On Soft Normed Quasilinear Spaces

Abstract

In this study, we investigate some properties of soft quasi-sequences and present new results. We then study the completeness of soft normed quasilinear space and present an analog of convergence and boundness results of soft quasi sequences in soft normed quasilinear spaces. Moreover, we define regular and singular subspaces of soft quasilinear spaces and draw several conclusions related to these notions. Afterward, we provide examples of these results in soft normed quasilinear spaces generalizing well-known results in soft linear normed spaces. Additionally, we offer new results concerning soft quasi subspaces of soft normed quasilinear spaces. Finally, we discuss the need for further research.

Keywords

• Soft set
• soft quasilinear space
• soft normed quasilinear space
• soft quasi vector
• convergence

References

1. S. M. Aseev, \emph{Quasilinear Operators and Their Application in the Theory of Multivalued Mappings}, Proceedings of the Steklov Institute of Mathematics 2 (1986) 23{--}52.
2. Y. Y\i lmaz, S. \c{C}akan, \c{S}. Aytekin, \emph{Topological Quasilinear Spaces}, Abstract and Applied Analysis 2012 (2012) Article ID 951374 10 pages.
3. Y. Y\i lmaz, S. \c{C}akan, \emph{Normed Proper Quasilinear Spaces}, Journal of Nonlinear Sciences and Applications 8 (2015) 816{--}836.
4. S. \c{C}akan, \emph{Some New Results Related to Theory of Normed Quasilinear Spaces}, Doctoral Dissertation \.{I}n\"{o}n\"{u} \"{U}niversity (2016) Malatya.
5. H. Levent, Y. Yılmaz, \emph{Translation, Modulation and Dilation Systems Set-Valued Signal Processing}, Carpathian Mathematical Publications 10 (1) (2018) 143{--}164.
6. H. Levent, Y. Yılmaz, \emph{Hahn-Banach Extension Theorem for Interval-Valued Functions and Existence of Quasilinear Functionals}, New Trends in Mathematical Sciences 6 (2) (2018) 19{--}28.
7. H. Bozkurt, Y. Y\i lmaz, \emph{Some New Properties of Inner Product Quasilinear Spaces}, Bulletin of Mathematical Analysis and Applications 8 (1) (2016) 37{--}45.
8. H. Bozkurt, Y. Y\i lmaz, \emph{Some New Results on Inner Product Qusilinear Spaces}, Cogent Mathematics 2016 (3) (2016) Article ID 1194801 10 pages.

9. Y. Y\i lmaz, H. Bozkurt, S. \c{C}akan, \emph{On Orthonormal Sets in Inner Product Quasilinear Spaces}, Creative Mathematics and Informatics 25 (2) (2016) 237{--}247.
10. H. Bozkurt, Y. Y\i lmaz, \emph{New Inner Product Quasilinear Spaces on Interval Numbers}, Journal of Function Spaces 2016 (2016) Article ID 2619271 9 pages.
11. Y. Y\i lmaz, H. Bozkurt, H. Levent, \"{U}. \c{C}etinkaya, \emph{Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces}, Journal of Mathematics 2022 (2022) Article ID 2466817 15 pages.
12. H. Levent, Y. Y\i lmaz, \emph{Inner-Product Quasilinear Spaces with Applications in Signal Processing}, Advanced Studies: Euro-Tbilisi Mathematical Journal 14 (4) (2021) 125{--}146.
13. H. Levent, Y. Y\i lmaz, \emph{Analysis of Signals with Inexact Data by Using Interval Valued Functions}, The Journal of Analysis 30 (2022) 1635{--}1651.
14. D. Molodtsov, \emph{Soft Set--Theory First Results}, Computers and Mathematics with Applications 37 (4-5) (1999) 19{--}31.
15. P. K. Maji, R. Biswas, A. R. Roy, \emph{Soft Set Theory}, Computers and Mathematics with Applications 45 (2003) 555{--}562.
16. S. Das, S. K. Samanta, \emph{On Soft Metric Spaces}, Journal of Fuzzy Mathematics 21 (2013) 707{--}734.
17. S. Das, S. K. Samanta, \emph{Soft Real Sets, Soft Real Numbers and Their Properties}, Journal of Fuzzy Mathematics and Informatics 6 (2) (2012) 551{--}576.
18. S. Das, S. K. Samanta, \emph{Soft Metric}, Annals of Fuzzy Mathematics and Informatics 6 (1) (2013) 77{--}94.
19. S. Das, P. Majumdar, S. K. Samanta, \emph{On Soft Linear Spaces and Soft Normed Linear Spaces}, Annals of Fuzzy Mathematics and Informatics 9 (1) (2015) 91{--}109.
20. S. Das, S. K. Samanta, \emph{Soft Linear Operators in Soft Normed Linear Spaces}, Annals of Fuzzy Mathematics and Informatics 6 (2) (2013) 295{--}314.
21. S. Das, S. K. Samanta, \emph{On Soft Inner Product Spaces}, Annals of Fuzzy Mathematics and Informatics 6 (1) (2013) 151{--}170.
22. M. I. Yazar, T. Bilgin, S. Bayramov, \c{C}. G\"{u}nd\"{u}z, \emph{A New View on Soft Normed Spaces}, International Mathematical Forum, 9 (24) (2014) 1149{--}1159.
23. M. I. Yazar, \c{C}. G. Aras, S. Bayramov, \emph{Results on Hilbert Spaces}, TWMS Journal of Applied and Engineering 9 (1) (2019) 159{--}164.
24. H. Bozkurt, \emph{Soft Quasilinear Spaces and Soft Normed Quasilinear Spaces}, Ad\i yaman University Journal of Science 10 (2) 2020 506{--}523.
25. H. Bozkurt, M. \c{S}. G\"{o}nci, \emph{Soft Inner Product Quasilinear Spaces}, TWMS Journal of Applied and Engineering Mathematics (2022) (Accepted).
26. H. Bozkurt, \emph{Soft Quasilinear Operator}, Mathematical Sciences and Applications e-Notes 10 (2) 2022 82{--}92.