Meenakshi a, Saurabh Prajapati a, Vijay Kumar b

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1 Volume 119 No , ISSN: (on-line version) url: convergence in Rom normed spaces Meenakshi a, Saurabh Prajapati a, Vijay Kumar b a Department of Mathematics, Chigarh University, Gharuan, Mohali, Punjab, India b Department of Mathematics, Panipat institute of Engineering Technology, Panipat, Haryana, India chawlameenakshi7@gmail.com, saurabhprajapati234@gmail.com, vjy_kaushik@yahoo.com Abstract: This paper defines the notion of convergence ( statistical convergence of order for Double Sequences ) in Rom Normed Spaces. Some examples are given within through which the notion can be seen more generalized. Further we have defined the Cauchy Sequences in these spaces established the Cauchy Criterion for the same. Keywords: Statistical convergence, double sequences, rom 2-normed space. AMS Subject Classification: 40A35, 46A70, 46A99, 60B INTRODUCTION Fast [13] has introduced a generalized method of convergence baptized as Statistical convergence in This generalized convergence method is based on the concept of natural density of subsets of where represents the set of positive integers. For a set, the natural density of is symbolized as is defined as provided the limit exists, where denotes the characteristic function of. Here represents the cardinality of the set Definition 1.1 [13] A sequence provided that, for every, of numbers is said to be statistically convergent to a number In this case, we write. Let denotes the set of all statistically convergent sequences. 475

2 The pioneer research works of Šalát [30], Fridy [14] Connor [8] have stimulated researchers to work on summability theory their works lead many developments in this direction. One of the significant generalizations related to summability theory entitled as statistical convergence was given by Mursaleen [24] where he used a non-decreasing sequence Çolak [5] generalized the concept of statistical convergence of real numbers as "statistical convergence of order " for some. Çolak Bektas[6] further extended this concept as " statistical convergence of order " for some. More generalzations of statistical convergence followed their works which can be seen in [9-12], [31]. Definition 1.2 [5] A sequence of numbers is referred to as statistically convergent of order to a number provided that, for every, Here, we write. Let be the set of all statistically convergent sequences. Bromwich [4] Hardy [16] have originally introduced the Double sequences. Following them, numerous researchers including Móricz [21], Patterson [28], Belen Yildirim [3], Tripathy Sarma [34] Kumar Mursaleen [17] have expressed their keen interests in double sequences related convergence problems. Mursaleen Edely [27] Mursaleen [26] extended the ideas of statistical convergence statistical convergence for double sequences respectively obtained some analogous results. However, Çolak Altin [7] introduced statistical convergence of order for these kind of sequences. Later on Meenakshi [18] have outlined the notion of statistical convergence of order for double sequences. Definition 1.3 [29] A double sequence of real numbers is said to be convergent in Priengsheim's sense or convergent if for every there exists such that whenever. The number is called Priengsheim limit of we write. Definition 1.4 [27] A double sequence of real numbers is said to be statistically convergent to if for every In this case, we write denotes the set of all statistically convergent double sequences. Karl Menger [20] generalized the concept of a metric space entitled as statistical metric space or probabilistic metric space in Probabilistic metric spaces are considerable in the situations when we cannot find the accurate distance between two points, but only able to find the probabilities of this distance. In 1962, Serstnev [33] defined the concept of probabilistic normed space (concisely, PN-space), which is a significant family of probabilistic metric space. The norms of the vectors in PN-spaces are considered as probabiistic distribution functions. 476

3 Definition 1.5 A function is known a distribution function if it is non-decreasing left continuous with Also denotes the collection of all distribution functions on One of the Example of a distribution function is, defined as Definition 1.6 A triangular norm or a norm is a binary operation on closed interval [0, 1], which is continuous, commutative, associative, non-decreasing has 1 as a natural element. For example, the operations defined by on [0, 1] are all s. Definition 1.7 A triangular co-norm or briefly a which is continuous, commutative, associative, non- decreasing has is a binary operation on the closed interval as a neutral element. In 1963, S. G hler [15] presented an important idea of -norm on a vector space. Golet defined rom/ probabilistic 2-normed spaces via the concept of -norm of S. G hler[15]. Definition 1.8 [15]. Let be a real vector space of dimension where A 2-norm on is a function which satisfies the following conditions, (i) if only if are linearly independent vectors. (ii) for all (iii), where is real. (iv) The pair for all is then called 2-normed space. Definition 1.9 Let be a linear space of dimension greater than one, a triangle function,. Then is called a probabilistic norm a probabilistic normed space if the following conditions are satisfied. (i) if are linearly dependent, where denotes the value of at. (ii) if are linearly independent. (iii) for all. (iv) for every. (v) whenever, If condition (v) is swapped with (vi) for all ; then is called a rom normed space (concisely, space). Rom normed spaces have been an interesting area of research for many researchers (see [1], [19], [22], [23], [32]). The concept of statistical convergence has defined by Mursaleen [25] in rom normed space Alotaibi [2] has extended this notion for the space of double sequences. 477

