Yeni $\Delta _{q}^{v}$-fark operatörü ve topolojik özellikleri

Year 2021, Volume: 23 Issue: 1, 141 - 150, 29.01.2021

Abdulkadir Karakaş Mahir Salih Abdulrahman Assafı

https://doi.org/10.25092/baunfbed.842141

Abstract

$\Delta_{q}^{v}$ fark operatörünü kullanarak $\Delta^{v}$’yi genişlettik. $l_{p} (\Delta_{q}^{v} )$ fark dizi uzayını oluşturduk ve bazı topolojik özelliklerini inceledik. Eğer $l_{p} (\Delta_{q}^{v} )$ uygun bir $\left\| . \right\|_{p,\Delta_{q}^{v} } $ normu verilirse bunun bir Banach uzayı olacağını gösterdik. Ayrıca $\left({l_{p} (\Delta_{q}^{v} ),\left\| . \right\|_{p,\Delta_{q}^{v} } } \right)$ ve $\left( {l_{p} ,\left\| . \right\|_{p} } \right)$ dizi uzaylarının lineer izometrik olduklarını gösterdik. Çalışmanın sonunda ise $l_{p}(\Delta _{q}^{v})\subset l_{p}\left( \cal M,\Delta _{q}^{v}\right) $ olduğu gösterildi. Orlicz fonksiyonlarının ailesi ${\cal M}$, $\Delta_{2}$ şartı ile örtüşmektedir.

Keywords

Fark dizi uzayları, izometrik dizi uzayları, dizi uzayları

New $\Delta _{q}^{v}$ -difference operator and topological features

Abstract

We extented $\Delta^{v}$ by using difference operator $\Delta_{q}^{v}$. We generated the difference sequence space $l_{p} (\Delta_{q}^{v} )$ and investigated some of their properties. We showed that, if $l_{p} (\Delta_{q}^{v} )$ is supplied with an proper norm $\left\| . \right\|_{p,\Delta_{q}^{v} } $ then it will be a Banach space. We further more showed that, the sequence spaces $\left({l_{p} (\Delta_{q}^{v} ),\left\| . \right\|_{p,\Delta_{q}^{v} } } \right)$ and $\left( {l_{p} ,\left\| . \right\|_{p} } \right)$ are linearly isometric. At the end of this studies, it was shown that $l_{p}(\Delta _{q}^{v})\subset l_{p}\left( M,\Delta _{q}^{v}\right) $. The family of the Orlicz functions ${\cal M}$ is coincides the $\Delta_{2}$ condition.

Keywords

Difference sequence spaces, isometric sequence spaces, sequence spaces

References

• Kizmaz, H., On certain sequence spaces, Canadian Mathematical Bulletin, 24, 169-176, (1981).
• Çolak, R. and Et, M., On some generalized difference sequence spaces and related matrix transformations, Hokkaido Mathematical Journal, 26, 3, 483-492, (1997).
• Karakaş, A., Altın, Y. Et, M., On some topological properties of a new type difference sequence spaces, Advancements In Mathematical Sciences, Proceedings of the International Conference on Advancements in Mathematical Sciences (AMS-2015), Fatih University, Antalya, 144, (2015).
• Karakaş, A., Altın,Y. and Çolak, R., On some topological properties of a new type difference sequence spaces, International Conference on Mathematics and Mathematics Education (ICMME-2016), Firat University, Elaziğ-Turkey, 167-168, (2016).
• Peralta, Isometry of a sequence space generated by a difference operator, International Mathematical Forum, 5, 42, pp. 2077-2083, (2010).
• Altay, B. and Polat, H., On some new Euler difference sequence spaces, Southeast Asian Bull. Mathematics., 30, 209-220, (2006).
• Altay, B., Başar, F., The matrix domain and the fine spectrum of the difference operator on the sequence space , Communications in Mathematics and Applications, 2, 2, 1-11, (2007).
• Qamaruddin and Saifi, A. H., Generalized difference sequence spaces defined by a sequence of Orlicz functions, Southeast Asian Bull. Mathematics, 29, 1125-1130, (2005).
• Kamthan, P. K., Gupta, M., Sequence Spaces and Series, Marcel Dekker Inc. Newyork, (1981).
• Krasnoselskii, M. A., and Rutickii, Y. B., Convex Functions and Orlicz Spaces, Groningen, Netherlands, (1961).
• Kamthan, P. K., Convex functions and their applications, Journal of Istanbul University Faculty of Science. A Series, 28, 71-78, (1963).
• Lindenstrauss, J. and Tzafriri, L., On Orlicz sequence spaces, Israel Journal. Mathematics, 10, 3, 379-390, (1971).