H-U-R Stability Results of Mixed-Type Additive-Quadratic Functional Equation in Fuzzy β-Normed Spaces by Two Different Approaches

Abstract

Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields. Also, stability problem of a functional equation was first posed by Ulam which was answered by Hyers and then generalized by Aokiand Rassias for additive mappings and linear mappings, respectively. Since then, several stability problems for various functional equations have been investigated, and various fuzzy stability results concerning Cauchy, Jensen and quadratic functional equations were discussed. In this work, the authors newly introduce the mixed-type additive-quadratic functional equation and obtain its general solution which shows that this mixed type additive-quadratic functional equation satisfies both additive and quadratic characteristics. Mainly, we examine the Hyers-Ulam-Rassias (H-U-R) stability results for this mixed-type additive-quadratic functional equation in fuzzy β-normed spaces with appropriate two different tools of direct method which was introduced by D.H. Hyers and fixed point alternative method which was introduced by V. Radu.