Let be a real normed linear space and be a real Banach space. The Hyers-Ulam stability of the quadratic functional equation
for on the restricted domain for a is investigated. Furthermore, the Hyers-Ulam stability of another quadratic functional equation
with condition for a and or for a and is treated, first on the whole domain and then on the restricted domain . The results are applied to the study of the asymptotic behaviour of equation (1) and (2).