Statistical convergence in intuitionistic fuzzy n-normed linear spaces

The aim in our article is to introduce the notion of statistical convergence and statistically Cauchy sequences in intuitionistic fuzzy n-normed linear spaces. The paper shows that some properties of statistical convergence of real sequences also hold for sequences in this space. Characterization for statistically convergent and statistically Cauchy sequences is also given. Further, the concept of statistical limit points and statistical cluster points are introduced and their relation with limit points of sequences have been investigated.     

On lacunary statistical convergence of order α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong N_θ^α(p)-summability. Furthermore, some relations between the spaces N_θ^αS_θ^α are examined.     

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Abstract

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ?. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δ^α and interpret them to those of previous works. Also, by using the convergence of Δ^α-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δ^α-statistical convergence of positive linear operators.     

Almost-convergent sequences with respect to polynomial hypergroups

Each polynomial hypergroup on ?₀ generates a family of generalized translation operators?T _m on sequence spaces. We introduce the concept of almost convergence for polynomial hypergroups (determined by the operators?T _m ), extending the notion of almost convergence introduced by Lorentz. Our investigations lead to two theorems characterizing almost convergent sequences on polynomial hypergroups.     

A globally linearly convergent method for pointwise quadratically supportable convex–concave saddle point problems

We study the Proximal Alternating Predictor–Corrector (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle [8] to solve nonsmooth structured convex–concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent. We illustrate the proposed method on total variation denoising problems and on locally adaptive estimation in signal/image deconvolution and denoising with multiresolution statistical constraints.     

On ideal convergence of double sequences in probabilistic normed spaces

The notion of ideal convergence is a generalization of statistical convergence which has been intensively investigated in last few years.For an admissible ideal ∮N× N,the aim of the present paper is to introduce the concepts of ∮-convergence and ∮*-convergence for double sequences on probabilistic normed spaces(PN spaces for short).We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide.We also define ∮-Cauchy and ∮*-Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces.We establish example which shows that our method of convergence for double sequences on PN spaces is more general.     

Almost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbers

The purpose of this paper is to introduce the concepts of almost lacunary statistical convergence and strongly almost lacunary convergence of sequences of fuzzy numbers. We give some relations related to these concepts. We establish some connections between strongly almost lacunary convergence and almost lacunary statistical convergence of sequences of fuzzy numbers. It is also shown that if a sequence of fuzzy numbers is strongly almost lacunary convergent with respect to an Orlicz function then it is almost lacunary statistical convergent.     

On A-invariant mean and A-almost convergence

The idea of A-invariant mean and A-almost convergence is due to J. P. Duran [8], which is a generalization of the usual notion of Banach limit and almost convergence. In this paper, we discuss some important properties of this method and prove that the space F(A) of A-almost convergent sequences is a BK space with ?? · ??_??, and also show that it is a nonseparable closed subspace of the space l _?? of bounded sequences.     

Lopsided PMHSS iteration method for a class of complex symmetric linear systems

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method, we introduce a lopsided PMHSS (LPMHSS) iteration method for solving a broad class of complex symmetric linear systems. The convergence properties of the LPMHSS method are analyzed, which show that, under a loose restriction on parameter α, the iterative sequence produced by LPMHSS method is convergent to the unique solution of the linear system for any initial guess. Furthermore, we derive an upper bound for the spectral radius of the LPMHSS iteration matrix, and the quasi-optimal parameter α ^? which minimizes the above upper bound is also obtained. Both theoretical and numerical results indicate that the LPMHSS method outperforms the PMHSS method when the real part of the coefficient matrix is dominant.     

Optimal transformations for scalar sequences

Summary We consider transformations which accelerate convergence in some specified classes of convergent sequences. As an asymptotic measure of acceleration we introduce the order of transformation. We find a sharp upper bound on the order and show the explicit form of transformations of maximal order. We consider also the efficiency of transformations for fast convergent sequences. As a special case we find that the Germain-Bonne version of Richardson extropolation has maximal order for linearly convergent sequences.     

Matrix characterization of A-statistical convergence of double sequences

We characterize the multiplier space of summability fields of four dimensional RH-regular matrices and show that the space of multipliers of a nonnegative RH-regular matrix over an algebra \(\mathcal{U} \) is the space of A-statistically convergent double sequences. For this purpose we prove a variant of the Brudno–Mazur–Orlicz bounded consistency theorem for a class of four dimensional matrices. Finally we give a matrix characterization of A-statistical convergence over the space of the Pringsheim A-uniformly integrable double sequences.     

About the relaxed cocoercivity and the convergence of the proximal point algorithm

The aim of this paper is to study the convergence of two proximal algorithms via the notion of (α, r)-relaxed cocoercivity without Lipschitzian continuity. We will show that this notion is enough to obtain some interesting convergence theorems without any Lipschitz-continuity assumption. The relaxed cocoercivity case is also investigated.     

Enhanced Efficiency in Multi-objective Optimization

For a given multi-objective optimization problem, we introduce and study the notion of α-proper efficiency. We give two characterizations of such proper efficiency: one is in terms of exact penalization and the other is in terms of stability of associated parametric problems. Applying the aforementioned characterizations and recent results on global error bounds for inequality systems, we obtain verifiable conditions for α-proper efficiency. For a large class of polynomial multi-objective optimization problems, we show that any efficient solution is α-properly efficient under some mild conditions. For a convex quadratically constrained multi-objective optimization problem with convex quadratic objective functions, we show that any efficient solution is α-properly efficient with a known estimate on α whenever its constraint set is bounded. Finally, we illustrate our achieved results with examples, and give an example to show that such an enhanced efficiency property may not hold for multi-objective optimization problems involving C ^∞-functions as objective functions.     

A quantified version of the Dirichlet-Jordan test inL 1-norm

We introduce the notion of bounded variation in the sense ofL ¹-norm for periodic functions and prove a version of the classical Dirichlet-Jordan test for the convergence of Fourier series inL ¹-norm. We also give an estimate of the rate of convergence.     

Higher order numerical methods for solving fractional differential equations

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h ^3?α ). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h ³) for α≥1 and O(h ^1+2α ) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

Limits of locally–globally convergent graph sequences

The colored neighborhood metric for sparse graphs was introduced by Bollobás and Riordan [BR11]. The corresponding convergence notion refines a convergence notion introduced by Benjamini and Schramm [BS01]. We prove that even in this refined sense, the limit of a convergent graph sequence (with uniformly bounded degree) can be represented by a graphing. We study various topics related to this convergence notion such as: Bernoulli graphings, factor of i.i.d. processes and hyperfiniteness.     

Vitali type convergence theorems for Banach space valued integrals

Let(Ω,Σ,μ)be a complete probability space and let X be a Banach space.We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem.We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on N.     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.