IJPAM: Volume 68, No. 1 (2011)

ON NON-STANDARD n-NORM ON SOME SEQUENCE SPACES

Hemen Dutta$^1$, B. Surender Reddy$^2$
$^1$ Department of Mathematics
Gauhati University
Kokrajhar Campus, Assam, INDIA
e-mail: [email protected]
$^2$Department of Mathematics
Post Graduate College of Science - PGCS, Saifabad
Osmania University
Hyderabad, 500004, AP, INDIA
e-mail: [email protected]


Abstract. In this paper, we construct some difference sequence spaces which we call the spaces of $\Delta_{(r)}^s$-convergent, $\Delta_{(r)}^s$-null and $\Delta_{(r)}^s$-bounded sequences with respect to $n$-norm on a real linear space $X$. We study these spaces by defining non-standard $n$-norm and $(n-r)$-norm for every $r=1,2\dots,n-1$. We show that under certain cases, convergence and completeness in the $n$-norm is equivalent to those in the $(n-r)$-norm. We also prove the fixed point theorem for these spaces, which are $n$-Banach spaces.

Received: October 22, 2009

AMS Subject Classification: 40A05, 46A45, 46B70

Key Words and Phrases: $n$-normed spaces, completeness, fixed point theorem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 68
Issue: 1