Differential and Integral Calculus for Fuzzy Number-Valued Functions with Interactivity

Abstract

This chapter presents the theory of differential and integral calculus for autoregressive fuzzy processes, that is, fuzzy number-valued functions F such that F(t) and \(F(t+h)\) are interactive fuzzy numbers for every |h| sufficiently small. Recall that interactivity between fuzzy variables are described in terms of joint possibility distributions and plays a similar role to that of dependence between random variables. We present a theory of differential calculus for certain autoregressive fuzzy processes based on a special type of interactivity called \(\mathcal {F}\)-interactivity. Next, we introduce a Banach space, denoted by \(\mathbb {R}_{\mathsf {F}(A)}\), which is composed by fuzzy numbers that are interactive with each other, such that the vector addition corresponds to an interactive operation. The concepts of differential and integral for mappings from \(\mathbb {R}\) to \(\mathbb {R}_{\mathsf {F}(A)}\) correspond to the classical notions of Fréchet derivative and Riemann integral in Banach spaces. Thus, the theory of differential and integral calculus for functions taking values in \(\mathbb {R}_{\mathsf {F}(A)}\) is different from those found in fuzzy literature that are based in (generalized) Hukuhara differentiability and fuzzy Aumann (or Riemann) integral, which involve non-interactive arithmetic. In contrast to the usual fuzzy Aumann integral, the diameter of a fuzzy function defined in terms of the Riemann integral in \(\mathbb {R}_{\mathsf {F}(A)}\) may decrease. We also study fuzzy ordinary and partial differential equations in \(\mathbb {R}_{\mathsf {F}(A)}\).