We prove the Hyers-Ulam stability of the following Jensen functional inequality∥f((x-y)/n+z)+f((y-z)/n+x)+f((z-x)/n+y)∥≤∥f((x+y+z)∥inp-Banach spaces for any fixed nonzero integern.

Let n be a given positive integer, G an n-divisible abelian group, X a normed space and f : G → X. We prove a generalized Hyers-Ulam stabitity of the following functional inequality f (x) + f (y) + n f (z) n f x + y n + z + ϕ(x,y,z), ∀x,y,z ∈ G, which has been introduced in [3], under some conditions on ϕ : G × G × G → [0,∞) .

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We establish the general solution of the functional inequality ‖ (−)+ (−)+ (−)−3 ()−3 ()−3 ()‖ ≤ ‖ (+ +)‖ and then investigate the generalized Hyers-Ulam stability of this inequality in Banach spaces and in non-Archimedean Banach spaces.

Abstract. In this article, we prove the generalized Hyers-Ulam stability of the following Cauchy additive functional equationin fuzzy Banach spaces for any fixed nonzero integer n .Mathematics subject classification (2010): 39B52, 46S40.