Abstract
The class of membership functions is restricted to trapezoidal one, as it is general enough and widely used. In the present paper since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct for a fuzzy-valued function via related trapezoidal membership function. We derive uniform convergence of fuzzy-valued function sequences and series with some illustrated examples. Also we study Hukuhara differentiation and Henstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, we introduce the power series with fuzzy coefficients and define the radius of convergence of power series. Finally, by using the notions of H-differentiation and radius of convergence we examine the relationship between term by term H-differentiation and uniform convergence of fuzzy-valued function series.
Highlights
The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase “convergence in a uniform way” when the “mode of convergence” of a series is independent of two variables
Later Karl Weierstrass, who attended his course on elliptic functions in 1839-1840, coined the term uniformly convergent which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894
One of them is in the form of interval valued fuzzy sets
Summary
The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase “convergence in a uniform way” when the “mode of convergence” of a series is independent of two variables. One of them is in the form of interval valued fuzzy sets. Many authors have developed the different cases of sequence sets with fuzzy metric on a large scale. Basarir [3] has recently promoted some new sets of sequences of fuzzy numbers generated by a nonnegative regular matrix A, some of which reduced to Maddox’s spaces l∞(p; F), c(p; F), c0(p; F), and l(p; F) for the special cases of that matrix A. The final section is completed with the concentration of the results on uniform convergence of fuzzy-valued sequences and series. Let u be a fuzzy number, whose membership function μ(x) can generally be defined as [9]. The generalized Hukuhara difference A ⊖ B of two sets A, B ∈ K is defined as follows: A. Where the limits are in the Hausdorff metric d for intervals
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