Abdullah Alotaibi

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.

Statistical summability and approximation by de la Vallée-Poussin mean

Abstract In this paper we define a new type of summability method via statistical convergence by using the density and ( V , λ ) -summability. We further apply our new summability method to prove a Korovkin type approximation theorem.

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.

Ideal convergence of double sequences in random 2-normed spaces

Quite recently, Alotaibi and Mohiuddine (Adv. Differ. Equ. 2012:39, 2012) studied the idea of a random 2-normed space to determine some stability results concerning the cubic functional equation. In this paper, we define and study the concepts of I-convergence and I∗-convergence for double sequences in random 2-normed spaces and establish the relationship between these types of convergence, i.e., we show that I∗-convergence implies I-convergence in random 2-normed spaces. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I∗-convergence in random 2-normed spaces.MSC:40A05, 46A70.

On(Δm,I)-Statistical Convergence of Orderα

The idea of I-convergence of real sequences was introduced by Kostyrko et al., (2000/01) and also independently by Nuray and Ruckle (2000). In this paper, we introduce the concepts of (Δm, I)-statistical convergence of order α and strong (Δp m, I)-Cesàro summability of order α of real sequences and investigated their relationship.

On the stability of a cubic functional equation in random 2-normed spaces

In this article, we propose to determine some stability results for the functional equation of cubic in random 2-normed spaces which seems to be a quite new and interesting idea. Also, we define the notion of continuity, approximately and conditional cubic mapping in random 2-normed spaces and prove some interesting results.

Coupled coincidence points for monotone operators in partially ordered metric spaces

Using the notion of compatible mappings in the setting of a partially ordered metric space, we prove the existence and uniqueness of coupled coincidence points involving a (ϕ, ψ)-contractive condition for a mappings having the mixed g-monotone property. We illustrate our results with the help of an example.

Korovkin second theorem via statistical summability (C,1)

Korovkin type approximation theorems are useful tools to check whether a given sequence (Ln)n≥1 of positive linear operators on C[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions, which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x and x2 in the space C[0,1] as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line. In this paper, we use the notion of statistical summability (C,1) to prove the Korovkin approximation theorem for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line and show that our result is stronger. We also study the rate of weighted statistical convergence.MSC:41A10, 41A25, 41A36, 40A30, 40G15.

Statistical convergence through de la Vallée-Poussin mean in locally solid Riesz spaces

The notion of statistical convergence was defined by Fast (Colloq. Math. 2:241-244, 1951) and over the years was further studied by many authors in different setups. In this paper, we define and study statistical τ-convergence, statistically τ-Cauchy and S∗(τ)-convergence through de la Vallée-Poussin mean in a locally solid Riesz space.MSC:40A35, 40G15, 46A40.

Fuzzy stability of a cubic functional equation via fixed point technique

The object of this article is to determine Hyers-Ulam-Rassias stability results concerning the cubic functional equation in fuzzy normed space by using the fixed point method.