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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 101782, 11 pages

http://dx.doi.org/10.1155/2014/101782
Research Article

On Lacunary Mean Ideal Convergence in Generalized Random -Normed Spaces

1Department of Mathematics, Faculty of Science and Arts, King Abdulaziz University (KAU), P.O. Box 80200, Khulais 21589, Saudi Arabia

2Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt

3Department of Statistics, Faculty of Science, Sudan University of Science & Technology, Khartoum, Sudan

Received 6 January 2014; Accepted 17 March 2014; Published 24 April 2014

Academic Editor: Jianming Zhan

Copyright © 2014 Awad A. Bakery and Mustafa M. Mohammed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An ideal is a hereditary and additive family of subsets of positive integers . In this paper, we will introduce the concept of generalized random -normed space as an extension of random -normed space. Also, we study the concept of lacunary mean ( )-ideal convergence and -ideal Cauchy for sequences of complex numbers in the generalized random -norm. We introduce -limit points and -cluster points. Furthermore, Cauchy and -Cauchy sequences in this construction are given. Finally, we find relations among these concepts.

1. Introduction

The sets of natural numbers and complex numbers will be denoted by and , respectively. Fast [1] and Steinhaus [2] independently introduced the notion of statistical convergence for sequences of real numbers, which is a generalization of the concept of convergence. The concept of statistical convergence is a very valuable functional tool for studying the convergence problems of numerical sequences through the concept of density. Afterward, several generalizations and applications of this concept have been presented by different authors (see [36]). Kostyrko et al. [7] presented a generalization of the concept of statistical convergence with the help of an ideal of subsets of the set of natural numbers , and more is studied in [811]. This concept of ideal convergence plays a fundamental role not only in pure mathematics but also in other branches of science concerning mathematics, mainly in information theory, computer science, dynamical systems, geographic information systems, and population modelling. Menger [12] generalized the metric axioms by associating a distribution function with each pair of points of a set. This system is called a probabilistic metric space. By using the concept of Menger, Šerstnev [13] introduced the concept of probabilistic normed spaces. It provides an important area into which many essential results of linear normed spaces can be generalized; see [14]. Later, Alsina et al. [15] presented a new definition of probabilistic normed space which includes the definition of normed space which includes the definition of Šerstnev as a special case. The concept of ideal convergence for single and double sequences of real numbers in probabilistic normed space was introduced and studied by Mursaleen and Mohiuddine [16]. Mursaleen and Alotaibi [17] studied the notion of ideal convergence for single and double sequences in random 2-normed spaces, respectively. For more details and linked concept, we refer to [1826]. In [27, 28], Gähler introduced a gorgeous theory of 2-normed and -normed spaces in the 1960s; we have studied these subjects and constructed some sequence spaces defined by ideal convergence in -normed spaces [29, 30]. Another important alternative of statistical convergence is the notion of lacunary statistical convergence introduced by Fridy and Orhan [31]. Recently, Mohiuddine and Aiyub [4] studied lacunary statistical convergence by introducing the concept -statistical convergence in random 2-normed space. Their work can be considered as a particular generalization of the statistical convergence. In [32], Mursaleen and Mohiuddine generalized the idea of lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, and Debnath [33] investigated lacunary ideal convergence in intuitionistic fuzzy normed linear spaces. Also, lacunary statistically convergent double sequences in probabilistic normed space were studied by Mohiuddine and Savaş in [34]. Jebril and Dutta [35] introduced the concept of random -normed space. In this paper, we firstly give some basic definitions and properties of random -normed space in Section 2. In Section 3, we define a new and interesting notion of generalized random -normed spaces; convergent sequences in it are introduced and we provide some results on it. In Section 4, we study lacunary mean ( )-ideal convergence and -ideal Cauchy for sequences of complex numbers in the generalized random -norm. Finally, in Section 5, we introduce -limit points and -cluster points. Moreover, Cauchy and -Cauchy sequences in this framework are given, and we find relations among these concepts.

2. Definitions and Preliminaries

For the reader’s expediency, we restate some definitions and results that will be used in this paper.

The notion of statistical convergence depends on the density (asymptotic or natural) of subsets of .

Definition 1. A subset of is said to have natural density if where denotes the cardinality of the set .

Definition 2. A sequence is statistically convergent to if, for every , In this case, is called the statistical limit of the sequence .

Definition 3. A nonempty family of sets is said to be an ideal on if and only if (a) , (b)for each one has ,(c)for each and , implies .

