Hüseyin Aktuğlu

Lacunary equi-statistical convergence of positive linear operators

In this paper, the concept of lacunary equi-statistical convergence is introduced and it is shown that lacunary equi-statistical convergence lies between lacunary statistical pointwise and lacunary statistical uniform convergence. Inclusion relations between equi-statistical and lacunary equi-statistical convergence are investigated and it is proved that, under some conditions, lacunary equi-statistical convergence and equi-statistical convergence are equivalent to each other. A Korovkin type approximation theorem via lacunary equi-statistical convergence is proved. Moreover it is shown that our Korovkin type approximation theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally the rates of lacunary equi-statistical convergence by the help of modulus of continuity of positive linear operators are studied.

A-statistical approximation of generalized Szász―Mirakjan―Beta operators

Abstract In this paper we prove a Korovkin type approximation theorem and obtain the rate of convergence of the generalized Szasz–Mirakjan–Beta operators by means of modulus of continuity and elements of Lipschitz class. Furthermore we give the A -statistical approximation theorem for these operators and investigate the case which provides the best estimation.

On the Solvability of Caputo -Fractional Boundary Value Problem Involving -Laplacian Operator

We consider the model of a Caputo -fractional boundary value problem involving -Laplacian operator. By using the Banach contraction mapping principle, we prove that, under some conditions, the suggested model of the Caputo -fractional boundary value problem involving -Laplacian operator has a unique solution for both cases of and . It is interesting that in both cases solvability conditions obtained here depend on , , and the order of the Caputo -fractional differential equation. Finally, we illustrate our results with some examples.

An Overall View of Stochastics in Colombeau Related Algebras

Stochastic analysis in Colombeau algebras was initiated by the works of Capar, Oberguggenberger and Russo in the early 1990s. The present communique is a comparative and critical account of the major research papers that have appeared so far in this particular field of study. It turns out that application of the nonlinear distribution theories of Colombeau and Rosinger to stochastic analysis provides new horizons in dealing with singular objects like nonlinear functions of white noise, Wiener functionals and nonlinear stochastic PDE, especially when the initial data is irregular or distributional.

Solvability of differential equations of order 2<α≤3 involving the p-Laplacian operator with boundary conditions

In this paper, we study the existence of solutions for non-linear fractional differential equations of order 2<α≤3 involving the p-Laplacian operator with various boundary value conditions including an anti-periodic case. By using the Banach contraction mapping principle, we prove that, under certain conditions, the suggested non-linear fractional boundary value problem involving the p-Laplacian operator has a unique solution for both cases of 0<p<1 and p≥2. Finally, we illustrate our results with some examples.

A new construction of random Colombeau distributions

In this paper a generalized random process is modeled through the randomization of a bilinear form between the space of test functions and the Colombeau generalized functions. This results in a theory akin to Gelfand-Vilankins random Schwartz distributions. An extension theorem in Bochner-Badrikian style is proved under some continuity assumptions. An important application is a natural representation of nonlinear functionals of white noise.

Quantitative Global Estimates for Generalized Double Szasz-Mirakjan Operators

We introduce the generalized double Szász-Mirakjan operators in this paper. We obtain several quantitative estimates for these operators. These estimates help us to determine some function classes (including some Lipschitz-type spaces) which provide uniform convergence on the whole domain .

Weighted αβ-statistical convergence of Kantorovich-Mittag-Leffler operators

Abstract In this paper we introduce Kantorovich variant of the Mittag-Leffler operators including the modified Kantorovich-Szász-Mirakjan operators. We give αβ-statistical approximation theorems for these operators in various function spaces. The results include the statistical, lacunary statistical and λ-statistical cases. Moreover, we compute the rate of convergence in different Lipschitz type spaces.