Showing papers by "Prasanna K. Sahoo published in 2016"

On the order of a derivation

Abstract: In this note we provide the solution to a problem posed by the first author in a previous paper. In particular, we prove a result relating the number of nonzero coefficients of a certain functional equation to the order of any derivation satisfying that equation.

A functional equation with involution related to the cosine function

Abstract: Let G be an abelian group, $${\mathbb{C}}$$ be the field of complex numbers, $${\alpha \in G}$$ be any fixed element and $${\sigma : G \to G}$$ be an involution. In this paper, we determine the general solution $${f, g : G \to \mathbb{C}}$$ of the functional equation $${f(x + \sigma y + \alpha) + g(x + y + \alpha) = 2f(x)f(y)}$$ for all $${x, y \in G}$$ .

On a functional equation arising from proth identity

Abstract: . We determine the general solutions f : R 2 → R of the func-tional equation f(ux−vy,uy+v(x+y)) = f(x,y)f(u,v) for all x,y,u,v ∈R. We also investigate both bounded and unbounded solutions of thefunctional inequality |f(ux−vy,uy+v(x+y))−f(x,y)f(u,v)| ≤ φ(u,v)for all x,y,u,v ∈ R, where φ : R 2 → R + is a given function. 1. IntroductionThe simple identity(1.1) x 4 +y 4 +(x +y) 4 = 2x 2 +xy +y 22 is known as the Proth identity and was first published in 1878 (see [2]). It canbe easily established by expanding the right hand side and using the binomialtheorem. If we denote the quadratic form on the right hand side of (1.1) by(1.2) f(x,y) = x 2 +xy +y 2 ,then it can be easily verified that(1.3) f(ux −vy,uy +v(x +y)) = f(x,y)f(u,v)for all x,y,u,v ∈ R. However, it is not so obvious that the function f given in(1.2) is the only solution of (1.3).The Proth identity means that if we represent a given positive integer n asf(x,y), where x and y are integers, then we have a representation of 2n

Stability of functional equations of n-Apollonius type in fuzzy ternary Banach algebras

Abstract: Using the fixed point method, we investigate the generalized Hyers–Ulam stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras for the additive functional equation of n-Apollonius type, namely $${\sum_{i=1}^{n} f(z-x_{i}) = -\frac{1}{n} \sum_{1 \leq i < j \leq n} f(x_{i}+x_{j}) + n f (z-\frac{1}{n^{2}} \sum_{i=1}^{n}x_{i}),}$$ where \({n \geq 2}\) is a fixed positive integer.

Stability of the Pexiderized quadratic functional equation in paranormed spaces

Abstract: The aim of the present paper is to investigate the Hyers-Ulam stability of the Pexiderized quadratic functional equation, namely of $f(x + y)+f(x-y)=2g(x)+2h(y)$ in paranormed spaces. More precisely, first we examine the stability for odd and even functions and then we apply our results to prove the Hyers-Ulam stability of the quadratic functional equation $f(x + y)+f(x-y)=2f(x)+2f(y)$ in paranormed spaces for a general function.

On a functional equation on groups with an involution related to quadratic polynomials in two variables

Abstract: Let G be a group and \({\mathbb{C}}\) the field of complex numbers. Suppose \({\sigma \colon G \to G}\) is an involution on G. In this paper, we determine the general solution \({f\colon G\times G \to \mathbb{C}}\) of the functional equation $$\begin{aligned}f(x_1 \sigma y_1 , x_2 \sigma y_2) - f(x_1 \sigma y_1 , x_2) - f(x_1 , x_2 \sigma y_2)\\ \quad = f(x_1 y_1 , x_2 y_2) - f(x_1 y_1 , x_2) - f(x_1 , x_2 y_2)\end{aligned}$$ for all \({x_1, x_2 , y_1, y_2 \in G}\). In Chung et al. (J Korean Math Soc 38:37–47, 2001), the solution of the above equation was determined assuming (a) f is central in each variable, (b) \({\sigma (x) = x^{-1}}\) for all \({x \in G}\), and (c) x2 = y has a solution on G for all \({x, y \in G}\). We do not require any such conditions to obtain its solution. No new solutions emerge on arbitrary groups.

Approximation of a generalized Euler-Lagrange type additive mapping on Lie $C^{ast}$-algebras

Abstract: Using fixed point method, we prove some new stability results for Lie $(alpha,beta,gamma)$-derivations and Lie $C^{ast}$-algebra homomorphisms on Lie $C^{ast}$-algebras associated with the Euler-Lagrange type additive functional equation begin{align*} sum^{n}_{j=1}f{bigg(-r_{j}x_{j}+sum_{1leq i leq n, ineq j}r_{i}x_{i}bigg)}+2sum^{n}_{i=1}r_{i}f(x_{i})=nf{bigg(sum^{n}_{i=1}r_{i}x_{i}bigg)} end{align*} where $r_{1},ldots,r_{n}in {mathbb{R}}$ are given and $r_{i},r_{j}neq 0$ for some $1leq i< jleq n$.