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• # Artykuł - szczegóły

## Open Mathematics

2006 | 4 | 4 | 648-655

## σ-asymptotically lacunary statistical equivalent sequences

EN

### Abstrakty

EN
This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and σ-convergence. Let ϑ be a lacunary sequence; Two nonnegative sequences [x] and [y] are S σ,8-asymptotically equivalent of multiple L provided that for every ε > 0 $$\mathop {\lim }\limits_r \frac{1}{{h_r }}\left\{ {k \in I_r :\left| {\frac{{x_{\sigma ^k (m)} }}{{y_{\sigma ^k (m)} }} - L} \right| \geqslant \in } \right\} = 0$$ uniformly in m = 1, 2, 3, ..., (denoted by x $$\mathop \sim \limits^{S_{\sigma ,\theta } }$$ y) simply S σ,8-asymptotically equivalent, if L = 1. Using this definition we shall prove S σ,8-asymptotically equivalent analogues of Fridy and Orhan’s theorems in [5] and analogues results of Das and Patel in [1] shall also be presented.

EN

648-655

wydano
2006-12-01
online
2006-12-01

### Twórcy

autor
• Yüzüncü Yil University
autor
• University of North Florida

### Bibliografia

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• [5] J.A. Fridy and C. Orhan: “Lacunary statistical sonvergent”, Pacific J. Math., Vol. 160(1), (1993), pp. 43–51.
• [6] G.G. Lorentz: “A contribution to the theory of divergent sequences”, Acta. Math., Vol. 80, (1948), pp. 167–190. http://dx.doi.org/10.1007/BF02393648
• [7] Mursaleen: “Some new spaces of lacunary sequences and invariant means”, Ital. J. Pure Appl. Math., Vol. 11, (2002), pp. 175–181.
• [8] Mursaleen: “New invariant matrix methods of summability”, Quart. J. Math. Oxford, Vol. 34(2), (1983), pp. 133, 77–86.
• [9] M. Marouf: “Asymptotic equivalence and summability”, Int. J. Math. Math. Sci., Vol. 16(4), (1993), pp. 755–762. http://dx.doi.org/10.1155/S0161171293000948
• [10] R.F. Patterson: “On asymptotically statistically equivalent sequences”, Demonstratio Math., Vol. 36(1), (2003), pp. 149–153.
• [11] R.F. Patterson and E. Savaş: “On asymptotically lacunary statistically equivalent sequences”, (in press).
• [12] P. Schaefer: “Infinite matrices and invariant means”, Proc. Amer. Math. Soc., Vol. 36, (1972), pp. 104–110. http://dx.doi.org/10.2307/2039044