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We introduce calculus and calculus for intuitionistic fuzzy values and prove some basic theorems by using multiplicative calculus which has useful tools to represent the concepts of introduced calculi. Besides, we construct some isomorphic mappings to interpret the relationships between calculus and calculus. This paper reveals also new calculi for fuzzy sets in particular. [ABSTRACT FROM AUTHOR]

We define weighted geometric mean method of convergence for sequences in R + by using multiplicative calculus and obtain necessary and sufficient conditions under which convergence of sequences in R + follows from convergence of their weighted geometric means. We also obtain multiplicative analogues of Schmidt type slow oscillation condition and Landau type two-sided condition for the convergence in particular. Besides, we introduce the concepts of ⊕ convergence, ⊗ convergence, (N ¯ , p) − ⊕ convergence, (G ¯ , p) − ⊗ convergence for sequences of intuitionistic fuzzy numbers (IFNs) and apply the aforementioned conditions to achieve convergence in intuitionistic fuzzy number space. Examples of sequences are also given to illustrate the proposed methods of convergence. [ABSTRACT FROM AUTHOR]

We define statistical Cesàro and statistical logarithmic summability methods of sequences in intuitionistic fuzzy normed spaces(IFNS) and give slowly oscillating type and Hardy type Tauberian conditions under which statistical Cesàro summability and statistical logarithmic summability imply convergence in IFNS. Besides, we obtain analogous results for the higher order summability methods as corollaries. Also, two theorems concerning the convergence of statistically convergent sequences in IFNS are proved in the paper. [ABSTRACT FROM AUTHOR]

We define the concept of Cesàro summability method in intuitionistic fuzzy normed spaces and prove a related Tauberian theorem. Also, we define slowly oscillating sequences in intuitionistic fuzzy normed spaces, prove related theorems and show that Cesàro summability of slowly oscillating sequences implies ordinary convergence in intuitionistic fuzzy normed spaces. Finally, we give an analogue of classical two-sided Tauberian theorem due to Hardy by using the concept of q-boundedness. [ABSTRACT FROM AUTHOR]

We introduce semicontinuous summation methods for series of fuzzy numbers and give Tauberian conditions under which summation of a series of fuzzy numbers via generalized Dirichlet series and via generalized factorial series implies its convergence. Besides, we define the concept of level Fourier series of fuzzy valued functions and obtain results concerning the summation of level Fourier series. [ABSTRACT FROM AUTHOR]

We give Tauberian conditions of slow decrease type under which statistical Cesàro summability and statistical logarithmic summability of sequences of fuzzy numbers imply convergence in fuzzy number space. Besides, we obtain a sufficient condition for statistically convergent sequences of fuzzy numbers to converge in the ordinary sense. [ABSTRACT FROM AUTHOR]

We prove a Korovkin-type approximation theorem via power series methods of summability for continuous 2 π -periodic functions of two variables and verify the convergence of approximating double sequences of positive linear operators by using modulus of continuity. An example concerning double Fourier series is also constructed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We introduce (𝑪, 𝒌) summability method of sequences of fuzzy numbers and compare (𝑪, 𝒌) summability method with Abel summability method, Hölder summability method and with higher order Cesàro method of sequences of fuzzy numbers. Some Tauberian conditions under which (𝑪, 𝒌) summability of sequences of fuzzy numbers imply convergence in fuzzy number space are also obtained as corollaries. [ABSTRACT FROM AUTHOR]

In this study we compare Cesàro and Euler weighted mean methods of summability of sequences of fuzzy numbers with Abel and Borel power series methods of summability of sequences of fuzzy numbers. Also, some results dealing with series of fuzzy numbers are obtained. [ABSTRACT FROM AUTHOR]

We define logarithmic summability method for sequences of fuzzy numbers and prove theorems dealing with the convergence behavior of logarithmic summable sequences of fuzzy numbers. The study also reveals slowly decreasing and Landau one-sided type Tauberian results analogous to those given by Móricz (Stud Math 219:109-121, 2013). [ABSTRACT FROM AUTHOR]

In the present study, we have introduced Cesàro summability of integrals of fuzzy-number-valued functions and given one-sided Tauberian conditions under which convergence of improper fuzzy Riemann integrals follows from Cesàro summability. Also, fuzzy analogues of Schmidt type slow decrease and Landau type one-sided Tauberian conditions have been obtained. [ABSTRACT FROM AUTHOR]

We introduce logarithmic summability of fuzzy-number-valued functions by using logarithmic averages and establish some Tauberian conditions which guarantee the existence of limits of logarithmic summable fuzzy-numbervalued functions at infinity. We also obtained analogous results in case of improper fuzzy Riemann integrals. [ABSTRACT FROM AUTHOR]

We introduce Euler summability method for sequences of fuzzy numbers and state a Tauberian theorem concerning Euler summability method, of which proof provides an alternative to that of K. Knoop[Über das Eulersche Summierungsverfahren II, Math. Z. 18 (1923)] when the sequence is of real numbers. As corollaries, we extend the obtained results to series of fuzzy numbers. [ABSTRACT FROM AUTHOR]

In the present study, we have introduced the concept of Abel summability for sequences and series of fuzzy numbers. Also, some tauberian results in classical analysis have been generalized to fuzzy analysis. [ABSTRACT FROM AUTHOR]

In this paper, we have given two theorems for |A, pn; d| summability which generalize recent theorems on |A, pn| summability. Study also reveals many factor theorems for other summability methods. [ABSTRACT FROM AUTHOR]