Definition 4. An ideal is an admissible ideal if for each .

Definition 5. An ideal is said to be nontrivial if and .

Definition 6. A nonempty family of sets is said to be a filter on if and only if (a) ,(b)for each one has ,(c)for each and , implies .

For each ideal , there is a filter corresponding to ; that is, .

Example 7. If we take , then is a nontrivial admissible ideal of and the corresponding convergence coincides with the usual convergence.

Example 8. If we get , where denote the asymptotic density of the set , then is a nontrivial admissible ideal of and the corresponding convergence coincides with the statistical convergence.

Definition 9. A sequence is said to be -convergent to a real number if In this case, we write .

Definition 10. By a lacunary sequence , , where , one will mean an increasing sequence of nonnegative integers with as , . The intervals determined by will be denoted by .

Definition 11. A sequence is said to be lacunary ( )-statistically convergent to the number if, for every , one has The notion of lacunary ideal convergence of real sequences is introduced by Tripathy et al. [36], and Hazarika [37, 38] introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some properties.

Definition 12. Let be a nontrivial ideal. A sequence is said to be -summable to a number if, for every , the set

Definition 13. Let and let be a linear space over the field of dimension , where and is the field of real or complex numbers. A real valued function on satisfies the following four conditions:(1) if and only if are linearly dependent in ;(2) is invariant under permutation;(3) for any ;(4) is called an -norm on , and the pair is called an -normed space over the field .

Definition 14. A probability distribution function is a function that is nondecreasing, left continuous on such that and . The family of all probability distribution functions will be denoted by . The space is partially ordered by the usual pointwise ordering of functions and has both a maximal element and a minimal element ; these are given, respectively, by There is a natural topology on that is induced by the modified Lévy metric [39, 40]; that is, for all , where denote the condition Convergence with respect to this metric is equivalent to weak convergence of distribution functions; that is, in converges weakly to (written as ) if and only if converges to at every point of continuity of the limit function . Therefore, one has Moreover, the metric space is compact.

Definition 15. A binary operation is said to be a continuous -norm if the following conditions are satisfied: (1) is associative and commutative,(2) is continuous,(3) for all ,(4) whenever and for each .

Definition 16. A binary operation is said to be a continuous -conorm if the following conditions are satisfied: (1) is associative and commutative,(2) is continuous,(3) for all ,(4) whenever and for each .

Definition 17. Let be a linear space of dimension greater than one, a continuous -norm, and a mapping from into . If the following conditions are satisfied: (1) if and are linearly dependent,(2) for every and in ,(3) for every ; and ,(4) ,

then is called a random 2-norm on and is called a random 2-normed space.

Definition 18. Let be a linear space of dimension greater than one over a real field, a continuous -norm, and a mapping from into . If the following conditions are satisfied: (1) are linearly dependent,(2) is invariant under any permutation of ,(3) for every ; ,(4) ,

then is called a random -norm on and is called a random -normed space.

3. Generalized Random -Normed Space

Throughout the paper let be an admissible ideal of . By generalizing Definition 18, we obtain a new notion of generalized random -normed space as follows.

Definition 19. The five-tuple is said to be generalized random -normed linear space or in short GR NLS if is a linear space over the field of complex numbers , is a continuous -norm, is a continuous -conorm, and , are two mappings on into satisfying the following conditions for every and for each : (1) ,(2) ,(3) if and only if are linearly dependent,(4) for each ,(5) ,(6) is continuous,(7) is invariant under any permutation of ,(8) ,(9) if and only if are linearly dependent,(10) for each ,(11) ,(12) is continuous,(13) is invariant under any permutation of .

In this case, is called generalized random -norm on and we denote it by .

Example 20. Let be an -normed linear space. Put and for all , , and . Then, is GR NLS.

Proof. For all , we have the following.(1)Evidently, .(2)Visibly, .(3)And (4)While is invariant under any permutation of , then is invariant under any permutation of .(5)Consider (6)Suppose that, without loss of generality,

As a result, However, (7)Evidently, is continuous.(8) .(9)And (10)As is invariant under any permutation of , then is invariant under any permutation of .(11)Consider (12)Presume, without loss of generality, that

Currently, By (17), In the same way, (13)Clearly, is continuous.

Remark 21. Let be GR NLS. Since is a continuous -norm and is a continuous -conorm, the system -neighborhoods of (the null vector in ) with respect to is where defined a first countable Hausdorff topology on , called the -topology. Hence, the -topology can be completely specified by means of -convergence of sequences.

