Hibrid Δ-Statistical Convergence for Neutrosophic Normed Space
This study is about neutrosophic structures, which is one of the popular topics of recent days. In this study, different types of convergence concepts were applied to difference sequences. With the help of the properties of double type sequences, the concept of difference sequences is combined with structures that are advantageous to work like Lacunary sequences.
The idea of difference sequences emerged in the days when researchers focused on the idea of constructing new sequence spaces. The difference sequences were given by Kizmaz in . After that, Basarir  applied this concept to statistical analysis. After statistical convergence was introduced in , different versions of statistical convergence have been defined in area of functional analysis, e.g., [4–9]. Fridy and Orhan gave this concept for Lacunary sequences in . Esi and Araci applied this concept to operator theory in . The properties of Lacunary A-convergence are given in . Then, Hazarika applied that definition of Basarir to Lacunary sequences in normed spaces in . Altundag and Kamber made an important study by evaluating Lacunary sequence and difference sequence structures together in n-dimensional intuitionistic fuzzy normed space, where , in . Colak introduced Lacunary strong convergence using difference sequences for modulus function in . Et et al. gave a generalization of difference sequences in . Statistical convergence is defined for double sequences in . Later, this concept attracted great attention of researchers. Tripathy and Sarma applied the idea of this work to difference sequences in . In , Patterson and Savas transferred Lacunary statistical convergence to double sequences. Fuzzy-intuitionistic fuzzy sets are generalizations of classical sets established for compelling reasons in daily life. Due to the inadequacy of fuzzy set and intuitionistic set concepts, a new set concept was needed. Thus, a new concept emerged with the help of neutrosophy, which is called a subdivision of philosophy that studies the structure of neutrals: neutrosophic sets. Neutrosophic sets are a concept introduced to investigate the degrees of correctness, wrongness, and uncertainty of the elements in the set in . While classical statistics uses precise data and inferences, neutrophic statistics uses methods that contain uncertain, contradictory, and partially unknown data. Kirisci and Simsek introduced classical statistical convergence in neutrosophic normed spaces in . After that, Khan et al. carried this work to Lacunary sequences in . Granados and Dhital adapted this worked for double sequences in . In 2022, Kisi and Gurdal introduced triple difference sequences in neutrosophic normed spaces . Sahin and Kargın worked neutrosophic tripled normed spaces in . Apart from this, the concept of neutrosophic set has many applications in many different fields, i.e., [26, 27].
Now, by evaluating all this information together, a study was prepared to fill the relevant gap in the literature. Here, statistical convergence will be applied to neutrosophic normed spaces single-double sequences, and also, Lacunary statistical convergence will be given for these two types of sequences. Many important results were given, especially the relations between these two concepts. The properties of statistical Cauchy and Lacunary statistical Cauchy sequences will be examined for these sequences.
First of all, some necessary definitions will be given.
Definition 1 (see ). Let us consider m crisp-components: . If all of them are 100% independent two by two, then their sum isOn the contrary, if all of them are 100% dependent, thenAs stated by Smarandache in , in this study, will be taken when two components are dependent, while the third one is independent from them.
Definition 2 (see ). Let and , and are the degrees of correctness, uncertainty, and falsity. A neutrosophic set is in the next form: , where, for all in , and , .
It should be noted that is an independent component and and are dependent components.
Definition 3 (see ). Let be a linear spaces and and show the continuous norm and continuous conorm on . The notation of neutrosophic normed is , where , , and demonstrate the degree of correctness, uncertainty, and falsity of on which satisfies the following conditions, for all :(i)For every , (ii)For every ,(iii)For every , , , and .(iv)For each , , , and(v) is a continuous nondecreasing function; and are continuous nonincreasing function,(vi), , and (vii)If , then , , and .In this case, is called neutrosophic normed spaces. Here, and are interdependent and is an independent components.
Definition 4 (see ). Let and denote the cardinality of . The density of the set is given by the following equation and is denoted by :
Definition 5 (see ). is called to be statistically convergent to , wherefor all . Then, it will be represented by . is demonstrated, set of statistical convergence sequences.
Definition 6 (see ). is called to be statistically convergent to , wherefor all and , i.e.,Then, it is demonstrated . is denoted, set of all statistical convergence sequences.
