A New Type Generalized Difference Sequence Space $m\left( \phi,p\right) \left( \Delta_{m}^{n}\right) $

Abstract

Let $\left( \phi_{n}\right) $ be a non-decreasing sequence of positive numbers such that $n\phi_{n+1}\leq \left( n+1\right) \phi_{n}$ for all $n\in \mathbb{N}$. The class of all sequences $\left( \phi_{n}\right) $ is denoted by $\Phi$. The sequence space $m\left( \phi \right) $ was introduced by Sargent [1] and he studied some of its properties and obtained some relations with the space $\ell_{p}$. Later on it was investigated by Tripathy and Sen [2] and Tripathy and Mahanta [3]. In this work, using the generalized difference operator $\Delta_{m}^{n}$, we generalize the sequence space $m\left( \phi \right) $ to sequence space $ m\left( \phi,p\right) \left( \Delta _{m}^{n}\right) ,$ give some topological properties about this space and show that the space $m\left( \phi,p\right) \left( \Delta_{m}^{n}\right) $ is a $BK-$space by a suitable norm$.$ The results obtained are generalizes some known results.

Keywords

• Difference sequence,BK-space,Symmetric space,Normal space.

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