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