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 equationsf(ax+y)+f(x+ay)=(a+1)(a−1)2[f(x)+f(y)]+a(a+1)f(x+y)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.a≠0,±1. In addition, we also present a counterexample that does not satisfy the stability based on Ulam's question.