Statistical equi-equal convergence of positive linear operators

Positivity - 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...     

On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space

The concept of statistical convergence was introduced by Fast [H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244] which was later on studied by many authors. In [J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993) 43–51], Fridy and Orhan introduced the idea of lacunary statistical convergence. Quite recently, the concept of statistical convergence of double sequences has been studied in intuitionistic fuzzy normed space by Mursaleen and Mohiuddine [M. Mursaleen, S.A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons Fractals (2008), doi:10.1016/j.chaos.2008.09.018]. In this paper, we study lacunary statistical convergence in intuitionistic fuzzy normed space. We also introduce here a new concept, that is, statistical completeness and show that IFNS is statistically complete but not complete.     

Stochastic convergence of weighted sums in normed linear spaces

Let X be a (real) separable Banach space, let {V_k} be a sequence of random elements in X, and let {a_nk} be a double array of real numbers such that lim_n→∞ a_nk = 0 for all k and Σ^∞_k=1 |a_nk| ≤ 1 for all n. Define S_n = Σⁿ_k=1 a_nk(V_k − EV_k). The convergence of {S_n} to zero in probability is proved under conditions on the coordinates of a Schauder basis or on the dual space of X and conditions on the distributions of {V_k}. Convergence with probability one for {S_n} is proved for separable normed linear spaces which satisfy Beck's convexity condition with additional restrictions on {a_nk} but without distribution conditions for the random elements {V_k}. Finally, examples of arrays {a_nk}, spaces, and applications of these results are considered.     

On the variational convergence of fuzzy sets in metric spaces

Kaleva [9] has studied the relationships between the metric convergencesH andD of fuzzy convex sets on Euclidean spaces. The distanceH between two fuzzy set is given by Hausdorff distance of their sendographs, whileD is the supremum of the Hausdorff distances of the level sets corresponding to the fuzzy sets. The aim of this paper is to compareH andD with the variational convergence, called γ-convergence (see De Giorgi and Franzoni [3]). Our analysis which is carried out in the setting of metric spaces (not necessarily locally compact or vector spaces), improves Kaleva's results.

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Sunto Kaleva ha investigato in [9] le relazioni esistenti tra due convergenze metriche, detteH eD, di sottoinsiemi fuzzy di spazi euclidei finito-dimensionali. In questo articolo le convergenzeH eD (la loro definizione dipende dalla distanza di Hausdorff tra insiemi compatti) sono confrontate con la convergenza variazionale, detta γ-convergenza, introdotta da De Giorgi and Franzoni in [3] nel contesto degli spazi topologici. Tale confronto con la γ-convergenza (vedi Teorema 3.7), svolto nell'ambito degli spazi metrici (non necessariamente, localmente compatti o lineari) migliora ed estende i precedenti risultati di Kaleva.

     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

Statistical convergence of double sequences in intuitionistic fuzzy normed spaces

Recently, the concept of intuitionistic fuzzy normed spaces was introduced by Saadati and Park [Saadati R, Park JH. Chaos, Solitons & Fractals 2006;27:331–44]. Karakus et al. [Karakus S, Demirci K, Duman O. Chaos, Solitons & Fractals 2008;35:763–69] have quite recently studied the notion of statistical convergence for single sequences in intuitionistic fuzzy normed spaces. In this paper, we study the concept of statistically convergent and statistically Cauchy double sequences in intuitionistic fuzzy normed spaces. Furthermore, we construct an example of a double sequence to show that in IFNS statistical convergence does not imply convergence and our method of convergence even for double sequences is stronger than the usual convergence in intuitionistic fuzzy normed space.     

On the Aleksandrov problem in linear n-normed spaces

The aim of this article is to generalize the Aleksandrov problem to the case of linear $n$-normed spaces.     

Fixed-point theorems in intuitionistic fuzzy metric spaces

Fixed-point theorems for complete intuitionistic fuzzy metric spaces are given.     

Fixed points in intuitionistic fuzzy metric spaces

The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst 1986;20:87–96], we define the notion of intuitionistic fuzzy metric spaces due to Kramosil and Michalek [Kramosil O, Michalek J. Fuzzy metric and statistical metric spaces. Kybernetika 1975;11:326–34]. Further the well-known fixed point theorems of Banach and Edelstein are extended to intuitionistic fuzzy metric spaces with the help of Grabiec [Grabiec M. Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 1988;27:385–9].     

Compactness in fuzzy convergence spaces

Lowen and Lowen [Applications of category theory to fuzzy subsets (Kluwer, 1992) p. 153] and Lowen et al. [Fuzzy Sets and Systems 40 (1991) 347] recently introduced the category FCS of fuzzy convergence spaces, a topological quasitopos which is a supercategory of FTS, the category of fuzzy topological spaces. In this paper, compactness in FCS is examined. Doing so we found that to define compactness as an absolute property we had to generalize the definition of fuzzy convergence space to fuzzy subsets. All basic theorems are proved including the Tychonoff product theorem. Based on the theory developed here, in a following publication, a Richardson compactification for fuzzy convergence spaces will be given.     

Statistical order convergence in Riesz spaces

In this paper, we introduce the concepts of statistical monotone convergence and statistical order convergence in a Riesz space, and establish some basic facts. We show that the statistical order convergence and the statistical convergence in norm need not be equivalent in a normed Riesz space. Finally, we introduce the statistical order boundedness of a sequence in a Riesz space.     

On convergence in fuzzy metric spaces

The concept of p-convergence in fuzzy metric spaces, in George and Veeramani's sense, has been recently given by D. Mihet in [D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007) 915-921]. In this note we study some aspects relative to this concept and characterize those fuzzy metric spaces, that we call principal, in which the family of p-convergent sequences agrees with the family of convergent sequences. Also a non-completable fuzzy metric space, which is not principal, is given.     

Common fixed point theorems in intuitionistic fuzzy metric spaces

The purpose of this paper, using the idea of intuitionistic fuzzy set due to Atanassov [2], we define the notion of intuitionistic fuzzy metric spaces (see, [1]) due to Kramosil and Michalek [17] and Jungck’s common fixed point theorem ([11]) is generalized to intuitionistic fuzzy metric spaces. Further, we first formulate the definition of weakly commuting and R-weakly commuting mappings in intuitionistic fuzzy metric spaces and prove the intuitionistic fuzzy version of Pant’s theorem ([21]).     

On fuzzy almost continuous convergence in fuzzy function spaces

In this paper, we study the fuzzy almost continuous convergence of fuzzy nets on the set FAC(X, Y) of all fuzzy almost continuous functions of a fuzzy topological space X into another Y. Also, we introduce the notions of fuzzy splitting and fuzzy jointly continuous topologies on the set FAC(X, Y) and study some of its basic properties.     

On stability of a cubic functional equation in intuitionistic fuzzy normed spaces

In this paper, we determine some stability results concerning the cubic functional equation

f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)