statistical limit set also plays a crucial role in the gradually \(\mathcal {I}\)-deferred statistical boundedness of order \(\alpha \) of a sequence in a GNLS. We end up proving a necessary and sufficient condition for the rough \(\mathcal {I}\)-deferred statistical convergence of order \(\alpha (0<\alpha \le 1)\) of a sequence in GNLS. Significance of the work in a broad context: Summability theory and convergence of sequences have many applications in mathematical analysis. The study of the convergence of sequences in GNLS has made little progress and is still in its early stages. In this paper, we introduce rough \(\mathcal {I}-\)deferred statistical convergence of sequences in GNLS which provides a new direction to the researchers.