Some new classes of paranorm ideal convergent double sequences of sigma-bounded variation over n-normed spaces

Abstract

The sequence space $B{V}_{\sigma }$, the space of all sequence of $\sigma$-bounded variation, was firstly defined and studied by Mursaleen. Later on, Vakeel and Tabassum developed the same space to double sequences. Recently, using the concept of I-convergence, Vakeel and Vakeel et al. and others introduced many sequence spaces related to the space we just mentioned above which are defined by different operators. In this article, we keep the same direction up introducing some new classes of I-convergent double sequences of $\sigma$-bounded variation over n-normed spaces. In addition, we study some basic topological and algebraic properties of these classes. Also, we prove some inclusion relations on these classes.

Keywords:

• Invariant mean
• sequence of $\sigma$-bounded variation
• n-normed space
• paranormed space
• Orlicz function
• ideal
• filter
• I-convergence double sequence over n-normed space
• I-null over n-normed space
• I-Cauchy over n-normed space
• I-bounded over n-normed space
• solid space
• sequence algebra
• convergence free space

AMS Subject Classifications:

• 40 A05
• 40 A35
• 40 C05
• 46 A45
• 46 E30
• 46 B20

Public Interest Statement

The notion of ideal convergence is the most important development of the notion of usual convergence which played a big role for modelling uncertainty and vagueness in so many various problems in the field of science and engineering. On the other hand, the concept of n-normed spaces was developed, so that it became the focus of the researchers, interest. Quite recently, Vakeel Khan defined the space of all ideal convergent sequence of $\sigma$-bounded variation in space of real numbers. It was also obvious to define some new n-normed spaces of ideal convergent sequence of $\sigma$-bounded variation by using Orlicz function and study some topological and algebraic properties and some inclusion relations of these spaces which is our aim in this article.

Acknowledgements

I would like to thank the referees and the editor for their careful reading and their valuable comments.

Additional information

Funding

This work was supported by Department of Mathematics, Amman Arab University, Amman, Jordan.

Notes on contributors

Vakeel A. Khan

Vakeel. A. Khan received his MPhil and PhD degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently, he is an associate professor at Aligarh Muslim University, Aligarh, India. A vigorous researcher in the area of sequence spaces, he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylors and Francis), Information Sciences (Elsevier), Applied Mathematics Letters (Elsevier), A Journal of Chinese Universities (Springer- Verlag, China).

Kamal M.A.S. Alshlool

Kamal M.A.S. Alshlool received MSc, from Aligarh Muslim University, and is currently a PhD, scholar at Aligarh Muslim University.

Sameera A.A. Abdullah

Sameera A.A. Abdullah received MSc, from Aligarh Muslim University, and is currently a PhD scholar at Aligarh Muslim University..

Rami K.A. Rababah

Rami K.A. Rababah is working as an assistant professor in the Department of Mathematics, Amman Arab University, Jordan.

Ayaz Ahmad

Ayaz Ahmad is working as an assistant professor in the National Institute Technology, Patna, India.