A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f1 （2x ＋ y） ＋ f2（2x - y） = f3（x ＋ y） ＋ f4（x - y） ＋ f5（x） without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 ： R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f（2x ＋ y） ＋ f（2x - y） = g（x ＋ y） ＋ g（x - y） ＋ h（x） and f（2x ＋ y） - f（2x - y） = g（x ＋ y） - g（x - y）. The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.

A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f ₁(2x + y) + f ₂(2x − y) = f ₃(x + y) + f ₄(x − y) + f ₅(x) without assuming any regularity condition on the unknown functions f ₁, f ₂, f ₃, f ₄, f ₅ : ℝ → ℝ. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x − y) = g(x + y) + g(x − y) + h(x) and f(2x + y) + f(2x − y) = g(x + y) + g(x − y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszú. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi.