# zbMATH — the first resource for mathematics

Generalized strongly $$(V,\lambda)$$- summable sequences defined by Orlicz functions. (English) Zbl 1088.40002
In 1981 H. Kizmaz [Can. Math. Bull. 24, 169–176 (1981; Zbl 0454.46010)] introduced the notion of difference sequence spaces and this concept was generalized by M. Et and R. Çolak [Soochow J. Math. 21, 377–386 (1995; Zbl 0841.46006)] in 1995. In this paper the concepts of $$\kappa ^m$$-statistical convergence and strongly $$(V,\kappa)(\Delta ^m)$$-summable sequence with respect to an Orlicz function are introduced. The paper consists of three sections. The first section is an introduction and gives the necessary definitions which are used in the latter two sections. The second section deals with $$\kappa ^m$$-statistical convergence where some inclusion relations are given. Some topological properties of three sequence spaces defined by using an Orlicz function $$M$$ are introduced and examined in the third section. It is shown that if a sequence is strongly $$(V,\kappa)(\Delta ^m)$$-summable with respect to an Orlicz function, then it is $$S_{\kappa ^m}$$-statistically convergent.

##### MSC:
 40C05 Matrix methods for summability 46A45 Sequence spaces (including Köthe sequence spaces)
Full Text:
##### References:
 [1] BHARDWAJ V. K.-SINGH N.: Some sequence spaces defined by Orlicz functions. Demonstratio Math. 33 (2000), 571-582. · Zbl 0966.46002 [2] CONNOR J. S.: The statistical and strong p-Cesaro convergence of sequences. Analusis (Munich) 8 (1988), 47-63. · Zbl 0653.40001 [3] ET M.-COLAK R.: On some generalized difference sequence spaces. Soochow J. Math. 21 (1995), 377-386. · Zbl 0841.46006 [4] ET M.-NURAY F. : Am-Statistical convergence. Indian J. Pure Appl. Math. 32 (2001), 961-969. · Zbl 1028.46033 [5] FAST H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244. · Zbl 0044.33605 [6] FREEDMAN A. R.-SEMBER J. J.-RAPHAEL M.: Some Cesaro type summability spaces. Proc. London Math. Soc. 37 (1978), 508-520. · Zbl 0424.40008 [7] FRIDY J. A.: On statistical convergence. Analysis (Munich) 5 (1985), 301-313. · Zbl 0588.40001 [8] GÜNGÖR M.-ET M.-COLAK R.: $$\bigtriangledown_v^m$$-Statistical convergence and strongly Cesaro convergence of difference sequences of order $$m$$ defined by Orlicz functions. [9] KIZMAZ H.: On certain sequence spaces. Canad. Math. Bull. 24 (1981), 169-176. · Zbl 0454.46010 [10] KOLK E.: The statistical convergence in Banach spaces. Acta Comment. Univ. Tartu Math. 928 (1991), 41-52. [11] LEINDLER L.: Über die de la Vallee-Pousinsche Summierbarkeit Allgemeiner Orthogonalreihen. Acta Math. Acad. Sci. Hungar. 16 (1965), 375-387. · Zbl 0138.28802 [12] LINDENSTRAUSS J.-TZAFRIRI L.: On Orlicz sequence spaces. Israel J. Math. 10 (1971), 379-390. · Zbl 0227.46042 [13] MURSALEEN: $$\lambda$$-Statistical convergence. Math. Slovaca 50 (2000), 111-115. · Zbl 0953.40002 [14] PARASHAR S. D.-CHOUDHARY B.: Sequence spaces defined by Orlicz functions. Indian J. Pure Appl. Math. 25 (1994), 419-428. · Zbl 0802.46020 [15] ŠALÁT T.: On statisticaly convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. · Zbl 0437.40003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.