Document Type : Research Paper

Authors

• A. Esi ¹
• M. Acikgoz ²

¹ Adiyaman University, Science and Art Faculty, Department of Mathematics, 02040, Adiyaman, Turkey

² Gaziantep University, Science and Art Faculty, Department of Mathematics, 27200, Gaziantep,Turkey

Abstract

This paper presents the following new definition which is a natural combination of the definition for asymptotically double equivalent, double statistically limit and double ${\lambda }^2$-sequences. The double sequence ${\lambda }^2=({\lambda }_{m,n})$ of positive real numbers tending to infinity such that
$${\lambda }_{m+1,n}\le {\lambda }_{m,n}+1,{\lambda }_{m,n+1}\le \lambda m,n+1,$$
$${\lambda }_{m,n}-{\lambda }_{m+1,n}\le {\lambda }_{m,n+1}{\lambda }_{m+1,n+1},{\lambda }_{1,1}=1,$$
and
$$I_{m,n}=\{(k,l):m-{\lambda }_{m,n}+1\le k\le m,n-{\lambda }_{m,n}+1\le l\le n.$$
For double ${\lambda }^2$-sequence; the two non-negative sequences $x=(x_{k,l})$ and $y=(y_{k,l})$ are said to be
${\lambda }^2$-asymptotically double statistical equivalent of multiple $L$ provided that for every $\epsilon >0$
$$P-\underset{m,n}{lim}\frac{1}{{\lambda }_{m,n}}|\{(k,l)\in I_{m,n}:|\frac{x_{k,l}}{y_{k,l}}-L\ge \epsilon \}|=0$$
(denoted by $x{\sim }^{S_{{\lambda }^2}^L}y$) and simply ${\lambda }^2$-asymptotically double statistical equivalent if $L=1$.

Keywords

• Pringsheim Limit Point
• P-convergent
• Double Statistical Convergence