Riesz Triple Probabilisitic of Almost Lacunary Cesàro C 111 Statistical Convergence of Γ 3 Defined by Musielak Orlicz Function

Transcription

1 Available online at WSN 96 (2018) EISSN Riesz Triple Probabilisitic of Almost Lacunary Cesàro C 111 Statistical Convergence of Γ 3 Defined by Musielak Orlicz Function N. Subramanian 1, A. Esi 2, M. Aiyub 3 1 Department of Mathematics, SASTRA University, Thanjavur , India 2 Department of Mathematics, Adiyaman University, 02040, Adiyaman, Turkey 3 Department of Mathematics, College of Science University of Bahrain, P.O.Box Manam, Kingdom of Bahrain 1-3 address: nsmaths@yahoo.com, aesi23@hotmail.com, maiyub@uob.edu.bh ABSTRACT In this paper we study the concept of almost lacunary statistical Cesa'ro of Γ 3 over probabilistic p- metric spaces defined by Musielak Orlicz function. Since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, we feel that the concept of almost lacunary statistical Cesàro of Γ 3 over probabilistic p- metric spaces defined by Musielak-Orlicz function in a PP-space would provide a more general framework for the subject. Keywords: Analytic sequence, Orlicz function, triple sequences, entire sequence, Riesz space, statistical convergence 2010 Mathematics Subject Classification: 40F05, 40J05, 40G05 ( Received 16 March 2018; Accepted 30 March 2018; Date of Publication 31 March 2018 )

2 1. INTRODUCTION Throughout denote the classes of all, entire analytic scalar valued single sequences, respectively. We write for the set of all complex triple sequences where the set of positive integers. Then, is a linear space under the coordinate wise addition scalar multiplication. Some initial work on double series is found in Apostol [1] double sequence spaces is found in Hardy [2], Deepmala et al. [8,9] many others. The initial work on triple sequence spaces is found in Sahiner et al. [10], Esi [3] Esi et al. [4-7], Subramanian et al. [11], Shri Prakash et al. [12] many others. Let be a triple sequence of real or complex numbers. Then the series is called a triple series. Then the triple series is said to be convergent if only if the triple sequence is convergent, where. A sequence is said to be triple analytic if The vector space of all triple analytic sequence is usually denoted by is called triple entire sequence if. A sequence as A sequence is called triple entire sequence if as The triple entire sequences will be denoted by. Consider a triple sequence The section of the sequence is defined by for all [ ] with 1 in the position zero otherwise. The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [14] as follows { } -97-

3 for where for all Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by { } where: for all Consider the triple difference sequence space is defined as 2. DEFINITIONS AND PRELIMINARIES Definition An Orlicz function ([see [13]) is a function which is continuous, non-decreasing convex with for as If convexity of Orlicz function is replaced by then this function is called modulus function. Lindenstrauss Tzafriri ([15]) used the idea of Orlicz function to construct Orlicz sequence space. A sequence defined by { } is called the complementary function of a Musielak-Orlicz function. For a given Musielak- Orlicz function [see [16]] the Musielak-Orlicz sequence space is defined as follows { } where: is a convex modular defined by We consider equipped with the Luxemburg metric ( ) is an exteneded real number. -98-

4 2. 2. Definition 0 if A triple sequence of real numbers is called almost convergent to limit That is, the average value of taken over any rectangle { } tends to 0 as both to this convergence is uniform in Let denote the set of sequences with this property as [ ] Definition Let ( ) ( ) be sequences of positive numbers [ ] [ ] [ ] -99-

5 is given by: is called the Riesz mean of triple sequence If then the sequence is said to be Riesz convergent to 0. If is Riesz convergent to 0, then we write Definition The four dimensional matrix is said to be RH-regular if it maps every bounded convergent sequence into a convergent sequence with the same limit Definition The triple sequence {( )} is called triple lacunary if there exist three increasing sequences of integers such that as as as Let is determine by { } Using the notations of lacunary Fuzzy sequence Riesz mean for triple sequences. {( )} be a triple lacunary sequence be sequences of positive real numbers such that ( ] ( ] Clearly, If the Riesz transformation of triple sequences is RH-regular, as as ( ] as then {( )} {( )} is a triple lacunary sequence. If the assumptions as as as may be not enough to obtain the -100-

6 conditions as as as respectively. For any lacunary sequences ( ) are integers. Throughout the paper, we assume that as as as Let { such that } If we take for all then reduce to Let be a real vector space of dimension where A real valued function on satisfying the following four conditions: (i). if only if are linearly dependent, (ii). is invariant under permutation, (iii). (iv). is called the product metric. 3. ALMOST LACUNARY CESÀRO STATISTICAL CONVERGENCE OF PP TRIPLE SEQUENCE SPACES Let [ ] be a triple infinite matrix of real number for forming the sum (( ) ) (3.1) is called a triple sequence space of summable to the limit i.e., (( )) -101-

