On approximate solutions of the linear functional equation of higher order (original) (raw)

We prove two general theorems, which appear to be very useful in the investigation of the Hyers-Ulam stability of a higher-order linear functional equation in single variable, with constant coefficients. We give several examples of their applications. In particular, we show that we obtain in this way several fixed point results for a particular operator. The main tool in the proofs is a complexification of a real normed (or Banach) space X, which can be described as the tensor product X⊗ℝ 2 endowed with the Taylor norm.

In this paper, we have improved some of the results in [C. Choi and B. Lee, Stability of Mixed Additive-Quadratic and Additive-Drygas Functional Equations. Results Math. 75 no. 1 (2020), Paper No. 38]. Indeed, we investigate the Hyers-Ulam stability problem of the following functional equations 2φ(x + y) + φ(x − y) = 3φ(x) + 3φ(y) 2ψ(x + y) + ψ(x − y) = 3ψ(x) + 2ψ(y) + ψ(−y). We also consider the Pexider type functional equation 2ψ(x + y) + ψ(x − y) = f (x) + g(y), and the additive functional equation 2ψ(x + y) + ψ(x − y) = 3ψ(x) + ψ(y).

We show that every approximate solution of the Hosszu's functionalequationf(x + y + xy) = f(x) + f(y) + f(xy) for any x; y 2 R;is an additive function and also we investigate the Hyers-Ulam stability of thisequation in the following settingjf(x + y + xy) 􀀀 f(x) 􀀀 f(y) 􀀀 f(xy)j   + '(x; y)for any x; y 2 R and  > 0.

In this paper we study the Hyers–Ulam–Rassias stability theory by considering the cases where the approximate remainder φ is defined bywhere (G, ∗ ) is a certain kind of algebraic system, E is a real or complex Hausdorff topological vector space, and f, g, h are mappings from G into E. We prove theorems for the Hyers–Ulam–Rassias stability of the above three kinds of functional equations and obtain the corresponding error formulas.

We introduce a new ↵-cubic functional equation and investigate the generalized Hyers–Ulam stability of this functional equation in 2-Banach spaces.