Discrete fuzzy complex-valued function and complex fuzzy Caputo fractional difference equations

Abstract

In this paper, we introduce a notion of complex compact intervals on which Minkowski addition and scalar multiplication operators are well defined to form the space of complex fuzzy numbers whose α-cuts are complex intervals. Some basic fuzzy partial ordering relations among the complex numbers with “≤C”, the complex intervals with “≪”, the complex fuzzy numbers with “⪯C˜” and the discrete fuzzy complex-valued functions with ≲ are studied. Based on these fundamental properties, we introduce a concept of Caputo fractional sum and difference for the discrete fuzzy complex-valued functions and establish some of their basic results. Moreover, by introducing the complex monotone iterative technique combined with the method of upper and lower solutions, the existence of extremal solutions for discrete complex fuzzy Caputo fractional difference equations is investigated. In each section, several examples are presented to show the feasibility of our obtained results.