7 Define the means (( )) ((( )) ) We say that ( ) is statistically lacunary equivalent summable to if the sequence is statistically convergent to that is, It is denoted by Definition Let be sequences of positive numbers, A triple be a space. Then a triple sequence is said to statistically convergent to with respect to the probabilistic metric provided that for every ({ [ ( ) ] }) or equivalently [ ( ) ] In this case we write -102-

8 3. 2. Definition A triple be a space. The two non-negative sequences are said to be almost asymptotically statistical equivalent of multiple in space if for every ({ [ ( ) ] }) or equivalently In this case we write Definition A triple be a space be a lacunary sequence. The two non-negative sequences are said to be a almost asymptotically lacunary statistical equivalent of multiple in space if for every or equivalently ({ }) (3.2) In this case we write Lemma A triple be a space. Then for every the following statements are equivalent: (1) (2) ({ }) -103-

9 (3) ({ }) (4) 4 MAIN RESULTS Theorem Let be a Musielak Orlicz function a triple be a space. If two triple sequences are almost asympototically lacunary statistical equivalent of multiple with respect to the probabilistic metric then is unique sequence Proof: Assume that Then, for any For a given choose such that define the following set: { } Then, clearly so If is non-empty set,since then we have for all which implies since is arbitrary, we get This completes the proof Theorem Let be a Musielak Orlicz function a triple be a space. For any lacunary sequence if Proof: If then there exists a such that for all Let Now we have to prove Set -104-

10 Then by definition, for given there exists such that for all Let { } let be any positive integer with Then { } This completes the proof Theorem Let be a Musielak Orlicz function a triple be a space. For any lacunary sequence if Proof: If then there exists a such that for sufficiently large which implies Let then for every for sufficiently large we have -105-

11 Therefore This completes the proof Corollary Let be a Musielak Orlicz function a triple be a space. For any lacunary sequence with then Proof: The result clearly follows from Theorem 4.2 Theorem CONCLUSIONS We introduced the concept of almost lacunary statistical Cesa'ro of Γ 3 over probabilistic p- metric spaces defined by Musielak Orlicz function.the authors feel that this concept in a PP-space would provide a more general framework for the subject, since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, ACKNOWLEDGEMENT The first author third author wish to thank the Department of Science Technology, Government of India for the financial sanction towards this work under FIST program SR/FST/MSI-107/2015. References [1] T. Apostol, Mathematical Analysis, Addison-Wesley, London, [2] G.H. Hardy, On the convergence of certain multiple series, Proc. Camb. Phil. Soc. 19 (1917) [3] A. Esi, On some triple almost lacunary sequence spaces defined by Orlicz functions, Research Reviews: Discrete Mathematical Structures 1(2) (2014) [4] A. Esi M. Necdet Catalbas,Almost convergence of triple sequences. Global Journal of Mathematical Analysis, 2(1) (2014) [5] A. Esi E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space. Appl. Math. Inf. Sci. 9(5) (2015) [6] A. Esi N. Subramanian, On some triple sequence spaces of χ³, World Scientific News 95 (2018) [7] A. Esi, N. Subramaian A. Esi, On Triple sequence space of Bernstein operator of Rough I convergence pre-cauchy, Proyecciones Journal of Mathematics 36(4) (2017)

12 [8] Deepmala, N. Subramanian V.N. Mishra, Double almost in Riesz space, Southeast Asian Bulletin of Mathematics 35 (2016) [9] Deepmala, L.N. Mishra N. Subramanian, Characterization of some Lacunary convergence of order with metric defined by sequence of moduli Musielak, Appl. Math. Inf. Sci. Lett. 4(3) (2016). [10] A. Sahiner, M. Gurdal F.K. Duden, Triple sequences their statistical convergence, Selcuk J. Appl. Math. 8(2) (2007) [11] N. Subramanian A. Esi, Some New Semi-Normed Triple Sequence Spaces Defined By A Sequence Of Moduli, Journal of Analysis Number Theory 3 (2) (2015) [12] T.V.G. Shri Prakash, M. Chramouleeswaran N. Subramanian, Lacunary Triple sequence of Fibonacci numbers over probabilistic metric spaces. International Organization of Scientific Research, Vol. 12, Issue 1, Version IV (2016) [13] H. Nakano Concave modulars, Journal of the Mathematical society of Japan, 5, [14] H. Kizmaz On certain sequence spaces, Canadian Mathematical Bulletin 24(2) [15] J. Lindenstrauss L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971) [16] J. Musielak, Orlicz Spaces, Lectures Notes in Math. 1034, Springer-Verlag,