On Some Properties of m  -Statistical Convergence in a Paranormed Space

Abstract

In this study, we introduce the concepts of strongly   m  ,p -Cesàro summability, m  -statistical Cauchy sequence and m  -statistical convergence in a paronormed space. We give some certain properties of these concepts and some inclusion relations between them. Fast [1] and Steinhaus [2] introduced the concept of statistical convergence for sequences of real numbers. Several authors studied this concept with related topics [3-5]. The asymptotic density of K N  is defined as,   n 1 (K) lim k n : k K  n     where K be a subset of the set of natural numbers N and denoted by  K. . indicates the cardinality of the enclosed set. A sequence xk  is called statistically covergent to L provided that  k  n 1 lim k n х L 0  n       for each  0 . It is denoted by lim k k st x L    . A sequence хk  is called statistically Cauchy sequence provided that there exist a number N N( )   such that

Keywords

• Statistical convergence
• statistical Cauchy
• paranormed space
• difference sequence

On Some Properties of m  -Statistical Convergence in a Paranormed Space

Abstract

In this study, we introduce the concepts of strongly  ($\Delta ^{m}$,p)-Cesàro summability,  $\Delta ^{m}-statistical Cauchy sequence and  $\Delta ^{m}-statistical convergence in a paronormed space. We give some certain properties of these concepts and some inclusion relations between them.

Keywords

• Statistical convergence
• statistical Cauchy
• paranormed space

References

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