Muharrem Özlük

In this paper, we introduce a novel extension of the Bernstein-Kantorovich-Stancu type operator of degree n with the help of multiple shape parameters. Voronovskaja and Grüss-Voronovskaja type approximation theorems are examined via Ditzian-Totik moduli of smoothness. We investigate basic statistical convergence properties with respect to a non-negative regular summability matrix. Moreover, using Ditzian-Totik moduli, local and global approximation properties associated to the proposed operator have been established. Finally, several illustrative examples are presented to demonstrate the efficiency, applicability and validity of the operator. The graphical and numerical results verify that the proposed operator gives better approximation as well as expand the previous Bernstein-Kantorovich type modifications including single parameter.

Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

In this paper, we essentially deal with Köthe‐Toeplitz duals of fuzzy level sets defined using a partial metric. Since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct some classical notions. In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some of their properties like completeness and duality. Finally, we obtain the Köthe‐Toeplitz duals of fuzzy level sets with respect to the partial metric based on a partial ordering.

In this paper, we investigate the classical sets of sequences of fuzzy numbers by using partial metric which is based on a partial ordering. Some elementary notions and concepts for partial metric and fuzzy level sets are given. In addition, several necessary and sufficient conditions for partial completeness are established by means of fuzzy level sets. Finally, we give some illustrative examples and present some results between fuzzy and partial metric spaces.