4 Definition 1.10 Let be a R2N space. Then, a double sequence is said to be convergent to with respect to the probabilistic norm if, for every,, a positive integer such that for all. It is symbolized by Definition 1.11[2] Let ) be space. A double sequence = ( is defined as statistically convergent to in ) or simply convergent to if for every,, Or Or,, In this case, we write 2. MAIN RESULTS In this section, our purpose is to establish statistical convergence of double sequences of order in Rom 2- normed space. All through we take or else specified. We will specify as an ordered pair (a, b) as an ordered pair of (c, d). As well, we specify 1 1 Definition 2.1 Let ) be a space. A double sequence = ( is said to be statistically convergent of order in ) or simply convergent to if for every, non-zero, Or equivalently,, In this case, we write 478

5 The collection of all statistically convergent double sequences of order., the notion given in Definition 2.1 agrees with statistical con- Remark. For the particular choice of vergence of double sequences in these spaces. in Rom 2- normed space is symbolized as Theorem 2.1 Let ) be a space be given. For a double sequence = ( if, then must be unique. Proof. Suppose that, where. Given choose such that. For, express Since. Therefore for given, we obtain Now suppose. Then clearly, which follows that Let then we have Since was arbitrary, it gives that for each Hence. not for We next give an Example which shows that the notion defined in Definition 2.1 is well defined for Example 2.1 Consider consisting the norm, where for all. Let, where each. We define a sequence = ( as follows:, but For we have ; 479

6 Which implies for Also, for we have where This shows that the sequence = ( is statistically convergent of order to both which is impossible., Theorem 2.2 Let be two sequences in space be given. Then If be any scalar, then If If, then If Proof. It is easy so omitted. Theorem 2.3 Let be a space be given. For a sequence in if then however the converse need not be true in general. Proof. Since, for given there exist a positive integer such that. Also the set This shows that Hence We give an Example to show that the converse is not true in general. Example 2.4 Consider consisting the norm, where for all. Let, where each. We define a sequence = ( as follows: For we have This shows that Hence, = ( is not convergent in But if we consider 480

7 which implies for Hence for ; Theorem 2.4 For a sequence in, if only if there exists a set with such that Proof. Firstly Suppose that. Then for any, if we define Then, for In order to prove the result, it is sufficient to prove that over Let if possible is not convergent over Then for let This implies that Which leads to a contradiction to. Hence Conversely, suppose that there exist a subset of N such that such that So for given there exist a positive in- such that teger Since the set is contained in the set therefore,. Hence 481

8 Theorem 2.5 Let be a rom 2-normed space, where Then the inclusion is strict for some such that. Proof. Let for given, we have As so we have which immediately implies the inclusion. Next to show the strict relation of the inclusion we give the following Example. Example 2.6 Same as in Example, one can easily see that for but for Definition 2.7 Let ) be rom 2- normed space. A sequence ) is said to be statistically Cauchy of order if for every, there exists positive integers such that for all we have or equivalently Theorem 2.8 Let be given. A sequence ) is said to be statistically convergent of order if only if it is statistical Cauchy of order. Proof. Let ) be statistically convergent sequence of order For, choose such that. We define then ; As we are given that therefore or Let then`. 482

9 If we take, then it is sufficient to prove Let which provide. Let if possible, then we have. Also it can be easily seen that Which is impossible. Hence Conversely, suppose that ) is statistical Cauchy sequence of order but not statistical convergent of order Then for every, there exist positive integers such that Choose, such that. Let } Let, then Since,. Therefore,. which leads to a contradiction. Hence is statistically convergent of order. Conclusion In this study the concept of statistical convergence of order for Double Sequences has been developed in Rom Normed Spaces. This idea is more generalized than statistical convergence for double sequences in these spaces. For the choice this generalized notion coincides with statistical convergence for double sequences statistical Cauchy of order for Double sequences coincides with statistical Cauchy for Double sequences in Rom Normed Spaces. REFERENCES 1. M. Aldhaifallah, K. S. Nisar, H. M. Srivastava, M. Mursaleen, Statistical Convergence in Probabilistic Normed Spaces, Journal of Function Spaces, vol. 2017, Article ID , 7 pages, A. M. Alotaibi, On Statistical convergence of double sequences in Rom2-Normed Spaces, Journal of Inequalities Special Functions, 1(2), 2010, C. Belen, M. Yildirim, On generalized statistical convergence of double sequences via ideals, Ann. Univ. Ferrara, 58(2012), 11-20, DOI /s T.J.I'A. Bromwich, An Introduction to the Theory of Infinite Series, Macmillan, New York, NY, USA, R. Çolak, Statistical convergence of order alpha, Modern Methods in Analysis Its Applications, New Delhi, India, Anamaya Pub, (2010), R. Çolak, Ç. A. Bekta, statistical convergence of order alpha, Acta Mathematica Scientia, 31 B(3) (2011),

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