Equivalence Of Convergence Research Articles

Fuzzy Korovkin type Theorems via deferred Cesaro and deferred Euler equi-statistical convergence

We establish a fuzzy Korovkin type approximation theorem by using \(eq-stat^{D}_{CE}\)(deferred Ces\'{a}ro and deferred Euler equi-statistical) convergence proposed by Saini et al. for continuous functions over \([a,b]\). Further, we determine the rate of convergence via fuzzy modulus of continuity.

A Certain Class of Equi-Statistical Convergence in the Sense of the Deferred Power-Series Method

In this paper, we expose the ideas of point-wise statistical convergence, equi-statistical convergence and uniform statistical convergence in the sense of the deferred power-series method. We then propose a relation connecting them, which is followed by several illustrative examples. Moreover, as an application viewpoint, we establish an approximation theorem based upon our proposed method for equi-statistical convergence of sequences of positive linear operators. Finally, we estimate the equi-statistical rates of convergence for the effectiveness of the results presented in our study.

Convergence of Iterative Algorithms in Cat(0) Spaces

In this article, results have been shown via using a general quasi contraction multi-valued mapping in Cat(0) space. These results are used to prove the convergence of two iteration algorithms to a fixed point and the equivalence of convergence. We also demonstrate an appropriate conditions to ensure that one is faster than others.

Approximation via equi-statistical convergence in the sense of power series method

Approximation via equi-statistical convergence in the sense of power series method

Applications of relative statistical convergence and associated approximation theorem

<abstract><p>In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.</p></abstract>

Korovkin type approximation theorem via lacunary equi-statistical convergence in fuzzy spaces

In the present paper, we establish relations between equi-statistical convergence and lacunary equi-statistical convergence of sequences of fuzzy number valued functions. We make an effort to prove Korovkin type approximation theorem via lacunary equi-statistical convergence in fuzzy spaces. Further, we study rates of lacunary equi-statistical fuzzy convergence by using fuzzy modulus of continuity.

Deferred Cesàro and deferred Euler equi-statistical convergence and its applications to Korovkin-type approximation theorem

The present paper emphasises on equi-statistical convergence, pointwise statistical convergence and uniform statistical convergence for a sequence of real-valued functions by using deferred Cesàro and deferred Euler statistical convergence and obtain various implicative results with supporting examples. We make an effort to demonstrate Korovkin-type approximation theorem via deferred Cesàro and deferred Euler equi-statistical convergence. We also present an example which shows that our Korovkin-type theorem is powerful than its classical version. Further, we study rates of deferred Cesàro and deferred Euler equi-statistical convergence via modulus of continuity.

Equi-statistical convergence of a sequence of distribution functions via deferred Nörlund summability mean and associated approximation theorems

In this paper, the notion of statistical point-wise convergence, equi-statistical convergence and statistical uniform convergence of a sequence of distribution functions via the deferred N?rlund summability mean has been introduced, and accordingly an inclusion relation between these interesting notions is established. Moreover, as an application point of view, a new Korovkin-type approximation theorem is proved via the deferred N?rlund equi-statistical convergence for the sequence of distribution functions. Also, some illustrative examples are considered to justify that the proposed theorem is a nontrivial extension of some well established Korovkin-type approximation theorems for sequence of real-valued functions. Finally, a number of interesting cases are highlighted in support of the definitions and outcomes.

A Certain Class of Relatively Equi-Statistical Fuzzy Approximation Theorems

The aim of this paper is to introduce the notions of relatively deferred Nörlund uniform statistical convergence as well as relatively deferred Norlund point-wise statistical convergence through the dierence operator of fractional order of fuzzy-number-valued sequence of functions, and a type of convergence which lies between aforesaid notions, namely, relatively deferred Nörlund equi-statistical convergence. Also, we investigate the inclusion relations among these aforesaidnotions. As an application point of view, we establish a fuzzy approximation (Korovkin-type) theorem by using our new notion of relatively deferred Norlund equi-statistical convergence and intimate that this result is a non-trivial generalization of several well-established fuzzy Korovkin-type theorems which were presented in earlier works. Moreover, we estimate the fuzzy rate of the relatively deferred Nörlund equi-statistical convergence involving a non-zero scale function by using the fuzzy modulus of continuity.

Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

Generalized deferred Cesàro equi-statistical convergence and analogous approximation theorems

The concept of deferred Nörlund equi-statistical convergence has recently been studied by Srivastaval et al. (see,[19]). In this paper, we have introduced the notion of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence in conjunction with the deferred statistical convergence and established a inclusion relation between them. Moreover, we have applied our idea (presumably new) of the deferred equi-statistical convergence to prove a Korovkin-type approximation theorem and demonstrated that our theorem is a non-trivial extension of some well-established Korovkintype approximation theorems which ware proved by some earlier authors. Furthermore, we have established the rate of deferred equistatistical convergence and accordingly proved a theorem. Finally, some concluding remarks and fascinating cases are shown here in support of our definitions and results.

