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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.

In this article, we study the stability of various forms for the general octic functional equation ∑i=099Ci−19−ifx+iy=0. We first find a special way of representing a given mapping as the sum of eight mappings. And by using the above representation, we will investigate the hyperstability of the general octic functional equation. Furthermore, we will discuss the Hyers–Ulam–Rassias stability of the general octic functional equation.

In this paper, we introduce a way of representing a given mapping as the sum of odd and even mappings. Then, using this representation, we investigate the stability of various forms of the following general nonic functional equation: ∑i=01010Ci(−1)10−if(x+iy)=0.

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 Ga˘vruta, we examine the stability of general septic functional equation ∑i=08C8i−18−ifx+i−4y=0 which considered. The method of Ga˘vruta as just mentioned was given in the reference Gavruta (1994).

In this paper, we will consider the generalized sextic functional equation ∑i=07 7Ci−17−ifx+iy=0. And by applying the fixed point theorem in the sense of Ca˘dariu and Radu, we will discuss the stability of the solutions for this functional equation.

The general quintic functional equation and the general sextic functional equation are generalizations of many functional equations such as the additive function equation and the quadratic function equation. In this paper, we investigate Hyers−Ulam−Rassias stability of the general quintic functional equation and the general sextic functional equation.

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x1 + y1, x2 + y2, …, xn + yn) = f(x1, x2, … xn) + f(y1, y2, …, yn). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations.

In this paper, we investigate the generalized Hyers-Ulam stability of the Pexider functional equation f ( x + y , z + w ) = g ( x , z ) + h ( y , w ) .

We prove general stability theorems for n-dimensional quartic-cubic-quadratic-additive type functional equations of the form ∑ i = 1 ℓ c i f a i 1 x 1 + a i 2 x 2 + ⋯ + a i n x n = 0 . by applying the direct method. These stability theorems can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

By applying a fixed point approach, we investigate the stability problems for an AQCQ functional equation of the form fx+2y+fx-2y-4fx+y-4fx-y+6fx-f2y-f-2y+4fy+4f-y=0.

We investigate the general functional equation of the form f(ax+by+cz)-abf(x+y)-bcf(y+z) − acf(x+z)-a((a+1)/2-b-c)f(x)-b((b+1)/2-a-c)f(y)-c((c+1)/2-a-b)f(z)-(a(a-1)/2)f(-x) − (b(b-1)/2)f(-y)-(c(c-1)/2)f(-z)=0, whose solutions are quadratic-additive mappings in connection with stability problems.

We prove the generalized Hyers–Ulam stability of a mean value type functional equation f ( x ) − g ( y ) = ( x − y ) h ( x + y ) by applying a method originated from fixed point theory.

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

In this paper, we prove the stability of the following functional equation ∑ i = 0 n n C i ( − 1 ) n − i f ( i x + y ) − n ! f ( x ) = 0 on a restricted domain by employing the direct method in the sense of Hyers.

We take into account the stability of the following functional equation f(x+ay)+f(x-ay)-2f(x)-a2f(y)-a2f(-y)=0 in non-Archimedean normed spaces.

We examine the generalized Hyers-Ulam stability of the following functional equation: 2fx+y+z+w+f-x-y+z+w+f-x+y-z+w+f-x+y+z-w+fx-y-z+w+fx-y+z-w+fx+y-z-w-5fx-3f-x-5fy-3f-y-5fz-3f-z-5fw-3f-w=0, in the fuzzy normed spaces with the fixed point method.

We investigate the generalized Hyers-Ulam stability of a functional equation f∑j=1n‍xj+(n-2)∑j=1n‍f(xj)-∑1≤i

We establish the hyperstability of -Jordan homomorphisms from a normed algebra to a Banach algebra, and also we show that an -Jordan homomorphism between two commutative Banach algebras is an -ring homomorphism.

We investigate the stability of a functional equation by applying the direct method in the sense of Hyers and Ulam.

We will prove the stability of the functional equation 2f(∑i=1nxi)+∑1≤i,j≤n,i≠jf(xi-xj)=(n+1)∑i=1nf(xi)+(n-1)∑i=1nf(-xi) in non-Archimedean normed spaces.

We investigate the stability problems for a functional equation 2f(∑j=1nxj)+∑1≤i,j≤n, i≠jf(xi-xj)=(n+1)∑j=1nf(xj)+(n-1)∑j=1nf(-xj) by using the fixed point method.

We investigate the stability problems for the n-dimensional mixed-type additive and quadratic functional equation 2f(∑j=1nxj)+∑1≤i,j≤n, i≠jf(xi-xj)=(n+1)∑j=1nf(xj)+(n-1)∑j=1nf(-xj) in random normed spaces by applying the fixed point method.

We investigate a fuzzy version of stability for the functional equation f(2x+y)+f(2x−y)+2f(x)−f(x+y)−f(x−y)−2f(2x)=0 in the sense of Mirmostafaee and Moslehian.

We investigate a fuzzy version of stability for the functional equation 𝑓(𝑥+𝑦+𝑧)+𝑓(𝑥−𝑦)+𝑓(𝑥−𝑧)−𝑓(𝑥−𝑦−𝑧)−𝑓(𝑥+𝑦)−𝑓(𝑥+𝑧)=0 in the sense of M. Mirmostafaee and M. S. Moslehian.

We investigate the stability of the functional equation 2𝑓(𝑥+𝑦)+𝑓(𝑥−𝑦)+𝑓(𝑦−𝑥)−3𝑓(𝑥)−𝑓(−𝑥)−3𝑓(𝑦)−𝑓(−𝑦)=0 by using the fixed point theory in the sense of Cădariu and Radu.

We obtain the Hyers-Ulam stability of a bi-Jensen functional equation: 2f((x+y)/2,z)−f(x,z)−f(y,z)=0 and simultaneously 2f(x,(y+z)/2)−f(x,y)−f(x,z)=0. And we get its stability on the punctured domain.

We establish the generalized Hyers-Ulam stability of a Pexider-type functonal equation f1(x+y+z)+f2(x−y)+f3(x−z)−f4(x−y−z)−f5(x+y)−f6(x+z)=0, which is mixed of a quadratic and an additive functional equations. Also, we obtain its general solution from the stability results.