S. A. Mohiuddine

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.

On ideal convergence in probabilistic normed spaces

An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces

Recently, the concept of statistical convergence of double sequences has been studied in intuitionistic fuzzy normed spaces by Mursaleen and Mohiuddine [Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos, Solitons Fractals, 41 (2009) pp. 2414-2421]. We know that ideal convergence is more general than statistical convergence for single or double sequences. This has motivated us to study the ideal convergence of double sequences in a more general setting. That is, in this paper, we study the concept of ideal convergence and ideal Cauchy for double sequences in intuitionistic fuzzy normed spaces.

An application of almost convergence in approximation theorems

Abstract The concept of almost convergence was introduced by Lorentz in 1948 [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190], and has various applications. In this work we apply this method to prove some Korovkin type approximation theorems.

Korovkin type approximation theorems obtained through generalized statistical convergence

Abstract The concept of λ -statistical convergence was introduced in [M. Mursaleen, λ -statistical convergence, Math Slovaca, 50 (2000) 111–115] by using the generalized de la Vallee Poussin mean. In this work we apply this method to prove some Korovkin type approximation theorems.

Statistical Convergence of Double Sequences in Locally Solid Riesz Spaces

Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statistical -convergence, statistical -Cauchy and -convergence of double sequences in a locally solid Riesz space.

Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space

The object of the present paper is to determine the stability of the Hyers-Ulam-Rassias type theorem concerning the Pexiderized quadratic functional equation in intuitionistic fuzzy normed spaces (IFNS).

Statistical summability (C,1) and a Korovkin type approximation theorem

The concept of statistical summability (C,1) has recently been introduced by Móricz [Jour. Math. Anal. Appl. 275, 277-287 (2002)]. In this paper, we use this notion of summability to prove the Korovkin type approximation theorem by using the test functions 1, e−x, e−2x. We also give here the rate of statistical summability (C,1) and apply the classical Baskakov operator to construct an example in support of our main result.MSC:41A10, 41A25, 41A36, 40A30, 40G15.

Regularly σ-conservative and σ-coercive four dimensional matrices

The idea of @s-conservative and @s-coercive matrices for single sequences was introduced and studied by Schaefer in 1972. The theory of matrix transformations for double sequences has been recently developed by various authors. In this paper we study the concepts of regularly @s-conservative and @s-coercive matrices for double sequences.

Statistical summability through a lacunary sequence in locally solid Riesz spaces

In this paper, we introduce the concepts of lacunary statistical τ-convergence, lacunary statistically τ-bounded and lacunary statistically τ-Cauchy in the framework of locally solid Riesz spaces. We also define a new type of convergence, that is, S∗(τ)-convergence in this setup and prove some interesting results related to these notions.MSC:40A35, 40G15, 46A40.