Investigation of some fixed point theorems in hyperbolic spaces for a three step iteration process

In the present paper, we investigate the convergence, equivalence of convergence, rate of convergence and data dependence results using a three step iteration process for mappings satisfying certain contractive condition in hyperbolic spaces. Also we give non-trivial examples for the rate of convergence and data dependence results to show effciency of three step iteration process. The results obtained in this paper may be interpreted as a refinement and improvement of the previously known results.

Relatively equi-statistical convergence via deferred Nörlund mean based on difference operator of fractional-order and related approximation theorems

In the proposed paper, we have introduced the notion of point-wise relatively statistical convergence, relatively equi-statistical convergence and relatively uniform statistical convergence of sequences of functions based on the difference operator of fractional order including (p, q)-gamma function via the deferred Norlund mean. As an application point of view, we have proved a Korovkin type approximation theorem by using the relatively deferred Norlund equi-statistical convergence of difference sequences of functions and intimated that our theorem is a generalization of some well-established approximation theorems of Korovkin type which was presented in earlier works. Moreover, we estimate the rate of the relatively deferred Norlund equi-statistical convergence involving a non-zero scale function. Furthermore, we use the modulus of continuity to estimate the rate of convergence of approximating positive linear operators. Finally, we set up various fascinating examples in connection with our results and definitions presented in this paper.

A Picard-type iterative algorithm for general variational inequalities and nonexpansive mappings

In this paper, a normal S-iterative algorithm is studied and analyzed for solving a general class of variational inequalities involving a set of fixed points of nonexpansive mappings and two nonlinear operators. It is shown that the proposed algorithm converges strongly under mild conditions. The rate of convergence of the proposed iterative algorithm is also studied. An equivalence of convergence between the normal S-iterative algorithm and Algorithm 2.6 of Noor (J. Math. Anal. Appl. 331, 810–822, 2007) is established and a comparison between the two is also discussed. As an application, a modified algorithm is employed to solve convex minimization problems. Numerical examples are given to validate the theoretical findings. The results obtained herein improve and complement the corresponding results in Noor (J. Math. Anal. Appl. 331, 810–822, 2007).

Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems

We introduce the notions of weighted lacunary statistical pointwise and uniform convergence and a kind of convergence which is lying between aforementioned convergence methods, namely, weighted lacunary equi-statistical convergence and obtain various implication results with supporting examples. We then apply our new concept of weighted lacunary equi-statistical convergence with a view to proving Korovkin and Voronovskaya type approximation theorems. We also construct an example with the help of generating functions type Meyer-Konig and Zeller which shows that our Korovkin-type theorem is stronger than its classical version. Moreover, we compute the rate of weighted lacunary equi-statistical convergence for operators in terms of modulus of continuity.

Statistical equi-equal convergence of positive linear operators

Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.

Korovkin type approximation theorems proved viaweighted αβ-equistatistical convergence for bivariate functions

Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted ??-statistical convergence of order ?, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted ??-statistical convergence of order ?, which is an extension of ??-statistical convergence of order ?. Our definition, with sk = 1, for all k, reduces to ??-statistical convergence of order ?. Moreover, we use this definition of weighted ??-statistical convergence of order ?, to prove Korovkin type approximation theorems via, weighted ??-equistatistical convergence of order ? and weighted ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Also we prove Korovkin type approximation theorems via ??-equistatistical convergence of order ? and ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted ??-equistatistical convergence of order ? is introduced and discussed.

Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems

Statistical convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the classical convergence. Furthermore, the notion of equi-statistical convergence is stronger than that of the statistical uniform convergence. Such concepts were introduced and studied by Balcerzak et al. (J Math Anal Appl 328:715–729, 2007). In this paper, we have used the notion of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence in conjunction with the deferred Norlund statistical convergence in order to establish several inclusion relations between them. We have also applied our presumably new concept of the deferred Norlund equi-statistical convergence to prove a Korovkin type approximation theorem and demonstrated that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were proven by earlier authors. Finally, we consider a number of interesting cases in support of our definitions and results presented in this paper.

Korovkin type theorem for functions of two variables via lacunary equistatistical convergence

Aktu?lu and Gezer [1] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. Recently, Kaya and G?n?l [11] proved some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence by using test functions 1, x/1+x, y/1+y, (x/1+x)2 +(y/1+y)2. We apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem for functions of two variables by using test functions 1, x/1?x, y/1?y, (x/1?x)2+(y/1?y)2.

Weighted Equi-Statistical Convergence of the Korovkin Type Approximation Theorems

In this paper, the concepts of weighted statistical pointwise convergence, weighted statistical uniform convergence and weighted equi-statistical convergence are introduced. Then, using weighted equi-statistical convergence, a general Korovkin type theorem is obtained. Also, an example such that our new approximation result works but its classical case does not work is constructed.