The function sequences and Taylor series with a fuzzy argument

Abstract

The main consideration subject is functional sequences f_n(A) with fuzzy number A for an argument. It is supposed that lim_n→∞f_n(x)=f(x) and lim_n→∞f_n’(x)=f’(x), and these convergences are uniform on each interval within supp A. It is also supposed that the equation f(x)=y with respect to x has finite number of solutions for each y on each interval within supp A. The paper proposes sufficient conditions for f_n(A) to converge in the sense that the sequence of membership functions μ_fn(A)(y): converges point-wise. It is proved that lim_n→∞ μ_fn(A)(y)= μ_f(A)(y) for all y ϵ P, except such y=f(x), that x is a discontinuity point of μ_A(x), or f‘(x)=0. As a particular case of sequence f_n(A), the generalization of Taylor series f(x)=∑_i=0^∞(f⁽ⁱ⁾(x₀)(x-x₀)ⁱ/(i!)) is considered for real analytical function f(x) for the case of fuzzy argument x=A. Convergence of the series is considered in the sense of point-wise convergence of the partial sum μ_Sn(A)(y), where S_n(x)=∑_i=0ⁿ(f⁽ⁱ⁾(x₀)(x-x₀)ⁱ/(i!)).