Abstract

The main objective of the study was to understand the notion of Λ-convergence and to study the notion of probabilistic normed spaces. The study has also aimed to define the statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. The concepts of these approaches have been defined by some examples, which have demonstrated the concepts of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces. Previous studies have also been used to understand similar terminologies and notations for the extraction of outcomes.

1. Introduction

The notion of statistical metric spaces [1–3], called probabilistic metric spaces, was introduced by Menger [4]; it is an important generalization of metric spaces. The concept of probabilistic normed (PN) spaces [5] is a key generalization of the concept of normed spaces. The idea of statistical convergence for sequences of real numbers was introduced by Fast [3] and Steinhaus [6] individualistically. This concept has been studied by many researchers in different set-ups such as in normed linear spaces, in probabilistic normed spaces, in intuitionist fuzzy normed spaces, and in locally convex spaces. The theory of probabilistic metric spaces was studied by several authors (see, for details, [2, 7–10]). Karakus [11] studied the concept of statistical convergence in PN-spaces. It was subsequently carried out in the works due to [12–14]. To define the concept of Λ-convergence, statistical Λ-convergence, and statistical Λ-Cauchy in PN-spaces, the subsequent definitions are needed. The terminology and notations used in this paper are standard as in the recent works [1, 4, 15, 16].

Definition 1. Let be the subset of set of . The natural density, , is characterized bywhere is the cardinality of the enclosed set. A number sequence is said to be statistically convergent to the number if, for each , the sethas natural density zero; that is,and it can be written as stat .

Definition 2. A function is a distribution function if it is nondecreasing and left-continuous withHere is the set of all distribution functions such that . If , then , whereclearly for all .
A -norm is a continuous mapping such that is an Abelian monoid with unit one and if and for all .

Definition 3. Let be a linear space of dimension ≥ 1. Suppose also that is a -norm and . Then is said to be a probabilistic norm and is referred to as a probabilistic normed space if it satisfies the following:(i) if and are linearly dependent, where is the value of at .(ii) for all if and only if .(iii).(iv) for all and Recently, in [17–19], some λ-sequence spaces were introduced and studied. In this paper, let be a strictly increasing sequence of positive real numbers tending to infinity; that is,In this case a sequence is -convergent to the number , which is known as -limit of , if as , whereHence it says that is λ-convergent to the number if and only if the sequence is convergent to . Here (and in the sequel), it will take the convention that any term with a negative subscript is equal to zero; for example, and  .
The sets of all Λ-bounded, Λ-convergent, and Λ-null sequences ,  , and , respectively, are defined as follows:In the present investigation, it is proposed to systematically study the idea of Λ-convergence in PN-spaces. In particular, statistical Λ-convergence and statistical Λ-Cauchy were investigated in PN-spaces and give some illustrative examples to demonstrate these concepts.

Definition 4. Let be a PN-space. A sequence is called convergent in or, simply, -convergent to if, for every and , there exists a positive integer whenever and it is written asand ζ is limit of .

Definition 5. Let be a PN-space. A sequence is called statistically convergent in or, simply,   -convergent to if, for and ,or equivalentlyand this case is stated byand is limit of .

Definition 6. Let be a PN-space. A sequence is known to be statistically Cauchy in or, simply, (stat)-Cauchy if, for , there exists a number ,In addition to the above definitions, the following definitions are given.

Definition 7. A sequence is called statistically Λ-convergent to the number if, for each , the set given byhas asymptotic density zero; that is,

Definition 8. A sequence is known to be statistically -Cauchy sequence if, for every , there exists a number ,Based on the previous definitions, the concept of Λ-convergence, statistical Λ-convergence, and statistical Λ-Cauchy in PN-spaces is defined.

Definition 9. Let be a PN-space. A sequence is said to be convergent in or, simply, -convergent to if, for and , there exists a positive integer whenever . In this case can be written and is called limit of the sequence .

Definition 10. Let be a PN-space. A sequence is known to be statistically Λ-convergent in or, simply, (stat)-convergent if, for every and ,Or,that is, sinceand is called (stat) limit of the sequence .

Definition 11. Let be a PN-space. A sequence is called statistically Λ-Cauchy in or, simply, (stat)-Cauchy to if, for , there exists a number ,

2. Main Results

By making use of the definitions given in the preceding section, it is proposed here to systematically investigate the notion of statistical Λ-convergence and statistical Λ-Cauchy in PN-spaces and apply our findings to the problem of approximating positive linear operators.

Theorem 12. Let be a PN-space. If a sequence is (stat)-convergent, then the (stat) limit is unique.

Proof. Suppose thatFor a given , let ,Then, for any ,sincefor . Furthermore, by usingit hasfor all .
LetClearly,which implies thatFor , by using (2) and (3), Since is arbitrary, by using (5),. This implies . Hence, (stat) limit is unique. This establishes Theorem 12.

Theorem 13. Let be a PN-space. Ifthenbut the converse is not necessarily true in general.

Proof. By hypothesis, for every and , there exists a positive integer such thatwhenever . This guarantees that the sethas at most finitely many terms. As every finite subset of the set of positive integers has density zero, which establishes Theorem 13.

Example 14. This example would show that the converse of the assertion in Theorem 13 needs not be true in general. Let be the space of real numbers with the usual norm. Letwhere . Here, it is noted that is a probabilistic normed space. If it takes a sequence whose terms arethen, and for any , letSinceit getswhich implies thatHence, by Definition 7,Nevertheless, as the sequence shown in (39) is not convergent in the space , by Remark of [11], it is clear that the sequence is not convergent with respect to the probabilistic norm.

Theorem 15. Let be a PN-space. Ifthen

Proof. (i) LetAlso let . Choose such thatThen,Sinceit hasfor all . Furthermore, by usingget. LetThenwhich implies Ifthen This shows thatSo (ii) Let and suppose that and . Firstly, consider . Then, So Now let   . Sinceit follows from Theorem 13 thatIf then . In this case, If , then for ().
This demonstrates thatsothereby completing the proof of Theorem 15.

Theorem 16. Let be a PN-space. Thenif and only if there exists a setwithsuch that