Definition 22. Let be GR NLS, and let and . The set is called open ball with center and radius with respect to .

Definition 23. Let be GR NLS. A sequence in is -convergent to with respect to the generalized random -norm if, for and every , there exists such that In this case, one writes .

Theorem 24. Let be an -normed linear space. Put and for all , , and . Then, for every sequence and nonzero , one has

Proof. Assume that . Then, for every and for every , there exists a positive integer such that and, therefore, for any given , which is the same as By letting , we have And since , then we have This means .

4. -Cauchy and Convergence in GR NLS

Remark 25. Let be GR NLS. Since is a continuous -norm and is a continuous -conorm, the system -neighborhoods of with respect to are where determines a first countable Hausdorff topology on , called the -topology. Thus, the -topology can be completely specified by means of -convergence of sequences.

Definition 26. Let be GR NLS, and let and . The set is called open ball with center and radius with respect to .

Definition 27. Let be GR NLS. A sequence in is -convergent to with respect to the generalized random -norm if, for and every , there exists such that In this case, one writes .

Definition 28. Let and let be GR NLS. A sequence of elements in is said to be -convergent to with respect to the generalized random -norm if, for every and , the set Then, one writes .

Example 29. Let be an -normed linear space; take and for all . For all and every , consider Then, is GR NLS. If we take , define a sequence as follows: Hence, for every and , we have So .

Definition 30. Let be GR NLS. A sequence in is said to be a Cauchy sequence with respect to the generalized random -norm if, for every and , there exists satisfying

Definition 31. Let be GR NLS. A sequence in is said to be an -Cauchy sequence with respect to the generalized random -norm if, for every and , there exists satisfying

Theorem 32. Let , let be GR NLS, and let be a sequence in ; then, for every and , one has the following: (1) ,(2) and ,(3) and ,(4) and ,(5) and .

The proof is easy, so it is omitted.

Theorem 33. Let be GR NLS and let be a sequence in . If exists, then it is unique.

Proof. Suppose that and with . Give and choose such that and . Then, for each , there exists such that Also, there exists such that Now, consider . Then, for , we find a such that Then, we get Since is arbitrary, we have for all . By using a similar technique, it can be proved that for all ; hence, .

Theorem 34. Let be GR NLS and let be a sequence in . Then, one has

Proof. Let , and, then, for all and given , there exists such that Since is an admissible ideal and we get . So .

Theorem 35. Let be GR NLS and let be a sequence in . If exists, then it is unique.

The proof follows by using Theorems 33 and 34.

Theorem 36. Let be GR NLS and let be a sequence in . If , then there exists a subsequence of such that .

Proof. Let . Then, for all and given , there exists such that Observably, for each , we can take an such that It follows that .

We create the following two results without proofs, since they can be easily recognized.

Theorem 37. Let be GR NLS. If a sequence in is Cauchy sequence with respect to the generalized random -norm , then it is -Cauchy sequence with respect to the same norm.

Theorem 38. Let be GR NLS. If a sequence in is Cauchy sequence with respect to the generalized random -norm , then there is a subsequence of which is ordinary Cauchy sequence with respect to the norm .

5. -Limit Point, -Cluster Point, and -Cauchy Sequence in GR NLS

Definition 39. Let be GR NLS, and if a sequence in , then one has the following. (1)An element is said to be -limit point of if there is a set (2)An element is said to be -cluster point of if, for every and , one has By we denote the set of all -limit points and the set of all -cluster points in , respectively.

Definition 40. Let be GR NLS. A sequence in is said to be -Cauchy sequence with respect to the generalized random -norm if (i)there exists a set such that ;(ii)the subsequence of is a Cauchy sequence with respect to the generalized random -norm .

Theorem 41. Let be GR NLS. For each sequence in , one has

Proof. Let ; then there exists a set such that , where and are as in Definition 39, satisfies . Thus, for every and , there exists such that Thus, we have Since is an admissible ideal, we have

Theorem 42. Let be GR NLS. For each sequence in , the set is closed set in with respect to the usual topology induced by the generalized random -norm .

Proof. Let . Take and . Then, there exists . Choose such that . We have Thus, and so . Hence, .

Theorem 43. Let be GR NLS and in . Then, the following statements are equivalent: (1) is a -limit point of ;(2)there exist two sequences and in such that where is the zero element in .

Proof. Let (1) hold; then, there exist sets and as in Definition 39 such that Define the sequences and as follows: Consider the case such that . Then, for each and , we get Thus, in this case,