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The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.

In this paper, we investigate the stability of an additive-quadratic-quartic functional equation$$\begin{align*}f(x+2y)& +f(x-2y)-2f(x+y)-2f(-x- y)-2f(x-y)-2f(y-x)\nonumber \\ &+4f(-x)+ 2f(x)-f(2y)-f(-2y)+4f(y)+4f(-y)=0 \end{align*}$$by the direct method in the sense of Găvruta.

We have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

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 will investigate the superstability of a generalized trigonometric functional equation from the Pexideried functional equation: f (x + y) − f (x − y) = λ · g(x)h(y), which is a trigonometric functional equation mixed by the sine and cosine function.

In this paper, by using fixed point theorem, we obtain the stability of the following functional equations \begin{align*} f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)f(p,q)h(r,s) \\ f(pr,qs)+g(ps,qr)&=\theta(p,q, r,s)g(p,q)h(r,s), \end{align*} where $G$ is a commutative semigroup, $\theta : G^{4} \rightarrow \mathbb{R}_{k}$ a function and $f,g,h$ are functionals on $G^{2}$.

In this paper, we investigate the generalized Hyers–Ulam stability for the generalized psi functional equation f ( x + p ) = f ( x ) + φ ( x ) by the direct method in the sense of P. Gǎvruta and the Hyers–Ulam–Rassias stability.

In this paper, we solve the Hyers-Ulam stability problem for the following cubic type functional equation f(rx + sy) + f(rx− sy) = rsf(x + y) + rsf(x− y) + 2r(r − s)f(x) in quasi-Banach space and non-Archimedean space, where r 6= ±1, 0 and s are real numbers.

In this study, the character amenability of Banach algebras is considered and some characterization theorems are established. Indeed, we prove that the character amenability of Lipschitz algebras is equivalent to that of Banach algebras.

This work aims to study of the stability of two generalizations of the functional equa- tion f(pr,qs)+f(ps,qr )= f(p,q) f(r,s), namely (i) f(pr,qs)+g(ps,qr )= h(p,q)h(r,s) ,a nd (ii) f(pr,qs)+g(ps,qr )= h(p,q)k(r,s) for all p,q,r,s ∈ G ,w hereG is a commutative semi- group. Thus this work is a continuation of our earlier works (15 )a nd (16), and the functional equations studied here arise in the characterizations of symmetrically compositive sum form distance measures.

The aim of this paper is to prove the superstability of the following functional equations $$f\left(\frac{x+y}{m}\right)^m \,=\, g(x)h(y),$$ where \({f, g, h: V^{2} \to A}\) are unknown mappings and m is a fixed positive integer. Here V is a vector space, and A is a unital normed algebra.

The present work extends the study on the stability of the functional equation f(pr,qs)+ f(ps,qr )= f(p,q) f(r,s), which arises in the characterization of symmetrically compositive sum-form distance measures, and as a products of some multiplicative functions. In this paper, we obtain the superstability of the functional equations f(pr,qs) − f(ps,qr )= f(p,q)g(r,s) f(pr,qs) − f(ps,qr )= g(p,q)f(r,s) f(pr,qs) − f(ps,qr )= g(p,q)g(r,s) f(pr,qs) − f(ps,qr )= g(p,q)h(r,s), for all p,q,r,s ∈ G ,w hereG is an Abelian group. These functional equations arise in the char- acterization of the nonsymmetrically compositive difference-form related to distance measures, products of some multiplicative functions. In reduction, they can be represented as exponential functional equations.

The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation which is bounded by the unknown functions or . As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equationf(x+ky)+f(x−ky)=f(x+y)+f(x−y)+(2(k+1)/k)f(ky)−2(k+1)f(y)for fixed integerskwithk≠0,±1in fuzzy Banach spaces.

Let n>1 be an integer, let A be an algebra, and X be an A-module. A quadratic function D:A→X is called a quadratic n-derivation if D(∏i=1nai)=D(a1)a22⋯an2+a12D(a2)a32⋯an2+⋯+a12a22⋯an−12D(an) for all a1,...,an∈A. We investigate the Hyers-Ulam stability of quadratic n-derivations from non-Archimedean Banach algebras into non-Archimedean Banach modules by using the Banach fixed point theorem.

Let 𝑛 be a fixed integer greater than 3 and let 𝜆 be a real number with 𝜆 ≠ ( 𝑛 2 − 𝑛 + 4 ) / 2 . We investigate the Hyers-Ulam stability of derivations on Banach algebras related to the following generalized Cauchy functional inequality ‖ ∑ 1 ≤ 𝑖 𝑗 ≤ 𝑛 1 ≤ 𝑘 𝑙 ≠ 𝑖 , 𝑗 ≤ 𝑛 𝑓 ( ( 𝑥 𝑖 + 𝑥 𝑗 ∑ ) / 2 + 𝑛 − 2 𝑙 = 1 𝑥 𝑘 𝑙 ∑ ) + 𝑓 ( 𝑛 𝑖 = 2 𝑥 𝑖 ) + 𝑓 ( 𝑥 1 ∑ ) ‖ ≤ ‖ 𝜆 𝑓 ( 𝑛 𝑖 = 1 𝑥 𝑖 ) ‖ .

We present a fixed point theorem for generalized contraction in partially ordered complete metric spaces. As an application, we give an existence and uniqueness for the solution of a periodic boundary value problem.

The present work aims to study the stability of the following three functional equations: (i) f ( p r , q s ) + f ( p s , q r ) = f ( p , q ) f ( r , s ) , (ii) f ( p r , q s ) + f ( p s , q r ) = f ( p , q ) g ( r , s ) , and (iii) f ( p r , q s ) + f ( p s , q r ) = g ( p , q ) f ( r , s ) for all p , q , r , s ∈ ( 0 , 1 ) . The first functional equation arises in the characterization of symmetrically compositive sum form distance measures.