In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, u t t ( x , t ) − c 2 ▵ u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.

In this paper, we will obtain the optimal Hyers-Ulam’s constant for the first-order linear differential equations $p(t)y'(t) - q(t)y(t) - r(t) = 0$ .

In this paper, we will consider the generalized sextic functional equation And by applying the fixed point theorem in the sense of Cdariu and Radu, we will discuss the stability of the solutions for this functional equation.

If a mapping can be expressed by sum of a septic mapping, a sextic mapping, a quintic mapping, a quartic mapping, a cubic mapping, a quadratic mapping, an additive mapping, and a constant mapping, we say that it is a general septic mapping. A functional equation is said to be a general septic functional equation provided that each solution of that equation is a general septic mapping. In fact, there are a lot of ways to show the stability of functional equations, but by using the method of Gvruta, we examine the stability of general septic functional equation which considered. The method of Gvruta as just mentioned was given in the reference Gavruta (1994).

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and cumbersomely prove uniqueness in stability studies.

The functional inequality ∥f(x) + 2f(y) + 2f(z)∥ ≤ ∥2f(x/2 + y + z)∥ + ϕ (x, y, z) (x, y, z ∈ G) is investigated, where G is a group divisible by 2, f : G → X and ϕ : G³ → [0, ∞) are mappings, and X is a Banach space. The main result of the paper states that the assumptions above together with (1) ϕ(2x, −x, 0) = 0 = ϕ(0, x, −x) (x ∈ G) and (2) lim_n→∞(1/2ⁿ)ϕ(2ⁿ⁺¹x, 2ⁿy, 2ⁿz) = 0, or lim_n→∞2ⁿϕ(x/2ⁿ⁻¹, y/2ⁿ, z/2ⁿ) = 0 (x, y, z ∈ G), imply that f is additive. In addition, some stability theorems are proved.