On the stability of Euler–Lagrange type cubic mappings in quasi-Banach spaces

In this paper, we solve the generalized Hyers–Ulam–Rassias stability problem for Euler–Lagrange type cubic functional equations f (ax + y)+ f (x +ay) = (a +1)(a − 1)2 f (x) +f (y) +a(a +1)f (x + y) for mappings f :X→Y in quasi-Banach spaces and for fixed integers a with a = 0,±1. In addition, we also present a counterexample that does not satisfy the stability based on Ulam’s question. © 2006 Elsevier Inc. All rights reserved.