Mursaleen, M.; Mohiuddine, S. A. Invariant mean and some core theorems for double sequences. (English) Zbl 1209.40003 Taiwanese J. Math. 14, No. 1, 21-33 (2010). A bounded double sequence \(x=(x_{jk})\) of real numbers is said to be \(\sigma-\)convergent to a number \(L\) if \(x\in V_2^\sigma\) with\[ V_{2}^{\sigma}= \bigg\{x: \lim_{p,q\to \infty}\frac{1}{(p+1)(q+1)}\sum_{j=0}^p\sum_{k=0}^q x_{\sigma^j(s),\sigma^k(t)}=L\text{ uniformly in }s,t;\;L=\sigma-\lim x\bigg\}, \]where \(\sigma^p(k)\) denotes the \(p\)th iterate of the mapping \(\sigma\) at \(k\), and \(\sigma^p(k)\neq k\) for all integer \(k\geq 0\), \(p\geq1\).In this paper the authors define and characterize the class \((V_2^{\sigma},V_2^{\sigma})\) and establish a core theorem. They determine a Tauberian condition for core inclusion and core equivalence. Reviewer: Umit Totur (Aydin) Cited in 6 Documents MSC: 40C05 Matrix methods for summability 40H05 Functional analytic methods in summability 40B05 Multiple sequences and series 40E05 Tauberian theorems Keywords:double sequence; \(p\)-convergence; invariant mean; \(\sigma\)-convergence; \(\sigma\)-core; core theorems PDFBibTeX XMLCite \textit{M. Mursaleen} and \textit{S. A. Mohiuddine}, Taiwanese J. Math. 14, No. 1, 21--33 (2010; Zbl 1209.40003) Full Text: DOI