The Cauchy problem for quaternion fuzzy fractional differential equations

Article type: Research Article

Authors: Yang, Zhan-Peng^* | Xu, Tian-Zhou | Qi, Min

Affiliations: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, PR China

Correspondence: [*] Correspondence to: Zhan-Peng Yang, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R. China. [email protected]

Abstract: We discuss the existence of a solution to the Cauchy problem for fuzzy fractional differential equations that accommodates the notion of fuzzy sets defined by quaternion-valued membership functions. We first propose definitions of quaternion fuzzy sets and discuss entailed results which parallel those of regular fuzzy numbers. Two existence results, relevant to the Cauchy problem for fuzzy fractional differential equations in the case of bounded integral operators, are given. The results require either Hölder continuous or Lipschitz continuous response functions.

Keywords: Quaternion-valued grades of membership, quaternion fuzzy fractional differential equation, Cauchy problem, existence theorem, uniqueness

DOI: 10.3233/IFS-151612

Journal: Journal of Intelligent & Fuzzy Systems, vol. 29, no. 1, pp. 451-461, 2015