Definition 7 (see ). Let be a sequence of increasing integers, and also . Then, is called to be Lacunary sequences. Let , and .is said to be the density of if limit is exhibited. Let ; for all , ifin this case, is called to be Lacunary statistical convergent to . Then, it is represented as . is a denoted set of every Lacunary statistical convergence sequences.
3. Materials and Methods
In this section where important properties for difference sequences will be given, two separate parts will be given where convergence studies will be made for single and double sequences.
3.1. Statistical Convergence in Neutrosophic Normed Spaces
Now, convergence, statistical convergence, and Lacunary statistical convergence will be defined in neutrosophic normed spaces. The relations between these concepts will be given.
Definition 8. Let be neutrosophic normed spaces and . is called to be convergent to according to neutrosophic normed if, for all and , there exists a such that, for every ,This sequences is shown with .
Definition 9. Let be neutrosophic normed spaces. If there exist and , for all where , , and , then is called bounded sequences in .
Definition 10. Let be neutrosophic normed spaces, is called to be statistical convergence with respect to if, for every and there exist such thathas natural density zero, i.e.,Therefore, it will be denoted as or , where . denotes set of all statistical convergence sequences.
Theorem 1. Let be neutrosophic normed spaces. If is statistically convergent in this case, is unique.
Lemma 1. Let be neutrosophic normed spaces and be a statistical convergence sequences. Then, for each the next properties are equivalent:(i)(ii)(iii), , and (iv), , and (v), , and .
Using Definition 10, the equivalence of statements is easily demonstrated.
Definition 11. Let be neutrosophic normed spaces, is called to be Cauchy sequences if, for every and , there exist a such that, for every , .
Now, a new type of convergence will be defined by including the Lacunary sequence structure in the investigations for difference sequences in neutrosophic normed spaces.
Definition 12. Let be neutrosophic normed spaces and be Lacunary sequence in this spaces. is named to be Lacunary statistical convergence with respect to if, for every and , there exist such thathas natural density zero, i.e.,orTherefore, it will denote or as . denotes set of all Lacunary statistical convergence sequences.
Lemma 2. Let be neutrosophic normed spaces, be a Lacunary sequence, and be a Lacunary statistical convergence sequences. Then, for every and , the following properties are equivalent:(i)(ii)(iii), , and (iv), , and (v), , and .
They are easily demonstrated using Definition 12.
Theorem 2. Let be neutrosophic normed spaces, and . If is Lacunary statistically convergent in this case, is unique.
Theorem 3. Let be neutrosophic normed spaces. If is statistically convergent, then this sequences is Lacunary statistically convergent.
Proof. Let be statistically convergent to and be density of A obtained with the help of the difference sequence. Then, for every and there exists a such that, for every , or , and . So,has finite number of terms. Hence, density of this set is zero, i.e.,Thus, is Lacunary statistically convergent to .
Definition 13. Let be neutrosophic normed spaces and be Lacunary sequence. is called to be Lacunary statistical Cauchy sequences when, for all and there exist a , for every
Theorem 4. Let be neutrosophic normed spaces and be a Lacunary sequence. is Lacunary statistical convergent is Lacunary statistical Cauchy in neutrosophic normed spaces.
Proof. Let be a Lacunary statistical convergent sequence and For a given , choose such that and For any ,can be written so . For , and LetIt is necessary to show that . So, to show this, let In this case, and especially So,However, this is not possible. Moreover, and . Especially, . Hence,which is impossible. With a similar technique can be applied for So, and Then, is Lacunary statistical Cauchy sequences in neutrosophic normed spaces.
Otherwise, let be Lacunary statistical Cauchy but not Lacunary statistical convergent on For a given , choose such that and Because is not Lacunary statistical convergent,So, for and also . Since is Lacunary statistical Cauchy, this is impossible. Thus, is Lacunary statistical convergent in .
3.2. Lacunary Statistical Convergence with Double Sequences
Now, lacunary statistical convergence will be applied to neutrosophic normed spaces using double sequences. First of all, let us remind a few definitions necessary for the section.
Let be numbers of , where . with double density is defined  as
In , statistical convergence is defined using double sequences: is called statistical convergence if, for all , .
Then, in , double Lacunary sequences are defined as follows. Let there exist two increasing integers sequences, where , , and ; then, is called double Lacunary sequences. Here, , , and ; also, , , and